The Axion is Going Dark
Markus Dierigl Dušan Novičić


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The Axion is Going Dark 

 

Markus Dierigl^1, Dušan Novičić^2



 
 
^1 Arnold-Sommerfeld-Center for Theoretical Physics,

Ludwig-Maximilians-Universität, 80333 München, Germany
 

^2 Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 
 85748 Garching bei München, Germany










In this work we explore the effect of a non-Abelian dark sector gauge group that couples to the axion field. In particular, we analyze effects that arise if the dark sector gauge group mixes topologically with the Standard Model. This is achieved by gauging a subgroup of the global center 1-form symmetries which embeds non-trivially in both the visible as well as the dark sector gauge groups. Leaving the local dynamics unchanged, this effect modifies the quantization conditions for the topological couplings of the axion, which enter the estimate of a lower bound on the axion-photon coupling. In the presence of a dark sector this lower bound can be reduced significantly, which might open up interesting new parameter regions for axion physics. We further determine the allowed exotic matter representations in the presence of topological mixing with a dark sector, explore the generalized categorical symmetries of such axion theories, and comment on other model-independent phenomenological consequences.


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§ INTRODUCTION


Axions are one of the most promising candidates for particles beyond the Standard Model. They can provide a dynamical solution for the strong CP problem <cit.> and generically contribute to the dark matter density in the universe <cit.>, see also <cit.>. Axions further have a very rich structure of symmetries, including various generalized symmetries, see <cit.> for reviews and in particular <cit.>. These include the mixture of various higher-form symmetries into higher groups <cit.> as well as non-invertible symmetries <cit.>, making axions the ideal laboratory for studying these structures and their physical implications.

Since axions couple to the gauge dynamics only topologically via the instanton density they are sensitive to subtle differences in the realization of the gauge group. Because of this, they are able to go beyond the implications of the gauge algebra which dictates the local interactions. This is important even for the Standard Model for which the gauge algebra is known to be composed of strong, weak, and hypercharge interactions:

    𝔤_SM = 𝔰𝔲(3) ⊕𝔰𝔲(2) ⊕𝔲(1)  ,

but the actual gauge group is not fixed. The allowed realizations of the Standard Model gauge group can be parameterized by 

    G_SM = SU(3) ×SU(2) ×U(1)/ℤ_k ,   with  k ∈{1 , 2 , 3 , 6 } ,

see, e.g., <cit.>. The value of k leads to quantization conditions for the axion coupling that affects its coupling to the electro-magnetic field after electro-weak symmetry breaking <cit.>. This quantization leads to a lower bound on the axion-photon coupling which enters various experimental searches and is therefore of great importance to define the experimentally interesting parameter space, see, e.g., <cit.>.

In this work we will demonstrate that the presence of a non-Abelian dark sector can drastically modify the conclusions about a lower bound on the axion-photon coupling. This happens in cases where it mixes topologically with the Standard Model fields, i.e., the total gauge group is given by[G_d denotes the simply connected group associated to the algebra 𝔤_d, e.g., SU(N) for 𝔰𝔲(N).]:

    G = SU(3) ×SU(2) ×U(1) ×G_d/ℤ_k .

For non-trivial realizations of G the effective coupling of the axion to the electro-magnetic field g_a γγ will generically be reduced compared to its Standard Model value, 

    g^G_a γγ≤ g^G_SM_a γγ .

In certain cases this reduction is significant and might open up new parameter regimes. For example, we find that in the case of a dark sector with 𝔰𝔲(3) algebra and ℤ_6 quotient the minimal value of the axion photon coupling differs by an order of magnitude in the presence of topological mixing. Specifically,

    g_a γγ^min,SM≃α/2 π f 0.72(4) ≫ g_a γγ^min,d≃α/2 π f( 0.07(6) + ρ)  ,

where we expect ρ, modifying the kinetic mixing of axions with pions in the dark sector, to be sub-leading.

The topological mixing in the presence of a dark gauge sector further has implications on the allowed representations of exotic matter fields, which we determine for general dark sector gauge group. We further consider scenarios in which the dark sector is confining or broken, which leads to different restrictions. This influences other phenomenological applications of the axion field as well and we comment on the corresponding possible effects. These effects can also be constrained by the categorical symmetries of the system which we explore systematically in the presence of a dark sector including anomaly inflow onto the defects of the axion field given by axion domain walls and axion strings. This has the potential to open up interesting possibilities and new corners of the parameter space.

Restricting to a single non-Abelian dark sector gauge group and a single axion field we will quantify the consistency conditions and restrictions depending on the choices of the dark sector gauge algebra 𝔤_d and the quotient ℤ_k with a focus on universal, model-independent properties. For that we use that the quotient by ℤ_k in (<ref>) can be understood as gauging part of the global center 1-form symmetries of the theory <cit.>, which allows for more general gauge backgrounds carrying fractional instanton number <cit.>, see also <cit.>. The periodicity of the axion field leads to consistency conditions for these fractional instanton backgrounds which can be phrased in terms of a quantization condition on the topological coupling constants <cit.>. Fractional instanton configurations in the dark sector gauge group need to be included in the evaluation of these consistency conditions which modifies the result compared to those obtained from the Standard Model alone.

The rest of the manuscript is organized as follows. In Section <ref> we explain the notion of topological mixing, its effect on allowed matter representation, as well as its formulation in terms of the gauging of a subgroup of the center 1-form symmetries. We further explain how this gauging allows for gauge configurations with fractional instanton numbers. In Section <ref> we determine how the realization of the total gauge group leads to quantization of the topological axion coupling and explore the influence of the presence of a dark sector gauge group on the minimal value of the axion photon coupling. The generalized categorical symmetries and defects of the axion system are discussed in Section <ref> with a focus on the inclusion of a dark sector. In Section <ref> phenomenological consequences of coupling axions to the dark sector are discussed. Finally, we point out possible generalizations and conclude in Section <ref>. In Appendix <ref> we discuss the fractional instanton number in cases in which one only mods out a subgroup of the center symmetry.



§ TOPOLOGICAL MIXING



In this section we discuss the consequences of the topological mixing of gauge groups. For this we note that being given the gauge algebra of a theory does not specify the gauge group, which requires more detailed information, for example the allowed representations or the spectrum of line operators <cit.>. For a gauge algebra 𝔤 we denote by G the associated maximal form of the gauge group. For a non-Abelian Lie algebra this is given by the simply-connected realization whereas for U(1) this fixes the charge quantization, see Table (<ref>). The group G has the maximal center subgroup 𝒵_G, i.e., the subgroup commuting with all group elements. 

As an example consider the special unitary group SU(3), whose center elements form the discrete subgroup ℤ_3 and are represented by elements of the form

    [ e^2 π i / 3           0           0;           0   e^2 π i/3           0;           0           0   e^2 π i/3 ] = e^2 π i /3 1∈SU(3)  .

Being diagonal matrices, these elements clearly commute with all the SU(3) group elements.

For the groups relevant to us in later sections the Lie algebra 𝔤, the associated maximal Lie group G, and its center subgroup are given by[Our convention for labeling the symplectic groups is such that 𝔰𝔭(1) ≃𝔰𝔲(2).]

    [ Lie algebra 𝔤   Lie group G    center 𝒵_G;         𝔰𝔲(n)         SU(n)           ℤ_n;      𝔰𝔬(2n+1)    Spin(2n+1)           ℤ_2;        𝔰𝔬(4n)     Spin (4n)      ℤ_2 ×ℤ_2;      𝔰𝔬(4n+2) Spin (4n + 2)           ℤ_4;         𝔰𝔭(n)         Sp(n)           ℤ_2;           𝔢_6           E_6           ℤ_3;           𝔢_7           E_7           ℤ_2;             ℝ          U(1)          U(1);               ]

The remaining Lie groups E_8, G_2, and F_4 all have trivial center and will not be relevant for our discussion.

In general, a gauge group associated to a single Lie algebra summand 𝔤 takes the form

    G = G/𝒵 ,   with 𝒵⊂𝒵_G ,

where we divide by 𝒵, a subgroup of the center. Whenever the gauge algebra contains more than one summand, the quotient by 𝒵 can affect each gauge group factor simultaneously. We will refer to this phenomenon as topological mixing, since the local dynamics of a theory with such a gauge group remains unchanged. For several gauge group factors G_i this can be written as

    G = ∏_iG_i/𝒵 .

To specify the resulting gauge group G one further needs to fix the group homomorphisms from 𝒵 to each 𝒵_G_i. These can be inferred from the image of a set of generators of 𝒵, which also includes the possibility of 𝒵 having more than one factor. This data can be captured by an integer ℓ_i, which specifies the maps

    𝒵⊃ℤ_n →ℤ_m_i⊂𝒵_G_i ,

by the image of the generator

    1  mod  n ↦ℓ_i  mod  m_i  .

The subgroup in the image of ℤ_n is given by ℤ_m_i/gcd(ℓ_i, m_i), where gcd denotes the greatest common divisor. Moreover, for this to be well-defined the generated subgroup cannot be bigger than ℤ_n, which restricts the allowed values of ℓ_i and imposes that n is mapped to a multiple of m_i. In particular, it requires that

    ℓ_i   n/m_i∈ℤ ,

where we regard the ℓ_i parameter mod m_i, such that the identity element in ℤ_n is mapped to the identity element in ℤ_m. Note, that since for 𝔤 = 𝔰𝔬(4n) the center is ℤ_2 ×ℤ_2 one needs to specify maps into both ℤ_2 factors. If G_i = U(1) we can specify the group homomorphism 

    𝒵⊃ℤ_n →U(1) = ℤ_G_i ,

by the map

    1  mod  n ↦ e^2 π i ℓ_i / n∈U(1)  ,

which produces a ℤ_n/gcd(ℓ_i,n) subgroup of U(1) = 𝒵_G_i.

