Paper Title: Half-sided translations broken to pieces.


[
    Bruno S. Dandogbessi^2
    August 5, 2024
==========================










§ INTRODUCTION


The half-sided translations <cit.>
is a group of unitary transformations
built from the algebraic structure of the underlying quantum field theory, that can translate operators through spacetime. 
Specifically, their definition requires a pair of von Neumann algebras ℳ and 𝒩, where 𝒩 is a subalgebra of ℳ, and when sufficiency condiitons are met, they are generated by a positive operator 𝒢 that satisfies 

    e^i 𝒢 sℳ e^-i 𝒢 s⊂ℳ,    s ≤  0  (or s ≥ 0).

Therefore, ℳ is preserved under the flow of 𝒢 for “half of the real line", and only for the remaining half, 𝒢 can translate operators outside ℳ; a property that earns half-sided translations their name. This way, the half-sided translations connect the algebraic structure of the quantum field theory to translations within the spacetime, and serve as a powerful tool for studying quantum field theories and quantum gravity. In particular, they have been used in AdS/CFT for understanding the algebras of gravitational theories, state dependence of black hole interiors, and for doing explicit bulk reconstruction in the black hole interior <cit.>.


The sufficiency conditions for their existence for a given pair ℳ, 𝒩 are (i) of existence of a common cyclic and separating state |Ω⟩ 
and (ii) thus derived modular operators[We review these notions briefly in section <ref>.]
 Δ_ℳ, Δ_𝒩 that generate automorphisms of the algebra, to satisfy the property

    Δ_ℳ^-it𝒩Δ_ℳ^it⊂𝒩,    t≤ 0  (or  t ≥ 0).

Then 𝒢 is given by

    𝒢 = 1/2π (logΔ_𝒩 - logΔ_ℳ).

The operator  K_ℳ = -logΔ_ℳ 

is called the modular Hamiltonian <cit.> for algebra ℳ, however, it acts on both ℳ and its commutant ℳ', generically in a non-factorizable way. (Likewise is true for K_𝒩). For theories with a UV cut off however, a factorization is possible, such that K_ℳ, K_𝒩 and thus 𝒢 can be written as a combination of logarithms of density operators.








To see the factorization, let Σ be a Cauchy slice corresponding to the state |ω⟩ of the quantum field.
Consider the density operator ρ_M for a spatial subregion M on Σ, whose causal diamond ♢_M defines the algebra ℳ.
Likewise, let ρ_M' be the density operator for the complement region M', whose causal diamond ♢_M' gives the commutant algebra ℳ'. 
These are well defined positive operators if the field theory has a UV cut off. 
Therefore, one can take their logarithms  H_ℳ = -logρ_M, H_ℳ' = -logρ_M'.[These operators have been studied
extensively in high energy physics and condensed 
matter physics communities 
<cit.> 
and are also often referred to as modular Hamiltonians,
but to avoid confusion,
we will call them entanglement Hamiltonians following the condensed matter community.]
If |ω⟩ is a cyclic separating state for ℳ, then <cit.> the modular operator Δ_ℳ is

    Δ_ℳ = ρ_M ⊗ρ_M'^-1,   or  
        K_ℳ = H_ℳ - H_ℳ'.





Likewise repeating for Δ_𝒩 permits us to write 
𝒢 = G + G', where 

    G = H_ℳ - H_𝒩,   
    G'= H_𝒩' - H_ℳ'

where G is made only from ℳ, 𝒩, and G' involves only their commutants. 
This observation naturally raises the question whether one could break down the action of 𝒢 into the action of its constituents G and G', 
but its answer is not straightforward, because [G, G']≠ 0, and thus action of  𝒢 does not split cleanly into those of G and G'. In fact, since a finite action is implemented through exp(i𝒢 s), an attempt at the split will require the  Zassenhaus formula, and depending on the severity of the non-commutativity, an infinite product may be needed <cit.>.

Despite this, however, causality suggests that action of 𝒢 should be indistinguishable from G  in the region where G' has no causal influence. Likewise, the action of 𝒢 should be same as that of G' in regions that G cannot influence causally. 
In our setup, these would be regions ♢_N' and ♢_M' respectively. Moreover, it would be interesting to see the deviations between actions of G and G' compared with 𝒢 in the remaining regions of spacetime, and whether highly nested commutators are needed to approximate the action of 𝒢 in these regions. This is one of the main questions we will address in this paper.

In this paper, we will study flow of local operators ϕ under G and G', and the all the nested commutators involving [G, G'], presenting a breakdown of the action of 𝒢 into the actions of G and G'. We will use the 1+1D massless scalar field in Minkowski spacetime for our work as it is the simplest example where computations can be done explicitly. Let us list our main results here:

    
  * For ℳ, 𝒩 corresponding to Rindler wedges related by translation of origin in Minkowski spacetime, we compute all the nested operators that appear in the Zassenhaus product, and show that
        
    e^is 𝒢 = e^is G e^is G'exp([G, G'] ·1-4s - e^-4s/16 s^2 )
            .

        
    
  * As one would expect, in regions where G' has no causal influence, the action of G is indistinguishable from that of 𝒢, and likewise for G' in regions where G has no causal influence. We show this, and more importantly, in the region where both G and G' together create 𝒢, 
        
    i [G, ϕ(x,t)] = (x-t) ∂_- ϕ(x,t)

        where x-t = 2a is the boundary of ♢_N. 
         This is to be compared with the known equation
        
    i [𝒢, ϕ(x,t)] = 2a ∂_- ϕ(x,t)

        The equation (<ref>) highlights, among other things, that without G', G cannot take the operator outside ♢_M.


Let us give the outline of the paper. In section 2, we give the essentials of half-sided translations, and the main content begins in section 3, where we study all the nested commutators of G and G' for the massless scalar field, including the Zassenhaus product. In section 4, we study the action of G on local operators within the regions where G has a causal influence. The action of G' can be obtained by a symmetry. We conclude in section 5, discussing how our results can be applied to the island paradigm, which originally inspired the work. Finally, in the appendix, we have given derivations of unequal time commutators we use repeatedly in the main text.













§ BASICS OF HALF-SIDED TRANSLATIONS



Let us consider a quantum field theory in flat spacetime in a state |Ω⟩ which is cyclic and separating for a von Neumann algebra ℳ of operators. Cyclicity of the state means that ℳ when acting on |Ω⟩ produces a dense subset of the full Hilbert space, and separability means that no operator in the algebra ℳ, except the trivial operator, annihilates |Ω⟩. These conditions are necessary and sufficient to define the Tomita operator S whose polar decomposition is given by


    S = J_ℳΔ_ℳ^1/2

where Δ_ℳ = exp (- K_ℳ) implements the automorphisms of the algebra ℳ, and J_ℳ is an anti-unitary operator satisfying J_ℳℳ J_ℳ=ℳ', the commutant of ℳ. Therefore, the S operator knows about ℳ'; in fact Δ_ℳ also generates the automorphisms of ℳ'. 

The one parameter group of automorphisms  defined by Δ_ℳ^it is called the modular flow of ℳ, and its generator K_ℳ is called the modular Hamiltonian.  In mathematical form,


    e^i K_ℳ tℳ e^-i K_ℳ t = ℳ, t ∈ℝ


This is a point where we can introduce the half sided modular translation. Quite similarly to the modular flow, ℳ may have an additional structure associated to it which may allow us to write



    e^i 𝒢 sℳ e^-i 𝒢 s⊂ℳ,    s ≤  0  (or s ≥ 0)
 
with 𝒢|Ω_0⟩ = 0, and 𝒢≥ 0, like K_ℳ. 

The operator G is generating a one parameter unitary group called the half sided translations, which we will denote as ℋst(ℳ)^- for s≤ 0, or ℋst(ℳ)^+ for s≥ 0.
Clearly, the difference between (<ref>) and (<ref>) is that the latter preserves the algebra only for s ≤ 0 (or s ≥ 0), and the former preserves it for all t ∈ℝ. 


Interestingly, it is impossible to have s ∈ℝ, because when combined with the properties 𝒢≥ 0 and G|Ω_0 ⟩, it would force 𝒢 = 0 <cit.>. Below we'll mostly focus on the case s ≤ 0.









 §.§ Theorems for half-sided translations


Reiterating the conditions for their existence, we need two von Neumann algebra ℳ, 𝒩 with 𝒩⊂ℳ that satisfies the following properties:



    
  * The state |Ω_0⟩ is cyclic and separating for 𝒩.
    
  * The half sided modular flow of 𝒩 with Δ_ℳ lies within 𝒩:
        
    Δ_ℳ^-it𝒩Δ_ℳ^it⊂𝒩,    t ≤ 0  (or t ≥ 0).



If satisfied, there are several theorems that guarantee the existence of the half sided translations and describe their properties <cit.>. Most notably for us, there are four theorems to highlight; for a more detailed discussion, see <cit.>.



    
  * There exists a unitary group U(s) = e^-i 𝒢 s, s ∈ℝ generated by a positive generator 𝒢, such that 
        
           
  * U(s) |Ω_0⟩  = |Ω_0⟩, s ∈ℝ, or equivalently 𝒢 |Ω_0⟩ = 0.
           
  * U^† (s) ℳ U(s) ⊂ℳ,     s≤  0 
           
  * 𝒩 = U^† (-1) ℳ U(-1).
        


  * The commutator [K_ℳ, K_𝒩] = - 2π i (K_ℳ - K_𝒩) and K_ℳ- K_𝒩 = 2π𝒢.

    
  * Given an ℳ, 𝒩 and  |Ω_0⟩ defined this way, U(s) is unique.

    
  * The von Neumann algebra ℳ needs to be type  III_1 for 𝒢 to be non-trivial.



Despite working with entanglement Hamiltonians H_ℳ, H_𝒩, whenever we refer to 𝒢,
we will mean the one in the UV limit, where the algebras are type III_1.



  §.§.§ Half sided translations in Minkowski spacetime


Let us introduce, as an example, the setup that will become the main focus of this paper.
Consider a quantum field theory in 1+1 dimensional Minkowski spacetime having coordinates (x,t). 
Let the system be in the vacuum state |0⟩ with respect to t, and let the algebras ℳ, 𝒩 correspond
to two Rindler wedges associated with different origins in Minkowski spacetime: 

    ♢_M = {x^+ > 0, x^- < 0},   ♢_N = {x^+ > 0, x^-<-2a}.

Their associated intervals are M=(0,∞) at t=0, and  N = (a, ∞)  at t=-a respectively. See figure <ref>.




There are several ways to derive H_ℳ, H_𝒩 and the corresponding K_ℳ, K_𝒩 for these algebras <cit.>. Explicitly, we note that 


    H_ℳ = ∫_0^∞ dx  x T_00(x,0),    H_ℳ' = - ∫_-∞^0 dx  x T_00(x,0),


    H_𝒩 = ∫_a^∞ dx  (x-a) T_00(x,-a),    H_𝒩'=  -∫_-∞^a dx  (x-a) T_00(x,-a).


The operators K_ℳ and K_𝒩 are the boost operators preserving (x,t) = (0,0) and (x,t)=(a,-a) respectively. One way to relate these two, which allows one to identify 𝒢 = K_ℳ - K_𝒩, is using spacetime translations like in <cit.>. Another is to use transformation properties of the stress tensor[This one is shown in the appendix for a conformal field theory.]. One finds that 

    𝒢 = 2 a P^+≡ a (H+P)

is the generator of translations in the x^- direction for the algebras (<ref>). It generates a flow in the null direction, and is an example of ℋst(ℳ)^-. [For the same ℳ, one can get a different 𝒢 = 2a P^- by considering a different algebra 𝒩, associated to the region {x^+ > 2a, x^-< 0}, of the type ℋst(ℳ)^+.]



 §.§ A non example


An arbitrary 𝒩⊂ℳ does not satisfy conditions of half-sided translations.  Regions of the type shown in figure <ref> do not satisfy the inclusion property (<ref>). This is because the modular flow mimics the action of a boost operator <cit.>, which means any local operator in the wedge associated to ℳ under modular flow will approach the upper and the lower vertices of the wedge, as t →∞ and t →  -∞ respectively. Both of the associated regions however are not covered by the algebra 𝒩, hence evolution of operators in 𝒩 under the modular flow of ℳ goes outside 𝒩 for both t≥ 0 and t≤  0. For more examples, see <cit.>.







§ COMMUTATORS FOR THE MASSLESS SCALAR FIELD


Now that we have defined our algebras (<ref>) and the entanglement Hamiltonians (<ref>), (<ref>), we can generally say that

    [G, G'] =    [H_ℳ - H_𝒩, H_𝒩' - H_ℳ'] 
    
        =    [H_ℳ, H_𝒩']

where the second line follows from the definitions (<ref>), (<ref>) and causality. See figure <ref>.
Thus by making a change of variables, we can write

    [H_ℳ, H_𝒩']  = ∫_0^∞ dx · x·∫_0^∞ dy · y· [T_00(x,0), T_00(a-y, -a)]


The y integral runs on thin line from -∞ to a, effectively, and the one on thick x axis from 0 to ∞. Now, clearly all the T_00(x,0) operators commute with T_00(a-y,a) for y<-a, and all those with x> 2a commute with all of them, so this operator becomes

    [H_ℳ, H_𝒩']  = ∫_0^2a dx · x ∫_0^2a dy · y·  [T_00(x,0), T_00(a-y,-a)]

but that is all we can say in general, without assuming a specific form of the stress tensor. 
Similarly, for studying the effect of entanglement Hamiltonians on local operators, we need a theory where computations can be done explicitly.
Here onwards, we will specialize to the massless scalar field in 1+1D Minkowski spacetime, and compute the quantities explicitly.





The massless scalar field ϕ(x,t) in 1+1D Minkowski spacetime satisfies the Klein-Gordon equation 

    ∂π/∂ t = ∂ ^2 ϕ/∂ x^2,   π≡∂_t ϕ

and the equal time commutation relations

    [ϕ(x,t), π(x', t)] = i δ(x-x')
    .

The stress tensor components are
 the Hamiltonian density ℋ = T^00 = T_00 and the momentum density 𝒫 = T^01 = -T_01,

    T_00 = 1/2 ( π^2 + (∂_x ϕ)^2),    T_01 = 1/2 (π·∂_x ϕ + ∂_x ϕ·π).



Our computations will involve studying the operators defined at different times, therefore we will need unequal time commutators. While I have discussed them in great detail in the appendix <ref>, for the clarity of the main text, I will summarize their computation here. The crucial points are to write the Heisenberg operator 

    ϕ(x,t) = exp(i H t) ϕ(x) exp(-i Ht)

using the Schroedinger operator, and then use Hadamard's lemma and the equation of motion (<ref>). That gives us two commutators we will need in next section:

    [T_00(y,0), ϕ(x,t)]    =  - i  ∂_+  ϕ(y,0) ·δ(x+t-y) - i ∂_- ϕ(y,0) ·δ(x-t-y)
    
        [T_01(y,0), ϕ(x,t)]   =  - i ∂_+ ϕ(y,0) ·δ(x+t-y) + i ∂_- ϕ(y,0) ·δ(x-t-y)


Note that the effect of evolution, because of the equation of motion, is coming from only two points: (x+t, 0) and (x-t, 0). 

Further, for computing [G, G'] and subsequent nested commutators, we will need commutators of stress tensors at different times as well. In particular, we will need

    [T_–(x), T_–(y)] = i D(x,y) (∂_y - ∂_x) δ(x-y)

where

    D(x,y) = π(x,0) π(y,0) + χ(x,0) χ(y,0) -{π(x,0),  χ(y,0)}.

To reach this expression, one has to recognize the commutator is of the form [A^2, B^2] for appropriate A,B, after which simplifications are easier. Note that in the left side the operators depend only on one variable since the theory is conformal. 



 §.§ The commutator [G, G']


Using the transformation properties of the stress tensor for the conformal field, 

    T_00(x + a, -a) = T_++(x) + T_–(x+ 2a)
 
we can write G = I_1 + I_2, where

    I_1 = 2a ∫_2a^∞ T_–(x) dx,   I_2= ∫_0^2a x · T_–(x)dx.

This derivation is given in the appendix <ref>. Further, we can convert the equation (<ref>) into



    [H_ℳ, H_𝒩']= ∫_0^2a dx · x ∫_0^2a dy ·  y  · [  T_– (x) ,  T_–(-y+2a)]
    .

Direct use of (<ref>) will yield 4 terms, in which only this [T_–,T_–] commutator will survive. Using (<ref>), we see this integral will split into two, one involving the derivative over x and another on y. One also has to carefully note that argument of second T_– is 2a-y. Let us write it as [H_ℳ, H_𝒩'] = I_x - I_y, then the integral with x derivative becomes 

    I_x = - i/4∫_0^2a du · (u-2a) ·   [D(u,u) + u ·∂_ (1) D(u,u)]

and the integral with y derivative becomes

    I_y = - i/4∫_0^2a du · u·[D(u,u)+ (u-2a) ∂_ (2)  D(u,u) ] 
    .

Here the ∂_(1), ∂_(2) denote derivatives of D with respect to first and second variable respectively. D symmetrically has same derivatives and thus when taking the difference this term gets cancelled. Ultimately we are left with
[
Following this method, one can show that

    [H_ℳ, H_𝒩] =  - 2 ia ∫_2a^∞ T_–(x) dx

giving us [H_ℳ, H_𝒩] = -i(H_ℳ-H_𝒩) +i I_2, which is in stark contrast with [K_ℳ, K_𝒩] = -i(K_ℳ - K_𝒩).
]

    [G, G'] = [H_ℳ, H_𝒩'] = i · 2a/4∫_0^2a dx ·  D(x,x)  = 2a i ∫_0^2a T_–(x) dx
    .





 §.§ The nested commutators


We shall now show that all the nested commutators are proportional to [G, G'] and find them explicitly. Before we do it, let us use C = [G, G'] notation to help readability. Proceeding, to get all the nested commutators, note that we only need to evaluate two terms: [G, C] and [G', C]. The same integrals that appeared above will make appearance here.

First we note that [G, C] =  [I_2, C], since I_1 is integrated over (2a,∞) and thus would contain operators spacelike to C. Therefore

    [G, C] = 2i a ∫_0^2a dy ∫_0^2a dx · x · [T_–(x), T_–(y) ]

Again, one can split this into two integrals involving derivatives over x and y, and like in (<ref>), (<ref>), the terms with ∂ D cancel, and we're left with

    [G, [G,G']] =-2a ∫_0^2a dx · D(x, x) = i^2 · 2a  · 4 ∫_0^2a dx · T_–(x)


    = 4 i [G, G'].

Now let us look at [G', C]. One can get G' = 𝒢- G, and from there it follows that

    [G', C] =(2ia) ∫_0^2a dx · (2a-x) ∫_0^2ady·[ T_–(x), T_–(y)]
    .

which is the same as [G, C], except for x → 2a-x in the integrand. When integrating, this just leads to a minus sign on the term that does not get cancelled, and we get

    [G', [G, G']] = -4i [G, G']
    .

Therefore, all the nested commutators are just (4i)^n(-1)^k [G, G'] for some n,k. 



 §.§ The Zassenhaus formula


The action of half sided translations formally can be written as an infinite product:

    exp(i s 𝒢)  = exp[ i s (G + G') ] = exp(is G) exp(i s G') · e^Z_2 e^Z_3 e^Z_4⋯

using the Zassenhaus formula <cit.>. The terms Z_2, Z_3, … involve nested commutators of  G and G', and the procedure of finding them is very tedious in general. Although there are some cases where a simplification can be made <cit.>, we will see that our situation doesn't satisfy their conditions, and hence we must compute the Z 's iteratively. Because all the nested commutators are simply proportional to  [G,G'] however, this is a doable task. 

To start, let us take A = i s G, B = i s G';  both  A, B are anti-hermitian, and so is [A, B]. Because all the Z's commute and are proportional to [G, G'], we will end up with the form

    exp(is 𝒢) = exp (i s G) exp(i s G') ·exp( [G, G'] f(s))

for some f(s). For this operator to be unitary, f(s) must obey f(s)= f(s). 
As noted earlier, [G, C] = -[G',C] = 4iC, thus we can say that [A, C] = α = - [B, C] for α = -4s. In this case we find using the recursive formulas <cit.> that

    Z_n = - · C α ^n-1/(n+1)! 
    .

Therefore the resultant series is easily summable, giving us

    ∑_n=2^∞ Z_n =  -C ·(e^α - 1 - α)/α^2


    e^is 𝒢 = e^is G e^is G'exp([G, G'] ·1-4s - e^-4s/16 s^2 )

where we've substituted back α = -4s, and obtained a real function f(s),

    f(s) = 1- 4s - e^-4s/16 s^2

ensuring unitarity of e^is 𝒢. 

This expression clearly reflects that if you're looking at action of 𝒢 on local operators within ♢_N, they would commute with C and G' and therefore only effect would be of G'. In the next section, we will see this point in more detail and further study the significance of this result.












§ THE FLOW OF G = H_ℳ - H_𝒩





Let us consider a local operator ϕ(x,t), where x,t belongs to any of the regions 1, 2, or 3, marked in the figure <ref>. Explicitly, the regions are

    R_1 = ♢_N, R_2 = ♢_M ∩♢_N,    R_3 = {x^+ > 0, x^- > 0}.

In this section we will study the action of the operator exp[i s G] on ϕ(x,t), where G =H_ℳ - H_𝒩, made out of the entanglement Hamiltonians (<ref>), (<ref>) associated to the algebras ℳ and 𝒩 (<ref>) respectively. 
As we mentioned, G = I_1 + I_2, where I_2 is integrated over [0, 2a], and thus has no effect on local operators in R_1. Hence, as far as the action of G on region 1 is concerned, we will ignore I_2 and focus only on I_1.

Before we proceed, we need to highlight an important result that will simplify the subsequent calculations enormously. 
Since ϕ is a scalar field, for a fixed point P with coordinates (x,t) and x^± = t± x,

    ϕ(x,t) = ϕ̂(x^+,x^-)

where we use ϕ̂ to denote the field in null coordinates. Further note, translations in x^- are generated by P^+, giving us

    ϕ(x^++a, x^-)    = e^i P^- aϕ(x^+,x^-) e^-i P^- a
       =  ϕ̂(x^+,x^-) + i a [P^-, ϕ̂(x^+,x^-)] + i^2 a^2/2 [P^-, [P^-, ϕ̂(x^+,x^-)]] + …
       =  ϕ̂(x^+,x^-) +  a ∂_+ϕ̂(x^+,x^-) + a^2/2∂_+^2 ϕ̂(x^+,x^-) + …
    .

Now, note a must not depend on x^+ for this to work. To connect with our problem, let P be (x,t),  Q be (x-t,0), and R be (x+t,0). The latter we will use in the next section. Then note that

    ϕ̂(x^+_Q,x^-_Q) = ϕ̂(x^+_P, x^-_P) + (Δ x^+) ·∂_+ϕ̂(x^+_P, x^-_P) + (Δ x^+)^2/2·∂_+^2 ϕ̂(x^+_P, x^-_P) + …

because x^-_P = x^-_Q = t-x. On the other hand, note Δ x^+ = x^+_Q - x^+_P = -2t does not change with x^-_Q, because Δ x^+ vector is orthogonal to x^-_Q variations. This is an important point because it allows for an enormous simplification:

    ∂_-ϕ̂(x^+_Q, x^-_Q)  = ∂_-ϕ̂(x^+_P, x^-_P) + ( Δ x^+ ) ·∂_- ∂_+ϕ̂(x^+_P, x^-_P) + (Δ x^+)^2/2∂_-∂_+^2 ϕ̂(x^+_P, x^-_P) + …


    = ∂_-ϕ̂(x^+_P, x^-_P)

where I've used the equation of motion in null coordinates: ∂_- ∂_+ ϕ = 0 and the commutativity of partial derivatives ∂_+, ∂_-. As a consequence, ∂_- ϕ(x-t,0) = ∂_- ϕ(x,t). The same analysis can be done in x^+ direction, and that tells us ∂_+ ϕ(x+t,0) = ∂_+ ϕ(x,t).

















 §.§ Flow in Region 1






To understand the flow in this region, let us define

    ϕ(x,t,s) = e^is I_1 ϕ(x,t) e^-is I_1

where I_1 is in (<ref>). Now note that T_–(x) = 1/2 (T_00(x) - T_01(x)) =  1/2 (ℋ(x) + 𝒫(x)). This is true for any conformal field theory with vanishing stress tensor trace. In other words,
    
    I_1 =  2 a ∫_2a^∞ dx ·   T_–(x) = 2a ∫_2a^∞ dx   1/2(ℋ(x) + 𝒫(x))

which makes it evident that if the range of integration were the whole line, integral would become 2 a P^+. As mentioned, this observation alone, in conjunction with causality, should convince the reader that the flow in region 1  generated by I_1 is same as that generated by 𝒢. 

Let us do an explicit computation now to show this, that will also help us set up the machinery for the more complicated cases of regions 2 and 3.
Using the Hadamard Lemma, we can write

    e^is I_1ϕ(x,t) e^-is I_1 = ϕ(x,t) + is [I_1, ϕ(x,t)] + (is)^2/2 [I_1, [I_1, ϕ(x,t)]] + …
    .
 
Now note, 
    
    [I_1, ϕ(x,t)] = a ∫_2a^∞ dy    [(T_0 0 - T_0 1) , ϕ(x,t)]
        .

For ϕ(x,t) to be in region 1, ♢_N = {x^+ > 0, x^-< -a}, by definition, both the delta functions in the commutators (<ref>) and (<ref>) contribute ordinarily, but because we are dealing with T_00 - T_01, the integral becomes 

    [I_1, ϕ(x,t)] = - 2ai ∫_2a^∞ dy ·∂_- ϕ(y,0) δ(x-t-y)
 

    = -2ia ∂_- ϕ(x-t,0)
    .

which proves that 
    
    ∂ϕ(x,t, s)/∂ s = 1/i [ϕ(x, t), I_1] = 2a ∂_- ϕ(x,t)
        .

where we have used (<ref>). The right side is independent of s, and therefore as long as the point x,t is in region 1, the flow is along the null direction x^-; we don't need to compute the higher order terms in the expansion of e^is I_1ϕ(x,t) e^-is I_1.








 §.§ Flow in Region 2


In region 2, R_2 = ♢_M ∩♢_N, the derivative is 

    ∂ϕ(x,t,s)/∂ s = i [I_1, ϕ(x,t)] + i [I_2, ϕ(x,t)].

As we make the transition from region 1 to 2, we're losing a part of the effect previously generated by I_1, while gaining an effect generated by a part of I_2. Below we will see this explicitly.

To begin the computation, we need to use the results (<ref>), (<ref>) to write

    [T_– (y), ϕ(x,t)] = - i∂_- ϕ(y,0) δ(x-t-y)
    .

Note that in region 2, delta function contributes at y = x-t, which must be less than 2a, and therefore I_1 contributes nothing to the derivative. Therefore

    ∂_s ϕ(x,t,s) = i [I_2, ϕ(x,t)] = i ∫_0^2a  dy · y · [T_–(y), ϕ(x,t)]


    = + ∫_0^2a  dy · y ·∂_- ϕ(y,0) ·δ(x-t-y)


    = (x-t) ·∂_- ϕ(x-t,0)  = (x-t) ∂_- ϕ(x,t)

where again we have used (<ref>).
Note that as x - t → 2a, the boundary of regions R_1, R_2, ∂_s ϕ is same as (<ref>), therefore the flow is continuous across the boundary.


This result would have been very surprising had we not known that all the non-commutativity of [G, G'], and the ensuing (infinite) Zassenhaus series can be tamed. Ultimately is shows that the effect of non-commutativity is to create an effect of a graded deceleration of the vector field, all the way to zero as one reaches the x=t line, the boundary of ♢_M. It means that the local operator localized in region ♢_M is never able to escape. Mathematically this seems very interesting,
    
    e^i s Gℳ e^-i s G⊂ℳ,    s ∈ℝ
        ,

because it is in stark contrast to (<ref>), which holds only for half of the line, and like mentioned in the discussion that followed, it is impossible to find a 𝒢 such that (<ref>) is true for s ∈ℝ, which is the distinguishing property of half sided translations from modular flows of ℳ.







 §.§ Flow in Region 3


In region 3, R_3 = {x^+ > 0, x^- > 0}, as we can again look back from [T_–, ϕ] commutators, the derivative is exactly zero. G operator, in other words, is unable to cause any effect in the region 3, despite it being a region where G has a causal influence.

Here we should mention that because of the symmetry between G and G' it is evident that same kind of results can be expected for the flow of G'. Therefore we can apply exactly the same analysis with ℳ→𝒩' and 𝒩→ℳ since ℳ' ⊂𝒩'.



















































































































































































 §.§ General evolution operator


Let us now consider the operator

    𝒪 = ∫_a^b dx ·  (f_0(x) T_00 (x,0) + f_1(x) T_01(x,0) )

where b > a >0. Given that our field is conformal, this is the most generic operator one can take for creating flows for local operators. Let us see whether this operator has the capacity to take a local operator outside the region ♢_M. In other words, let us compute [𝒪, ϕ(x,t)]. This is fairly easy now since we've already obtained the tools for it.  First let us write down

    F(x)  = f_0(x) T_00 (x,0)+ f_1(x) T_01(x,0),

then again using the results (<ref>), (<ref>), we get

    [F(x), ϕ(x,t)] = i [f_1(x)  - f_0(x)] ∂_- ϕ(y,0) ·δ(x-t-y)  - i[f_1(x)+f_0(x)] ∂_+ ϕ(y,0) ·δ(x+t-y)
    .

To see the meaning of this result,[Post integration one would also have to use (<ref>) to connect derivatives at (x± t, 0) to (x,t). ]  we will take the limit x → t so that we are on the boundary, as we want to see if the operator can take a local operator outside ♢_M. Well, if x - t = 0, then δ(-y) is left, and that means the integral better pass the origin for delta function to click and give possibility of motion along x^-. Clearly however, for any f_1, f_0, this will never happen. On the other hand, there will be generic motion along x^+, given by ∂_+ ϕ(x,t). So, no 𝒪 will be able to lead to an exit for the local operator. 
It doesn't matter what a,b are, so long as a >0. And δ(x+t-y) will click if x+t lies in (a,b), otherwise it will be zero.  

Since (<ref>) is the most general operator one could take that is made up of degrees of freedom of system contained in the  ♢_M region, or the algebra  ℳ, this result implies that any shouldn't permit the flow of local operator ϕ(x,t) outside ℳ.
The generality of this result may make one suspect whether it can be derived with just the properties of the von Neumann algebras, however, I am not aware of any such argument, and the proof here is based on the explicit computation of the commutator for the massless scalar field. 






§ CONCLUSION


In this work, we have studied the flow of local operators in a 1+1 dimensional conformal field theory, using the massless scalar field as an example. We have shown that for the regions ℳ, 𝒩 chosen in the Minkowski spacetime, the effect of the half-sided translation generator 𝒢 can be understood in terms of the entanglement Hamiltonians H_ℳ, H_𝒩, H_ℳ', H_𝒩'. The non-commutativity of [H_ℳ, H_𝒩'] creates the central challenge in splitting the effect of 𝒢 in two parts, which introduces the need for the infinite Zassenhaus product. Here we have been able to compute this product exactly.

Further, we have shown that the flow generated by the entanglement Hamiltonian difference H_ℳ - H_𝒩 is unable to take a local operator outside the region ♢_M, which is the region of spacetime accessible to the algebra ℳ. This result is in contrast to the half sided translations, which can take a local operator outside the region ♢_M. We have also shown that the flow generated by H_ℳ - H_𝒩 is continuous across the boundaries of the regions ♢_M, ♢_N, and that the flow is exactly zero outside the region ♢_M, even though causally H_ℳ - H_𝒩 is permitted to have an influence there. We have generalized this result to a generic operator 𝒪 made out of ℳ 's degrees of freedom, and have shown that it cannot take a local operator outside the region ♢_M.


 

 §.§ Application to AdS/CFT and the island formula


In the work <cit.>, the authors considered a thermal bath in equilibrium with an AdS_2 black hole, which lead to unbounded growth of entanglement entropy of outgoing radiation, leading to a Hawking like paradox. The unitarity was restored using the island formula <cit.>, which concluded that after the Page time, the bath degrees of freedom can access the black hole interior. 

The island formula asserts that before the Page time, the left and right CFTs alone are enough to reconstruct the black hole interior while after the Page time, the bath degrees of freedom are needed to the reconstruction. In <cit.>, the authors showed that a half-sided translation protocol using the left and right CFTs succeeds to do interior reconstruction before Page time, but fails to do so after the Page time. 

In the same spirit, the island formula implies that before the Page time the bath degrees of freedom are not needed, This part should follow easily by using the methods in this paper.
For instance, by considering the algebras ℳ, 𝒩 shown in the figure <ref> depicting AdS_2 in equlibrium with the left and right baths, and asking the question whether the bath operators can be brought into the AdS region is exactly the problem we've studied in this paper, because bath is a flat spacetime, and the field theory considered in the work is also conformal. 










We will try to make this above connection with island paradigm more explicit in a future work. We also suspect that several of these features may extend beyond the massless scalar, which is an interesting direction for us. 



  §.§.§ Acknowledgements


I'd like to thank Budhaditya Bhattacharjee and Adarsh Sudhakar for helpful discussions. 









































































§ APPENDIX


















































































 §.§ Unequal time commutators for massless scalar field




For our calculations, we will need the commutators


    [T_00(y, 0), ϕ(x, t)],   [T_0 0(y,0), π(x, t)]
 

which we'll compute using the equal time commutation relations of the field operators, and the equations of motion. Below we use ∂_y ϕ(y,0) ≡χ(y,0).


    [T_00(y, 0), ϕ(x,t)]  = 1/2 [ϕ̇ (y,0)^2 + ∂_y ϕ(y,0)^2, ϕ(x,t)]


    =  1/2{π (y,0),   [π(y,0), ϕ(x,t)] } + 1/2{χ (y,0),   [χ(y,0), ϕ(x,t)] }

Now, let us write

    ϕ(x,t) = exp (i H t) ϕ(x,0) exp(-i H t)
    .
 

    = ϕ(x,0) + i t [H, ϕ(x,0)] + t^2/2 [H, [H, ϕ(x,0)]] + …

using the Hadamard's lemma. Here however we note that from Heisenberg equation of motion, [H, ϕ(x,0)] = -i π(x,0), and [H, π(x,0)] = -i ∂_t^2 ϕ(x,0), and so on, which makes this simply a Taylor series expansion, so that we can write

    [π(y,0), ϕ(x,t)]  = ∑_k t ^k/k![π(y,0),  ∂_t ^kϕ(x,0)]

This we will find using the equation of motion ∂_x^2ϕ = ∂_t π = ∂_t^2 ϕ. It is easy to see that k=0 gives -iδ(x-y). Then k=1 makes it a commutator between two π' s, and we get 0. Then k=2 gives ∂_t^2ϕ which equals ∂_x^2 ϕ and we can bring ∂_x^2 outside commutator, and so on, to get

    [π(y,0), ϕ(x,t)]  
       = ∑_k=0^∞t ^2k/(2k)!∂_x^2k[π(y,0),  ϕ(x,0)]
       = -i ∑_k=0^∞t ^2k/(2k)!∂_x^2kδ(y-x) 
       = -i cosh(t ∂_x ) (δ(x-y) )

Now spending a few minutes staring at it, one may think perhaps we can write it as some δ(x + f - y) because it does look like some Taylor series. Well, kind of; we can write it as sum of two Taylor expansions:

    [π(y,0), ϕ(x,t)] = -  i/2(δ(x+t - y) + δ(x-t - y))


The opposite signs of t lead to only even terms, which we can divide by 2, to get our answer. Likewise, a similar analysis for χ term yields sinh (t∂_x) acting on the delta function, giving us

    [χ(y,0), ϕ(x,t)] = - i/2 (δ(x+t-y)  - δ(x-t-y))


which only picks up odd terms from the full expansion. Thus

    [T_00(y,0), ϕ(x,t)] =  - i  ∂_+  ϕ(y,0) ·δ(x+t-y) - i ∂_- ϕ(y,0) ·δ(x-t-y).


Likewise we can compute [T_00(y, 0), π(x,t)], if we wish, which may be needed for computing nested commutators of the form [H, [H, ϕ]]. However we will not need them. The only other commutator we need would be [T_01(y,0), ϕ(x,t)], which simplifies to 


    [T_01(y,0), ϕ(x,t)] =  ∂_y ϕ(y,0) ·   [π(y,0), ϕ(x,t)]  + π(y,0) · [∂_y ϕ(y,0), ϕ(x,t)]

and by using (<ref>) and (<ref>) can be written as

    [T_01(y,0), ϕ(x,t)]    =  - i  ∂_+  ϕ(y,0) ·δ(x+t-y) + i ∂_- ϕ(y,0) ·δ(x-t-y).

Note that compared to (<ref>), the only difference is a sign for ∂_- ϕ term.






 §.§ Simplifying H_ℳ - H_𝒩 for Minkowski spacetime




Note that

    H_𝒩 = ∫_a^∞ (x-a) T_00(x,-a)dx

since its a CFT, T_00(x, -a) = T_++(x-a) + T_–(x+a), so

    H_𝒩 = ∫_a^∞ (x-a) ( T_++(x-a) + T_–(x+a))dx


    = ∫_a^∞ (x-a)  T_++(x-a) dx + ∫_a^∞ (x-a)T_–(x+a)dx


    = ∫_0^∞ x  T_++(x) dx + ∫_2a^∞ (x-2a)T_–(x)dx


    = ∫_0^∞ x  T_++(x) dx + ∫_0^∞ (x-2a)T_–(x)dx - ∫_0^2a (x-2a)T_–(x)dx


    = ∫_0^∞ x  T_00(x) dx -2a ∫_0^∞ T_–(x)dx - ∫_0^2a (x-2a)T_–(x)dx

where we use T_00(x) = T_++(x)+ T_–(x), then note that second and third integrals combine to form

    H_𝒩= H_ℳ-2a ∫_2a^∞ T_–(x)dx - ∫_0^2a x · T_–(x)dx

so

    H_ℳ - H_𝒩 = 2a ∫_2a^∞ T_–(x) dx + ∫_0^2a x · T_–(x)dx


Similarly one can get H_𝒩' - H_ℳ' and that way verify that 𝒢 = a P^+.




JHEP
 






 Paper Title: Black bounces in Cotton gravity.






	 ednaldobarrosjr@gmail.com
Faculdade de Física, Universidade Federal do Pará, Campus Universitário de Tucuruí, CEP: 68464-000, Tucuruí, Pará, Brazil

     
    tarcisojunior17@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill

	 fslobo@ciencias.ulisboa.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal
Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     esialg@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill
Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil

   drubiera@ucm.es
Departamento de Física Téorica and IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain

      
        fc53497@alunos.fc.ul.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     henriquefisica2017@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill





§ ABSTRACT 


Recently, J. Harada proposed a theory relating gravity to the Cotton tensor, dubbed as “Cotton gravity" (CG). This is an extension of General Relativity such that every solution of the latter turns out to be a solution of the former (but the converse is not true) and, furthermore, it is possible to derive the cosmological constant as an integration constant within it. In this work we investigate CG by coupling it to both non-linear electrodynamics (NLED) and scalar fields.  We study static and spherically symmetric solutions implementing a bouncing behaviour in the radial function so as to avoid the development of singularities, inspired by the Simpson-Visser black bounce and the Bardeen model, both interpreted as magnetic monopoles. We identify the  NLED Lagrangian density and  the scalar field potential generating such solutions, and investigate the corresponding gravitational configurations in terms of horizons, behaviour of the metric functions, and regularity of the Kretchsman curvature scalar. Our analysis extends the class of non-singular geometries found in the literature and paves the ground for further analysis of black holes in CG.



