The loop equations for noncommutative geometries on quivers
Carlos Perez-Sanchez


§ ABSTRACT 
We define a path integral over Dirac operators
that averages over noncommutative geometries on a fixed graph,
as the title reveals, partially using quiver representations.
We prove algebraic relations that are satisfied by the
expectation value of the respective observables, computed  in terms of
integrals over unitary groups, with weights defined by the spectral action. These equations generalise the
Makeenko-Migdal equations—the constraints of lattice gauge theory—from lattices to arbitrary graphs.
As a perspective, these constraints are combined with
positivity conditions (on a matrix of parametrised by
composition of Wilson loops). A simple example
of this combination known as `bootstrap' is fully worked out.

RAG based Question-Answering for Contextual Response Prediction System
    Nian Yan
    5th September 2024
======================================================================





§ INTRODUCTION

Before discussing our problem in its due context,
we describe it aridly
and postpone its motivation for Section <ref>. For integers n, N > 1,
fix a polynomial
S∈_⟨ 2 n⟩=⟨ u_1, u_1^*, u_2, u_2^*,…, u_n,u_n^*⟩
in noncommutative u-variables satisfying u_ju_j^*=1=u_j^* u_j for j=1,…, n.
Consider a family of integrals of the type

    I_β=  ∫_(N)^nβ(U_1,U^*_1,…, U_n,U_n^*)
    ^N  S(U_1,U^*_1,…, U_n,U_n^*) U_1   U_2 ⋯ U_n,   β∈_⟨ 2 n ⟩,

with each factor U_i being the Haar measure on (N).
Assuming that S is real-valued over the whole integration domain,
 we derive the loop equations, that is to say,
algebraic relations among the integrals {I_β}_β∈ℐ parametrised by a certain family ℐ⊂_⟨ 2n  ⟩.
This type of integrals has been considered by physicists in the context of
lattice gauge field theory. In mathematics, integrals over the unitary group
are relevant in the context of
Weingarten-calculus <cit.>, developed mainly by
Collins and collaborators (e.g. <cit.>).



 §.§ Motivation: Random matrix theory and noncommutative geometry

Our interest in integrals of the type (<ref>) emerges from
Connes' noncommutative geometrical <cit.> approach to fundamental interactions, in which geometric notions are mainly governed by a
self-adjoint operator D named after Dirac. In this setting,
the physical action
S(D) is claimed to depend only
on (the spectrum of) D and is known as spectral action <cit.>.
Our problem is to evaluate the moments
that the spectral action yields via

    𝔼 [ h(D) ] = 1/𝒵∫_Dirac h(D) ^-S(D) D,     𝔼 [1 ] =1,   h(D)∈,

for an ensemble of  Dirac operators D (the normalisation condition defines 𝒵). Of course, this requires to have defined
the measure D over such ensemble of
Dirac operators D.
(The original problem,
as it appears in <cit.>,
considers also fermions, as it has  been recently
addressed in <cit.>, but which we do not include  yet.)
Part of the relatively vivid interest in
the problem (<ref>) during the last decade is
due to the reformulation   <cit.>
of fuzzy spaces[We do not
aim at a comprehensive review here, for fuzzy spaces see e.g. <cit.>
and the works of Rieffel
<cit.> (and references therein)
that address, from diverse mathematical angles, the rigorous convergence of matrix algebras
to the sphere. We are also not reviewing all the quantisation approaches either;
for a Batalin-Vilkovisky approach: cf
<cit.>
for Tate-Koszul resolutions applied to a model of 2× 2-matrices
and
<cit.> for the
homological-perturbative approach to
Dirac-operator valued integrals.
]
as finite-dimensional spectral triples. This  led to the application of
random matrix theory tools
<cit.> that followed to the first
numerical results <cit.>.
All these works deal with multimatrix interactions that include
a product of traces (as opposed to
the ordinary interactions that are single a traces of a noncommutative polynomial).

Independently,
  <cit.>
Taylor-expands the spectral action as a matrix model of the form





V(M) = ∑_l=1^∞∑_i_1,i_2,…,i_l F_i_1,i_2,…, i_l M_i_1,i_2
M_i_2,i_3⋯ M_i_l,i_1. This expansion was shown in <cit.>  to be
convergent
and, taking some elements of  <cit.> and with own techniques, to possess a neat reorganization in terms of universal Chern-Simons forms
and Yang-Mills forms integrated against (B,b)-cocycles that do depend on the geometry. The resulting model
V(M) above goes beyond the solved <cit.> generalisations
of the Kontsevich matrix model <cit.> (in which only the unitary invariance
of the quadratic term is broken) known as Grosse-Wulkenhaar model <cit.>.