Let us illustrate this in more detail for four different examples:


    
  * 𝒵 = ℤ_6 and G_i = SU(3): The homomorphisms are specified by the maps of the generator of ℤ_6 into 𝒵_G_i = ℤ_3
    
    1  mod  6 ↦ℓ_i  mod  3  .

     In both non-trivial cases the generated subgroup of 𝒵_G_i is the full ℤ_3, but the specific homomorphism differs, captured by ℓ_i ∈{ 1, 2}.
    
  * 𝒵 = ℤ_6 and G_i = SU(12): With the group homomorphism into 𝒵_G_i = ℤ_12 specified by
    
    1  mod  6 ↦ℓ_i  mod  12  .

    For this to be a group homomorphism ℓ_i needs to be even. And we find the generated subgroup and remaining parameters of the non-trivial homomorphism 
    
    [                      ℓ_i         m_i/gcd(ℓ_i,m_i) ℤ_m_i/gcd(ℓ_i,m_i)⊂𝒵_G_i;                        2                        6                      ℤ_6;                        4                        3                      ℤ_3;                        6                        2                      ℤ_2;                        8                        3                      ℤ_3;                       10                        6                      ℤ_6;                          ]

    We see that there are many different possibilities parametrized by the generated subgroup in 𝒵_G_i.
    
  * 𝒵 = ℤ_2 and G_i = Spin(8): For this we need to specify two maps, since 𝒵_G_i = ℤ_2 ×ℤ_2 and we find
    
    1  mod  2 ↦ (ℓ^1_i  mod  2  , ℓ^2_i  mod  2)  ,

    allowing for three different non-trivial embeddings, see also Appendix <ref>.
    
  * 𝒵 = ℤ_6 and G_i = U(1): The map is specified by
    
    1  mod  6 ↦ e^2 π i ℓ_i / 6 ,

    which as for SU(12) leads to various subgroups of U(1) and group homomorphisms. To be specific, one finds ℤ_6 for ℓ_i ∈{1  ,5 }, ℤ_3 for ℓ_i ∈{ 2 , 4 }, and ℤ_2 for ℓ_i = 3.

For the quotient in (<ref>) to lead to topological mixing we demand that 𝒵 embeds non-trivially in a product of at least two different gauge group factors G_i. This quotient by part of the center symmetry has various consequences:

    
  * Restriction of the allowed representations
    
  * More general gauge theory backgrounds
    
  * Fractional instantons

We will discuss these in the following subsections and in particular choose 

    ∏_i G_i ⊃G_SM = SU(3) ×SU (2) ×U(1)  ,

the form of the Standard Model gauge group with the largest center symmetry.

First, let us recall that even without additional gauge factors in (<ref>) there can be topological mixing, parametrized by the different forms of the Standard Model gauge group G_SM in (<ref>). With the center of G_SM given by

    𝒵_G_SM = ℤ_3 ×ℤ_2 ×U(1)  .

The specific embedding of the possible discrete quotients ℤ_k with k ∈{1  , 2  , 3  , 6 }, where ℤ_1 is the trivial group, is given by

    ℤ_3:      1  mod  3 ↦(1  mod  3  , 0  mod  2  , e^2 π i 2/3) ∈𝒵_G_SM ,    (ℓ_3  , ℓ_2  , ℓ_1) = (1  , 0  , 2)  , 
    ℤ_2:      1  mod  2 ↦(0  mod  3  , 1  mod  2  , e^2 π i 1/2) ∈𝒵_G_SM ,    (ℓ_3  , ℓ_2  , ℓ_1) = (0  , 1  , 1)  ,

Here, ℓ_3, ℓ_2, and ℓ_1 specify the maps to 𝒵_SU(3), 𝒵_SU(2), and U(1) of 𝒵_G_SM, respectively. The transformation under ℤ_6 can be deduced by the fact that ℤ_6 = ℤ_3 ×ℤ_2 and is generated by, e.g.,

    ℤ_6:    1  mod  6 ↦(1  mod  3  , 1  mod  2  , e^2 π i 1/6) ∈𝒵_G_SM ,    (ℓ_3  , ℓ_2  , ℓ_1) = (1  , 1  , 1)  .

In the following we further include a single additional non-Abelian gauge group factor G_d and interpret it in terms of a dark sector gauge group. Here, we focus on a single non-Abelian dark sector in order to avoid the strong additional constraints for dark U(1) gauge fields imposed by kinetic mixing effects, see, e.g., <cit.> and <cit.>, and not to clutter notation by multiple additional gauge factors. The total gauge group is then given by

    G = G_SM×G_d/𝒵 ,

where 𝒵 embeds non-trivially in both factors.

We will see that the presence of topological mixing with the dark sector has several important consequences for exotic matter representations as well as in the presence of an axion field. To allow for a sensible phenomenology we further assume a mass gap for the dark sector gauge fields. In particular we will distinguish between the following two scenarios:



  §.§.§ Confining dark sector


In this scenario the non-Abelian dark sector gauge group is unbroken at low energies and a mass gap is generated dynamically via confinement. We leave the associated energy scale Λ_d unspecified, since it depends on the details of the model, but discuss several interesting consequences below. In order not to alter the interactions of Standard Model particles they need to be singlets under the the dark sector gauge group. This constrains the allowed quotients 𝒵 and can be realized by demanding that 𝒵 embeds into 𝒵_G_SM as described in (<ref>) or (<ref>). Depending on the scale Λ_d there can be important modifications to the axion potential induced by the strong coupling dynamics in the dark sector. 

Note that such models also lead to interesting dark matter candidates in terms of dark baryons and glueballs, see, e.g., <cit.>, and are very common in string theory compactifications <cit.>. 



  §.§.§ Broken dark sector


The second possibility is that the dark sector gauge group is broken completely at low energies, which means that in principle the Standard Model gauge fields can transform as non-trivial representations under G_d. This however, leads to a multiplicity of the Standard Model matter fields according to the dimension of the dark sector representation, which needs to be implemented carefully to allow for a reasonable phenomenology in the visible sector. All interactions coming from the dark sector would be suppressed by the breaking scale Γ_d, which can be very high and therefore does not change the measured interactions. In this case 𝒵 does not have to embed as a ℤ_k as in (<ref>) and (<ref>) discussed above and can be more general. Moreover, the modifications of the axion potential will be suppressed by the breaking scale.

This scenario is typical in Grand Unified theories where the Standard Model embeds into a larger gauge group at high energies, see, e.g., <cit.> for a recent discussion. It is also generically realized in phenomenological applications of the heterotic string <cit.> with its large 10d gauge groups.




In the following we do not want to analyze individual models, with a certain realization of the matter fields, which are also subject to other consistency constraints such as anomaly cancellation. Instead we focus on the model independent restrictions imposed by topological mixing and argue how they alter the general properties of the theory. 

Before we go into the investigation of the axion dynamics we will explain what the topological mixing implies for the allowed representations of matter fields as well as consistent gauge theory backgrounds.



 §.§ Representations and exotic matter



The quotient by a subgroup of the center symmetry in (<ref>) has consequences for the allowed representation of matter fields in the theory. In particular, only representations that are invariant under 𝒵 in (<ref>) survive the quotient. The fact that all matter representations of the Standard Model are invariant under ℤ_k, that acts as described in (<ref>) and (<ref>), allows for the different global realizations G_SM = G_SM / ℤ_k of the Standard Model gauge group.

For example consider k = 6, all the fields of the Standard Model, as well as the right-handed neutrinos, are invariant under ℤ_6, e.g., the left-handed quarks transform as

    Q_L = ( 3 , 2 )_1/6:    e^2 π i/3× (-1) × e^2 π i / 6 = 1  .

Here, the subscript denotes the hypercharge normalized to be in 16ℤ as is common in the literature.[Note, however, that the ℤ_6 ⊂U(1) still acts on the charge 16 state as the phase e^2 π i/6 as is expected for an embedding with ℓ_1 = 1.] However, for k = 6 representations of the form 

    Q_exotic = (3 , 1)_0:    e^2 π i/3≠ 1  ,

are not invariant under ℤ_6 and therefore would not be allowed, even though they are well-defined for the maximal gauge group G_SM. Once the quotient by 𝒵 involves a dark sector gauge group the allowed representations change accordingly as we will discuss next.

Including the dark sector gauge group we specify the particle representations as

    𝐑 = (𝐑_3  , 𝐑_2  , 𝐑_d)_q  ,

where the charges of U(1)_Y once more normalized to be elements in 16ℤ. Next, we define the center charge r_i of a representation 𝐑_i as given by its phase under center transformations 

    𝒵_G_i⊃ℤ_m_i:   𝐑_i → e^2 π i/m_i r_i 𝐑_i  ,

e.g., r_3 = 1  mod 3 for the fundamental representation 3 of SU(3). Thus, one obtains a consistency condition for each ℤ_n ⊂𝒵 factor, which can be phrased in terms of the ℓ_i as 

    𝒵⊃ℤ_n:   𝐑→exp( 2 π i ∑_i ℓ_i/m_i r_i + 2 π i ℓ_1/n6q ) 𝐑 = 𝐑 ,

which can be rephrased as

    ∑_i ℓ_i/m_i r_i + ℓ_1/n6q ∈ℤ .

As usual these formulas must be modified if G_i = Spin(4n), since then 𝒵_G_i = ℤ_2 ×ℤ_2 and one needs to specify two charges (r_i^1  , r_i^2), and two embedding parameters (ℓ_i^1  , ℓ_i^2) as discussed above.