04.50.Kd,04.70.Bw



Black bounces in Cotton gravity
    Henrique A. Vieira
    August 5, 2024
===============================


HMS










 








§ INTRODUCTION




Recently, Harada <cit.> proposed a theory that generalizes General Relativity (GR), known as “Cotton gravity" (CG). It is based on the use of the Cotton tensor, assumed to encapsulate the effects describing gravity, and leading to third-order field equations in the derivatives of the metric tensor  <cit.>. Part of the interest in this theory comes from the fact that  every solution of  Einstein's equations of GR (including a cosmological constant term) is a solution of CG gravity via a particular Cotton tensor. Indeed, the cosmological constant itself arises as an integration constant of the theory. Furthermore, in <cit.> it was shown that a particular new solution of CG gravity not present within GR could be able to explain the galactic rotation curves without the need for dark matter, further raising the interest in the theory. As a consequence, different applications of this theory have been studied very recently in the literature <cit.>. In fact, there is a current debate on the theory as a viable extension of GR <cit.>, which highlights the need for further work to determine the potential of Cotton gravity to supersede GR.

Among the many gravitational configurations explored in the literature beyond the canonical (Kerr) black hole of GR, those aimed to solve the singularity problem are of particular interest. Such a problem is related to the unavoidably development of space-time singularities (as given by the incompleteness of at least one geodesic trajectory) inside black holes within the GR framework (see <cit.> for a discussion of this topic), and which typically correlates with the presence of infinities in some sets of curvature scalars. A canonical strategy in the field at this regard is to imbue the radial function with a non-trivial bouncing behaviour in order to prevent the focusing of geodesics appearing in the singularity theorems, this way allowing geodesic congruences to re-expand at a minimum radial distance. Examples of this are the well known Bardeen model <cit.> and, more recently, the class of black bounces introduced by Simpson and Visser in <cit.>. Typically, models of this kind introduce one (or more) parameters associated to a minimum length scale, which is responsible for the bouncing behaviour. Furthermore, the bounce can be hidden behind event horizons thus representing regular black holes, or be naked instead, interpreted as a kind of traversable wormhole allowing free communication between far-away distances <cit.>. 

The various forms of the black bounce proposal have led to the development of several additional classes of solutions <cit.> and the analysis of their properties. Among them, we can underline regularity, causal structure and energy conditions <cit.>, gravitational lensing  <cit.>, and observational  signatures <cit.>, besides generalizations to the rotating case <cit.>. Typically, such bouncing configurations are supported by a combination of a non-linear electrodynamics (NED) and a scalar field   <cit.> working under a reverse procedure: the geometrical line element incorporating the bouncing behaviour is set first, and then the field equations are driven back to find the explicit matter sources treading the geometry.

The main aim of this work is to implement suitable generalizations of both Simpson-Visser and Bardeen solutions within CG gravity according to the new features of the theory. In particular, we shall generalize the exact vacuum solution of CG of <cit.>, characterized by a new parameter γ, towards bouncing geometries supported by NLED and scalar fields. The geometries obtained this way depend on the mass of the black hole M, on the magnetic charge q, and on the CG parameter γ, besides the cosmological constant itself Λ. Given the four-parametric nature of these CG solutions, we shall take different choices for them in both the Simpson-Visser and Bardeen solutions, hopefully covering the most relevant conceptual classes of configurations in terms of types of horizons and global shape of the corresponding metric functions. We shall then obtain the functional shapes of the NLED Lagrangian density and the scalar field potential. 









This work is organized as follows. In Sec. <ref>, we briefly present the field equations of CG coupled to NLED and scalar fields, introduce the class of geometries we are interested in, and find the equations allowing to reconstruct the matter (NLED supported by magnetic charge) and scalar field potential. In Sec. <ref>, we  generalize the black bounce geometries and study in detail their properties, and in Sec. <ref> we do the same with the Bardeen one. In Sec.  <ref>, we summarize and discuss our results. We work in units G=c=1.







§ COTTON GRAVITY COUPLED TO NON-LINEAR ELECTRODYNAMICS AND SCALAR FIELDS






 §.§ Field equations of Cotton gravity



In CG the gravitational field equations are given by  <cit.> 
    C_αμν=κ^2∇_βΘ^β_βαμν,

where κ^2=8π, while the tensors C_αμν and Θ_αμν are defined as

    C_αμν   ≡   ∇_μR_αν-∇_νR_αμ
          -1/6(g_να∇_μR-g_μα∇_νR),
    	 
    ∇_βΘ_μαμν^β   ≡   1/2(∇_μΘ_αν-∇_νΘ_αμ)
          -1/6(g_να∇_μ-g_μα∇_ν)Θ .

respectively, where C_μν stands for the Cotton tensor, R_μν for the Ricci tensor and Θ_μν  for the energy-momentum tensor, and their traces are represented by R and Θ, respectively. Moreover, Einstein's equations of GR are solutions of Eq. (<ref>), and therefore, GR can be seen as a particular case of CG. Note that if we contract the equation of motion  (<ref>) with g^αν, the conservation law g^ανC_αμν=κ^2∇_βT_βμ^β=0 is naturally  satisfied.

For the matter sector of the theory, in this work we are interested in space-times sourced by NLEDs and scalar fields, whose energy-momentum tensors can be written as

    ∇_βΘ_μαμν^β=∇_βFΘ_μαμν^β+∇_βφΘ_μαμν^β.

Next, we will explicitly show the forms of these matter components described by the energy-momentum tensor:

    ∇_βFΘ_μαμν^β   ≡   1/2(∇_μFT_αν-∇_νFT_αμ)
          -1/6(g_να∇_μ-g_μα∇_ν)FT,
    ∇_βφΘ_μαμν^β   ≡   1/2(∇_μφT_αν-∇_νφT_αμ)
          	-1/6(g_να∇_μ-g_μα∇_ν)φT ,

respectively, with

    FT_μν   =g_μν L_ NLED(F)- L_FF_μαF_ν^να,
    FT   =4 L_ NLED(F)-4 L_FF,
    φT_μν   =2 ϵ ∂_μφ∂_νφ-ϵ 
    g_μν∂^σφ∂_σφ+g_μνV(φ),
    φT   =-2 ϵ ∂^νφ∂_νφ+4V(φ),

where  ϵ=+1 is the scalar field, while ϵ=-1 represents the phantom scalar field, φ is the scalar field, while V(φ) denotes the scalar potential,  L_ NLED(F) is the Lagrangian density describing a NLED that depends on the electromagnetic scalar

    F=1/4F^μνF_μν,

 and the  Maxwell-Faraday antisymmetric tensor is defined by F_μν = ∂_μ A_ν -∂_ν A_μ,

  

 where A_α is the vector potential. 
 
 The field equations for these two matter contributions can be found by the usual variational procedure as 

    ∇_μ ( L_F F^μν)   =1/√(-g)∂_μ (√(-g) L_F F^μν)=0,
    
         2∇_ν∇^μφ   =-dV(φ)/dφ ,

respectively, where g is the determinant of the metric and we have denoted  L_F=∂ L _ NLED(F)/∂ F.





 §.§ Spherically symmetric solutions



For the sake of our considerations we shall work in the static, spherically symmetric, line element 

    ds^2=A(r)dt^2-B(r)dr^2-Σ^2(r)   dΩ^2,

where A(r), B(r) and Σ(r) are functions of the radial coordinate r and dΩ^2≡ dθ^2+sin^2(θ)dϕ^2



is the line element on the two-spheres. We point out that in this line element we have included a non-trivial radial function Σ^2(r) given the fact that we are interested in geometries implementing a bouncing behaviour in the radial sector, for which Σ^2(r) is non-monotonic. On the other hand, it is always possible to perform a change of variables of the form B(r)dr^2=dx^2/A(r) to write the line element (<ref>) under a more canonical form with g_tt=g^xx, meaning that one of these two metric functions is redundant. Nonetheless, for the sake of generality, we shall work in the line element above and later fix one of the metric functions. 

In Ref. <cit.>, Harada found the following exact vacuum solution of the CG field equations  (<ref>) as

    A(r)=1/B(r)=1 - 2M/r - Λ/3r^2 + γ r.

where M is the standard mass term of the Schwarzschild solution (to which the solution above reduces when Λ=0 and the parameter γ resulting from the theory goes to zero, γ→ 0). Furthermore,  Λ arises as an integration constant, and given the fact that the term -Λ r^2/3 denotes a de Sitter term makes Λ naturally interpreted as a cosmological constant term. 

Our aim is to generalize the above solution in CG to implement bouncing behaviours in the radial function. To this end we shall take the canonical choice, due to Ellis <cit.>, as

    Σ(r)=√(L_0^2+r^2) ,

where the parameter L_0 ∈ is a regularization parameter and possesses the dimension of a length. This constant can be consistently interpreted as a magnetic charge q, provided that we source the line element with a NLED and scalar fields. This means that the coordinate r extends over the whole real line, r ∈ (-∞,+∞), while the function Σ(r) which gives the area of the two-spheres as S=4πΣ^2(r) takes a minimum value Σ(r)=q at r=0.

The invariants associated to such fields are

    F_23=q sinθ ,

for the electromagnetic field, and 

    F=q^2/2 Σ^4(r).

for the scalar one. In addition, to assess the consistency of our solutions we shall also use the following useful relationship

    L_F=∂ L _ NLED/∂ r(∂ F/∂ r)^-1 .


The field equations (<ref>) are third-order coupled gravitational equations for the metric, which makes it very difficult to obtain a solution by direct integration when the matter content is included. To overcome this difficulty and find new solutions for these equations, we use the following approach in our work: we propose the spherical symmetry for the metric as well as a functional form that already contains the new characteristic term of the vacuum solution in CG while at the same time implementing the bouncing behaviour given by Eq. (<ref>), and then integrate the equations of motion to determine the Lagrangian of the NLED as well as its derivative in order to check the consistency equation (<ref>). In this way, we will be able to generalize some known bouncing solutions found within GR (namely, the Simpson-Visser and Bardeen solutions), and obtain the specific shapes of the matter (electromagnetic and scalar),

To run this procedure we first solve the (0,0,1) and (2,1,2) components of the CG field equations (<ref>). This provides cumbersome yet analytic expressions for both the NLED Lagrangian density and its derivative as


    L_ NLED(r)= f_0+2f_1q^2/√(q^2+r^2)+∫1/2κ^2q^2A(r)^3B(r)^3(q^2+r^2)^5/2{2B(r)^2(q^2+r^2)^3A'(r)^3
          
    +(q^2+r^2)^2A(r)B(r)A'(r)[A'(r)(2(q^2+r^2)B'(r)-rB(r))-4B(r)(q^2+r^2)A”(r)]
    
          	
    +2A(r)^3[-2rB(r)^3(q^2+r^2)-2r(q^2+r^2)^2B'(r)^2+2rB(r)^2(κ^2ϵ(q^2+r^2)^2φ'(r)^2+3q^2+r^2)
    	.
    
          
    +B(r)(q^2+r^2)(r(q^2+r^2)B”(r)+3q^2B'(r))]
    -A(r)^2(q^2+r^2)[-2(q^2+r^2)^2A'(r)B'(r)^2
          	
    +B(r)(q^2+r^2)(3(q^2+r^2)A”(r)B'(r)+A'(r)((q^2+r^2)B”(r)+rB'(r)))
    
          
    +2B(r)^2(A'(r)(κ^2ϵ(q^2+r^2)^2φ'(r)^2+q^2+2r^2)-(q^2+r^2)(A^(3)(r)(q^2+r^2)+2rA”(r)))] } dr
          +∫1/κ^2A(r)^3B(r)^3(q^2+r^2)^3{-2B(r)^2(q^2+r^2)^3A'(r)^3
          	-(q^2+r^2)^2A(r)B(r)A'(r)[A'(r)(2(q^2+r^2)B'(r)-rB(r))-4B(r)(q^2+r^2)A”(r)]
    	
          	-A(r)^3[-4r(q^2+r^2)^2B'(r)^2+B(r)^3(q^2+r^2)(κ^2(q^2+r^2)^2V'(r)-4r)
    	.
          	+B(r)(q^2+r^2)(2r(q^2+r^2)B”(r)+B'(r)(κ^2ϵ(q^2+r^2)^2φ'(r)^2+6q^2))
    	
          .+2B(r)^2(2r(3q^2+r^2)-κ^2ϵ(q^2+r^2)^3φ'(r)φ”(r))]
    	
          	+A(r)^2[-2(q^2+r^2)^2A'(r)B'(r)^2-2B(r)^2(q^2+r^2)(A^(3)(r)(q^2+r^2)+2rA”(r))
          	+B(r)(q^2+r^2)(3(q^2+r^2)A”(r)B'(r)+A'(r)((q^2+r^2)B”(r)+rB'(r)))
    	
          	+B(r)^2A'(r)(3κ^2ϵ(q^2+r^2)^2φ'(r)^2+2(q^2+2r^2))](q^2+r^2)]} dr,

and 
 
    L_F(r) = (q^2+r^2)^3/2{f_0+∫1/2κ^2q^2A(r)^3B(r)^3(q^2+r^2)^5/2{2B(r)^2(q^2+r^2)^3A'(r)^3
          
    +(q^2+r^2)^2A(r)B(r)A'(r)[A'(r)(2(q^2+r^2)B'(r)-rB(r))-4B(r)(q^2+r^2)A”(r)]
    
          	
    +2A(r)^3[-2rB(r)^3(q^2+r^2)-2r(q^2+r^2)^2B'(r)^2+2rB(r)^2(κ^2ϵ(q^2+r^2)^2φ'(r)^2+3q^2+r^2)
    	.
    
          
    +B(r)(q^2+r^2)(r(q^2+r^2)B”(r)+3q^2B'(r))]
    -A(r)^2(q^2+r^2)[-2(q^2+r^2)^2A'(r)B'(r)^2
          	
    +B(r)(q^2+r^2)(3(q^2+r^2)A”(r)B'(r)+A'(r)((q^2+r^2)B”(r)+rB'(r)))
    
          
    +2B(r)^2(A'(r)(κ^2ϵ(q^2+r^2)^2φ'(r)^2+q^2+2r^2)-(q^2+r^2)(A^(3)(r)(q^2+r^2)+2rA”(r)))] } dr}.


respectively, where f_0 and f_1 are integration constants.

As for the scalar field φ, we choose the corresponding expression of the GR case which is given by <cit.>

    φ(r)=tan ^-1(r/q)/√(κ ^2 (-ϵ )).

After we substitute Eq. (<ref>) into Eqs. (<ref>) and (<ref>) and then into the condition (<ref>) with the choice (<ref>) we arrive at the equation

    (q^2+r^2)/2r[-2A'(r)/κ^2-(q^2+r^2)^2V'(r)/q^2]=0 ,

which yields the following solution for the potential:

    V(r)= V_0 + ∫ -2  q^2 A'(r)/κ ^2(q^2+r^2)^2 dr.


This completes our framework to finding new solutions in CG gravity. In order to achieve it so we just need to set a functional form for the metric coefficient A(r) (since as mentioned above we can always choose B=A(r)^-1 without any loss of generality). In this sense, in the following sections  <ref> and <ref> we present the bouncing solutions for the generalizations of Simpson-Visser and Bardeen solutions. In this sense, we use the set of equations   (<ref>) and (<ref>)  for the NLED Lagrangian density and its derivative, and (<ref>) and (<ref>)  for the scalar field potential, in order to reconstruct the functional form of the matter field Lagrangians.





§ SIMPSON-VISSER TYPE SOLUTION






 §.§ Metric and solutions



A natural generalization of the solution (<ref>) is to imbue it with the choice (<ref>) for the radial function, that is, to propose the metric coefficient

    A(r)= 1-2M/(q^2+r^2)^1/2-1/3Λ(q^2+r^2)+γ√(q^2+r^2).

This is the counterpart of the Simpson-Visser black bounce introduced in <cit.>, which is recovered when the CG parameter γ=0 (and for vanishing cosmological parameter). The black bounce proposal via the choice (<ref>) has become canonical in  the literature given its ability to regularize curvature singularities in any spherically symmetric space of interest, like ours. Furthermore, it has been shown that it is generically supported by combinations of NLED and scalar fields, see e.g. <cit.>. A similar approach is possible in our case, as shall be shown later in this section. 


In order to analyze the properties of the metric function (<ref>) we shall characterize the presence of horizons via the condition g_rr^-1(r_H)=0 (which coincides with the Killing horizon g_tt=0), that is:

    A(r_H)=0,

where the radius r_H denotes the presence of the horizon. In order to analyze the number and nature of such horizons, we add the condition 

    d A(r_H)/dr|_r=r_H=0,

which is capable to detect the extreme black holes (characterized by a degenerate horizon). Such a relation will numerically provide the critical values of the parameters of the model, namely, M, q, Λ or γ for which such extreme black holes exist. Elaborating from them we shall be able to characterize the remaining solutions within each choice of the parameters.  Given  the fact that there are four free parameters, in our approach we set three of them and take the remaining one (either M, q, or γ)  as a free parameter, and investigate the corresponding solutions for both positive and negative values of the cosmological constant Λ. This leads to six classes of solutions in order to explore all the possible geometrical structures within this setting.  For each of them we numerically determine the value of the critical values of the free parameter (either M, q, or γ) from the simultaneous resolution of Eqs. (<ref>) and (<ref>).








(i) Type-I solution: {q = 0.5, Λ = -0.2, γ=0.01}

For these values the critical mass, found by solving simultaneously Eqs.(<ref>) and (<ref>),  takes the value M_c = 0.255. Fig. <ref> shows the behavior of the metric function (<ref>) in terms of the radial coordinate r, for three mass scenarios: M > M_c, M = M_c, and M < M_c. When the mass exceeds the critical value, M > M_c, we observe the presence of two event horizons. This is possible given the fact that at r=0 we have a bounce from the region r > 0 to the region r < 0, and conversely, due to the symmetry with respect to r, a bounce from the region r < 0 to the region r > 0 is also possible. In the case where the mass is equal to the critical mass, M = M_c, we observe a degenerate double horizon at r=0, and a bounce is still present at r=0. Finally, when the mass is less than the critical mass, M < M_c, there is no horizon formation, and the geometry we have is a wormhole with a traversable throat at r=0. 



























(ii) Type-II solution: {q = 0.5, Λ = 0.2, γ=0.01}

In this case the critical mass is given by M_c=0.770. In the case where M < M_c, we see in Fig. <ref> the formation of four horizons throughout the entire region r∈(-∞,+∞), two on each side of the bounce. The innermost horizons on each side correspond to the event horizons, and the outermost are the cosmological horizons, denoted here by r_Λ. This is due to the fact that outside the cosmological horizons the metric function A(r) is negative with signature for the metric (-,+,-,-). The bounce thus connects two black hole space-times both surrounded by a cosmological horizon. In the case where the mass is equal to the critical mass, M = M_c, we observe the existence of degenerate horizons on each side of the bounce, made up of the event and cosmological horizons. When the mass exceeds the critical value, M > M_c, there is no horizon of any kind, and the metric function remains negative throughout the entire r region, which can thus be interpreted as a traversable wormhole embedded in a de Sitter space-time.







(iii) Type-III solution: {γ=0.01,M=2.0,Λ=-0.2}

Now the combined resolution of Eqs.(<ref>) and (<ref>) provides instead a critical value for the charge as  q_c=2.665. The behavior of A(r) is shown in Fig. <ref> for the scenarios of q > q_c, q=q_c and q<q_c.
Now, when q>q_c, there is no event horizon, and the space-time is described by a traversable wormhole with its throat at r=0. In the configuration where q=q_c, we have a case similar to Fig. <ref> for M=M_c with a degenerate horizon at r=0. And when q<q_c, we find a situation similar to the dashed green curve represented in Fig. <ref>, which corresponds to the scenario in which M > M_c for which there are event horizons on both sides of r=0. 







(iv) Type-IV solution: {γ=0.5,M=2.0,Λ=0.2}

The critical charge for these conditions reads q_c=2.240. The behavior of A(r) as a function of r is shown in Fig. <ref>, where the charge takes  the values q > q_c, q=q_c, and q < q_c. When q>q_c, we observe the presence of two event horizons and two cosmological horizons; for q=q_c event horizons and cosmological horizons merge to form a single degenerate horizon; while for q<q_c then we have a traversable wormhole. This scenario is thus analogous to the case represented in Fig. <ref> with M<M_c. 







(v) Type-V solution: {q=0.75,M=1.0,Λ=-0.2}

In this case the joined resolution of Eqs.(<ref>) and (<ref>) provides a critical value for the constant γ_c=2.172. The behavior of A(r) is shown in  Fig. <ref> for the scenarios γ < γ_c, γ=γ_c and γ>γ_c. Now, when γ>γ_c, there is no event horizon and a traversable wormhole arises, similarly as the cases of M<M_c and q>q_c described in Figs. <ref> and <ref>, respectively.  
In the configuration where γ=γ_c, we have a case similar to Fig. <ref> for M=M_c and also to the case of q=q_c described in  Fig. <ref>  (middle left). And when γ<γ_c, we find a situation similar to the curve represented in Fig. <ref>, which corresponds to the scenarios of M > M_c and also to the case of q<q_c described in  Fig. <ref>.







(vi) Type-VI solution: {q=0.52,M=1.2,Λ=0.2}

Under these conditions, we obtain γ_c=0.133. The behavior of A(r) as a function of r is shown in  Fig. <ref>,  where the parameter takes the values γ> γ_c, γ=γ_c and γ < γ_c. In the case that γ>γ_c, we observe a similar behaviour as in  Fig. <ref>,  for M<M_c and also for the case that q < q_c, as described in  Fig. <ref>. In the case that γ=γ_c, we have a similar scenario as in  Fig. <ref>,  for M= M_c. Finally, for γ < γ_c, we have a wormhole geometry, analogous to the case in  Fig. <ref>, but with M>M_c. 





 §.§ Kretschmann scalar



On the other hand we can study the regularity properties of our model by computing the  Kretschmannn scalar as


    K=	
    1/3(q^2+r^2)^5{36 M^2 (3 q^4-4 q^2 r^2+4 r^4)+	12 M q^2[q^2 (r^2 (4 Λ√(q^2+r^2)-9 γ)-8 √(q^2+r^2))
    	
       +8 r^2 √(q^2+r^2)-2 r^4 (γ -Λ√(q^2+r^2))+q^4 (2 Λ√(q^2+r^2)-7 γ)]
    	
       +(q^2+r^2)^[4Λ^2q^8+8r^6(3γ^2-3γΛ√(q^2+r^2)+Λ^2r^2)+q^6(27γ^2-20γΛ√(q^2+r^2)+16Λ(Λ r^2-1))
    	. 
       ..4q^2r^4(12γ^2-13γΛ√(q^2+r^2)+2Λ(3Λ r^2-1))+q^4(-48γ√(q^2+r^2)(Λ r^2-1)+51γ^2r^2+4(7Λ^2r^4-6Λ r^2+9))]} 	.


which, at asymptotic infinity takes the value R(r →∞) →8 Λ ^2/3, while being regular at the bouncing region r=0, as is transparent from Eq. (<ref>).





 §.§ Matter Lagrangian reconstruction



The above solutions can be shown to be supported by a NLED plus a scalar field matter field. To obtain the explicit shapes of their Lagrangian densities, we use Eq. (<ref>)  to obtain L_ NLED as

    L_ NLED(r)    =  f_0+2f_1q^2/√(q^2+r^2)-2q^2/6κ^2√(q^2+r^2)(q^2+r^2)^2×
       ×[-45M-3γ(q^2+r^2)+4√(q^2+r^2)]+
    
       2q^2/κ^2(-6M/(q^2+r^2)^5/2-γ/3(q^2+r^2)^3/2+1/2(q^2+r^2)^2),

and its derivative L_F as

    L_F(r)    =   (q^2+r^2)^3/2[f_1-(-45M-3γ(q^2+r^2)
    	..
          .. +4√(q^2+r^2))/6κ^2(q^2+r^2)^2],

while for the scalar field  φ(r) we use Eq.  (<ref>) to find the potential V(r) as

    V(r)= 2 q^2 [6 M+5 (q^2+r^2) (γ -Λ√(q^2+r^2))]/15 κ ^2 (q^2+r^2)^5/2 .


On the other hand, inverting numerically the expressions above to obtain  r(F) and r(φ), we determine

    L _ NLED (F)     =   	f_0+2√(2)f_1√(F)q^3/2-2F/3κ^2
           +6√(2)F^5/4M/κ^2√(q)+2^3/4γ F^3/4√(q)/3κ^2,
   
and

    V(φ)    =   2(3Mcos(2√(κ^2(-ϵ)))+3M)/15κ^2(q^2^2(√(κ^2(-ϵ))))^3/2
          +2q^2(γ-Λ√(q^2^2(√(κ^2(-ϵ)))))/3κ^2(q^2^2(√(κ^2(-ϵ))))^3/2.
  
Considering f_0=0, f_1=0, γ=0 and Λ=0 in the Lagrangian (<ref>) we obtain the same expression as the one of GR as found in Ref. <cit.>. On the other hand, in expression (<ref>) the scalar field must necessarily be a phantom field, i.e. 
ϵ=-1, as is transparent in Eq. (<ref>). Note that in the limit F≫ 1, the NLED Lagrangian behaves as

    L _ NLED (F)∼    f_0+2√(2)f_1q^3/2F^1/4+6√(2)F^5/4M/κ^2√(q)
       +2^3/4γ q^1/2F^3/4/3κ^2-2F/3κ^2.

which is a non-Maxwellian behavior in the central region of the field. The bottom line of this analysis is that, similarly as in the case of GR <cit.>, the Simpson-Visser solution can be modelled as a solution of a NLED and a scalar field, though with a more general functional form.

In Fig. <ref> we depict the behaviour of the NLED Lagrangian density given by Eq. (<ref>) using the blue dashed curve for f_0=0 and f_1=0.2 and for the case similar to GR with the red dotted-dashed curve with the values f_0=0 and f_1=0. Similarly, in Fig. <ref>. we illustrate the behaviour of the scalar field potential in Eq. (<ref>) for some choices of the model parameters. Note that the amplitude of this potential changes depending on the charge values. We illustrate this for three different values of the charge: q = 0.2, q = 0.3 and q = 0.4.














§ BARDEEN-TYPE SOLUTION






 §.§ Metric and solutions



Consider now the following metric function:

    A(r)= 1-2Mr^2/(q^2+r^2)^3/2-1/3Λ(q^2+r^2)+γ√(q^2+r^2).

This is a generalization, within CG, of the so-called Bardeen magnetic monopole <cit.> via the CG parameter γ, using a similar trail of thought as in the Simpson-Visser solution of the previous section with the replacement of the radial function with the choice (<ref>). Bardeen's monopole avoids the presence of a singularity via its replacement by a de Sitter core, a canonical procedure within GR to achieve this end <cit.>, here generalized to display a bouncing behaviour on its radial sector. In this section we shall study the properties of this metric function using again to this end Eqs. (<ref>) and (<ref>) for the radius of the horizon and the critical parameters of the solution, and split the discussion also into six sub-cases.




























(i) Type-I solution: {q = 0.5, Λ = -0.2,γ=0.01}

In this case the critical mass reads M_c = 0.686. The behaviour of the metric function (<ref>) is shown in Fig. <ref> for three mass scenarios: M > M_c, M = M_c and M < M_c. When the mass exceeds the critical mass, M > M_c, we observe a total of four horizons in the entire r region, the innermost horizons being the Cauchy horizons and the outermost being the two event horizons.  At r=0 we have the throat of the wormhole, which is hidden behind these horizons. For the case where M=M_c, we have two degenerate horizons, one at r > 0 and the other at r < 0,  connected by the throat at r=0. In the case where M < M_c, we find the geometry of a traversable wormhole, a scenario similar to that in Fig. <ref> for M < M_c, Fig. <ref> for q > q_c and Fig. <ref> for γ>γ_c. 







(ii) Type-II solution: {q=0.5,Λ=0.2,γ=0.01}

In this case we find the critical mass value M_c=0.811. The scenarios M > M_c, M = M_c and M < M_c are illustrated in Fig. <ref>. When  M > M_c, two event horizons are formed, one in each r region, connected at r=0 by a bounce in the radial function. When M=M_c, we observe the formation of four horizons, the innermost being Cauchy horizons, while the outermost ones are degenerate horizons that arise from the union of cosmological and event horizons. And when  M < M_c, we observe the formation of six horizons: the two innermost ones (on each side) are Cauchy horizons, the  two intermediate ones are the event horizons, and the two outermost ones are the cosmological horizons.
 






(iii) Type-III solution: {γ=0.01,M=2.0,Λ=-0.2 }

In this case we find the  critical charge value of q_c=1.204. The behavior of A(r) is shown in Fig. <ref> for the cases q > q_c, q=q_c, and q < q_c. When q > q_c, we observe a similar behavior to the curve described in Fig. <ref> for the case where M < M_c, finding a traversable wormhole geometry. Similarly, when q=q_c we find the same behaviour as that of M=M_c in Fig. <ref>. Finally, when  q < q_c the metric function exhibits the same behavior as the one of Fig. <ref> with M > M_c. 







(iv) Type-IV solution: {γ = 0.01,M = 0.5,Λ = 0.12}

Now the critical charge reads q_c = 0.389. The behavior of A(r) is shown in Fig. <ref> for the cases q > q_c, q = q_c, and q < q_c. When q > q_c we observe a similar behavior as in Fig. <ref> for the case q > q_c; for q=q_c we note the presence of degenerate horizons and two cosmological horizons, one each in the regions r>0 and r<0; and when  q < q_c, it shows the same behavior as in Fig. <ref>, but with M < M_c.







(v) Type-V solution: {q=0.75,M=2.0,Λ=-0.2}

Now we obtain the critical value of γ_c=0.792. The behavior of A(r) is shown in Fig. <ref> for the scenarios γ < γ_c, γ=γ_c and γ>γ_c. When γ>γ_c we have a similar behavior to the cases of M<M_c and q>q_c as described in Fig. <ref> and <ref>; when γ=γ_c a similar behaviour as Fig. <ref> for M=M_c and Fig. <ref> for q=q_c; and when γ<γ_c a similar situation as in Fig. <ref> when  M > M_c.







(vi) Type-VI solution: {q=0.75,M=1.0,Λ=0.2 }

Under these conditions we obtain γ_c=0.107. The behavior of A(r) as a function of r is shown in Fig. <ref> for the values γ> γ_c, γ=γ_c and γ < γ_c. When  γ>γ_c we observe a similar behavior as in Fig. <ref> for the case of q > q_c; when γ=γ_c we also have a similar scenario as in Fig. <ref> for q= q_c; and for γ < γ_c a similar behaviour as is Fig. <ref> for q<q_c.





 §.§ Kretschmann scalar



As for the regularity properties of this family of solutions, we compute the Kretschmann scalar as


    K=	
    1/3(q^2+r^2)^7{ 12M^2(4q^8-44q^6r^2+169q^4r^4-68q^2r^6+12r^8)+4Mq^2(q^2+r^2).	
    	
       +r^6(2Λ√(q^2+r^2)-30γ)-3q^2(16r^2√(q^2+r^2)+15γ r^4)+24r^4√(q^2+r^2)
        +q^6(4Λ√(q^2+r^2)-6γ) +q^4r^2(2Λ√(q^2+r^2)-21γ)	
       +(q^2+r^2)^3[4Λ^2q^8+8r^6(3γ^2-3γΛ√(q^2+r^2)+Λ^2r^2)+q^6(27γ^2-20γΛ√(q^2+r^2)+16Λ(Λ r^2-1))+.	
       ..4q^2r^4(12γ^2-13γΛ√(q^2+r^2)+2Λ(3Λ r^2-1))+q^4(-48γ√(q^2+r^2)(Λ r^2-1)+51γ^2r^2+4(7Λ^2r^4-6Λ r^2+9))]} 	.


which provides a finite value at r→0 and, therefore, likewise in the previous case the geometry is everywhere regular.





 §.§ Matter Lagrangian reconstruction



Using Eq. (<ref>), we obtain the NLED Lagrangian density as

    L_ NLED(r)    =    f_0+2f_1q^2/√(q^2+r^2)+q^2/21κ^2(q^2+r^2)^7/2×
          ×[M(41q^2+231r^2)+7(q^2+r^2)×
          ×(γ(q^2+r^2)-√(q^2+r^2))],

its derivative as

    L_F(r)   =   (q^2+r^2)^3/2{f_1-[190Mq^2-3γ(q^2+r^2)^2
          -165M(q^2+r^2)+4(q^2+r^2)^3/2]/[6κ^2(q^2+r^2)^3]} ,
 
and the scalar field potential as







    V(r)= 2 q^2 /3 κ ^2(6 M (7 r^2-8 q^2)/35 (q^2+r^2)^7/2+γ/(q^2+r^2)^3/2-Λ/q^2+r^2),


respectively. Expressing r(F) and r(φ), we determine

    L _ NLED (F)  = 	f_0+2 √(2) f_1 √(F) q √(q/√(F))-2 F/3 κ ^2
        -380 2^3/4 F^2 M √(q/√(F))/21 κ ^2+2^3/4γ  F √(q/√(F))/3 κ ^2
        +22 √(2) F^3/2 M √(q/√(F))/κ ^2 q.
  
and

    V(φ)    =    -2 γcos ^4(φ√(κ ^2 (-ϵ ))) √(q^2  ^2(φ√(κ ^2 (-ϵ ))))/3 κ ^2 q^2
           +4 M sin ^2(φ√(κ ^2 (-ϵ ))) cos ^6(φ√(κ ^2 (-ϵ ))) /5 κ ^2 q^4×
          ×√(q^2  ^2(φ√(κ ^2 (-ϵ )))) -2 Λcos ^2(φ√(κ ^2 (-ϵ )))/3 κ ^2
            -32 M cos ^8(φ√(κ ^2 (-ϵ ))) √(q^2  ^2(φ√(κ ^2 (-ϵ ))))/35 κ ^2 q^4.
 

Considering again f_0=0, f_1=0, λ=0 and Λ=0 in the Lagrangian of equation (<ref>), we obtain the same expression as the Lagrangian of GR, found in Ref. <cit.>. The central limit, F≫1, of the corresponding NLED Lagrangian reads as 

    L _ NLED (F)  ∼  f_0+2√(2)f_1q^3/2√(F)+22√(2)F^5/4M/κ^2√(q)
    
    -2F/3κ^2   
       -380M2^3/4√(q)F^7/4/21κ^2+2^3/4γ√(q)F^3/4/3κ^2.

which is again a non-Maxwellian behaviour there.

Fig. <ref> shows the behavior of the NLED Lagrangian density using Eq. (<ref>) for the values {f_0=0,f_1=0.2} (dashed blue curve), and {f_0=0,f_1=0} (dotted-dashed red curve). Similarly, in Fig. <ref> we depict the behaviour of the scalar field potential following  Eq. (<ref>).













§ SUMMARY AND CONCLUSION





In this paper we have studied spherically symmetric geometries implementing a bouncing in the radial function within a recently proposed theory of gravity known as Cotton Gravity (CG). Such geometries are inspired by a recently found vacuum solution of CG which generalizes the Schwarzschild solution of GR via a new parameter γ, and are further extended towards the black bounce mechanism to avoid the focusing of geodesics ascribed to the singularity theorems.  We have investigated such solutions by coupling the field equations of CG to a combination of a NLED and a scalar field. Solving the corresponding field equations we have found the free functions L_ NLED (r) and L_F(r) that characterize the NLED sector, and the scalar field potential V(r) associated to a specific scalar field found within the context of GR. 

Subsequently, we have studied two generalizations of bouncing solutions found within GR, namely the  Simpson-Visser and Bardeen type solutions, both understood to be supported by magnetic NLED plus scalar fields. Such geometries differ from their GR counterparts  via an additional term depending on the parameter γ of CG, and further supplemented by a cosmological constant term given the fact that the latter naturally arises as an integration constant in CG. Given the presence of four constants in our model, {M,q,Λ,γ}, we used an approach in which we set three of them and vary the remaining one (keeping always fixed the cosmological constant term Λ to both positive and negative values), thus leading to six types of solutions on each case. 


For the first model we used a generalized version of the bouncing metric of the Simpson-Visser type and simultaneously solved the equations (<ref>) and (<ref>) to find the critical parameters for {M_c,q_c,γ_c}. For Λ < 0, in the case of the critical mass M_c,  when M > M_c we found an event horizon on both sides of the bounce r=0, which for M = M_c merge into a degenerate horizon at r=0, while if M < M_c a traversable wormhole is formed. In the case of Λ > 0, no horizons are formed for M > M_c, while degenerate horizons are observed for M = M_c, and M < M_c, two horizons are observed on each side of the bounce, the outermost ones being cosmological horizons and the two innermost ones event horizons. On the other hand, for Λ < 0, in the case of critical charge, we observe that there is no horizon for q > q_c, when q = q_c we have degenerate horizons, whereas for q < q_c we observe two horizons. In the scenario where Λ > 0, we identify the presence of two cosmological horizons in all cases, whether q > q_c, q = q_c, or q < q_c.  We also analyzed the Kretschmann scalar of this model finding its regularity everywhere and, in particular, at the bounce location r=0. Finally, we verified via direct computation of their specific shapes that such bouncing geometries are supported by a specific combination of a NLED Lagrangian and a scalar field potential. Our setting recovers the GR one when we choose f_0=f_1=γ=0.


For the second model, we adopted Bardeen's regularized metric function and also determined the solutions for the horizons numerically. In the case of critical mass, for Λ < 0, when M > M_c we observe four horizons, where the innermost horizons are Cauchy horizons on each side of the bounce; when M = M_c, degenerate horizons are observed; and if M < M_c we find a traversable wormhole. For Λ > 0, if M > M_c we observe two horizons, when M = M_c we find  four horizons, the innermost of which are the Cauchy horizons. In the scenario where M < M_c, we discover six horizons, the innermost of which is the Cauchy horizon and the outermost being the cosmological horizons. For the critical charge in which Λ < 0, if q > q_c there is no horizon, if q = q_c we observe a degenerate horizons on each side of the bounce, while if q < q_c we have two horizons on each side (Cauchy and event). For Λ > 0, we also observe that there are cosmological horizons in all scenarios (q > q_c, q = q_c, q < q_c) and, in particular, for q < q_c a total of three horizons on each side are observed. With the critical parameter γ for Λ<0, if γ > γ_c, we have a wormhole, for γ =γ_c, degenerate horizons occur, and if γ < γ_c we have four horizons in total, with the innermost ones being Cauchy horizons. In the case where Λ > 0, two horizons are formed for γ > γ_c, degenerate horizons are formed for γ = γ_c, and for γ < γ_c we have six horizons. Similarly as in the Simpson-Visser geometry we verified the regularity of the  Kretschmann scalar and found the explicit shapes of the NLED Lagrangian density and the scalar potential.

The bottom line of our analysis is that CG naturally allows for suitable generalizations of bouncing geometries of GR in terms of the new CG parameter γ, extracted out of the class of vacuum solutions found within this theory. The geometries built this way display promising features to be studied within the context of black hole thermodynamics, the analysis 
 the stability of solutions through the analysis of perturbations, gravitational lensing and black hole shadows, topics in which we hope to report soon. 
 







MER thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Brazil, for partial financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
FSNL acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT) Scientific Employment Stimulus contract with reference CEECINST/00032/2018, and funding through the research grants UIDB/04434/2020, UIDP/04434/2020 and PTDC/FIS-AST/0054/2021.
DRG is supported by the Spanish Agencia Estatal de Investigación Grant No. PID2022-138607NB-I00, funded by MCIN/AEI/10.13039/501100011033, UE, and ERDF A way of making Europe.





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 Paper Title: Instantons and the Large N=4 Algebra.