These two independent approaches portend a symbiosis between
random matrix theory and noncommutative geometry.
Both the multiple trace interactions  and  the unitary broken interactions
could motivate (if they have not yet) new developments in random matrix theory.
And vice versa,  the
path-integral quantisation  (<ref>) of noncommutative
geometries  seems
hopeless without the intervention of random matrix theory[The  only
alternative known to the author is
the use of Choi-Effros operators systems <cit.> (cf. also <cit.>) that emerge
when one assumes (or rather, when one accepts) that only
a finite part of the Dirac spectrum is measurable. The price to pay is nonassociativity.
].


 §.§ Ensembles of unitary matrices in noncommutative geometry

The interaction between these two disciplines has taken place in hermitian grounds.
In this letter,
integrals over Dirac operators boil down to
ensembles of unitary matrices (they are also unitary-invariant,
like ordinary hermitian matrix ensembles, but
unitary ensembles integrate over unitary random matrices). These
can be considered as an approach to average over
 `noncommutative geometries
on a graph'.
When the graph is provided with
additional structure,
it might be grasped as a discretisation of space.
For instance, edges would carry a representation, and vertices
equivariant maps, in a
  spin network approach; here, we refrain from including information that
  associated to  gravitational degrees of freedom and
  address exclusively the  problem of gauge interactions.
  The background geometry is therefore fixed
  and the finiteness of the unitary groups appearing is not
  a shortage of the theory; as a caveat,
  they are not to be interpreted as a truncation of infinite-dimensional
  symmetries (but to be compared with the unitary structure group of
  Yang-Mills, for example).Representation theory does still play a role, but rather
in the context of quiver representations in  a certain category
that emerges from noncommutativity geometry as
exposed in <cit.> after the fundamental ideas of <cit.>. 

We can now restate the aim of this letter as follows:

Define a partition function
for noncommutative geometries on a graph—that is, define a measure
over all Dirac operators—and prove the algebraic relations that the respective
observables  shall satisfy.  Such quantities
have the form I_β as in eq. (<ref>) and are called  Wilson loops (although
not each I_β is a Wilson loop).

(Proper definitions follow in the main text.)
Such relations generalise the Makeenko-Migdal equations, the loop equations in
lattice gauge theory.  After
introducing the setting in
Section <ref>, we prove the main result
in Section <ref> and conclude with
a fully worked-out application
that mixes the loop equations with positivity conditions
of a certain matrix (`bootstrap') in Section <ref>.




§ QUIVER REPRESENTATIONS AND NONCOMMUTATIVE GEOMETRY
 
We call quiver Q  a  directed multigraph.
Since Q is directed, there are maps  s,t: Q_1 ⇉ Q_0
(from the  edge-set  Q_1 to the vertex-set Q_0) determining the vertex  s(e) at which
an edge e begins, and the one  t(e) where it ends.
Multiple edges e,e' ∈ Q_1 and self-loops o_v∈ Q_0 at a
certain vertex v∈ Q_0 are allowed, namely { s(e) , t(e) } = {s(e'),t(e')} as sets,
and s(o)=t(o) =v, respectively.  

One interprets a quiver Q as a category
whose objects are Q_0. The morphisms _Q(v, w)
are the paths from v to w, namely edge-sequences
γ=(e_1 e_2⋯ e_n) with e_1,…, e_n∈ Q_1  and  s(γ)=s(e_1)=v,
and t(γ)=t(e_n)=w as well as t(e_j)=s(e_j+1) for j=1,…, n-1.
We shall write γ:v → w if v=s(γ) and w=t(γ)
and call ℓ(γ)=n  the length of γ.
The path γ with reversed order is denoted by γ̅=(e_n e_n-1⋯ e_2e_1)
(not to be confused with the inverse morphism of γ).
Obviously, unless otherwise stated, paths are directed, but it will prove useful to
consider also paths in Γ Q,
the underlying graph of the quiver (Q with
forgotten orientations). If s(γ)=t(γ)
we call γ a loop. The space of loops[We comment for sake of completeness,
that the space of endomorphisms Ω_v(Q)=_Q(v,v) has as identity the constant zero-length path, which does play a role in the theory
of path algebras while constructing an equivalence between the category of representation
and modules of the path algebra <cit.>, but here we do not need this explicitly.] at v, is denoted here Ω_v(Q),
that is  Ω_v(Q)=_Q(v,v),
and   Ω Q will denote the space ∪_v ∈Q_0Ω_v(Q)
of all loops.
A quiver exists essentially to be  represented (otherwise one would say multidigraph)
in a category  𝒞.
A 𝒞-representation of Q is by definition a functor from Q
to 𝒞.