Let us exemplify that for the left-handed quarks Q_L that additionally behave as singlets under the dark sector and 𝒵 = ℤ_6 embedded as in (<ref>), i.e. (ℓ_3  , ℓ_2  , ℓ_1  , ℓ_d) = (1  , 1  , 1  , 0). One finds

    ∑_i ℓ_im_i r_i + ℓ_1n6q = 13 + 12 + 16 + 0 = 1 ∈ℤ ,

which indeed satisfies the quantization condition. If instead we consider Q_exotic in (<ref>), again transforming as singlet under the dark sector gauge group, one finds

    ∑_i ℓ_im_i r_i + ℓ_1n6q = 13 + 0 + 0 + 0 = 13∉ℤ ,

which does not satisfy the quantization condition.

Next, we introduce topological mixing with the dark sector and see how this changes the allowed matter representations. In case the dark sector confines we demand that all Standard Model fields are singlets under G_d, i.e., 

    𝐑_SM = (𝐑_3 , 𝐑_2 , 1_d)_q ,

since otherwise they might be confined in dark baryons. This is automatically satisfied if we demand that 𝒵 embeds into 𝒵_G_SM as the ℤ_k groups defined in (<ref>) and (<ref>), and there is a non-trivial homomorphism of ℤ_k to 𝒵_d to generate the topological mixing. This allows for the appearance of more general representations. 

For example for 𝒵 = ℤ_6 and starting with Q_exotic in (<ref>) under the Standard Model one has

    𝐑 = (3 , 1 , 𝐑_d)_0 :   ∑_i ℓ_im_i r_i + ℓ_1n6q = 13 + 0 + 0 + ℓ_d m_d r_d∈ℤ ,

from which one finds the consistency constraint

    ℓ_d m_d r_d = 23 mod  1  .

This can be achieved for example for 𝔤_d = 𝔰𝔲(3) and the embedding specified by

    (ℓ_3  , ℓ_2  , ℓ_1  , ℓ_d) = (1  , 1 , 1  , ℓ_d)  ,

for which we find

    𝐑 = (3 , 1 , 3)_0:     ∑_i ℓ_im_i r_i + ℓ_1n6q = 13 + 0 + 0 + ℓ_d 3× 1 ∈ℤ ,

which is allowed for ℓ_d = 2. Similiarly,

    𝐑 = (3 , 1 , 3)_0:   ∑_i ℓ_im_i r_i + ℓ_1n6q = 13 + 0 + 0 + ℓ_d 3× 2 ∈ℤ ,

which is allowed for ℓ_d = 1.

For the broken dark sector scenario we do not have to demand that the Standard Model fields are singlets under G_d and one can take more general quotients. The allowed matter representations of the UV gauge group will still be influenced by the topological mixing as discussed above.

For that let us consider another example with 𝒵 = ℤ_6 and 𝔤_d = 𝔰𝔲(3), but this time the embedding differs from (<ref>) in the Standard Model factors and instead is given by

    (ℓ_3  , ℓ_2  , ℓ_1  , ℓ_d) = (1  , 1  , -1  , 1) ∼ (1  , 1  , 5  , 1)  .

In this case the the left-handed quarks need to transform non-trivially under G_d in order to absorb the phase

    Q_L = (3, 2)_1/6:    Q_L → e^2 π i 2/3  Q_L  ,

under 𝒵 = ℤ_6. This can be ensured by transforming as 3 under G_d = SU(3), leading to the consistency condition

    𝐑 = (3 , 2 , 3)_1/6:   ∑_i ℓ_im_i r_i + ℓ_1n6q= 13 + 12 - 16 + 13 = 1 ∈ℤ ,

being satisfied. Note that the exotic matter representation (<ref>), realized via (3 , 1 , 3)_0 is also allowed for this particular embedding.

Similar conclusions about the realization of allowed exotic matter states can be applied for all the possible gauge groups and follow the same logic as explained above. This demonstrates that the global form of the gauge group leads to significant restrictions on the allowed exotic matter states.



 §.§ Non-trivial 1-form symmetry backgrounds



 One can interpret taking the quotient in (<ref>) as gauging part of the center 1-form symmetry <cit.>. This center 1-form symmetry acts on Wilson lines labeled by a representation 𝐑 of the gauge group with the charge r defined as in (<ref>)

    𝒲_𝐑 = tr( P  exp( i ∮ A_𝐑) )  ,

where P denotes a path-ordering in the exponential. Since the charged objects are one-dimensional, the associated symmetry is called a 1-form symmetry. The existence of dynamical particles transforming under the center breaks this symmetry explicitly, since now the Wilson lines can end on a charged local operator. For the Standard Model the discrete ℤ_6 is not broken and one can gauge (subgroups of) it. Once one includes the dark sector gauge group the gauging and breaking depends on the specific matter representations.[In quantum gravity it is expected that all global symmetries are either gauged or broken, e.g., <cit.>, so in our scenario this would mean that either there are exotic matter representation that break the 1-form symmetry or it necessarily needs to be gauged.]

To perform such a gauging we couple the 1-form center symmetry of G = G_SM×G_d given by

    𝒵_G = ℤ_3 ×ℤ_2 ×U(1) ×𝒵_d ,

to background gauge fields, which are discrete 2-form fields B_i. We will further make the assumption that 𝒵 maximally embeds as ℤ_6 in the Abelian U(1)_Y of the hypercharge sector. Reason is that one otherwise generically ends up with fractionally charged particles that are additionally charged under the dark sector and are experimentally heavily constrained, see, e.g., <cit.>, similar to kinetic mixing with another Abelian factor that can also produce such states <cit.>. In this case the background fields are given by elements

    (B_3  , B_2  , B_1  , B_d) ∈ H^2 (M; ℤ_3) × H^2 (M; ℤ_2) × H^2 (M; ℤ_6) × H^2 (M; 𝒵_d)  ,

where M denotes the spacetime manifold. The gauging proceeds by promoting a linear combination of these background fields to be dynamical, which for a discrete fields merely means that we sum over its configurations in the partition function. The specific form of linear combination which is gauged is determined by the embedding of 𝒵 into the individual factors of 𝒵_G, i.e., by the embedding parameters ℓ_i.

For 𝒵 = ℤ_6 with embedding parameters (ℓ_3  , ℓ_2  , ℓ_1  , ℓ_d) we can parametrize the `dynamical' linear combination by summing over ω∈ H^2 (M; ℤ_6) and identifying

    (B_3  , B_2  , B_1  , B_d) = (ℓ_3  ω_3  , ℓ_2  ω_2  , ℓ_1  ω ,  ℓ_d ω_d)  .

Here, ω_i is the reduction of ω modulo the order of the different factors of 𝒵_i. For embedding of subgroups into U(1) and 𝒵_i we use the fact that we can rephrase a background in H^2(M; ℤ_k) as a background in H^2(M; ℤ_m k) by multiplication by m, see Appendix <ref>. Again, we note the usual caveats in case 𝔤_d = 𝔰𝔬(4n), for which one needs to specify two independent background gauge fields in H^2(M;ℤ_2).  

For example, for (<ref>) one finds the linear combination

    (B_3  , B_2  , B_1  , B_d) = (ω_3  , ω_2  , ω , ℓ_d ω_3)  .

Similarly, for (<ref>) the appropriate linear combination is given by

    (B_3  , B_2  , B_1  , B_d) = (ω_3  , ω_2  , - ω , ω_3)  ,

where one could equally well use 5 ω instead of - ω, since 6 ω is trivial in H^2(M;ℤ_6).

Since for the maximal gauge group G we effectively need to set all ω_i to zero we see that the gauge group G = G/𝒵 has more gauge configurations than G.[This can also be seen from classifying the gauge configurations via transition functions where in G the co-cycle condition on triple overlaps needs to produce the identity in G, it can be a non-trivial element in 𝒵 for G/𝒵, see e.g. <cit.>.] These new gauge field backgrounds, however, lead to a restriction on allowed matter representations, which have to be compatible with the gauging procedure and precisely lead to the restrictions discussed in Section <ref>. Moreover, after gauging part of the center 1-form symmetry the quantization of instanton numbers is modified as we will discuss next.



 §.§ Instanton numbers



Instantons are topological gauge field configurations with a non-trivial value of the integral 

    n_i = 1/8 π^2∫tr(F_i ∧ F_i)  ,

where F is the non-Abelian field strength of 𝔤_𝔦 in differential form notation, see, e.g., <cit.>. Since they are characteristic classes of the gauge theory their number, n_i, is quantized.[They are the pullbacks of non-trivial elements in group cohomology with integer coefficients, i.e., originate from H^4 (BG; ℤ), with classifying space BG.] In our conventions of the trace the instanton numbers for simply-connected non-Abelian Lie groups on 
Spin manifolds are integers. For Abelian gauge groups the instanton number is given by

    n_1 = 1/8 π^2∫ F_1 ∧ F_1  ,

with hypercharge U(1)_Y field strength F_1 = d A_1. Once more n_1 is an integer on Spin manifolds. It is important to stress that the spacetime is Spin, since four-dimensional Spin manifolds have an even intersection form for H_2(M;ℤ). This, together with the charge quantization condition [F_1/2π] ∈ H^2(M;ℤ), is why we can divide by an additional factor of 12 in (<ref>). For non-Spin manifolds, e.g., M = ℂℙ^2, this is generally not the case and (<ref>) can be half-integer. However, this is modified in the presence of non-trivial background fields B_i for the center 1-form symmetries.

