Noninvertible symmetries and anomalies from gauging $1$-form electric centers

We devise a general method for obtaining $0$-form noninvertible discrete chiral symmetries in $4$-dimensional $SU(N)/\mathbb Z_p$ and $SU(N)\times U(1)/\mathbb Z_p$ gauge theories with matter in arbitrary representations, where $\mathbb Z_p$ is a subgroup of the electric $1$-form center symmetry. Our approach involves placing the theory on a three-torus and utilizing the Hamiltonian formalism to construct noninvertible operators by introducing twists compatible with the gauging of $\mathbb Z_p$. These theories exhibit electric $1$-form and magnetic $1$-form global symmetries, and their generators play a crucial role in constructing the corresponding Hilbert space. The noninvertible operators are demonstrated to project onto specific Hilbert space sectors characterized by particular magnetic fluxes. Furthermore, when subjected to twists by the electric $1$-form global symmetry, these surviving sectors reveal an anomaly between the noninvertible and the $1$-form symmetries. We argue that an anomaly implies that certain sectors, characterized by the eigenvalues of the electric symmetry generators, exhibit multi-fold degeneracies. When we couple these theories to axions, infrared axionic noninvertible operators inherit the ultraviolet structure of the theory, including the projective nature of the operators and their anomalies. We discuss various examples of vector and chiral gauge theories that showcase the versatility of our approach.


Introduction
Symmetry is the bedrock upon which quantum field theory (QFT) is constructed.In the past decade, a seismic shift has occurred in our understanding of symmetries, transcending their conventional application to mere point-like particles.In the contemporary paradigm, a p-form symmetry in 4 dimensions is linked to operators residing on (3−p)-dimensional topological manifolds that act on p-dimensional objects charged under the symmetry.Moreover, over the last couple of years, symmetries have expanded their domain to encompass operators that defy the conventional notion of inversion.These are known as noninvertible symmetries.While noninvertible symmetries initially found their roots and applications in the realm of 2-dimensional QFT, see, e.g., [1,2], their significance in the context of 4-dimensional QFT sparked a deluge of research endeavors in this area (a non-comprehensive list is .Also, see [30,31] for reviews.) It is well known that quantum electrodynamics has a classical U (1) χ axial symmetry that breaks down because of the Adler-Bell-Jackiw (ABJ) anomaly.However, it was realized in [10,11] that the axial symmetry does not completely disappear.Instead, it resurfaces as a noninvertible symmetry for each fractional element of the classical U (1) χ .This profound reinterpretation of symmetries prompted a compelling quest to unearth analogous structures in QFT.In [32], one of the authors established a technique for unveiling noninvertible 0-form symmetries within SU (N ) × U (1) gauge theories in the presence of matter in representation R.This approach employed the Hamiltonian formalism, where the theory was put on a three-dimensional torus T 3 , subjecting it to Z N magnetic twists along all three spatial directions.Taking the matter to be a single Dirac fermion, this theory is endowed with invertible Z χ 2gcd(T R ,d R ) 0-form chiral symmetry, where T R and d R are the Dynkin index and dimension of R, respectively.Yet, it was shown that the theory also possesses a noninvertible Zχ 2T R 0-form chiral symmetry 1 .Such symmetry acts on the Hilbert space projectively by selecting special sectors characterized by certain magnetic numbers.New noninvertible symmetries were also revealed in [33] in theories with mixed anomalies between Z (1) 2 1-form and 0-form discrete chiral symmetries.
The topological essence of symmetries, encompassing the noninvertible variants, underscores their sensitivity to the global structure of the gauge group.Consequently, the inquiry arises: how do we identify these noninvertible symmetries within a general gauge group, characterized as either SU (N )/Z p or SU (N ) × U (1)/Z p where Z p is a subgroup of the center symmetry?In this work, we answer this question by devising a general method that applies to any theory with a direct multiplication of abelian and semi-simple nonabelian gauge groups quotiented by a discrete center, whether the theory is vector-like or chiral.This is achieved by putting the theory on T 3 and turning on magnetic fluxes in a refined subgroup of Z N , depending on the matter content as well as the global structure of the gauge group.
