Noninvertible anomalies in SU ( N ) × U (1) gauge theories

: We study 4-dimensional SU ( N ) × U (1) gauge theories with a single massless Dirac fermion in the 2-index symmetric/antisymmetric representations and show that they are endowed with a noninvertible 0-form ˜ Z χ 2( N ± 2) chiral symmetry along with a 1-form Z (1) N center symmetry. By using the Hamiltonian formalism and putting the theory on a spatial three-torus T 3 , we construct the non-unitary gauge invariant operator corresponding to ˜ Z χ 2( N ± 2) and find that it acts nontrivially in sectors of the Hilbert space characterized by selected magnetic fluxes. When we subject T 3 to Z (1) N twists, for N even, in selected magnetic flux sectors, the algebra of ˜ Z χ 2( N ± 2) and Z (1) N fails to commute by a Z 2 phase. We interpret this noncommutativity as a mixed anomaly between the noninvertible and the 1-form symmetries. The anomaly implies that all states in the torus Hilbert space with the selected magnetic fluxes exhibit a two-fold degeneracy for arbitrary T 3 size. The degenerate states are labeled by discrete electric fluxes and are characterized by nonzero expectation values of condensates. In an Appendix, we also discuss how to construct the corresponding noninvertible defect via the “half-space gauging” of a discrete one-form magnetic symmetry.


1 Introduction
Symmetries are the backbone of quantum field theory (QFT); the spectrum of local and extended operators are organized in symmetry representations.The modern way to define symmetry is via its action on topological surfaces.A p-form symmetry acts on p-dimenstional objects and is generated by operators supported on p + 1-codimensional surfaces [1].Traditionally, symmetries are connected to groups, and the operator that generates the symmetry is unitary.However, recent developments have highlighted the need to broaden the definition of symmetries to include actions generated by nonunitary operators.These symmetries are noninvertible since the corresponding operators do not have an inverse.Consequently, these operations do not form groups but can be comprehended as categories, offering a new perspective on the nature of symmetries.Noninvertible symmetries were first identified and applied in 2-dimensional QFT, see, e.g., [2,3].The appreciation of the role of this new development in 4-dimensional QFT resulted in an avalanche of works on this topic (a non-comprehensive list is ).
One significant development in the field is the recognition that 4-dimensional quantum electrodynamics (QED) possesses a noninvertible symmetry [11,12].The idea is that QED with a single Dirac fermion has a classical U (1) χ chiral symmetry broken to the Z 2 fermion number by the ABJ anomaly.The Noether current corresponding to U (1) χ is not conserved as it receives a contribution from the anomaly.However, one may define a conserved chiral current by subtracting a Chern-Simons term that encodes the anomaly.Next, an operator corresponding to U (1) χ is constructed by exponentiating the modified current and integrating it on a 3-surface.This operator is not gauge invariant.However, dressing it with a TQFT makes it gauge invariant.The resulting operator can be shown to generate a noninvertible symmetry for every rational value of the U (1) χ parameter.The construction in [11,12] was further developed in [17,18] by showing that the definition of the noninvertible operator can be extended to any real parameter of U (1) χ by coupling the theory to a scalar field living on the 3-surface.
Comprehending this novel structure in gauge theories is important in pursuing a deeper understanding of QFT.The present work examines SU (N ) × U (1) gauge theory with a single massless Dirac fermion in a representation R. The theory has a classical U (1) χ symmetry broken by the ABJ anomaly in SU (N )-and U (1)-instanton backgrounds.Does this theory exhibit noninvertible symmetries, and if so, can they be utilized to establish exact nonperturbative statements?We demonstrate that the answer to this query is affirmative.