 §.§ The spectral triple associated to a quiver representation


We restrict the discussion to finite dimensions
and introduce the setting of <cit.>.
By definition, an object in  the category  of prespectral triples  is a pair (A,H) of a *-algebra A
faithfully *-represented,  λ: A ↷ H, in an inner product  -vector space H  (*-represented means here,
that λ(a^*) is the adjoint operator of λ(a) for all a∈ A).
A morphisms in  _ (  A_s,H_s   ;  A_t,H_t )
is a couple (ϕ,U) of an involutive algebra map ϕ: A_s  → A_t as well as
a unitary map U: H_s → H_t. As part of the definition,
a morphism should in addition satisfy U λ_s (a)
 U^* = λ_t [ ϕ (a) ] for all a∈ A. 







.74
In other words, a -representation of Q
associates with each vertex v of Q a prespectral triple (A_v,H_v) ∈ and
with any path γ: v→ w a morphism (ϕ_γ,γ ) :
(A_v,H_v)  → (A_w,H_w)
in such a way that if γ = (e_1⋯ e_2), then
γ= U_e_n⋯ U_e_1 and ϕ_γ= ϕ_e_n∘ϕ_e_n-1∘⋯∘ϕ_e_1,
where ϕ_e_j  : A_s(e_j)→ A_t(e_j) and U_e_j: H_s(e_j)→ H_t(e_j)
form a -morphism.
We refer to γ as the holonomy of γ.


 
.23 


    < g r a p h i c s >




If  two vertices are connected by a path γ, notice that
γ is a unitarity and H_s(γ) = H_t(γ).
If Q is connected there might be no (directed) path between two given vertices v and w;
it is however easy, if necessary by inverting some subpaths of a path γ̃ in Γ Q
that connects v with w, to establish the constancy of the map Q_0∋ v↦ H_v:=N;
we call such constant N= R,  the dimension of the representation R, somehow abusively. A spectral triple (A,H,D)
is a prespectral triple (A,H) together with a self-adjoint element D ∈(H), referred to as Dirac operator.
(This terminology comes from the hard fact that, if certain operators are added
to the [in that case, infinite-dimensional] spectral triple and if, together with D,
such operators, satisfy a meticulous list of axioms, then D is the spin geometry Dirac operator <cit.>; see also
<cit.> for an introduction geared to physicists).


As a side note, it is possible to compute the space of all
-representations of Q. It was proven in <cit.>
that such space—which in fact forms the category of representations—can be described in
terms of products of unitary groups subordinated to
combinatorial devices called Bratteli networks. This will not be recalled here,
since we will focus only on the spectral action.
Nevertheless, it is important to observe that, in stark contrast with ordinary 𝖵𝖾𝖼𝗍_-quiver representations, providing labels to the vertices is not enough to determine a -quiver representation.
The lifts of whole paths should exist, and this needs the compatibility of the maps ϕ_v
at all vertices v, which in turn is what the so-called
Bratteli networks  guarantee (concretely
*-algebra maps for M_m() → M_n() for m>n do not exist, and
if a representation yields A_s(e) = m and A_t(e) = n for some edge e,
a lift fails, cf. <cit.>).
Despite this, we denote representations of quivers as
R={ (A_v,H_v), (ϕ_e,U_e) } _v∈ Q_0,e∈ Q_1
instead of R={ (A_v,H_v), (ϕ_γ,γ) } _v∈ Q_0,γ∈Ω Q,
under the tacit assumption that lifts of whole paths exist.



We associate now a spectral triple to a given -representation
R={ (A_v,H_v), (ϕ_e,U_e) } _v∈ Q_0,e∈ Q_1 of a
 connected quiver Q.
We define the Dirac operator associated to R as the matrix
D_Q(R) ∈  M_# Q_0 ()   ⊗   M_N()
with matrix entries [D_Q(R)]_v,w∈ M_N() in the second factor given by

    [D_Q(R)]_v,w = (
    ∑_e ∈ s (v) ∩ t (w)
    U_e) +(
    ∑_e ∈ t (v) ∩ s (w)
    U_e^* )       (v,w∈ Q_0).

By construction, this operator is self-adjoint, and crucially for our purposes,
the next collection

    (A_Q(R), H_Q(R), D_Q(R)   ) =
     ( ⊕ _ v∈ Q_0 A_v, ⊕ _ v∈ Q_0 H_v, D_Q(R) )

is a spectral triple.