In the context of SU (N ) gauge theory, the introduction of matter characterized by an N -ality n has the effect of breaking the Z N center of the group down to a subgroup Z q , where q is the greatest common divisor (gcd) of N and n.Our focus is on understanding the noninvertible 0-form symmetries present in the SU (N )/Z p gauge theories, where Z p is a subgroup of the remaining center Z q .These theories exhibit both electric Z (1) q/p and magnetic Z (1) p 1-form global symmetries2 .To identify the noninvertible symmetries, we initiate the process starting from SU (N ) theory endowed with a single Dirac fermion in representation R, which possesses an invertible Z χ 2T R chiral symmetry.We then subject this theory to electric and magnetic twists characterized by elements of Z p .If the theory exhibits a mixed anomaly between its chiral and electric Z (1) p 1-form symmetries, the act of gauging Z p effectively reveals the chiral symmetry as noninvertible.The construction of a gauge-invariant operator corresponding to the noninvertible symmetry Zχ 2T R involves several steps.First, we create a topological operator by integrating the anomalous current conservation law over T 3 .The resulting operator is not invariant under Z p gauge transformations.Yet, we can restore gauge invariance by summing over all possible Z p gauge-transformed operators.This process results in a noninvertible chiral symmetry operator that projects onto specific sectors in the Hilbert space, each characterized by certain 't Hooft lines charged under the magnetic Z (1) p 1-form symmetry.Zχ 2T R can exhibit further anomalies when subjected to twists by the electric Z (1) q/p 1-form symmetry, implying that states within the Hilbert space of the SU (N )/Z p gauge theory will display multiple degeneracies.
We employ a similar approach to identify noninvertible symmetries in SU (N ) × U (1)/Z p gauge theories, where Z p is a subgroup of the electric Z (1) N 1-form center symmetry.Unlike in SU (N ) theories, the introduction of matter does not reduce the Z N center.This is due to the presence of an abelian U (1) sector, which ensures that all matter representations adhere to the cocycle condition.In addition to the 1-form electric center symmetry, this theory is also endowed with a magnetic U (1) m (1) 1-form symmetry.SU (N ) gauge theory with matter exhibits an anomaly between its chiral and U (1) baryon-number symmetries.Gauging the latter transforms the theory into an SU (N ) × U (1) gauge theory and reveals the chiral symmetry Zχ 2T R as noninvertible.Placing the theory on T 3 enables us to construct the corresponding noninvertible chiral operator by summing over large U (1) gauge transformations with distinct winding numbers.Furthermore, since the theory exhibits a 1-form electric center symmetry, we can decorate the noninvertible operator with Z N magnetic twists.If we choose to further gauge a Z N symmetry, thereby resulting in the SU (N ) × U (1)/Z p theory, we must ensure that the noninvertible operator remains invariant under Z p gauge transformation.This is accomplished by summing over all Z p gauge-transformed chiral operators.Once again, we discover that the resultant operator projects onto specific sectors within the Hilbert space, distinguished by the presence of 't Hooft lines charged under U m (1).The noninvertible symmetry also exhibits a mixed anomaly with the remaining electric Z Placing the theory on T 3 offers a distinct advantage: it presents a systematic approach for computing the 't Hooft anomalies inherent to a given theory.Simultaneously, it provides a means to construct the Hilbert space explicitly.In our work, we put a significant emphasis on this Hilbert space construction, shedding light on the intricate relationship between Wilson's lines, 't Hooft lines, and the noninvertible operator.Specifically, through several illustrative examples, we showcase how the noninvertible chiral operator, within the framework of the Hilbert space and Hamiltonian formalism, acts to annihilate the minimal 't Hooft lines.
We also introduce couplings of gauge theories to axions.The underlying renormalization group invariance of the noninvertible symmetries, along with their associated anomalies, guarantees that the infrared (IR) axion physics faithfully inherits all the characteristics of the theory at the ultraviolet (UV) level.We substantiate this by explicitly constructing noninvertible chiral operators, commencing from the IR anomalous axion current conservation law.In our exploration, we offer concrete illustrations of various UV theories and their corresponding IR axion physics manifestations.This paper is organized as follows.In Section 2, we provide a concise overview of the essential elements required for the development of noninvertible symmetries.This section encompasses the introduction of our notation, a review of the path-integral formalism on the 4-torus (T 4 ), 't Hooft twists, and the Hamiltonian formalism on T 3 .Moving on to Section 3, we proceed to construct noninvertible symmetries within the context of SU (N )/Z p theories while also identifying their associated anomalies.This section concludes with the presentation of specific examples of noninvertible symmetries in both vector and chiral gauge theories.In Section 4, we replicate the same analysis, this time focusing on SU (N ) × U (1)/Z p .Two examples are discussed, including the Standard Model (SM), and we demonstrate that the SM lacks noninvertible symmetries within its non-gravitational sector.Finally, our paper culminates in Section 5, where we explore the coupling of gauge theories to axions.We show that noninvertible symmetry operators can also be constructed using the axion anomalous current.

Preliminaries
In this section, we review the path integral and the Hamiltonian formalisms of gauge theories put on a compact manifold with possible 't Hooft twists, both in space and time directions.Additionally, we examine the global symmetries and anomalies in both formalisms, providing an exploration of these key aspects.We base our formalism and notation on [32,[35][36][37], and set the stage for constructing the noninvertible operators we carry out in the subsequent sections.While some results in this section are new, many are a mere review of previous results.Moreover, some details are avoided, referring the reader to the literature for an in-depth discussion.Yet, the information encapsulated here is necessary to make this paper self-contained.