Unlike QED, when our theory is put on a general manifold, the chiral symmetry is reduced to Z χ 2gcd(T R ,d R ) , where T R and d R are the Dynkin index and dimension of R. Interestingly, we also show that the theory possesses a noninvertible Zχ 2T R discrete chiral symmetry, wherein Z χ 2gcd(T R ,d R ) is an invertible part. 1 We establish the noninvertibility through a sequential process starting from the nonconservation of Noether's current of the chiral symmetry.Employing the Hamiltonian formalism and putting the theory on a 3-dimensional spatial torus T 3 , we construct a noninvertible symmetry operator.This setup provides a simple and explicit route to select the states on which the operator Zχ 2T R acts nontrivially.SU (N )×U (1) QCD-like theories are naturally endowed with an electric 1-form Z (1) N symmetry acting on the Wilson loops as well as a 1-form U (1) m magnetic symmetry that characterizes sectors with definite magnetic fluxes.Then, it is natural to ask whether the theory exhibits a 't Hooft anomaly as we perform a Zχ 2T R transformation in the background of Z N .To address this question, for definiteness 2 we consider a theory with fermions in the 2-index symmetric/anti-symmetric representation, which possesses a Zχ 2(N ±2) noninvertible symmetry.We subject the theory to Z N twists (2form background fields for Z (1) N ) along the non-trivial cycles of T 3 .In the presence of the Z N twists, the noninvertible symmetry projects onto sectors in Hilbert space with definite magnetic fluxes, easily identified in the Hamiltonian formalism.For N even, the algebra of Zχ 2(N ±2) and Z N fails to commute by a Z 2 phase inside these sectors, 1 Throughout this paper, we use a tilde to distinguish noninvertible symmetries and operators.Note also that after gauging the vector U (1), the Z 2 part (fermion number) of Zχ 2T R is part of the gauge symmetry and not a global symmetry.With this in mind, we continue to denote the noninvertible symmetry by Zχ 2T R , as this subtlety does not affect our considerations of the anomaly and spectral degeneracy. 2 The conclusions of the work described can be generalized to fermions in higher representations.
revealing a mixed anomaly between Z N and Zχ 2(N ±2) symmetries (anomalies involving noninvertible symmetries have previously been considered in 1 + 1 dimensions, see [30][31][32][33][34][35] and references therein).This anomaly implies that the states in these special sectors must be 2-fold degenerate3 on arbitrary size T 3 .Such degeneracies could be seen by examining the condensates in realistic lattice simulations.
This paper is organized as follows.In Section 2, we demonstrate the origin of the noninvertible symmetry by carefully examining SU (N ) × U (1) QCD-like theories put on T 3 with general flux backgrounds and build the noninvertible operator of Zχ 2(N ±2) .Then, we show that this symmetry has a 't Hooft anomaly with the Z (1) N symmetry.In Section 3, we discuss the implication of this anomaly in the magnetic sectors selected by Zχ 2(N ±2) and exhibit the exact degeneracy.In the Appendix, we show the equivalence of our operator construction to the "half-gauging" of a discrete subgroup of the magnetic one form symmetry U (1) m , used in [11,12] to construct a properly normalized defect yielding consistent Euclidean correlation functions.We conclude with a brief discussion in Section 4.

Noninvertible symmetries and their anomalies
Consider SU (N ) × U (1) gauge theory with a single-flavor massless Dirac fermion in a representation R. We use n, T R , and d R to denote the N -ality, Dynkin index, and the dimension of R, respectively, and focus mostly on T R = N ± 2 for the two-index symmetric (S)/antisymmetric (AS) representations of N -ality n = 2 and dimension d R = N (N ±1) 2 for S/AS.Yet, our construction can be easily generalized to theories with several flavors and fermions in higher representations.Classically, the theory is endowed with a U (1) χ global chiral symmetry.The fermion charges (both are left-handed Weyl) under (SU (N ), U (1), U (1) χ ) are The theory with the gauged U (1) has a Z N 1-form electric symmetry, acting on both SU (N ) and U (1) Wilson loops, as well as a U (1) (1) m 1-form magnetic symmetry which distinguishes the different U (1)-flux sectors.
We use A and a for the 1-form gauge fields of SU (N ) and U (1), respectively.The corresponding field strengths are F and f .The anomaly equation for the chiral where K µ SU (N ) is the SU (N ) topological current.Its normalization is such that the integral of K 0 SU (N ) ≡ K CS over a three-dimensional manifold changes by an integer under large gauge transformations.In other words, the operator is invariant under large gauge transformations.This operator (which shifts the θ-angle by 2π) will be important in what follows.
