Coupling a Cosmic String to a TQFT

A common framework of particle physics consists of two sectors of particles, such as the Standard Model and a dark sector, with some interaction between them. In this work, we initiate the study of a qualitatively different setup in which one of the sectors is a topological quantum field theory (TQFT). Instead of particles, the physics of a TQFT only manifests itself in non-trivial spacetime topologies. Topological defects provide a natural place to investigate such effects. In particular, we consider two possible ways in which axionic cosmic strings can interact with a Zn TQFT. One of them, by extending the structure of the axion coupling, leads to specific predictions for the localized degrees of freedom on the cosmic string, which can in turn effect their evolution and leave observable signals. The second approach, by gauging a discrete subgroup of the axionic shift symmetry, leads to dramatic changes in the string spectrum. We stress that the scenario considered here should be regarded as a plausible way for new physics to arise since it can be the low energy effective field theory for quite generic scenarios at high energies. To demonstrate this point and further illustrate the physical implications, we constructed such UV completions for both of the cases of couplings to TQFTs. The detailed prediction for observable signals of such scenarios needs further investigation. At the same time, our results demonstrate that there are rich new phenomena in this scenario.


Introduction
There are many proposed scenarios of physics beyond the Standard Model. One universally adopted framework to incorporate new physics is to couple a known particle physics sector, such as the full Standard Model, to a "new physics" sector. The new physics sector includes new particles and these extra local degrees of freedom together with new interactions which may introduce novel dynamics and lead to solutions to existing problems in particle physics.
In studying these new theories, symmetry provides an extremely powerful tool.
Historically, new understandings of symmetry in physics has almost always led to clarifications of existing puzzles and provided new insights. In this paper, we initiate the study of a new class of couplings and analyze them by means of new symmetries. Specifically, we study the effect of coupling a particle physics theory described by a local relativistic quantum field theory (QFT) to a topological quantum field theory (TQFT). This "coupling a QFT to a TQFT" was introduced in [1] (see also [2,3]). Yet, to the best of our knowledge, our current work is the first in considering such possibilities in the context of particle physics. Since TQFT is not characterized by any local excitations, understanding the physics of TQFT and TQFT-couplings requires a new set of tools which is afforded by generalized global symmetry [3].
Our goal is to demonstrate through a couple of simple examples that such TQFTcouplings can lead to non-trivial and interesting phenomenological implications, including possible observable effects, which are more difficult to analyze using traditional methods. We also hope to emphasize that TQFT-couplings, which may appear somewhat exotic, in fact arise as IR discrete remnants of familiar examples of local QFTs. For instance, as described in detail in Appendix B, a U (1) gauge theory with charge a n scalar field flows to a non-trivial Z n TQFT with non-trivial physical observables. This suggests that the UV completions of many theories in particle physics may also have effects from such discrete remnants. In these cases, it is crucial to be able to identify their physical implications and formulate appropriate experimental search strategies. Using generalized global symmetry seems to be the best available technique to accomplish this goal.
Ever since the notion of generalized global symmetry was introduced [3], it has been a very active field of research and has lead to many insights in theoretical QFT (see [4] for a summary and references there-in). Accordingly, it has become increasingly important to determine how to effectively implement generalized global symmetry in the study of particle physics (for example see recent works [5][6][7][8][9]).
In this paper we will apply the techniques of generalized global symmetries to study the effects of coupling a TQFT to axion-Maxwell theory 1 . Axion-Maxwell theory is described by an action 2 (1. 1) and appears frequently in the literature. Here, K A ∈ Z is a discrete coupling constant that matches the U (1) PQ [U (1)] 2 Adler-Bell-Jackiw (ABJ) [10,11] anomaly coefficient of any UV completion -we will have in mind a completion by a KSVZ-type theory [12,13]. Here we will distinguish between the axion-Maxwell sector and TQFT sector by using a subscript A for axion-Maxwell sector and B for the TQFT sector.
For the TQFT sector, we consider a gauge theory associated with a Z n discrete gauge group whose action is given by 3 where, B (2) is a 2-form gauge field (hence two antisymmetric indices) and F (2) (1) is the field strength of a 1-form gauge field associated with a gauge group U (1) B which is restricted to Z n ⊂ U (1) B by the form of the above action. A review of generalized global symmetry is presented in Appendix A and a detailed discussion of this Z n gauge theory can be found in Appendix B. As we mentioned already, this TQFT can arise as the IR limit of the Higgs phase of an abelian Higgs model with a charge n Higgs field.
There are many ways to couple axion-Maxwell sector to a Z n TQFT, each of which lead to distinct physical effects. In Section 2, we discuss the TQFT-coupling via axion-portal given by S TQFT-coupling I = − iK AB 4π 2 f a aF (2) A ∧ F For this coupling, we show that • [ Section 2.1 ] On an axion string, there must exist a set of chiral fermion zero modes to cancel gauge anomalies from the bulk topological interactions (via anomaly inflow).
In particular, the couplings to the TQFT sector implies that those chiral modes must carry Z n charges as well as U (1) A charges.
• [ Section 2.2 ] This theory can be an IR effective field theory of a extended version of KSVZ theory where the KSVZ fermions charged under an additional U (1) B that is spontaneously broken U (1) B → Z n .
• [ Appendix C ] The TQFT-coupling in (1.3) enriches the generalized global symmetry structure and in particular, modifies 3-group structure. A non-trivial 3-group symmetry implies constraints on the renormalization group flow in the form of (para-metric) inequalities among symmetry emerging energy scales of different higher-form symmetries. 4 • [ Section 2 and 4 ] These features may lead to multiple interesting observable effects. Our theory predicts not only axion strings but also strings coming from the TQFT sector. It may also include "coaxial hybrid strings" whose existence and properties have not been studied. The fact that the chiral zero modes living on the string core Z n charges may additionally lead to significant changes in the evolution of the axion string. These in turn might have far reaching consequences such as for "cosmological plasma collider" effects and vorton stability.
In Section 3, we discuss a different TQFT-coupling which can be obtained by gauging a discrete symmetry. Concretely, we describe a gauging of Z M subgroup of (0-form) axion shift symmetry, a → a + 2πfa K A . This leads to a TQFT-coupling of the form where C (1) is the dynamical Z M gauge field of an Z M TQFT and ω 3 (A (1) ) is 3d U (1) Chern-Simons action as defined in eq. (2.6).
The implications of this TQFT-coupling via discrete gauging can be summarized as follows.
• [ Section 3.1.1 ] In order to single out and clarify the important physical features of discrete gauging, we first discuss discrete gauging of free U (1) Goldstone boson theory, i.e. K = 0 case. We show two consequences of discrete gauging of shift symmetry of Goldstone boson φ(x) → φ(x) + c: (i) it projects out some of the local operators I(q, x) = e iqφ(x) and (ii) it adds additional cosmic strings with fractional winding numbers. Specifically, if we gauge a Z M subgroup of the axion shift symmetry, local operators with q = M Z are removed since they are not invariant under Z M gauge transformations. Simultaneously, surface operators (i.e. cosmic strings) with fractional winding dφ 2π ∈ 1 M Z (1.5) are included; these objects are identified as cosmic strings of the TQFT sector.
• [ Section 3.1.2 ] Next, we describe Axion-Maxwell theory which couples to a TQFT via discrete gauging of a Z M subgroup of the axion shift symmetry. In this case, we show how the spectra of local operator and cosmic string are modified in the presence of TQFT-coupling.
• [ Section 3.2 ] We present a KSVZ-type UV field theory which flows in the IR limit to the axion-Maxwell theory coupled to a Z M TQFT as described above. This re-sults further illustrates our claim that non-trivial TQFT-couplings can appear from standard UV field theories when discrete remnants are left out in the IR.
• [ Section 3.3 ] We show how Z M discrete gauging changes 3-group symmetry structure of the theory. In particular, in addition to above described features, the 1-form electric symmetry is broken Z (1) K/M , thus altering the spectrum of Wilson lines.
• [ Section 3.4 ] It is possible to systematically classify the possible ways to couple axion-Maxwell theory to TQFTs via discrete gauging. We list possible discrete gaugings of the axion-Maxwell theory and briefly discuss the resulting symmetry structure for each case.
In summary, the analysis in this paper demonstrates the importance of understanding couplings to TQFTs and tracking them along RG flows in particle physics settings. Further, the fact that the coupled models we consider in this paper have such simple UV completions illustrates that such couplings to TQFTs are not at all exotic or unrealistic. Instead, they can generically arise in the long distance behavior of standard QFTs. We hope that our work demonstrates the utility of the techniques provided by generalized global symmetries in a concrete setup and helps to initiate a broader effort in the application of generalized global symmetry techniques in particle physics.

TQFT-Coupling I: Axion-Portal to a TQFT
In this section, we discuss an axion-Maxwell theory coupled to a Z n topological quantum field theory. The relevant actions are presented in eqns. (1.1), (1.2), and (1.3). For convenience of the discussion, we reproduce them below. The axion-Maxwell sector without a TQFT-coupling is described by an action Here, a is a periodic scalar field (axion) with a ∼ a + 2πf a (which is the remnant of a U (1) PQ global symmetry that is spontaneously broken at a scale set by the axion decay constant f a ) and A (1) and F A are U (1) A one-form gauge field and its two-form field strength, respectively. The coupling constant K A ∈ Z is quantized, which is necessary in order to ensure the periodicity of a, and furthermore is the coefficient of a perturbative U (1) PQ U (1) 2 A Adler-Bell-Jackiw (ABJ) anomaly [10,11]. We would now like to study what happens when we couple axion-Maxwell theory to a TQFT. One can imagine a various of ways to achieve this. A simple choice of TQFT to couple to is a Z n gauge theory [1,3,[19][20][21] which can be viewed as the low energy limit of a spontaneously broken U (1) B gauge theory. We can couple such a Z n TQFT to the axion-Maxwell theory via an axion portal coupling 5 (1) is the field strength of a one-form Z n gauge field B (1) and B (2) is a two-form gauge field associated with one-form Z (1) n gauge invariance. 6 The first term in eq. (2.2) is the action for Z n TQFT, often also called a BF theory. It describes a Z n gauge theory and a brief review is presented in Appendix B (see [1,3,[19][20][21] for more discussion). The BF theory sector admits two gauge invariant "electric" operators, a Wilson line and a Wilson surface operators: which act as sources for B (2) and B (1) respectively due to form of the equations of motion.
The second and third terms in eq. (2.2) describe (local) interactions between the axion-Maxwell sector and the TQFT sector. The goal of the rest of this section is to investigate the implications of this TQFT-coupling. In particular, we are interested in properties of the IR effective theory described by S 0 + S 1 which are universal, i.e. independent of specific UV completions.
In the rest of this section we study the implication of the TQFT on the 2d QFT living on the axion-string and how this physics can be realized in a simple UV model. The coupling discussed in this section has the possibility to produce interesting, observational signals. We relegate these details to Section 4 where we more broadly discuss the phenomenological implications of coupling axion-Maxwell theory to TQFTs.