 §.§ The spectral action and the partition function


Given a polynomial f(x) = f_0 + f_1 x^1 +f_2 x^2 +…+ f_d x^d in real variables f_0,f_1,…, f_d ∈ℝ,
and a quiver representation, the spectral action on a quiver reads
S( D) = _H f(D), where we abbreviate D=D_Q(R).
It is possible to compute the spectral action as a loop expansion in terms of generalised
plaquettes γ as follows

    _H f(D)=   ∑_k=1^d f_k  ∑_v ∈ Q_0  γ∈Ω_v(Q) 
    ℓ(γ)  =k γ,

where  in the rhs is the trace of M_N() with 1= R=N.
The proof of eq. (<ref>) is given in <cit.>, but the reader will recognise this
formula as a noncommutative
generalisation of the following well-known fact in graph theory:
and  C_G denotes the adjacency matrix of a graph G,
the number of length-n paths in G between two of its vertices, i and j, is the entry
[C_G^n] _i,j of the matrix (C_G)^n.



Now we turn to the space of -representations of Q.
Let
A_v=⊕_j=1^l_vn_v, j
denote the algebra associated by R to the vertex v (so l_v is the number
of simple subalgebras of A_v).
Let r_v,j be the multiplicity of the action of the factor
n_v,j⊂ A_v on the Hilbert space H_v,
that is H_v= ⊕_j=1^l_v^ r_v,j⊗^n_v,j
where n_v,j only acts non-trivially on ^n_v,j via the fundamental
representation.
Once labels to the vertices are consistently[`Consistently'
is quantitatively described in terms of certain transition matrices in <cit.>, and qualitatively means that the labels
are such that involutive maps A_s(e)→ A_t(e) always exist for each e∈ Q_1, and
several dimension-matching checks. The reader will note that we do not
include the minimal amount of information in each group at the edges.
The origin of the projective groups (n) is that
 (n) acts via the adjoint action.
] assigned,
the possible labels of an edge e are parametrised by
 ∏_j=1,…, l_t(e)(n_t(e),j)
 <cit.>,
 and at each vertex it holds ∑_j   r_v,j×  n_v,j = N,
since each  vector space H_v is isomorphic to ^N.
This in turn allows us to embed
each (n_t(e),j) as a subgroup of the unitarities
(N) of H_t(e) and motivates the following measure.






Given an N-dimensional representation R={ (A_v,H_v), (ϕ_e,U_e) } _v∈ Q_0,e∈ Q_1
of a connected quiver Q, we define the Dirac operator measure on Q
as

    D  := ∏_(v,w) ∈ Q_0× Q_0  [D_Q(R)]_v,w,    where    [ D_Q(R)]_v,w  := ∏ _ e ∈ s (v) ∩ t (w) ∏_j=1^l_t(e) u_e,j,

being u_e,j the Haar measure on  ( n_t(e),j ),
where u_e,j sits in the matrix U_e associated to e by R  in the respective block-diagonal entry in
U_e = (
        1_r_t(e),1 ⊗ u_e,1
,      1_r_t(e),2⊗ u_e,2 ,⋯,
      1_r_ t(e) ,l_t(e)⊗ u_e, l_t(e) ).


   Therefore D
is the product Haar measure on
∏_e ∈ Q_1 ∏_j=1^l_t(e) ( n_t(e),j)
seen as subgroup of (N)^# Q_1.





The partition function of a given -representation R of a quiver Q

reads

    𝒵_Q,R  = ∫^-  S( D ) D,     
    N=  R  and  D=D_Q(R).

For any  β∈Ω Q,
a Wilson loops[We refer both to β and to 𝔼 [ β]
ambiguously as Wilson loops.] is by definition

    𝔼 [  ( β) ]:=
    1/𝒵_Q,R∫ (β ) ^- N S(D) D.




For fixed N, the partition function
𝒵 _Q= ∑_R -rep of Q^ R=N𝒵_Q,R
 seems to be a more interesting
quantity, or even more so the sum over
a class of quivers Q encoding different
background geometries, 𝒵=∑_Q ∑_R -rep of Q^ R=N𝒵_Q,R. For the moment we content ourselves with the partition function (<ref>)
for a fixed representation R and a fixed quiver Q.