In the case of a U(1) gauge symmetry and 𝒵 = ℤ_n this can be understood as follows. The background field B_1 ∈ H^2(M; ℤ_n) can be rephrased as a flat 2-form gauge field b_1 ∈ H^2(M; U(1)) which is roughly related to B_1 as

    b_1 = 2 πn B_1  .

In particular this means that if B_1 integrates to q ∈ℤ mod n on a certain 2-cycle, b_1 integrates to 2 π q/n mod 2π. In the presence of this background field the instanton number is shifted to

    n_1 = 1/8 π^2∫ (F_1 - b_1) ∧ (F_1 - b_1)  .

In general this gives rise to fractional instanton contributions of the form 

    1/4 π^2∫ F_1 ∧ b_1 ∈1nℤ ,   and  1/8 π^2∫ b_1 ∧ b_1 ∈1n^2ℤ .

At the same time these new configurations violate the quantization condition for the magnetic flux F_1 and would require electric charges to be multiples of n, which is precisely the U(1) version of the restriction of representations discussed in Section <ref>. Equivalently, since the action is quadratic in F_1 these results can been seen explicitly by noticing that gauging the ℤ_n 1-form symmetry effectively replaces F_1 with 1n F_1 in the Lagrangian. 

A similar conclusion holds for non-Abelian gauge groups where the non-trivial 1-form symmetry backgrounds B_i lead to fractional contributions to the instanton numbers. To extract their fractionality one modifies the field strength to include contributions of B_i and evaluates their effect. This has been done in detail in <cit.> to which we refer for a detailed derivation. 

Here, we briefly recall the derivation in the special case G = SU(n) and 𝒵 = ℤ_k and embedding specified by

    1  mod  k ↦ℓ mod  n  .

For the resulting gauge group to be SU(n)/ℤ_k we further have to demand the stronger condition compared to (<ref>) that k divides n to generate a proper subgroup, where in general we would allow for a subgroup of 𝒵 to map non-injectively to 𝒵_G_i. The non-trivial background field B ∈ H^2(M; ℤ_n) is defined by

    B = ℓ ω  n,

where ω∈ H^2(M;ℤ_k), is the parameter we sum over in the partition function, and B is well defined because of (<ref>). One also has

    ngcd(ℓ,n) = k  .

Because of this there is a way to interpret B as a background in H^2(M;ℤ_k) as one would expect from a quotient by ℤ_k, see Appendix <ref>. To see how this non-trivial background B modifies the instanton number we promote the SU(n) field strength F to a U(n) field strength F', see <cit.>, which can be done by introducing a U(1) gauge field f = da and defining

    F' = F + 1n f  1 ,

with unit matrix 1. One then identifies the U(1) part with the background gauge field, which after using the identification (<ref>) in terms of b ∈ H^2(M;U(1)) takes the form

    tr(F') = f = n    b  .

This can be implemented via a Lagrange multiplier in the action, see <cit.>. One then replaces the instanton density (<ref>) with

    1/8 π^2∫tr( (F' -    b  1) ∧ (F' -    b  1) )  ,

which contains the U(n) field and the background b. Since the second Chern class of F' integrates to integer values

    ∫ c_2(F') = ∫( 1/8 π^2tr(F' ∧ F') - 1/8 π^2tr(F') ∧tr(F') ) ∈ℤ ,

we identify the fractional part of (<ref>) as the second term in 

    1/8 π^2∫tr( (F' -   b  1) ∧ (F' -   b  1) ) = ∫ c_2 (F') + 1/8 π^2∫  n(n-1)   b ∧ b  .

Going back to the discrete gauge field ω, which is summed over,  using b=2π/n B=2π/nℓω this can be written as

    n^frac =  (n-1)/2 n∫𝒫(ℓω)  .

Here, 𝒫 is the Pontryagin square, see, e.g., <cit.>, which is a cohomological operation

    𝒫:    H^2 (M; ℤ_n) → H^4 (M; ℤ_p)  ,

where p=n for odd n and p=2n for even n. It refines the bilinear form given by the cup product

    ∪: H^2(M;ℤ_n) × H^2(M;ℤ_n) → H^4 (M;ℤ_n)  .

of B with itself. It is only relevant for n even, since for n odd one has 𝒫(B) = B ∪ B ∈ H^4(M;ℤ_n). For even n it introduces an additional factor of 12 in the fractional part of the instantons. In cases where B ∈ H^2(M;ℤ_n) has a well-defined lift to integer cohomology B_ℤ∈ H^2(M;ℤ), which we will always assume and which is the case if H^3(M;ℤ) is torsion free[In case this lift does not exist, one can still perform the calculations above using the definition of the Pontryagin square on the co-chain level, see <cit.>. ], it is given by 𝒫(B) = B_ℤ∪ B_ℤ mod 2n.

For example for G = SU(2)/ℤ_2, where ℓ = 1 since we assume a non-trivial embedding, the fractional instanton number in the presence of B ∈ H^2(M;ℤ_2) is given by

    n_2^frac = 14∫𝒫 (B) = 14∫ B_ℤ∪ B_ℤ mod  1 ∈{ 0  , 12} ,

where we once more used the evenness of the intersection form on Spin manifolds.

For other simple Lie algebras in the presence of the background B ∈ H^2(M; 𝒵_G) one can determine fractional instanton contributions by utilizing different embeddings of SU(n) groups <cit.>. The result is presented in the table:

    [                                  gauge algebra 𝔤 fractional instanton contribution for  G = G/𝒵_G;                                            𝔰𝔲(n)                                       n-12n∫𝒫(B);                                         𝔰𝔬(2n+1)                                          12∫𝒫(B);                                           𝔰𝔬(4n)                  n4∫𝒫(B_L + B_R) + 12∫ B_L ∪ B_R;                                         𝔰𝔬(4n+2)                                      2n+1/8∫𝒫(B);                                            𝔰𝔭(n)                                          n4∫𝒫(B);                                              𝔢_6                                          23∫𝒫(B);                                              𝔢_7                                          34∫𝒫(B);                                                  ]

Indeed, setting ℓ =1, and therefore gcd(ℓ,n) = 1 and n = k, in (<ref>), we find

    n^frac = n-1/2n∫𝒫(B)  ,

which conincides with the table. 

In case 𝒵 is only a subgroup of the center of the dark sector G_d, i.e., if gcd(ℓ_d, m_d) ≠ 1, this affects that fractionality condition. Note that this is only relevant for SU(n) with n not prime and Spin(2n), since otherwise the center does not have non-trivial subgroups. We discuss the various cases in Appendix <ref>.

In the models above the overall quotient 𝒵 embeds into several gauge factors. As we discussed in Section <ref> this correlates the various background fields for the individual 1-form center symmetries. It also implies that fractional instanton contributions in one gauge factor necessarily are accompanied by fractional contributions in a different sector. For 𝒵 = ℤ_6 and 𝔤_d = 𝔰𝔲(3) with embedding (<ref>) the fractional parts are given by

    n_3^frac   = 13∫𝒫(ω_3)  ,  n_2^frac = 14∫𝒫(ω_2)  ,  n_d^frac = ℓ_d^23∫𝒫(ω_3)  , 
    
    n_1^frac   = - 16∫ c_1 (F_1) ∪ω + 172∫𝒫(ω)  ,

where c_1(F_1) ∈ H^2(M;ℤ) is the first Chern class of the hypercharge gauge field represented by [12π F_1]. The different embedding (<ref>) leads to positive sign for the first term for n_1^frac, but is otherwise identical after setting ℓ_d = 1.

In the next section we will see how the fractional instanton charges induced by the quotient 𝒵 enters the dynamics of axion physics.



§ AXIONS AND TOPOLOGICAL MIXING



Now that we have analyzed how a topological mixing influences the possible representations and leads to more general gauge backgrounds that can carry fractional instanton numbers we analyze how this influences the behavior of an axion field.

For us, an axion field a is a real, pseudoscalar particle that is periodic

    a ∼ a + 2 π ,

Its kinetic term in differential form notation, see <cit.> for an introduction, is given by

    ℒ^a_kin = 12 f^2 da ∧∗ da  ,

with dimension-full parameter f known as the axion decay constant. The canonically quantized axion field, of mass dimension 1, would therefore be fa and have periodicity 2 π f. Importantly, the axion couples topologically to the instanton density of the gauge groups, for which we include the dark sector. This coupling is given by

    ℒ^a_top = i a/8 π^2( κ_3  tr(F_3 ∧ F_3) + κ_2  tr (F_2 ∧ F_2) + κ_1   F_1 ∧ F_1 + κ_d tr(F_d∧ F_d) )  ,

with the topological coupling constants κ_i. This interaction must not violate the periodicity condition for the axion (<ref>), which can be interpreted as a gauged shift symmetry. More specifically, thinking of a as a real field it has ℝ shift symmetry and by gauging a discrete subgroup 2πℤ⊂ℝ one imposes a periodicity condition. 

For G = G_SM×G_d, i.e., trivial quotient, this simply implies that the constants κ_i are all integers since all instantons are integer quantized. However, once there is non-trivial topological mixing of the gauge theory factors, there are fractional contributions to the instanton numbers, leading to a correlation of the quantization condition of the coupling constants κ_i. In <cit.> this was used to show that the axion-photon coupling takes a minimal value that depends on the topological mixing in the Standard Model. Here, we include the influence of mixing with a dark sector gauge group and find that the constraints are modified.