Anomaly Inflow and 2d String Worldsheet QFT
In this section, we discuss the axion strings in the theory described by S 0 + S 1 shown in eq. (2.1) and (2.2), with a special emphasis on the universal IR features.
Imagine a cosmic string placed in the spacetime M 4 . We would like to study the effects on the world volume induced by the axionic couplings The factor of 2 difference between KAB and KB (and KA) terms comes from the fact that the anomaly polynomial is given by ch2(F (2) ) = Tr e iF (2) 2π restricted to the degree 4 differential form where the trace is over the total bundle. The cross term comes with a factor of 2 due to the fact that a U (1) In fact, it is possible and interesting to consider coupling to a broader class of Zn TQFTs such as [1] To simplify expressions, we often use the language of descent equation of chiral anomaly (see for e.g. [22][23][24]) We comment that both 3d Chern-Simons (CS) action ω 3 and 2d chiral anomaly ω 2 are defined as cohomology classes (shifting by an exact term corresponds to shifting by local counter terms) and expressions shown are canonical choices (see appendices of [24] for a comprehensive review). This allows us to rewrite S axion as In the presence of the axion vortex, neither a nor da is well-defined at the vortex worldsheet r = 0. 7 However, as demonstrated in [25] we can obtain an action well-defined everywhere in M 4 even in the presence of an axion string. This can be achieved by "smoothing out" the axion string singularity by inserting a bump function ρ(r) into S axion : so that ρ(r) = 0 for r > 1/f a and (1 + ρ(r)) → 0 smoothly as r → 0. This insertion of 1 + ρ(r) regularizes the the action in the string core r = 0, extending the validity of our description there. In addition, we recover the original action well outside the vortex due to the second BC. As we discuss below, another advantage of this formulation is that the implementing a bump function regulator allows us to switch from covariant to consistent anomalies on the axion string world sheet so that the anomaly cancellation can be understood straightforwardly [25,26]. In other words, ρ(r) can be thought of as a smooth generalization of a δ-function that describes the embedding of the cosmic string world-sheet M st 2 into M 4 . These conditions on ρ(r) imply that for a cosmic string with winding number m m = Here, δ (2) (M st 2 ) is the two-form δ-function, that is only non-vanishing only on the (1 + 1)d cosmic string world-sheet M st 2 .
7 By this, we really mean that at distance scale smaller than fa, the axion winding is not anymore protected by topology, i.e. there is enough energy fluctuation to unwind the axion. In the deep IR, the region r < f −1 a is represented as a singularity around which axion winds, but in the UV it is a smooth field configuration.

Anomaly Inflow
Now let us consider the axion interaction term S axion . Under a U (1) A gauge transformation, A (1) → A (1) + dλ A , the action varies as Here we see that the variation leads to an anomalous term that is localized on the cosmic string worldsheet. Similarly, the variation of S axion under Z n gauge transformations: Since our theory is well defined everywhere, the anomalies localized on the cosmic string world-sheet in eqns. (2.11) and (2.12) must be canceled which furthermore implies that there must be degrees of freedom living on the cosmic string worldsheet that cancels these anomalies. In terms of 2d dynamics, these anomalies must be reproduced by consistent gauge anomalies [25,26]: a U (1) 2 A gauge anomaly with coefficient −K A , a U (1) A × Z n , mixed anomaly with coefficient −K AB , and a Z 2 n gauge anomaly with coefficient −K B . It is also illuminating to reproduce our discussion on the existence of charged matter on the cosmic string world volume in terms of currents. The current is defined by taking a functional derivative with respect to A (1) , * J 1 (a, A (1) B . (2.13) which satisfies the conservation equation (2.14) in the presence of an axion string with unit winding. This shows that there are indeed charged matter fields that are localized on the world sheet of the cosmic string.
Here there is a subtlety to the matching of the gauge anomalies involving the Z n gauge symmetry. The point is that at short distances, the Z n gauge symmetry can be enhanced to a U (1) B gauge symmetry. In these cases, the Z n anomaly cancellation is only required mod n. We can see this reduction of the anomaly coefficients mod n as follows. First note that any terms of the form with z 1,2 ∈ Z are gauge invariant. This term is clearly U (1) A invariant, so we only need to discuss Z n invariance. Under Z n gauge transformations, the above counter terms transform as Comparing with eq. (2.5) and (2.15), we see that the local counter terms eq. (2.15) allow us to reduce the anomaly coefficients for Z n mod n, which of course is what we expect for discrete anomalies [27].

Anomaly Cancellation by Fermion Zero Modes
As we have shown, the consistency of IR EFT requires that anomalies in eq. (2.11) and (2.12) need to be canceled by 2d QFT on the cosmic string world sheet. These anomalies can be matched by 2d massless chiral fermion that are localized on the string. And indeed, such 2d fields often arise in UV completions as pertrubations around zero-modes of bulk 4d fermions in the presence of cosmic strigs [28]. We will demonstrate this feature of anomaly matching in a particular UV completion in Section 2.2.
Here, we will argue for the existence of 2d chiral fermions on the cosmic string world sheet purely based on IR consistency and determine conditions for their quantum numbers. For a set of (1 + 1)d Weyl fermions {α i (z, t)} living on the core of the cosmic string, if their charges under U (1) A × Z n are (Q i , k i ), the anomaly cancellation conditions are written as This is a straightforward but interesting result. Adding the TQFT-coupling, while not altering the 4d bulk QFT, has modified the 2d QFT on the string worldsheet: anomaly cancellation requires that the chiral degrees of freedom carry Z n as well as U (1) A charges.
In particular, as shown in the second requirement in eq. (2.17), at least some of the zero mode fermions localized on the string must be charged under both U (1) A and U (1) B . As we discuss in Section 4, this property can lead to interesting features in cosmic string physics, which could in principle be observable.

UV Field Theory Completion
In this section, we introduce UV completions that reduce in the IR to the effective theory described by eq. (2.1) and (2.2). We solve the Dirac equation in the vortex background and determine the spectrum of fermion zero modes. Using this, we will check the anomaly cancellation discussed in Section 2.1.2 explicitly.

UV theory
Here we take a U (1) A ×U (1) B gauge theory coupled to scalars and fermions. The Lagrangian is Table 1. Quantum numbers of the fields of the UV completion eq. (2.18). This theory admits the limit Λ f a but not f a Λ.
The scalar potential is such that both Φ 1 = f 1 and Φ 2 = f 2 are non-zero. The quantum numbers of the fields are summarized in Table 1.
The vacuum expectation value (vev) f 2 spontaneously breaks U (1) B → Z n , producing a Z n BF theory, while f 1 spontaneously breaks a combination of global U (1) PQ and U (1) B , which leads to a low energy axion. The fermion covariant derivatives contain U (1) A and U (1) B gauge fields with proper charges.
To get an effective theory below f 1 and f 2 by integrating out heavy fermions, it is convenient to rotate away the phases of scalar fields from the Yukawa terms. We first write the scalar fields as Φ i = ϕ i e iθ i , i = 1, 2 where ϕ i is classical configurations. For vacuum solution ϕ i = f i , while for a string solution (with winding number n i ) the Higgs vev has a non-trivial profile The phases θ i are (would-be) Goldstone bosons and can be removed from the Yukawa terms by the following field redefinitions: which are associated to the global symmetry transformations U (1) 1 and U (1) 2 respectively. These are, however, anomalous transformations and which will generate axionic terms in the effective action.
where the prefactors 1, q and q 2 are respectively anomaly coefficients for B anomalies (i = 1, 2). After these field redefinitions, the Yukawa interactions become just fermion mass terms at energies below f 1 and f 2 . At these energies, the effective action is eq. (2.1) and (2.2) with the matching The other orthogonal combination of θ's is the would-be Goldstone boson eaten by the U (1) B gauge boson. In the IR, the eaten Goldstone boson, leads to a Z n gauge theory since it has charge n > 1. See Appendix B for more details.
In the limit f 2 f 1 , this theory describes the case where U (1) B is Higgsed down to Z n gauge theory below f 2 generating BF strings. Provided that λ 2 is not too small, the pair ψ 2 and χ 2 are integrated out around f 2 , and the effective theory at f 2 E f 1 is a KSVZ-type theory [12,13] coupled to a Z n BF theory. This latter coupling is in the form of anomalous terms proportional to θ 2 in eq. (2.21). Here, θ 2 encodes the 2-form BF degree of freedom B (2) via the 4d duality relation dθ 2 ∼ * dB (2) so that the "BF"-sector of the theory is described by At lower energies with E f 1 , the U (1) PQ is broken by the vev of Φ 1 , resulting in the physical axion field in S axion (see eq. (2.22)).
In cosmology, at temperature f 1 < T < f 2 , a Z n BF theory arises and BF strings can form according to the Kibble-Zurek mechanism [29][30][31][32]. The BF string is supported by non-zero B (2) magnetic flux through its core which corresponds to the winding of the θ 2 in the dual picture. Equivalently, the BF string defect can be thought of as the Wilson surface operator defined in eq. (2.4).
We make a few remarks about the Wilson operators of the BF theory. BF theory has that are charged under a 1-form Z n global symmetry which acts as B (1) → B (1) + 2π n λ (1) , where λ (1) is a 1-form with integer periods. These operators can be cut by Z n charged fermions -in our model {ψ 1 , χ 1 }. Therefore, the topologically protected line operators are characterized by Z GCD(q,n−q) .
Additionally, the theory has Z n -classified string/surface operators W 2 = e i Σ 2 B (2) that are charged under a 2-form Z (2) n global symmetry which acts as B (2) → B (2) + 2π n λ (2) where λ (2) is a 2-form with integer periods. They may be cut and broken by creation of monopole-anti-monopole pair. In the absence of monopoles in the theory, all Z n BF strings are topologically stable. 8 These Z n BF strings are quasi-Aharonov-Bohm strings according to the classification by Polchinski [21,33]. Indeed, the light fermions at f 1 < E < f 2 give rise to discrete Aharonov-Bohm phases when they circle around the BF strings. These strings are "quasi" because the probe states are not only charged under Z n ⊂ U (1) B , but they are also charged under low energy unbroken gauge group U (1) A .
At energies below f 1 , all charged fermions are integrated out. Since there are no charged states, at this scale BF strings become local strings which do not have any topological charges probable by observers at spatial infinity. In addition, due to the U (1) PQ breaking, we get global strings which is measured by a topological charge: the axion winding. 9 Our theory with charge assignment given in Table 1, however, does not allow for the Z n TQFT to emerge below PQ symmetry breaking. Note that as we try to take the other limit f 1 f 2 , U (1) B is still broken to Z n since both scalars carry U (1) B charge n. However, it is possible to construct models which have f a Λ BF .