§ THE MAKEENKO-MIGDAL LOOP EQUATIONS FOR THE SPECTRAL ACTION
 



 §.§ Notation
 We now derive the constraints on the set of  Wilson loops.
We root an edge ∈ Q_1 with s() ≠ t(), and consider its associated
unitarity U_ as given by a fixed -representation R={ (A_v,H_v) ,(ϕ_e, U_e)}_v,e
of Q. Now consider the infinitesimal variation of the spectral action
by the change of variable exclusively
for the unitarity U_
at the edge  as follows. Let

    U_↦ U_'=  ^ Y U_ ,       Y ∈𝔰𝔲(N) ,    =√(-1),

where Y is given in terms of arbitrary matrices y_k∈𝔰𝔲(n_t(e),k) for k=1,2,…,
l_t()=: L by

    ^ Y := [ 1_r_t(e),1 ⊗exp( y_1  ),
     1_r_t(e),2 ⊗exp( y_2 ), …,
     1_r_t(e),  L   ⊗exp ( y_ L  )   ] .

One should keep in mind that this implies also the substitution
 U_ ^*
 ↦ (U_')^*
 = U_^* ^- Y, as it follows from the change (<ref>).
 This rule defines a new representation R' differing from R
 only by the value of the unitarities at the edge , that is
 R'= {(A_v,H_v) ,(ϕ_e, exp( δ_e,e_0 Y ) U_e}_v ∈ Q_0,e ∈ Q_1,
 where δ_e,e' is the indicator function on
 the edge-set. 

 Assume that along a given path γ
 the combinations e e̅ and e̅ e are absent
for each edge e ∈γ. We call this type of paths reduced (Fig. <ref>)
and it is trivial to see that reduction of a path (i.e. removing those pairs) yields a new one
with unaltered holonomy.
Consider then a reduced loop γ that appears in the spectral action and
contains the rooted edge .
This assumption allows (w.l.o.g. due to cyclic reordering) the decomposition

 

    γ = ^ϵ_1α_1 ^ϵ_2α_2 ⋯^ϵ_mα_m

 (cf. Fig. <ref>) where each
 of ϵ_1,ϵ_2,…,ϵ_n ∈{+1,-1} is  a sign.
 This convention means that
 ^ϵ= is the edge
   backwards if ϵ=-1, while of course  ^ϵ is  itself if ϵ=1.
  (The condition that γ starts with  implies ϵ_1=1 above, but leaving
  this implicit is convenient.)
  By asking that each subpath α_1,…, α_m ⊂γ does not
  contain neither  nor , one
  uniquely determines them.
  For another loop β, which also starts with ,
  under the same assumption that  and  do not appear consecutively
  in β in any order,
  a similar decomposition holds

    β   = ^σ_1μ_1 ^σ_2μ_2 ⋯^σ_pμ_p


in terms of signs σ_j ∈{-1,+1}
and paths μ_j not containing neither the rooted edge 
nor . The only difference in notation —which we will keep throughout—
is that γ will refer to generalised plaquettes (i.e. contribution to the spectral action)
while β will be the path of a Wilson loop.







Take again the polynomial f(x)=f_0+ f_1 x + f_2 x^2 + … + f_d x^d,
and rephrase the spectral action of eq. (<ref>) as

    S(D_Q)=  f(D_Q) =  γ∈Ω Q 
    γ reduced   g_γγ.

Now g_γ is a function of f_ℓ(γ) but possibly also
of f_ℓ(γ) + 2 , f_ℓ(γ) + 4 ,…, whenever these last
coefficients are non-zero. The reason for the contribution
of the higher coefficients is the possible appearance of a contiguous pair of edges
e, e̅, for which the respective unitarities will satisfy U_e U^*_e=1 =U_e^* U_e.
These cancellations are not detected
by the holonomy, which is the criterion used in (<ref>) to collect all terms
(instead of using, as in eq. (<ref>), the f_0,…, f_d coefficients and
performing directly the sum over paths).
For instance, if γ is the path in
Fig. <ref> (b),
then g_γ depends on f_ℓ(γ) and f_ℓ(γ)+8, since
 Fig. <ref> (a) contributes the same to the spectral action.
The function g_γ=g_γ(f_0,f_1,…, f_d)
 is of course quiver-dependent.



 §.§ Main statement

The  Makeenko-Migdal or loop equations
we are about to generalise appeared first in lattice quantum chromodynamics
<cit.>. They have been a fundamental ingredient in
  the construction of
Yang-Mills theory in two dimensions <cit.> in rigorous probabilistic terms.