 §.§ Quantization of topological terms



In the following we analyze the quantization condition for the constants κ_i in the presence of non-trivial topological mixing involving a non-Abelian dark sector gauge group. Thus, we assume a gauge group of the form 

    G = G_SM×G_d/𝒵 ,

with a choice of embedding specified by (ℓ_3  , ℓ_2  , ℓ_1  , ℓ_d). As discussed in Section <ref> this embedding identifies a linear combination of the 1-form symmetry background fields which are summed over in the partition function, gauging the corresponding subgroup of the center 1-form symmetries. Assuming for simplicity that 𝒵 = ℤ_k has a single factor, which can be easily generalized, the dynamical gauge field takes the general form

    (B_3  , B_2  , B_1  , B_d) = (ℓ_3  ω_3  , ℓ_2  ω_2  , ℓ_1  ω , ℓ_d  ω_d)  ,

with ω∈ H^2(M;𝒵) and ω_i its mod m_i reductions as discussed around (<ref>). The last entry ω_d is the reduction of ω mod the order of 𝒵_d, with the usual caveat for Spin(4n) with its two factors of the center. With this one can evaluate the fractional contribution to the instanton number and their coupling to the axion 

    ℒ^a_top,frac = a ( α_3  𝒫 (ω_3) + α_2  𝒫(ω_2) + 
    α_1  𝒫(ω) - α_1  ω∪ c_1(F_1) + α_d 𝒫(ω_d) )

The fractional prefactors α_i depend on the specific form of the embedding and are given by

    α_3 = ℓ_3^23 κ_3  mod  1  ,   α_2 = ℓ_2^24 κ_2  mod  1  ,   α_1 = ℓ_1^22 k^2 κ_1  mod  1  ,   α_1 = ℓ_1k κ_1  mod  1  ,

and for the dark sector

    α_d = c^fracℓ_d^2  κ_d ,

where c^frac is the fractional coefficient in Table (<ref>), which depends on the choice of 𝔤_d. Since the axion periodicity is fixed to 2π the coupling to the topological quantities in (<ref>) has to respect that.[If this is not the case this can be interpreted as a mixed anomaly between the gauged 2πℤ shift symmetry of the axion and the gauged center 1-form symmetries presenting an inconsistency <cit.>. This is also common for topological terms involving higher form fields in larger dimensions, see, e.g., <cit.>.] This leads to the quantization condition

    ∫( α_3  𝒫 (ω_3) + α_2  𝒫(ω_2) + 
    α_1  𝒫(ω) - α_1  ω∪ c_1 (F_1) + α_d 𝒫(ω_d) ) ∈ℤ ,

for all backgrounds of the form (<ref>) which are summed over. However, note that not the individual pieces but only their linear combination has to satisfy this constraint.

Let us first set α_d to zero and reproduce the known result for 𝒵 = ℤ_6, (ℓ_3  , ℓ_2  , ℓ_1) = (1  , 1  , 1), in the Standard Model. One has

    ∫( 13κ_3  𝒫(ω_3) + 14κ_2  𝒫(ω_2) + 172κ_1  𝒫(ω) - 16κ_1  ω∪ c_1(F_1) ) ∈ℤ .

Note that for this condition to be satisfied κ_1 needs to be a multiple of 6 and one has

    κ_1 ∈ 6 ℤ ,

which makes the integral of the last term in (<ref>) an integer. Furthermore the term 

    172κ_1 ∫𝒫(ω)  ,

is well-defined modulo integers since κ_172∈112ℤ and ∫𝒫(ω) is defined mod 12. Integer instantons in the non-Abelian group factors, which are present for any value of 𝒵, further provide the conditions

    κ_3  , κ_2 ∈ℤ .

The remaining condition can be deduced by taking an integer lift of ω which we always assume to exist, see however footnote <ref>, i.e., ω_ℤ∈ H^2 (M;ℤ) with

    ∫_C ω_ℤ mod  6 = ∫_C ω ,   for all C ∈ H_2 (M; ℤ)  ,

and notice that this is also a good integer lift of ω_i

    ∫_C ω_ℤ mod  n = ∫_C ω mod  n = ∫_C ω_i  ,   for all C ∈ H_2 (M; ℤ)  .

With this we find 

    ( 13κ_3 + 14κ_2 + 172κ_1 ) ∫ω_ℤ∪ω_ℤ∈ℤ .

Recalling that ω_ℤ∪ω_ℤ always integrates to an integer number on Spin manifolds the remaining quantization condition for the topological couplings κ_i is given by

    13κ_3 + 14κ_2 + 172κ_1 ∈12ℤ  ↔   24 κ_3 + 18 κ_2 + κ_1 ∈ 36 ℤ ,

This precisely reproduces the result of <cit.>.

After the inclusion of 𝔤_d this quantization can change, which we will demonstrate for 𝒵 = ℤ_6 and general dark sector gauge algebra. The embedding is defined by the parameters (ℓ_3  , ℓ_2  , ℓ_1  , ℓ_d). As above, integer instantons in all of the non-Abelian gauge groups give rise to the condition

    κ_3  , κ_2  , κ_d∈ℤ ,

leaving a single condition from fractional instantons. For cases with 𝒵 having several factors one gets a condition for each generator of 𝒵. It reads

    ∫( ℓ_3^23κ_3 𝒫(ω_3) + ℓ_2^24κ_2  𝒫(ω_2) + ℓ_1^272κ_1 𝒫(ω) - ℓ_16κ_1  ω∪ c_1 (F_1) + α_d 𝒫(ω_d) ) ∈ℤ .

For this to possibly be an integer we need to demand that

    κ_1 ∈6gcd(ℓ_1, 6) ℤ ,

which also ensures that the κ_1 𝒫(ω) term is well-defined modulo integers. Lifting ω and ω_i to H^2(M; ℤ) the consistency condition is

    ( ℓ_3^23κ_3 + ℓ_2^24κ_2 + ℓ_1^272κ_1 + α_d) ∫ω_ℤ∪ω_ℤ∈ℤ ,

which we express as

    ℓ_3^23κ_3 + ℓ_2^24κ_2 + ℓ_1^272κ_1 + α_d ∈12ℤ  ↔   24 ℓ_3^2  κ_3 + 18 ℓ_2^2  κ_2 + ℓ_1^2  κ_1 + 72 α_d∈ 36 ℤ .

To go through the various possibilities, we provide a list of α_d for all 𝔤_d and embeddings of ℤ_6 specified by ℓ_d on Spin manifolds

    [      gauge algebra 𝔤_d gauged subgroup of 𝒵_d                    ℓ_d                    α_d;               𝔰𝔬(2n+1)                    ℤ_2                      1                  12κ_d;                 𝔰𝔬(4n)                    ℤ_2                  (1,0)                  n4κ_d;                                           ℤ_2                  (0,1)                  n4κ_d;                                           ℤ_2                  (1,1)                  12κ_d;               𝔰𝔬(4n+2)                    ℤ_2                      2                  12κ_d;                  𝔰𝔭(n)                    ℤ_2                      1                  n4κ_d;                    𝔢_6                    ℤ_3               1  or  2                  23κ_d;                    𝔢_7                    ℤ_2                      1                  34κ_d;                        ]

Since α_d is only relevant modulo 12 we find that the only non-trivial embeddings happen for 𝔰𝔬(4n), 𝔰𝔭(n) for odd n as well as 𝔢_6 and 𝔢_7. The most flexible embeddings happen for 𝔰𝔲(n) dark sector gauge groups for which we find (up to 𝔰𝔲(6))

    [      gauge algebra 𝔤_d gauged subgroup of 𝒵_d                    ℓ_d                    α_d;                  𝔰𝔲(2)                    ℤ_2                      1                  14κ_d;                  𝔰𝔲(3)                    ℤ_3               1  or  2                  13κ_d;                  𝔰𝔲(4)                    ℤ_2                      2                  12κ_d;                  𝔰𝔲(6)                    ℤ_2                      3                  34κ_d;                                           ℤ_3                      2                  23κ_d;                                           ℤ_6               1  or  5                 512κ_d;                        ]

One sees that the additional contribution of κ_d generated by the topological mixing with the dark sector changes the quantization conditions.

In the next section we explore how this can influence the axion-photon interactions.



 §.§ Axion-photon coupling



At energies below electro-weak symmetry breaking the weak SU(2) and the hypercharge U(1)_Y combine to the U(1) of electro-magnetism, whose field strength we denote by F. The topological part of the action below this breaking scale is given by

    ℒ'_a,top = i a ( N/4 π^2 tr (F_3 ∧ F_3) + E/8 π^2  F ∧ F + κ_d/8 π^2 tr (F_d∧ F_d) )  ,

where the constants N and E are often used in phenomenological discussions <cit.> and are defined in terms of the κ_i as

    N = 12κ_3  ,    E = 136 (κ_1 + 18 κ_2)  .

From this one can also determine the axion-photon coupling which is given by (α is the fine-structure constant)

    g_aγγ = α N/π f( E/N - 1.92(4) + ρ)  .

Note that the numerical result 1.92 arises from the mixing of the axion with the neutral pion, π_0, see the derivation in <cit.>. The particular realization of the dark sector might alter this numerical coefficient. However, since at leading order this mixing proceeds via Standard Model fields and we do not alter the electro-weak symmetry breaking, these effects will be suppressed with respect to the result stated above. Since further we do not want to include sensibility of the particular dark sector spectrum, we simply parameterize this modification with the constant ρ, which we expect to be sub-leading.

The important conclusion of (<ref>), as pointed out in <cit.>, is that there is a minimal axion-photon coupling, since one has to cancel the mixing contribution (-1.92(4) + ρ) with a quantized EN, which cannot be done with arbitrary precision. For example for the ℤ_6 quotient in the Standard Model from (<ref>) one has

    E/N = κ_1 + 18 κ_2/18 κ_3 = - 4 κ_3 + 6 m/3 κ_3 ,   with  m ∈ℤ .