Pure SU (N ) theory
We begin by reviewing 't Hooft twists on a compact 4-dimensional Euclidean manifold with nontrivial 2-cycles.We consider SU (N ) pure Yang-Mills (YM) theory on T 4 , where T 4 is a 4-torus with periods of length L µ , µ = 1, 2, 3, 4 3 .The SU (N ) gauge fields A µ are taken to obey the boundary conditions upon traversing T 4 in each direction.The transition functions Ω µ are N × N unitary matrices in the defining representation of SU (N ), and êν are unit vectors in the x ν direction.The subscript µ in Ω µ means that the function Ω µ does not depend on the coordinate x µ .Then, the compatibility of (2.1) at the corners of the x µ − x ν plane of T 4 gives the cocycle condition The exponent e i 2πnµν

N
, with anti-symmetric integers n µν = −n νµ , is the Z N center of SU (N ).The freedom to twist by elements of the center stems from the fact that both the transition function and its inverse appear in (2.1).This is also equivalent to the fact that the Wilson lines in pure SU (N ) gauge theory are charged under the electric N 1-form center symmetry.The fundamental (defining representation) Wilson lines wind around the 4 cycles and are given by Aµ Ω µ , where □ denotes the defining representation of SU (N ) and the insertion of the transition function Ω µ ensures the gauge invariance of the lines.It will be useful to break n µν into spatial (magnetic) m i and temporal (electric) k i twists: and i, j = 1, 2, 3 or x, y, z.We also use bold-face letters, e.g., k ≡ (k 1 , k 2 , k 3 ), to denote 3-dimensional vectors.When applied to the gauge fields on T 4 , the twists induce a background with fractional topological charge 4 [35]: 3 YM theory on T 4 with Z N 't Hooft twists dates back to the original work by 't Hooft [35].The groundbreaking paper [38] unveiled a novel mixed anomaly, specifically involving the electric Z (1) N 1form symmetry.The background of the 1-form symmetry is a 2-form field that can be implemented via a 't Hooft twist.This fact led to a wave of enthusiasm to understand the semi-classical limit of gauge theories on T 4 or T 2 × R 2 [39][40][41][42]. 4The simplest way to find the topological charge is by activating the electric and magnetic 't Hooft fluxes along the Cartan generators of SU (N ); see, e.g., [43].We set L3L4 ν a H a along the 1-2 and 3-4 planes (and similar expressions in the rest of the planes), where H a are the Cartan generators, ν a are the weights of the defining representation, a = 1, ..., N − 1, with summation over repeated indices.Plugging into Q = 1 8π 2 T 4 tr[F ∧ F ], and using tr[H a H b ] = δ ab and where F is the field strength of A. Notice that the twists (m, k) ∈ (Z Mod N ) 6 .Adding multiples of N to m or k leaves the cocycle condition intact.However, this has the effect of changing the topological charges by integers.Hence, from here on, we shall take the twists m i , k i ∈ Z, not Mod N .The partition function of the SU (N ) gauge theory with given twists (m, k) is Here, S Y M is the YM action, and the subscript (m, k) indicates that the path integral is to be performed with a given set of twisted boundary conditions.Summation over the integer-valued topological sectors, ν ∈ Z, is necessary so that the theory satisfies locality (cluster decomposition).

SU (N ) theory with matter
Next, we add matter fields in a representation R under SU (N ).The matter representation has N -ality n.Then, the full Z N center breaks down to Z q , q = gcd(N, n), i.e., the Wilson lines are charged under Z q 1-from center symmetry 5 .Putting the matter, which, from now on, will be assumed to be fermions, on T 4 modifies the cocycle conditions.Let ψ be a left-handed Weyl fermion transforming under R of SU (N ).Then, the fermion obeys the boundary conditions (2.7) The matrix R(Ω µ (x)) is built from Ω µ , transforming in the defining representation of SU (N ), with suitable symmetrization or anti-symmetrization over n indices (the Nality of the representation) according to the specific representation R. Thus, schematically (ignoring symmetrization over indices) 5 For example, SU (2M ) gauge theory with matter in the 2-index (anti)symmetric representation has a Z which, via Eq.(2.2), reveals that the allowed values of the twists m and k are N q , 2N q , .... Twisting by the center subgroup Z q induces a background field with fractional topological charge and the partition function in the presence of matter reads In the presence of matter, the theory is endowed with classical nonabelian and abelian flavor symmetries.The U (1) baryon-number symmetry survives the quantum corrections.In contrast, the chiral part of the abelian symmetry, denoted by U (1) χ , will generally break down to a discrete symmetry because of the Adler-Bell-Jackiw (ABJ) anomaly of U (1) χ in the background of color instantons (which have integer topological charges).To fix ideas, we consider a single flavor of a Dirac fermion with classical U (1) baryon number and U (1) χ chiral symmetries.We take the U (1) baryon charge of the Dirac fermion to be +1.The ABJ anomaly breaks U (1) χ down to invertible Z χ 2T R chiral symmetry, where T R is the Dynkin index of the representation.Generalizing the theory to include many flavors is straightforward, and we shall work out examples of this sort later in the paper.In the presence of the twists (m, k), there can be an anomaly of Z χ 2T R in the background of Z q .The anomaly is a non-trivial phase acquired by Z[m, k] SU (N )+matter as we apply a transformation by an element of Z 2T R : and ℓ = 0, 1, 2, .., T R − 1 are the elements of Z χ 2T R .For the smallest twists m j = k j = N q in the j-th direction, we obtain [44] Z Bearing in mind that N/q ∈ Z, we can generally absorb the integral part of N/q 2 by adding an integer topological charge, which cannot change the anomaly.Nevertheless, we will retain the phase as indicated in Eq. (2.13).The phase is nontrivial, and hence there is an anomaly, if and only if ℓ N q 2 ̸ ∈ Z.In the next section, we show how to obtain the same anomaly using the Hamiltonian formalism.