When the theory is defined on a spatial manifold with nontrivial 2-cycles, e.g., on T 3 , gauging U (1) breaks Z χ 2T R of the SU (N ) theory further down to Z χ 2gcd(d R ,T R ) .The easiest way to see that is by compactifying the time direction, so we consider the Euclidean version of the theory on T4 .Under a chiral transformation, (ψ R , ψ R) → e iα (ψ R , ψ R), the measure changes by where c 2 (F ) ≡ f ∧f 8π 2 ∈ Z are the second Chern classes of SU (N ) and U (1).Then, demanding that the phase (2.4) is trivialized, and using Bézout's identity, which states that integers of the form az 1 + bz 2 are exact multiples of gcd(a, b), we arrive at our conclusion. 4For example, when gcd(d R , T R ) = 1, there is no residual chiral symmetry and the only symmetry left over is the Z 2 fermion number symmetry which stays intact, assuming the Lorentz symmetry is unbroken.
It is important to emphasize that Z χ 2gcd(d R ,T R ) is a genuine invertible symmetry, which is represented by a unitary operator acting on the Hilbert space of the theory.What is the fate of Z χ 2T R that is broken because of the U (1) instantons?Below, following the approach of [17,18], we argue that this symmetry becomes noninvertible.We shall exhibit this and study the consequences using Hamiltonian quantization on T 3 in a very explicit manner.
Consider the Hamiltonian quantization of the theory on a rectangular T 3 of sides L 1 , L 2 , L 3 , in the A 0 = a 0 = 0 gauge.The Hilbert space is constructed in terms of gauge fields A = A i dx i and a = a i dx i , where i = 1, 2, 3 is the spatial index (below, we use êi to denote a unit vector in the i-th direction).We often use x, y, z for x 1 , x 2 , x 3 when writing the components explicitly.
The physical Hilbert space is obtained from field-operator eigenstates after appropriate gauge averaging and imposing Gauss's law; a detailed description can be found in [36], see also the earlier works [38,39].The gauge fields obey boundary conditions on T given in terms of SU (N ) and U (1) transition functions, Γ i and ω i , respectively [40].We work in a gauge where the SU (N ) transition functions Γ i are constant unitary N × N matrices [41].The fermions obey similar boundary conditions (for brevity, we write these for the N -ality two case, n = 2): The U (1) and SU (N ) transition functions obey cocycle conditions assuring that the fields satisfying (2.5, 2.6) are single-valued on the torus.The cocycle conditions obeyed by the transition functions are or Here, the (mod N ) integers n ij represent topological classes of 2-form Z N background fields in the respective 2-planes and m ij label integer U (1)-flux sectors, distinguished by their magnetic flux through the various 2-planes.The integer m ij are charges under the global magnetic U (1) (1) m 1-form symmetry [1].It is easily seen that all gauge and matter fields are single-valued on T 3 when (2.7) are obeyed.The integers n ij and m ij label different flux sectors of the torus Hilbert space.
We next consider the global Z N symmetry [1].The generators of the 1-form Z (1) N symmetry act on the transition functions.We label by t j and T j the U (1) and SU (N ) group elements representing the action of the generators of Z N in the j-th direction on the U (1) and SU (N ) transition functions, respectively.The action of Z N is given by N transformations are represented by "improper" gauge transformations on the Hilbert space (in the first line, since both t j and Ω i are abelian, the Ω j 's can be dropped).