Setting κ_3 = 1, which avoids potential domain wall problems, see Section <ref> below, the best cancellation with the pion mixing is achieved for m = 2, leading to EN = 83 and

    g_aγγ^min≃α/2 π f 0.72(4)  ,

and one obtains a lower bound of this coupling.

After the inclusion of the topological mixing with the dark sector, which we take to be induced by 𝒵 = ℤ_6 as above, one instead finds the relevant quantities to be

    E/N = κ_1 + 18 κ_2/18 κ_3 = 6 m - 12 α_d + 3(ℓ_1^2 - ℓ_2^2) κ_2 - 4 ℓ_3^2 κ_3/3 ℓ_1^2 κ_3 ,   with m ∈ℤ ,

where we used (<ref>) to eliminate κ_1. We already see that depending on the topological couplings κ_i and the embedding parameters ℓ_i this equation can be tuned more finely and the minimal value for the axion-photon coupling can in turn be reduced significantly.

We will exemplify this for the example 𝔤 = 𝔰𝔲(3) with embedding given in (<ref>), in which case α_d = ℓ_d^23κ_d = 13κ_d and one finds

    E/N = 6m - 4 κ_d - 4 κ_3/3 κ_3 ,

again setting κ_3 = 1 one can obtain a significantly better cancellation in cases where κ_d = - 1 for which we can choose m = 1 to find

    E/N = 2  ,

and the minimal value of the axion-photon coupling is given by

    g_a γγ^min,d≃α/2 π f( 0.07(6) + ρ)  .

We see that while the direct dynamical consequences of a topological mixing with the dark sector are rather mild, it can have interesting consequences to the axion sector which is sensitive to these topological properties. In fact, assuming the corrections ρ are sub-leading, the axion-photon coupling can be reduced by an order of magnitude in the presence of a dark 𝔰𝔲(3) with non-trivial mixing with the Standard Model fields. Note further that even in situations where ρ modifies the Standard Model calculations the additional parameter κ_d leads to more freedom in tuning the parameters in order to achieve a better cancellation of the terms among each other.



§ SYMMETRIES AND DEFECTS



Systems with axionic degrees of freedom are very rich in terms of their symmetries. In particular they realize so-called higher-group structures, where various higher-form symmetries mix among each other <cit.>, and non-invertible symmetries, which are symmetries that do not follow a group law <cit.>. Some of these symmetries still exist after the coupling to the dark sector. Moreover, since the axion is a periodic scalar field there can be configurations with a non-trivial winding number, i.e., axion strings. Due to the topological coupling the axion string needs to host localized degrees of freedom accounting for the anomaly inflow from the bulk <cit.>. For non-trivial coupling to the dark sector these localized degrees of freedom are also charged under the dark gauge group.

We will briefly discuss these properties of the symmetries and defects in the system in the following.



 §.§ The symmetries of the axion system


In the absence of the topological coupling (<ref>) there are four different types of symmetries. The axion (<ref>) possesses a U(1) 0-form shift symmetry

    a → a + σ ,  with σ∈ [0, 2 π)  .

and a winding U(1) 2-form symmetry, which measures the winding number of the axion string configuration. 
The corresponding currents are

    ∗ j_shift = i f^2 ∗ da  ,   ∗ j_wind= 12πda
  
which are conserved, i.e., d ∗ j = 0, prior to coupling of the axion to the gauge fields. In the modern language we would say that the shift symmetry is related to a 3-dimensional topological operator, which can be written in terms of the conserved current

    U_σ (Σ_3) = e^i σ∫_Σ_3∗ j_shift

where Σ_3 is the 3-dimensional submanifold of M, the topological operator is located on, see also <cit.>.

On the other hand, as discussed in <ref> the gauge theory sector contributes an electric center 1-form symmetry. It is the subgroup of 

    𝒵_G/𝒵 = ℤ_3 ×ℤ_2 ×U(1) ×ℤ_d/𝒵 ,

which is not broken by the presence of dynamical charged matter states, as for example provided by the Standard Model matter fields. On the other hand the gauging process introduces a dual magnetic 1-form symmetry 𝒵^∨, the Poincaré dual of 𝒵, which acts non-trivially on 't Hooft lines that cannot be screened by dynamical magnetic monopoles <cit.>.

In the presence of the interaction (<ref>) the equations of motion are modified and the current for the shift symmetry is not conserved anymore, instead one has
 
    d∗ j_shift = κ_3/8π^2tr(F_3∧ F_3) + κ_2/8π^2tr(F_2∧ F_2) + κ_1/8π^2 F_1∧ F_1 + κ_d/8π^2tr(F_d∧ F_d)  ,

and the continuous symmetry is broken. However, it might not be broken completely. To see that, consider a shift (<ref>) and define the modified operator

    U_σ (Σ_3) = e^i σ∫_Σ_3 (∗ j_shift - κ_3/2πΩ_3-κ_2/2πΩ_2 - κ_1/8π^2A_1∧ F_1 - κ_d/2πΩ_d) ,

with non-Abelian Chern-Simons 3-forms Ω_i = 1/4πtr(A_i∧ dA_i+2/3A_i^3), which satisfy

    dΩ_i=14πtr(F_i∧ F_i)  .

Since the Chern-Simons terms are not invariant under large gauge transformations the terms σ κ_i Ω_i are in general not well-defined, even though the resulting operator in (<ref>) is topological. An alternative way to define (<ref>) is to define a 4-dimensional submanifold Σ_4 of M with boundary ∂Σ_4 = Σ_3 and write

    U_σ (Σ_3) = e^i σ( ∫_Σ_3∗ j_shift - ∫_Σ_4 ( κ_3/8π^2tr(F_3∧ F_3) + κ_2/8π^2tr(F_2∧ F_2) + κ_1/8π^2 F_1 ∧ F_1 + κ_d/8π^2tr(F_d∧ F_d)) ) .

For that to be a well-defined 3-dimensional topological operator, we need to demand that U_σ does not depend on Σ_4, which means that it should be trivial whenever Σ_4 does not have a boundary. The special values for σ for which this is the case, for given κ_i defines the unbroken part of the U(1) shift symmetry. The requirement reads

    σ ∫_Σ_4( κ_38π^2tr(F_3∧ F_3)+κ_28π^2tr(F_2∧ F_2)+κ_18π^2F_1∧ F_1+κ_d8π^2tr(F_d∧ F_d))∈ 2πℤ .

This precisely corresponds to the discussion of fractional instantons in Section <ref> and the quantization condition in (<ref>). As we saw there in the presence of the nontrivial 1-from background ω∈  H^2(M; 𝒵) we can rewrite condition (<ref>)

    σ(   κ_3 n_3^int++κ_2 n_2^int + κ_1 n_1^int + κ_d n_d^int+ 
       ∫_Σ_4(α_3  𝒫 (ω_3) + α_2  𝒫(ω_2) +  α_1  𝒫(ω) - α_1  ω∪ c_1(F_1) + α_d 𝒫(ω_d) )) ∈ 2πℤ ,

where we have split the expression into the integral instanton contributions n_i^int and the fractional contributions with α_i defined as in (<ref>). From the consistency of the axion we know that the second line has to be an integer. There is an unbroken shift symmetry in case this integer as well as the prefactors κ_3, κ_2, κ_1/6, and κ_d are divisible by a common integer. The remaining shift symmetry is given by ℤ_p with shifts σ = 2πp and p in the case of 𝒵 = ℤ_6, defined in (<ref>), given by

    p = gcd( κ_3  , κ_2  , κ_1/6 , κ_d , 24 ℓ_3^2 κ_3 + 18 ℓ_2^2 κ_2 + ℓ_1^2 κ_1 + 72 α_d/36)  .

As discussed in <cit.>, for other special values of shifts σ it is possible to resurrect part of the broken shift symmetries as non-invertible symmetries. For that one compensates the non-invariance under gauge transformation of the topological operators in (<ref>) by including a topological field theory sector, that couples to the background gauge fields to make the whole system gauge invariant. Consider a U(1) gauge field c living on the topological operators localized on Σ_3 with the topological level k Chern-Simons action

    S=ik/4π∫_Σ_3( c∧ dc +2 c ∧ b )  ,

where b = 2 πkω is the U(1) realization of ω∈ H^2 (M;ℤ_k) as discussed in (<ref>). Since c appears only quadratically in the effective action (<ref>), one can integrate it out using its equation of motion dc+b=0. Substituting this back in the action and using the extension to Σ_4 as in (<ref>) one finds

    S= -ik/4π∫_Σ_4 b∧ b= -2π i /2k∫_Σ_4𝒫(ω)

compensating precisely for a fractional instanton in one of the gauge sectors. In our case this allows the discussion of a non-invertible shift symmetry with shifts σ = 2 πq, where now

    q = gcd (κ_3  , κ_2  , κ_d)  .

Note that the coefficient κ_1 for the Abelian gauge factor does not show up any more, since for U(1) one can recover similarly, utilizing (<ref>) as above, the full shift symmetry, or at least its rational part ℚ/ℤ, see <cit.>. Since the cancellation of the fractional instanton contribution happens separately for each gauge sector, we need Chern-Simons terms of the form above for each gauge group factor, including G_d that allows for fractional instanton contributions, whose gauge fields c_i couple to the non-trivial background ω_i, respectively. In general we notice that in the presence of topological mixing with a dark sector the unbroken invertible and non-invertible shift symmetries are smaller than what would be inferred from the Standard Model couplings alone.

The topological coupling of the axion further modifies the electric 1-form symmetries, as discussed in <cit.>, since it modifies the equation of motion for the gauge field as well. The specific form of the (non-invertible) electric 1-form symmetries, however, depends on the spectrum of dynamical particles, including the exotic matter representations discussed in Section <ref> and is therefore model dependent. For a detailed account of this in the context of the Standard Model we refer to <cit.>.