We can do more regarding turning on fractional fluxes in SU (N ) with matter.Instead of limiting ourselves to Z q twists, we can twist with the full Z N center symmetry or any subgroup of it provided we also turn on backgrounds of U (1) baryon number symmetry [43,44].Let ω µ denote the U (1) transition functions such that for the U (1) gauge field a µ , we have Then, Ω µ and ω µ obey the cocycle conditions: where the N -ality of the matter representation is incorporated in the abelian transition functions.The topological charges of both the nonabelian center and abelian backgrounds read 6 (2.15) Here, A, B are arbitrary integral magnetic and electric quantum numbers that we can always turn on since they leave the cocycle condition intact.

SU (N ) × U (1) theory with matter
We may also choose to make U (1) dynamical, which entails summing over small and large gauge transformations of U (1), with the latter implementing integer winding.This results in SU (N ) × U (1) gauge theory with a Dirac fermion in representation R, with N -ality N and baryon-charge +1.In this case, the U (1) instantons reduce Z χ 2T R down to the genuine (invertible) symmetry Z χ 2gcd(T R ,d R ) , and d R is the dimension of R. The easiest way to see that is by recalling the partition function under a U (1) χ transformation acquires a phase: where f is the field strength of the U (1) field.Recalling that for the dynamical SU (N ) and U (1) fields we have 6 Here, we use Footnote 4 along with abelian field strengths survises the chiral transformation.The theory admits Wilson's lines: which are charged under an electric Z N 1-form center symmetry.In addition, the theory is endowed with a magnetic U (1) m (1) 1-form symmetry because of the absence of magnetic monopoles.For the sake of completeness, we also give the partition function of SU (N ) × U (1) theory with matter in the background of given (m, k) fluxes: and in addition to the SU (N ) integer topological charges ν, we included a sum over integer topological charges ν U (1) of the U (1) sector.

Twisting in the Hamiltonian formalism
Pure SU (N ) theory Let us repeat the above discussion using the Hamiltonian formalism, starting with pure SU (N ) YM theory (we use a hat to distinguish an operator in this section.)To this end, we put the gauge theory on a spatial 3-torus T 3 and apply the magnetic m twists along the 3-spatial directions.The transition functions in the defining representation along the spatial directions, denoted by Γ i , can be chosen to be constant N ×N matrices obeying the cocycle condition Then, one can construct the states of the physical Hilbert space using the temporal gauge condition A 0 = 0.The states can be written using the "position" eigenstates of the gauge fields A j , j = 1, 2, 3 (or i = x, y, z) as follows: and the subscript m emphasizes that the Hilbert space is constructed in the background of the magnetic twists.In writing Eq. (2.20), we have put many details under the rug, and the reader is referred to [35][36][37] for details.For example, notice that the gauge fields A i need to respect the twisted boundary conditions (2.19), i.e., they transform according to (2.1) as we traverse any spatial direction on T 3 .The theory admits 3 fundamental Wilson lines wrapping the three cycles of T 3 ; these are given by (2.3) by restricting µ to the spatial directions.The Wilson lines are charged under the Z N 1-form symmetry generated by three symmetry generators Tj , the Gukov-Witten operators, supported on co-dimension 2 surfaces.Thus, we have and there are Ŵ e j j distinct Wilson's lines with N distinct N -alities e j = 0, 1, .., N − 1.The center-symmetry generators Ti are hard to construct explicitly.However, their explicit form is not important to us.What is important is that they commute with the YM Hamiltonian Ĥ, and thus, Ĥ and Ti can be simultaneously diagonalized.The physical states of the theory |ψ⟩ phy ,m are designated by the eigenvalues of Ti .It can be shown that the action of Ti on |ψ⟩ phy ,m is given by where e j , m j ∈ Z N and the θ term ensures that T N i |ψ⟩ phy ,m = e −iθm j |ψ⟩ phy ,m , and hence, T N i works as a large gauge transformation.The combination e j − θ 2π m j is the Z N electric flux in the j-th direction.This is justified as follows.Consider the state Ŵj |ψ⟩ phy ,m , obtained from |ψ⟩ phy ,m by the action of Ŵj .Using Eqs.(2.21, 2.22), we find Tj Ŵj |ψ⟩ phy ,m = e i 2π N (e j +1)−iθ m j N Ŵj |ψ⟩ phy ,m .Therefore, acting with Ŵj on the state |ψ⟩ phy ,m increases e j by one unit in the j-th direction.Since Ŵj inserts an electric flux tube winding in the j-th direction, the interpretation of e j as electric flux follows.Notice also that because Tj and Ĥ can be simultaneously diagonalized, we may label the states by the energy and the electric flux: It is worth spending some time to explain our notation in Eq. (2.23), as we shall use this notation extensively in our paper.The physical state is labeled by the eigenvalues of a set of commuting operators, here the energy and the electric flux.The SU (N ) theory does not admit a 1-form magnetic symmetry, and thus, we cannot label the states by magnetic fluxes.Yet, we can turn on a background magnetic flux m, indicated as a subscript; all physical quantities are calculated in this magnetic background.Also, we use the letter m to denote the set of magnetic fluxes we can consistently turn on.Here, we have m ∈ Z 3 .