Fermion Zero Modes
In this section, we solve the Dirac equation of the UV theory eq. (2.18) with charge assignment Table 1, and determine spectrum of fermion zero modes, thereby demonstrating the anomaly cancellation. Our analysis below closely follows [34]. The equations of motion for the pair {ψ 1 , χ 1 } are given by We look for a solution in the background of string Φ 1 = F 1 (r)e iφ in a form 10 where α 1 and η 1 are zero modes localized on the string core and β 1 and ξ 1 are transverse zero modes. We split the Dirac operator into string-and transverse-part i / D = i / D s (z, t) + i / D T (x, y) and we also define chirality operators Γ int = γ 0 γ 3 and Γ ext = iγ 1 γ 2 that act on the string-core and transverse space, respectively.
Let us first solve for the transverse zero modes, which satisfy For φ-independent solution, the transverse Dirac operator can be written as and the Dirac equations become in the background of winding number one string The angular dependence requires that and with this the radial part can be straightforwardly integrated to obtain of domain walls bounded by the axion string. This then destabilizes the axion strings because such a domain wall will tend to shrink the string or a hole of closed string can be nucleated inside of the wall. 10 The case with arbitrary winding number can be studied by adopting the method of [28]. Table 2. Quantum numbers of string zero modes in the Φ 1 -string of winding number one (for α 1 ) and the Φ 2 -string of winding number one (for α 2 ).
Using transverse zero mode equations, the equation for the string zero mode can be shown to be and this shows that non-trivial string zero mode exists if and only if α 1 = η 1 , that is there is only one string zero mode per pair {ψ 1 , χ 1 }. The string zero mode equation becomes (2.34) Recalling γ 5 ψ 1 = −ψ 1 , eq. (2.31) implies that α 1 should be 2d LH chiral fermion, Γ in α 1 = −α 1 . The zero mode equation is then solved by for an arbitrary function g. This means that 2d LH chiral zero mode propagates in the −ẑ direction at the speed of light. Overall, in the background of Φ 1 -string of winding number one, the pair {ψ 1 , χ 1 } coupled to Φ 1 gives rise to a single LH 2d zero mode traveling in the −ẑ direction at the speed of light. On a anti-Φ 1 string, the same procedure shows that β 1 should be a negative helicity state while α 1 is now a positive helicity mode running in the +ẑ direction. The analysis for {ψ 2 , χ 2 } coupled to Φ 2 is follows similarly with one exception: while {ψ 1 , χ 1 } couples to Φ † 1 , {ψ 2 , χ 2 } couples to Φ 2 (see eq. (2.18)). This has an effect that the pair {ψ 2 , χ 2 } coupled to winding number one Φ 2 -string behaves like they couple to anti-Φ 2string (winding −1). Practically, we need to do the replacement e iφ → e −iφ in eq. (2.29) and (2.30) with 1 → 2 relabeling. The final result is the pair {ψ 2 , χ 2 } in the background of winding number one Φ 2 -string has a single RH zero mode localized at the core which propagates in the +ẑ direction.
The quantum numbers of string zero modes are read off from those of original fermions after taking into account the field redefinitions eq. (2.20) (note that the Dirac equations are in the basis where all the phase degrees of freedoms are removed from the Yukawa interaction). 11 The final results are summarized in Table 2.

Anomaly Cancellation
Recall from eq. (2.21) that at E < f 1 , f 2 , the phase factor appearing in the anomalous term is the combination θ 1 − θ 2 = a/f a . Here, θ 1 measures the winding of Φ 1 -string and θ 2 measures that of Φ 2 -string. Let us denote the winding numbers as {n 1 , n 2 }. We are interested in checking the anomaly cancellation in the background of winding number {n 1 , n 2 } string. In this case, the bump function satisfies Using this, anomalous variation of S axion is found to be (2.38) One immediately notices that for n 1 = n 2 , there is no localized anomaly on the composite cosmic string. The reason is that the 2d fermions α 1 and α 2 form a vector-like pair so that when n 1 = n 2 (i.e. there are equal number of α 1 and α 2 fields) the 2d QFT is completely vector-like. In fact, for n 1 = n 2 , one may identify Φ 1 = Φ 2 in the action and realize that the theory is just Witten's vector-like theory of superconducting string [34], augmented by a coupling to U (1) B . For n 1 = n 2 , there are non-vanishing anomalies from the bulk term. The 2d chiral anomalies from the string zero modes are given by Here, we can see that the 2d anomalies cancel the bulk anomalies exactly mod n which, taking into account the gauge invariant local counterterms eq. (2.15), leads to full anomaly cancellation from the bulk variation eq. (2.11) and (2.12).

TQFT-Coupling II: Gauging a Discrete Subgroup
In this section, we discuss alternative ways to couple the axion-Maxwell theory to a TQFT. In Section 3.1 we consider a coupling of axion-Maxwell theory to a Z M BF theory which effectively gauges the Z M subgroup of axion 0-form shift symmetry. As we show below, this TQFT-coupling modifies the cosmic string spectrum in an interesting way, which can be in principle observable. This coupling also changes the 3-group symmetry structure as we show in Section 3.3. We then further generalize this discussion to classify all possible TQFT-couplings via discrete gauging in Section 3.4. We begin with a charge K axion-Maxwell theory The global symmetry structure of this simple theory, without gauging the subgroup K , is quite rich. It is discussed in detail in [5,14,15] and follows from the analysis of Appendix C by removing all terms involving K B and K AB . For reader's convenience we give a quick summary below.
The symmetry structure of the charge K axion-Maxwell theory is comprised of K axion shift symmetry: without the axion coupling to gauge fields, theory has a U (1) (0) shift symmetry a → a + cf a , c ∈ S 1 with a corresponding current (2) axion winding symmetry: this symmetry is dual to the 0-form shift symmetry and has the corresponding current * J 3 = 1 2πfa da. This symmetry is a consequence of the Bianchi idenentity d 2 a = 0 and acts on 2d cosmic/axion strings.  4. 1-form U (1) (1) magnetic symmetry: this symmetry is dual to the electric 1-form symmetry and has the corresponding current * J 2 = 1 2π F . This symmetry is a direct consequence of the Bianchi identity dF (2) = 0 and acts on 't Hooft lines which can be thought of as the IR-realization of massive, stable monopoles of the UV theory.

1-form Z
In the following, we will refer to Z (0) K as "electric symmetries" -since they shift local fields -and the U (1) (1) , U (1) (2) as "magnetic symmetries" because they are both dual to an electric symmetry.
In order to analyze these symmetries, we will couple to background gauge fields of the global symmetries listed above. The naive coupling leads to where, in the last line, we have written the axion-coupling on an auxiliary 5-manifold N 5 bounding our 4d spacetime M 4 : ∂N 5 = M 4 . This presentation makes the theory manifestly invariant under background gauge transformations up to terms that are independent of the dynamical fields. 13 However, the above presentation is in fact dependent on the choice of N 5 due to the fact that background gauge fields A (1) e and B (2) e are Z K -valued, hence the integrand in the last line evaluated on a closed 5-manifold is not 2πZ.
The theory can be made independent of N 5 by modifying the "magnetic" symmetries so that instead of the standard U (1)-transformations, the background gauge fields A additionally transform under the electric symmetries: Note that there is a Z (1) K that participates in the 3-group global symmetry rather than the Z which is a genuine 1-form global symmetry.
These modified symmetries then have modified field strengths G (4) , H (3) : so that the action is written and is invariant under the choice of N 5 . The modified transformation rules in eqs. (3.3) - (3.4) show that the axion-Maxwell theory possesses a 3-group symmetry. Loosely speaking, an n-group is constructed from 0-, 1-,· · · , (n − 1)-form global symmetries where the p-form symmetries mix non-trivially with the q-form symmetries where q > p. In our case, eq. (3.5) shows that the Z K symmetry turns on field strength of U (1) (2) symmetry and similarly, eq. (3.6) shows that the Z K has no ABJ-anomaly, it is a good quantum symmetry and can be gauged 14 . Here, we will study the gauging of a subgroup Z K and its implications. 13 The additional terms that are generated by background gauge transformations that are independent of the dynamical fields can be interpreted as 't Hooft anomalies. We will discuss these 't Hooft anomalies in Section 3.3.
14 Here we assume that there is no cubic 't Hooft anomaly. This is allowed, but highly constrains the UV physics.
First, note that gauging the Z M discrete subgroup is equivalent to coupling the original theory to a Z M TQFT. Indeed, this is not specific to discrete gauging and is familiar from gauging continuous symmetries. Imagine a theory with a continuous global symmetry with a current J µ coupled to a background gauge field V µ Gauging this global symmetry is achieved by supplementing the theory with a dynamical gauge field. The above term then describes nothing but the coupling between the original theory and newly born gauge theory sector. In our case, the gauge theory sector is a Z M TQFT and it couples to the theory analogously to the case of a continuous gauge field because the Z (0) M symmetry results from an explicitly broken U (1) (0) symmetry. In order to understand how the discrete gauging effects the axion theory, it is instructive to first study the discrete gauging of free U (1) Goldstone boson.