The 0-form, 1-form, and 2-form symmetries of the system mix into a higher group structure as was analyzed in <cit.>. Once more, the specific realization of this depends on the matter representations of the system and needs to be analyzed on a case by case basis. For the Standard Model the result was derived in  <cit.>. This higher group structure is also reflected in the anomaly inflow on various defects of the axion system which we will briefly discuss in the following. 
 


 §.§ The defects of the axion system


Theories with axions have extended objects. Among them are codimension-one objects, i.e., domain walls, across which the value of the axion field jumps, and codimension-two objects, axion strings, around which the axion value winds a → a + 2π. Comparing this behavior of the axion field with the definition of the generalized symmetries in (<ref>) we see that dynamical axion strings break the 2-form winding symmetry.

As discussed above, if q = (κ_2, κ_3, κ_d)>1, there is an exact ℤ_q symmetry which, even though it could be non-invertible, still acts on axions as the standard shift symmetry a → a + 2π/q. This implies that axion potential has to be 2 πq-periodic. Another way to argue for this 2 πq-periodicity of the potential is by considering instanton contribution to the computation of the axion potential. Since it is expected that the axion potential comes from summing over all instantons e^iκ_i n_i a for integral instanton numbers n_i. The Abelian and fractional instantons do not contribute to the axion potential for Minkowski spacetime <cit.>, since they need non-trivial spacetime topology.[This changes once one includes topology changes that are expected to occur for quantum gravity.] Thus, below the electro-weak phase transition the only contributions come from the strong interactions, SU(3), and potentially the dark sector G_d in case it confines. If it does, one expects that there will be contributions of order Λ_d^4 to the potential and the potential has approximate 2 πgcd(κ_3,κ_d)-periodicity imposed by the non-invertible shift symmetry. Therefore there are degenerate vacua and domain walls in case gcd(κ_3, κ_d) > 1. If, on the other hand, the dark sector gauge group is broken the axion potential is approximately  2πκ_3-periodic and one obtains |κ_3| vacua between the axion domain walls interpolate. In both cases the vacuum expectation value of the axion spontaneously breaks the non-invertible shift symmetry <cit.>.

This further demonstrates the close relation between axion domain walls and the symmetry operators for the shift symmetry. As we have seen, in case the shift symmetry is non-invertible this requires the existence of a topological field theory sector on the worldvolume of the domain wall. These topological field theories, related to the fractional quantum Hall effect, contain anyons which feel the influence of the background fields of both the Standard Model and dark sector 1-form symmetries. This can also be interpreted in terms of an anomaly inflow of the 1-form symmetries, see, e.g., <cit.>.

There is also anomaly inflow, this time of the 0-form gauge symmetries, in the presence of axion strings <cit.>, see also <cit.> for a recent discussion. This implies that the string hosts local degrees of freedom that cancel this contribution. To see this inflow explicitly we can perform a 0-form gauge transformation and 1-form transformations in the presence of an axion string localized on a 2-dimensional manifold Σ_2. This means that the value of the axion field on paths winding around this surface undergoes the monodromy a → a + 2 π. Locally, one can write that as

    d^2a= 2 π δ_Σ_2 ,

where δ_Σ_2 is a 2-form which is localized on Σ_2, i.e., the Poincaré dual of Σ_2, for a more explicit treatment with bump functions see <cit.>. Considering the topological interaction of the axion (<ref>), we find locally

    ℒ^a_top = - i   da ∧( κ_32 πΩ_3 + κ_22 πΩ_2 + κ_18 π^2 A_1 ∧ F_1 + κ_d2 πΩ_d)  .

Under infinitesimal gauge transformations λ_i the action transforms as

    Δ S = - ∫_Mi8π^2  da ∧( κ_3  tr(d λ_3 ∧ F_3) + κ_2  tr(d λ_2 ∧ F_2) + κ_1   d λ_1 ∧ F_1 + κ_d tr(d λ_d∧ F_d) )  .

Integrating this expression by parts and using (<ref>) one finds a gauge variation localized to the worldvolume of the axion string Σ_2

    Δ S = - ∫_Σ_2i4 π( κ_3  tr(λ_3 F_3) + κ_2  tr(λ_2 F_2) ) + κ_1  λ_1 F_1 + κ_d tr(λ_d F_d) )  ,

due to anomaly inflow. This needs to be cancelled by the existence of charged chiral fields on Σ_2. Assuming the existence of two-dimensional chiral fermions ψ transforming in the representation 𝐑 = (𝐑_3  , 𝐑_2  , 𝐑_d)_q of G they contribute to the gauge variation due to their perturbative anomaly as 

    Δ S_f =   ∫_Σ_2∑_𝐑i/4π( dim(𝐑_2,𝐑_d) I(𝐑_3)  tr (λ_3 F_3) + dim(𝐑_3,𝐑_d) I(𝐑_2)  tr(λ_2 F_2 ) + 
       dim(𝐑_3,𝐑_2) I(𝐑_d)  tr(λ_d F_d) + dim(𝐑_3, 𝐑_2, 𝐑_d) 36 q^2  λ_1 F_1 )  ,

where I(𝐑) is the Dynkin index such that tr (T^a_𝐑 T^b_𝐑) = 12 I(𝐑) δ^ab and the sum runs over representations of all chiral fermions on Σ_2. Note that these localized fermions can also arise as localized zero modes of Dirac fermions in four dimensions and therefore might be sensitive to even heavy exotic matter particles, see also <cit.>. Demanding that

    Δ S + Δ S_f = 0  ,

one finds the consistency conditions

    ∑_𝐑dim(𝐑_3, 𝐑_2, 𝐑_d)/dim(𝐑_i) I(𝐑_i) = κ_i  ,    36 ∑_𝐑dim(𝐑_3, 𝐑_2, 𝐑_d) q^2 = κ_1  ,

for non-Abelian and Abelian variations, respectively. Therefore, we see that if κ_d does not vanish the axionic strings necessarily host chiral degrees of freedom that are charged under the dark sector.

Similarly, we can perform a 1-form gauge transformation in the topological action. Let us perform this analysis explicitly with the example of a single gauge factor SU(n) on which 𝒵 = ℤ_k acts non-trivially. Expressing the H^2(M;𝒵) gauge field as continuous b, see (<ref>), one can rewrite the topological coupling using (<ref>) as

    ℒ^a_top⊃i κ8 π^2 a  tr((F'-b1) ∧ (F'-b 1))= i κ8 π^2 a ( 
    tr(F' ∧ F')-n b∧ b ) ,

where the F' is a U(n) gauge field as in (<ref>) . Moreover, locally we can express the closed 2-form field b in terms of a U(1) 1-form gauge field c as

    k b = d c  .

The 1-form gauge symmetry transformations are then implemented by

    A' → A' + Λ1 ,    b → b + d Λ ,    c → c + k  Λ ,

with 1-form gauge parameter Λ, see <cit.>. After expressing b in terms of c we can aim to use the same trick as for the 0-form symmetries above, i.e., integrating by parts then doing the gauge variation and integrating by part once more to find the localized anomaly. However, we find that after integration by parts

    S ⊃ - i κ/2π∫ da ∧( Ω - n4 π k^2  c ∧ dc )

the action is invariant under 1-form gauge transformations and there is no inflow of an anomaly for the 1-form symmetries at the local level. Analogously, with a product of multiple simple gauge groups like in our setup, one has a cancellation for each separate gauge sector, analogous to (<ref>). 

Once more this analysis shows what a diverse system, in terms of its symmetries, anomalies, and defects, the axion model is.



§ PHENOMENOLOGICAL CONSEQUENCES



The fact that the axion now couples to more than the Standard Model gauge groups also has consequences on the phenomenological implications, which we want to mention in the following. This is particularly important for the confining scenario, since it leads to major modifications to the axion potential of order Δ V_a∼Λ_d^4, which is the model-dependent dynamical scale of the dark sector. In the scenario of a broken dark sector gauge group all corrections are suppressed by the symmetry breaking scale Γ_d and therefore can be made small.



 §.§ Solution to the strong CP problem


Famously, the QCD axion offers a dynamical solution to the strong CP problem <cit.>, i.e., its vacuum expectation value minimizes the CP-violation generated by a bare QCD θ-angle, whose effective value is experimentally constrained to be

    θ≲ 10^-10 .

This works if the axion potential is generated by the strong QCD dynamics but can be modified after the inclusion of further interactions, this is also known as the axion quality problem, see, e.g., <cit.>.

For a broken dark sector gauge group the corrections are suppressed by Γ_d and it depends on the details of the setup, whether the corrections are small enough to allow for a dynamical solution for the strong CP problem. For a confining dark sector the situation is significantly worse, since there will be corrections to the axion potential of order Λ_d^4, for Λ_d a strong coupling scale of the dark sector,  which will generically shift the minimum of the potential away from θ = 0. The only scenario for which this does not seem to happen is if the contributions to the axion potential from both sectors align in such a way that the minimum remains at the CP preserving value for QCD. For example this would require an alignment in the bare theta angles θ and θ_d to a very high precision, which seems not to be natural without additional symmetries demanding that.