 Let R be a representation of a connected quiver Q and let N= R.
Root an edge  of Q and abbreviate by
  U=U_ the unitarity  that R determines for . Then
for any reduced loop β
  the following relation among Wilson loops holds : 
    𝔼[
    j =1  
    σ_j =+1  ^p  1/N  ( U^σ_1μ_1 ⋯  U^σ_j-1μ_j-1 )   1/N   (U^σ_j μ_j⋯ U^σ_pμ_p )  
          -
     j =1 
    σ_j =-1 ^p   1/N (  μ_1 U^σ_2μ_2⋯  U^σ_j-1μ_j-1        ) 1/N  (μ_j U^σ_j+1⋯ U^σ_pμ_p )
    ]  
        = γ∈ S(D) 
    γ reduced  g_γ𝔼[
    j =1  
    ϵ_j =+1  ^m
    1/N (
     β· U^ϵ_jα_j   ⋯α_m U^ϵ_1α_1 ⋯ U^ϵ_j-1α_j-1 )
     
                   -
    j =1  
    ϵ_j =-1   ^m 1/N ( β·α_j  U^ϵ_j+1⋯α_m U^ϵ_1α_1 ⋯α_j-1   U^ϵ_j )  ] .



Notice that the second line takes the expectation value of
1N  ( U^σ_1μ_1 ⋯  U^σ_j-1μ_j-1     U^σ_j    ) ×1/N  (μ_j ⋯ U^σ_pμ_p ),
but σ_j being -1 allows for a cancellation, hence the
apparent lack of harmony between the two first lines of the lhs.
We also stress that the first term in the lhs, which corresponds to j=1=σ_1,
yields the input Wilson loop β in the first trace and a constant path in the second;
the latter yields a factor of N, which is cancelled by its prefactor.

The loop or Dyson-Schwinger or Makeenko-Migdal equations follow from the identity

    ∫∑_a,b=1^N(∂_Y)_a,b{ (_R' )β_b,a×^- N S ( D'  ) } D =0     D'=D_Q(R')

(The entries of the matrix derivative are (∂_Y)_a,b  = ∂/∂ Y_b,a.) This identity
follows from the invariance of the Haar measure  at the rooted edge
 under the transformation
(<ref>), implying
D'= D. Below, we show that this implies

    𝔼[  ( 1/N⊗1/N) (∂_Y  β ) ]
    =𝔼[ 1/N (∂_Y S β ) ],

and compute each quantity inside the trace(s). On the lhs, the matrix derivative is then the Rota-Stein-Turnbull noncommutative derivation

    ∂_Y  Y^k+1  =∑_l=0^k    Y^l ⊗ Y^k-l     (Y^*=Y ∈ N, k=_≥ 0)

while on the rhs, the matrix yields Voiculescu's cyclic derivation, since the quantity
it derives, S, contains a trace. Recall that  is multiplicative,
 one has
γ = U ^ϵ_1α_1
 U^ϵ_2α_2 U^ϵ_3⋯α_m-1
  U ^ϵ_mα_m.
With respect to the transformed representation R'
we can compute the holonomy _R'δ of any path δ. This depends on Y and  but we use a prime
in favor of a light notation and write only ' δ.
Since none of the subpaths α_j contains
the transformed edges  and , one has
' α_j = α_j, so
 
    ' (γ)
     =U ^'ϵ_1α_1
     U^'ϵ_2α_2 U^'ϵ_3⋯
      U ^'ϵ_mα_m

 where U^'ϵ is ^ Y U if ϵ=1
 and U^*^- Y if ϵ=-1.
Therefore the variation of the loop γ writes

    [∂_Y ' γ ]  |_Y=0    =
     j =1 
    ϵ_j = +1  ^m
    U  α_j U^ϵ_j+1α_j+1⋯
    U^ϵ_nα_n
     U^ϵ_1α_1 ⋯
    U^ϵ_j-1α_j-1
       
    -
    j =1 
    ϵ_j = -1 ^m
     α_j U^ϵ_j+1α_j+1⋯
    U^ϵ_nα_n
     U^ϵ_1α_1 ⋯
    U^ϵ_j-1α_j-1 U^* .

The cyclic wandering of any fix holonomy, say α_1, in the rhs of
the main result is due to Voiculescu's cyclic derivation. We now compute the variation of the Wilson line β, whose
holonomy writes for the representation R' as
' β =∏_j=1^p (U')^σ_j' μ_j
=
∏_j=1^p (U')^σ_jμ_j.
 To take the variation observe that 'β is not inside a trace.
For any matrices A,B∈ N, and a,b,c,d=1,…, N,

    (∂_Y)_b,a
    [ A exp(    Y ) B ]_c,d|_Y=0    = ∑_k=0^∞^k/k!∑_l=0^k   A_c,r  [ Y^l ⊗ Y^k-1-l] _b,s | a,r |_Y=0  B_s,d
       = A_c,rδ_b,sδ_a,r
    B_s,d.

Using this rule for the previous expression of  ' β, one obtains each summand for each occurrence of U^±1
and the result follows after equating the indices c=a, and b=d, which
is the initial situation in the initial identity (<ref>).
