Discrete Gauging of Free U (1) Goldstone Boson
Consider a U (1)-valued Goldstone boson φ. This field has the standard action 15 This periodic identification of φ can be thought of as a gauge redundancy that transforms a naturally R-valued scalar field to a U (1)-valued scalar field. This theory has a U (1) (0) × U (1) (2) global symmetry corresponding to shift symmetry and winding symmetry. These symmetries have currents Here * J 1 is the momentum density -generates shifts of φ -and * J 3 measures the winding of φ. The charged operators are I(q, x) = e iqφ(x) and S 2 ( , Σ 2 ) (which is the cosmic string operator of charge ) respectively. Now let us consider gauging the Z In this case, we modify the action by coupling to the 1-form Z M gauge field C (1) where D (2) is a 2-form Z M gauge field. This gauging modifies the spectrum of operators in the theory in opposite ways: it projects out local operators (charged under Z (0) M ) and adds "dual" string operators (Z (2) M BF strings). 15 We can alternatively write the action in terms of a canonically normalized field Φ = f φ which satisfies Φ ∼ Φ + 2πf analogously to the axion a ∼ a + 2πfa.
First, note that the gauging identifies This means that the gauging projects out local operators because under the shift φ → φ + 2π M , the local operator shifts as so that it is not gauge invariant unless q ∈ M Z. Alternatively, such a charged operator can be made a gauge invariant operator by attaching a (discrete) Wilson line to it For q = M Z, however, the operatorĨ(q, x) becomes a well-defined local operator because the attached Wilson line is trivial andĨ(q, x) reduces to I(q, x). 16 Similarly, due to the identification (3.12) we are now allowed to have fractional winding numbers: These fractional winding numbers are in a sense "added" to the spectrum -they are constructed by φ passing through 2π/M of the 2π period which is then accompanied by a Z M gauge transformation. This Z M gauge transformation requires a non-trivial gauge background given by an insertion of the BF string operator , (3.17) which are charged under the 2-form Z M global symmetry that shifts by D (2) → D (2) + 2π M λ 2 with λ 2 ∈ 2πZ. These string operators are classified by the Z M charges = 0, 1, · · · , (M − 1).
We can see that the insertion of the BF string operator leads to the fractional φ-winding configurations explicitly by noting that in the presence of the BF string the equations of motion for C (1) are given by: which implies that C (1) 2π = M in the presence of the BF string. Now we can use the fact that the combination dφ − C (1) is Z M -gauge invariant to see that in the presence of the BF 16 As discussed in detail in Appendix B, the line operators satisfy W M = 1 and so for q ∈ M Z, one sees thatĨ(q, x) reduces to I(q, x).
See Appendix B for a detailed discussion.
As far as local physics are concerned, this theory is IR equivalent to the original free periodic scalar. This may be seen explicitly as follows. Consider rescaling: It is easy to show that the theory with discrete gauging eq. (3.11) becomes The second term is recognized as a trivial theory (it is a Z M gauge theory with M = 1) and it effectively gauges none of U (1) shift symmetry ofφ. Accordingly,C (1) is a background (as opposed to dynamical) gauge field of U (1) shift symmetry. One can easily check thatφ has 2π periodicity and its winding number is given by dφ ∈ 2πZ.
One may then wonder if Z M ⊂ U (1) discrete gauging has no physical consequences. The answer is: discrete gauging does have important physical implications and in current example it is captured by cosmic string physics. In terms of original variables, we showed that in addition to the original global strings, the discrete gauging adds Z M -classified local (BF) strings. While the tension of the former is of the order ∼ f 2 , the tension of the latter is the square of the scale at which the BF theory is born Λ 2 BF . In general, this second scale can be parametrically larger than f and therefore, the varying tension of the cosmic string spectrum can encode the effect of Z M gauging.

Discrete gauging of Axion-Maxwell Theory
Now let us return to axion-Maxwell theory and discuss the partial gauging of discrete K . After gauging, the action becomes where C (1) is the dynamical gauge field for the Z K is a non-linearly realized discrete symmetry and the gauging, therefore, corresponds to gauging a Higgsed discrete symmetry. This is in general a result of the fact that the axion is the pseudogoldstone field for a spontaneously broken (anomalous) U (1) symmetry of which Z (0) However, even though we are gauging the axion shift symmetry, we still have a continuous field with dynamical excitations. While gauging of U (1) global symmetry would have removed all of the fluctuations of periodic scalar, here the gauging is applied only to a discrete subgroup and it turns only "measure-zero" part into gauge degrees of freedom, leaving most of the continuous excitations intact.
As in the case of the free periodic scalar, the discrete gauging removes some of the local operators and increases the number of surface operators. First, consider the allowed local operators. Without gauging, the charged objects under the 0-form Z (0) K is local operators I(q, x) = e iq a(x)/fa with a charge q ∈ Z (recall that the axion is 2πf a -periodic). It shifts as under global Z because they are now gauge equivalent to winding configurations that traverse a complete period. Cosmic strings with integral winding are global axion strings, and the ones with fractional Z M -valued windings are BF strings. As discussed in the previous section, this can be seen explicitly by noting that the equation of motion for a implies that which is activated by an insertion of BF surface operator W 2 (Σ 2 , m) = e im Σ 2 D (2) . At this point we recall that the scale at which the Z M gauge theory emerges, 17 Λ BF , should be generically higher than the scale where the axion is emerges from a spontaneously broken, anomalous U (1) symmetry, f a . The reason for the hierarchy Λ BF f a is simply because, in any UV completion, a discrete global symmetry in the low energy EFT (nonlinearly realized or otherwise) can only be gauged if the gauge field is also discrete at or above the scale where the symmetry emerges: Λ sym Λ EFT . Therefore, one expects on a general grounds that Λ BF f a . This hierarchy has a measurable effect on the spectrum of cosmic strings in the theory. In the Z (0) M -gauged theory, cosmic strings consists of global axion strings with tension T ∼ f 2 a and Z-valued winding numbers, and BF strings with tension T ∼ Λ 2 BF and "winding 17 Here we insist on the existence of a UV completion which only contains continuous gauge symmetries. The scale ΛBF is the scale at which the continuous gauge symmetry is broken to ZM . Table 3. Quantum numbers of the fields of the UV completion eq. (3.27). Below E < Φ I , this theory matches to our effective theory.

UV Field Theory Completion
In this section, we show that the axion-Maxwell theory coupled to a TQFT as in the previous section can arise as a long distance description of a local QFT. This result demonstrates that TQFT couplings and their associated remarkable features can appear rather ubiquitously in a broad class of particle physics models.
In order to formulate a UV completion, we recall that coupling to the TQFT can be thought of as gauging a discrete subgroup of the axion shift symmetry. If we take the KSVZ UV completion of axion-Maxwell theory, the discrete gauging corresponds to gauging We can then try to couple to the Z M TQFT via an Abelian Higgs Model which breaks a gauge symmetry U (1) B → Z M . Requiring that the emergent Z M gauge symmetry act on U (1) PQ requires that we have the UV symmetry structure: where we used the same notations for gauge groups as in Section 2: U (1) A for unbroken electromagnetism and U (1) B for the gauge group that is spontaneously broken The second factor means that Z M ⊂ U (1) PQ is redundant and can be undone by means of This means that the true global symmetry of the theory is given by projecting out the Z M gauge symmetry from the U (1) PQ so that Below, we will present a theory that realizes this symmetry structure. 18 The local strings of BF theory are not really defined by a winding number as they do not possess any topological charge [33]. Rather, they are measured by a conserved magnetic flux m = 1 2π C (1) . In an Abelian Higgs model, however, we have (dϕ − C (1) ) ∈ Z and the magnetic flux is transfered to the would-be Goldstone boson as a winding number.
Consider a theory whose Lagrangian is The scalar potential is such that both Φ 1 , Φ 2 , and Φ 3 are non-zero. We will assign gauge and global symmetry charges to the fields which are summarized in Table 3. These choices of quantum number are tightly constrained by multiple requirements: (i) In order to construct a well defined UV theory, we first choose the charges of the gauge symmetries U (1) A and U (1) B so that they are anomaly free. The choices above ensure that cubic gauge anomalies vanish.
(ii) In order to produce an axion, we demand that U (1) PQ is the only sponatneously broken global symmetry with any ABJ anomalies and that it only has an ABJ anomaly with U (1) A . The ABJ anomalies for U (1) PQ are given by 19 By a simple counting argument, there are three remaining U (1) global symmetries which are parametrized by U (1) F and two vector-like symmetries U (1) V 1 and U (1) V 2 (not present in Table 3). The vector-like symmetries U (1) V i act only on the fermions and hence are not spontaneously broken by Higgs condensation. On the other hand, the global symmetry U (1) F is spontaneously broken and has vanishing ABJ anomalies.
(iii) In order to realize non-trivial overlap described by , we need the charges of U (1) PQ to reduce to those of U (1) B mod M . This can be easily verified upon inspection of Table 3.
The theory described by the Lagrangian in eq. (3.27) with charge assignments in Table 3 manifestly has the symmetry structure in eq. (3.25).
To reproduce the axion-Maxwell coupled to a Z M TQFT in the IR, we choose the parameters of the scalar potential such that   In this UV completion, we find that Λ BF = Φ 3 and f a ∼ Φ 1,2 and that we respect the hierarchy Λ BF f a . However, there are other limits of this UV theory we can consider -for example by trying to "invert" the hierarchy. We will find that when we invert the hierarchy, the RG flow will lead to a different IR theory, thus side-stepping the constraints from EFT considerations.
Let us first consider the limit is completely broken. Now the intermediate theory is given by a KSVZ theory. In this KSVZ theory, the U (1) PQ and U (1) F symmetries have become degenerate as shown in Table 4 Instead, the physics of condensing Φ 1 mandates that we redefine the global symmetries U (1) PQ , U (1) F so that they are orthogonal to the broken gauge symmetry U (1) B , such as U (1) P Q , U (1) F with charges defined in Table 4. 20 The reason we should redefine the global symmetries is that by breaking U (1) B , we are required to implicitly project onto orthogonal global symmetries so that the transformations of the remaining fields do not move along the broken, gauged direction. Now consider the other limit where is broken down to Z M +1 and the intermediate theory is described by a KSVZ model coupled to a Z M +1 BF TQFT. The charges of the resulting KSVZ model are given in Table 5.
Due to the symmetry structure: we know that Φ 2 breaks the symmetry to

30)
20 Note that this differs from the case where Φ3 Φ1,2 where the ZM ⊂ U (1)B gauge symmetry, which overlaps with ZM ⊂ U (1)PQ × U (1)F . In this case, we are not required to project onto the orthogonal symmetry because of the overlapping, preserved gauge symmetry.
which completely decouples from Z M +1 . Thus, when we flow to the deep IR Z M +1 is spontaneously broken by Φ 1,3 , and there is a remaining axion with charge 2 and decoupled U (1) goldstone boson. Therefore in our UV completion, we find that if we tune the scales of the theory so that σ = Λ BF /f a → 0, that there will be a phase transition near σ ≈ 1 where the IR theory will have different IR dynamics as described above. Note that we can physically think of the Z M gauging as a being a consequence of the fact that we "chose our symmetries" in a particular way. To see this, note that if we lift condition (iii) above, it is more natural to parametrize the global symmetries by Here U (1) 1,2 correspond to U (1) V 1 ,V 2 which are not spontaneously broken when the Φ I condense. The global symmetries U (1) PQ and U (1) F (generated by Q PQ and Q F respectively) are related to U (1) B and U (1) 3,4 (generated by Q B , Q 3,4 ) as In other words, we have chosen U (1) PQ and U (1) F so that they overlap with U (1) B mod M by construction -this redundancy is in some sense artificially engineered. In this construction, it is clear how the Z M ⊂ U (1) B gauge symmetry effects the resulting IR theory. First, note that after condensing Φ 3 , the charges of remaining fields under Z M are given by By shifting by Z M ⊂ U (1) A , we can see that Z M gauges U (1) 4 and does not act on the U (1) 3 . This implies that the Z M gauging can be decoupled from the axion by a field redefinition although the BF strings are indeed physical.
In summary, we find that in the parameter space spanned by the vevs Φ 1,2,3 , there is only one hierarchy that flows to the axion-Maxwell theory coupled to a Z M BF theory. This hierarchy of scales is given by Φ 3 Φ 1,2 and leads to Λ BF f a , thus reproducing the expectation from EFT and higher-group symmetry considerations.