How can we make sense of the fractional topological charge (2.10) on T 3 ?We consider the product of T 3 and the time interval [0, L 4 ] and consider the boundary conditions Âi (t = L 4 ) = C[k] • Âi (t = 0), where C[k] is an "improper gauge" transformation implementing a twist k ∈ Z 3 on the gauge fields by an element of the center 7 .In the presence of the magnetic twists m, it can be shown that an application of C[k] results in the topological charge (Pontryagin square) [35][36][37]: where K( Â) is the topological current density operator

SU (N ) theory with matter
Adding fermions of N -ality n changes the center from Z N to Z q , q = gcd(N, n), and the twists (m, k) are now in (N Z/q) 6 .Otherwise, all the steps used to put the theory on T 3 and construct the Hilbert space carry over.In particular, Ti now are the generators of the Z (1) q 1-form symmetry, and their action on the physical states in the Hilbert space is given by 8 (now we turn off the θ angle as we can rotate it away via a chiral transformation acting on the fermion) Tj |ψ⟩ phy ,m = e i 2π q e j |ψ⟩ phy ,m , ( and the theory has e j = 0, 1, 2, .., q − 1 electric flux sectors in each direction j = 1, 2, 3. The operators Tj act on the spatial Wilson lines in the defining representation of SU (N ) as Tj Ŵj = e i 2π q Ŵj Tj , and there are q distinct Wilson's lines W e j j .The physical states |ψ⟩ phy ,m are simultaneous eigenstates of the Hamiltonian and Tj since both operators commute.Thus, we can write the physical states in the magnetic flux background m ∈ (N Z/q) 3 as and N e j /q is the amount of electric flux carried by the state in direction j.We may also say that e j is the number of electric fluxes in units of N/q.For matter with N -ality 7 In fact, C should be designated by both k and the integral instanton number ν; see [35].However, ν does not play a role in this work. 8It is conceivable to introduce an additional label to signify the distinct symmetries generated by different operators Tj .For instance, we could designate TN,j as the generator of Z q .Nonetheless, this approach may lead to increased complexity in our expressions, and we opt not to pursue it.Instead, we will explicitly specify the symmetry in question when discussing these distinct operators.n = 0, e.g., in the adjoint representation, q = N and we recover what we have said about pure SU (N ) gauge theory.
The partition function (2.11) can be written in the Hamiltonian formalism as a trace over states in Hilbert space: where the subscript m in the trace means that we are considering the states in the background of the magnetic flux m ∈ (N Z/q) 3 .We also used Eqs.(2.25, 2.26), the fact that the states are eigenstates of both the energy and the 1-form center operators.