The explicit expressions for U (1) and SU (N ) group elements t j (x) and T j (x) obeying (2.9) and generating the center symmetry can be worked out.The explicit form of T j (x) can be found in the literature 5 .The explicit form of T j (x) depends on both the choice of gauge for the transition functions and on the n ij 2-form background (this is because T j (x) have nontrivial winding numbers for n ij ̸ = 0 (modN ) [45]).We will not need the expression for T j , but only the commutation relation of the SU (N ) Z (1) N center symmetry generators and the gauge invariant operator (2.3), which is nontrivial in the presence of a 't Hooft twist; see [36] for a derivation.On the other hand, the center symmetry generators' action on the U (1) can be taken Next, we construct the generator of the noninvertible chiral symmetry.To this end, we integrate the anomaly equation (2.2) and use it to define a conserved (but not gauge-invariant) U (1) χ symmetry operator on T 3 .Because of the boundary conditions in the space directions for the U (1) fields, whose transition functions (2.8) necessarily depend on x i , there are a-dependent boundary terms. 6We find, denoting K 0 (a) ≡ . (2.12) Next, we note that ∂ 0 a k is periodic on T 3 , while a k itself obeys (2.5) with transition function ω i of (2.8).It is easy to see that the second term above is also a total time derivative.Integrating (2.12), we finally obtain that Q χ (x 0 = L 0 ) = Q χ (x 0 = 0), where (2.13) The last line above indicates that there are two more terms obtained from the term on the second line by cyclic rotation of x, y, z.Again, this is the operator Q χ in the sector of Hilbert space with U (1) fluxes m ij and Z N fluxes n ij .Exponentiating (2.13), we find the (non-gauge-invariant) operator representing Qχ . (2.14) Let us now study the gauge transformation properties of X 2T R .First, we note that because of the gauge invariance of (2.3), the invariance of (2.14) under SU (N ) (large and small) gauge transformations is manifest.Next, consider U (1) gauge transformations with periodic 7 e iλ : To study the transformation properties of (2.14) under U (1) transformations (2.15), we note that, with λ from (2.15), −2π T 3 d 3 xK 0 (a) transforms as Since, recalling (2.5, 2.7), we find −2π (2.18) 7 For use below, we also recall from (2.11) that the generator t j of the global Z N in the j-th direction acts on the U (1) gauge field as a nonperiodic gauge transformation, i.e. is obtained from (2.15) upon replacing n j → − n N , where, we remind the reader, n is the N -ality of the matter representation.
Likewise, we also find the U (1) gauge transformation of the boundary terms in (2.13) along with two identical relations obtained by cyclic permutations of x, y, z.Thus, combining the U (1) gauge transformations (2.18, 2.19), with the expression for Q χ from (2.13), we find the transformation of showing explicitly that the operator (2.14) is not U (1) gauge invariant.Thus, following [17,18], to make X 2T R gauge invariant, we sum over n x , n y , n z , obtaining the noninvertible Zχ with Q χ from (2.13).This equation shows that the operator is noninvertible and determines the sectors not annihilated by X2T R .To see this, we use the Poisson resummation formula For X2T R to act nontrivially, i.e., not be set to zero, it must be that in each two-plane, the fluxes m ij , n ij , i < j, have to obey It is easy to see that such integer-valued combinations of fluxes always exist (we shall see examples below).In the Appendix, we construct the operator X2T R using the "half-gauging" procedure of refs.[11,12].
To summarize, here we have constructed a symmetry operator X2T R of the noninvertible chiral symmetry.The operator of the noninvertible symmetry acts as a projection operator: it annihilates sectors of the torus Hilbert space whose fluxes do not obey (2.23) and acts as unitary operator in each flux sector obeying (2.23). 8e next study the commutator of the Z (1) N center symmetry transformation with the noninvertible X2T R .We denote the Z (1) N generator in the x-direction by T x t x (for brevity omitting both hats over operators and the tensor product sign, with T x , t x obeying (2.9)).We use the commutation relation of (2.10) of T i with the operator (2.3), as well the U (1) transformation law derived above, eqn.(2.20), with only n x nonzero and with the integer n x replaced by − n N (as we reminded the reader in footnote 7).We find Equation (2.24) and its two cyclically permuted versions, constitute our main result.It shows that-provided both the phase on the r.h.s. is nontrivial and X2T R is nonzero (i.e.(2.23) holds)-there is a mixed anomaly between the Z (1)

N center symmetry and the noninvertible Zχ
2T R chiral symmetry of the SU (N )×U (1) theory.As both operators act nontrivially in the chosen integer-l i T 3 Hilbert space, the algebra (2.24) will be seen to imply exact degeneracies for any size T 3 .