3-Group
We now return to the 3-group symmetry structure of axion-Maxwell coupled to the Z M TQFT and its associated 't Hooft anomalies. Understanding the higher-group symmetry structure of this theory is essential to understanding all gaugable symmetries, hence all possible non-trivial TQFT-couplings that arise from discrete gauging. We will discuss the set of possible gaugings in the next section. Here, we will focus on the 3-group symmetry of charge K axion-Maxwell theory and the effect of Z In axion-Maxwell theory, the theory contains couplings of the form Since the background gauge fields transform as δA e , the anomalous variation is captured by the 5d anomaly inflow action The first term describes a mixed anomaly between Z K and U (1) (2) , which due to the 3-group symmetry, also leads to a mixed anomaly between Z (0) K and Z (1) K . Similarly, the second term describes a mixed anomaly between Z (1) K and U (1) (1) , which also leads to a mixed anomaly between Z   First, let us consider the effect of the gauging on U (1) (2) . To study this, let us consider turning off B m . Now we can write the anomaly action as In this case, we find that the anomalous variation of the action is given by This implies that gauging Z M extends the periodicity of U (1) (2) to 2πM . This is a consequence of the fact that U (1) (2) is the dual symmetry of Z K global symmetry. The correct way to show that the Z (1) K symmetry is modified is to first choose counter terms so that the action is invariant under Z M -gauge transformations. Then, we see that the variation of the action from inflow is given by (3.39) Because the Z M gauging sums over A 21 Note that the gauging of Z (0) K does not affect the U (1) (1) magnetic 21 Using the fact that B (2) 2π ∈ 1 K Z for B (2) a ZK 2-form gauge field, we see that charge K axion theory has a genuine Z

Other TQFT Couplings via Discrete Gauging
As we have seen, one way to couple a theory to a TQFT is by a gauging a discrete symmetry. It is clear from our analysis above that this process is non-trivial and can lead to interesting features [1]. Given a theory described by a local QFT, potential TQFT couplings of this sort can be systematically analyzed by studying higher-group and associated 't Hooft anomalies. Here, we demonstrate this analysis on charge K axion-MW theory as a concrete example.
As we have discussed in the previous section, charge K axion-Maxwell theory posses a 3-group global symmetry structure. This 3-group involves U (1) (2) axion winding symmetry that is intertwined with Z This is what we have focused so far. From the first term in the inflow action eq. (3. 36) we see that such a gauging turns some of the 't Hooft anomalies among global symmetries into ABJ-type anomalies between background gauge fields of global symmetry and dynamical gauge fields, hence resulting in quantum mechanical breaking of global symmetries. It leads to the modification of U (1) (2) and breaks Z  M ⊂ U (1) (2) . The anomalies then imply that Z (0) K has an ABJ anomaly that completely breaks and U (1) (1) is modified so that it has periodicity 2πM . 22 3. 2-form axion winding U (1) (2) or its subgroup Z which produces K = 2M q 2 . 22 Although Z K is broken as a global symmetry, it participates in a non-invertible symmetry structure. See [16,35,36] for related discussions.
Here the anomaly becomes an ABJ anomaly for Z (0) K that breaks the symmetry completely while leaving the other parts of the 3-group untouched.

1-form magnetic U (1) (1) or its subgroup Z
(1) Here the anomaly becomes an ABJ anomaly for Z (1) K that breaks the symmetry completely while leaving the other parts of the 3-group untouched.
It would be interesting to analyze each of these cases and understand the observable consequences of these discrete gaugings (e.g. local, line, and surface operator spectrum). Furthermore, it is an interesting question if some of all of these TQFT couplings can arise as a long-distance effective description of more fundamental QFTs at short distance scales. We leave this analysis to future investigations.

Brief Comments on Phenomenological Implications
We conclude this paper by briefly commenting on the potential phenomenological implications of the scenarios discussed in this paper. Further studies are certainly required make precise predictions, which we leave for a future work. Our discussion here will be brief and qualitative, highlighting the potential difference with well studied signals of the cosmic strings (see [37] for a review).
We begin with the discussion of the implications which are largely independent of the UV completions. The main focus of this paper is a Higgsed U (1) B gauge symmetry as the origin of a TQFT. However, there is another unbroken U (1) A gauge symmetry in the story. To be more concrete, we will proceed by first assuming it is the SM U (1) EM . A TQFT does not have low energy excitations, hence it is "absent" in the IR. Yet, it still leaves some imprints.
As described in detail in this paper, a main portal can be an axion with coupling to a TQFT, as shown in eq. (2.2). Such a coupling does lead to important differences in comparison with the "usual" well studied axion strings. Potential signals for axion strings with localized fermionic zero modes have been studied [26,34,[38][39][40][41]. However, the emphasis has been on the fermion which is charged under the SM U (1) EM in which case axion strings will be charged up by passing through regions of magnetic field in the universe. In our case, the fermionic zero modes localized on the string are required to carry specific U (1) B charges in addition to the U (1) A charge. This leads to some remarkable differences.
First of all, since U (1) B is Higgsed, there are no macroscopic regions with non-zero U (1) B field. However, if we assume the unbroken U (1) A is the SM U (1) EM , then the axion strings will be similarly charged up by the magnetic fields in the universe. One of the interesting consequences of the U (1) B charges carried by the zero mode fermions is that they can potentially affect the fate of the string loops.
As pointed out in [41][42][43][44][45][46][47][48], charged fermions present on the string can provide a pressure which prevent the string from shrinking, leading to potentially stable final string loops (vorton). However, fermions with EM charges can be expected to decay into SM charged particles which leads to the decay of the vorton [41,47]. Such a decay, however, would not be possible in our case with the absence of the light particles charged under U (1) B , leading to stable vortons with potentially distinct signals.
We emphasize that, from the point of view of the IR theory, the stability could be attributed to a global symmetry which is imposed "by hand". However, in the scenario discussed here, we see that it is a consequence of the TQFT coupling and requirement of anomaly matching. It is possible that the U (1) PQ is also broken explicitly, such as by the intanton effect in the QCD. In this case, a string domain-wall network will form and collapse. The anomaly on the string then implies a certain Chern-Simons theory living on the domain wall [14,49]. Additionally, it has been shown that a charged axion string can influence the evolution of the string network and the dynamics of string domain-wall network [41]. It would be interesting to investigate further such phenomena in our case.
As discussed in Ref. [39], it is expected that axion string is approximated electrically neutral due to the Schwinger pair production of SM light charged particles in the vicinity of the string, in the strong electromagnetic field produced by the charged particles localized on the string. Such an effect would not be effective for the BF charge on the string, since the BF-charged fermions are heavy and the U (1) B is Higgsed outside of the string . Hence, if the axion string is charged up with BF-charged particles, it would not be neutral in the BF charge. At the same time, we do not expect this would lead to macroscopic effect since the U (1) B field is short ranged.
If BF strings are present in the universe, they can in principle lead to different signals as well. It is generically expected that their tension will be different from the axion string. A more unique feature of the BF string is the non-trivial holonomy of the U (1) B gauge field around the string. In principle, BF-charged particles (for example DM candidate), passing around the BF string could experience AB effect which changes their distributions. It remains to be investigated whether this can lead to observable effects.
Thus far we have discussed the possible IR (universal) signals. However, signals that depend on UV completion can be equally important. In addition to enhancing the discovery potential, they also provide complementary information which may eventually lead to a more complete picture. We will briefly mention a couple such possibilities in the following.
The mechanisms discussed in this paper are largely independent of the absolute scales of UV symmetry breaking. At the same time, the potential signals are sensitive to the scales. A large class of signatures of the cosmic strings are through their gravitational interactions, which is highly sensitive to the string tension µ, with the current limit roughly in the range of Gµ < 10 −8 to 10 −9 [50][51][52] and may potential reach Gµ < 10 −10 to 10 −12 in the near future [53]. If the cosmic string discussed here can give rise to such signals, it will certainly provide highly valuable information. This is especially important for the second case of TQFT coupling discussed in the paper, in which the main feature is the varying tension of the string spectrum.
If both the axion string and the BF string are present in the universe, there is potential for richer dynamics. In particular, the existence of two different kinds of strings differs from well studied scenarios. For example, if the symmetry breaking dynamics are such that it is energetically favored for the strings to overlap, they would tend to align rather than cut through each other. This allows for the possibility of producing co-axial strings with different properties and can also potentially affect the evolution of the string network. There can also be interesting differences with the standard axion string story depending on the relative scale of the U (1) B and U (1) PQ breaking. Either one can happen at higher scale in the UV theory for the first type of TQFT coupling. In particular, if U (1) PQ breaks at a higher scale (earlier in the evolution of the universe), there could be an epoch in which the universe is filled with a background of primordial (unbroken) U (1) B field. This can charge up the axion string through the interaction with the zero mode fermions. The influence of the primordial electromagnetic field on the evolution of the axion string network has been studied [41]. It would interesting to generalize it to the case of a primordial U (1) B background field.
We have been assuming that the unbroken U (1) A is the SM U (1) EM . However, U (1) A could instead be a dark photon. Another intriguing possibility is that the gauge boson of the U (1) B is the dark photon. Instead of (or in addition to) coupling to the SM model via a kinetic mixing with the photon, it couples through the TQFT portal described in this paper. Since it is common to assume the dark photon mass is small, this could be an extreme example of the case in which PQ symmetry breaking happens at a much higher scale. It has been pointed out recently [54] that cosmic string associated with dark photon can be produced in a broad range of dark photon production scenarios, such as through the axion coupling in eq. (2.1). Hence, this would be a natural stage to study the interplay between these two kinds of strings and implication of the couplings in eq. (2.2).