To detect the anomaly between Z χ 2T R and Z (1) q in the Hamiltonian formalism, we first define the operator that implements the discrete chiral symmetry.To this end, we recall that under a chiral U (1) A rotation, the presence of the ABJ anomaly indicates non-conservation of the corresponding symmetry rotation: Yet, we can define a conserved current: and correspondingly a conserved charge: Therefore, it is natural to define the operator for ℓ = 0, 1, .., T R − 1, which implements the action of the Z χ 2T R chiral symmetry.ÛZ 2T R is invariant under both small and large SU (N ) gauge transformations (with integer winding).To find the mixed anomaly between Z χ 2T R and Z q , we compute the commutation between Tj , which implements the action of the electric center symmetry in the j-th direction, and ÛZ 2T R : remembering that the theory is in the background of a magnetic twist m j ∈ N Z q in the j-th direction 9 .First, Tj commutes with the current j 0 A since the latter is a color singlet operator.However, K0 fails to commute with Tj ; the commutation between the two operators is found by recalling that the action of Tj is implemented on the gauge fields Âj as Âj = C[k j ] • Âj .Thus, we find, after making use of (2.24), Tj exp i2πℓ noting the restriction m j , k j ∈ N Z q 2 due to the presence of matter; otherwise, we would not satisfy the cocycle condition.Collecting everything and using the minimal twists which is exactly the mixed anomaly between the Z χ 2T R chiral and the Z q 1-form center symmetries found in (2.13) from the path integral formalism.The anomaly along with the commutation relations (remember that both Z (1) q and Z χ 2T R are good symmetries of the theory, and hence, the corresponding operators commute with the Hamiltonian) furnishes a finite-dimensional space with a minimum dimension of q 2 /gcd(q 2 , N ).This means that sectors in Hilbert space exhibit a q 2 /gcd(q 2 , N )-fold degeneracy.

SU (N ) × U (1) theory with matter
Next, we discuss the Hamiltonian quantization of SU (N ) × U (1) gauge theory with matter fields on T 3 in the background of twists.In this case, we may twist with the full 9 Similar to the Footnote 8, we could use a label that denotes the specific magnetic flux background when we are dealing with the operator ÛZ 2T R ,ℓ .This background can be taken in sets such as N Z q or N Z p , among others.However, adopting this approach may introduce unnecessary complexity to our notation.As a result, we have chosen to adopt a more transparent approach: we will explicitly mention the magnetic flux background whenever we discuss this operator.
Z N center symmetry provided we also turn on a background of U (1).Thus, we replace the cocycle conditions (2.19) with and we included the N -ality of the matter representation n in the cocycle condition of the abelian field.This guarantees that the combined transition functions satisfy the correct cocycle conditions in the presence of matter.Here, we can allow background center fluxes with (m, k) ∈ Z 6 for all matter representations, thanks to the U (1) gauge group.We also introduce the operators Tj for SU (N ) and tj for U (1), j = 1, 2, 3.
The combinations Tj tj are the generators of the electric Z N 1-form global symmetry and act on the spatial Wilson lines in (2.17) as: Tj W j,SU (N ) = e i 2π N W j,SU (N ) Tj and tj W j,U (1) = e −i 2π N W j,U (1)) tj .The action of tj is implemented on the gauge fields, as usual, by improper gauge transformations of âj as âj = c[k j ] • âj , and amounts to applying nk j (Mod N ) electric twists (notice the appearance of the N -ality).Unlike Tj , the explicit form of tj is simple: Since Z N is a good global symmetry, we can choose the states in Hilbert space to be eigenstates of the Z where e j = 0, 1, ..., N − 1.Notice that the states are constructed in the "fractional" background magnetic flux m ∈ Z 3 (remember that in principle m i ∈ Z Mod N , and thus, it implements the fractional magnetic twist.However, we can always add multiples of N to m i without affecting the cocycle conditions, and hence, we drop the Mod N restriction.)In addition, the theory has a magnetic U (1) m 1-form global symmetry, which can be used to characterize the physical states by an "integer" value of the magnetic flux.Therefore, a state in the physical Hilbert space can be labeled as (2.40) We also build the operator that corresponds to the chiral transformation.This construction was detailed in [32], and we do not repeat it here.Instead, we only give a synopsis of the derivation, which is needed in this work.The anomaly equation of the chiral current is Then, the chiral symmetry operator in the background of the m j magnetic flux is given by where the conserved charge Q5 is given by Q5 = The last term comes from carefully treating the boundary term implied from the transition functions ω j (x), since, unlike Γ j , they depend explicitly on x j , see [32] for details.
In addition to the background flux n j , which introduces the fractional winding number, we also allow integer magnetic winding N j .Under a transformation with tj , the integral of the abelian Chern-Simons term K0 (â) = ϵ ijk âi ∂ j âk in the background of the integral M j and fractional m j magnetic fluxes transforms as (recall (2.37)) tj exp i The reader will notice that we switched from the letter m, which we use to signify the set of fractional fluxes we can activate, e.g., here we have m ∈ Z 3 , to the letter n, which is the actual number of fractional magnetic fluxes we turn on.We shall use the same labeling throughout the paper.
In the next sections, we use these constructions to argue that SU (N )/Z p , Z p ⊆ Z q as well as SU (N ) × U (1)/Z p , Z p ⊆ Z N enjoy a noninvertible 0-form chiral symmetry, with a possible mixed anomaly with the 1-form center symmetry.