Suppose that the
plaquettes in the action S(D) = ∑_γ^ reducedg̃_γ [ γ + γ̅], intersect each either  and  exactly
once. Notice that  this time we have rewritten it
as sum over pairs γ and γ̅ (which is always possible
since the paths are in Γ Q and the spectral action is real valued). Then
 
    𝔼[
    j =1  
    σ_j =+1  ^p 1/N
    ( U  μ_1 ⋯  U^σ_j-1μ_j-1 ) 1/N   (U^σ_j μ_j ⋯ U^σ_pμ_p )  
         -
     j =1 
    σ_j =-1 ^p 1/N (  μ_1 ⋯  U^σ_j-1μ_j-1        )  1/N  (μ_j ⋯ U^σ_pμ_p )  ]  
        = γ∈ S(D) 
    γ=(U,α)  reduced  g̃_γ𝔼[
     (
     β· U  α  )
     -
      (
     βα̅· U^*   ) ] .




 §.§ Graphical representation of the Makeenko-Migdal equations

We illustrate graphically the meaning of the Makeenko-Migdal equations.
Let us place  and the reversed edge  along a fixed axis of the picture.
To represent a  Wilson loop β or a reduced
generalised plaquette γ, we choose the following notation.
In order to avoid drawings with several intersections,
for each time that γ or β walks along either  or ,
we jump to the next `plane' in anti-clockwise direction around the fixed axis.
Thus each of these planes
represents abstractly the subpath μ_j ⊂β
or α_j⊂γ according to the decompositions
 (<ref>)
and  (<ref>), that is:

    < g r a p h i c s >

We kept a rectangular appearance for sake of visual simplicity,
but the subpaths μ_j and α_j are arbitrary (as far as
they have positive length).
In fact, the depicted situation is still to some extent oversimplified, since the theorem
describes the more general case that μ_j or α_j
might be loops themselves  (as α_3 in Fig. <ref>),
but this would render the pictures unreadable. The
representation of the Makeenko-Migdal equations reads then as follows:


    1/N^2×(-.5
        < g r a p h i c s >
    )  =  ∑_γ  g_γ(
    -.5
        < g r a p h i c s >
    )

In the rhs, the very similar terms need a word of notation.
The blue arrow is executed right after the green part of the path,
while the red arrow follows only after the purple ones.



§ APPLICATIONS AND OUTLOOK
 
This last section has as  aim to illustrate the power
of the equations derived here when combined with the positivity conditions.
This combination, sometimes known as `bootstrap',
 appeared in <cit.> for
lattice gauge theory
and <cit.> in a string context (for hermitian multimatrix models).





 §.§ Positivity constraints

Let v∈ Q_0 be fixed for this subsection and fix a representation R of Q
of dimension N.
Consider a complex variable z_β for each
loop β based at a fixed vertex v ∈ Q_0, z={ z_β :  β∈Ω_v( Q) },
as well as the matrix

    P( z ) :=  β∈Ω_v(Q) z_ββ,      P( z ) ∈ N.

It follows that [ P(z)  P(z)^* ] =
∑_β, α z_β z^*_αβ·  (α)^*
=
∑_β, α∈Ω_v(Q) 
z_β z^*_α( βα̅) ≥ 0
independently of the z-tuple; this is preserved by
expectation values, i.e.

    ∑_β, α∈Ω_v(Q) 
    z_β z^*_α𝔼[ ( βα̅) ]  ≥ 0,     for all  z ∈^Ω_v(Q),

which is an equivalent way to state the positivity ℳ≽  0 of the matrix
ℳ∈ [[N,f_0,f_1,…,f_d ]]
whose entries are given by

    (ℳ )_i,j := 𝔼[ ( β_i  β̅_j ) ]

for any ordering of the loops {β_1,β_2,…} at the fixed vertex v.
The positivity of ℳ is clearly independent of the way we order these loops,
as a conjugation by a permutation matrix (which is a unitary transformation) will not change the eigenvalues of ℳ.  





The choice for the matrix (<ref>) is originally from <cit.>,
who pushed forward the bootstrap for lattice Yang-Mills theory.
The techniques of <cit.> were implemented for fuzzy spectral triples
for an interesting kind of hermitian matrix <cit.>
and  a hermitian 2-matrix model <cit.>.
The loop equations of <cit.> have been extended here to include
arbitrary plaquettes that go along the rooted edge more than once, and Wilson loops
that are allowed to do the same. 