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    ===#1#1AdSSTGSUUSOCStrdimıiSymPUadOSp𝒦K3 X ASDSchool of Natural Sciences, Institute for Advanced Study,
      1 Einstein Drive, Princeton, NJ 08540 USAWe investigate the differential geometry of the moduli space of instantons on ^3 ×^1. Extending previous results, we show
    that a sigma-model with this target space can be expected to possess a large =4 superconformal symmetry, supporting speculations that this
    sigma-model may be dual to Type IIB superstring theory on _3×^3×^3×^1.   The sigma-model is parametrized by three integers – the rank of the gauge
    group, the instanton number, and a “level” (the integer coefficient of a topologically nontrivial B-field, analogous to a WZW level).    
    These integers are expected to correspond to two five-brane charges and
    a one-brane charge.   The sigma-model is weakly coupled when the level,  conjecturally corresponding to one of the five-brane changes, becomes very large, keeping the
    other parameters fixed.  The central charges of the large =4 algebra agree, at least semiclassically, with expectations from the duality.Instantons and the Large =4 Algebra
        Edward Witten
        
    ===================================
    
    
    
    § INTRODUCTION
    
    
     §.§ Overview
     
    
    Among the original examples of AdS/CFT duality <cit.> were Type IIB superstring theory on _3 ×^3×^4 and _3×^3×, which are
    believed to be dual to two-dimensional sigma-models in which the target space is the moduli space of instantons on ^4 or , respectively. Some of the arguments were recently
    assessed and extended in  <cit.>, where one can also find detailed references. These examples have
    =4 supersymmetry, which greatly facilitates their understanding.
    
    However, the superficially similar example of Type IIB superstring theory on _3×^3×^3×^1 has been less well understood, despite having an
    even larger superconformal symmetry.    This model has been studied extensively
     <cit.> and in particular it is known that a dual conformal field theory should possess a “large”=4 superconformal algebra
     (in fact, two copies of this algebra, for chiral and antichiral modes, respectively), 
     as opposed to the more familiar “small”=4 superconformal algebra relevant to strings on _3×^3×^4 or _3×^3×.   The large =4 superconformal algebra is an extension of the small one with
    an additional (2)R-symmetry and  some additional free fields <cit.>.  
    The simplest indication that _3×^3×^3×^1 leads to a large superconformal
    algebra is simply that it has many (2) symmetries, acting on the left and right on the two copies of
    ^3=(2).  These all turn out to be R-symmetries of the left or right superconformal algebra.  The detailed analysis demonstrating the existence of a large
    =4 algebra was made in <cit.> from a worldsheet point of view
    (for the case that the fluxes on _3×^3×^3×^1 are  of Neveu-Schwarz type) and in <cit.> in supergravity.  
    A simple example of a two-dimensional conformal field theory with the “large”=4 symmetry
    is an (2)_k×(1) supersymmetric WZW model, which is the same thing as a supersymmetric sigma-model with target ^3×^1 and k units of flux 
    of  H=Ḅ (where B is the sigma-model B-field).   
    
    The Type IIB supergravity solutions on _3×^3×^3×^1, which were analyzed in <cit.>, depend on three parameters, namely the three-form
    fluxes on _3 and on the two ^3's.   Supersymmetry requires that all fluxes are of the same type, Neveu-Schwarz or Ramond-Ramond or a mixture. 
    The three fluxes can be parametrized by integers 
    Q_1,Q_5,Q_5', where Q_5 and Q_5' have been interpreted as the numbers of fivebranes wrapped in two different ways  that produce the  fluxes on the two
    ^3's, and Q_1 is similarly interpreted[The integrality of Q_1 is not visible in the supergravity solution.]  
    as the number of one-branes, related to the flux on _3.
    
    Proposals for a dual of superstring theory on _3×^3×^3×^1 have  been mainly of two types.  One idea is that, at least for some values of Q_1, Q_5, and
    Q_5',  the dual might be a symmetric product
    of N copies of the (2)_k×(1) model, for some N<cit.>.   One motivation for this proposal is simply that this symmetric product is one of the relatively few known examples
    of a model with the large =4 superconformal symmetry.  Another motivation is that the duals of _3×^3×^4 and _3×^3×
    are related to similar symmetric products (if the integers Q_1 and Q_5 that characterize those models are relatively prime).   An obvious limitation of this proposal is that the symmetric product only depends on two integers, k and N, while strings on
    _3×^3×^3×^1 depend on three integers Q_1,Q_5,Q_5'.  
    It is believed that strings on _3×^3×^3×^1 do not have
    dualities that would make one of the three integers irrelevant.
      However, if Q_5=1 (or Q_5'=1), there is reasonably strong evidence that strings on _3×^3 ×^3×^1  are indeed dual to a symmetric product of
      many copies of the (2)_k×(1) WZW model <cit.>.
      
      A second idea has been that the dual of strings on _3×^3×^3×^1 might be a sigma-model with target the moduli space  of instantons on ^3×^1<cit.>.
      A question about this idea has been whether it is true that this sigma-model does have the large superconformal symmetry.    There are important 
      partial results in this direction 
      <cit.>.   Roughly, by combining previous results, 
      it is known that  is a generalized hyper-Kahler manifold, also known as a bi-HKT or (4,4) manifold. 
    Assuming conformal invariance,
       this is enough to show that the model has =4 supersymmetry with the small =4 algebra.   The main goal of the present article is to complete this story
      and show that this sigma-model  actually  possesses  large =4 superconformal symmetry.   For this,  one must show
      that Killing vector fields on  that come from the symmetries of ^3×^1 are covariantly constant for  appropriate connections on  with torsion.
    It is also necessary to show that the sigma-model with target  is conformally-invariant and not just scale-invariant, something that  is non-trivial  for sigma-models with
      =4 supersymmetry <cit.>. We address this point by
      applying arguments of <cit.>.  
      
      If Q_1 and Q_5 are relatively prime, the
       moduli space of instantons on ^4 or  is a deformation of a symmetric product of copies of ^4 or .    If a similar statement were true for ^3×^1,
      then potentially the two proposals about the dual of string theory on _3×^3×^3×^1 could both be correct.   However, generically (except for
      gauge group (1)) it is not true that the moduli space  of instantons on ^3×^1 is deformation equivalent to a symmetric product.   This will be explained
      in section <ref>.
      
      
    
     §.§ Motivation For The Conjecture
    
      
      The idea that a sigma-model with target  is dual to Type IIB string theory on _3×^3×^3×^1 can be motivated by a simple brane construction.
       In describing this, we slightly amplify the discussion in <cit.> (see scenario 3 in section 3 of that article) as well as  <cit.>.  We also assume that the fluxes considered
       are of Ramond-Ramond (RR) type, so the corresponding branes are D-branes.  We denote the RR two-form field as C_2 and its three-form field strength as G_3=C̣_2.  
       For the starting point, we consider
       Type IIB superstring theory on X=^2×^1×^*^3×, where ^*^3 is a noncompact Calabi-Yau manifold (the deformed conifold) with Q'_5 units
       of RR flux  on ^3⊂^*^3.   The deformed conifold with this flux is a supersymmetric configuration, studied originally in <cit.>. Then we wrap Q_5 D5-branes on ^2×^1 ×^3× p ⊂ X, where p is a point in the last factor of X=^2×^1×^*^3×.   These D5-branes support a (Q_5) gauge theory,
       with gauge connection A and  field strength  
       F=Ạ+A∧ A. Because of the assumed G_3 flux, the effective action of the
       gauge theory (in Euclidean signature) 
       contains a coupling[With G_3=C̣_2, integration by parts puts this  in the form ı/8π^2∫ C_2∧  F∧ F, which may be more familiar.]-ı∫_^2×^1×^3(A) ∧G_3/2π
    involving the  Chern-Simons three-form (A)=1/4π (AẠ+2/3 A^2). 
       This configuration, in which Q_1 has not yet been introduced, has actually been discussed previously  <cit.>. (For related examples, see for instance
       <cit.>.)  In everything that we have said up to this point,  ^2×^1 could be replaced with any three-manifold W, or simply with W=^3.
            As explained in <cit.>, the low energy physics on W is a three-dimensional topological field theory, a  (Q_5) Chern-Simons theory at level Q_5' (where the Chern-Simons coupling comes
       from (<ref>)).
       We will return to this Chern-Simons theory momentarily. 
     
     We can now add Q_1 D1-branes wrapped on ^2× p_i ⊂^2×^3×^1, where p_i are points in ^1×^3.   However, we assume that these
     D1-branes “dissolve” into instantons in the (Q_5) gauge theory.   Generically, a (Q_5) instanton on ^3×^1 completely breaks the (Q_5) gauge symmetry
     down to the center (1). (This is true for all positive values of the instanton number.)   The  (1) gauge field  becomes massive because of the coupling (<ref>), and is described at low energies by a (1)BF theory at level
     Q_5' (which can be understood as the reduction to ^2 of a (1)_Q_5' Chern-Simons theory on W=^2×^1).   Being a topological field theory, this level Q_1BF theory
     plays no role in the analysis of the large =4 algebra, though  it is undoubtedly important in some subtle aspects of the story.[Since (Q_5)=((1)×(Q_5))/_Q_5
     is only locally a product group,
     there will be a subtle coupling between the (1)BF theory and the sigma model associated to (Q_5) instantons, which we concentrate on in this
     article.]
     The rest of the low energy physics is described, by standard arguments, by a supersymmetric sigma-model with target the moduli space  of instantons on ^3×^1, except that
     we must understand the role of the interaction (<ref>).  As this interaction depends on an integer Q_1, is odd under reflection of ^2, and cannot be written
     as the integral of a gauge-invariant local density, it should come as no surprise that in the sigma-model this interaction becomes the coupling to a topologically non-trivial B-field,
     that is, a B-field whose field strength H=Ḅ has nonzero periods that must satisfy a Dirac quantization condition.   This will be explained in section <ref>.
     
     So far we have arrived at a sigma-model with target space the moduli space  of instantons on ^3×^1.   But what does this have to do with string theory on
     _3×^3×^3×^1?    To answer this question, we just follow the original analysis of holographic duality of the D1-D5 system <cit.>.   If the flux
     Q'_5 is small compared to Q_1 and Q_5, so that its local effects are small, then we can simply borrow the original analysis.   The normal bundle to ^2×^1×^3× p⊂^2×^1
     ×^*^3× is, of course, locally a copy of ^4.    When we take the near horizon geometry, the zero-section of the normal bundle (that is,
     the origin in this ^4) is omitted, the radial direction in the normal bundle
     combines with the first factor of  ^2×^1
     ×^*^3× to make a copy of _3, and the angular directions in the normal bundle simply survive in the near horizon geometry as a factor of ^3.   In general,
     this ^3 would be fibered over the worldvolume of the D5-branes, which in our present discussion is ^2×^1×^3. But as ^3 is parallelizable, the fibration 
     is trivial and the angular variables just give another factor of ^3.   Thus the near horizon geometry is _3×^3×^3×^1.
     
    Particularly if Q_5' is not assumed to be small compared to Q_1 and Q_5, it is not entirely clear from this analysis
     that the near horizon geometry will be the standard maximally symmetric _3×^3×^3×^1 
     geometry.  If, however, this is the case (which is plausible but will not be proved here), then we get a reasonable basis for expecting that Type IIB superstring theory on  _3×^3×^3×^1 is
     dual to a sigma-model with target the instanton moduli space .   We also learn the dictionary in this relationship: Q_5 maps to the rank of a (Q_5) gauge group;
     Q_1 is the instanton number; and Q_5' is the “level,” that is, the coefficient of a topologically non-trivial B-field in the sigma-model.  
     
     A puzzle here is that the _3×^3×^3×^1 geometry, with Q_5 and Q_5' understood as the flux of G_3/2π over the two ^3's, has a manifest
     symmetry between Q_5 and Q_5'.   It is not at all clear why the low energy limit of the D-brane system would have that property.   However, an encouraging observation was made in
     <cit.>.  As remarked earlier, for Q_1=0, the low energy limit is  a (Q_5) Chern-Simons theory at level Q'_5, which has a symmetry
     Q_5↔ Q_5', usually called level-rank duality <cit.>.   If it is true that string theory on _3×^3×^3×^1 is dual to the sigma-model,
     then for any value of Q_1, the low energy limit must have the same Q_5↔ Q_5' symmetry.   The symmetry is only predicted in the low energy limit because
     on the gravity side, it only emerges in the near horizon limit of the geometry.  Since level-rank duality is rather subtle, this example suggests that understanding the 
     Q_5↔ Q_5' symmetry of the sigma-model may not be easy.
     
     A final remark is that conformal invariance of the sigma-model depends crucially on a renormalization group flow.    The sigma-model whose target is the instanton moduli space 
     is somewhat analogous to a sigma-model with target a compact Lie group : it is constructed from a target space metric  as well as a B-field.   Conformal invariance
     will hold only if these are properly related.   In the case of the group manifold, if the target space radius is large compared compared to the “level,” 
     then the model is asymptotically free but flows in the infrared to a conformally invariant fixed point at which the radius is determined in terms of the level <cit.>.  We anticipate a
     similar behavior for the sigma-model with target .   In the preceding discussion, we assumed no relation between the radius of the 
     ^3⊂^*^3 on which the D5-branes were wrapped and the sigma-model level Q'_5.  If the radius is too large, 
     the sigma-model is definitely not conformally invariant; it has a target space metric that is large
     compared to the level.   What we will argue in the body of this article is that with a correctly adjusted radius, 
     the sigma-model has a conformally invariant fixed point with large =4 superconformal symmetry.  This will be an infrared stable fixed point, since a theory with large =4 
     symmetry does not have relevant couplings <cit.>.   Hopefully, for any (or perhaps any sufficiently large)
      initially assumed radius, there is a renormalization group flow to this fixed point.
      
      Just as in the case of the WZW model, the metric of the target space at the critical point is proportional to the level, which here is Q_5'.  Hence at its critical point, the sigma-model
    becomes weakly coupled if Q'_5 is taken to be large, for fixed Q_1 and Q_5.  By contrast, the supergravity description becomes reliable when Q_1≫ Q_5,Q_5'≫1.
     
     
    
     §.§ Differential Geometry Of A Four-Manifold and Its Instanton Moduli Space
    
     
    In many cases,  if a four-manifold M has a differential geometric structure which ensures that a sigma-model with target M possesses a certain
     supersymmetry algebra, then the moduli space  of instantons on M has the same differential geometric structure, so a sigma-model with target 
     possesses the same supersymmetry algebra.
     
     The most familiar results of this kind arise in the absence of a  B-field.   If M is  Kahler, so that a sigma-model with target M has
      global (2,2) supersymmetry, then  is also  Kahler, and a sigma-model with target  also has global (2,2) supersymmetry.
      If M is hyper-Kahler, so that a sigma-model with target M has 
      =4 superconformal symmetry with the small =4 algebra for both left- and right-movers, 
      then  is also hyper-Kahler, again leading to small =4 superconformal symmetry for both chiralities.
      
        
    Some results along these lines that are relevant to sigma-models with a B-field are as follows: 
    
    
      *  If M satisfies the conditions for (0,2) supersymmetry – it is a complex manifold with a hermitian metric whose torsion is closed in a sense reviewed in section <ref>–
    then  is also a complex manifold[The theorem of <cit.> actually identifies  as a moduli space of
    stable bundles on M.]<cit.>, with a natural hermitian metric that also has closed torsion <cit.>, so the sigma-model with target
     also has (0,2) supersymmetry,
    
      *  If M is a generalized Kahler manifold (the geometry that leads to (2,2) supersymmetry with a B-field) then so is <cit.>.
    
      *   If M is an HKT manifold (the geometry that leads to (0,4) supersymmetry, with a small =4 algebra), then so is <cit.>.
    
      *  If M is generalized hyper-Kahler  or bi-HKT (leading to (4,4) supersymmetry with the small =4 algebra), then so is .   This follows on combining results in
    <cit.> and <cit.>; see section <ref>.
    
      *  Finally, if M has the properties that lead to invariance under the large =4 algebra, then so does .   This is shown in  section <ref>.
    
    
     §.§ Organization Of The Paper
    
     
     This article is organized as follows.
     
     In section <ref>, we describe basic aspects  of the moduli space  that will be important in this article.   In section <ref>,
     we review the geometry required for extended supersymmetry in a two-dimensional sigma-model and the relevant geometry of ^3×^1.  
     In section <ref>, we explain how to prove that the moduli space  is a hypercomplex manifold, and in section <ref>
     we extend that and prove that it has the geometry  associated with small =4 symmetry.   These two sections are 
     primarily based on previous results <cit.>, with some details added.
      In section  <ref>, we show that the sigma-model
     with target  actually has the geometry associated to the large =4 algebra, not just the small one.  For this, one has to show that the Killing vector
     fields on  associated to the symmetries of ^3×^1 are covariantly constant for appropriate connections with torsion.  We also determine the central
     charges of the =4 algebra.  
     In section <ref>, we complete the story by arguing that the sigma-model with target , with appropriate metric and B-field,
      is conformally-invariant and not just scale-invariant.   This is argued in several
     ways, using considerations in  <cit.>.   Putting all this together, it follows
     that the sigma-model with target  is a conformal field theory with large =4 superconformal symmetry for both chiralities.
     
     
     In section <ref>, in the starting point, we could have replaced ^* ^3 with ^*(^3/_n)  for some integer n≥ 2, where _n acts on ^3=(2) on,
     say, the left.   Then the same logic as before
     would motivate the idea that Type IIB superstring theory on _3×^3×^3/_n×^1 is dual to instantons on ^3/_n×^1.  Most
     of our considerations carry over directly to that case, but there is an interesting novelty, discussed in section <ref>, and related to the possibility of turning on a discrete NS B-field
     (assuming the background is of Ramond type).
    A gauge bundle on ^3/_n×^1 has a _n-valued discrete
     topological invariant that is absent for ^3×^1, and this makes it possible to construct examples that may be interesting  purely from a geometrical point of view.
      Hyper-Kahler manifolds can be generalized to include
     a non-flat B-field, leading to the concept of a strong HKT manifold, a notion that we will review in section <ref>.  
       In general, a σ-model with target a compact  strong HKT manifold will  have a large =4 superconformal algebra for one chirality.
      (To get large =4 algebras for both chiralities, one needs a pair of strong HKT structures with equal and opposite torsion.)  However,
     known examples of manifolds of this type   are very limited; apart from  hyper-Kahler manifolds, one has only  homogeneous examples <cit.>, of which the
     simplest is  ^3×^1, and products of hyper-Kahler and homogeneous manifolds.      As we explain in section <ref>, for certain values of the instanton number and the
     discrete topological invariant, the instanton moduli spaces on ^3/_n×^1 are smooth and compact.   These may potentially give the first examples of compact strong HKT
     manifolds that are not merely products of hyper-Kahler manifolds and homogeneous spaces.
     
    
    We conclude this introduction with a note on notation and terminology.   Concerning notation, we generally denote a four-manifold on which we study the instanton equation as M
    and the corresponding instanton moduli space as .  The metric on M is denoted as g and tangent space indices to M are denoted i,j,k; the metric on  is
    denoted G and tangent space indices to  are denoted α,β,γ.   Given a geometric structure on M, a corresponding structure on  is denoted with a hat.  For example, if  is a complex structure and V is a vector field
    on M, then the corresponding complex structure and vector field on  are denoted as  and V.
     The target
    space of a general sigma-model is denoted ; the metric on  is denoted as G and tangent space indices are denoted I,J,K.   Lie algebra indices of (2) and
    (related to this) tangent space indices of ^3 are denoted a,b,c.
    At some points, it is hard to be completely consistent with these conventions; for example,  although we usually consider ^3×^1  as a four-manifold on which we study the instanton
    equation, it is also considered in section <ref> as the target space of a sigma-model.
    
    
    Concerning terminology, the geometry that leads to a small =4 algebra (for both left-movers and right-movers) has been
    called twisted generalized hyper-Kahler geometry, where “twisting” means that  the three-form H is topologically nontrivial (closed but not exact).  
    Similarly, the analog with =2 supersymmetry has been called twisted generalized Kahler geometry <cit.>.
     The generalized
    hyper-Kahler manifolds of primary interest in the present article are  twisted in that sense[There are, however, interesting complete but not compact examples
    in which H is non-zero but exact.  See <cit.> for examples.  The analysis in this article should apply to those examples, with minor changes.]    
    and we will take the liberty of sometimes omitting the word “twisted.”  Somewhat
    similarly, the geometry that leads to a small =4 algebra for, say, right-movers only (or left- and right-movers both) has been called strong HKT geometry (or strong bi-HKT
    geometry), where HKT stands for a generalization of hyper-Kahler
    geometry to allow torsion,  “strong” means that Ḥ=0, and “bi” means that there are two separate HKT structures with opposite torsion. 
    Likewise, the analogs  for  (0,2) (or (2,2)) have been called strong KT
    (or strong bi-KT) geometry.
      In this article, we only consider geometries with Ḥ=0, appropriate to Type II superstrings, and we sometimes
    omit the word “strong.”
    
    § SOME PROPERTIES OF THE INSTANTON MODULI SPACE
    
     
    
    
    In discussing general properties of the moduli spaces described in the introduction, we will take the gauge group to be (Q_5), since the center of (Q_5) leads to a BF topological field theory
    that generally decouples.    We view the Lie algebra of (n) as the algebra of traceless antihermitian n× n matrices and we define the invariant
    quadratic form
       (a,b)=- a b, 
    where the minus sign is needed for positivity.
    
      Most of what we will say will apply with minor modifications for other compact semi-simple gauge groups, such as orthogonal and symplectic groups that one would
    encounter in an orientifold construction. Sometimes we consider a general compact simple Lie group .
     
     
    
     §.§ Cohomology: Some Simple Observations
    
     
     Some low-dimensional cohomology classes of the instanton moduli space  will be important.    One thing that will emerge in the following analysis is that some statements
     are only true, or are only known to be true, if Q_1 and/or Q_5 is sufficiently large.  For small values of the charges, there may be exceptional behavior not seen in supergravity.
     
     
    
      §.§.§ More on the B-Field
    
     
     First let us explain in more detail the statement that the interaction
        - ı∫_^2×^3×^1(A) ∧G_3/2π
        can be interpreted at low energies in terms of a B-field on .      The fact that the model is at low energies a sigma-model with target 
        means that, if m^α, α=1,⋯, are local coordinates on , then at low energies the model can be expressed in terms of fields m^α(x),  x∈^2,
        along with their supersymmetric partners.   This implies in particular that after integrating out massive fields,
         the gauge field A– both its components A_μ, μ=1,2 along ^2 and its components
        along ^3×^1– can be expressed in terms of the m_α.  The A_μ  vanish if the m's are constant along ^2 (a constant set of m^α's simply
        describes an instanton on ^3×^1 with no dependence on ^2).  So the A_μ are proportional to the derivatives of the m's.  The general form, modulo
        irrelevant terms of higher order,  
       is A_μ(x,y) =∑_α f_α(m(x),y) ∂/∂ x^μ m^α, where  y∈^3×^1  and
     f_α(m,y) are Lie algebra valued functions of  m and y.   So we have ∂_μ A_ν-∂_ν A_μ =
        ∑_α,β (∂_α f_β-∂_β f_α) ∂_μ m^α∂_ν m^β, and [A_μ,A_ν]=∑_α,β[f_α,f_β]∂_μ m^α∂_ν  m^β.   Since (A) depends on A_μ through ∂_μ A_ν-∂_ν A_μ and [A_μ,A_ν],
        the form of these expressions shows that the interaction (<ref>) reduces at low energies, after integrating over ^3×^1,  to something of the  general form
        ı∫_^2^̣2 x ϵ^μν B_αβ(m)∂_μ m^α∂_ν m^β,
        with B_αβ=-B_βα.   This is the standard form of the contribution of a B-field to the sigma-model action.
        
        If the m^α are constant at infinity along ^2, so that A_μ vanishes there, then in evaluating the expression (<ref>) we can compactify ^2 to ^2.
        If we then view ^2 as the boundary of a ball U and extend the gauge field from ^2×^3×^1 over U×^3×^1, then we can replace
        eqn. (<ref>) with
        -ı∫_U×^3×^1 F∧ F/4π∧G_3/2π.
        This is an improved formula because the integrand is gauge-invariant, but it does potentially depend on the choice of U and of the extension of the gauge field over U.  To compare
        two different choices with different extensions over possibly different manifolds U and U', we glue together U and U' along their boundaries to make a closed oriented manifold
        W and  learn that the difference in the two evaluations of the action is
        -ı∫_W×^3×^1 F∧ F/4π∧G_3/2π.
        We can explicitly do the integral over ^3, using the fact that in the setup described in section (<ref>), 
        G_3 is a pullback from ^3 and ∫__3G_3/2π=Q'_5.  So the coupling turns out to be
        -ı Q'_5 ∫_W×^1 F∧ F/4π.
        The integral is 2π, where  is the instanton number of the gauge field on W×^1.   Thus the action is
         -2πı Q'_5.
        As  is in general an arbitrary integer, this formula explains the interpretation of Q'_5 as the sigma-model “level.”   In other words,
        the formula shows that the effective B-field is Q'_5 times a minimal B-field that would satisfy the appropriate Dirac quantization.
        
        This characterizes the sigma-model B-field topologically, but the reader may wonder if it is possible to write an explicit formula for the gauge-invariant field strength
        H=Ḅ as a three-form on .  Indeed, to understand the supersymmetry of the sigma-model, we will need to understand explicit formulas.   We explain
        some necessary tools in section <ref> and eventually arrive at the final formula in section <ref>.
        
        
    
      §.§.§ Theta-Angles And The Second Betti Number
    
        
        Another important question is whether the σ-model admits a theta-angle.   In general, a (continuous)  theta-angle is associated to a term in the Euclidean action of the 
        form -ıθΓ, where Γ is an integer-valued topological invariant and therefore θ is an angular variable, 
        θ≅θ+2π.  In the present context, there is exactly one  suitable invariant, namely the
        integral of the third Chern class,[For Q_5=2, the third Chern class vanishes and the sigma-model with target the (2) moduli space
        does not have a continuous theta-angle. (It does have a discrete one because π_5((2))=_2.)   For Q_5=2, possibly
        the parity-odd supergravity modulus found in <cit.> decouples from the low energy physics. We note that the gauge group of the brane system is really not (Q_5) but (Q_5),
        or (2) for Q_5=2.  The third Chern class is nonzero for (2), but at least for most purposes, the center (1)⊂(2) decouples at low energies because of a  Chern-Simons coupling, as reviewed in the introduction. For Q_5=2, perhaps some subtle effects depend on the coupling to the third Chern class.]Γ=1/24π^3∫_^2×^1×^3  F∧ F∧ F. 
        Thus a single theta-angle will appear as a modulus of the sigma-model with target .  
       Precisely the same coupling appears in the gauge theory description of Type IIB superstrings on ^4 or .   In that context, this coupling
       is interpreted as the dual to a Ramond-Ramond parameter that appears in the supergravity solution <cit.>.
       We expect that this coupling has the same interpretation in the case of Type IIB on _3×^3×^3×^1.   Indeed, in that case
       the supergravity solution has precisely one Ramond-Ramond modulus <cit.>. 
        
        
        It is straightforward to compare the moduli space of sigma-model parameters to supergravity.
         Since Γ is odd under a charge conjugation symmetry of (Q_5) that exchanges the
        fundamental and anti-fundamental representations,
         the angle θ  contributes a factor ^1/_2 to the moduli space.  
         In this article, we will always assume that the metric on ^3×^1 is a product of standard round metrics on the two factors, in general with arbitrary radii r,r'.
        Since the instanton equation is conformally invariant, only the ratio r'/r is relevant.  It is a positive number, taking values in the positive half-line _+.   Including also the theta-angle,
        the sigma-model moduli space is _+×^1/_2.   This is precisely the moduli space of Type IIB supergravity on _3×^3×^3×^1,
        as determined in <cit.>.    That gives some preliminary support to the duality conjecture relating the  the sigma-model with target  to string theory on
        _3×^3×^3×^1.   We have assumed that no unknown dualities are present; this was argued in <cit.> based on 
        an examination of the ends of the moduli space. 
        
           The parameter θ  has a qualitative effect on the sigma-model spectrum.   
          For any oriented  four-manifold M, the moduli space   of instanton solutions on M is compact and smooth except for the small instanton singularity as well
          as singularities associated with “un-Higgsing”– that is, singularities associated with instanton solutions that do not completely break the gauge symmetry. These singularities
          are “universal”– the small instanton singularity does not depend on the four-manifold in which the small instanton is embedded, and the un-Higgsing singularity depends only
          on the unbroken gauge group and the spectrum of massless charged hypermultiplets, not on details of the four-manifold.[These statements fail in some exceptional cases described in section <ref>.]  In the case of ^4 or , 
          there are four parameters, related to each other
           by supersymmetry, that control the singularities.   Three of these are parameters associated to 
           Neveu-Schwarz (NS) B-field modes,
            which resolve the small instanton singularity via a noncommutative deformation of the instanton
       equation <cit.>.  The same parameters also resolve the un-Higgsing singularities as long as Q_1 and Q_5 are relatively prime.   (If Q_1 and Q_5 are not relatively
       prime, the model has  unavoidable un-Higgsing singularities.)
        The fourth is a Ramond-Ramond mode, which in the sigma-model becomes the theta-angle associated to the third Chern class <cit.>.  It resolves
        the same singularities that the NS parameters resolve in the abstract sense of giving the sigma-model a discrete spectrum (not in the sense of classically resolving the 
        singularities of ).   
       In the case of ^3×^1, the B-field modes that could resolve the singularity in a classical sense are absent globally, since H^2(^3×^1;)=0.   
       However, the theta-angle is still present.  One expects the small instanton singularity to be resolved as long as θ≠0, and (though this point deserves a more careful
       study) one expects the un-Higgsing singularities
       to be resolved as long as the triple Q_1,Q_5,Q'_5 has no common divisor.   
          
    The reader might notice the following gap in our reasoning.   In the sigma-model with target , we are not interested in arbitrary gauge fields on ^2×^3×^1
        but only in those that can be interpreted in terms of maps of ^2 to .   Concretely, these are gauge fields on ^2×^1×^3
        that when restricted to p×^1×^3, for any point p∈^2, satisfy the instanton equation on ^1×^3.   With this restriction, is it still true that Γ
        can take arbitrary integer values?   General results that will be described in section <ref> imply  that, for any Q_5, this is true for sufficiently large Q_1.
        This allows the possibility that, for example, when we approximate the six-dimensional gauge theory by a two-dimensional sigma-model, Γ might vanish
        identically for some small values of Q_1.   If this happens (and the duality conjecture that we are discussing is correct), 
        it would mean that the modulus that is seen in supergravity and is related to the theta-angle of the sigma-model
        decouples from the low energy physics for those
       particular values of Q_1.
       
       A similar question concerns the coupling (<ref>) that we interpret in terms of a two-form field B on .   If A is a completely general gauge field on
       W×^1×^3, then the expression in eqn. (<ref>) can equal an arbitrary integer multiple of 2π Q'_5, and therefore the coupling (<ref>) is well-defined
       precisely mod 2π Q_5'.      
         That is the basis for interpreting Q_5' as the sigma-model “level.”   However, if we constrain 
       A so that its restriction to p×^1×^3 satisfies the instanton equation for every p, then is it still true that  eqn. (<ref>) can equal an arbitrary integer
       multiple of 2π Q_5'?   Again, for sufficiently large Q_1, this is true by the general results that will be explained in section <ref>.
       
       One can also consider the possibility of discrete theta angles associated to torsion in H^2(;(1)).   There is no evidence in supergravity that any such discrete parameters should exist.
       
    
        
    
      §.§.§ First Betti Number
    
        
        We will also need to understand H^1(;) and its dimension, which is the first Betti number b_1().   In general, a generator of H^1(;) is a closed 1-form λ
        that is not exact.   A closed 1-form λ can always be written locally as λ=φ̣ with some function φ, but φ may not be single-valued; it may be
        well-defined only modulo a constant.  Conversely, if φ is a function on  that is well-defined modulo a constant, then λ=φ̣ is a nonzero
        element of H^1(;).
        
        For the case that  is the instanton moduli space on ^3×^1, a real-valued function that is only well-defined modulo a constant is
        φ=1/2π∫_^3×^1(A) ∧ϕ̣/2π   ,
        with ϕ an angular variable on ^1.   Here we have chosen to integrate over ϕ, but for the purpose of finding a generator of H^1(;), it would not matter if
        we instead set ϕ to a specific value.   The function φ defined in eqn. (<ref>)  is multi-valued because of the usual multi-valuedness of the Chern-Simons form.
        
        Therefore, the exterior derivative of φ  is a generator of H^1(,). 
        Let  be the space    of all gauge connections on
        ^3×^1 (not just the ones that satisfy the instanton equation), and let  be the group of gauge equivalences (locally, this    is simply the group of maps of M to the gauge group (Q_5)). 
        We can view φ as a multi-valued function defined on the space / of gauge fields
      modulo gauge equivalences, and likewise we can define the exterior derivative of φ as a 1-form
        on /.   We will reserve the symbol $̣ for the exterior derivative on a finite-dimensional manifold such^3×^1or, and writeδfor the exterior derivative
        on the infinite-dimensional manifoldor its quotient/.   We also define
        ψ(y) = δ A(y),     y∈^3×^1.
        We  can compute an explicit formula forλ=δφ:
        λ=1/(2π)^2∫ _^3×^1  F∧δ A ∧ϕ̣/2π  =1/(2π)^2∫_^3×^1  F∧ψ∧ϕ̣/2π
        
         One might wonder how to explicitly  describe a loop inon whichλhas a nonzero integral.   This can be done as follows.  Letudenote a point in^3and letA(u,ϕ)be any instanton solution on^3×^1of instanton numberQ_1, representing a point in.   Introduce a second circle^1parametrized by another angular variableϕ.   ThenA(u,ϕ+ϕ)is, for each fixedϕ, an instanton solution describing a point in; asϕvaries over^1,A(u,ϕ
         +ϕ)varies over a loopγ∈.     It is also true that for fixedϕ,A(u,ϕ+ϕ)can be viewed as an instanton solution, again of instanton numberQ_1,
         on^3×^1.   Using this, we can evaluate the integral that defines∮_γλ:
              ∮_γλ = Q_1. 
    
     §.§ Symmetries Of 
    
        
    Here we will discuss properties of the instanton moduli spacethat are associated to   symmetries of^3×^1.
    
    First of all,^3×^1has discrete symmetries that act by a reflection of^3and/or^1.   But separate reflections of the two factors reverse the
    orientation of^3×^1, so they are not symmetries of the instanton equation and do not lead to symmetries of.   Only a combined reflection
    of^3and^1leads to a symmetry of.  We will denote such a joint reflection of the two factors asρ.   This only characterizesρup to a rotation of the
    two factors, but the precise choice ofρwill never be important.
    
    The  restriction to a joint reflection of the two factors agrees with what one would expect based on a presumed duality with Type IIB superstrings on_3×^3×^3×^1. 
      As analyzed originally in <cit.>, the superstring
     solution on_3×^3×^3×^1has a nonzero three-form field on each of the first three factors,_3and the two^3's.    This three-form is invariant under all continuous symmetries of_3×^3×^3×^1and is  of the same type (Neveu-Schwarz, Ramond-Ramond, or a combination) on all three factors.S-duality implies that the
    type of three-form does not affect the following remarks, and for definiteness we will assume a Neveu-Schwarz three-formH.
     A reflection of one of the^3's will
    reverse the sign ofH.   To get a symmetry, we can compensate for this by reversing the worldsheet orientation, which also reverses the sign ofH.
    But as there isHflux on each of the first three factors of_3×^3×^3×^1, to get a symmetry that involves reversing the world-sheet orientation, we must make a simultaneous reflection of all three of those factors.
    Finally, as Type IIB superstring theory does not have a symmetry that reverses the spacetime orientation, to get a symmetry we must also simultaneously reflect the^1.
    This explains that in a dual description involving gauge fields on the product^3×^1of the last two factors, we should only expect to see a joint reflection of the two factors as a  symmetry, not a reflection of just one factor.
    But we also see that this discrete symmetry of^3×^1is accompanied in the full string theory by a reflection of_3, which will reverse the boundary orientation.
    So therefore the joint reflectionρof the two factors of^3×^1will be a parity symmetry of the dual CFT – a symmetry reverses the orientation and  exchanges left- and right-moving modes.
    
    Now we move on to discuss continuous symmetries of^3×^1and their action on.
    For a convenient model of^3×^1, describe^3with real variablesy_λ, λ=0,⋯, 3satisfying∑_λ=0^3 y_λ^2=1, and parametrize^1by a periodic variableτwithτ≅τ+Tfor someT.    Choose a metricgon^3×^1that is described by the line element
    ṣ^2=∑_λ=0^3 ỵ_λ^2+τ̣^2.
    In what follows, indicesi,j,kare tangent to^3×^1, while 
    indicesa,b,c=1,⋯,3are tangent to^3, and an indexτis tangent to^1. 
    We denote the orientation of^3via the antisymmetric Levi-Civita tensorϵ_abcand orient^3×^1so that the instanton equation reads
     F_ab+ϵ_abcg^cc'F_c'τ =0.
    
    
    
    
    Let us look at the vector fieldV=∂/∂τthat generates a rotation of^1, and the corresponding vector field Von.   In general, the action of a vector fieldVon a gauge fieldA, with field strengthF=Ạ+A∧ A, is only uniquely determined
    up to an infinitesimal gauge transformation and takes the form
    δ A_i=V^j F_ji-D_iσ, 
    whereσis the generator of a gauge transformation.   Let us prove that the particular vector fieldV=∂/∂τacts without fixed points on the instanton
    moduli space(assuming that the instanton number is nonzero).   A zero or fixed point of the vector field Vonthat corresponds to the vector fieldVon^3×^1would be an instanton solution such that, for someσ, eqn. (<ref>) reduces to
    0= F_τ i -D_iσ. 
    Squaring, taking a trace, and integrating, we get
    0=-∫_^3×^1^̣4 x√(g)∑_i  (F_τ i-D_iσ)^2 .
    The term linear inσvanishes after integrating by parts, since an instanton connection satisfies the second order Yang-Mills equationD^i F_ji=0.   Hence
     0=-∫_^3×^1^̣4 x√(g)∑_i  ( F_τ i^2+D_iσ^2). 
    In particular, a fixed point satisfiesF_τ i=0.   Since the instanton equation (<ref>) then implies thatF_ij=0for alli,j, it follows that a fixed point is actually a flat connection
    and can only exist if the instanton number is zero.  
    
    
    Note thatVhas constant length as a vector field on^3×^1.
    In section <ref>, after defining the metric of, we will show that the vector field Vonthat is associated toValso has a  constant length, which
    will determine one of the central charges in the large=4algebra.  Of course, the assertion that Vhas (nonzero) constant length is  much more precise  than the statement that we just proved
    showing that Vhas no zeroes or fixed points.    Similarly,  writing(2)_ℓand(2)_rfor the left and right actions of(2)on^3=(2), a generatorT_ℓorT_rof(2)_ℓor(2)_ris a vector field on^3×^1of constant length.
    In section <ref>, we will learn that the corresponding vector fields T_ℓand T_ronlikewise have (nonzero) constant  length, and hence act without fixed points.
       More generally, ifT_ℓandT_rhave unequal
    lengths (as vector fields on^3), then T_ℓ+ T_racts  onwithout fixed points.
    It  will turn out as well that any linear combinationu  T_ℓ+v  T_r +w  V withw≠0has no fixed point.
     The fact thathas many symmetry generators that act without fixed points is likely to mean that
    some attempts at supersymmetric localization will give a trivial result.
    
    However, ifT_ℓandT_rhave equal length, then T_ℓ+ T_rdoes have fixed points in, as analyzed in 
     <cit.>.   We discuss  some consequences  in section <ref>.   For now, we just note
    that an example of a vector fieldT=T_ℓ+T_ron^3such thatT_ℓandT_rhave equal length is
     T= y_2∂/∂ y_3- y_3∂/∂ y_2. 
    The corresponding vector field T= T_ℓ+ T_rondoes have fixed points.
    
    
    
        
    
     §.§ Is The Moduli Space A Symmetric Product?
    
        
        As explained in the introduction, in the literature there are primarily two proposals for the dual of Type IIB superstring theory on_3×^3×^3×^1:
      the dual might be a symmetric product ofncopies of^3×^1for somen, denoted^n(^3×^1), or it might be  a sigma-model with target the instanton moduli spaceon^3×^1.
        