3 SU (N )/Z p , Z p ⊆ Z q theories, noninvertible symmetries, and their anomalies In this section, we direct our attention to YM theories featuring matter fields residing in a particular representation R and characterized by an N -ality n.Building upon the discussion in the preceding section, it is established that SU (N ) gauge theories, when coupled to matter, exhibit an electric Z (1) q 1-form center symmetry (recall q = gcd(N, n)).A notable maneuver within this framework involves the gauging of Z or a subgroup of it, leading to SU (N )/Z p theory, Z p ⊆ Z q , whose partition function is obtained by summing over integer and fractional topological charge sectors.Thus, gauge transformations with fractional winding numbers are part of the gauge structure, and well-defined operators should be invariant under such gauge transformations.Here, we would like to emphasize that there are p distinct theories: (SU (N )/Z p ) n , n = 0, 1, ..., p, which differ by the admissible genuine (electric, magnetic, or dyonic) line operators.In this paper, we limit our treatment to (SU (N )/Z p ) n=0 , and whenever we mention SU (N )/Z p , we particularly mean (SU (N )/Z p ) 0 .What happens to the invertible Z χ 2T R discrete chiral symmetry of this theory?As we shall discuss, this symmetry can stay invertible or become noninvertible, depending on whether it exhibits a mixed anomaly with Z (1) p symmetry in the original SU (N ) theory.

SU (N )/Z q
We start by discussing noninvertible 0-form chiral symmetries in SU (N )/Z q theories, i.e., theories obtained by gauging the full electric Z (1) q 1-form center symmetry.Such theories do not possess global electric 1-form symmetry; hence, there are no genuine Wilson's lines.This can be understood as follows.We start with pure SU (N ) gauge theory, which has an electric Z (1) N 1-form symmetry and admits the full spectrum of Wilson lines, i.e., it admits Wilson's lines with all N -alities n = 0, 1, 2, .., N −1.Gauging a Z q subgroup of Z N , we obtain SU (N )/Z q gauge theory.Now, the spectrum of allowed Wilson lines must be invariant under Z q , forcing us to remove those lines with N -alities that are not multiples of q.The remaining lines in pure SU (N )/Z q theory are charged under an electric Z (1) N/q 1-form symmetry; these are W qe j j , with e j = 0, 1, .., N/q − 1 and W j is Wilson's line in the defining representation of SU (N ).Finally, introducing matter with N -ality q means that those remaining lines can end on the matter and must also be removed from the spectrum.This deprives SU (N )/Z q gauge theory with matter from all genuine Wilson's lines.
Despite that SU (N )/Z q theory with matter does not possess an electric 1-form symmetry, it is endowed with a magnetic Z m(1) q 1-form global symmetry.This can be understood, again, starting from the pure SU (N )/Z q theory.As we discussed above, the pure theory has an electric Z (1) N/q 1-form symmetry.The magnetic dual of SU (N )/Z q is SU (N )/Z N/q , which admits a magnetic Z m(1) q 1-form symmetry.The pure SU (N )/Z q theory has q distinct magnetic fluxes ('t Hooft lines) in its spectrum.Let T j be the 't Hooft line winding around direction j in the defining representation of SU (N ), i.e., it has N -ality 1.Then, the pure SU (N )/Z q theory possesses the following set of 't Hooft lines T n j N/q j , n j = 0, 1, .., q − 1 for j = 1, 2, 3, which are mutually local with the set of Wilson's lines W qe j j , e j = 0, 1, .., N/q − 1 10 .Introducing electric matter removes all Wilson's lines (as stated above) but does not alter the magnetic symmetry.Thus, we conclude that SU (N )/Z q theory with matter possesses a magnetic Z m(1) q 1-form global symmetry acting on a set of 't Hooft lines T n j N/q j , n j = 0, 1, .., q − 1 for j = 1, 2, 3.
We can label the states in the physical Hilbert space of SU (N )/Z q theory with matter by both energy and magnetic fluxes since the Hamiltonian commutes with the generators of the magnetic Z m(1) q 1-form symmetry 11 : The partition function of these theories involves summing over sectors with fractional topological charges N Z/q 2 (use Eq.(2.10) and set k i = m i = N/q), which can be written in the path-integral formalism as (we set the vacuum angle θ = 0) 10 This can be easily seen since T nj N/q j and W qej j satisfy the Dirac quantization condition. 11Recall that the allowed magnetic twists in the SU (N ) theory with matter are m ∈ (N Z/q) 3 .