Consider the triangle quiver Q=-1ex
    < g r a p h i c s >
with a rooted edge ,
and let ζ = μ be the only loop of length 3 starting
with  (μ is the path v_2→ v_3→ v_1, of course).  Consider the action S(D) =  f(D) for
f(t)=f_0 + f _1  t+f_2 t^2+ f_3 t^3. This implies

    S(D) =(f_0+2 f_2)N +
    x [
    ζ + ζ ]

where we set x =3 f_3. Since the terms in the even coefficients are just constants
that when evaluating the Wilson loops expectation values, they
disappear when dividing by the partition function
and play no role in integration we set f_0=f_2=0. Now pick a  loop β= ζ^n for positive
n∈. According to the loop equations (<ref>), one has

    𝔼[ ∑_k=0^n-1  (1/N⊗1/N) (ζ^k ⊗ζ^n-k)]
    = x/N (𝔼ζ^n+1 -
    𝔼ζ^n-1 ) .

Defining the large-N moments by m_j:= lim_N→∞𝔼 [  1Nζ^j] for each j∈,
this means

    ∑_l=0^n-1
    m_l  · m_n-l
    =
    x (m_n+1
    -m_n-1  ) ,      (N→∞)

since large-N factorisation holds, N^-2𝔼[ ζ^i ζ^j]  →
m_i · m_j, as N→∞.
For the loop β= ζ^-n
with n∈, one has

    -∑_j=0^n-1
    m_-(n-j)· m_-j
    =x (  m_-(n-1)  -  m_-(n+1)) ,

Finally,  going through the derivation of the MM
equations for the constant Wilson loop, one obtains the vanishing of the lhs, so 0= x (m_1 - m_-1),
hence m̅_1 = 𝔼 [ ζ ]  =
  𝔼 [ ζ̅] 
= 𝔼 [ ζ ] =m_-1 = m_1, so m_1 is real (this can be derived
by other means, but the loop equations  yield this explicitly).  Together
with the last equation this implies m_-j = m_j for all j=1,2,… and the
moments can be arranged in the following (Toeplitz) matrix.

    ℳ =
    [    1  m_1  m_2  m_3    …     ; m_-1    1  m_1  m_2    …     ; m_-2 m_-1    1  m_1    …     ; m_-3 m_-2 m_-1    1    …     ;    ⋮    ⋮    ⋮    ⋮    ⋱ ]=
    [    1  m_1  m_2  m_3    …     ;  m_1    1  m_1  m_2    …     ;  m_2 m_ 1    1  m_1    …     ;  m_3  m_2 m_ 1    1    …     ;    ⋮    ⋮    ⋮    ⋮    ⋱ ]

Thanks to Theorem <ref>, ℳ can be computed recursively in terms of y:=m_1 and
the coupling x,

    m_1     = y           m_4    = 4  y/x + 3  y^2/x^2 + 1/x^2 + y/x^3 + 1 
     m_2      = y/x + 1
              
      m_5     = y + 3  y^2/x + 2  y^3/x^2 + 3/x + 9  y/x^2 + 6  y^2/x^3 + 1/x^3 + y/x^4
     m_3    = y + y^2/x + 1/x + y/x^2          m_6     =9  y/x + 18  y^2/x^2 + 10  y^3/x^3 + 6/x^2 + 16  y/x^3 + 10  y^2/x^4 + 1/x^4 + y/x^5 + 1 .



The positivity condition
ℳ(x,y) ≽ 0
can be plotted on the first-moment–coupling plane
in terms of the simultaneous positivity of the minors.
The observed situation for a large class of hermitian matrix integrals
are tight constraints for the first moment
in terms of the coupling, which, by increasing
the size of the minors,  typically
determine a curve y=y(x)—and by the respective loop equations, all
the moments and thus the solution of the model.
In this letter we do not claim convergence, but
the results of this academic example encourage us to
explore this combination in future works, including also a hermitian
scalar (`Higgs') field
that self-loops yield.











§ ACKNOWLEDGEMENTS

This work was mainly
 supported by the European Research Council (ERC) under the European
 Union’s Horizon 2020 research and innovation program (grant agreement
 No818066) and also by the Deutsche Forschungsgemeinschaft (DFG,
 German Research Foundation) under Germany’s Excellence Strategy
 EXC-2181/1-390900948 (the Heidelberg Structures Cluster of
 Excellence).  I thank the
 Erwin Schrödinger International Institute for Mathematics and Physics
 (ESI) Vienna, where this article was finished, for optimal working conditions
 and hospitality.
I acknowledge the kind answers
by the group of Masoud Khalkhali at Western U.,  specially by  Nathan Pagliaroli,
on a question about bootstrapping.


alpha


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