        One can also ask – and this question has been raised as well – whether the two conjectures are different.   
        Or is, which according to an index theorem has real dimension4 Q_1 Q_5,  the same as, or possibly deformation equivalent to,^Q_1Q_5(^3×^1)?   A motivation for this question is that the instanton moduli spaces on^4orare indeed
    such  symmetric products.   Is the same true for^3×^1, at least for some values of the charges?
        
        The answer to this question is that in fact (forQ_5≠1)
        the instanton moduli spaces on^3×^1are not symmetric products of copies of^3×^1or deformation equivalent to such  products.
       One can show this  by considering the fixed points of the(1)symmetry of^3×^1that is generated by
        the vector fieldTof eqn. (<ref>).   It is shown in <cit.> that  instantons on^3×^1that are invariant under this(1)are equivalent to BPS monopoles on a hyperbolic
        three-manifold_3/, where_3is hyperbolic three-space (the Euclidean version of_3), and a generator ofacts by a hyperbolic element of the symmetry group of_3; thus_3/is simply the Euclidean version of a BTZ black hole.  
        For=(2), the magnetic charge is a single integerq, which as shown in <cit.> must be a divisor of the instanton numberQ_1.   For every divisorq, the monopole moduli space is non-empty
        according to Corollary 5.3 in <cit.>.  For any[For (2) gauge theory with instanton number 1, the instanton modulli space
        is described explicitly in <cit.> and does not appear to be a symmetric product.]Q_1≥ 2, there are at least two divisors, namely 1 andQ_1.    Thus the fixed point set always has at least
        two topological components.  ForQ_5>2, the magnetic charge of a monopole is classified byQ_5-1integers, and the number of topological components of the fixed point set 
        grows rapidly.   
        
        By contrast, the zeroes of the vector fieldTon^3comprise the circley_2=y_3=0, so its zeroes on^3×^1make up the connected manifold^1×^1,
        a two-torus.
        The zeroes ofTon^Q_1 Q_5(^3×^1)make up^Q_1 Q_5(^1×^1).
        This is still a connected manifold, and will remain connected after any deformation or resolution.  So the instanton moduli space on^3×^1is not deformation
        equivalent to^Q_1Q_5(^3×^1).
        
        We should remark that forQ_5=1, there is reasonable evidence that the dual of string theory on_3×^3×^3×^1is a sigma-model
        with target^Q_1(^3×^1), with each factor taken at levelQ_5'<cit.>.  Of course, forQ_5=1, the gauge group is(1)and instantons only exist after a noncommutative deformation <cit.>.   In the case of^4or, there are four parameters associated to such a deformation.  Three of these parameters arise from  Neveu-SchwarzB-field modes and are studied in <cit.> and the fourth,
        related to these by supersymmetry, is the expectation value of a certain Ramond-Ramond  field.   The NSB-field modes have no analog for^3×^1, becauseH^2(^3×^1;)=0.   But the RR mode does have an analog <cit.>, and it is plausible that with this mode turned on, the instanton moduli space on^3×^1forQ_5=1is a symmetric product,  potentially providing a framework for the results of <cit.>.
        
        
        
        
        
        
        
        
        
    
     §.§ Cohomology: Systematic Approach
    
        
       At the end of section <ref>, we  asked whether certain topological statements about gauge fields on^2×^1×^3(with prescribed behavior at infinity
       along^2) are modified if one requires
       that the restriction of the gauge field top×^1×^3, for anyp∈^2, is an instanton solution.   These are questions about the relationship between
       the topology of the space/of gauge fields modulo gauge transformations on a given-bundleE→^3×^1,
        and  the topology of the moduli
        spaceof instanton connections onE. Hereis the space of all connections onEandis the group of gauge equivalences, sois a subspace of/, namely the subspace that parametrizes gauge equivalence classes of instanton solutions.
        
        
    The original mathematical statement about questions of this nature was     
        the Atiyah-Jones conjecture <cit.> about(2)gauge fields on^4.  It asserts, roughly speaking, that in the limit that the instanton number is
        large,and/are topologically equivalent. 
       This equivalence means, for instance, that for anyd≥ 0, the restriction mapj^*:H^d(/;)→ H^d(;)associated to the embeddingj:→/is an isomorphism.
        The original Atiyah-Jones conjecture was proved in <cit.> and its generalization to(n)bundles on^4was proved in <cit.>.   For a short survey of related results and questions, see <cit.>.
        
       In this article, we are primarily interested in instantons on^3×^1, rather than^4.   Hence a generalization of
       the Atiyah-Jones conjecture is relevant.  
      A plausible generalization  would say that for any compact gauge groupand any oriented compact  four-manifoldM,and/are topologically equivalent in the limit of large
      instanton number.   What has actually been proved for generalandMis a somewhat weaker statement (Theorem 2^*in <cit.>), which says in particular
      that for anyMand, the map on homologyj_*:H_d(;)→ H_d(/;)is a surjection if the instanton number is large enough compared tod.   This implies that for any manifoldN,
      constraining a gauge field onN×^1×^3to satisfy the instanton equation when restricted top×^1×^3for anyp∈ Nplaces no topological restriction if the instanton number is sufficiently large.  
     So it answers the questions raised at the end of
     section <ref>, which involved the case thatNis of dimension 2 or 3.
     
      However, several issues that arise in the present article require further information concerning the relation between the topology ofand of/.   In particular, it would
      be nice to know that the restriction mapsj^*:H^i(/;)→ H^i(;)are isomorphisms, at least fori=1,2and suitableQ_1,Q_5.
     The relevance to arguments in this paper is as follows.   In general, the number of continuous theta-angles in a sigma-model with targetis the dimension ofH^2(;).   The invariant defined in eqn. (<ref>) can be viewed as a generator ofH^2(/;)and therefore (forgetting its integrality) ofH^2(/;).
    As we explain shortly,
     this invariant generatesH^2(/;). It can be restricted toand this restriction is nonzero according to <cit.>.  In comparing the sigma-model moduli space to the supergravity moduli space, we assumed that this restriction generatesH^2(;), leading to a unique theta-angle.   This is true if the mapj^*is an isomorphism on the degree two cohomology. 
       Somewhat similarly, in eqn. (<ref>), we defined a closed one-formλthat is a generator ofH^1(/;).
    As we explain shortly,λgeneratesH^1(/G;). 
    We proved that the restriction ofλtois nonzero by exhibiting a curve inon whichλhas a nonzero integral.
      In section <ref>, in one approach to proving conformal invariance
      of the sigma-model with target, we will want to know thatλactually generatesH^1(;).   This is true 
      if the mapj^*is an isomorphism on the degree one cohomology.   In the present article, we will assume that 
    some sort of analog of the Atiyah-Jones
      conjecture is true and justifies these statements,  at least for suitableQ_1,Q_5.     Hopefully, such a statement holds at least in the regime in which
      supergravity is valid, namelyQ_1≫ Q_5≫ 1.    
    
      The context for some statements in the last paragraph is the following.     
          In <cit.>, Atiyah and Bott explicitly described a set of generators
          of the real cohomology ring  of/, along lines that we will explain shortly.[Atiyah and Bott were studying
         the case that M is a two-manifold.   However, as long as one considers only the real cohomology of /, not the integer cohomology, their considerations carry over to
         the case that M is of any dimension.]   
          For the case ofM=^3×^1,
         in  degree 1 and degree 2,
         the generators that they describe are precisely the ones that we considered in eqns. (<ref>) and (<ref>).  
         
             
        We will need some knowledge of the explicit generators of the real cohomology of/constructed by Atiyah and Bott, mainly because we will in section <ref> need
        to recognize a certain explicit three-form onthat represents the three-dimensional cohomology class that was described somewhat implicitly in section <ref>.
        The following discussion is in the spirit of <cit.>, as later refined and extended for applications to four-manifolds <cit.> and  interpreted more physically <cit.>.
        Recall that we denote the exterior derivative onasδand we also defineψ=δ A.  
    The componentsA_μ^a(x)(xis a point inMandμandaare tangent space and Lie algebra indices, respectively) are understood as functions on, whileψ_μ^a(x) is a one-form on.   A general functionF(A,ψ)that is homogeneous
        of degreekinψrepresents ak-form on.      We will extend the definition ofδwith other fields included so thatδ^2will generate a gauge transformation and therefore   will vanish on gauge-invariant functions.
            
        If we were interested in differential forms on,  we would simply defineδψ=0and thenδwould represent the exterior derivative acting on functions ofAandψ.   
        But  in order to construct differential forms on/, it is necessary to take into account
        the gauge group.   For this purpose, we introduce a Lie algebra valued scalar fieldσonM.  The action ofδon the three fieldsA,ψ,σis defined by
        
        δ A     = ψδψ   =-_̣Aσδσ   =0.
    
        Here_̣A=+̣[A,·]is the gauge-covariant extension of the exterior derivative.   These formulas imply thatδ^2is equivalent to a gauge transformation
        generated byσ; for example,δ^2 A=δψ=-_̣Aσis the infinitesimal transformation ofAunder a gauge transformation generated byσ, and similarlyδ^2ψ=[σ,ψ],δ^2σ=[σ,σ]=0.    Since we wantδto increase the degree of a differential form by 1, and we have assigned degree 0 toAand
        degree 1 toψ, we have to assign degree 2 toσ.   We will call this degree theδ-degree; it has also been called ghost number.   In keeping with the fact thatσhasδ-degree 2, we will eventually learn that it can be converted for many purposes  to a two-form on/.
        
        Since in generalδ^2is the generator of a gauge transformation, it follows that acting on gauge-invariant
        functionsF(A,ψ,σ),δ^2=0.   We will explain explicitly how to convert a gauge-invariant functionF(A,ψ,σ)that is homogeneous of degreekto ak-form on/.
    A functionFthat isδ-closed, that is, one that  satisfiesδ F=0, will map  to a closed  form on/.   What we have described is in fact the Cartan model of the equivariant cohomology
        ofacting on.   Becauseacts freely[Except for the center of  and except for subtleties involving gauge connections on M that are reducible, that is, those that do not 
       completely break the gauge symmetry down to the center of .   For our purposes, these subtleties are unimportant, because reducible instantons on ^3×^1
    arise only in very high codimension, and the center of  acts trivially on gauge fields.  However, a more careful treatment of the cohomology
        might be important for more delicate questions beyond the scope of the present article.] 
         on, this equivariant cohomology is the same as  the cohomology of/. 
        
        Before explaining[The reader might prefer to return to the following  discussion
    after reading section <ref>.]  how to map functionsF(A,ψ,σ)to differential forms on/, we will describe the important gauge-invariant andδ-invariant functions on.
        The most obvious possibility is to pick a pointx∈ Mand a homogeneous polynomialPof degreeson the Lie algebraof, and consider the functionP(σ(x)).  This is gauge-invariant andδ-invariant and is ofδ-degree2s, so it will map to a closed2s-form on/.   This form is notδ-exact, sinceP(σ(x))is notδØfor anyØ, so it will potentially lead to a nontrivial cohomology class on/.   
        
        To minimize the notation that is required in the following discussion, let us assume that=(Q_5), in which case the ring of invariant polynomials onhas a simple
        set of generators that we will denoteP_s^(0)(σ)= σ(x)^s,s=2,⋯, Q_5.  
         Now with$̣ as the exterior derivative on M, we compute P̣_s^(0)= s  σ^s-1_̣A σ.   This is not annihilated by δ, but we find s  σ^s-1_̣Aσ=-
         δ P_s^(1) with P_s^(1)=s  σ^s-1ψ.    Thus δ P_s^(1) does not vanish, but rather
         δ P_s^(1) =-P̣_s^(0). 
         In this formula, P_s^(0) is a scalar function on M, which can be defined at any point x∈ M, and has δ-degree 2s; on the other hand P_s^(1) is a one-form on M
         of δ-degree 2s-1. 
         
          The formula (<ref>) can be
         read in two ways.   First of all,  reading the formula from right to left, 
         it says that although P_s^(0) is not δ-exact, its derivative along M is δ-exact.   Hence, once we learn how to interpret
         σ(x)^s as a 2s-form on /, this 2s-form will be independent of x, up to an exact form.   
         
        On the other hand, reading the formula from left to right, it says that although the 1-form P_s^(1) on M is not δ-closed, its variation under δ is $̣-exact.
        Hence ifγ⊂ Mis a closed loop, then∮_γ P_s^(1)isδ-closed, since
        δ∮_γ P_s^(1)=-∮_γP̣_s^(0)=0. 
        Hence, once we learn to convert functionsF(A,ψ,σ)that are gauge-invariant andδ-invariant to closed differential forms on/, the function∮_γ P_S^(1)ofδ-degree2s-1will
        correspond to a closed differential form of degree2s-1on/.
        
        What we have described so far is the first step in a “descent” procedure.   Form≥ 0, one finds inductively gauge-invariant polynomialsP_s^(m)(A,ψ,σ)such thatP_s^(0)= σ^s(x)andP̣_s^(m)=-δ P_s^(m+1).   These relations imply that ifΣ⊂ Mis anym-cycle, then∫_Σ P_s^(m)isδ-closed (indeed,δ∫_Σ P_s^(m)=-∫_ΣP̣_s^(m-1) =0).    So∫_Σ P_s^(m), which hasδ-degree2s-m, will correspond to a closed form on/of degree2s-m.    The cohomology class of this form on/only depends on the homology class ofΣ,
        since ifΣ=∂ Bis a boundary, then∫_Σ P_s^(m) =∫_B P̣_s^(m)=-δ∫_B P_s^(m+1)isδ-exact.
        
            The construction just sketched
         has been important in the theory of smooth four-manifolds <cit.>.   As follows from the arguments in  <cit.>, the cohomology classes corresponding to∫_Σ P_s^(m)for variousΣandmgenerate the (real)  cohomology ring of/.  All these classes can be restricted from/to.   An analog
          of the Atiyah-Jones conjecture would potentially say that these restrictions generate the real cohomology of, but at any rate, the cohomology classes ofthat
          will be important in this article can be obtained as such restrictions.
          
               
          For the particular caseM=^3×^1, we can only make one generator ofH^1(;)this way, namely∫_^3× q P_2^(3)for an arbitrary 
          pointq∈^1, and one generator
          ofH^2(M;), namely∫_^1×^3P_3^(4).    These  correspond to the generators described in more direct terms in sections <ref> and <ref>.
          In degree 3, there are two possible generators, namely∫_p×^1P_2^(1), withp∈^3, and∫_^3× q P_3^(3),q∈^1.  The first
          of these corresponds as we will see to the field strength  of theB-field on, and the second will not play an important role in the present
          article.   
          
          Before explaining how to concretely interpret∫_Σ P_s^(m)as a closed differential form on, we will make the descent procedure more explicit in the
          case that is actually important for understanding theB-field on.   Settings=2and changing the normalization, we setP^(0)=1/8π^2 σ^2.
          Inductively solvingP̣^(m)=- δ P^(m+1), we find
          
        P^(0)    =1/8π^2 σ^2   
               P^(1)   =  1/4π^2 σψ
               P^(2)   = 1/4π^2 (1/2ψ∧ψ - σ F) 
                P^(3)    = -1/4π^2 ψ∧ F 
                P^(4)    = 1/8π^2 F∧ F.
    
      SoP^(4)is the four-form whose integral overMis the instanton number∫_Σ c_2(E), wherec_2is the second Chern class of the gauge bundleE→ M.
        This statement can be generalized as follows.   Consider a family of gauge connections onE→ M, parametrized by some parameter spaceS.   These will fit together[For a more complete description, see section <ref>. We disregard a potential
      obstruction involving the center of the gauge group,
    which can be circumvented by considering the bundle ad(E) associated to E in the adjoint representation of  and is inessential for our purposes.  Note that rather
    than a single -bundle E→ M, we really should begin this construction with a family of  bundles E→ M, parametrized by S. ]  
         into
        a gauge connection in theMdirection on a bundle E→ M× S(for a points∈ S, Erestricts ons× Mto the originalE→ M, up to isomorphism).  In section <ref>, in discussing theB-field on, withM=^3×^1, we made this construction forS=^2, and
        subsequently forSa three-manifoldW.  The universal choice ofSis simply/, which parametrizes all possible gauge fields onM, up to gauge transformation.
       Any otherSis obtained by mapping some other space into this one.
       
       The bundle E→ M×/Ghas a second Chern class of degree 4.   We will  consider this second Chern class as a 
       real cohomology class, valued inH^4(M×/;).   This cohomology group has a decompositionH^4(M×/;)=⊕_m=0^4 H^m(M;)⊗ H^4-m(/;).   The geometrical interpretation of the descent procedure <cit.> is
       thatP^(m)corresponds to the part ofc_2( E)that is valued inH^m(M;)⊗ H^4-m(/G;).    Then if we are given anm-cycleΣ⊂ M,
       we can integrateP^(m)overΣto get aδ-closed function ofδ-degree4-m, namely∫_Σ P^(m), representing an element ofH^4-m(/;).
       This element can be restricted from/to⊂/to get aδ-closed function on, which will correspond to a closed form of degree4-m.
       
       To get a three-dimensional cohomology class on/, we have to implement this procedure withm=1.   So we pick a pointp∈^3and integrateP^(1)overp×^1to get1/4π^2∮_p×^1 σψ.   Once we learn how to interpretσas a two-form on/, this will indeed become a closed
       three-form on/, normalized so that its periods (its integrals over closed three-cycles in/) are integers.  That integrality will follow from  the integrality of the second Chern class.
       The cohomology class of1/4π^2∮_p×^1 σψdoes not depend onp, as explained in the discussion of eqn. (<ref>).   Any choice ofpwill give a representative of the same cohomology class, but we can get a particularly nice representative by averaging overp.  If^̣3Ωis a “round”
       volume form on^3normalized so that the total volume is 1, then the average is just∫_^3×^1^̣3Ω P^(1).   After convertingσto a two-form, this will become a three-form on/– and therefore, by restriction, on– with integer periods.
      Since the field strength of theB-field is supposed to have periods that are2πtimes integers, a minimal three-form that satisfies Dirac quantization will be2πtimes this or
      H_0=2π∫_^3×^1^̣3Ω  P^(1). 
    
    
      
      In a moment, we will explain how to turn the expression (<ref>)
       into a concrete three-form on/, by eliminatingσin favor ofAandψ.   
      But for now, let us compare the expression (<ref>)  to eqn. (<ref>).   Since eqn. (<ref>) is a formula for∫_W H(rather than a description of a three-formH), to make this comparison
      we will integrate eqn. (<ref>) over a three-cycleW⊂/, corresponding to a family of gauge connections on^3×^1.   We 
      get
      ∫_W H_0=2π∫_W×^3×^1^̣3Ω P^(1).
      Since∫_^3^̣3Ω=1, we can do the integral over^3, giving∫_W H_0 =2π∫_W×^1 P^(1).But sinceP^(1)just represents the part of1/8π^2 F∧ F∼ c_2( E)that is of degree 1 on^3×^1and degree 3 on/, we have∫_W H_0=2π∫_W×^1 F∧ F/8π^2.    Comparing this to eqn. (<ref>), we see that the relation betweenH_0and the field strengthH=Ḅof
      the sigma-model is
      H = Q'_5 H_0+(̣⋯).
    The derivation of this statement involved integrating over an arbitrary cycleW, so the statement only holds modulo an unknown exact form, denoted here as(̣⋯). 
      
      The reader who has gotten this far may well feel that we have not learned much, since we already asserted at the end of section <ref> that theB-field of the sigma-model
      isQ'_5times a minimalB-field that obeys Dirac quantization.   However, by turningσinto an explicit differential form, we can now do what was not available previously
      and describe  explicit three-form representatives ofH_0.
      
      Actually, though the formula forH_0that we will get makes sense and is correct as a three-form on/, it is possibly easier to get some intuition if we restrict to, which is in any event
      what we want to do for our application.   In this analysis, we make use of the way this construction fits into an actual quantum field theory, as explained in <cit.>.
      
      When we restrict to, the gauge fieldAon^3×^1satisfies the instanton equation[We denote the selfdual and anti-selfdual parts of a two-form
      b as b^+ and b^-.]F^+=0.  (In the construction in <cit.>, the restriction tois made  in the sense of supersymmetric localization.)
       Thereforeψ=δ Asatisfies the linearization of the
      instanton equation, namely(_̣A ψ)^+=0.   However, being a one-form in four-dimensions,ψhas four components, and (since the bundle of selfdual two-forms has
      rank three) the equation(_̣Aψ)^+=0is only three equations.  Therefore in any physical model in which this construction is embedded,ψwill obey a fourth equation.
      That equation is model-dependent, and different choices will actually lead to different three-forms all representing the same cohomology class on.
      
      In <cit.>, the theory considered was a twisted version of=2super Yang-Mills theory.   In that theory, in addition toA,ψ,σ, the important fields for our present
      purposes are a pairσ̅,η. These  are  spin zero fields in the adjoint representation;σ̅is a boson ofδ-degree-2andηis a fermion ofδ-degree-1.  They transform underδas
      
        δσ̅    = ηδη    = [σ,σ̅].
    
       We note that these formulas are consistent withδhavingδ-degree 1, and withδ^2being a gauge transformation generated byσ(thus,δ^2σ̅=[σ,σ̅], and similarly forη).
       
       The action of the theory isδ-invariant,  and the part of the action that is relevant for our purposes isδ-exact.   There is much arbitrariness in theδ-exact part of
       the action.   The minimal choice made in <cit.> amounted to
       -δ∫_M^̣4x√(g)  σ̅D_iψ^i=∫_M ^̣4x √(g) ( σ̅D_i D^iσ  -σ̅[ψ_i,ψ^i] -η D_iψ^i ). 
       Assuming that these are the only terms by whichσ̅andηappear in the action (or the only terms that are relevant in lowest order, which is good enough
       in the localization argument), we can read off the equations of motion:
       
        D_iψ^i     = 0 
                                     D_i D^i σ    = [ψ_i,ψ^i].
    
        We have simultaneously accomplished two things.   First, we have found the desired extra equation of motion forψ, namelyD_iψ^i=0.   And second,
        we have learned how to expressσin terms ofψ, namely by solving the equationD_i D^iσ =[ψ_i,ψ^i], implying that
        σ(x) =∫_^3×^1 B(x,y)[ψ_i,ψ^i](y)^̣4y √(g_y) , 
        whereB(x,y) is the Green's function of the Laplace operator:
        D/D x^iD/D x_i B(x,y)=δ^4(x,y) Id. 
        Here√(g_y)is just√(g)regarded as a function ofy, andIdis the identity operator acting on the fiber  of the adjoint bundle(E).   
         
        Using in eqn  (<ref>) the explicit formula forP^(1)from eqn. (<ref>) and using eqn. (<ref>) to eliminateσ, 
        we get an explicit formula forH_0as a three-form on:
        H_0=1/2π∫_^3×^1×^3×^1^̣3Ω_x τ̣_x ^̣4 y√(g_y)⊗ ψ_τ(x) B(x,y) [ψ_μ,ψ^μ](y). 
        Here the two copies of^3×^1are  parametrized respectively  byxandy.
       We have used the fact that the only part ofψ(x)that contributes in the evaluation of eqn. (<ref>) isψ_τ(x)τ̣.
        
        
    The preceding derivation can be generalized by replacingD_iψ^ion the left hand side of 
     eqn. (<ref>) with another  expression.  Then the auxiliary conditionD_iψ^i=0in  eqn. (<ref>)
    will be replaced by another condition.  As we will learn in section <ref>,
    to understand the moduli spaceof instantons on^3×^1, a different auxiliary condition is more useful, namely D_iψ^i+⋆(H∧ψ)=0
     (where⋆is the Hodge star).
    To modify the preceding derivation to incorporate this condition, we merely add anotherδ-trivial term to the action, taking it to be
       
        -δ∫_M^̣4x√(g)  σ̅( D_iψ^i+⋆(H∧ψ))  =∫_M   ^̣4x√(g) (σ̅(D_i D^i σ
           +⋆(H∧_̣Aσ) ) .   .-η( D_iψ^i+⋆(H∧ψ))-σ̅[ψ_i,ψ^i])
        
     The equation of motion forηgives the desired condition (<ref>).   But the equation of motion forσ̅also changes and is
     now
      Wσ=[ψ_i,ψ^i], 
     where the operatorWis defined by
      Wσ =D_i D^i σ +⋆(H∧_̣Aσ). 
     The formula (<ref>) is still valid, except thatB(x,y)is now the Green's function ofW, obeying
      W_x B(x,y) =δ^4(x,y)Id,
     whereW_xis the operatorWacting on the variablex.  An important detail, though, is that asWis not self-adjoint, its Green's functionB(x,y)is not symmetric
     inxandy, and the equation that it obeys in theyvariable is actually
      W_y^† B(x,y)=δ^4(x,y)Id,
     whereW^†is the adjoint operator
       W^†σ =D_μ D^μσ -⋆(H∧_̣Aσ). 
     One way to prove eqn. (<ref>)  is to observe that eqn. (<ref>) implies that
     ∫_M^̣4 y √(g_y)  B(x,y) W_y B(y,z) = B(x,z). 
     On the other hand, we can write the left hand side of eqn. (<ref>) as 
     ∫_M^̣4 y √( g_y)  W_y^† B(x,y) · B(y,z), 
     and eqn. (<ref>) follows by comparing this with eqn. (<ref>).
     
     Regardless of whether we take B(x,y) to be the Green's function of the Laplace operator, the operator W, or some other operator related to another valid condition for ψ,
     eqn. (<ref>) is an explicit formula for a three-form on  in the appropriate cohomology class.   But as will eventually become clear, the torsion of the generalized hyper-Kahler metric of  is the representative of this cohomology class that  we get if B(x,y) is the Green's function of the operator W.  
     
        
        
    
    
        
         
        
        
        
        
      
    
    § SOME BACKGROUND
    
     
     In this section, we will review relevant background material on three topics:   supersymmetric sigma-models with a B-field; the relevant geometry of ^3×^1;
     and the large =4 superconformal algebra in two dimensions.   The first two topics were reviewed recently in <cit.>;  more information can be found there on some aspects,
     along with further references.
     
     
    
     §.§ Supersymmetric Sigma-Models With a B-Field
    
     
     The most basic type of  two-dimensional supersymmetric sigma-model with a B-field is a model with (0,1) supersymmetry.[We will treat these models
     classically, not worrying about sigma-model 
     anomalies <cit.>, as they will be absent in the =4 models that we consider eventually.   Likewise we will not worry about a possible 
     anomaly in the (1) chiral symmetry in the (0,2) models that we discuss presently; this is again absent when the model is extended to =4 and the global
     symmetry is extended from (1) to (2).]
     We  will briefly review such models following
     <cit.>.   We consider a model with a general target space , which we describe by local coordinates X^P, P=1,⋯,, and endow with a metric
     G_PQ and a B-field B_PQ, with field strength H_PQR=∂_P B_QR+∂_Q B_RP+∂_R B_PQ.   On a superspace Σ of dimension 2|1 with
     even coordinates x^-, x^+ and one odd coordinate θ, we introduce a  supersymmetry generator Q_+=ı(∂/∂θ-ıθ∂/∂ x^+), satisfying Q_+^2=ı∂_+ and commuting with D=∂/∂θ+ıθ∂/∂ x^+, which is used in writing Lagrangians.
     The sigma-model map Φ:Σ→ can be described concretely in terms of superfields ^P=X^P+ıθψ^P.  The supersymmetry variation of these fields is
     is δ^P=Q_+^P or 
     δ X^P=ıψ^P,  δψ^P=∂_+ X^P.
     A natural supersymmetric action for these fields  is
     S=-ı/2∫^̣2 x θ̣ (G_PQ+B_PQ) D ^P ∂/∂ x^-^Q.    
     After integrating over θ, the bosonic part of the action is the standard 
      S_b=1/2∫^̣2x (G_PQ+B_PQ)   ∂_+X^P ∂_- X^Q.  The fermion action turns out to be
      S_f =ı/2∫^̣2x  G_PQψ^P(δ^Q_S∂/∂ x^- + ∂ X^R/∂ x^-Γ^Q_RS)ψ^S,
     where  Γ^Q_RS is not the usual Riemannian affine connection Γ^Q_RS but has also a torsion term proportional to H:
     Γ^Q_RS =Γ^Q_RS +1/2G^QTH_TRS. 
     Using this new affine connection, we
      define a connection ∇ with torsion that differs from the Riemannian connection D. The covariant derivative of a vector field V with respect to
     ∇ is 
     ∇_P V^R=∂_P V^R+Γ^R_PQV^Q=D_P V^R+1/2 G^RSH_SPQV^Q. 
     This connection has completely antisymmetric torsion H.  Accordingly it is metric compatible, meaning that the Riemannian metric G_PQ is covariantly constant.
     
     As an example of the usefulness of this connection, let us ask a question whose importance in relation to the large =4 algebra was explained in <cit.>.
     What condition should a vector field V on  satisfy so that the sigma-model field Λ=∑_K V_Kψ^K obeys ∂_-Λ=0 and thus is a chiral free fermion?  
     Since the equation of motion for ψ is ∂_- ψ^P=-∂_- X^R Γ̃^P_RSψ^S, we have
     ∂_-Λ=∂_-  X^R∂_R V_S ψ^S  - V_P∂_- X^R Γ̃^P_RSψ^S.
     The condition for this to vanish is ∂ _R V_S-Γ̃^P_RS V_P=0, or in other words ∇_R V_S=0, that is, V is covariantly constant for the connection ∇.  
     If this condition is satisfied, then in addition to the chiral free fermion Λ of spin 1/2, we get a free chiral current of spin 1:
      J={Q,Λ}= V_K∂_+ X^K+ı∂_L V_K ψ^Lψ^K. 
    
     
     Up to this point, G_PQ and B_PQ were not subject to any particular constraint.   That changes if we ask for the sigma-model to have (0,2) supersymmetry <cit.>.
     The =2 superconformal algebra has a (1)R-symmetry under which the two supercharges have charge ± 1.  We do not want to restrict the target
     space   to have a  (1) symmetry,
     so we assume the bosons to have R-charge 0 and the fermions to have R-charges ± 1.
    Thus we  assume a (1) symmetry that acts on fermions only, generated by 
     δψ^P=^P_Q ψ^Q ,
     with some linear operator .   In a generic local coordinate system on , the matrix elements of  are position-dependent, but as the components of ψ  are supposed
     to have R-charges ± 1, we can normalize  so that ^2=-1, making   an almost complex structure on . This implies in particular that  has
     even dimension 2p.  We denote the +ı and -ı eigenmodes
     of  as ψ^α and ψ^β̅, α,β=1,⋯ ,p.   The condition that the fermion action S_f of eqn. (<ref>)
     actually does have the symmetry of eqn. (<ref>) gives two conditions.   First, the metric tensor G must be of type (1,1) with respect to , meaning that
     its nonzero components are G_αβ̅=G_β̅α.  (The metric G is therefore said to be hermitian with respect to .) 
      Second, since S_f is constructed in terms of the covariant derivative of ψ with
     respect to the connection ∇, invariance of S_f requires that the tensor ^P_Q should be covariantly constant with respect to this connection:
     ∇ = 0. 
     Thus both G and  are covariantly constant with respect to ∇, implying that ithe structure group of ∇ reduces to (p).  
    The R-symmetry conjugates the original supersymmetry (<ref>) to a second one that acts by 
    δ X^P=^P_Q ψ^Q,   δψ^P= -^P_Q∂_+ X^Q.
    
     Requiring that this squares to a translation generator ı∂_+ gives a condition that is quadratic in  and its first derivative.   This condition is precisely the
     vanishing of the Nijenhuis tensor.  Thus  is actually an integrable complex structure on , and one can introduce on  local holomorphic coordinates x^α, x̅^β̅,
     α,β =1,⋯, p.   In such a coordinate system, the nonzero components of  are simply ^α_β=ıδ^α_β, 
     ^α̅_β̅=-ıδ^α̅_β̅, and the  condition (<ref>) simplifies to Γ̃^α_βγ̅=Γ̃^α_β̅γ̅=0 (and the complex conjugate of this).   From this we learn that H_αβγ=H_α̅β̅γ̅=0, saying that  H is of type (2,1)⊕ (1,2) with respect to , and that 
      H_αβγ̅=-∂_α G_βγ̅+∂_β G_αγ̅. 
     For H to be the field strength of some two-form field  B,
     one requires[Here the remark in footnote  <ref> is relevant. As sigma-model anomalies will eventually be canceled
     by adding left-moving fermions, we are not interested in canceling them by a Green-Schwarz mechanism with Ḥ≠0, as is appropriate in the context of the heterotic string.]Ḥ=0, which here tells us that
     ∂^2/∂ x^α∂x̅^β̅G_γδ̅x̣^αx̣^γx̣̣̅^β̅x̣̣̅^δ̅=0.
     So locally 
      G_αβ̅=∂_α K_β̅+∂_β̅ K_α,
      where the one-form K is the closest analog in this context of a Kahler potential.
     This observation is natural in a certain superspace construction of this class of models <cit.>.   
     
     Associated to this data we can define a hermitian form ω by ω_PR=G_PQ^Q_R, or in local holomorphic coordinates ω_αβ̅=-ı G_αβ̅
     =-ω_β̅α, with other components of ω vanishing. In particular ω is of type (1,1); moreover, ω  is covariantly constant with respect to ∇,
     since G and  are.     Note that the two-form ω associated to an antisymmetric tensor ω_PR is defined as[The factor of 1/2 ensures that if A is a one-form and F=Ạ, then F=1/2∑_P,RX̣^P X̣^R F_PR where the definition of F_PR is the
     standard F_PR=∂_P A_R-∂_R A_P. Note that as =̣∑_P X̣^P∂_P and A=∑_R X̣^R A_R, we have Ạ=∑_P,RX̣^P X̣^R∂_P A_R
     =1/2∑_P,RX̣^PX̣^R F_PR.]ω=1/2∑_P,RX̣^P X̣^R ω_PR.
       If ω̣=0, then  is called a Kahler manifold and eqn. (<ref>) implies that H=0.
     Indeed, if as usual we expand the exterior derivative in parts of type (1,0) and (0,1) by writing =̣∂+∂̅, then eqn. (<ref>)   tells us that
     H=-ı(∂-∂̅)ω. 
     An alternative way to write this formula is the following.   One defines an action of  on differential forms by (f(X) X̣^P)=f(X) ^P_Q X̣^Q and more generally
     (f(X) X̣^P_1⋯X̣^P_s)=f(X) ^P_1_Q_1X̣^Q_1⋯^P_s_Q_sX̣^Q_s. 
     Then, since ω is of type (1,1) and H is of type (2,1)⊕ (1,2),   the formula (<ref>) for H is equivalent to
      H=-ω̣=-ω. 
     The formulas (<ref>) and (<ref>) are useful because they express H just in terms of the gauge-invariant data ω, , as opposed to the usual formula
     H=Ḅ, where B is gauge-dependent.     For completeness, we also include a formula for the Lee form (which appears in the one-loop beta function of a model
     of this type <cit.>)
     θ_K=-1/2^L_K H_LPQ^PQ,
     though we will not actually use this formula in the present article.
     
     Having gotten this far, the further extension to (0,4) supersymmetry is straightforward.   In this case, one wants an (2)R-symmetry that acts only on the fermions.
     This group has three symmetry generators ,,𝒦, and as the fermions should all transform as spin 1/2 under (2), the generators can be normalized so that
     they obey the quaternion relations ^2=^2=𝒦^2=-1, =-=𝒦.  This implies in particular that  must have dimension D=4q for some integer q.
     Using  or  instead of  in eqn. (<ref>) will give two more supersymmetries, making four supersymmetries in all.
       The preceding considerations apply to each of ,,𝒦 separately, so in particular each of ,,𝒦
     is an integrable complex structure.   Being endowed with three complex structures that satisfy the quaternion relations,  is called a hypercomplex manifold.[A hypercomplex manifold actually has a family of complex structures parametrized by ^2, since if a,b,c are real numbers
    satisfying a^2+b^2+c^2=1, then the quaternion relations imply that a+b+c is an almost complex structure, and it is integrable if , , and  are integrable.] Moreover, the metric G is of type (1,1) for each of ,,𝒦, and is therefore said to be hyper-hermitian.    Each of ,, and 𝒦
    must be covariantly constant for the connection ∇ constructed as in eqn. (<ref>) in terms of G and H.  This implies that the structure group of ∇
    reduces to (q). For each complex structure, we define a corresponding hermitian form ω^()_ PR=G_PQ^Q_R, ω^()_ PR=G_PQ^Q_R,
    ω^()_PR=G_PQ^Q_R.  The derivation of eqn. (<ref>) applies equally to any of ,,,  so we must have
    H=-ω̣^()=-ω̣^()=-ω̣^().   
    For the same three-form H to satisfy these three different formulas is a very strong constraint. 
    
    So far we have discussed (0,n) worldsheet supersymmetry with n=1,2,4.   Now we discuss the extension to (n,n) supersymmetry.   Roughly speaking, one just
    adds additional fermions of opposite chirality and one requires two copies of the structure that has just been described, one copy for positive chirality and one for negative
    chirality.   Since both chiralities of fermions will be present, we write henceforth ψ^P_+ for the fermions that we have been considering so far and ψ^P_- for
    the fermions of opposite chirality.  For n=1, the main point to consider is that exchanging the two chiralities involves reversing the worldsheet orientation, and under
    this operation H is odd.   Therefore, fermions of the two chiralities  will see  connections that differ in  the sign
    of the torsion, which is proportional to H and therefore is odd under exchanging the two chiralities.  Since two different connections with torsion will play a role,
    we will be more precise in the notation.  
    Previously we defined a connection ∇ 
    that appears in the kinetic energy of the positive chirality fermions and is described by the affine connection (<ref>).   Reversing the sign of H, we get the affine connection 
    Γ'^Q_RS =Γ^Q_RS -1/2G^QTH_TRS
    associated to the connection ∇' that appears in the kinetic energy of negative chirality fermions.
     The respective definitions of covariant derivatives are
      ∇_P V^R=D_P V^R+1/2 G^RSH_SPQV^Q,   ∇'_P V^R=D_P V^R-1/2 G^RSH_SPQV^Q. 
      It will be important that these definitions depend only on the metric G and the torsion H, not on the complex structure or hermitian form.
     
     To get (2,2) supersymmetry, we want separate (1)R-symmetries for both positive chirality and negative chirality fermions.   We denote the
     corresponding symmetry generators as  and '.   An important insight of <cit.> is that in general no relation between  and '
     must be assumed.   In particular, in general they need not commute, though models in which they do commute are simpler in some respects
     (the models considered in the present article do not have that property).  The derivations that we have explained up to this point apply separately
     for positive chirality fermions and  and for negative chirality fermions and '.  In particular, both  and ' are integrable complex
     structures;  is covariantly constant for the connection ∇ and ' is covariantly constant for the connection ∇'. So both
     connections have holonomy in (p).   Defining the
     two hermitian forms ω_PR=G_PQ^Q_R,   ω'_PR=G_PQ'^Q_R, the derivation of eqn. (<ref>) applies for
     each, with a sign change:
     H=-ω̣=+' ω̣'. 
     The structure just described has been interpreted as generalized Kahler geometry <cit.>.   It has also been called strong bi-KT geometry (KT refers to
     Kahler geometry generalized to allow torsion; the prefix “bi” refers to the presence of separate KT structures for the two fermion chiralities; and “strong” means that
     Ḥ=0).  
     
     To get (4,4) supersymmetry, we need separate hypercomplex structures ,, for positive chirality fermions, with closed torsion H,  and ',','
     for negative chirality fermions, with torsion -H.   Here ,, must be covariantly constant for ∇, and ',',' must be covariantly
     constant for ∇'.   So both connections have holonomy in (q).   Defining the various hermitian forms by ω^()_PR=G_PQ^Q_R,
     etc., the derivation of the relation between H and the hermitian forms goes through as before to give formulas for H in terms of any one
     of , , or ', ',':
      H=-ω̣^()=-ω̣^(
    J)=-ω̣^( ) =+' ω̣^( ')='ω̣^(')
      =+'ω̣^('). 
     Such geometry has been called generalized hyper-Kahler geometry, or strong bi-HKT geometry (where HKT refers to hyper-Kahler geometry generalized to allow torsion). 
     