In Section 3, we shall show that there are cases where both the phase in (2.24) and X2T R are nontrivial, i.e., (2.23) holds in the appropriate Hilbert space.As already familiar from [36], this will be seen to imply degeneracies in the Hilbert space between different "electric flux" states (eigenstates of Z N , i.e., of T x t x ).Before we continue with analyzing the implications of (2.24) for the finite-volume spectrum, let us make a connection of (2.24) with the Euclidean path integral.We denote the T 3 Hilbert space by H l i , with the understanding that U (1) (m ij ) and Z (1) N (n ij ) fluxes are chosen to yield an integer-l i , so that X2T R acts nontrivially.We now define the T x t x -twisted partition function via the Hamiltonian formalism as a trace9 over states in H l i with a Z (1) N x-direction generator inserted in the partition function, i.e.Z ≡ tr H l i e −βH T x t x .Then, we use the fact that X2T R acts as an invertible unitary operator in H l i , as well as the commutation relation (2.24), to obtain We conclude that the chiral symmetry X2T R implies that Z = e −i2π[ nyz N − n N lx] Z, so that, if the phase is nontrivial, Z vanishes, unless fermion fields (ψ Rψ R ) k are inserted to make Z nonzero. 10To obtain a path integral interpretation of (2.25), we note that the twisted partition function (2.25) sums over SU (N ) and U (1) gauge fields which obey the boundary conditions (2.5) in the T 3 spatial directions, determined by n ij and m ij .The boundary conditions in the Euclidean time direction (of extent β) are twisted, by the insertion of T x , leading to a nontrivial SU (N ) twist n x4 = 1 in the x-time plane.Thus, the twisted partition function (2.25) sums over SU (N ) field configurations with topological charges − n 14 nyz N +k = − nyz N +k, with all possible integer k [46].On the other hand, the insertion of t x implies that the U (1) background obeys a(x+ê 4 β) = a(x)+ 2πn N dx L 1 and thus f 14 = − 2πn N L 1 L 4 .Recalling that the U (1) field strength in the yz plane is 2π Applying a chiral transformation with α = 2π 2T R , and using the measure transformation (2.4) (with c 2 (F ), c 2 (f ) substituted by the fractional topological charges just mentioned) we obtain the phase e , which, after using (2.23), is seen to be the same as in (2.25), as expected.
3 Hilbert space, magnetic sectors, and the 2-fold degeneracy Here, we analyze the consequences of condition (2.23) and the algebra of (2.24).Condition (2.23) selects sectors in Hilbert space with definite U (1) magnetic fluxes in the 2-3, 3-1, and 1-2 planes proportional to the integers l x , l y , l z , respectively.In the following, we always set l y = l z = 0 (with n xy = m xy = n zx = m zx = 0) to reduce complexity and examine the theory for various values of l x = d R T R (m yz + nnyz N ).We also choose n yz ∈ {0, 1, 2, ..., N − 1}.Since we are mainly concerned with symmetric (S)/antisymmetric (AS) fermions, we set the N -ality n = 2.
We start with sectors with a vanishing magnetic flux, i.e., we set l x = 0, which translates into m yz + 2 N n yz = 0.When N is odd, the only solution is the null solution m yz = n yz = 0.This yields a trivial phase in the algebra of (2.24).However, when N = 2M is even, there are two solutions.First, the null one m yz = n yz = 0, giving a trivial phase in (2.24).The second is the new solution (we denote the phase in (2.24) by e −iα = e −2πi( nyz The Z 2 phase in (2.24) implies that some electric flux states in a sector with a 0magnetic flux are 2-fold degenerate.
Numerical tests reveal that this pattern continues in sectors with nonzero magnetic flux, l x ̸ = 0. First, when N is odd, all allowed sectors have a trivial phase in the algebra of (2.24), indicating no kinematical constraints in these theories.For N even, N = 2M , one can use l x = d R T R (m yz + nnyz N ) to see that there exists integers m yz ∈ Z and n yz ∈ {0, 1, 2, ..., N − 1} that satisfy the relation for S/AS fermions, respectively.Sectors with definite m yz and n yz that satisfy (3.2) will also yield at most a Z 2 phase in the algebra (2.24), if they additionally satisfy The Z 2 phase implies that the states in these sectors have double-fold degeneracy.