A Brief Introduction to Generalized Global Symmetries
The modern notion of global symmetry, in addition to group-like symmetries, also includes non-group-like symmetries such as higher-group symmetry, non-invertible symmetries, and subsystem symmetries. In this section, we provide a short introduction to this notion of generalized global symmetries, with main focus on higher-form symmetries. We refer to [4] and references therein for more extensive and detailed discussion.

A.1 Ordinary symmetry
We begin our discussion by recalling the general properties of ordinary (i.e. 0-form) symmetries a the language that admits straightforward generalizations to higher-form symmetries.
According to Noether's theorem, an ordinary continuous symmetry corresponds to a conserved current ∂ µ j µ 1 = 0 (A.1) In differential-form notation, the conserved current is written in terms of a co-closed 1-form: Here * j 1 is a 3-form that is the Hodge dual of j 1 . By definition, the existence of a symmetry means there are charged objects which transform under the symmetry. In the case of the ordinary (0-form) symmetry, charged objects are local (i.e. 0-dimensional, hence the name 0-form symmetry) operators O(x). For example, they can be elementary or composite field operators. In general, O(x) transform under a symmetry transformation g as where R(g) denotes the representation of g. A set of ordinary symmetry transformations {g} often forms a group G, which can be either continuous or discrete. The group G can be either abelian or non-abelian. More recently, it has become clearer that there exist symmetries whose mathematical structures are not a group such as higher-group [5,14,15,55,56] (as we also discuss in Section 3.3 and Appendix C.2) and non-invertible [7, 16-18, 35, 36, 57-69] symmetries which strictly speaking are described by category theory. 23 In the case of continuous symmetry, it is often useful to analyze the behavior of the theory coupled to background gauge fields. For an ordinary global symmetries, we can couple the conserved current to a background gauge field A µ Here, we used ψ to denote collectively all quantum fields of the theory. 24 In terms of differential forms, we can couple a 0-form continuous symmetry to a background gauge field A (1) as Since a 0-form symmetry has a conserved current that is a 1-form (its dual * j 1 is (d−1)-form in d spacetime dimension), the background gauge field A (1) is a 1-form gauge field. The conservation of j 1 (A.2) then implies that the above partition function is invariant under background gauge transformations A (1) → A (1) + dλ 0 , where λ 0 is a 0-form transformation parameter. 25 23 For applications of higher-group and non-invertible symmetries in particle physics model building, see [7,9]. 24 We emphasize that the theory under consideration needs not have a Lagrangian description. We simply imagine having an action here to streamline the discussion. 25 Here, we imagine an abelian symmetry such as U (1) for simplicity sake. It can be easily generalized to In a symmetry preserving vacua, all non-vanishing expectation values O(x) · · · are invariant under G transformations. The invariance under infinitesimal G transformation implies the Ward identity where R(Q) is the generator for the infinitesimal transformation in the representation R.
Using the Ward identity, one can then define a charge operator that generates the G symmetry in the quantum theory by integrating the dual current * j 1 over a closed (d − 1)- This charge operator Q(Σ d−1 ) is topological in a sense that any correlation function containing Q(Σ d−1 ) it is unchanged under continuous deformations of the manifold Σ d−1 so long as such a deformation does not cross a charged operator.
In the more familiar case of d = 4, we often choose Σ 3 to be a spatial slice at a fixed time on which the Hilbert space is defined, and the above expression becomes However, in general we can choose any closed (d − 1)-manifold to define the charge operator due to the topological nature of the definition in eq. (A.7).
This allows us to define a topological operator for any group element g = e iλ by exponentiating the charge operator called a symmetry defect operator. This operator is topological in the same way as Q(Σ 3 ). More explicitly, suppose Σ d−1 is a small continuous deformation of Σ d−1 without crossing any charged local operators, then the difference of the two symmetry defect operators with the same group element is given by whereΣ is 4 dimensional manifold whose boundary is the union of Σ and Σ . Here we used the fact that g −1 = e −iλ and the conservation equation d * j 1 = 0.
Using similar manipulations, it is easy to show that the symmetry defect operators non-abelian cases for which the background gauge transformation takes the familiar form δ λ Aµ = [λ, Aµ] + ∂µλ. Since higher-form group symmetries are abelian, it is sufficient to focus on the U (1) case.
satisfy the G multiplication law The Ward identity eq. (A.6) then implies that these symmetry defect operators (SDO) implement the G action on charged operators that cross its world volume. For example, if we consider Σ d−1 that links the point x, then wrapping a SDO for the element g on Σ d−1 will act on a charged operator O(x) as 26 We can similarly define symmetry defect operators for discrete 0-form global symmetries. These are again topological operators that implement symmetry transformations on charged local operators. This structure of a global symmetry corresponding to the existence of topological operators can be taken as a definition or, if the reader prefers, we can think of discrete abelian symmetries as being part of some continuous abelian symmetry which is broken to a discrete subgroup (explicit or otherwise), in which case the structure of symmetry defect operators is inhereted from the continuous completion.

A.2 Higher-form symmetry
We can generalize the discussion in the previous subsection to higher-form symmetries. A p-form global symmetry in a QFT defined on d-dimensional spacetime, denoted G (p) , acts on charged objects supported on p-dimensional manifolds (obviously p ≤ d). For continuous p-form symmetry, it has p + 1-form conserved current d * j p+1 = 0 (A. 13) and one can construct the charge and symmetry defect operators as before where now the charge operator and symmetry defect operators are defined on codimension p + 1 manifolds. 27 The conservation of j p+1 ensures that the associated symmetry defect operator is topological -the argument follows analogously to the case of 0-form symmetry described in Section A.1. If the collection of symmetry transformations {g} forms a group G, the products of symmetry defect operators furnish the group multiplication law. For any p > 0, a p-form symmetry group is necessarily abelian -this follows from the fact that there is 26 A (d − 1)-manifold Σ d−1 that wraps a point x is one that can be contracted to a point by passing through x. The relation (A.12) then follows from the topological property of U (g, Σ d−1 ) by contracting it to a point (the trivial operator) and acting on O(x) as Σ d−1 passes through x. 27 For d space-time dimensional space, a codimension p + 1 manifold has d − p − 1 dimensions.
no consistent ordering of co-dimension p + 1 > 2 manifolds to accomodate non-abelian multiplication. 28 A p-form symmetry acts on p-dimensional operators W p (m, Σ p ) where we take the notation that m is the charge. Again, there is an associated Ward identity, which leads to an action of the symmetry defect operator on W p (m, Σ p ) 15) similar to the case of the 0-form symmetry, where Link(Σ d−p−1 , Σ p ) is the linking number of the two manifolds. This equation holds for both continuous (e.g. λ ∈ S 1 for U (1)) and discrete (e.g. λ = 2π/n for Z n ) symmetries.
In the rest of this section, we will demonstrate these points with a simple and tractable example with continuous global symmetries. We will an analogous example with discrete higher-form symmetry (Z n gauge theory) in Appendix B.

1-form electric symmetry and U (1)
(1) m 1-form magnetic symmetry. The action is written where F (2) = dA (1) is the field strength of 1-form gauge field A (1) . The equation of motion and Bianchi identity are written as These equations can be thought of as current conservation equations, where the former is interpreted the conservation of a 1-form electric symmetry and the latter is interpreted as the conservation of a dual magnetic 1-form symmetry. Note that for d = 4, both F (2) and its dual * F (2) are 2-forms, and these are 2-form currents for a pair of dual 1-form global symmetries. The 1-form electric symmetry U (1) e has a 2-form current and symmetry defect operators for g = e iλ ∈ U (1). This symmetry is an "electric" symmetry because Σ 2 * F (2) measures the electric flux through Σ 2 and it acts on the dynamical electric gauge field by a shift e where the transformation parameter λ (1) e itself is a closed 1-form (i.e. a flat gauge connection) normalized as λ (1) e ∈ U (1). 29 28 The reason is that any pair of marked manifolds Σ (1) , Σ (2) of codimension>2 can be freely deformed Σ (1) , Σ (2) −→ Σ (1) , Σ (2) so that Σ (1) ∼ = Σ (2) and Σ (2) ∼ = Σ (1) . For example, any two loops in R 3 can be exchanged by smooth deformations. 29 The relation of the 1-form parameter λ (1) e and λ ∈ 2πZ appearing in eq. (A.18) is the following.
The gauge invariant operators that are charged under U (1) (1) e are the Wilson line , where m ∈ Z. It is acted on by a non-trivial linking with the symmetry defect operator where Link(Σ 2 , Σ 1 ) denotes the linking number between Σ 2 and Σ 1 (see [20] for an explanation of linking number in the language of QFT).
The 1-form electric symmetry can be explicitly broken by coupling to electrically charged fields. The presence of those electrically charged particles modifies the equation of motion to which violates the conservation law for J e 2 . Here, j charge is the Hodge dual of the "usual" 1form momentum density for the charged particles. If the particles have charge n, the source in eq. (A.21) breaks U (1) (1) e → Z n . Physically, this breaking occurs because a dynamical field with charge n can pair produce to break n Wilson lines whose charge is a multiple of n. This means that the Wilson line charge is only preserved mod n and consequently U (1) m has a 2-form current and associated symmetry defect operator for g = e iλ ∈ U (1). We say that this symmetry is "magnetic" because Σ 2 F (2) measures the magnetic flux through Σ 2 and consequently this symmetry acts on the dual magnetic photonÃ (1) by a shiftÃ (1) →Ã (1) + λ , ∈ Z: The magnetic 1-form global symmetry can be broken by dynamical monopoles, similar to the way the 1-form electric symmetry is broken by electrically charged particles. Such dynamical states modify the Bianchi identity to As with the Wilson lines and charged particles, dynamical monopoles can break 't Hooft lines by monopole-anti-monopole pair production, thereby breaking U (1) We can couple the Maxwell theory to background gauge fields of these two 1-form global symmetries. The action with such couplings are given by 25) where B (2) e and B (2) m are the 2-form background gauge fields of the electric and magnetic 1-form symmetries. They transform under the respective symmetry as background gauge transformations.
Note that in addition to coupling B (2) e,m to their respective currents -analogous couplings i A (1) ∧ * j 1 -we have also added additional background counter terms to make the theory explicitly invariant under U (1) However, now we see that the action is not invariant under U (1) We can make a different choice for local counterterms that makes the action invariant under m background gauge transformations so that the theory is given by However, we now see that the theory is not invariant under U (1) In fact, one can show that no choice of local counterterms makes the theory invariant under both electric and magnetic 1-form symmetries with generic background gauge fields B (2) e and B (2) m turned on. This means that we can not gauge both global symmetries. Such a "tension" among different global symmetries is indicative of a 't Hooft anomaly involving both U (1) It is often useful to organize anomalies of d dimensional quantum field theory in terms of a d + 1 dimensional topological quantum field theory. This is known as anomaly inflow [70,71] (see also [24] for a recent discussion of anomaly inflow in the context of AdS/CFT duality and with relevance to particle phenomenology). In our current example, the U (1) (1) m mixed anomaly is described by a 5d anomaly TQFT: 30) where ∂N 5 = M 4 , i.e. the boundary of the auxiliary 5d manifold N 5 is the 4d spacetime.
In fact, one easily sees that under a magnetic transformation B