     
     
    
    
     §.§ Large And Small Algebras
    
    
    
    A sigma-model with any (complete) generalized hyper-Kahler target M of dimension D=4q  has superconformal symmetry with what is known as the small =4 algebra.   Obvious
    generators are the stress tensor T and the four supercurrents G_a associated to the four supersymmetries that were constructed in section <ref>.  The other generators of the chiral algebra, for generic M, are the currents
    of an (2) current algebra at level q that acts only on the fermions.
    These (2) current algebras exist, for both left- and right-moving modes, because the fermion kinetic energy is constructed using connections ∇ or ∇'
    whose (q) holonomy groups each commute with a corresponding (2) action on the tangent bundle of M.   For both left-movers and 
    right-movers of the sigma-model, it is possible to define
    fermion bilinears that generate the corresponding (2).  Along with T and the G_a, these bilinears generate the small =4 superconformal algebra.  
    
    In the small =4 algebra,  the G_a are primaries of the (2) current algebra; they transform in the spin 1/2 representation of (2), viewed as a
    four-dimensional real representation.  There is a natural (4) action on the four real fields G_a.   Here (4)=((2)× SU(2))/_2, where the (2) that
    is part of the small =4 algebra is one of the two factors.    In particular, this  (2) action on the G_a commutes with a second (2).  
    This second (2) is a group of outer automorphisms of the
    small =4 algebra; it commutes with all the algebra generators except the G_a.     In many familiar models with small =4 superconformal symmetry, the outer automorphism group is not realized as a symmetry.  
    For example, a sigma-model with target ^4 or K3 has small =4 superconformal symmetry, and the outer automorphism group is not realized as a symmetry.
    
    A simple example of a theory with small =4 superconformal symmetry in which the outer automorphism group is realized is the theory of a single free hypermultiplet,
    or in other words a sigma-model with target the flat hyper-Kahler manifold M=^4.   There are four massless free bosons ϕ_AX, A,X=1,2 with a reality condition
    ϕ̅^AX=ϵ^ABϵ^XYϕ_BY (here ϵ is the antisymmetric tensor with ϵ^12=1); similarly there are four massless free chiral fermions ψ_RX, R, X=1,2, with – in Lorentz signature – a similar reality condition.
    (For simplicity we consider fermions of one chirality only.)   The four supercurrents are G_AR=∂ϕ_AXψ_RYϵ^XY and the generators of the
    (2) current algebra are J_RS=ϵ^XYψ_RXψ_SY.   Clearly, those currents generate a current algebra that acts on the fermions only, by the natural (2)
    action on the first index of ψ_RX,  and just as clearly, the G_AR
    transform as spin 1/2 under this (2).  This current algebra is at level 1; the analogous current algebra in a model with q hypermultiplets 
     is at
    level q.      The outer automorphism group of the small =4 algebra, in this presentation, is an (2) symmetry that acts on the first
    index of G_AR.   It is realized in this model by a global (2) symmetry that acts on the first index of the scalar field ϕ_AX, with trivial action on ψ_RX.   
    However, it is not possible to define a holomorphic current that generates this symmetry, and therefore the currents that generate it are not part of the superconformal algebra
    of the theory.   
    
    To extend the small =4 algebra associated to a hypercomplex structure ,, on some generalized hyper-Kahler manifold to
    a large =4 algebra, this manifold   should have an (2) symmetry that rotates , , and  and that is generated by holomorphic currents.  The simplest example
    is M=^3×^1, to which we turn next.    One of the main goals of the present article is to show that the moduli space  of instantons on M is another example.
        
     
    
     §.§ Geometry of ^3×^1
    
     
     To understand the unusual geometry of ^3×^1, it is convenient to start with ^4 minus the origin with the scale invariant metric
     ṣ^2=̣⃗Y^2/Y⃗^2, 
     where Y⃗=(Y_0,Y_1,Y_2,Y_3),Y⃗^2=∑_I  Y_I^2.   Because the metric is scale-invariant, we can divide by the equivalence relation Y⃗≅ e^TY⃗ for any
     fixed T>0.   Setting τ=log |Y⃗|, the  quotient is ^3×^1 with metric
     ṣ^2=Ω̣^2+τ̣^2,   τ≅τ+T,
     where Ω̣^2 is the metric of a round three-sphere of radius 1.  If Y_i= y_i e^τ, ∑_i y_i^2=1, then Ω̣^2=∑_i ỵ_i^2.
     
     To see a complex structure on ^3×^1, let Z_1=Y_0+ı Y_1, Z_2=Y_2+ı Y_3.   Then we can parametrize ^3×^1 by complex variables (Z_1,Z_2)
     with (Z_1,Z_2)≅ e^T(Z_1,Z_2), exhibiting a complex structure that we will call[In discussing ^3×^1, to
     take advantage of the symmetries and write some formulas more economically, we denote a triple of complex structures that obey the quaternion relations
     as _1,_2, _3 rather than ,,.]_1.   In this complex structure, the metric of ^3×^1 is hermitian (that is, of type (1,1))
     but not Kahler.   Indeed ^3×^1 cannot be Kahler as its second Betti number vanishes.   ^3×^1 with this complex structure is known as a Hopf surface, a prototype
     of a compact manifold that is complex but not Kahler.   
     
     Complex structure _1 is not invariant under all of the symmetries of ^3×^1, so those symmetries can be used to generate many more complex structures.
     Parametrize ^3=(2) as
     g=[        z_1 -z̅_2  z_2       z̅_1 ],  |z_1|^2+|z_2|^2=1,  Z_i = z_i e^τ. 
    We denote  (2) acting on itself on the left or right 
    as (2)_ℓ or (2)_r.   Clearly,  is invariant under (2)_ℓ, which maps z_1 and z_2
    to complex linear combinations of themselves, but not under (2)_r.   Rather, (2)_r maps _1 to a family of complex structures on ^3×^1
    that make up a hypercomplex
    structure (explicit formulas are given presently).  Moreover the discrete symmetry ρ that acts by a joint reflection of the two factors of ^3×^1 exchanges (2)_ℓ and (2)_r,
    and so maps  this hypercomplex structure to a second one that is invariant under (2)_r and rotated by (2)_ℓ. 
    
    To write explicit formulas,   view ^3×^1 as the group manifold K=(2)×(1).  The metric  (<ref>)
      is invariant under the left and right action of K on itself, so it is possible to choose orthonormal bases consisting of left-invariant or right-invariant one-forms.
      As (1) is abelian, L^0=R^0=τ̣ is both left-invariant and right-invariant.   To find left-invariant forms on (2), we simply note that g^-1g̣ is
      left-invariant.  
    Expanding this in components, a basis of left-invariant one-forms is given by
      L^1=y_0ỵ_1-y_1ỵ_0+y_2 ỵ_3-y_3ỵ_2,
      and two more forms  L^2 and L^3 that differ from this by cyclic permutation of indices 1,2,3.   The L^a are an orthonormal basis of the cotangent bundle of ^3,
      since  ∑_a=1^3 L^a⊗ L^a=Ω̣^2.
      Because of the (2)×(2) symmetry of ^3, to verify this and similar statements later, it suffices to verify that the statement is true at the
      point p with (y_0,y_1,y_2,y_3)=(1,0,0,0); this is immediate.  The L^a obey
      Ḷ^1=2 L^2∧ L^3
      and cyclic permutations of this statement.  Of course, we also have 
      Ḷ^0=0.  Similarly, g̣ g^-1 is right-invariant and can be expanded in terms of the right-invariant one-form
       R^1=y_0ỵ_1-y_1ỵ_0-y_2 ỵ_3+y_3ỵ_2,
       and two more right-invariant forms  R^2 and R^3 differing from this by cyclic permutation of indices 1,2,3.  
       Another way to see that the  L^a and R^b are respectively left-invariant and right-invariant is to observe that
       they are constructed from 4× 4 matrices that are respectively selfdual or anti-selfdual.  The R^b satisfy
       Ṛ^1=-2R^2∧ R^3,
       and cyclic permutations, again with
       Ṛ^0=0.   As the L^a and R^b for a,b=1,2,3 are  orthonormal bases of the cotangent bundle of ^3, L^1∧ L^2∧ L^3
       and R^1∧ R^2∧ R^3 are equal up to sign.   A short check at the point p shows that they are equal.
       
       Including L^0 and R^0, we get orthonormal bases L^i and R^j, i,j=0,⋯, 3, of the cotangent bundle of M=^3×^1:
       ṣ^2=∑_i=0^3 L^i⊗ L^i =∑_j=0^3 R^j⊗ R^j. 
       We can define a connection
       ∇ on the tangent bundle of M by saying that the L^i are covariantly constant, and another connection ∇'
       by saying that the R^j are covariantly constant.[In <cit.>, these connections are called ∇̂ and ∇̆.  A similar remark applies for the complex structures and
       hermitian forms introduced below. In the present article, the notation is different, as we reserve the “hat” (as in ∇̂) for a structure on the moduli space  as opposed
       to the four-manifold M.]     Both of these connections are metric compatible, since the L^i and R^j are orthonormal bases.  
       
         To place ∇  in the sigma-model context  of
     section <ref>, we need to find a three-form H such that for any one-form V,    ∇_P V_Q = D_P V_Q-1/2H_PQRV^R, where D is the
     Riemannian connection.   It suffices to verify this condition if V is one of the L's.
     In this case ∇ V=0, so we need D_P V_Q=1/2H_PQRV^R.  If this equation is symmetrized in P and Q, the right hand side  vanishes
     because H is totally antisymmetric, 
     and the left hand side also vanishes since the vector fields dual to the L^i are Killing vector fields, implying that  D_P V_Q+D_Q V_P=0.    The antisymmetric
     part of the equation is ∂_PV_Q-∂_Q V_P=H_PQRV^R, and this is satisfied for H=2 L^1∧ L^2∧ L^3 by virtue of eqns. (<ref>) and (<ref>).
     
     To place ∇' in the sigma-model context, we need the same equation to hold with H replaced by -H and ∇ replaced by ∇'.
     In other words, for every one-form V, we need ∇' _P V_Q =D_P V_Q +1/2 H_PQR V^R.  
     It suffices to verify this if V is one of the R's; this verification, again for H=2 L^1∧ L^2∧ L^3=2 R^1∧ R^2∧ R^3,  can be carried out 
    using eqns. (<ref>) and
     (<ref>).  
     
     Since a three-sphere of radius 1 has volume 2π^2, the fact that H=2L_1L_2L_3 is twice the Riemannian volume form of the sphere implies
     that ∫_^3H=4π^2.   This means that to construct a model in which  ∫_^3H=2π k, we  have to multiply the metric of ^3 by[The formula
     (<ref>) shows that rescaling G will rescale H by the same factor. Since the radius of ^1 does not affect ∫_^3H, it does not matter whether
     we rescale it or not. Only the rescaling of ^3 is relevant.]k/2π.   The line
     element of ^3 will then be
     ṣ^2=k/2πΩ̣^2. 
     This normalization will be important in section <ref>, but  for now we continue with the case of unit radius.
     
    The complex structure _1 that we defined at the outset acts on the L_i by
     _1(L^0)=L^1,  _1(L^1)=-L^0,  _1(L^2)=L^3,  _1(L^3)=-L^2. 
     Because the L_i are left-invariant, it suffices to verify this formula at the point p.
     The group (2)_r can act, in particular, by a cyclic permutation of L_1,L_2,L_3, thus conjugating _1 to two more complex structures 
     _2, _3 that differ  by cyclic permutation of indices 1,2,3.      A short calculation at the point p
     shows that the quaternion relations are obeyed.  And since the _a act by constant linear transformations on the one-forms L^i which are covariantly constant
     with respect to ∇, they are also covariantly constant.  
     
     The corresponding hermitian forms are
     ω^(1) =L^0∧ L^1+L^2∧ L^3,
     with ω^(2), ω^(3) obtained by cyclic permutation of indices 1,2,3.   To establish the =4 supersymmetric structure for positive
     chirality modes, it remains only to verify that eqn. (<ref>) is satisfied.
     We have ω̣^(1)=2L^0 ∧ L^2∧ L^3, and indeed _1(2L^0∧ L^2∧ L^3)=2L^1∧ L^2∧ L^3=H.
    
    
     To define a similar structure for negative chirality modes, we need a hypercomplex structure that is defined by  constant linear transformations of the R^j and hence  
     is covariantly constant for ∇'.   Here we have a sign choice to make as we can define either
     '_1(R^0)=R^1, '_1(R^1)=-R^0, '_1(R^2)=R^3, '_1(R^3)=-R^2,
     or 
      ”_1(R^0)=R^1, ”_1(R^1)=-R^0, ”_1(R^2)=-R^3, ”_1(R^3)=R^2,
     with  _2',_3' and ”_2, '_3 obtained by cyclic permutations.  The hermitian forms are then
     ω'^(1)=R^0∧ R^1+R^2∧ R^3,
    or
     ω”^(1)=R^0∧ R^2-R^2∧ R^3,
     along with cyclic permutations.
     
     The origin of this sign choice is as follows.   Starting with one hypercomplex structure defined by the _a, to get a second hypercomplex structure with equal and opposite
     torsion, one way is to apply a reflection of ^3, doing nothing to ^1, and a second way is to apply a joint reflection to both ^3 and ^1.  Under either operation, the
     three-form H=2 L^1∧ L^2∧ L^3=2R^1∧ R^2∧ R^3  changes sign, so either operation gives a second hypercomplex structure with opposite torsion.
     The joint reflection (which is unique only up to a rotation of ^3×^1)
     can be chosen to lead to eqn. (<ref>) and the reflection of only ^3  can be chosen to lead to eqn. (<ref>).   We have seen in section <ref> that only the joint
     reflection of ^3×^1 is natural in our problem, so we will use the hypercomplex structure defined by the '_a, not the one defined by the ”_a.
     
     In section <ref>, when we explain how a hypercomplex structure on a four-manifold leads to a hypercomplex structure on the instanton moduli space, we will
     see that (assuming instantons are defined to have anti-selfdual curvature) the construction only works if the hermitian forms associated to the hypercomplex structure are selfdual.
     A look back at the previous formulas shows that, if we orient ^3×^1 using the four-form L^0∧ L^1∧ L^2∧ L^3,
      the ω^(k) and ω'^(k) are selfdual but the ω”^(k) are anti-selfdual.   Therefore, the _k and '_k are the choices for which the construction will work.
       
        A consequence of asking for the ω^(k) and ω'^(k) to be all selfdual is that the _k and '_k do not commute.   Supersymmetric sigma-models with
     torsion are simpler in some respects when the left and right complex structures do commute <cit.>, but we will not be in that situation.
     
       While invariant under (2)_ℓ, the _k and ω^(k) transform in the adjoint representation of (2)_r.  Conversely,
       while invariant under (2)_r, the '_k and ω'^(k) transform in the adjoint representation of (2)_ℓ.
       
       
       
    
     
       
     
    
     §.§ The Large =4 Algebra
    
     
     The model we have arrived at – a sigma-model with a “round” metric on the target space ^3×^1, and with H a multiple of the volume 
     form of ^3– has a very special structure.   First of all, it is an exactly soluble conformal field theory, since the ^3 model with H a multiple of the volume
     form is simply an (2) WZW model, and the ^1 factor leads to a free superconformal field theory with abelian symmetry.   Beyond that, this
     particular model is actually a prototype of a model that exhibits the “large”=4 superconformal algebra <cit.>.   
    We will follow the presentation of  <cit.>.   (We also follow their notation, which differs slightly from notation used in the rest of the present article, and their conventions. Note
    that in those conventions, currents are antiholomorphic, which accounts for some minus signs in the following formulas.)
      
     The supersymmetric WZW with target (2) is actually equivalent to a purely bosonic (2) WZW model with three decoupled free fermions (of each chirality)
     in the adjoint representation of (2)<cit.>.[To be more precise, the  (2) WZW model has (2)_ℓ×(2)_r symmetry,
     with the two factors corresponding to the left and right action of (2) on itself.  With an appropriate choice of orientation, the  (2)_r currents are chiral
     (holomorphic) and are accompanied by three positive chirality fermions in the adjoint representation of (2)_r, and similarly the (2)_ℓ currents
     are antichiral (antiholomorphic) and are accompanied by three negative chirality chiral fermions in the adjoint representation of (2)_ℓ.]  
      Likewise the supersymmetric model with target ^1 is equivalent to a purely bosonic model with target ^1
     with a decoupled free fermion.  For generic radius of the ^1, the chiral algebra of the ^1 model is just generated by a single abelian current ∂ U.
     (For special radii, that is for particular values of the circumference of the ^1, the ^1 theory has additional chiral fields – exponentials of U.  We will
     not consider that case.)   Thus overall the supersymmetric model with target ^3×^1 has a chiral algebra that includes  (2) currents J^i, i=1,⋯,3
     of some integer level[The case κ=0 is possible <cit.>.   In that case, the bosonic WZW model with target (2) becomes trivial and drops out;
     the supersymmetric WZW model with target (2) is then just equivalent to three free fermions in the adjoint representation of (2).]κ≥ 0 associated with the bosonic (2) WZW model, four free fermions ψ^a,  a=0,⋯ 3 (one can consider ψ^0 to come from the
      ^1 model and the others from the (2) WZW model), and an abelian current J^0 that we can normalize to satisfy J^0(z)J^0(w)∼-1/2(z-w)^2. 
     The chiral algebra generated by these fields is actually the large =4 algebra.
     
      To describe the realization of this algebra, it is  convenient to introduce selfdual and anti-selfdual 4× 4 matrices
     α_ab^±,i=1/2(±δ_iaδ_b0∓δ_ibδ_a0+ϵ_iab),  i=1,⋯,3,
     where ϵ_iab is antisymmetric with ϵ_123=1. These matrices obey
     [α^±,i,α^±,j]=-ϵ^ijkα^± k,  [α^±,i,α^∓,j]=0,   {α^± i,α^± j}=-δ^ij/2.
     The generators of the large =4 algebra in this model are then[The discussion in section <ref> would make one anticipate the presence of generators
     T, G_a, and A_±,i, but does not make clear why U or Q^a are needed.   In fact <cit.>, it is possible to construct a “smaller” version of the large =4 algebra
     in which U and Q^a are omitted.  This algebra is a generalized W-algebra (the short distance singularity
     in a product of generators is in general a  nonlinear function of the generators,
     in contrast to simple chiral algebras like the Virasoro algebra – or the version of the large =4 algebra with U and Q^a included –
     that can be presented in such a way that OPE singularities are linear in the generators).
     The symmetry of string  theory on _3×^3×^3×^1 is, however <cit.>, the “large” version of the large =4 algebra, with extra generators U and Q^a and a simpler structure of the OPE's.      So that is the relevant version for our application.  The existence of the extra generators is a consequence of the symmetries of
    _3×^3×^3×^1.]
        T    =-J^0J^0-∑_i=1^3J^i J^i /κ+2-∑_a=0^3 ∂ψ^aψ^a 
                 G_a    = 2 J^0ψ_a+4α_ab^+,i J^iψ^b/√(κ+2)  -2ϵ_abcdψ^bψ^cψ^d/3√(κ+2)
                   A^-,i    = α_ab^-,iψ^aψ^b 
                    A^+,i   =J^i+ α^+,i_abψ^aψ^b 
                          U     = -√(κ+2) J^0
                           Q^a    =√(κ+2)ψ^a.
    
    The operators A^-,i and A^+,i generate (2) current algebras respectively at level 1 and level κ+1.
    A^-,i is constructed  from fermions only and its existence depends only on the fact that the relevant connection on
    the tangent bundle of the sigma-model target space M=^3×^1 has holonomy
     (2)⊂(4), so in particular A^-,i would have an analog, also at level 1, in a sigma-model with target a hyper-Kahler four-manifold such
     as ^4 or K3.   By contrast, A^+,i contains the currents that generate bosonic symmetries of ^3×^1 (namely the
    right action of (2)=^3 on itself) and U∼ J^0 generates a rotation of the second factor of ^3×^1.  Those generators, and similarly
    the Q^a, which are multiples of the free fermions ψ^a, have no analog if the target space is . (A ^4 target has  translation symmetries
    that lead to the existence of holomorphic currents and free fermions in the chiral algebra of the sigma-model.   Relative to ^3×^1, the important difference is that
    the  continuous symmetries of ^4 are commutative and do not enhance the small =4 algebra to a large one.)
    
    
    In general, a representation of the large =4 algebra is characterized by two integers k^+ and k^-, which are the levels of the A^+,i and A^-,i current 
    algebras.  The Virasoro central charge is
    c=6 k^+k^-/k^++k^-.
    What we see in eqn. (<ref>) is a realization of the algebra with (k^+,k^-)=(κ+1,1).   By taking the tensor product of several such realizations, possibly
    with different values of κ, one can get realizations of the large =4 algebra with any k^+≥ k^-.   The tensor product procedure is not straightforward
    and is described in section 4.7 of <cit.>. Since the =4 algebra has an outer automorphism that exchanges A^+,i and A^-,i, and therefore
    exchanges k^+ and k^-, there is no essential loss of generality in assuming k^+≥ k^-.   
    For the record, though we will not use these formulas in any detail, the singular 
    OPE's of the large =4 algebra, apart from standard OPE's involving the stress tensor T,
    are as follows, with γ=k^-/k^++k^-:
    
        G^a(z) G^b(0)     = 2cδ^ab/3z^3-8γα^+,i_abA^+,i(0)+8(1-γ)α^-,i_ab A^-,i(0)/z^2      -4γα_ab^+,i∂ A^+,i(0) +4(1-γ)α^-,i_ab∂ A^-.i(0)/z+2δ^ab T(0)/z+⋯
                                                 A^±,i(z)A^±,j(0)    = -k^±δ^ij/2z^2+ϵ^ijk A^±, k(0)/z+⋯   
                                                      Q^a(z) Q^b(0)    =-(k^++k^-)δ^ab/2z+⋯
                                                          U(z) U(0)    = -k^++k^-/z^2+⋯
                                                            A^±,i(z) G^a(0)     = ∓2 k^±α^±, i_ab Q^b(w)/(k^++k^-)z^2+α^±,i_ab G^b(0)/z+⋯
                                                                 A^±, i Q^a(0)    =α^±, i_ab Q^b(0)/z+…
                                                                  Q^a(z) G^b(0)    = 2α^+,i_ab A^+,i(0) -2α^-,i_ab(0)+δ^ab U(0)/z
                                                                    U(z) G^a(0)    = Q^a/z^2+⋯.
        
     
    
     
     
     
     
      In the case of the moduli space  of instantons on ^3×^1, 
     we will explain enough in sections <ref> and <ref> 
    to show that the sigma-model with target  has =4 supersymmetry with the small =4 algebra.  To extend this result to get a large
     =4 algebra, we need to “find” the extra generators U, Q^a,  A^+,i for both positive and negative chirality.  The strategy for finding them  is simple.
     ^3×^1 has many Killing vector fields, associated with the rotation symmetries of the two factors, 
     and associated to these are Killing vector fields on the instanton
     moduli space .   We will show that these Killing vector fields on  are covariantly constant for the 
     appropriate connections with torsion, and this, as in the analysis
     of eqn. (<ref>), leads to the existence of additional chiral operators in the sigma-model with target .   In that way we will find the extra generators that
     are needed to fill out the large =4 algebra.  This analysis will be the topic of section <ref>, but first we need to understand why a sigma-model with
     target  has at least small =4 supersymmetry.
     
     
    
    § HYPERCOMPLEX STRUCTURE OF THE INSTANTON MODULI SPACE
    
     
    In this section, largely following <cit.>, 
    we will show that if the oriented four-manifold M has a hypercomplex structure, with selfdual hermitian forms and closed torsion,
    then the moduli space  of instantons on M also carries a hypercomplex structure (which we will show in section <ref> to have closed torsion).   This is true 
    for every simple compact  gauge group ,  and every component of instanton moduli space.   To be more precise, we define these structures on the smooth part of .
    That means in particular that we always consider instanton solutions that are irreducible in the sense that the equation
    _̣Aσ=0,
    where σ is a section of (E), has no non-zero solutions.   Such solutions, which generate unbroken gauge symmetries (continuous automorphisms of E), arise only at singularities of .
    
    If  M has two hypercomplex structures compatible with the same metric, both with selfdual hermitian forms, and
    with opposite torsion, then  we will see that  likewise has two hypercomplex structures 
    with the same properties.   (The proof that the two hypercomplex structures of   have opposite torsion is given in section <ref>.)  
    In our application, we have M=^3×^1 and =(Q_5),
    but presenting the arguments in greater generality poses no difficulty.[Actually, a compact hypercomplex four-manifold that is not hyper-Kahler is
    locally isomorphic to ^3×^1<cit.>.  But the following analysis can likely be extended to other examples that are complete but not compact.]
    
    To begin with, we assume that the oriented four-manifold M has simply a strong KT  structure, namely a complex structure , a metric g that is of type (1,1)
    and (therefore) a hermitian form ω_IL =g_IK^K_L that is also of type (1,1), and that we assume to be selfdual,[The definition ω_IL =g_IK^K_L implies that ω is  selfdual or anti-selfdual, depending on how M is oriented.   Assuming that the instanton equation is going to be F^+(A)=0, we require ω to be selfdual so that
    the (1,1) part of the instanton equation can be written as in eqn. (<ref>) below.]  and with closed torsion H=ı(∂̅-∂)ω.   
    Given this, we will define a similar hermitian structure on  (the closedness of the torsion will be shown in section <ref>).
    
    First of all, the space  of all connections on the -bundle E→ M itself has a natural metric, with the length squared of a variation δ A of a connection A being
    |δ A|^2=-1/4π^2∫_M δ A∧⋆δ A=-1/4π^2∫_M^̣4 x √(g)   g^ij δ A_iδ A_j. 
    The factor 1/4π^2 is arbitrary for now; when we compute the torsion on the moduli space, it will be convenient.
    The space  also has a natural complex structure , defined by saying that the (0,1) part of  is holomorphic and the (1,0) part is antiholomorphic.  
    Thus if δ A is a variation of A, and  δ A=δ A^1,0+δ A^0,1 is its expansion  in parts of type 
    (1,0) and (0,1), then 
    (δ A)=ıδ A^(0,1)-ıδ A^(1,0).
    
     The
    reason to define the (0,1)– rather than (1,0)– part of A to be holomorphic is as follows.   First of all, if we decompose the Yang-Mills curvature F(A)
    in pieces of type (p,q) with respect to the complex structure  of M, then the instanton equation reads
    F^(0,2)=F^(2,0) = ω∧ F^(1,1)=0.
    In particular, vanishing of F^(0,2) means that 
     ∂̅_A =∂̅+[A^(0,1), · ]
    satisfies ∂̅_A^2=0. This makes  the bundle E–  or more precisely its complexification E_– a holomorphic  bundle over the complex manifold M.   We would
    like to define the complex structure of  in a way that ensures that, after eventually reducing to the instanton moduli space ,  the holomorphic  bundle
    E_ varies holomorphically over . For this we have to define the complex structure of  so that ∂̅_A=∂̅+A^(0,1) varies holomorphically with A. This is true with the sign choice that we made.
    In fact, a theorem of <cit.>  identifies  as the moduli space of stable holomorphic structures on E (where here the notion
    of stability is a generalization of the usual stability condition for a holomorphic  bundle over  a Kahler manifold).  We will not need this  difficult theorem. For our purposes
    it will suffice to know how to define the complex structure of , which is  much easier. 
    
    
    Having a metric and a complex structure,  acquires  a hermitian form:[Here we include a factor of 1/2 that was introduced in eqn. (<ref>).]ω=-1/8π^2∫_M ω∧δ A∧δ A.
    Viewing δ A as a 1-form on  (valued in 1-forms on M), we see that ω is a two-form on , and it is of type (1,1) because ω is of type (1,1).
    
    ASD
    Within the infinite-dimensional space , there is an infinite-dimensional submanifold ^ that parametrizes gauge fields that solve the instanton equation.
    The instanton moduli space is =^/, where  is the group of gauge equivalences (if E is trivialized, then  is the group of
    maps from M to ).  
    There is no problem to restrict the metric |δ A|^2 or  the two-form ω from  to ^.  However, there is no immediate
    way to extract from these objects a corresponding metric or two-form on . The difficulty is as follows.   Let A be an instanton solution describing a point in , and let ζ be a tangent
    vector to  at that point.    Up to gauge transformation, ζ corresponds to a solution of the linear equation obtained by linearizing the instanton equation F^+(A)=0.
    This linear equation is
    (_̣Aδ A)^+   =0,
    where _̣A=+̣[A, · ] is the gauge-covariant extension of the exterior derivative.
    This equation possesses a gauge invariance
    δ A→δ A-_̣Aσ,
    where σ, a section of the adjoint bundle (E),  is an arbitrary generator of a gauge transformation.
    The metric and hermitian forms defined in (<ref>) and eqn. (<ref>) are not invariant under such a change[ To be more precise, the metric
    is not invariant under δ A→δ A-_̣Aσ.   The hermitian form ω is invariant if and only if ω̣=0, that is, if and only if M is Kahler.
    To prove this, substitute δ A→δ A-_̣Aσ in eqn. (<ref>), integrate by parts, and use F^+=(_̣Aδ A)^+=0, giving a result proportional to ω̣.]
     of δ A.   Therefore, if we want to use
    eqns. (<ref>) and (<ref>) to define a metric and a hermitian form on the tangent space T_A to  at A, we need to impose some sort of gauge condition on δ A.
    We will only consider gauge conditions that are invariant under gauge transformations of the background gauge field A (but not under the transformation (<ref>) of δ A).
    For example, though it is not the gauge condition that we will ultimately use,
    an obvious gauge condition that is invariant under gauge transformations of the background 
    is the Landau gauge condition D_i δ A^i=0, where D_i=∂_i+[A_i,· ].
    
    Once a gauge condition on δ A is picked, the tangent space T_A to A in 
     becomes a subspace of the tangent space  T_A to A in .   
    If and only if the conditions we impose on δ A– the equation (<ref>) plus the gauge condition that we have not yet chosen – are invariant under δ A→δ A,
    it will make sense to restrict the complex structure   defined in (<ref>) from T_A to T_A.   This will define an almost complex structure on , which we will eventually show to 
    be integrable. 
    
    With this aim, let us discuss the  Hodge decomposition
    of the instanton equation F^+(A)=0, and its linearization (_̣Aδ A)^+=0.  
    The equation F^+(A)=0 has parts of type (2,0) and (0,2), which are complex conjugates of each other, and a part of type (1,1).
    The (0,2) part of the instanton equation is explicitly
    ∂̅A^(0,1)+A^(0,1)∧ A^(0,1)=0 
    and the (2,0) part is just the complex conjugate of this. 
    Similarly the (0,2) part of the linearized equation is
    ∂̅δ A^(0,1)+[A^(0,1),δ A^(0,1)]=0. 
    Since the complex structure on  is defined by saying that A^(0,1) is holomorphic, the eqn. (<ref>) is holomorphic.   Similarly, eqn. (<ref>) is linear in 
    δ A^(0,1) with no dependence on δ A^(1,0), so under δ A→δ A, it is just multiplied by ı=√(-1).  Hence the equation (_̣Aδ A)^(0,2)=0 is invariant
    under .   Its complex conjugate, namely (_̣Aδ A)^(2,0)=0, is of course also invariant.
    
    However, the (1,1) part of the instanton equation is not holomorphic or antiholomorphic and its linearization is not invariant.   That (1,1) part is
    ω∧ F =0. 
    As ω is of type (1,1), it is equivalent to write ω∧ F^(1,1)=0, and because ω is selfdual, it is also equivalent to write ω∧ F^+(A)=0.
     It is to ensure that the (1,1) part of
    the instanton equation can be written as in (<ref>) that we require ω to be selfdual.
    Eqn. (<ref>) is neither holomorphic nor antiholomorphic, and similarly its linearization, namely
    ω∧_̣Aδ A=0,
    depends on both δ A^(1,0) and δ A^(0,1) and is not invariant under δ A→δ A.
    
    The only hope of getting a set of conditions on δ A that is invariant under  is to interpret whatever eqn. (<ref>) transforms into under δ A→δ A as
    the gauge condition that we are going to impose.   In other words (as in section 5.3 of <cit.>), the gauge condition will have to be
    ω∧_̣A(δ A)=0.
    If this does make sense as a gauge condition, then this gauge condition plus eqn. (<ref>) is a set of equations that is invariant under δ A→δ A, since
    if we substitute δ A→δ A in eqn. (<ref>), we get back eqn. (<ref>).   Imposing on δ A the linearized instanton equation together with eqn.
    (<ref>), we can then restrict  from  to  to provide an almost complex structure on , 
    and eqns. (<ref>) and (<ref>) define a metric and hermitian form on ,  both of type (1,1).   
    
    In general, to show that a proposed gauge condition is valid, we have to demonstrate two facts: (1) it should be possible to impose this gauge condition,
    in the sense that by a suitable gauge transformation, it can always be satisfied; (2) it fully fixes the gauge, in the sense that the gauge transformation  that
    ensures that the condition is satisfied is unique.   We have to decide whether the gauge condition (<ref>) satisfies those two conditions.
    
    As a preliminary, we will put the candidate gauge condition (<ref>) in a more convenient form.
     More explicitly, the left hand side of eqn. (<ref>) is
     1/2ϵ^ijklω_ij D_k(^m_l δ A_m)
     =1/2ϵ^ijklω_ij^m_l D_kδ  A_m +1/2ϵ^ijklω_ij(D_k ^m_l)
     δ A_m. 
     Being selfdual, ω satisfies 1/2ϵ^ijklω_ij=ω^kl, so the first term on the right in eqn. (<ref>) is ω^kl^m_l  D_k δ A_m
     = G^kmD_kδ A_m=D_kδ A^k.  Using ω_ln=G_lp^p_n, the last term on the right can be written  1/2ϵ^ijklω_ij (D_k ω_ln)
     δ A^n.   To express this in terms of the torsion, we recall that ω is covariantly constant for the connection ∇ with torsion, ∇_kω_ln=0.
     Using this and the definition (<ref>) of ∇,  we get 
     1/2ϵ^ijklω_ij (D_k ω_ln)
     δ A^n=1/4ϵ^ijklω_ij(H_kl^p ω_pn-H_kn^pω_pl)δ A^n. 
     The right hand side  as written is homogeneous and quadratic in ω, but rather surprisingly it actually turns out to be entirely independent of ω.
     We will see that this  surprising fact is a necessary input in defining a hypercomplex structure on .  To show that the right hand side of eqn.  (<ref>) is
     really independent of ω is a matter of doing some group theory in the tangent space to an arbitrary point p∈ M at which we want to prove the statement.
     The expression in question is a bilinear function of  a one-form δ A and a three-form H.  This bilinear function  depends on the metric tensor g of M at p (used to raise
     and lower indices in eqn. (<ref>)) and on ω, but nothing else (the antisymmetric tensor ϵ^ijkl is determined by the metric and  orientation of M;
     the orientation is determined by ω, which is assumed to be selfdual).   So this expression is invariant under linear transformations of the tangent space to M
     at p that preserve g and ω.   The group of such linear transformations is (2).   However, the expression that we are trying to analyze is an even function
     of ω, invariant under ω→ -ω.  So this expression  is also invariant under  linear transformations of the tangent space that map ω to -ω.
    Including such linear transformations extends (2) to a double cover 
    that we will call ^*(2) (this group is the group of 2× 2 unitary complex matrices extended by the operation of complex conjugation,
     which reverses the sign of ω).   The one-form δ A transforms in a four-dimensional representation of ^*(2) that is irreducible even over the complex
     numbers (this statement would not be true if we consider (2) instead of  ^*(2)).   Because the Hodge duality map ⋆ from three-forms to one-forms commutes with ^*(2), the three-form H transforms
     in the same irreducible four-dimensional representation.   Therefore, the right hand side of eqn. (<ref>) is an invariant bilinear form on an irreducible representation
     of ^*(2).    Such an invariant bilinear form is unique up to a constant multiple, so the expression in question must be a multiple of any conveniently chosen ^*(2) invariant that is
     bilinear in H and δ A.   A convenient invariant is ⋆( H∧δ A)=1/6ϵ^ijklH_ijkδ A_l.   So the right hand side of eqn. (<ref>)
     must be a multiple of this.   By checking an example,[In local coordinates x^1,⋯, x^4, one can take the metric of M at the point p to be δ_ij, the hermitian form to
     have non-zero coefficients ω_12=-ω_21=ω_34=-ω_43=1, and H and δ A to have nonzero elements H_123=δ A_4=1.]
      one can verify that the coefficient is 1.  Putting these facts together, the candidate gauge condition is[This assertion is Lemma 8 in <cit.>, where a rather
      different proof is given.]D_mδ A^m+⋆( H∧δ A)=0. 
      
      It is now relatively simple to determine whether this is a satisfactory gauge condition.    Under δ A→δ A-_̣Aσ, the gauge condition
      transforms by
       D_mδ A^m+⋆ (H∧δ A)→ D_mδ A^m+(⋆ H∧δ A) -D_i D^iσ -⋆( H∧_̣Aσ).
      Assuming that the right hand side of eqn. (<ref>) vanishes  for some σ, the condition that this choice of σ is unique is that there is
      no non-zero solution of the equation 
      -D_i D^iσ -⋆ (H∧_̣Aσ)=0.    
      This equation can be written
       Wσ=0,
      where the linear operator W was introduced in eqn. (<ref>).
      Given any solution of this equation, we can multiply by σ, take a trace,
      and integrate to get
      0=-∫_M^̣4x√(g)  σ(-D_i D^iσ) -∫_M  σ H∧_̣Aσ.
      Here the second term is 
       -∫_M H∧ σ_̣Aσ =-1/2∫_M H∧ σ^2=0,  where in the last step we integrate by parts and use Ḥ=0.
      Integrating by parts in the first term we then learn that
        0=-∫_M^̣4x√(g)  D_iσ D_jσ g^ij,
      which implies that D_iσ=0.  As remarked earlier (see eqn. (<ref>)), at a smooth point of , this implies that σ=0.  That establishes the desired uniqueness.
      
      It remains to verify that the right hand side of eqn. (<ref>) always does vanish for some σ, which is true if the equation 
        -D_i D^iσ -⋆( H∧_̣Aσ)= w 
       has a solution for every section w of (E).   If this is not the case, then  sections of (E) 
       of the form  -D_i D^iσ -⋆( H∧_̣Aσ) generate a proper 
       subspace  of the Hilbert space $̋ of allL^2sections of(E), and there is some sectionwthat is orthogonal to this subspace.
       The orthogonality condition is
       0=-∫_M^̣4x √(g)  w (-D_i D^iσ -⋆( H∧_̣Aσ)),
       and the condition that this is true for allσis
       -D_i D^i w +⋆ (H∧_̣A w)=0.
       This is the same equation (<ref>) that we have already analyzed, but withH→ -H.  The same argument as before shows that there are no nonzero solutions.
          Actually, the operator that appears in eqn. (<ref>) is simply the adjointW^†of the operatorWintroduced earlier, so to prove that the gauge condition is satisfactory,
          what we have had to prove is thatWandW^†both have trivial kernel.  SinceWandW^†are related byH↔ -H, this pair of statements is invariant under reversing the sign ofH.  Instead of saying thatWandW^†have trivial kernel,
          an equivalent statement is that either one of them has trivial kernel and cokernel.
       
       
    At this point, having verified that the gauge condition is a good one, we have defined an almost complex structureon.  It remains to verify
       that this structure is integrable.  A simple argument is available.   First of all, the equationF^0,2(A)=0is  holomorphic inA, so if we impose
       only this equation, we get an (infinite-dimensional) complex submanifoldof.   However, the conditionsω∧_̣Aδ A=ω∧_̣A(δ A)=0appear to spoil holomorphy.  To prove thatis integrable, we will show that imposing these nonholomorphic equations is equivalent to a certain holomorphic operation.
       