Let us flesh this out in detail.For definiteness, we consider S/AS fermions, which yield a nontrivial phase in (2.24) as well as involve only a minimal SU (N ) 't Hooft twist, n yz = 1 (as opposed to the l x = 0 solutions (3.1), which must have n yz = N/2 to produce an anomaly).For both S/AS fermions, these minimal-twist solutions, yielding an integer l x and a Z 2 phase in (2.24), must have , examples of m yz that give the Z 2 phase are: Thus, we now focus on the flux backgrounds (3.4), enumerate the degenerate "electric flux" sectors11 in the corresponding Hilbert spaces H lx,ly=lz=0 , and discuss some of their properties.The transition functions for SU (N ) and U (1) obeying the cocycle conditions with n yz , m yz given in (3.4) are The center symmetry generators for SU (N ) and U (1), obeying (2.9), can be taken to be 12 Since T x t x is a symmetry, eigenstates of the Hamiltonian can be labeled by its eigenvalues, T x t x |E, e x ⟩ = |E, e x ⟩e i 2π N ex , with e x ∈ Z (modN ).On the other hand, the algebra (2.24), Thus, X2T R maps an eigenstate of the Hamiltonian of energy E and flux e x to another eigenstate of the same energy, but with flux e x + N 2 (modN ) (hence, a T x t x eigenvalue differing by e iπ , as per (2.24); we also note that the phase in the action of X also depends on whether the state is bosonic of fermionic). 13 To further characterize the degenerate flux states, we will show that the degenerate states (3.7) have nonvanishing expectation values of a condensate, which we write schematically as (ψ Rψ R ) N ±2 2 .These expectation values take opposite values in the two degenerate flux states.To this end, we now go back to our twisted partition function (2.25).Since in H lx,ly=lz=0 the phase is in Z 2 , in order to obtain a nonzero phase we 12 For SU (N ), these are the ones from [47].Briefly, we remind the reader that T x cannot be taken to be constant, since, as already discussed, a twist of the partition function in the time direction by T x , recall (2.25), leads to fractional topological charge on the T 4 , equal to 1 N + integer.This implies that T x has fractional winding number [45], as a map from T 3 to the gauge group, with its N -th power being a large gauge transformation.Thus, on physical states T x obeys T N x = 1.Explicit expressions for T 1 can be found in the literature (see [42,43] for SU (2) and [44] for SU (N )) but are not needed here. 13As promised, on a small T 3 , the lowest flux states can be worked out classically.For the bosonic backgrounds, the lowest-energy gauge field backgrounds are The fundamental SU (N ) winding Wilson loops then take values W y = W z = 0, W x = e i 2π N l .Using the backgrounds (3.8), solving for the fermions, imposing Gauss's law, and averaging over gauge transformations, one can construct the N classically-degenerate states in H lx,ly=lz=0 .It is already clear from (3.8) that these N states are obtained by the action of T x t x from each other.The electric flux states from eqn. (3.7) are a discrete Fourier transform thereof.The anomaly implies that the Nfold degeneracy will be lifted and that only the pairwise degeneracy will remain quantum mechanically.Extending this small-torus explicit analysis further, along the lines of [48,49], as well as similar studies for other backgrounds, e.g., (3.1), are left for the future.

must insert (ψ
2 , a gauge invariant object which transforms with a Z 2 phase under ψ R, R → e i 2π 2(N ±2) ψ R, R. We have Next, recall that X † 2T R X2T R = 1 in H l i , remembering that we are in a definite magnetic flux sector, where the chiral symmetry operator X2T R acts in an invertible manner, and using (3.7), we find that gauge-invariant condensates obey In particular, (3.10) shows that the condensate appearing in (3.9) takes opposite values in the degenerate states.In the twisted partition function (3.9), this minus sign is cancelled by change of the phase e i 2π N e 1 (from the action of T x t x ).Thus, we can restrict the evaluation of (3.9) by summing over half the e 1 sectors: These nonzero expectation values can be computed semiclassically at a small torus and shown not to vanish, similar to [50].
To summarize, above we constructed the doubly-degenerate states using the background (3.4), in the N = 4p+2 theory, as an example.However, based on the Z 2 -valued anomaly, similar descriptions involving degenerate states with opposite values of the relevant condensate hold in all even-N cases.In particular, the doubly-degenerate flux states corresponding to (3.1) can also be explicitly worked out.