B Z n TQFT
In this appendix, we review an example of a topological field theory, also known as a BF theory, which is given by Z n gauge theory. introduced in [19,20]. This theory appears ubiquitously in the literature on generalized global symmetry (see [1,3,21,72] for a useful introduction). 30 The action for 4d Z n TQFT is given by 31 It can also be written as B (2) ∧F (2) , hence the name BF theory. While this theory has many subtleties associated to the fact that it describes a discrete gauge theory, it is illuminating to keep in mind a particularly simple UV completion.
Consider an Abelian Higgs model with a charge n Higgs field Φ.
where V (Φ) is chosen so that Φ condenses in the the IR. It is clear that the condensation of Φ will break the U (1) gauge group down to Z n since the vev of Φ is invariant under Z n ⊂ U (1) gauge transformations. Thus, in the IR the theory will flow to a Z n gauge theory.
To see this, note that the radial mode of Φ has a mass similar to the scale of the symmetry breaking and is integrated out. Hence, we can decompose Φ = Λe iϕ where Λ is the scale of the symmetry breaking and ϕ is a periodic scalar field of charge n. Substituting this into the above action yields In the low energy limit, which effectively sends Λ → ∞, imposes A (1) = dϕ n (i.e. pure gauge configuration) and sets the gauge kinetic term to zero -leaving no local degrees of freedom. This is not a surprising statement at all since all but discrete part of scalar degrees of freedom are "eaten" by the gauge boson. The key point however, is that there is still an important discrete remnant.
Let us study the action (B.3) a bit more closely. To proceed further, it is useful to dualize ϕ. This can be achieved by introducing a new 3-form field H and rewriting the Lagrangian as (1) ).

(B.4)
One can check that by integrating out H using its equation of motion, * H = 4πiΛ 2 (dϕ − nA (1) ), we recover the Lagrangian in eq. (B.3). Additionally, the equation of motion for ϕ, dH = 0, which means we can locally introduce a 2-form B (2) with dB (2) = H. The Lagrangian then becomes Taking the limit Λ → ∞, we arrive at the BF theory in eq. (B.1).
Having shown that eq. (B.1) describes a Z n gauge theory, we now study it in more detail. First note that the theory is completely independent of metric and hence is a "topological" field theory. 32 Additionally, the equations of motion set the two field strengths to vanish This eliminates all local degrees of freedom in the IR, and confirms once again that the theory is topological.
This theory also has two U (1) gauge symmetries The A (1) gauge symmetry follows from the UV theory while the B (2) gauge symmetry is a consequence of the fact that B (2) is dual to ϕ. 33 These two gauge fields have corresponding Z n higher-form global symmetries under which fields transform as = 2πZ. (B.8) The existence of these symmetries is also manifested by the presence of gauge invariant Wilson line and surface operators: Since there are no dynamical degrees of freedom in the theory, there is no dynamical screening of these operators and they are absolutely stable.
In other words, the operators in eq. (B.9) are protected by the Z n global symmetries 32 In the language of differential forms, the metric only enters via the Hodge star operation. 33 More explicitly, the vortices of ϕ correspond to the Wilson-like surface operators of B (2) . The fact that ϕ is periodic means that these vortices are integer quantized which requires that B (2) is a 2-form gauge field with the transformation properties above.
above: the lines are charged under the 1-form global symmetry Z (1) n and the surface operators are charged under the 2-form global symmetry Z (2) n . This is seen by checking that these Wilson operators transform under the global symmetries as In order to see that the spectrum of these operators are consistent with Z n , we first recall that A (1) = 1 n dϕ. This shows Therefore, the Wilson line operators are classified by a charge = 0, · · · , (n − 1), as we expect for Z (1) n symmetry.
For the surface operator, we further dualize A (1) to the dual photon fieldÃ (1) . To this end, we view the field strength F (2) = dA (1) as an independent field and add a Lagrange multiplier term to impose the Bianchi identity In this presentation, one views the dual photon field as a matter field that Higges the 2-form gauge field B (2) with charge n: U (1) 1-form gauge symmetry is broken down to Z (1) n . This is linked to the fact that the gauge symmetry of eq. (B.13) is (1) (B.14) where we see thatÃ (1) also comes with its own 0-form gauge symmetry with parameterλ (0) . This is an emergent gauge symmetry as a result of the duality transformation (similarly to the 1-form gauge symmetry of B (2) ). The equation of motion for F (2) sets dÃ (1) + nB (2) = 0. This is all we need to prove the Z n spectrum of surface operators. For the sake of completeness, we repeat the exercise: In order to understand the correlation function of Wilson operators, we first show that an insertion of surface operator W 2 (m, Σ 2 ) in the path integral (equivalent to introducing a source term) modifies the equation of motion of B (2) and effectively turns on non-trivial Z n valued F (2) flux localized on the worldvolume of the surface operator: Here, δ (2) (Σ 2 ) is a 2-form delta function which is non-zero only on Σ 2 and is normalized as where Γ 2 is a 2-manifold that transversely intersects Σ 2 once. The modified equation of motion means that the holonomy of W 1 = e i A (1) around Σ 2 evaluates to a phase e 2πi n . 34 Similarly, an insertion of a line operator modifies the equation of motion of A (1) to dB (2) = 2π n δ (1) (Σ 1 ), inducing holonomy for W 2 = e i B (2) which Z n valued. This means that where Link(Σ 1 , Σ 2 ) is the linking number of Σ 1 and Σ 2 .
Note that the Z n theory does not have any non-trivial 't Hooft operators (dual to Wilson operators for U (1) gauge symmetries) nor any local operators (dual of string operators for U (1) symmetries). The reason is that the local operators constructed from the dual of B (2) are written in terms of ϕ: I(m, x) = e imϕ(x) , m ∈ Z which is not gauge invariant. However, it can be made by attaching a Z n Wilson line operator to it (see eq. (B.3)) at the expense that the operator has now become non-local. , ∂Σ 2 = Σ 1 (B.20) Again, the equation of motion makes this a trivial operator.
Before we conclude, we would like to point out that there exist interesting variations of the BF theory by adding a discrete θ-parameter term.
Here, p ∈ Z. The discrete θ-parameter ipn 4π B (2) ∧B (2) is also known as "discrete torsion" or as a "Symmetry Protected Topological phase" (SPT phase) associated with Z

2-form axion string
Field Strength of Magnetic Symmetries 2-form axion string G (4) = dA m + · · · Table 6. List of generalized symmetries, their currents, and background gauge fields in decoupled axion-Maxwell and Z n BF theory.
theory (see for e.g. [1,3,73]) and non-invertible symmetry as a way of gauging Z n subgroup of bulk 1-form magnetic symmetry (see for e.g. [35]). This addition introduces several interesting and subtle effects. It makes the 1-form gauge field A (1) charged under the 1-form gauge symmetry of B (2) , it modifies the global symmetry to Z J , where J = 1 2 gcd(p, n)for p, n even and J = gcd(p, n) otherwise. This clearly changes the spectrum of conserved operators and may have interesting signals different from those discussed in our paper. For a very nice and detailed discussion, we recommend highly Section 6 of [1].

C Global Symmetries of TQFT-Coupling I
In this appendix, we discuss in detail symmetries of the theory studied in Section 2: axion-Maxwell theory coupled to a Z n gauge theory. 35 Since there are many different symmetries and we need to introduce a background gauge field for each of them, we list these symmetries and associated current and background gauge fields in Table 6, setting up our notations. 36 The action of the theory is B . (C.1) The above theory has six generalized global symmetries: four 'electric' symmetries and two 'magnetic' symmetries. Electric symmetries are obtained from EoMs of the fields appearing in the action and magnetic symmetries come from Bianchi identities.
The procedure for analyzing the symmetries of this theory is to couple the classical symmetries to background gauge fields and study the transformation of the action under background gauge transformations. Then, we can probe the anomalies and higher symmetry 35 The generalized symmetry and associated higher-group structure of the axion-Maxwell were studied in [5,14,15]. 36 Strictly speaking, discrete symmetries do not have associated currents. In Table 6, by the conserved current generating a discrete symmetry, we mean current of an associated U (1) symmetry that is broken down to the appropriate discrete subgroup.
structures by coupling all such symmetries to background gauge fields simltaneously and studying their gauge transformations.
A complimentary viewpoint to study the symmetry groups of the theory is to look at the conservation equation (take for example a U (1) symmetry). When there are anomalous terms such that j is a quantized charge density j ∈ KZ, the U (1) symmetry is broken U (1) → Z K . In a sense, this is a quick and easy way to detect symmetry breaking interactions.
The theory can be coupled to background gauge fields as In general we choose conventions so that all background gauge fields satisfy the analog of Dirac quantization: As we will see below, some of the electric symmetries are mixed up with magnetic symmetries which leads to a higher-group structure. Practically, this happens when the theory only admits field strengths of the magnetic symmetry which are shifted by nonlinear terms in electric symmetry gauge fields.