       The idea is to exploit the fact that, given the complex structure ofM, the action onof the groupof gauge transformations can be analytically continued
       to a holomorphic action of the complexification_of this group.  A generatorσof_is a section of(E)⊗_, the complexification of(E).
       More explicitly, such a generator isσ=σ_1+ıσ_2whereσ_1,σ_2are sections of(E).  We define an action ofσonAbyA→ A-_̣Aσ_1-_̣Aσ_2.   In particular, the holomorphic variableA^(0,1)tranforms byA^(0,1)→ A^(0,1)-∂̅_A σ_1-∂̅_Aσ_2
       =A^(0,1)-∂̅_A σ, where we use the fact thatacts asıon the(0,1)-form∂̅_A σ_2.   SinceA^(0,1) -∂̅_Aσis
       holomorphic inAandσ, this does define a holomorphic action of_on.   
       
       The claim now is that the conditionsω∧_̣Aδ A=ω∧_̣A(δ A)=0can be viewed as gauge-fixing conditions that fix the action of_on, so thatcan be interpreted as/_, making obvious the complex structure ofand thus the integrability of.   We already know
       that the conditionω∧_̣A(δ A)=0can be viewed as a gauge-fixing condition for the ordinary gauge symmetryA→ A-_̣Aσ_1; it remains to
       verify that the other conditionω∧_̣Aδ A=0can be viewed as a gauge-fixing condition for the imaginary gauge symmetryA→ A-_̣Aσ_2.  
       Following the same logic as before, we have to show that the operatorσ→ -ω_̣A(_̣Aσ)has trivial kernel and cokernel.  But this is actually up to sign
       the same operator that we have already studied so indeed the kernel and cokernel are trivial.
       
       At this point, then, we know thatis a complex manifold with metric and hermitian forms given by equations (<ref>) and (<ref>).   Now as in <cit.>,  let us assume
       thatMhas not just the single complex structurethat we have assumed so far, but a hypercomplex structure, with three complex structures,,that satisfy
       the quaternion relations and the conditions described in section <ref>.  In particular, this means that the metricgofMis of type(1,1)for each of,, and,
       and that,,andas well asgare all covariantly constant for the same connection∇whose torsion is a closed three-formH.  The three hermitian
       formsω_=g,ω_=g, andω_= g are, of course, different.    However, because of the crucial fact that the gauge condition (<ref>) depends only
       onHbut not onω, the construction we have described, whether carried out for, for, or for, leads to the same metric onand the same
       description of the tangent spaceT_pfor any pointp∈in terms of the common gauge condition (<ref>).
         Therefore,  for anyq∈, restricting,, andfromTtoT_q, we get almost 
       complex structures,,onT_q. They are all integrable by the argument that was just given, and  they obey the quaternion relations because those were obeyed before restricting toT_q.  Eqn. (<ref>)
       gives a metric that is of type(1,1)for each of,,, and eqn. (<ref>), withωreplaced byω_,ω_, orω_,
       gives three hermitian formsω_,ω_,ω_, each of which is of type(1,1)for the corresponding complex structure.   In section <ref>,
       we will show that each of,,is associated to the same torsion, since-ı(∂_-∂̅_)ω_=-ı(∂_-∂̅_)ω_
       =-ı(∂_-∂̅_)ω_.
       
     Now we specialize to  the case that[It follows from the classification of compact hypercomplex four-manifolds <cit.> that ^3×^1 is the unique compact four-manifold with 
     the properties specified in this sentence.]M=^3×^1, which admits a second hypercomplex structure',','such that the same metricgis
     of type(1,1)for each of',',', and the three hermitian formsω'_I,ω'_J, andω'_Kare all selfdual, but the torsion is-H, as opposed to the
     previously assumed+H.  
     Of course, we can carry out the same construction as before, endowingwith a new hypercomplex structure',',', and a compatible metric and hermitian 
     forms.   The only problem is that since the torsion is now-H,  we have to reverse the sign ofHin the gauge condition, which is now
       D_Mδ A^M-⋆( H∧δ A)=0. 
    Therefore, it seems that the hyperhermitian metric  that we will define onwill be different.  If so, then  no one metric
    onwill be consistent with all of the structures predicted by the duality conjecture with strings on_3×^3×^3×^1. One metric onwill
    lead to a sigma-model with(0,4)supersymmetry and another metric onwill lead to a sigma-model with(4,0)supersymmetry.   This would disprove
    the duality conjecture.
    
    What saves the day is another striking fact (Lemma 9 in <cit.>, where the following proof can be found): reversing the sign ofHin the gauge condition actually does
    not   change the resulting metric on.    Consider an instanton connectionArepresenting a pointq∈ M, and consider a tangent vector toqinthat
    can be represented  by a deformationδ AofAthat satisfies 
      D_mδ A^m+⋆( H∧δ A)=0, 
    and also by another deformationδ A'that satisfies a similar condition withHreplaced by-H:
      D_mδ A'^m-⋆( H∧δ A')=0, 
    The statement thatδ Aandδ A'represent the same tangent vector toqinmeans that they are gauge equivalent,
    δ A=δ A'-_̣Aσ
    for some generatorσof a gauge transformation.  
    Based on this information, we want to show that
    |δ A|^2=|δ A'|^2,
    where these expressions are defined via eqn. (<ref>).    This will show that the length of a tangent vector at any pointq∈is the same
    regardless of which of the two gauge conditions is used to compute it, or in other words that the two gauge conditions lead to the same metric on.
    The difference between the left and right of eqn. (<ref>), in view of the relation (<ref>) betweenδ Aandδ A', is
    -1/4π^2∫^̣4 x√(g) (D_mσ D^mσ +2 D_mσδ A^m),
    and we will show that this vanishes.
    Using eqn. (<ref>) to solve forδ A'in eqn. (<ref>),
    we get
    D_mδ A^m+ D_m D^m σ-⋆(H∧δ A) - ⋆(H∧_̣Aσ)=0.  
    Adding  eqn. (<ref>) to this, we find 
     2 D_mδ A^m +D_m D^mσ -⋆(H∧_̣Aσ)=0.
    Multiplying byσ, taking a trace, and integrating, we have
     0=∫_M^̣4x √(g) (2 σ D_mδ A^m  +σ D_m D^mσ) +∫_M H∧ σ_̣A σ. 
    The last term on the right hand side vanishes, as we have already seen in eqn. (<ref>).   Dropping this term and integrating by parts in the 
    remaining terms,  eqn. (<ref>) becomes equivalent to the desired result
    (<ref>),  completing the proof.
    
    The theory of generalized Kahler reduction gives a possibly more conceptual explanation of the success of this calculation <cit.>, at least in the untwisted case (the case thatHis exact). Similarly,
    a more conceptual explanation of why the relevant gauge condition turned out to depend only onHand notωmight conceivably be found in the theory of
    generalized hyper-Kahler reduction.  
    
     
    
    § B-FIELD ON THE INSTANTON MODULI SPACE
    
    
     For a complex structureonMsatisfying certain conditions, 
     we defined in section <ref>
     a corresponding complex structureon, along with a metric and hermitian form defined in eqns. (<ref>) and (<ref>).   In particular,
     the hermitian form is
     ω_=-1/8π^2∫_M  ω_∧δ A∧δ A,
     whereδ Asatisfies the gauge conditionWδ A=0, as in eqn.  (<ref>).   
    
    In the present section, 
     we will compute the corresponding torsion H=-ω̣_.    A formula for Hwas computed in <cit.>, section 5.3; another useful reference is <cit.>.
     The important properties of Hfor the strong HKT  geometry ofwere pointed out in <cit.>.
     
     It turns out that Hhas  several key properties: (i) it depends only on the metricgand torsionHofMand not onorω_; (ii) it satisfiesH=0, assumingḤ=0;  (iii) it is homogeneous and linear inH, and therefore changes sign underH→ -H. 
     All three conditions are needed to ensure that forM=^3×^1,has the properties that are predicted by the conjectured duality with Type IIB superstrings.
     Condition (i) means that the same torsion form His compatible with the three complex structures,,of, defined in section <ref>,  that generate the hypercomplex
     structure  of.  Hence one can define ona metric compatible connection∇, with torsion H,  for which, , andare all covariantly constant.
     We say that∇corresponds to the connection∇onM.   This givesan HKT structure, which according to condition (ii) is strong if the HKT structure ofMis strong.
     To extend this structure to a (twisted) generalized hyper-Kahler structure on, we need to define ona second HKT structure with equal and opposite torsion.
     The ability to do this is what we get from condition (iii), which enables us to define a metric compatible connection∇'on, with torsion- H, such
     that',', and'are all covariantly constant.
     
     Furthermore, we will show that with the normalization that was chosen in  eqn. (<ref>), the periods of Hare valued in2π, so His the curvature of aB-field Bon. Bis uniquely determined up to the possibility of adding to it a flatB-field.   Shifting Bby a flatB-field that is not pure gauge
      is a modulus of the sigma-model with target; it amounts to shifting the theta-angle that was introduced in section <ref>.
    
    
    
     
    Before getting into too many details, let us note that
     it may seem surprising thatω̣_≠0.   Viewed as a differential form on the infinite-dimensional manifold,ω_is a differential
    form with constant coefficients on a linear space and it certainly satisfiesω̣_=0.   We can in a completely natural way restrictω_fromtoA^, the subspace consisting of all connections that satisfy the instanton equationF^+(A)=0, and it remains closed.   However, there is no natural way to pushω_down
    to a two-form on the moduli space=^/.   The key point is that, assuming[As explained in footnote <ref>, if ω̣_=0, meaning that M is
    Kahler, then ω_ is a pullback from  after all.   This can be used to prove that if M is Kahler, then ω̣_=0 and  is Kahler.  The
    fact that ω̣_=0 if ω̣_=0 will be clear from the formula obtained below.]Mis not Kahler,ω_does not vanish if we substituteδ A=-_̣Aσ.
    This substitution amounts to contractingω_with a vector field tangent to the fibers of^→, and the fact that it does not vanish means thatω_is not the pullback of a form on.   Because of this, in section <ref>, to defineω_as a form on, we had to stipulate that the
    formula (<ref>) should be evaluated only for variationsδ Athat satisfy the gauge condition.
    In what follows, we will work  through  in more detail 
     what is involved in interpretingω_as a form onwith the help of the gauge condition; this will hopefully make it clearer whyω̣_≠0and 
    how to computeω̣_.   
    
    The following considerations are local alongso we can work in a small
     neighborhood of a general  pointr∈.  We pick local coordinatesm^α,α=1,⋯, on, withm^α(r)=0.    Let us first formulate
     a condition that, if it could be satisfied, would suffice for proving thatω̣_=0.  Then we will describe what happens instead.
     For any choice of them's,  there is a gauge fieldA(x;m)that, in its dependence onx∈ M, satisfies the instanton equation.  Up to gauge transformation,A(x;m)is the solution of the instanton equation determined bym.   Of course,A(x;m)is only uniquely determined up to a gauge transformation,
     and this gauge transformation can depend onm.   Suppose that we could make a gauge choice such that, at any point inand for any choice
     ofβ, the quantity∂ A(x;m)/∂ m^βobeys the gauge condition that is used in defining the metric of:
     W ∂ A(x;m)/∂ m^β=0. 
     If so, then in the definition (<ref>) ofω_, we could interpretδ Aas an explicit one-form on(valued in one-forms onM), namelyδ A=∑_βṃ^β∂ A(x;m)/∂ m^β.   Substituting this in eqn. (<ref>), we would get an explicit formula forω_as
     a two-form on, namely
     ω_?=-1/8π^2∑_α,βṃ^α ṃ^β∫_Mω_∧  ∂ A(x;m)/∂ m^α∧∂ A(x;m)/∂ m^β.   
     Then using=̣∑_γṃ^γ∂_m^γ, we would get
      ω̣_?=-1/4π^2∑_α,βṃ^α ṃ^βṃ^γ∫_Mω_∧ ∂^2 A(x;m)/∂ m^α∂ m^γ∧∂ A(x;m)/∂ m^β=0, 
      where the vanishing holds because∂^2 A(x;m)/∂ m^α∂ m^γis symmetric inαandγ.  
      
      However, eqn. (<ref>) is unrealistic.  It represents gauge conditions, one for each choice ofβ, while in actuality at each point inM×we are only entitled to impose one gauge condition.   An example of a gauge condition that is not too restrictive is
      ∑_i[∂/∂ x^i+A_i(x;0), A^i(x;m)]+⋆(H∧ (A(x;m) -A(x;0)))=0  , 
      whereA(x;0)is simplyA(x;m)evaluated atm=0.   The condition (<ref>) is trivial atm=0, so it places
      no gauge condition onA(x;0).   To first order inm, eqn. (<ref>) says that.∂ A(x;m)/∂ m^α|_m=0obeys the gauge condition (<ref>) that was used in  
    definingω_as a differential form on.   Therefore, assuming thatA(x;m)satisfies eqn. (<ref>), eqn. (<ref>) is actually
    correct atm=0(but only atm=0, as we will see).    The analysis in section <ref> implies that the gauge condition (<ref>) is a valid one to first order inm, and this argument
    can be slightly extended to show that this gauge condition is a good one to all orders inm(we will only need the result in second order).  
    
    Since it is not possible to satisfy eqn. (<ref>) beyond first order inm, 
    for the purpose of evaluatingω_, the vector field∂/∂ m^βcannot be taken to act onAin the obvious wayδ A = ∂/∂ m^βA.   We have to accompany this with a gauge transformation chosen to ensure thatδ Aobeys the desired gauge condition. 
    Thus we define
    D A/D m^β=∂ A/∂ m^β-_̣Aε_β, 
    whereε_βis the infinitesimal generator of a gauge transformation chosen to satisfy the gauge conditionW D A/D m^β=0, that is
    [∂/∂ x^i+A_i(x;m),∂ A^i/∂ m^β-D^iε_β] +⋆(H∧(∂ A/∂ m^β-_̣Aε_β))=0.D A/D mis defined intrinsically as the variation ofAunder a change inm, accompanied if necessary by a gauge transformation to ensure that the gauge condition is satisfied.
               
    We can put eqn. (<ref>)  in a more convenient form by subtracting the  condition (<ref>) that we  assume is satisfied for allm.  The result can be conveniently written:
    [A_i(x;m)-A_i(x;0),∂ A^i/∂ m^β] -W ε_β=0.  
    Nearm=0, we have
     A(x;m)=A(x;0) + ∑_α m^α.∂ A(x;m)/∂ m^α|_m=0+Ø(m^2). 
    Eqn. (<ref>) then tells us that
     Wε_β = ∑_α m^α[∂ A_i/∂ m^α,∂ A^i/∂ m^β]_m=0+Ø(m^2). 
    We can solve this forϵ_β:
    ϵ_β(x)=-∑_α m^α∫_M_y_y   B(x,y) [∂ A_i(y;m)/∂ m^α,∂ A^i(y;m)/∂ m^β]^̣4y√(g_y)+
    Ø(m^2), 
    whereB(x,y), introduced in eqn. (<ref>), is the Green's function of the operatorW,M_yis a oopy ofMparametrized byy,
    and_yrepresents a trace in theyvariable.  Here and in subsequent formulas,∂ A/∂ mis evaluated atm=0, and we save space by writing justA_i(y)rather thanA_i(y;m)and similarly for other quantities.
    
    
    It is now straightforward to computeω̣_.  In eqn. (<ref>), we just have to replace∂ A/∂ m, which in general does not satisfy the
    gauge condition,  withD A/D m, which does.  Using eqns. (<ref>)
    and (<ref>), we can make this quite explicit up toØ(m^2):
    
        ω   _=  -1/8π^2∑_α,βṃ^α ṃ^β(∫_Mω_∧  ∂ A(x)/∂ m^α∧∂ A(x)/∂ m^β.
           . -2∑_γ m^γ∫_M_x× M_yω_(x) ∧ ⊗ ∂ A(x)/∂ m^α∧_̣A  B(x,y)  [∂ A^i(y)/∂ m^γ,
                      ∂ A_i(y)/∂ m^β] ^̣4y√(g_y))        +Ø(m^2).
     
    Now we can compute.ω̣_|_m=0,just by acting with=̣∑_γṃ^γ∂_m^γand then settingm=0:
    
        ω_.|_m=0=-1/4π^2∑_α,β,γṃ^αṃ^βṃ^γ×   ∫_M_x× M_yω_(x) ∧ ⊗ ∂ A(x)/∂ m^α∧_̣A  B(x,y) [∂ A^i(y)/∂ m^γ,
                      ∂ A_i(y)/∂ m^β] ^̣4y√(g_y)     .
    
     This can be simplified significantly.   First, integrate by parts in thexvariable, using the linearized instanton equationω_∧_̣A∂ A(x;m)/∂ m^α  =0.  This gives  
       
        -.5cm   ω̣_.|_m=0=∑_α,β,γṃ^αṃ^βṃ^γ×   1/4π^2∫_M_x× M_yω̣_(x) ∧ ⊗ ∂ A(x)/∂ m^α  B(x,y) [∂ A^i(y)/∂ m^γ,
                      ∂ A_i(y)/∂ m^β]  ^̣4y√(g_y)    .
     
               Atm=0,D A/D m^αcoincides with∂ A/∂ mα, so eqn.  (<ref>) remains valid atm=0if we replace∂ A/∂ mα,
               everywhere withD A/D m^α.   But once we make that replacement, the formula is valid for allm, not just asm=0.  To prove the formula at some given
               value ofm, we just do the same computation as before, expanding about that value ofmrather than aboutm=0.  
            Finally, the formula can be written more economically  in terms of 
                 ψ=∑_αṃ^αD A/Dm^α.
     The final version of the formula is 
        ω̣_ =1/4π^2∫_M_x× M_yω̣_∧⊗ ψ(x)  B(x,y) [ψ_i(y),ψ^i(y)]^̣4 y√(g_y) . 
    
    
    Now it is straightforward to determine the torsion H=-ω̣_, where the action of an almost complex structure on differential forms was defined in eqn. (<ref>). 
    To computeω̣_, we  just have to substituteψ→ψfor each of the three occurrences ofψin eqn. (<ref>).   Asacts by ψ^(0,1)→ıψ^(0,1),  ψ^(1,0)→ -ıψ^(1,0),
     and[ψ_i,ψ^i]is of type(1,1), we have[ψ_i,ψ^i]=[ψ_i,ψ^i], 
    so we only need to consider the action ofonω̣_∧ψ.    As ω̣_∧ψ=(ω̣_)^(2,1)∧ψ^(0,1)
    +(ω̣_^(1,2))∧ψ^(1,0), we have
     ω̣∧ψ= ı(ω̣_)^(2,1)∧ψ^(0,1)
    -ı(ω̣_^(1,2))∧ψ^(1,0)=(ω̣)∧ψ=-H∧ψ.
    So
     H=-ω̣_=1/4π^2∫_M× MH∧⊗ ψ(x) B(x,y) [ψ_i(y),ψ^i(y)]^̣4y √(g_y). 
                
      
     Now we can verify the three key points that were mentioned at the outset of this section.  Point (i)  is immediate: eqn. (<ref>) makes it evident
     that Hdepends only onHand not onω_.   Point (ii) is the assertion that ifḤ=0, thenH=0.   Indeed, in the language of section <ref>,
     and bearing in mind the descent formulas (<ref>) and the formula (<ref>) forσ,
     the formula (<ref>) can be written
      H =∫_M H∧ P^(1). 
     So following the logic of section <ref>, 
     we have
     H=∫_M H ∧δ P^(1) =-∫_M H∧P̣^(0)=0,
     where the vanishing in the last step follows by  integrating by parts and usingḤ=0.   Similarly,  
     the cohomology class of Honly depends on the
     cohomology class ofH, since if we changeHbyH→ H+Ḅ, then the change in His
     Δ H =∫_M Ḅ∧ P^(1)=-∫_M B∧P̣^(1)=δ∫_M B∧ P^(2),
     and thus is exact.   Finally, concerning point (iii),  the formula (<ref>) shows that reversing the sign ofHreverses the sign of H.
     
     Properties (i) and (ii) show that ifMhas a (strong) HKT structure than so does.   The addition of property (iii) tells us that ifMhas a (strong) bi-HKT structure,
     or equivalently a (twisted) generalized hyper-Kahler structure, with all of the hermitian forms being selfdual, then so does.   The relation ofP^(1)to the second Chern
     class implies that periods of Hare integer multiples of periods ofH, and therefore that Hobeys Dirac quantization ifHdoes.
     
     Of course, we are mainly interested in the case thatM=^3×^1, withHthe pullback from^3of a rotationally invariant form with total flux2π Q_5'.
     In other words,H=2π Q_5'^̣3Ω, where^̣3Ωis a rotationally invariant volume form on^3of total volume 1.
     In this case
     
        H   =
         Q_5'/2π∫_M× M^̣3Ω_x ∧⊗ δ A(x) B(x,y) [δ A_i(y),δ A^i(y)]^̣4y √(g_y)   =Q_5'/2π∫_M× M^̣3Ω_x ∧⊗ ψ(x) B(x,y) [ψ_i(y),ψ^i(y)]^̣4y√(g_y),
    
     in agreement with eqn. (<ref>).
      
    Finally, we can describe the usual mathematical setting for these formulas and in the process explain claims that were made at the end of section <ref>.
    By definition of the instanton moduli space, each pointm∈parametrizes a-bundleE_m→ Mthat carries an instanton connectionA.
    Asmvaries in, theE_mvary as fibers of a-bundle E→ M×(sometimes called the universal instanton bundle). Ehas
    a natural connection in theMdirection, namely the instanton connection itself, but it does not have a natural connection in thedirection.  To define a connection
    on Ein thedirection, one has to make some sort of a choice.   We made such a choice by placing the gauge conditionWδ A=0on variations ofA.
    With the aid of this choice, we defined in eqns. (<ref>) and (<ref>) a connectionD/D m^α=∂/∂ m^α+[ε_α,·]on the bundle Ein thedirection.   
    Taken together,  the original instanton connectionD/Dx^i=∂/∂ x^i+[A_i,· ]in theMdirection and the connectionD/D m^α=∂/∂ m^α+[ϵ_α,·]in thedirection give a full-fledged connection Aon E→ M×.   This connection has a curvature F =A+ A∧ A, and a second Chern class, valued inH^4(M×;), that at the level
    of differential forms is represented by the four-form1/8 π^2  F∧ F.   The curvature Fhas components of typesij,α i,andαβ(where indicesi,jare tangent toMand indicesα,βare tangent to).  Theijpart of Fis simply the curvatureF_ij=[D_i,D_j]of
    the original instanton bundle.  Theα i part of Fis[D_α, D_i]= ∂ A_i/∂ m^α -D_i ϵ_α, a quantity that was introduced in eqn.
    (<ref>).   The corresponding part of the curvature two-form Fis∑_α iṃ^αx̣^i[D_α,D_i].   This is the(1,1)-form onM×(a two-form onM×with one index tangent to each factor) that was calledψin section <ref>. Indeed,ψwas characterized as a general variation ofAthat obeys the gauge condition (as well as the linearized instanton equation),
    or in other words as∑_α iṃ^αδ_α A_i  x̣^i, where the variationsδ _α A_iare constrained to be annihilated by the operatorW; these variations are precisely what we now call[D_α, D_i].   Finally, theαβpart of the curvature is[D_α,D_β].   In eqn. (<ref>),
    we determinedϵ_αin an expansion around a base-point atm=0and in a gauge with.ϵ_α|_m=0=0.   In that gauge, 
    theαβpart of the curvature reduces atm=0to F_αβ=∂_αϵ_β-∂_βϵ_α, which is a quantity that we evaluated as a step in the computation ofω̣_.  The result can be stated 
    1/2∑_αβṃ^αṃ^β F_αβ(x)=-∑_αβṃ^αṃ^β∫_M_y_y   B(x,y) [∂ A_i(y)/∂ m^α,∂ A^i(y)/∂ m^β]^̣4y√(g_y).  
    We derived this formula atm=0, but the formula becomes valid for allmif  we replaceṃ^α∂ A_i(y)/∂ m^αwithψ_i=ṃ^α F_α i(which is a gauge-covariant expression that reduces toṃ^α∂ A_i(y)/∂ m^αatm=0). 
    After this replacement, eqn. (<ref>) agrees (up to sign) with the formula forσas a two-form onthat was deduced in eqn. (<ref>), confirming the interpretation ofσas part of the curvature of a natural connection on the bundle E→ M×.   This also confirms the interpretation that was claimed in section <ref> of
    the observablesP^(n)defined via the descent procedure: they represent various components of1/8 π^2  F∧ F.  
      
     
    
    § LARGE =4 ALGEBRA 
    
    
     §.§ Overview
    
     
     By now, we know that ifMis a generalized hyper-Kahler four-manifold, then similarly the  moduli spaceof instantons onMhas a generalized hyper-Kahler
     structure, leading[If we assume conformal invariance, which is discussed more critically in section <ref>.] 
      to=4superconformal symmetry with the small=4algebra.   Our next task is to show that,
    in the specific case of the instanton moduli space on^3×^1,  the small=4algebras actually are extended to large ones.
      For this, we need to find new holomorphic and antiholomorphic fields
    in the sigma-model with target.   From the discussion of eqns. (<ref>) and (<ref>), we know where to find them: vector
    fields onthat are covariantly constant for the connection∇or∇'onwill lead to holomorphic or antiholomorphic currents and free fermions
    in the sigma-model, such as  we need.
    
    Let us discuss what it means for a vector field to be covariantly constant for one of these connections.  In general, on a (strong) HKT manifold with connection∇and torsionH,
    the condition for a vector fieldVto satisfy∇_R V_S=0amounts to the following.  The part of the equation that is symmetric inRandSis
     D_R V_S+D_S V_R=0,
    saying thatVis a Killing vector field.
    The antisymmetric part of the equation is
    ∂_R V_S-∂_S V_R=H_RST V^T. 
    This can equivalently be written
    Λ̣=ι_V H
    whereι_Vis the operation of contracting a differential form with the vector fieldV, and the left hand side is the exterior derivative of the one-formΛ=X̣^I G_IJ V^J, which we call the one-form dual toV.
    
    Eqn. (<ref>)  implies in particular that the Killing vector fieldVgenerates a symmetry ofH. Indeed, the change in a differential formHunder an infinitesimal diffeomorphism generated byVis given by the Lie derivativeℒ_Vacting onH.  Concretely the definition ofŁ_V isŁ_V H=ι_V Ḥ +ι̣_V H.  In the present case,Ḥ=0and eqn. (<ref>) implies thatι̣_V H=0, soŁ_V H=0.   (BecauseΛis required to be dual toV,
    this is not the full content of eqn.  (<ref>).) 
    
    So  to extend the small=4algebra of the sigma-model with targetto a large one, we need in particular 
    Killing vector fields onthat are symmetries of the torsion Hof.   This should come as no surprise, since extending the small=4algebra to a large one involves finding a holomorphic(2)×(1)current algebra,
    and the obvious way to find one is to find Killing vector fields ongenerating a suitable(2)×(1)symmetry of the sigma-model.^3×^1itself has Killing vector fields generating the group((2)_ℓ×(2)_r)/_2×(1)of isometries connected to the identity,
    where(2)_ℓand(2)_ract on^3=(2)on the left and right, and(1)is the group of rotations of^1.
    Moreover, these symmetries transform the complex structures of^3×^1in a particular way, described in section <ref>.
    The((2)_ℓ×(2)_r)/_2×(1)symmetries of^3×^1are all symmetries of the instanton equation on^3×^1,
    so automatically they are symmetries[The group that acts on  may in general be an extension of
      ((2)_ℓ×(2)_r)/_2×(1) by the center of the gauge group.]  of.   Moreover, because the complex structures  ofare directly inherited from the corresponding complex structures of^3×^1,((2)_ℓ×(2)_r)/_2×(1)transforms
    the complex structures ofthe same way that it transforms the complex structures of^3×^1.
    
    So to show that the sigma-model with targethas large=4symmetry, all we need to show is that the Killing vector fields onthat generate the(2)_ℓ×(1)symmetry are covariantly constant for∇, and those that generate the(2)_r×(1)symmetry are covariantly constant for∇'.    The statements about(2)_ℓ×(1)and the statements about(2)_r×(1)are exchanged by the discrete symmetryρof^3×^1that acts by reflection on each factor, so it suffices to analyze one of the two
    cases.   
    
    
    
    Actually,^3×^1has only one(1)symmetry, which contributes both 
    a holomorphic and an                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         antiholomorphic(1)current algebra. This is possible because the Killing vector fieldVthat generates this(1)is covariantly constant for both∇and∇',
    leading to both holomorphic and antiholomorphic conserved currents.   As we will see, something similar happens on the moduli space.
    
    
    
       In section <ref>, we analyze the Killing 
    vector field Vonthat corresponds to the Killing vector fieldVon^3×^1.  We show that it is covariantly constant for both∇and∇',  implying that the sigma-model with targethas the expected holomorphic and
    antiholomorphic(1)current algebras and free fermions.    It also follows that this sigma-model has a scalar field that is free, at least locally.
    In section <ref>,  we show that the vector fields that generate the(2)_ℓsymmetry ofare covariantly constant for∇, leading to the expected holomorphic(2)current algebra that is needed to complete the holomorphic  large=4algebra.
    The analogous statement for(2)_rsymmetry and∇'is an immediate consequence.
     In section <ref> we compute, at least in the semiclassical limit of largeQ_5',
     the levels or central charges of the(2)current algebras that are related to the(2)_ℓand(2)_rsymmetries.   
     We get the expected   resultsQ_1 Q_5andQ_1 Q_5', as predicted by the duality conjecture  described in the introduction.
     In section <ref>, we show that certain symmetry generators have no fixed point on. This is potentially relevant to supersymmetric localization 
     of the sigma-model with target.   In  section
     <ref>, we show that ifMis such that a sigma-model with targetMhas a large=4algebra only for  one chirality, then
     the same is true of.
      
     
    
     §.§ Rotations of ^1
    
     
     In general, a vector fieldVon a four-manifoldMacts on gauge fields onMby
     δ A_i = V^j F_ji-D_iσ,
     whereσis an arbitrary generator of a gauge transformation. IfVis a Killing vector field onM,
     then  eqn. (<ref>) describes in a gauge-invariant sense the corresponding Killing vector field Von the instanton moduli space.
     However, to use eqn. (<ref>) together with the differential geometric formulas of sections <ref> and <ref>, we must
     pickσso thatδ A_isatisfies the gauge conditionWδ A=0.    Actually, there are two gauge conditions of interest, differing in the sign ofH.
      
    Here we consider  the vector fieldV=∂/∂τthat acts on^3×^1by rotating the second factor.
     It turns out that in this specific case,  the relevant gauge conditions are both satisfied withσ=0.
    Withσassumed to vanish, eqn. (<ref>) becomes
    
        δ A_i   = F_τ i,    i=1,⋯, 3 δ A_τ    = 0.
    
    The relevant gauge conditions areD_i δ A^i ±⋆ (H∧δ A)=0.   In the particular case of the(1)generatorV, we want
    this equation to be satisfied for each choice of sign, since we expect to get both a holomorphic and an antiholomorphic(1)current algebra.
    Indeed,H∧δ A=0becauseHis a pullback from^3andδ A_τ=0, andD_i δ A^i=0because the instanton equationF^+=0implies
    the second order Yang-Mills equationD_i F^ij=0, which forj=τgivesD_i F^iτ=0.    
    
    
    
    
    We expect to show that Vis covariantly constant for both connections∇and∇'on.  This means that eqn. (<ref>) or (<ref>) must hold
    with either sign of the torsion H, so we really need to establish two conditions
    Λ = 0=ι_ V H, 
    whereΛis the one-form dual to V.   
    
    The first of the two desired relations is almost immediate.   From  the definition (<ref>) of the metric ofand the formula
    (<ref>) forδ A(which we are entitled to use since we have verified thatδ Aas defined in this formula  satisfies the gauge condition that
    was assumed in defining the metric),  it follows that the one-dual dual to Vis
    Λ=-1/8π^2∫_M^̣4 x √(g)∑_i   F_τ iδ A^i =-1/8π^2∫_M  F∧δ A ∧τ̣,
    where the instanton equation was used in the last step.   Up to a constant multiple,Λis the basic example discussed in eqn. (<ref>)
    of a one-form onthat is closed but not exact.   
     In particular,Λis closed, as desired.
    
    To verify the second relationι_ V H=0requires a more detailed analysis.   We use the formula (<ref>) for H.   We note that in this formula,^̣3Ω ψ=^̣3Ω δ Acan be replaced by^̣3Ω δ A_ττ̣, since the part ofδ Atangent to the first factor of^3×^1does not contribute
    when multiplied by the volume form^̣3Ωof^3.  So asι_ Vδ A_τ=0according to eqn. (<ref>), we haveι_ V(^̣3Ω δ A)=0.
       Hence  in computingι_ V H, we only have to contract Vwith[δ A_i,δ A^i], giving 
    ι_ V H= Q_5'/π∫_M× M^̣3Ω_x τ̣ ⊗ δ A_τ(x) B(x,y) [δ A^i(y),F_τ i] ^̣4y√(g_y).
    To proceed farther, we need to study the second order differential equation obeyed byδ A.   This equation can be efficiently found by first observing
    that an instanton solution certainly obeys the Yang-Mills equationD^i F_ij=0.   The variation of this equation is 
     D^i (D_iδ A_j- D_j δ A_i)+[δ A^i,F_ij]=0. 
    To simplify this, we use
     D_i D_j δ A^i= D_j D_i δ A^i +[D_i,D_j]δ A^i=[F_ij,δ A^i]+ R_jkδ A^k -D_j⋆(H∧δ A),
    whereR_jkis the Ricci tensor of^3×^1and we used the gauge condition (<ref>) satisfied byδ A.   Settingj=τ, we haveR_τ k=0for^3×^1,
    so the term
    involving the Ricci tensor drops out.  Sinceδ A_τis the only component ofδ Athat contributes toH∧δ A, andD_τcommutes withH, forj=τwe 
    haveD_j ⋆ (H∧δ A)=⋆ (H∧_̣Aδ A_τ)and (<ref>) becomesD_i D_τδ A^i =[F_iτ,δ A^i]-⋆ [H∧_̣A δ A_τ].   
    Substituting this in eqn. (<ref>), we getD^i D_iδ A_τ+2[δ A^i,F_iτ]+⋆ [H∧ D_τδ A_τ]=0or in other words
         [ δ A^i,F_τ i]= 1/2W δ A_τ, 
     whereWis the operator 
     Wϕ=D_i D^i ϕ+⋆ [H∧_̣Aϕ] 
     whose Green's functionB(x,y)appears in the formula (<ref>).   
     So eqn. (<ref>) can be written 
     ι_ V H= Q_5'/2π∫_M× M^̣3Ω_x τ̣ ⊗ δ A_τ(x) B(x,y) W_yδ A_τ(y)^̣4y √(g_y).   
     Integrating by parts and using eqn. (<ref>) forW_y^† B(x,y), this simplifies to
      ι_ V H= Q_5'/2π∫_M^̣3Ω_x τ̣  δ A_τ(x)δ A_τ(x)=0.
      The vanishing in the last step holds by fermi statistics, sinceδ A_τ(x)=ψ_τ(x)is a fermionic object – a scalar function onMvalued in one-forms on–
      while δ A_τ(x)δ A_τ(x)is symmetric in the two factors ofδ A_τ(x).
     
     This completes the proof that the vector field Vis covariantly constant for both connections∇and∇'. 
    Therefore, the holomorphic and antiholomorphic chiral algebras of the sigma-model with targetwill contain, at a minimum,
    a(1)current algebra and a free fermion, beyond the small=4algebra that is guaranteed by the generalized hyper-Kahler structure.
     
     
      Since Vis a Killing vector field, it satisfiesD_i V_j+D_j V_i=0, and since the dual one-form is closed, we have alsoD_i  V_j-D_j V_i=0.
      SoD_i V_j=0and Vis covariantly constant for the Riemannian metric of.   In Riemannian geometry, if a compact Riemannianhas a nonzero vector
      field Vthat is covariantly constant, then locally='×^1where the two factors are orthogonal and Vacts by rotating the^1.   In general
      this may be true only locally and globally one may have=('×^1)/_nfor somen≥ 2, where_nacts by a2π/nrotation of^1and by 
      some symmetry of ordernof'.   In the present context, we also have a torsion field Hon; sinceι_ V H=Ł_ V H=0, His a pullback from'.
      This means that ifφis an angle-valued field that parametrizes the^1, thenφis decoupled from theB-field of(in some gauge, at least locally,
      theB-field is independent ofφand has no component in theφdirection).  Soφis a free field, and the holomorphic and antiholomorphic current algebras
      whose existence we have deduced are generated by the holomorphic and antiholomorphic conserved currents∂φand∂̅φ.
      
      Since Vis covariantly constant, its length squared| V|^2is constant.   We can now easily determine the constant value.  In this
      calculation,  we  use the definition (<ref>) of the metric of.  If we take the metric of^3×^1to be the usualΩ̣^2+τ̣^2, then with Vas described in eqn. (<ref>), we get
         | V|_0^2=-1/4π^2∫_^3×^1^̣4x√(g)  F_iτ F^iτ=-1/8π^2∫_^3×^1 F∧ F
      =Q_1,
      where  we used the instanton equation andQ_1is the instanton number.
      
      
      The reason for the subscript in| V|_0^2in eqn. (<ref>) is that we actually want to use a different normalization for this calculation.
    In sections <ref> and <ref>, we will 
     use the value of| V|^2as an ingredient in determining one of the central charges of the large=4algebra.
      For this purpose, bearing in mind the brane construction described in section <ref> that motivates the duality,
      it is important to normalize the metric of^3×^1so that∫_^3H=2π Q_5'.   According to eqn. (<ref>), we can ensure
      this by taking the metric of^3×^1to beQ_5'/2π(Ω̣^2+τ̣^2).  This rescaling of the metric of^3×^1does not affect the instanton equation, but according to eqn. (<ref>), it does multiply the metric ofbyQ_5'/2π.  
    Therefore with this normalization, we have 
      | V|^2=Q_1 Q_5'/2π.
    
     §.§ Rotations of ^3
    
     
    Now we consider the case of a vector field that is a generator of, say, the right action(2)_rof^3=(2)on itself.(2)_rleaves fixed the right-invariant one-formsR^ithat were introduced in section <ref>.  On the other hand, the left-invariant one-formsL^itransform
    in the adjoint representation of(2)_r.
    
    It is convenient to introduce a basisT_1,⋯, T_3of the Lie algebra of(2)_r, normalized so that the Lie algebra takes the canonical form
     [T_a,T_b]=ϵ_abc T_c,    a,b,c=1,⋯, 3. 
    In the notation of section <ref>, we  can take
    T_1=-1/2( y_0 ∂/∂ y_1-y_1 ∂/∂ y_0+y_2 ∂/∂ y_3-y_3 ∂/∂ y_2),
    withT_2andT_3obtained by cyclic permutations of indices1,2,3.
     The one-form dual to  the vector fieldT_ais  a multiple ofL_a:
     T_a^i G_ijx̣^j=-1/2 L_a.
     We recall that the metricGand the one-formsL_aare covariantly constant for the connection∇onM. 
     From eqn. (<ref>), it follows  that also theT_aare covariantly constant for this connection.
     
     SinceT_agenerates a symmetry of the instanton equation on^3×^1, it corresponds to a Killing vector field T_aon. The T_agenerate an(2)symmetry ofsince theT_agenerate an(2)symmetry of^3×^1. The complex structures_aoftransform in the adjoint representation
     of this(2)since the complex structures_aof^3×^1, from which the_aare deduced by restriction fromTtoT, transform in this representation.
     