Before we discuss these cases, let us contrast the findings in the SU (N ) × U (1) theory on T 3 with those in the SU (N ) theory, also on T 3 .Consider the SU (4p + 2) theory (i.e., with even N not divisible by 4).It has a Z (1) 2 center symmetry and an invertible discrete chiral symmetry Z χ 2(N ±2) .These have a Z 2 -valued mixed anomaly in appropriate Z (1) 2 backgrounds on T 3 .This anomaly implies, as in [36], an exact two-fold degeneracy in the twisted Hilbert space of the SU (4p + 2) theory, on any torus size.This is similar to the degeneracy of the SU (4p+2)×U (1) theory on T 3 discussed in this paper.We stress, however, that the latter theory has gcd(d R , T R ) = 1, and hence no genuine chiral symmetry.The degeneracy we found is, thus, due to the noninvertible chiral Zχ 2T R symmetry.
Consider now the case when N is divisible by 4. In the SU (N ) theory this mixed anomaly is trivial on T 3 and hence one cannot use the Z (1) 2 1-form symmetry to argue for an exact degeneracy on a finite-size torus.In the SU (N ) × U (1) theory, however, for any even N , we showed that there is an anomaly between the Z (1) N center and noninvertible chiral symmetries in the (3.1) background, leading to an exact two-fold degeneracy at any size T 3 .
The case of minimal dimension condensate occurs if we take N = 4 and an antisymmetric tensor.Here, the anomaly predicts equal and opposite values of the bilinear fermion condensate ψ Rψ R in the two states that are degenerate at any finite volume, in the appropriately twisted background.Since the degeneracy is present at any finite volume, should the condensate remain nonzero in the infinite volume limit, this predicts the Z 4 → Z 2 (noninvertible) chiral symmetry breaking in the thermodynamic limit.
We stress that the use of appropriate twists-the m ij , n ij with integer l i , i.e. the ones that reveal the anomaly-at finite volume is simply a tool to probe the gauge dynamics.At least in the theory with a nonzero mass gap, the infinite volume limit is expected to be independent of the boundary conditions and the degeneracies revealed are expected to persist in the thermodynamic limit.

Discussion
Here, we found that the anomaly establishes the 2-fold degeneracy on arbitrary-size T 3 .Yet, one eventually wants to see what happens as we take the thermodynamic limit by sending the volume of T 3 to infinity.
To speculate on what could happen in SU (N ) × U (1) theory, let us again return to its cousin, the SU (N ) gauge theory with S/AS fermions.The latter has a global U (1) baryon number and U (1) χ chiral symmetry.As usual, quantum effects break U (1) χ down to, now, the invertible Z χ 2(N ±2) .The 0-form faithful global symmetry of this theory is U (1) , with p = gcd(N, 2).The fact that the quotient group is nontrivial means that we can activate a 't Hooft flux in the center of SU (N ) accompanied by a flux in U (1) such that the cocycle conditions are always obeyed on general fourdimensional manifolds.The authors constructed these fluxes in vector-like theories, dubbed as the baryon-color (BC) fluxes, in [51,52] (also see [53][54][55][56][57] for applications, and [58] for the construction and applications of these fluxes in chiral gauge theories).The partition function acquires a Z N ±2 phase as we apply a Z χ 2(N ±2) rotation in the background of the BC flux.This phase was interpreted as an anomaly of Z χ 2(N ±2) in the BC background.Assuming that the theory has a mass gap and forms hadrons in the IR, the anomaly is interpreted to imply the existence of N ± 2 degenerate vacua [53].One expects to see this degeneracy emerge on a finite-volume manifold (larger than the inverse strong-scale) and persist in the thermodynamic limit.
As we argued in this work, gauging the U (1) baryon symmetry endows the theory with a Z 2 anomaly phase when N is even, implying that at any finite volume there is exact 2-fold degeneracy.Thus, 2 degenerate vacua are guaranteed to survive the infinite volume limit.If the SU (N ) × U (1) theory also has N ± 2 degenerate vacua in the infinite volume limit (as the SU (N ) theory is believed to) the exact N ± 2-fold degeneracy should be revealed in the thermodynamic limit.