C.1 Symmetries of Uncoupled Theories
Now we will discuss the symmetries of the theory. We find that it will be simplest first analyze the decoupled theory where K AB = K B = 0 (i.e. the charge K A axion-Maxwell theory and Z n BF theory are decoupled) starting with the magnetic symmetries before discussing the electric symmetries. We will discuss the symmetry structure of the full theory in the following section.
U (1) (2) 2-form axion string symmetry: The axion, being a smooth field obeys a Bianchi identity d 2 a = 0. This implies the existence of a two-form U (1) (2) symmetry with a current * J 3 = 1 2πf a da . (C.5) This couples to a background 3-form gauge field A which is consistent with Dirac quantization condition and the gauge redundancy a → a + 2πf a . The charged objects of U (1) (2) are 2d world-sheet of axion strings V (m, Σ 2 ). The charge and symmetry defect operators, and their action on an axion string are given by Such symmetry defect operators act on the axion strings with which they have non-trivial linking This symmetry is a U (1) (2) and it is broken in the presence of dynamical strings Indeed, in the presence of string, the Bianchi identity d 2 a = 0 is violated and the axion carries non-zero winding around the string, see eq. (2.10). A dual version of this appeared in the BF theory case, i.e. electric 2-form BF string symmetry discussed above.
This symmetry is a direct consequence of the Bianchi identity dF A = 0 and the current is given by * J 2m = 1 2π F A . (C.10) If there were dynamical monopoles present in the theory, they would lead to a source term in the conservation equation This symmetry couples to a background 2-form U (1) (1) gauge field B (2) m as A . (C.12) K A 0-form axion shift symmetry: The equation of motion of the axion is .
(C. 13) In the absence of anomalous/instanton terms on the right hand side, the axion has U (1) (0) shift symmetry a → a + f a c, c ∈ S 1 and its current is * J 1 = if a * da. The above equation shows that the axion coupling breaks the continuous shift symmetry to a discrete subgroup K A . This can also be seen explicitly by studying the shift of the action under a → a + f a c: Clearly, the action is invariant only if where we used the identity We can define a topological symmetry defect operator for Z (0) where = 0, 1, · · · , K A − 1 and ω 3 (A (1) ), ω 3 (A (1) , B (1) ), ω 3 (B (1) ) are Chern-Simons terms that satisfy B . (C.18) The gauge invariant operators that are charged under this symmetry are the vertex operators I(m, x) = e im fa a(x) , m ∈ Z. It is invariant under a gauge transformation a → a + 2πf a and U aE ( , Σ 3 ) acts on it when it has non-trivial linking where on the right-hand side we used the topological invariance of the defect operator U aE ( , V 3 ) to shrink it away after it passes through the charged operator I(m, x).
In order to couple a background gauge field to this symmetry, we in principle need to couple to a Z K A gauge field A (1) e . However, since Z K A ⊂ U (1) we can treat A (1) e as a U (1) gauge field that has been restricted so that: This has the consequence of imposing the condition where [dλ (1) ] is a trivial cohomology class. Additionally, there is a "discrete" version of a Z n gauge field which is simply given by 37 by which we mean β A (1) 2π = 0, 1, ..., n − 1. See [1] for more in depth discussion of discrete gauge theories.
There is a mixed 't Hooft anomaly between 'electric' Z (0) K A shift symmetry and 'magnetic' U (1) (2) symmetry. To see this, we couple the theory to background gauge fields where we added a local counterterm consisting of only background gauge fields to make the action manifestly invariant under electric symmetry. Now we can see that there is a non-trivial 't Hooft anomaly by noting that the action is not invariant under U (1) (2) transformations: Alternatively, if we chose instead to omit the counterterm, the magnetic symmetry would be preserved but the action would manifestly break axion shift symmetry.
As discussed in Appendix A, we can identify this anomaly with a 5D TQFT which is 37 This operation is called the Zn Bockstein map.
given by K A 1-form A-electric symmetry: The 1-form electric symmetry associated to the U (1) A gauge field is a bit trickier. The equation of motion for A (1) is .
(C. 26) In the absence of coupling terms, the pure Maxwell sector has a U (1) (1) e 1-form electric symmetry with a current * J e 2A . Naively, the interaction terms would appear to break However, let us couple the theory to a 2-form U (1)-gauge field B (2) e and restrict to the case where K AB = 0: Now, due to the fact that (2) e 8π 2 ∈ 1 K 2 Z , (C. 28) we see that the axion interaction term appears to break U (1) (1) → Z k A where K A = k k 2 A . However, we can actually extend the electric 1-form global symmetry to include a Z (1) K A as follows. Recall that the 2-form axion string symmetry U (1) (2) also couples to the axion as If we modify the transformation laws for A m , we can cancel the fractional contribution 38 30) so that the associated gauge invariant field strength is e , (C.31) 38 Note that here we are taking B (2) e to be a ZK -valued 2-form gauge field. This can be achieved in analogy with the ZK a 1-form gauge field for the Z

(C.32)
This modified transformation law (C.30) is the hallmark of a 3-group global symmetry.
However, now that the 1-form electric symmetry has mixed with U (1) (2) , it also acquires a mixed 't Hooft anomaly with Z (0) Ka axion shift symmetry. However, in order to probe these anomalies, we really need to determine how to turn on the background gauge field for Z K A simultaneously. This requires picking a 5-manifolds N 5 that bounds M 4 . In this case we find that The requirement that our action describe a well-defined, local 4d theory then becomes demanding that the theory is independent of the choice of N 5 . One can explicitly check that the action is not independent of this choice due to the terms A . (C.34) However, we see that we can actually cancel this variation by also modifying the U (1) (1) magnetic 1-form global symmetry: so that the gauge invariant field U (1) (1) field strength is given by 36) and the U (1) (1) coupling now appears as This is the hallmark of a 2-group global symmetry which is then interlaced with the 3-group global symmetry indicated by the mixing of Z K A with U (1) (2) .

Z
(1) n 1-form B-electric symmetry: Now let us turn to the symmetries of Z n BF theory. The equations of motion for B (1) is given by Here we see that the BF source term j e 3B (B (2) ) breaks U (1) transforms electrically under Z n , we can couple the theory to a Z (1) n background gauge field by n 2-form BF string symmetry: Similarly, the equation of motion for B (2) is the same as in BF theory Here, the source term j string 2 , breaks U (1) (2) down to Z n gauge transfrmations. This is indicative of a mixed 't Hooft anomaly which we will now discuss.

C.1.1 Anomalies
Now we are able to discuss the 't Hooft anomalies of the decoupled theory. Here we will turn on all of the background fields except for Z (2) n -recall that we can only turn on a background gauge field for Z (1) n or Z (2) n -and analyze the variation of the partition function under the background gauge transformations. Here we see that there are terms in the action that are explicitly not invariant under the electric symmetries: where here δa = f a λ (0) e , δA (1) = λ (1) e , δB (2) =λ (1) e , δD (3) e = dλ (2) e , δB (1) =λ (1) e , δC (2) e = dλ (1) e .

C.2 Symmetries with TQFT Coupling
Now let us consider how turning on the coupling between axion-Maxwell theory and the Z n BF gauge theory: K AB , K B = 0. This has many effects on the symmetry structure. However, there are several symmetries that are not effected by turning on the coupling: • U (1) (2) 2-form axion string symmetry: Adding the new axionic couplings to the theory does not affect the symmetry structure of U (1) (2) . This is evident from the fact that the normal axion coupling does not affect the winding 2-form symmetry of a U (1)-valued scalar field.
(C. 46) We now see that there are additional source terms that can further break the Z

1-form A-electric and B-electric symmetry
Now let us consider the effect of adding the coupling K AB to the 1-form electric symmetry for U (1) A and 1-form electric symmetry for Z n . As we saw, the axion coupling for generic K A broke the free theory U (1) (1) to a discrete Z B does indeed satisfy the Dirac quantizatoin condition and hence there is no appreciable difference for the purpose of analyzing symmetries. Now the symmetry structure of the theory follows straightforwardly from the analysis of the 3-group symmetry structure of axion-Maxwell theory. In particular, the K A and K B coupling break U (1) The mixed axion coupling term is a bit more tricky. Let us denote L = LCM(K A , k B ). The Z (1) L enveloping group. We then see that the K AB coupling now breaks Z (C. 47) In our UV complete model in Section 2.2, K A = 1, K AB = q, and K B = q 2 so that κ A = 1 and κ B = q and there is only a remanining Z (1) q 1-form global symmetry remaining.
However, when we define the 5d gauge invariant axionic coupling, we also need to cancel the 5-dimensional dependence of the terms B ∧ C (2) e .

(C.48)
These additionally need to be cancelled by modifying the transformations of B Now we see that the coupled theory has a 3-group symmetry involving U (1) (2) , U (1) (1) , Z so that the gauge invariant field strengths are given by e is now replaced by its Z Here we can clearly see that the effect of coupling the axion-Maxwell theory to the BF TQFT is that the 3-group structure has been dramatically modified. In summary, the effect of the adding the coupling is given by: • The axion shift symmetry is reduced Z (0) Ka for K a = GCD(K A , K B , K AB ), • The 1-form U (1) A eletric symmetry is reduced Z (1) • The 1-form BF electric symmetry is reduced Z • The 1-form BF electric symmetry now participates in a 3-group that mixes with U (1) (2) , U (1) (1) magnetic symmetries as well as the electric symmetries Z (0) n .

Electric Symmetries
Magnetic Symmetries 0-form axion shift Z m + · · · 1-form A-magnetic H (3) = dB (2) m + · · · Table 7. List of generalized symmetries, their currents, and background gauge fields in the full coupled axion-Maxwell and Z n BF theory. Additionally, D e is defined in eq. (C.51) and G (4) , H (3) are defined in eq. (C.50), indicating that the 0-and 1-form electric symmetries all participate in 3-groups, mixing into the magnetic symmetries and 2-form BF string symmetry.

C.2.1 Anomalies
The full action with all background field strengths turned on is given by The full set of background gauge fields and their associated symmetries are summarized in Table 7. 39 39 7, by the conserved current generating a discrete symmetry, we mean current of an associated U (1) symmetry that is broken down to the appropriate discrete subgroup.

C.2.2 Constraints from Symmetry
As discussed in [5], one of the physical consequences of having an EFT with 3-group global symmetry is that any UV completion that gives rise to it must satisfy an inequality of scales at which the different components of the 3-group emerge. In particular, since the 0-and 1-form component symmetries turn on the 2-form U (1) (2) background gauge fields, we must have the inequality (C. 56) In terms of physical quantities, this is given by where T string is the tension of the axion string and m ψ is the mass of the lightest charged particle, which must be charged under both U (1) A , Z n (or U (1) A , U (1) B where U (1) B breaks to Z n at a scale E Zn m ψ ) that breaks the 1-form electric symmetries. See [5] for further discussion.

C.2.3 Other TQFT Couplings via Discrete Gauging
As discussed in Section 3.4, we can get many new couplings to TQFTs by gauging discrete subgroups of the 3-group global symmetry. Due to the similarity of the structure of the 3-group, we find that most of the possible discrete gaugings follow straightforwardly. The main difference is that now we can additionally gauge 1.) Z  Ka , extends the periodicity of U (1) (1) and (at least partially) breaks Z (2) n . It would be interesting to study the theories produced by these disrete gaugings in more detail.