      In a moment, we will show
      that T_ais covariantly constant for the conection∇onthat corresponds to the connection∇onM.
     As explained in eqns. (<ref>) and (<ref>), this implies that associated to the T_aare holomorphic currents with associated free fermions.
    Being holomorphic, the T_awill generate not just a global action of(2)_rbut   a holomorphic(2)_rcurrent algebra.  
    Together with the free fermions and(1)modes discussed in section <ref>,
     these holomorphic fields will extend the holomorphic chiral algebra of theσ-model with targetfrom a small=4algebra to a large one.  
     
    Actually, there is a trivial way to find vector fields onthat are covariantly constant for the connection∇and that, like the T_a, transform in the adjoint
    representation of(2)_r.   The complex structures_a,a=1,2,3, are covariantly constant with respect to∇, and we showed in <ref> that the(1)generator Vis covariantly constant with respect to∇(as well as∇').   So the vector fields_a  Vare covariantly constant with respect to∇.
    We claim that
     T_a=-1/2_a  V,  so T_ais likewise covariantly constant. This
     will complete the proof that the currents associated to the T_aare holomorphic on the sigma-model worldsheet and generate an(2)_rcurrent algebra.[Without knowing eqn. (<ref>), we would
    still know that, as the vector fields _a are covariantly constant with respect to ∇, they are associated to holomorphic currents
    that transform in the adjoint representation of an (2) symmetry group.  We would not know that they are actually the (2) generators.]
    
    
    First, let us verify the statement on^3×^1that is analogous to (<ref>), namely
    T_a=-1/2_a V. 
    From the definition (<ref>) of the_a, we have_a L_0=L_a.Vis the vector field dual toL_0andT_ais-1/2times the vector field dual toL_a, so
    eqn. (<ref>) is valid.
    
    The corresponding statement on, namely T_a=-1/2_a V, is actually an immediate consequence.   First of all, from the statementT_a=-1/2_a Von^3×^1, it immediately 
    follows
     that the corresponding statement T_a=-1/2_a Vis valid on the spaceof all connections.   To deduce from that a statement
    on the moduli space, we have to take into account that in general the action of a vector field or a complex structure on a tangent vector tocannot be trivially deduced from
    the action on; the natural formula in general 
     must be accompanied by a gauge transformation.   However, the gauge condition that we used in embedding the tangent space toin that towas
    chosen to be invariant under the action of the_a, so the natural action of_adoes not need to be accompanied by a gauge transformation.  And we showed in 
    section <ref> that the same is true for V.   So the formula T_a=-1/2_a Vcan be just restricted from the tangent space toto the tangent space to.
    As a bonus, this argument also shows that the naive action of T_a byδ A_j=T_a^i F_ijsatisfies the gauge condition, with no accompanying gauge transformation needed. 
    
    
    
    
    In section <ref>, to  compute the level of the(2)_rcurrent algebra, we will need to know the length squared| T_a|^2 = T_a^i G_ij T_a^jof the vector fields T_a.   The relation T_a=-1/2_a Vgives an easy way to do this.   As a complex structure,_ais in particular a length-preserving linear transformation of the tangent bundle of.  So the relation T_a=-1/2_a  V implies that| T_a|^2 =1/4 | V|^2.
    In eqn. (<ref>), we showed that| V|^2=Q_1 Q_5'/2π, so
     | T_a|^2 =Q_1 Q_5'/8π. 
    
    By now, we have shown that the sigma-model with targethas a holomorphic(2)×(1)current algebra, extending the small=4holomorphic superconformal
    algebra to a large one.  A similar analysis shows a similar enhancement of the antiholomorphic chiral algebra of the sigma-model.  Indeed, the two analyses are
    exchanged by the discrete symmetry of^3×^1that acts as a joint reflection on each factor.
    
    
    
     
     
    
     §.§ The Central Charges
    
     
     Finally, we will analyze the central charges of the(2)current algebras in the large=4holomorphic (or antiholomorphic) chiral algebra.
     
     One(2)is contained in the small=4algebra.   This is the(2)symmetry of the worldsheet fermions that exists because
     the connection∇has symplectic holonomy. 
    The worldsheet fermions all transform as spin 1/2 under this(2). A single set of four real fermions in the spin
     1/2 representation contributes 1 to the(2)current algebra level.   The number of such multiplets in the sigma-model is  the quaternionic dimension of, or equivalently
     one fourth of its real dimension.   By an index theorem, the real dimension ofis4Q_1Q_5, so the level of the current algebra in the small=4algebra isQ_1Q_5, in accord with expectations from supergravity and the duality conjecture <cit.>. This formula for the current algebra level 
     is exact, by the usual nonrenormalization theorem for fermion anomalies.
     
     The second(2)is the one that we analyzed in section <ref>, which acts by isometries of.    As remarked in section <ref>, the semiclassical limit of the sigma-model with targetis the limit thatQ_5'is large,
     keepingQ_1andQ_5fixed (or at least sufficiently small compared toQ_5').   In general, in a sigma-model with aB-field, anomalies in target space symmetries receive classical contributions that are large in the semiclassical regime. The WZW model is an example of this. (The semiclassical regime of the WZW model is the regime that the anomaly coefficientkis large, and in this regime,kcan be computed classically from the WZW action.)   In a sigma-model with fermions as well as aB-field, in addition to a large classical contribution, 
     the anomaly can receive  anØ(1)contribution due to the fermions.  In the present context, since we are studying a supersymmetric sigma-model with fermions, such anØ(1)contribution to the anomaly is possible, and it would be desirable  to know how to compute it.   But here we will only evaluate the classical contribution.  In an expansion in powers of1/Q'_5, there can be no further correction
     beyond theØ(1)contribution, because the anomaly coefficient is always an integer.
     
     For a vector field Tonthat generates part of a holomorphic chiral algebra, the semiclassical limit of the anomaly coefficient is easy to evaluate, because
     it depends only on the length squared of the vector field.  In the present context, we will take Tto be one of the T_a. Its length squared is
     known from  eqn. (<ref>), and we know that it corresponds to a holomorphic current in the sigma-model.
     
     For largeQ'_5, the metric of the sigma-model target spaceis scaled up, and in the vicinity of any assumed classical ground state,
       local coordinatesϕ^1,⋯, ϕ^ can be chosen such that  the metric
     becomes the flat metricG_αβ=δ_αβplus a correction of order1/Q'_5.  
      This means that the classical contribution to the anomaly can be computed in a theory of free scalar fields with the action 
      I=1/2∫^̣2 x ∑_μ=1,2 ∑_α=1^ ∂_μϕ ^α∂^μϕ_α. 
      For largeQ_5', the vector field Tbecomes a vector field with constant coefficients, generating a symmetryδϕ^α= T^α.
     In the free field
     theory, this symmetry can be generated by the canonical current[The canonical 
     current is defined by J^μ= 2π∑_αδϕ^αδ I/δ∂_μϕ^α, where here δϕ^α= T^α.  
      In two-dimensional conformal field theory, the factor of 2π in the canonical current, and a similar factor in the definition of the stress tensor,
      is conventionally included  to avoid factors of 2π in operator product coefficients.]J_μ=2π T^α∂_μϕ_α.
        However, if (as in our present application) 
     we know that in the full sigma-model, the current
     generating the symmetry is holomorphic, then in the free field approximation that we are analyzing, the current will have an extra term ensuring this holomorphy and will be
     J_μ=2π T^α(∂_μϕ_α+ıϵ_μν∂^νϕ_α).
     This extra term, which concretely will originate from theB-field coupling of the sigma-model,[The reason that we do not write any explicit formula for this
     coupling is that there is no canonical local formula; any local formula involves the local form of H=B, which does not actually affect the coefficient that we are evaluating.
     Any choice of B and H that is consistent with the assertion that the current J_μ is holomorphic will lead to the result analyzed in the text.]
      does not affect the conservation ofJ_μor the fact that
     it acts onϕbyδϕ^α= T^α.  Ifzis a local complex coordinate on the worldsheet such that^̣2x=|ẓ|^2, then eqn. (<ref>) is equivalent to
      J_z=4π T^α∂_zϕ_α,   J_z̅=0.
     The two-point function of the free scalar fieldϕ_αis⟨ϕ_α(z)ϕ_β (0)⟩=-δ_αβlog |z|/2π,which leads to
     ⟨ J_z(z) J_z(0)⟩ =-4π T^2/ z^2.  
    
    The levelkof an(2)current algebra is characterized by the statement that if symmetry generators are canonically normalized to satisfy (<ref>),
    then the two-point functions of the corresponding currents have short distance behavior
    ⟨ J_z(z) J_z(0)⟩ =-k/2 z^2. 
    In the present context, that means thatk=8π| T|^2.   From  eqn. (<ref>), we see that if Tis any of the T_a, then| T|^2=Q_1 Q_5'/8π.
    So  the level of the(2)current algebra for largeQ_5'is
    k=Q_1 Q_5',  
    as predicted by the duality conjecture. 
    
      As already explained, this formula is conceivably subject to a correction that isØ(1), that is, independent ofQ_5'.
    To compute this correction actually requires among other things
    a more precise definition ofQ_5'than we have given, because a fermion anomaly can affect theB-field flux by anØ(1)amount. 
    
    
    
     §.§ Vector Fields on  And Their Fixed Points
    
    
    In section <ref>, we proved that the(1)generator Vhas no fixed points on– something we now understand more deeply, having proved that Vactually has
    constant length.   We can also now justify certain analogous assertions that were made in section <ref>.
    
    Consider a linear combinationT =∑_a c_a T_aof the(2)_rgenerators.    The
    corresponding symmetry generator ofis T =∑_a=1^3 c_a  T_a=-1/2∑_a=1^3 c_a _a .    Since1/√(∑_b c_b^2)∑_a c_a_ais a complex structure and acts on  the tangent bundle ofas an orthogonal transformation,  preserving lengths, it follows  that Thas constant length1/2√(∑_a c_a^2)| V|.   In other words, any(2)_rgenerator has a (nonzero) constant length and hence
    acts onwithout fixed points. 
    
    Now consider an(2)_ℓgeneratorT=∑_a c'_a T'_a.   The corresponding vector field onis T' =∑_a=1^3 c_a'  T'_a=-1/2∑_a=1^3 c_a' _a' .
    Reasoning as  before, its length is1/2√(∑_a c'_a^2)| V|and  it acts onwithout fixed points.
    
    
    What about a sum T+ T'of generators of(2)_rand(2)_ℓ?    This vector field does not have constant length, as in general the angle between the two vector
    fields Tand T'is variable.  However, the triangle inequality gives a lower bound on the length of T+ T'which shows that T+ T'acts without fixed points
    as long as| T|≠| T' |, in other words, as long as the original vector fieldsTandT'on^3×^1have unequal length.
    If for example∑_a c_a^2>∑_a c'_a^2, then| T|>| T'|and the triangle inequality gives
    | T+ T'|≥  | T|-| T'|>0.
    So a vector field T+ T'can only have fixed points onif the underlying vector fieldsTandT'on^3×^1have equal length.   In that case,
    there actually are fixed points, as analyzed in <cit.> and discussed in section <ref>.
    
    
    Finally,  consider a linear combination U=u T+v  T' +w  V.   The formulas T =-1/2∑_a=1^3 c_a _a and T'=-1/2∑_a=1^3 c_a' _a' show that Tand T'are both everywhere orthogonal to V.  Therefore the inner product of Uwith Vis the constantw| V|^2. Ifw≠0, the nonzero constant value of that inner product implies that Uhas no zeroes.
    
    
    
     §.§ Large =4 Algebra for One Chirality OnlyM=^3×^1is the unique compact four-manifold such that a sigma-model with targetMhas two copies of the large=4algebra (for left-
    and right-movers, respectively).   For this choice ofM, we have shown that
     a sigma-model with target the corresponding instanton moduli spacelikewise has two copies of the large=4algebra.
    
    What ifMis  such that the sigma-model with targetMhas just one copy of the large=4algebra, say for right-movers only? What choices ofMare
    possible?Mshould have a single hypercomplex structure, with an(2)symmetry that rotates the three complex structures, and an additional1)symmetry.  
    As shown in  <cit.>,  a compact hypercomplex manifold of dimension four, if not Kahler, is a Hopf surface, which means in particular that it is locally isomorphic to^3×^1.   A convenient way to exploit
    this fact is to observe that such anMhas the same universal cover as that of^3×^1, namely^3×.  So we can return to the starting point of section <ref>.^3×with the familiar metricΩ̣^2+τ̣^2is equivalent to^4minus the origin with the metric
    ṣ^2=̣⃗Y^2/Y⃗^2,
    as in eqn. (<ref>).   The isometry group of^3×isH=((2)_ℓ×(2)_r)×^*where^*acts byY⃗→λY⃗,λ>0.
    Any manifold locally isomorphic to^3×^1is the quotient of^3×by a discrete subgroupΓ⊂ H.   However, since we want to preserve one(2)symmetry,
    say(2)_r, we should pickΓto commute with(2)_r, which means that we wantΓ⊂(2)_ℓ×^*.  The quotientM=(^3×)/Γis a manifold ifΓis a discrete subgroup of(2)_ℓ×^*, and is compact as long asΓ⊄(2). Moreover, as long asΓ⊄^*,
    a sigma-model with targetMwill support precisely one copy of the large=4algebra.
    
    It is then true that ifis the instanton moduli space onM, a sigma-model with targetalso supports precisely one copy of the large=4algebra.  We do not need
    any essentially new calculation to show this.   The following structures on^3×areΓ-invariant, and therefore descend toM:
    the(2)_ℓ-invariant hypercomplex structure studied in this paper and its associated connection∇,  and the symmetry group[The group that acts faithfully on M
    may be  a quotient of (2)_r×^*, depending on Γ.](2)_r×^*, 
    where(2)_rrotates the complex structures that make up the hypercomplex structure.  Since the local geometry is the same as it is for^3×^1, the computations that we have done apply without change to show that the vector fields generating the(2)_r×^*action are 
    covariantly constant for the connection∇onthat corresponds to∇. This extends the small=4algebra 
    that we would expect based on the analysis in section <ref> to a large one.
    
    The example studied in section <ref> withM=^3/_n×^1corresponds toΓ=_n×, with_n⊂(2)_ℓand⊂^*.
    Another simple example, withΓ=,  can be obtained if we describeMby complex variablesZ_1, Z_2with an equivalence relation(Z_1,Z_2)≅ e^T(Z_1,Z_2)for some constantTwith Re T>0.   If Im T=0,
    this gives  back the original exampleM=^3×^1with a metric of the standard form, leading to two copies of the large=4algebra. But taking Im T≠0gives a deformation that allows only one copy
    of the large=4algebra. 
    
     
    
    § CONFORMAL INVARIANCE
    
    
     
     There is one more issue to consider.   It has long been understood that for two-dimensional sigma-models
     with=4supersymmetry,  there is a potential gap between scale invariance and conformal invariance <cit.>.
     In general, the condition for scale invariance is weaker than the condition for conformal invariance.   Scale invariance says that the trace of the stress tensor is 
     the divergence of a current, and conformal invariance says that the trace of the stress tensor vanishes <cit.>.    
     
     In a sigma-model with torsion, the one-loop beta functions for the metric and theB-field are conveniently combined together to make R_KL,
     the Ricci tensor of the connection∇with torsion.   Strong HKT geometry then leads to[If the same metric and B-field 
     on the σ-model target space are compatible with more than one strong HKT structure – as happens in the case of the instanton moduli space
     on ^3×^1– then a formula like this holds for each of them.  In what follows, we will not need an explicit formula for the Lee form θ.  It is difficult
     to get a useful explicit expression for θ  in the case of instanton moduli space , because the definition (<ref>) of the Lee form
     involves the inverse of the metric on , and although there is a simple formula for this metric,
     it is difficult to give a useful description of its inverse.] R_KL=∇_Kθ_L, 
     whereθis a 1-form called the Lee form.   For conformal invariance, one wants instead
      R_KL=-2∇_K∇_L Φ,
     whereΦis a scalar field (the dilaton) on the  sigma-model target space.    Unlessθis such that eqn. (<ref>) can actually
     be put in the form (<ref>), the model is only scale-invariant and not conformally-invariant.
     
    
     
     There are several ways to argue that the sigma-model with target the instanton moduli spaceis actually conformally-invariant.  However, some of the following
     arguments have technical gaps or are limited, in the form presented here,  to lowest order in
     sigma-model perturbation theory. 
     
     First of all, we can simply invoke the general result <cit.>  that a scale-invariant quantum field theory in two dimensions with a discrete spectrum of
     operator dimensions is always conformally invariant.  A sigma-model with smooth compact target space will have a discrete spectrum of operator dimensions.
     The instanton moduli spaceis not smooth and compact, because of singularities arising from  small instantons and un-Higgsing.   However, when the theta-angle is
     non-zero (see section <ref>), the model is expected to have a discrete spectrum of operator dimensions (if the charges are relatively prime), and therefore the general theorem applies.
     
     Even without invoking the theta-angle, we can note that the general argument for conformal invariance <cit.> only requires that the theory have a discrete
     spectrum of operator dimensions near dimension zero.   The small instanton and un-Higgsing singularities do produce a continuous spectrum of operator dimensions,
     but only above a positive threshold (which moreover is large ifQ_1andQ_5are large).   So one would not expect these singularities to cause difficulty in the proof
     of conformal invariance.
     
     
     A number of more explicit alternative arguments are available, though  in the form that will be presented here these arguments have technical gaps or are limited to lowest order of
     sigma-model perturbation theory (that is,  lowest order in an expansion in powers of1/Q'_5).   Hopefully these arguments can be sharpened and extended.
     
     One approach is a Perelman-style argument, using an auxiliary 
     Schrödinger equation.  This approach was explained in <cit.>, following <cit.>.    For this purpose, one 
     considers onthe Schrödinger operatorH=-4Δ^2+R-1/12H^2.
     The proof of conformal invariance requires that this operator should have a unique, positive ground state.   For example, this is true ifis compact and smooth except for conical
     singularities which are at finite distance and at which the potentialR-1/12H^2is bounded below.   (It is also true under some hypotheses ifhas ends at infinite distance.
     In the related context of thec-theorem, this case was studied in <cit.>.)  In the case of the instanton moduli space, the small instanton and un-Higgsing singularities
     are indeed conical singularities at finite distance.   Generically these singularities are hyper-Kahler singularities at whichRandH^2remain bounded.   However,
     there are a few exceptional cases.    For example, consider the small instanton singularity on a moduli space of instantons with instanton numberQ_1>1.   When a single instanton bubbles or in other words becomes small and shrinks to a point,acquires the same universal small instanton singularity as in the case of instantons
    on any other four-manifold.  This universal singularity is a conical hyper-Kahler singularity, and is harmless for our present purposes, since bothRandHvanish near such a singularity.
     This is true as long as the remaining instanton solution after instanton bubbling remains irreducible: the details of the
     manifold in which the small instanton is embedded do not matter.   However, if  allQ_1instantons become small, so that the remaining
     instanton solution is simply a flat connection on^3×^1, we get a singularity at which the global structure is relevant.  Likewise, generic un-Higgsing produces
     a hyper-Kahler singularity similar to those for instantons on^4or, but complete un-Higgsing down to an abelian subgroup is a special case in which the global
     structure becomes relevant.   These special cases need more study before claiming to prove conformal invariance based on a Perelman-style argument.
     
     Another approach to proving conformal invariance uses more detailed facts about the instanton moduli spaceas well as results of <cit.>. However, this argument,
     in the form we will present, is only valid to lowest order in sigma-model perturbation theory.  We have constructed
     on the instanton moduli spacea hypercomplex structure,,, covariantly constant with respect to a connection∇, and a second hypercomplex structure',',', covariantly constant with respect to a second connection∇'.   Associated to the first hypercomplex structure is a Lee formθ(though this is not
     obvious from the definition  in
    eqn. (<ref>), the Lee form  depends only on the hypercomplex structure and not on the specific
     choice of,, orin writing the formula).  Similarly there is a Lee formθ'associated to the second hypercomplex structure. The potential obstruction
     to scale invariance can be expressed in terms of the tensor∇_Kθ_L, which is also equal to∇'_Lθ'_K.   The equality∇_Kθ_L =∇'_Lθ'_Kgives (as in eqn. (3.16) of <cit.>) 
      D_K(θ-θ')_L+D_L(θ-θ')_K=0,    (θ+θ')_KL= H^P_KL(θ-θ')_P, 
     whereDis the Riemannian connection.
     The obstruction to conformal invariance vanishes if and only if there is a functionΦon(the dilaton) such that
     ∇_Kθ_L=-2∇_K ∂_LΦ.
     Let us see why such aΦexists in the case of the instanton moduli space.
     
    Isometries of^3×^1that preserve the orientation of each factor  rotate,, andamong themselves.  Sinceθis determined by any of these (or any linear combinationa+b+cwitha^2+b^2+c^2=1), it is invariant under such isometries.
     The same argument applies forθ'.   However, as discussed in section <ref>,  we can also define an isometryρthat
     acts as a reflection on each factor of^3×^1, reversing the orientation of each factor and preserving the overall orientation.
     Such an isometry exchanges the two hypercomplex structures, so it exchangesθandθ'.
    Soθ+θ'is even underρ, andθ-θ'is odd.
     Eqn. (<ref>) tells us thatθ-θ'is the one-form dual to a Killing vector field.   This Killing vector field must be invariant under rotations of either factor of^3×^1,
     but odd under the joint reflectionρ.   The only Killing vector field  onwith this property is the Killing vector field Vassociated with  the Killing vector
     fieldVon^3×^1that generates a rotation of^1.  Soθ-θ'=uλ, whereλis the 1-form dual to Vandu∈.
      In section <ref>, we have proved that H_PKL V^L=0.   Consequently, the second
     equation in (<ref>) reduces to(̣θ+θ')=0.   Thusθ+θ'is a closed 1-form on, defining an element ofH^1(;).   However,θ+θ'is even under the joint reflectionρ, and we have seen in section <ref> that (assuming something along the lines of the Atiyah-Jones
     conjecture)H^1(;)is one-dimensional, generated by the dual of V, which is odd underρ.  Hence the closed formθ+θ'is actually exact:θ+θ'=4Φ̣for some functionΦ.
     Putting these statements together, we haveθ=1/2((θ+θ')+(θ-θ'))=2Φ̣+ 1/2 uλ.
     Inserting this in∇_Kθ_L, the term proportional toλdoes not contribute, since∇λ=0.   So∇θ=2∇Φ̣and the condition for conformal invariance holds with dilatonΦ.
     
     A noteworthy fact is that each of these arguments for conformal invariance of the sigma-model with targetmakes use of some global information (compactness, relevant 
     for Polchinski's argument,  some
     knowledge of the possible singularities, relevant for analyzing the effective Schrödinger equation, or some more detailed information that entered in the discussion of the Lee forms).
     Local considerations alone do not suffice.
     
     
     
      
            
        
     
     
    
    § SEARCHING FOR INTERESTING GEOMETRIES
    
    
     §.§ Preliminaries
    
     
     The moduli spaceof instantons on^3×^1has an interesting differential geometric structure that, as we have seen,  leads to(4,4)supersymmetry in two dimensions
     (with large superconformal symmetry) 
     despite the presence onof a topologically non-trivialB-field.   However,is not a smooth compact manifold; it has small instanton singularities and also 
    un-Higgsing singularities – singularities that arise at points incorresponding to instantons with a non-trivial automorphism group.
    
     
     In the case of instantons on^4or, the instanton moduli spaces are hyper-Kahler, again with singularities associated to small instantons and un-Higgsing.
     In this case, however, it is possible to resolve the singularities by turning on a Neveu-SchwarzB-field on^4or.  This leads to a noncommutative deformation
     of the instanton equation <cit.>, and resolves all of the singularities, provided thatQ_1andQ_5are relatively prime.  (IfQ_1andQ_5are not relatively
     prime, certain un-Higgsing singularities are unavoidable.)    In this way one can construct smooth,
     compact hyper-Kahler manifolds that are genuinely new – they cannot be constructed by taking products of tori andsurfaces.  In fact, these are the main known 
     examples of compact hyper-Kahler manifolds.
     
     No comparable examples are known of strong HKT manifolds, even if one only asks for a single strong HKT structure, rather than a pair of  strong HKT
     structures, as  the instanton  moduli space on^3×^1possesses.   If one asks for a strong HKT manifold that is compact (and smooth), the known examples
     are compact hyper-Kahler manifolds (in which the torsionHvanishes) and homogeneous spaces (which can have strong HKT structures withH≠0; there are many examples
     <cit.>, of which
     the simplest is^3×^1).   It would be interesting to find genuinely new examples of compact  strong HKT manifolds.   The instanton moduli spaces on^3×^1do not qualify, since they have unavoidable singularities associated to small instantons and to unbroken gauge symmetries.   The NSB-field modes that
     can resolve these singularities in the case ofor^4have no analog here, asH^2(^3×^1;)=0.   It is true that there is a theta-angle that,
     as discussed in section <ref>, resolves the singularities in a quantum mechanical sense (ifQ_1andQ_5are relatively prime), but this does not give
     a classical deformation or resolution of the singularities of the instanton moduli space.
     
       Aiming to find a classical example of a compact (smooth) manifold with the geometry that leads to large=4symmetry – at least for one chirality in the sigma-model – we will
       replace^3×^1withM=^3/_n×^1for some integern≥ 2.   Here we pick_nto act on, say, the left on^3≅(2), thus
       breaking the rotation group(2)_ℓto(1)_ℓand leaving(2)_runbroken.  Instanton moduli space onM, which we will again denote as, 
       now carries just a single strong HKT structure, which will lead to a large=4superconformal algebra for right-moving modes of the sigma-model.  Left-moving
       modes will see a single KT structure (Kahler with torsion), leading to an ordinary=2algebra for left-movers.   
       
       To take advantage of the torsion in the cohomology ofM, we will take the gauge group to be(n)=(n)/_n, the quotient of(n)or(n)by its center,
       rather than(n)as we have assumed up to this point.[More generally, we could  consider (n) bundles
       over ^3/_m×^1 with m≠n.  The ideas would be similar but some statements would become slightly more complicated as we would have to consider
       the cohomology of ^3/_m with _n coefficients.]   
       In fact, replacing(n)with(n)/_nwould have  little effect[The instanton moduli space on ^3×^1 for (n)/_n is obtained by dividing
       the (n) moduli space by a _n symmetry that multiples the holonomy around the second factor of ^3×^1 by an element of the center of (n).]  on the differential geometry of the
       instanton moduli space on^3×^1.   The reason that^3/_n×^1is different is that the there is torsion in the cohomology of^3/_n×^1and as a result there are more choices in the topology of a(n)bundle.
       
       An(n)bundle over   four-manifoldEis  classified topologically by its second Chern classc_2(E), which we usually think of as the second Chern class of the
       vector bundle associated toEin the fundamental representation.(n)also has, of course, an adjoint representation.   We will denote the associated vector bundle
       in the adjoint representation as(E).  As a real
        vector bundle, it has  a first Pontryagin classp_1((E)).   The relation between the two invariants is
        p_1((E)) = 2n c_2(E).
        Thus, for an(n)bundle,p_1((E))is always an integer multiple of2n.
        
    The classification of a(n)bundleE→ Mis not quite so simple.  
    As(n)has no  analog of then-dimensional fundamental representation of(n), there is no ranknvector bundle
        associated toE.  But(n)does, of course, have an adjoint representation, and we can again consider the adjoint bundle(E)associated toE.
        It has an invariantp_1((E))which is still an integer-valued invariant, in the sense that it is valued inH^4(M;)and gives an integer upon integration overM.
        However, in general it is no longer divisible by2n.
        
    However, this is not the whole story. A(n)bundle has an additional invariantζ(E)taking values inH^2(M;_n).   Forn=2,(2)is the same as(3), andζis the same as the second Stieffel-Whitney classw_2(E).
        As in that example,ζis the obstruction to lifting a(n)bundle to a bundle with structure group(n), the universal cover of(n).
        
        A(n)bundleEover a four-manifoldMis determined topologically byp_1 ((E))andζ(E), but these cannot be specified independently.  They are subject to one
        relation, which was determined in <cit.>:
         p_1((E))= (n+1)C(ζ) mod 2n      n  evenn+1/2 C(ζ)  mod 2n     n odd.
        HereCis a cohomology operation that generalizes the Pontryagin square (which was first considered in the physics of gauge theory in <cit.>).
        Forn=2, withζ=w_2(E),Cactually is the Pontyagin square and the formula is
        p_1((E)) =C(ζ) mod 4.
           For oddn,C(ζ)=2ζ^2according to <cit.> so
        in that case
         p_1((E)) =(n+1)ζ(E)^2  mod 2n. 
        The details for evenn>2are slightly more complicated and we omit them.
        
        One has[See for example <cit.> or Prop 3.8 in <cit.>.]H^i(^3/_n;_n)=_nfori=1,2,3, with generatorsx∈ H^1(^3/_n),y∈ H^2(^3/_n;_n),xy∈ H^3(^3/_n;_n).   One also hasH^1(^1;_n)≅_n, say with generatorz.  The cohomology ring of^3/_n×^1with_ncoefficients is generated byx,y,zwith the
        following relations:y^2=z^2=0for dimensional reasons; moreover, for oddn,x^2=0and for evenn,x^2=     n/2y.   
        
      In particular, sinceH^2(^3/_n×^1;_n)=_n×_nwith generatorsyandxz, we have in generalζ= a y + b xz, witha,b∈_n.    
        We can then make eqn. (<ref>) explicit for oddn:
         p_1((E)) = 2 ab mod 2n. 
        Thus in general, for oddn,p_1((E))can take any even value mod2n.
        
        We will also make eqn. (<ref>) explicit forn=2,(2)=(3).   First of all, forn=2, the cohomology of^3/_nsimplifies slightly asy=x^2, and accordinglyx^3zgeneratesH^4(^3/_2×^1;_2)=_2.     The groupH^2(^3/_2×^1;_2)=_2×_2has three nonzero elements, namelyx^2,xz, andx^2+xz.   Two of these possible nonzero values ofζcan be realized by flat(3)bundles.   Indeed, since a real line bundleε→ Mis classified byw_1(ε)∈ H^1(M,_2), overM=^3/_2×^1, there exists a real line bundleε→ Mwithw_1(ε)=x, and another real line bundleε'→ Mwithw_1(ε')=z.   Then with𝒪representing a trivial real line bundle, there is a flat(3)bundle overMwith(E)=Ø⊕ε⊕εand hencew_2(E)=x^2, and another flat(3)bundle overMwith(E')=ε⊕ε'⊕ε⊗ε'andw_2(E)=x^2+xz.
        As these bundles are flat, we havep_1((E))=p_1((E'))=0.   Therefore, by virtue of eqn. (<ref>), in general an(3)bundleE→^3/_2×^1withζ=x^2orx^2+xzhasp_1((E)) divisible by 4, implying via eqn.  (<ref>) thatC(x^2)=C(x^2+xz)=0.
          To see what happens forζ=xz, we use the general relation for the Pontryagin square:C(u+v)=C(u)+C(v)+2uv mod 4.
        In the present case, takingu=x^2,v=xz, this tells us thatC(xz)= 2x^3 z mod 4=2mod 4.   Hence eqn. (<ref>) tells us that in general an(3)bundle withζ=xzwill havep_1((E))=2mod 4.  
        
        We can see this more explicitly by constructing an elementary example.   We begin with a simple construction of an(2)bundle  of instanton number 1 over^3×^1.   We start with a trivial(2)bundle over the cylinder^3× I, whereIis an interval.   To build an(2)bundleE→^3×^1,
        we glue the top of the cylinder to the bottom after making a gauge transformation by a mapΦ:^3→(2).   To get a bundle of instanton number 1,
        we can takeΦto be the identity map from^3=(2)to itself.    This gives a bundle withc_2(E)=1and hencep_1((E))=4.    To get an(3)bundle over^3/_2×^1, we make the same construction starting with a trivial bundle over the cylinder^3/_2× I.   Then we make an(3)bundle
        over^3/_2×^1by gluing the top of the cylinder
        to the bottom after making a gauge transformationΦ:^3/_2→ SO(3).   ForΦwe choose the identity map from^3/_2=(3)to itself.   This
        gives an(3)bundleE→^3/_2×^1.  A double cover of this construction gives the previous one, so the curvature integral is half of what it was before
        andp_1( (E))=2.   As for the invariantζ(E), since the bundleEis trivial when restricted to^3/_2× pfor a pointp∈^1, there is nox^2contribution inζ(E).  On the other hand,ζ(E)must be nonzero sincep_1((E))is not a multiple of 4.  Soζ(E)=xz.      
      
      
    
     §.§ Instanton Moduli Spaces
    
      
      Now we will begin a more detailed discussion of instanton moduli spaces onM=^3/_n×^1.   The first question to ask is whether these moduli spaces are nonempty.
      The usual existence theorem for instantons <cit.> applies for any simple and simply-connected gauge groupand any compact oriented smooth four-manifoldM.  The
      theorem  shows that the moduli space is non-empty if the instanton number is large enough; ifMis simply-connected, it must be
      at leastb_2^+(M)-b_1 +1(b_2^+is the dimension of the space of selfdual harmonic two-forms onMandb_1is the first Betti number).
      The proof is made by starting with a trivial flat-bundle overM.  Then, after picking a suitable embedding of(2)in, one
      glues in some numberkof small(2)instantons to get an approximate instanton solution overM, and one proves that ifk≥ b_2^+(M)-b_1+1, the approximate solution
      can be corrected to get an exact solution.
      
      This proof is not sufficient for our purposes.   We want to consider instantons with structure group the non-simply-connected group(n)and withζ(E)≠0.
      This means thatEcannot be a trivial flat bundle.   Actually, the proof in <cit.> can easily be modified to start with any flat bundle, not necessarily trivial.   
      But for genericζ, eqn. (<ref>) shows thatp_1((E))cannot vanish and thereforeEcannot be flat.
      
      Luckily, there is another existence theorem for instantons that applies for arbitraryζ<cit.>.    This theorem says that for any compact simpleand anyζ, the instanton moduli space is non-empty if the instanton number is sufficiently large.   In our context, this means that  for anyζ(E)the instanton moduli space is nonempty
      for any sufficiently largep_1((E))that is consistent with eqn. (<ref>).   (Actually this statement is a special case of a more general theorem
      that was already cited in section <ref>.)  The proof proceeds roughly as follows.   Starting with any-bundleE→ Mwith the desired value ofζ(E), and any connection on this bundle,  one glues in many small instantons.  One shows that the gluing parameters can be chosen
      so as to reduce theL^2norm of the selfdual part of the Yang-Mills field strength, making the field  strength more nearly anti-selfdual.   Once one gets close enough
      to anti-selfduality, one then shows (as in the original proof <cit.>) that the connection can be modified to achieve full anti-selfduality.
      
      A drawback of this proof is that it does not tell us what is, for givenζ(E), the smallest value ofp_1((E))at which the instanton equation can be solved.   We only
      learn that the moduli spaces are non-empty for all sufficiently large values ofp_1((E)).   (In fact, there are no “gaps”: if the moduli space is non-empty for one value
      ofp_1((E)), it remains non-empty at larger values consistent with eqn. (<ref>).) 
      
      In short, for allζand all sufficiently largep_1((E)), we do get an instanton moduli spacethat will have the differential geometric properties that lead to
      a single copy of  large=4superconformal symmetry.   Our goal now is to argue that with some further choices,will be smooth and compact.   If so, these moduli spaces
      are candidates as the first examples of smooth, compact strong HKT manifolds that are not products of hyper-Kahler  manifolds and homogeneous spaces.
      
      Forto be smooth and compact, we have to avoid singularities associated to bubbling of small instantons, and also un-Higgsing singularities associated to instanton solutions
      with non-trivial automorphism groups. First let us discuss how to eliminate the small instanton singularity.  There is a  natural
      way to do this.   Pick a value ofζsuch that for a(n)bundleE→ Mwith the givenζ, eqn. (<ref>) implies thatp_1((E))is not a multiple of2n.
      Ifp_1((E))is negative, then the instanton moduli space is empty.  But  ifp_1((E))is sufficiently positive, then the instanton
      moduli space is non-empty according to the proof that we just sketched.  Therefore, there is a smallest value ofp_1((E))with the givenζsuch that the instanton moduli
      spaceis nonempty.   Sincep_1((E))is not an integer multiple of2n, it does not vanish at the minimum.    So the componentwith the minimum value
      ofp_1((E))parametrizes instanton solutions that are not merely flat bundles.  Let us call this the minimal component of instanton moduli space for the givenζ.
      This minimal component
      has no small instanton singularity,
      since such a singularity would connectto another component with a smaller value ofp_1((E))and the sameζ.
    
       The question now arises of whether it is also true, perhaps after putting a further restriction onζ, that the minimal component of instanton moduli space has
       no un-Higgsing singularity.     
    Such a singularity  occurs at a pointp∈that corresponds to an instanton solution whose structure group is a proper subgroup of(n)– and more specifically, 
    a proper subgroup that has a nontrivial commutant in(n).
       Forn=2, it is clear that the minimal component can have no such singularity.   A proper subgroup of(2)=(3)is a either a  finite group or is isomorphic to(2)or to(2),
       and no such group is the structure group of a non-flat instanton solution on^3/_2×^1.  
       It is not clear what is the dimension of the minimal component for(2), and it is not clear  whether the minimal component might be the product of a hyper-Kahler manifold and a homogeneous strong 
       HKT manifold.   However, the minimal component for(2)is a candidate as a genuinely new strong HKT manifold that is compact and smooth.
       
       To make a similar argument forn=3, we observe that a connected proper  subgroup of(3)that is the structure group of a non-flat instanton bundle on^3/_3×^1is
       either(2)or its subgroup(2)or else(3)(here(2)is embedded in(3)so that the fundamental representation of(3)transforms as2⊕1, and is then projected to(3), and similarly(3)is embedded in(3)so that the fundamental representation remains irreducible  and is then projected
       to(3)).   A routine check shows that any(3)bundleEwhose structure group reduces to(2)or(3)hasp_1((E))a multiple of 6, and therefore
       such reductions are not possible for anyζsuch thatp_1((E))is not such a multiple.   (In any event, a reduction of structure group to(3)would
       not produce a singularity of the moduli space, since the stabilizer of(3)in(3)is trivial.)   
       
       
       For generaln,  consider a component of the moduli spacecharacterized by some given values ofp_1((E))andζ(E)at a point inat which the structure groupEreduces from(n)to a proper subgroupK.Kis not necessarily connected, and it may contain(1)factors.
       Gauge fields of a finite group or an abelian group or a product or  extension of these can contribute toζ(E), but they do not contribute top_1((E))(for a group containing factors of(1), this statement depends upon the fact thatb_2(M)=0).   Sop_1((E))is unchanged if we simply replaceEby another
       instanton bundle whose structure group reduces to a maximal connected semi-simple subgroupH⊂ K.    What are the possible values ofp_1((E))for
       a bundle with such a reduction?  The embeddingH⊂(n)gives a homomorphismφ:π_1(H)→π_1((n))=_n.   Suppose first for simplicity thatnis prime.  In that case, the only subgroups
        of_nare_nitself and the trivial group containing only the identity.   ForHa proper subgroup of(n)andnprime,π_1(H)does not have an element
        of order[Indeed, (n) itself is the only semisimple Lie group of rank ≤ n-1 whose fundamental group contains an element of order n.  This follows from the classification of simply-connected simple Lie groups and the explicit description of their centers.]nand therefore the image ofφis trivial.   This means that, after restricting the structure group ofE→ Mfrom(n)toH, it can be lifted to a bundle
        with structure group(n).  Hencep_1((E))is divisible by2n.   Therefore, fornprime, ifζ(E)is such thatp_1((E))is not a multiple of2n, the instanton
        moduli space has no un-Higgsing singularity related to reduction of structure group. 
        Hence the minimal component of instanton moduli space for the given value ofζ(E)is compact and smooth.  
        