 §.§ Axion domain wall problem


As discussed above, if the axion only couples to the Standard Model, there are stable axion domain walls for κ_3 > 1. This is due to the fact that the axion potential has periodicity 2 π/κ_3 and thus there are |κ_3| distinct minima. Axion profiles that interpolate between two of such minima form stable domain walls.[These domain walls become unstable once the total shift in the axion value is a multiple of 2π for which they can form holes bounded by axion strings.] The energy density of a single domain wall is so high that it would lead to inconsistencies in the cosmological evolution of our universe, which is called the axion domain wall problem <cit.>, which also can be phrased in terms of domain wall networks see, e.g., <cit.>. A solution to this problems is to set κ_3 = 1, which we have also done for the analysis in Section <ref>, in which case there are no stable domain walls. Alternatively, one can dilute the density of these objects by mechanisms like inflation, if they form early enough.

Once there is a confining dark sector there are important further contributions to the axion potential and one finds various parameter regimes according to the hierarchy between the two confinement scales Λ_QCD and Λ_d:

    
  * Λ_QCD≫Λ_d: In this regime the axion potential is dominated by the dynamics of the strong force. This also applies for the number of (approximate) minima and we expect to have a potential domain wall problem for κ_3 > 1. Nevertheless the dark sector contributions to the potential generically lift the individual minima, which might extend the allowed parameter space for phenomenologically viable models.
    
  * Λ_d≫Λ_QCD: The axion potential is dominated by the dark sector potential and therefore κ_3 > 1 does not necessarily lead to a domain wall problem. However, there can be a domain wall problem in case κ_d > 1.
    
  * Λ_QCD∼Λ_d: In this regime the form of the potential depends on many of the details, such as the bare θ angles of the system, and one needs to analyse the specific models.

We see that the presence of a dark sector can weaken the axion domain wall problem and open up new parameter regimes. It even offers a solution to the axion domain wall problem without requiring κ_3 = 1.



 §.§ Dark matter


Axions also have appeared as viable dark matter candidate <cit.>, but it is not clear what the fraction of dark matter formed by axions is. This contribution of axion dark matter will be highly sensitive to the specific scenario and therefore requires a model dependent analysis, however, the presence of a dark sector typically opens up new interesting regions of parameter space. Moreover, in the case of a confining dark sector one further has natural other candidates for dark matter at mass scales set by Λ_d, which can correspond to dark baryons, or dark glueballs in the absence of matter states, <cit.>. Of course for that it is important that the confined dark objects making up dark matter are neutral under the Standard Model gauge group and are produced appropriately.




Summarizing, the presence of the dark sector poses a challenge for the dynamical solution of the strong CP problem, but might open up new phenomenologically interesting regimes for viable model building. 



§ CONCLUSIONS



In this manuscript we explored several effects of having a non-Abelian dark sector gauge group mixing topologically with the Standard Model fields. The specification of the global form of the gauge group imposes restrictions on the allowed matter representations and therefore influences the spectrum of potential exotic matter particles. Moreover, the topological mixing extends the allowed gauge field configurations which need to be included in the partition function, these include the appearance of fractional instantons that influence the dynamics of axion fields that couple to the instanton density. The periodicity of the axion leads to quantization conditions of these couplings which in turn modifies the possible values of the axion-photon coupling. In this way the presence of a dark sector can lead to a significant reduction of such a coupling compared to the Standard Model analysis, which opens up a larger parameter space. Finally, the topological mixing further influences the symmetries and defects of the axion system which further influences its phenomenological implications.

The setup we study allows for various generalizations. For example one could include several dark sector gauge groups that mix topologically with the Standard Model. This would further modify the quantization conditions and can even further reduce the axion-photon coupling. Nevertheless, since the fractional instanton numbers are rather constrained, we expect that mixing with a single gauge group already captures most of the allowed reduction. Of course the freedom to let part of the dark sector confine and break another part further increases the flexibility in phenomenologically interesting scenarios.

A more drastic modification is the inclusion of more than one axion field, which is well motivated from a string theory perspective, see, e.g., <cit.>. In this case the periodicity condition for each of the axion fields will lead to quantization conditions for its topological coupling. If the dark sector confines at a high scale, one can integrate out the heavy axion combination coupling to its instanton density, with the other light linear combinations only coupling to the Standard Model gauge fields. Since κ_d for the remaining axion vanishes, they are only sensitive to the topological mixing within the Standard Model gauge group, consistent with decoupling.[We thank Matt Reece for emphasizing this aspect.] Moreover, the now larger dimensional field space for the axion potential allows for a richer structure of minima and might save axions as a dynamical solution to the strong CP problem. Thus, it would be interesting to see the effect of topological mixing, which is very common at least in a large number of dimensions <cit.>, see also <cit.>, in explicit string theory constructions with axions, which also allows for a statistical analysis <cit.>. This might also shed some light on the fate of the rich variety of generalized symmetries of axion systems in the context of a quantum gravity theory.

While in this work we focused on the general properties and implications of topological mixing with a dark sector, the next step would be to analyze promising specific models in more detail. This includes an estimation of the dark matter abundance in a confined dark sector scenario <cit.> as well as estimates for the stability of axion domain walls <cit.>. Also a hierarchy of breaking scales, now involving the characteristic energy scales of the dark sector, along the lines of <cit.> offers the possibility of constraints on allowed generalized symmetries for axion systems with viable phenomenology.

Throughout our analysis we have restricted to spacetime manifolds that allow for a Spin structure. While for the Standard model this seems to be the natural choice, one might ask which conclusions change in more general setup. In particular, theories for which spacetime does not have to be orientable, see <cit.>, or in which the Spin structure is replaced by a more general tangential structure that allows for (charged) fermions <cit.> are natural places to look for extensions of the models with Spin structure considered here.

Finally, it would be very interesting to explore a top-down explanation of when the bare θ-angles of different gauge sectors have to be aligned, or whether there exist mechanisms to guarantee such an alignment. Such a mechanism might also point towards alternative solutions to the strong CP problem which does not need the introduction of dynamical matter fields, see, e.g., <cit.>.



§ ACKNOWLEDGEMENTS


MD thanks the ESI in Vienna, in particular the program “The Landscape vs. the Swampland”, for hosting him during part of the time in which this work was completed. MD would like to thank Jakob Moritz for fruitful discussions. We are especially grateful to Matt Reece, for very valuable comments on the draft.







§ FRACTIONAL INSTANTONS FOR EMBEDDING OF SUBGROUPS



In this appendix we investigate the fractional instanton number in situations in which the map from the quotient group 𝒵 into the center of the gauge group factor G_i only produces a subgroup of 𝒵_G_i.

If G_i is non-Abelian the only relevant groups for which 𝒵_G_i admits non-trivial subgroups are, see Table (<ref>),

    𝒵_SU(n) = ℤ_n  ,   𝒵_Spin(4n) = ℤ_2 ×ℤ_2  ,   𝒵_Spin(4n+2) = ℤ_4  ,

for which we analyse the fractional instanton numbers individually.

For G_i = Spin(4n) there are three different ℤ_2 subgroups of the center generated by

    (1,0)  ,    (0,1)  ,    (1,1)

as elements in ℤ_2 ×ℤ_2. Identifying the background fields of the two ℤ_2 factors with B_L and B_R, and introducing ω∈ H^2 (M;ℤ_2) as a summation index in the partition function, we can use Table (<ref>) to find

    (1,0):      n^frac = n4∫𝒫(ω)  mod  1  , 
    
    (0,1):      n^frac = n4∫𝒫 (ω)  mod  1  ,

which on Spin manifolds can only be fractional, with non-trivial values 12, for n ∈{ 1 , 3} mod 4. For n = 1 this has a simple explanation, since Spin(4) ≅SU(2) ×SU(2) and the ℤ_2 factors appear as center symmetries for the SU(2)'s. For the last embedding we find

    (1,1):    n^frac = 12∫ω∪ω ,

which is not fractional on Spin manifolds.

For G_i = Spin(4n+2) one has non-trivial subgroup ℤ_2 and the summation is given by the embedding

    B = 2 ω ,

where B ∈ H^2(M;ℤ_4) and ω∈ H^2(M;ℤ_2). Assuming ω has a lift to an element in H^2(M;ℤ), denoted by ω_ℤ one has

    ∫𝒫(2 ω) = 4 ∫ω_ℤ∪ω_ℤ mod  8  ,

which vanishes on Spin manifolds. So there are no fractional instanton numbers for gauge group Spin(4n+1)/ℤ_2.

Finally, we consider G_i = SU(n). For that to have non-trivial subgroup n cannot be prime and we want to consider fractional instanton numbers for gauge group SU(n)/ℤ_k. This happens in case the embedding parameter ℓ_i satisfies

    gcd(ℓ_i, n) ≠ 0  ,

and we can identify

    k = n/gcd(ℓ_i,n) .

We can define the 1-form symmetry background via an element ℓ_i  ω∈ H^2(M; ℤ_n) with ω∈ H^2(M; ℤ_k).
 With ℓ_i  ω we can use the usual formula as indicated in Table (<ref>) 

    n^frac = n-1/2n∫𝒫 (ℓ_i  ω) = ℓ_i^2  n-1/2n∫𝒫 (ω) = ℓ_i^2/(gcd(ℓ_i,n))^2 n (n-1)/2 k^2∫𝒫 (ω)  ,

for which the fractional parts match. The first 𝒫 is considered with respect to a ℤ_n class, while in the latter two with respect to ℤ_n class.[One should lift ℓ_i ω to its integral representative and use co-chains instead of co-cycles to make sense of the transition <cit.>.]

For Abelian G_i = U(1) gauge groups for which the center is the group itself, it is clear that the image of 𝒵 is just a subgroup defined by

    𝒵⊃ℤ_n:    1  mod  n ↦ e^2 π i ℓ_i/n .

The calculation follows that around (<ref>) and shows that the fractional part of the Abelian instanton is given by

    n^frac = ∫( ℓn c_1(F) ∪ω + ℓ^2n^2𝒫(ω) )  .












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