One might, of course, wonder whether a stronger phase (and stronger constraints, as in [59]) can be exhibited if we subject the SU (N ) × U (1) theory to a gravitational background.In other words, it would be interesting to investigate whether there is a mixed anomaly between the noninvertible Zχ 2(N ±2) and gravity.If the gravitational anomaly does not produce a stronger phase beyond Z 2 , the implications in the thermodynamic limit of the SU (N ) × U (1) theory are of interest.By weakly gauging U (1) in the SU (N ) theory, one anticipates the presence of N ± 2 nearly degenerate vacua.However, the dynamics of the U (1) gauge field may affect the precise degeneracy.This intriguing investigation remains open for future study.
Finally, we comment on the SU (N ) × U (1) theory at finite temperature.The presence of the identified mixed anomaly implies that it is necessary for either the 1-form symmetry Z N , the 0-form symmetry Zχ 2(N ±2) , or both symmetries to be broken [60][61][62].Actually, since Z N acts on the U (1) Wilson lines, we expect it to be broken at zero and finite temperature.Consequently, the anomaly is consistently matched, leading us to only expect the restoration of the broken Zχ 2(N ±2) symmetry and the breaking of the Z (1) 2 subgroup of the 1-form symmetry for even-N at some finite temperature.
Acknowledgments: M.A. is supported by STFC through grant ST/T000708/1.E.P. is supported by a Discovery Grant from NSERC.We thank Aleksey Cherman, Theo Jacobson, and Emily Nardoni for discussions.We also thank an anonymous referee for the suggestion to consider the "half-gauging" construction of the noninvertible defect and study its relation to the Hilbert space construction, see Appendix A.
A The noninvertible defect via "half-gauging" of Z Here, we consider the construction of the noninvertible defect for14 Zχ 2T R using the "halfgauging" procedure of refs.[11,12] instead of the sum over gauge orbits of [17,18].
The advantage of the former construction is that it gives rise to well-defined Euclidean correlation functions involving the noninvertible defect, considered more generally than as an operator inserted at a particular time.In particular, this allows inserting the defect at a particular location in space, giving rise to well-defined Hilbert spaces twisted by the noninvertible symmetry.
Our discussion below makes use of the techniques described explicitly in ref. [11] and is restricted to the case gcd(d R , T R ) = 1.For our SU (N ) × U (1) theories with a two-index S/AS Dirac fermion, this is the case of even-N not divisible by 4, leaving the generalization to the more general case for future work (although a generalization to gcd(d R , T R ) > 1 should be possible [11]).In order to define the defect, we consider the gauging of the Z (1) T R subgroup of the magnetic U (1) (1) m 1-form symmetry.As shown in [11], this gauging produces the same theory but with a discrete shift of the U (1) theta angle θ → θ− 2πd R T R .This shift, as per our Eqn.(2.4), is undone by a Z 2T R chiral rotation of the fermions (ψ R , ψ R).We conclude that the SU (N )×U (1) theory is invariant under the above gauging of Z (1) m .As in [11,12], the upshot of the half-gauging procedure is to define a defect, replacing our eqn.(2.21) for the operator X2T R by the following object, which we label, for brevity, by the same letter X2T R = e where, as in the main text, a is the dynamical U (1) gauge field and A is the SU (N ) gauge field with K CS [A] entering as in (2.3).The 3d defect TQFT A T R ,d R [ da T R ] is defined via an integral over a 4d bulk with a boundary, which is here taken to be the t = 0 plane: 2) ∧b (2)   . (A. 2) The fields (b (2) , c (1) ) define the 2-form Z T R gauge field (used in the half-gauging procedure) and obeying the Dirichlet boundary condition b (2) = 0 at t = 0, and k is the modular inverse of d R , i.e. kd R = 1 (modT R ).Defining the defect via the "halfgauging" procedure replaces the sum over gauge copies of the gauge noninvariant U (1) Chern-Simons term by the TQFT A T R ,d R [ da T R ] and produces a well defined Euclidean defect.
The A T R ,d R [ da T R ] theory has a Z

( 1 )
T R global symmetry with an anomaly d R , and the partition function e t=0 A T R ,d R [ da T R ] of (A.2) has the transformation properties of