         Ifnis not prime, it is possible to find a proper subgroupH⊂(n)such that the homomorphismφis surjective
        (for example, taken=pqwith relatively primep,q, and chooseH=P((p)×(q)), with(p)×(q)embedded in(n)such that the fundamental representation of(n)is
        the tensor product of the fundamental representations of(p)and(q); this can be generalized ifnis the product of any number of relatively prime factors).  
        In this case, however, the commutant ofHin(n)is trivial, so
        reduction of structure group toHproduces no singularity in.
        If the image ofφis a proper subgroup_k⊂_nwithk<n, then a(n)bundle overMwhose structure group restricts toHcan
        be lifted to an(n)/_kbundle, whiich implies thatp_1((E))is an integer multiple of2n/k.    Ifζ(E)is chosen so that this is not the case for any proper
        divisorkofn, then the minimal component of instanton moduli space again has no un-Higgsing singularity and is compact and smooth.
        For example, ifnis odd, then according to eqn. (<ref>), fora=b=1,p_1((E))=2 mod 2n, and is therefore not a multiple of2n/kfor anyk<n.
        A similar choice is possible for evenn.
    
        
    
     §.§ String Theory Interpretation
    
        
        The construction that we have described actually has a string theory interpretation.
        
        Assuming that the underlying branes considered are D-branes,   the two-form fieldBand three-form curvatureH=Ḅin the_3×^3×^3×^1or_3×^3×^3/_n×^1geometry are of Ramond-Ramond type.   
        However, we can also turn on the Neveu-Schwarz  two-form fieldB_ NS.   Topologically, a two-form fieldB_ NSon a spacetimeYis classified
        by a characteristic classξvalued inH^3(Y;).    At the level of differential forms,ξis represented byH_ NS/2π, but here it will be important
        to considerξas an integral cohomology class.  
        
        For our present purposes, the interesting case is thatξis a torsion class, which can be represented by a topologically non-trivialB_ NSfield withH_ NS=0.
        The reason that this is the interesting case is that if we assume thatH_ NSis nonzero, and impose the global symmetries and supersymmetries
        of the_3×^3×^3×^1or_3×^3×_3/_n×^1geometry, we will just end up with the same spacetime geometry
        that we have already studied, up to anS-duality rotation that replacesH_ RRwith a linear combination ofH_ RRandH_ NS.
       
      But we get something essentially new ifB_ NSis flat but topologically nontrivial.
        The reason that this is possible is that there is torsion in the three-dimensional cohomology of_3×^3×_3/_n×^1.
        This torsion is pulled back fromH^3(^3/_n×^1;)=⊕_n.   
        
        We have encountered in the preceding analysis(n)bundles over^3/_n×^1that have non-integer values of the instanton number and therefore
        cannot be lifted to(n)or(n)bundles over^3/_n×^1.    In general, a(n)bundleEover any spaceYcan be lifted to a(n)bundleE'→ Yif and only if the characteristic classζ(E)∈ H^2(Y;_n)that has been important in our analysis
        can be lifted to a classζ'∈ H^2(Y;)(which will then be the first Chern classc_1(E')).  The obstruction to
        this lifting can be understood by considering the long exact cohomology sequence associated to the short exact sequence0→n→r→_n
        → 0, where the first map is multiplication bynand the second is reduction modn.  The associated long exact cohomology sequence reads in part 
                ⋯ H^2(^3/_n×^1;)r→ H^2(^3/_n×^1;_n)β→ H^3(^3/_n×^1;)⋯,
                whereβis called the Bockstein map.  Thusζ∈ H^2(^3/_n×^1;_n)is not the modnreduction of an integer classζ'– and so is not in the
                image ofr– if and only ifβ(ζ)is a nonzero element ofH^3(^3/_n×^1;_n).
                
                
               Ifβ(ζ)≠0, it is not possible to lift the(n)bundleE→^3/_n×^1to a(n)bundle  or in other words to a ranknvector
               bundle.   But it can be lifted to what is called a twisted vector bundle, twisted by the classβ(ζ).   Such a twisted vector bundle, rather than an ordinary one,
               is precisely what one gets in D-brane physics in the presence of a background fieldB_ NSwhose characteristic classξis a torsion class.
               Indeed,  a system ofnD-branes interacting with a backgroundB_ NSfield such thatξ=β(ζ)supports not an ordinary vector bundle but a twisted vector bundle
               associated to a(n)bundle with characteristic classζ<cit.>.   Thus in the presence of a suitableB_ NSfield, the slightly exotic
               instanton moduli spaces studied in this section do actually appear in D-brane physics.
                
                
        
        
        
    
    
    
       
     
    
    
    
    
    1cm
     Acknowledgements   I thank O. Aharony, L. Eberhardt, J. Maldacena, G. W. Moore, R. Moraru, S. Murthy,  N. Nekrasov,
     G. Papadopoulos, A. Tomasiello, and M. Verbitsky for discussions.
      Research supported in part by NSF Grant PHY-2207584.
      
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     §.§ String Theory Interpretation
    
        
        The construction that we have described actually has a string theory interpretation.
        
        Assuming that the underlying branes considered are D-branes,   the two-form fieldBand three-form curvatureH=Ḅin the_3×^3×^3×^1or_3×^3×_3/_n×^1geometry are of Ramond-Ramond type.   
        However, we can also turn on the Neveu-Schwarz  two-form fieldB_ NS.   Topologically, a two-form fieldB_] NSon a spacetimeYis classified
        by a characteristic classξvalued inH^3(Y;).    At the level of differential forms,ξis represented byH_ NS/2π, but here it will be important
        to considerξas an integral cohomology class.  
        
        For our present purposes, the interesting case is thatξis a torsion class, which can be represented a topologically non-trivialB_ NSfield, withH_ NS=0.
        The reason that this is the interesting case is that if we assume thatH_ NSis nonzero, and impose the global symmetries and supersymmetries
        of the_3×^3×^3×^1or_3×^3×_3/_n×^1geometry, we will just end up with the same spacetime geometry
        that we have already studied, up to anS-duality rotation that replacesH_ RRwith a linear combination ofH_ RRandH_ NS.
        Instead, we get something essentially ifB_ NSis flat but topologically nontrivial.
        
        The reason that this is possible is that there is torsion in the three-dimensional cohomology of_3×^3×_3/_n×^1.
        This torsion is pulled back fromH^3(^3/_n×^1)=+_n.   
            
        A  useful description of a generstor of this_nis the following.
        Consider the long exact cohomology sequence
            ⋯ H^2(^3/_n×^1;)r→ H^2(S^3/_n×^1;_n)β→ H^3(^3/_n×^1;)⋯.
            The mapris reduction modnand the mapβis called the Bockstein.   The long exact sequence shows that a classζ∈ H^2(^3/_n×^1;_n) can be lifted to an integral class – that is a class inH^2(^3/_n×^1;)– if and only ifβ(ζ)=0.
            
        
            
        The reader might notice some  gaps in our explanation of the underlying topology.   
        First, in general, the number of theta-angles in a sigma-model with some target spaceis the second Betti number of, orb_2(), 
        the dimension of the real cohomology groupH^2(;).   
        Withbeing the moduli space of instantons on^3×^1,b_2()is at least 1, withΓas a generator ofH^2(;).
        But how do we know thatb_2()is precisely 1?   Could there be other generators ofH^2(,)that  are not given by such obvious gauge theory formulas?
        The answer is that, for anyQ_5, at least for sufficiently large values ofQ_1, there are no other generators; this follows from  general results summarized
         in section <ref>.       These results do not rule out the possibility of exceptional behavior, not visible in supergravity, for some small values ofQ_1.
      A similar comment applies to the discussion in section <ref> below  of the first Betti numberb_1(): we will describe one generator ofH^1(;), and 
        we will eventually need to know that this cohomology group is 1-dimensional, with no additional generators.  For sufficiently largeQ_1, this is ensured by the same general arguments.


 Paper Title: Dynamical System Analysis Of Chameleon Mechanism in Brans-Dicke Scalar-Tensor Model.



Article Title]Dynamical System Analysis of Chameleon Mechanism in Brans-Dicke Scalar-Tensor Model










[1,2]Azwar Sutionoazwarsu@gmail.com

1,3]Agus Surososuroso@itb.ac.id
These authors contributed equally to this work.

1,3]Freddy Permana Zenfpzen@itb.ac.id
These authors contributed equally to this work.

*[1]Theoretical Physics Laboratory, THEPI Division Indonesia, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, Indonesia

[2]Department of Physics, Universitas Hasanuddin, Jl. Perintis Kemerdekaan KM. 10, Makassar, 90245, Indonesia

[3]Indonesia Center for Theoretical and Mathematical Physics (ICTMP), Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, Indonesia





We investigated the stability of the chameleon screening mechanism in the Brans-Dicke scalar-tensor model. We define a constraint on the Brans-Dicke parameter ω_BD^* identifying two stability groups, ω_BD>ω_BD^* and 0<ω_BD<ω_BD^*. The first group achieves stability with both appropriate eigenvalues and a density profile consistent with dark energy dominance. The second exhibits eigenvalue stability but contradicts conditions for a stable universe. We explore the impact of variations in the scalar field potential and matter coupling by analyzing different parameter sets. Each unique set of parameters results in a distinct ω_BD^*. Dynamic analysis reveals that stability is achieved when the scalar field dominates, highlighting the importance of the kinetic and potential terms while minimizing the influence of matter density. In high matter density regions, the scalar field's negligible presence aligns with standard gravitational behavior, whereas in low matter density regions, the scalar field grows exponentially, driving dark energy and cosmic acceleration.







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    August 5, 2024
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§ INTRODUCTION

Modified gravity refers to theoretical modifications or alternative theories to Einstein's general theory of relativity, which describes the force of gravity as the curvature of spacetime caused by mass and energy <cit.>. While general relativity has been extremely successful in explaining various gravitational phenomena, there are some observations on cosmological and galactic scales that it doesn't fully account for. Modified gravity theories attempt to address these issues by introducing new concepts, fields, or modifications to the gravitational force. The field of modified gravity was still an active area of research, with scientists exploring various theoretical frameworks and observational tests. One of the most significant observations in cosmology is that the expansion of the universe is accelerating. This discovery, based on observations of distant supernovae, led modified gravity to the proposal of dark energy as a hypothetical form of energy that permeates space and counteracts the attractive force of gravity <cit.><cit.><cit.><cit.><cit.><cit.>. Some examples of modified gravity theories include Scalar-Tensor Theories, f(R) Gravity, Extra Dimensions and Braneworlds, and the Tensor-Vector-Scalar (TeVeS) Theory <cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.>. 

Screening mechanisms come into play within these modified gravity theories. They are mechanisms that suppress or screen the effects of the modified gravity in environments where standard gravity predictions are well-established. This screening allows the theory to be consistent with local tests of gravity (such as those conducted in our solar system) while still producing noticeable effects on cosmic scales. Common types of screening mechanisms include the chameleon mechanism, Vainshtein mechanism, and symmetron mechanism <cit.><cit.><cit.><cit.>. These mechanisms allow modified gravity theories to evade detection in regions where gravitational effects are well-understood, ensuring agreement with experimental tests conducted in our immediate cosmic neighborhood.  The background of screening mechanisms is rooted in the quest to reconcile cosmological observations, particularly the accelerated expansion of the universe, with modified gravity theories that can explain these phenomena without relying on the introduction of dark energy. These mechanisms enable consistency with local tests of gravity while allowing for deviations at cosmic scales <cit.><cit.><cit.><cit.><cit.>.

The Chameleon mechanism, embedded in the Brans-Dicke scalar-tensor model <cit.><cit.><cit.><cit.>, adeptly conceals gravitational modifications in regions of high density. This mechanism marks a progression from the conventional chameleon mechanism, intricately coupling the scalar field with matter <cit.><cit.>. As a consequence, the scalar field assumes a variable mass contingent upon the density of matter in the relevant region. In regions characterized by extremely low matter density, the scalar field within the model reverts to its role as dark energy, contributing to the universe's expansion acceleration <cit.>. 

Equally crucial, the pursuit of screening mechanism stability in modified gravity originates from the need to understand the behavior and viability of these theories in different environments, particularly on cosmological and astrophysical scales. The indispensability of stability becomes evident in ensuring the viability of modified gravity theories. An unstable mechanism could manifest observable deviations from general relativity in regions subject to experimental and observational constraints. Stability ensures that the modified gravity theory remains consistent with empirical evidence and can provide a viable alternative explanation for gravitational phenomena. The exploration of screening mechanism stability serves as a crucial means to anticipate and understand how these theories behave in diverse cosmic environments <cit.><cit.><cit.>. It offers insights into the interplay between gravity and matter at different energy scales and provides a testing ground for the compatibility of modified gravity theories with the principles of general relativity. Analogous to conventional dark energy models, the Chameleon mechanism must satisfy stringent stability conditions to establish itself as a trustworthy dark energy model. Hence, a thorough analysis of the dynamic system is conducted to assess the stability of the mechanism.

The structure of the paper is outlined as follows. Section <ref> give review of the thin-shell solution of the Brans-Dicke chameleon.  Section <ref> delves into the Brans-Dicke chameleon model, presenting the system of equations formulated as an autonomous system. Moving to Section <ref>, a comprehensive system dynamics analysis is conducted to assess the stability of the mechanism. Section <ref> elucidates the correlation between the Brans-Dicke chameleon and the universe's expansion acceleration. The final section encapsulates a summary and draws conclusions based on the findings of the study.



§ THIN-SHELL SOLUTION OF BRANS-DICKE CHAMELEON

In our previous work, we investigate the thin-shell solution of the Brans-Dicke Chameleon mechanism <cit.>. The Brans-Dicke Chameleon action is given as follows:

    S=∫ d^4x√(-g)[ϕℛ-ω_BD/ϕ(∂ϕ)^2-2V(ϕ) ] + 2S_m(A^2(ϕ)g_μν, ψ),

where (∂ϕ)^2=g^μν∂_μϕ∂_νϕ, V(ϕ) is the scalar field self-interaction potential, w_BD is the Brans-Dicke dimensionless coupling parameter, A(ϕ) is the coupling function to matter, g_μν is the metric in Jordan Frame and g̃_μν=A^2(ϕ)g_μν is the metric in Einstein Frame. The potential V(ϕ) is assumed to be of the runaway form that monotonically decreasing and satisfies: V, V_,ϕ/V, V_,ϕϕ/V_ϕ...→∞ as ϕ→∞, as well as V, V_,ϕ/V, V_,ϕϕ/V_ϕ...→∞ as ϕ→ 0 <cit.><cit.>. By appropriately selecting V(ϕ) and A(ϕ), it is feasible to create a scenario where the scalar propagates unrestrictedly and facilitates a fifth force in regions characterized by a low Newtonian potential. However, this force is deactivated in high-density areas, such as those found in the solar system. Variation of action (<ref>) with respect to ϕ give the equation of motion for ϕ,

    ∇^2ϕ=2/3+2ω_BD[ϕ V_,ϕ-2V+(A_,ϕϕ-A/2)ρ],

with ρ is density in general. In order to understand how the chameleon force is suppressed in the presence of high density region, we solve for the field profile in the presence of a massive compact object. We consider a spherically symmetric object characterized by a constant radius R and density ρ_c for simplicity. Additionally, we envision that this object exists in the presence of a homogeneous background of density ρ_b. We choose the thin-shell case that refers to a region around a massive object where the chameleon field experiences a significant change in its behavior. This thin shell arises due to the interaction between the chameleon field and matter, particularly in regions of high density. Within the thin shell, which typically extends to a certain distance from the surface of the massive object (its distance is R_c from the center of the massive object), the chameleon field transitions from its light, weakly interacting state to its heavier, strongly interacting state. This transition zone effectively shields the effects of the chameleon field from being observable at larger distances.


We find the approximate  exterior solution of the thin-shell case,
 
    ϕ_thin(r)=-ηe^-m_b(r-R)/r+ϕ_b,
 
with,

    η   =   e^-(R_c+R)√(Qβρ_c/M_pl)(M_pl-2β) /2β(e^2R_c √(Qβρ_c/M_pl)M_plR_c(-R_c+√(M_pl/Qβρ_c))( 1+R√(Qβρ_c/M_pl))/-R_c(Qβρ_c+M_pl)+M_pl(√(M_pl/Qβρ_c)- √(Qβρ_c/M_pl)).
          .+e^2R √(Qβρ_c/M_pl)R_c(R+√(M_pl/Qβρ_c))(Q R_cβρ_c+√(Qβρ_c/M_pl))/√(βρ_c/M_pl)(R_c(Qβρ_c-M_pl)+M_pl( √(M_pl/Qβρ_c)+ √(Qβρ_c/M_pl)))),

where Q=2/3+2ω_BD, M_pl, β, ϕ_b and m_b  are Planck mass, coupling constant, a minimum of the effective potential in the background, and the mass of small fluctuations about ϕ_b, respectively. Note that η is a constant that depends on Planck mass, theory parameters, and the compact object radius and density.

The solution is subsequently implemented on earth by choosing the suitable constants. As observed, the chameleon field undergoes exponential growth once it surpasses the Earth's radius, eventually stabilizing at an infinite distance. In regions with high matter density, where the chameleon field has a large effective mass, the associated fifth force becomes short-ranged. The force acts effectively only over small distances, confined to the immediate vicinity of the massive object. The short-range nature of the fifth force leads to a screening effect. On the contrary, in low-density regions far from the Earth, where a lighter scalar field operates, a long-range force is mediated, highlighting the impact of modified gravity on the cosmological background.



§ AUTONOMOUS SYSTEM OF BRANS-DICKE CHAMELEON ACTION

Considering the variation of the Chameleon Brans-Dicke action with respect to both the metric g_μν and the scalar field ϕ in the flat Friedmann–Lemaître–Robertson–Walker background, we arrive at the following Friedmann equation and dynamic equation:

    3H^2   =   -3Hϕ̇/ϕ+ω_BD/2(ϕ̇/ϕ)^2-V/ϕ+Aρ_m/2,
    
    2Ḣ+3H^2   =   -2Hϕ̇/ϕ-ω_BD/2(ϕ̇/ϕ)^2-ϕ̈/ϕ-V/ϕ,
    ϕ̈+3Hϕ̇   =   2/3+2ω_BD[ϕ V_,ϕ-2V+(A_,ϕϕ-A/2)ρ_m].

where ρ_m represents the matter density. The matter fluid is considered as pressureless dust (p_m = 0). In cosmology, pressureless dust is a good approximation for non-relativistic matter, such as cold dark matter and baryonic matter, at large scales. These components dominate the matter content of the universe in the current epoch, making pressureless dust a relevant model for studying the chameleon mechanism <cit.>. In the absence of a screening effect,  without any coupling between matter and the scalar field, the analysis reduces to the standard Brans-Dicke theory <cit.>. 
Combining equation (<ref>), (<ref>) and (<ref>), we get modified continuity equation as,

    ρ̇_m+3Hρ_m=-3/4A_,ϕϕ̇ρ_m/A
The effective equation of state of the system is ω_eff=-1-2/3Ḣ/H^2 <cit.>. In order to study the cosmological dynamics of the system, it is convenient to introduce the following dimensionless variables: 

    x=ϕ̇/Hϕ,
          y=1/H√(V/3ϕ),
          z=1/H√(ρ_mA/3ϕ).

The equation (<ref>) can be simplified and written as a constraint,

    ω_BD/6 x^2 -x - y^2 +z^2 = 1.

We also define the energy fraction of dark energy and matter:

    Ω_ϕ=ω_BD/6x^2-x - y^2;      Ω_m=1-Ω_ϕ.

From equation (<ref>) and (<ref>), we get

    Ḣ/H^2   =   -x/2-ω_BDx^2/4+3( Q-1/2)y^2+3Qz^2/4-Q/2H^2(V_,ϕ+A_,ϕρ_m)-3/2,
    ϕ̈   =   -3Hϕ̇-6Qy^2H^2ϕ-3Qz^2H^2ϕ/2+Qϕ( V_,ϕ+A_,ϕρ).


By choosing V(ϕ)=V_0ϕ^-n and A(ϕ)=A_0ϕ^α, with n and α are positive dimensionless constant, then differentiating each dimensionless variable in equation (<ref>) with respect to the number of e-foldings N= ln a, obtain the autonomous equations


    dx/dN   =   ω_BDx^3/4-3x^2/2-3x/2-3(n+2)Qy^2+3( α-1/2)Qz^2
          -3[(n/2+1)Q-1/2]xy^2+3/2( α-1/2)Qxz^2,
    dy/dN   =   -3[(n/2+1)Q-1/2]y^3+3y/2 +ω_BDx^2y/4-(n/2+1 )xy
          +3/2( α-1/2)Qyz^2,
    dz/dN   =   3/2(α-1/2)Qz^3 +ω_BDx^2z/4-3(n/2+1 )Qy^2z+3y^2z/2
          +(α/8-1)xz,

and effective equation of state,

    ω_eff   =   -1-2/3[-x/2-ω_BDx^2/4+3( Q-1/2)y^2+3Qz^2/4. 
           .+3Q/2(ny^2 -α z^2 )-3/2],

with Q=2/3+2ω_BD and d/dN=1/Hd/dt. By setting dx/dN=dy/dN=dz/dN=0, fixed points can be obtained. Solving the above equations we find the fixed points that some explicitly depend on ω_BD, n, and α as illustrated in Table <ref>. The dynamics of the autonomous system can be analyzed by finding the exact constant fixed points of the system and examining the stability around these points <cit.>.





§ STABILITY ANALYSIS OF THE BRANS-DICKE CHAMELEON


The stability of a fixed point in a dynamical system is characterized by how the system behaves in the vicinity of that point. Specifically, stability analysis involves understanding the response of the system to small perturbations or deviations from the fixed point. In the context of autonomous systems, the stability of fixed points is often determined by analyzing the eigenvalues of the Jacobian matrix evaluated at the fixed point <cit.><cit.>. The eigenvalues provide information about the behavior of perturbations around the fixed point. Consider a system of three autonomous differential equations:

    dx/dN=f(x,y,z),    dy/dN=g(x,y,z),   dz/dN=h(x,y,z),

with f, g, and h are functions describing the dynamics of the system. We consider linear perturbations (δ x,  δ y, δ z) as follows 

    x=x_c+δ x,       y=y_c+δ y,      z=z_c+δ z.

Linearizing the autonomous equation above leads to the first-order differential equations

    d/dN[ δ x; δ y; δ z;     ] = M  [ δ x; δ y; δ z;     ],

where M is 3× 3  Jacobian matrix composed of partial derivatives of  f, g, h with respect to 
x, y, z.  It is given by

    M=
     [  ∂ f∂ x  ∂ f∂ y  ∂ f∂ z;  ∂ g∂ x ∂ g ∂ y ∂ g ∂ z;  ∂ h∂ x ∂ h ∂ y ∂ h ∂ z;         ]_(x=x_c,   y=y_c, z=z_c),


The stability of the fixed point is determined by the real parts of the eigenvalues. Depending on the signs of the real parts: 1) Stable node, if all eigenvalues are real and have negative real parts. 2) Unstable node, if all eigenvalues are real and have positive real parts. 3) Saddle point, if there is a mix of positive and negative real parts among the eigenvalues. 4) Stable spiral, If the real parts of the eigenvalues are negative and at least some of the eigenvalues have non-zero imaginary parts. 5) Unstable spiral, if the real parts of the eigenvalues are positive and at least some of the eigenvalues have nonzero imaginary parts <cit.><cit.><cit.>.

In order to obtain the eigenvalues of each fixed point, we need the explicit form of the potential and the matter coupling function. The choice of potential in dark energy models is crucial because it directly affects the dynamics of the scalar field responsible for the accelerated expansion of the universe. One crucial condition that must be fulfilled is the slow-roll condition <cit.>. The slow-roll conditions are more easily satisfied with smaller values of n. This choice balances simplicity and the ability to produce a slow-roll regime that can match the observed accelerated expansion of the universe, making it a suitable and popular choice in dark energy models. As a starting step, we choose n=2, and for simplification, we also choose the same value for α. Then note that there is a special case regarding the fixed points listed in Table  <ref> for α=2. The number of critical points will be reduced to 9 points. Specifically, point P2 coincides with point P3, point P4 coincides with point P5, and points P8 and P9 coincide with point P1. We can classify the stability of a fixed point based on the limitations on the value of the parameter ω_BD. The eigenvalues of each fixed point for case n=α=2 can be seen in Table <ref>.


Based on the eigenvalues, we can determine the characteristics of each fixed point based on the limits of the parameter ω_BD. Point P1, P8 and P9 will always be a saddle point because it does not depend on the parameter ω_BD, regardless of any choice of constant values n and α. Points P2-P7 will be will be always saddle points for ω_BD>23/6. Points P10 and P11 will be stable under condition ω_BD>23/6 and will be saddle point for ω_BD<23/6. Points P12 and P13 will always be unstable when ω_BD>0 and will never be stable.

There is one conditions that permit the existence of stable fixed points. To assess the stability of our mechanism, we introduce ω_BD^* as the threshold value of the Brans-Dicke parameter. We will review one condition, focusing on the limitations of the Brans-Dicke parameter for ω_BD>ω_BD^*, with ω_BD^*=23/6  for the specific case we have chosen. The stability characteristics of each fixed point for the case ω_BD>ω_BD^* can be seen in Table <ref> for ω_BD=4. Based on the eigenvalue analysis, there are nine saddle points, namely points P1-P9, two stable points, namely points P10 and P11, and two unstable points, namely points P12 and P13. 



Points P1, P8, and P9  represent the matter-dominated era with Ω_m=1. These points indicate the early universe was dominated by matter. Points P2 to P7 are saddle points but have non-physical density parameters and do not correspond to any physical era. The density parameter values for both the scalar field and matter must not be greater than one and must not be negative. These points highlight non-physical solutions, emphasizing constraints on the model. Points P10 and P11 represent the era of dark energy domination, with Ω_ϕ=1. The stability of these points suggests that the universe will eventually evolve into a phase dominated by dark energy, leading to accelerated expansion. Points P12 and P13 can be considered as kinetic points of the scalar field with density parameter values approaching 1. The instability indicates that these states are not sustainable and the system will evolve away from these points. 





We can also identify the phase in which a fixed point is situated by analyzing the ω_eff parameter. For the universe to be in an accelerating phase, the effective equation of state parameter ω_eff must be less than -1/3. It is evident that points P2-P7 and P10, P11 satisfy the conditions for being in the accelerated phase of the universe. This is further supported by the dominance of the scalar field in these points, as indicated by their density parameter values, particularly for the stable points P10 and P11.

To gain a deeper understanding of the stability characteristics, we can analyze the trajectory of each fixed point in the phase space. Visualizations were performed using Mathematica <cit.>. Figure <ref> on the left shows the trajectory paths around points P10 and P11 in 3D phase space. To further confirm the attractor properties of the fixed point, we can set one of the critical point components as constant and visualize it in a 2D plane. Figure <ref> on the right shows the trajectory around point P10 by setting y=-1.128i, demonstrating the point's attractor nature in xz-plane. A similar plot would be observed for point P11 with y=1.128i. In Figure <ref>, we can see the trajectory paths at points P1 (and also P8,P9), P12, and P13. In the xy-plane with set z=0, the characteristics of each point are clearly visible, with P1 being a saddle point and P12 and P13 being unstable points. Points P2-P7 are not visualized due to their non-physical nature. These points may represent theoretical solutions that do not manifest in real-world scenarios or under specific model constraints, thus not contributing to the stability analysis of the system.

As a comparison, we will examine the case where n≠ 2
so that all fixed points exist with different values. Table <ref> shows the eigenvalues of each critical point for the case n=3. Point P1 still be a saddle point. Points P2, P3, P4, and P5 will be stable under the condition 1/192(107+27√(417))≤ω_BD<25/6, stable spiral for 0<ω_BD<1/192(107+27√(417)) and will be saddle points for ω_BD>25/6. Points P6 and P7 will always be saddle points under the condition ω_BD>19/6. Points P8 and P9 will be always saddle points when ω_BD<-53/24 or ω_BD>-7/4 and can be stable under certain conditions, specifically when -53/24<ω_BD<-7/4. Points P10 and P11 will be stable under condition ω_BD>25/6 and will be saddle point for ω_BD<25/6. Points P12 and P13 still be unstable when ω_BD>0 and will never be stable.




There are three conditions that permit the existence of stable fixed points, but not all of them will be examined. In the Brans-Dicke theory, the scalar field ϕ contributes to gravitational interactions that the kinetic term of the scalar field in action has the correct sign, corresponding to positive kinetic energy. Negative kinetic energy would lead to instabilities, such as ghost fields, which are unphysical. A crucial step in achieving this is by ensuring that the parameter ω_BD is positive. We will review only two conditions, focusing on the limitations of the Brans-Dicke parameter for ω_BD>ω_BD^* and 0<ω_BD<ω_BD^*, with ω_BD^*=25/6  for the specific case we have chosen.



 §.§ Case ω_BD>ω_BD^*


The stability characteristics of each fixed point for the case ω_BD>ω_BD^* can be seen in Table <ref> for ω_BD=5. Based on the eigenvalue analysis, there are nine saddle points, namely points P1-P9, two stable points, namely points P10 and P11, and two unstable points, namely points P12 and P13. 



Point P1 represents the matter-dominated era with Ω_m=1. These points indicate the early universe was dominated by matter. Points P2-P7 have non-physical density parameters and do not correspond to any physical era. Points P8 and P9 represent an era where the scalar field starts moving but matter still dominates, with Ω_m=0.922. These points suggest transitional eras where both matter and scalar fields are significant. Points P10 and P11 represent the era of dark energy domination, with Ω_ϕ=1. The stability of these points implies that the universe will eventually transition into a dark energy-dominated phase, resulting in accelerated expansion. Points P12 and P13 can be considered as kinetic points of the scalar field with density parameter values approaching 1. The instability shows that these states are not sustainable, and the system will move away from these points over time. 



We can also determine the phase in which a fixed point resides by examining the ω_eff parameter. For the universe to be in an accelerating phase, the effective equation of state parameter ω_eff must be less than -1/3. It is clear that points P2-P7 and P10, P11 meet the criteria for being in the accelerated phase of the universe. This conclusion is reinforced by the dominance of the scalar field at these points, as reflected in their density parameter values, especially for the stable points P10 and P11.



To better comprehend the stability properties, we can analyze the behavior of each fixed point in the phase space. Figure <ref> on the left shows the trajectory paths around points P10 and P11 in 3D phase space. To further confirm the attractor properties of the fixed point, we can set one of the critical point components as constant and visualize it in a 2D plane. Figure <ref> on the right shows the trajectory around point P10 by setting y=-1.109i, demonstrating the point's attractor nature in xz-plane. A similar plot would be observed for point P11 with y=1.109i. Figure <ref> shows the phase space plot of points P8 and P9, with figures on the left providing 3D visual representations. The saddle nature of point P8 is further confirmed in figures on the right when the system is observed from an alternative viewpoint in the xy-plane with set z= -0.303i. Figure <ref> shows the trajectory paths at points P1, P12, and P13. In the xy-plane with set z=0, the characteristics of each point are clearly visible, with P1 being a saddle point and P12 and P13 being unstable points. Points P2-P7 are not visualized due to their non-physical nature. These points might be theoretical solutions that fail to appear in real-world scenarios or under certain model constraints, thereby not contributing to the stability analysis of the system.





 §.§ Case 0<ω_BD<ω_BD^*


We have chosen the value ω_BD=4 which falls within the range 0<ω_BD<ω_BD^*, to examine the characteristics of each fixed point in the context of the stability of the dark energy model. According to the eigenvalues listed in Table <ref>, there are 4 stable points, 2 unstable points, and the remainder are saddle points. Interestingly, some fixed points that were initially stable have become saddle points, and vice versa. Points P1, P6-P9, P12, and P13 retain the same characteristics as they did with the previous selection of ω_BD.  Points P2-P5, previously identified as saddle points, are now stable with all negative eigenvalues and remain in the accelerated phase based on their ω_eff value. However, their density parameter values do not align with the expected properties of stable points, specifically the absolute dominance of the scalar field. Although the scalar field remains dominant with Ω_ϕ=0.93, there is still a non-zero matter density, typically a characteristic of saddle points. Conversely, points P10 and P11, previously stable, have transitioned to saddle points based on their eigenvalues, yet they still remain in the accelerated phase. Despite this, their density parameter values indicate the total dominance of the scalar field with Ω_ϕ=1 which is usually a feature of stable fixed points.  





 §.§ Variation of ω_BD


To gain a clearer understanding of the stability changes in two groups of fixed points that can potentially be stable, namely the groups of points (P2-P5) and (P10, P11), we can examine various conditions with different values of ω_BD values within the two ranges mentioned above. Table <ref> shows the stability of the fixed points for ω_BD=2 and ω_BD=3, which fall within the range 0<ω_BD<ω_BD^*, and for ω_BD=10, ω_BD=10^3 and ω_BD=10^5 which fall within the range ω_BD>ω_BD^*. In the range 0<ω_BD<ω_BD^*, points P2-P5 consistently show stability, as stable spiral points, while points P10 and P11 consistently exhibit saddle behavior. However, this stability is not supported by their density parameter values. In fact, as ω_BD decreases, the matter density becomes more dominant, leading to a contradiction between stability based on eigenvalues and stability based on the density parameters of dark energy and matter. In the range ω_BD>ω_BD^* points P2-P5 consistently exhibit saddle behavior, and points P10 and P11 consistently show stability. This is corroborated by their density parameter values, where the scalar field dominates stable conditions. This analysis sheds light on how the stability properties of fixed points in the dark energy model change with different values of ω_BD, providing insight into the complex dynamics and stability conditions influenced by the Brans-Dicke parameter.



 §.§ Variation of n and α

In the previous work, we reviewed the stability of fixed points for specific values of n and α, which led to certain constraints on the Brans-Dicke parameter. In this section, we aim to vary the values of  n and α, exploring different forms of the potential and the matter coupling function. Table <ref> presents several combinations, specifically for  n=2, n=3, n=4 and α=4, α=5, α=6. Each combination of n and α results in a unique value of ω_BD^*. As the values of both of them increase, the value of ω_BD^* also increases.  This implies that each choice of the potential and matter coupling function imposes different constraints on the Brans-Dicke parameter ω_BD. These constraints are crucial in ensuring the stability of each fixed point in the dark energy model under consideration.






§ BRANS-DICKE CHAMELEON AND THE ACCELERATION OF THE UNIVERSE'S EXPANSION

In the case examined above, we identified several stable fixed points for specific values of ω_BD. For the special case when α=2, stability is achieved when ω_BD>ω_BD^*. For the case α≠ 2, and as an example α=3 as mentioned above, the stable fixed points are divided into two groups: one where ω_BD>ω_BD^* and another where 0<ω_BD<ω_BD^*, with ω_BD^*=23/6. Both groups of fixed points are in the accelerated expansion phase of the universe with ω_eff<-1/3. In the first group, we found fixed points that are stable based on their eigenvalues and are supported by density parameter values for dark energy and matter that match the actual conditions of the universe. In the second group, we also found stable fixed points based on their eigenvalues, which contradict the density parameter values. Therefore, we will focus on the first group, which can be considered as physically stable points.
It is established that the variable x denotes the kinetic term of the scalar field, y represents the potential term of the scalar field, and z signifies the coupling term between the matter density and the scalar field. The obtained results lead to the inference that the stability of the Chameleon mechanism in the Brans-Dicke model manifests when the scalar field is dominant especially the potential terms, accompanied by the minimum coupling of matter and scalar field.

In the discourse surrounding modified gravity, specific models posit the scalar field as a potential contender for dark energy, as seen in Quintessence, Gauss-Bonnet, and similar frameworks <cit.>. Essentially, the attractor solutions derived earlier suggest that stability is achieved when the scalar field takes on a dominant role, signifying the prevalence of dark energy. Upon examining other dark energy models that also put forth the scalar field as a candidate, it becomes apparent that the attractor solutions in those models align with the dominance of the scalar field, consistent with the aforementioned results.

The dominance of the scalar field’s kinetic and potential terms in the universe’s acceleration is a concept rooted in the dynamics of specific cosmological models, particularly those involving scalar fields. The kinetic term represents the rate of change of the scalar field for a time. It is associated with the field’s motion or velocity. When the kinetic term dominates, it implies that the scalar field is dynamically evolving, and its motion contributes to the overall energy density of the universe. The potential term represents the energy associated with the scalar field at a given point in space. It characterizes the field’s potential to influence cosmic evolution. When the potential term dominates, it suggests that the scalar field has settled into a particular energy state, potentially leading to the acceleration of the universe's expansion. 

In the specific context of the Chameleon mechanism within the Brans-Dicke model and similar scenarios, the dominance of the scalar field signifies stability. It indicates that the scalar field’s dynamics are primarily influenced by its motion and energy state, with minimal interaction with matter. This dominance contributes to the stability of the Chameleon mechanism in regions of low matter density. Fundamentally, the role of the scalar field in propelling the universe’s acceleration stems from their collaborative capacity to produce an effective energy density characterized by negative pressure, resembling the attributes of dark energy. When appropriately balanced, these terms can lead to the observed accelerated expansion, as seen in certain cosmological models.

A prior investigation identified thin-shell solutions for the Chameleon Brans-Dicke field <cit.>. The findings indicated that the scalar field maintains a negligible value in proximity to regions with high matter density (near the matter source). It then experiences an exponential increase, ultimately converging to a stable value in areas of low matter density situated far from the matter source (r→∞). In locales with high matter density, the Chameleon mechanism conceals the gravitational modification effects. Conversely, in regions with low density, the scalar field reassumes its role as dark energy, instigating the acceleration of the universe's expansion. Crucially, the scalar field reverts to functioning as dark energy when it attains a very large, stable, and constant value, coinciding with very low matter density. Implicitly, these conditions lend support to the attributes of the attractor solutions outlined in the dynamic system analysis discussed earlier.




§ CONCLUSION


The results of this study provide crucial support for the stability of the Chameleon mechanism, especially within the Brans-Dicke model. By examining the eigenvalues obtained, we define a constraint on the Brans-Dicke parameter, referred to as ω_BD^*. This constraint divides the stability points into two distinct groups: one for ω_BD>ω_BD^* and another 0<ω_BD<ω_BD^*, except for the special case of α=2 which only satisfies the conditions of the first group. In the first group, the fixed point is stable and is supported by suitable eigenvalues and a density parameter that represents the era of dark energy dominance. In the second group, the fixed point is stable according to its eigenvalues, but the density parameter does not correspond to the expected conditions of a stable universe. We then explore variations in the explicit forms of the scalar field potential and the matter coupling function by selecting different values of n and α. It is shown that each unique constant value results in a distinct ω_BD^*. This demonstrates that the stability of the dark energy model is sensitive to the specific values of these parameters. The exploration reveals that the stability conditions for dark energy dominance are closely linked to the chosen values of n and α. These parameters influence the behavior of the scalar field potential and the matter coupling function, thereby affecting the overall dynamics of the universe. By identifying the unique ω_BD^* for different parameter values, we gain insights into the stability characteristics of various dark energy models, which is crucial for understanding the long-term evolution of the universe under the influence of dark energy.

From the dynamic analysis performed, it was also discovered that stability is achieved when the scalar field assumes a dominant role, with a particular emphasis on the significance of the kinetic and potential terms. This dominance is highlighted while simultaneously minimizing the influence of matter density in the cosmic environment. In regions of high matter density, the scalar field’s negligible presence effectively conceals the gravitational modification effects, aligning with standard gravitational behavior. However, in areas of low matter density, far from massive objects, the scalar field experiences exponential growth. This growth contributes to the manifestation of dark energy, ultimately leading to the acceleration of the universe. These findings not only affirm the stability of the Chameleon mechanism in the Brans-Dicke model but also provide a deeper understanding of how this mechanism operates in different cosmic environments. The interplay between the scalar field, matter density, and the terms in the gravitational equations contributes valuable insights into the intricate dynamics of the universe on both large and small scales.





§ DECLARATIONS





  * Funding

This work was supported by Riset Unggulan ITB 2023

  * Competing interests 

The authors have no relevant financial or non-financial interests to disclose.

  * Author contribution

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Azwar Sutiono, Agus Suroso and Freddy Permana Zen. The first draft of the manuscript was written by Azwar Sutiono and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript




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