or in the Hamiltonian formalism as Z SU (N )/Zq+matter = tr e −L 4 Ĥ = physical states phy ⟨ψ|e −L 4 Ĥ |ψ⟩ phy . (3.3) Our main task is to build a gauge invariant operator that implements the Z χ 2T R chiral transformation in SU (N )/Z q theory with matter.To this end, we use the Hamiltonian formalism of Section 2.2, dropping the hats from all operators to reduce clutter.We also use x, y, z to label the three spatial directions.For ℓ ∈ Z χ 2T R , the chiral symmetry operator is given by: This operator is invariant under large gauge transformations with integer winding numbers.We will now gauge the Z q one-form symmetry.In SU (N )/Z q gauge theory with matter, we sum over arbitrary Z q twists with fractional topological charges N Z/q 2 .We consider the operator U Z 2T R ,ℓ in the presence of magnetic fluxes m ∈ (N Z/q) 3 (these are the magnetic fluxes that label the physical states in Eq. (3.1).)Let T x be the generator of an electric Z q center twist along the x direction (i.e., a Z q gauge transformation), and we take it to have the minimal twist of N/q.It acts on U Z 2T R ,ℓ via (recall the discussion around Eq. (2.33)) n x counts the magnetic fluxes inserted in the y-z plane in units of N/q.Identical relations to (3.5) hold in the y and z directions.As we saw in the previous section, if ℓ N q 2 ̸ ∈ Z, there is a mixed 't Hooft anomaly between the electric Z (1) q 1-form center and the discrete chiral symmetries of SU (N ) theory with matter.Eq. (3.5) implies that the operator U Z 2T R ,ℓ is not gauge invariant under a Z q gauge transformation as we attempt to gauge Z (1) q .We can remedy this problem and reconstruct a gauge-invariant operator, denoted by ŨZ 2T R , by summing over all Z q gauge transformations generated by T x , T y and T z : In the first line, we included a sum over arbitrary powers of T x , T y , T z to enforce the gauge invariance.Then, we used Eq.(3.5) in going from the first to the second line and the Poisson resummation formula in going from the second to the third line.Even though ŨZ 2T R ,ℓ is gauge invariant, it has no inverse; it is, in general, a noninvertible operator that implements the action of Zχ 2T R , and we use a tilde to denote the noninvertible nature of symmetries and their operators.The noninvertibility stems from the fact that ŨZ 2T R works as a projector: the insertion of this operator in the path integral of SU (N )/Z q theory with matter projects onto specific topological charge sectors of SU (N )/Z q , depending on ℓ.This can be seen from the second line in (3.6), which is a sum over Fourier modes that projects in and out sectors, depending on their topological charge, upon acting on them.One can see the projective nature of ŨZ 2T R ,ℓ by inserting it into the partition function (3.3): and then using the physical states defined in Eq. (3.1).We find that ŨZ 2T R ,ℓ annihilates sectors with nxℓN q 2 / ∈ Z, etc.We remind that nxN q 2 is the topological charge (see Eq. (2.10)), which we can write as and, as we mentioned earlier and emphasize now, n x is the number of magnetic fluxes in units of N/q.The same applies to the magnetic sectors in the y and z directions.We conclude that ŨZ 2T R ,ℓ selects sectors in Hilbert space with certain magnetic fluxes.
We can make the following observations about ŨZ 2T R ,ℓ : 1.If ℓ ∈ qZ, ŨZ 2T R ,ℓ is invertible since in this case nx,y,zℓN 2. If gcd(ℓN/q, q) = 1, then we must have n x , n y , n z ∈ qZ.In other words, ŨZ 2T R ,ℓ projects onto untwisted flux sectors.In particular, in the sector given by n x , n y , n z ∈ qZ, the symmetry operator ŨZ 2T R ,ℓ act invertibly for all elements of the chiral symmetry ℓ = 1, 2, .., T R .
sectors that have Z q ′ twists.
4. The noninvertibility of ŨZ 2T R ,ℓ can be seen by multiplying the operator by its inverse to find C is known as the condensation operator, which can be thought of as a sum over topological surface operators exp[−i T 2 ⊂T 3 B (2) ] = exp[−i2πZ/q] wrapping the three 2-cycles of T 3 , and B (2) is the 2-form field of the Z (1) q 1-form symmetry.
We use the fact that SU (N )/Z q theory possesses a magnetic Z m(1) q 1-form global symmetry to make one more observation.Let T j be 't Hooft line of N -ality 1 in direction j.Then, the minimal 't Hooft line in SU (N )/Z q theory is T N/q j , i.e., it has N -ality N/q.The minimal line acts on a physical state by increasing its magnetic flux by one in units of N/q 12 .Now, let us take a theory with gcd(N/q, q) = 1 so that ŨZ 2T R ,ℓ=1 acts projectively on certain states.Then, |E, (n x = q, n y = q, n z = q)N/q⟩ is one of the physical states that survive under the action of ŨZ 2T R ,ℓ=1 .We have T N/q x |E, (q, q, q)N/q⟩ = |E, (q + 1, q, q)N/q⟩.Thus, we immediately see from Eq. (3.6) that ŨZ 2T R ,ℓ=1 T N/q x |E, (q, q, q)N/q⟩ = ŨZ 2T R ,ℓ=1 |E, (q + 1, q, q)N/q⟩⟩ = 0 . (3.10) We write this result as ŨZ 2T R ,ℓ=1 T N/q j = 0 , j = x, y, z .(3.11)In other words, the operator ŨZ 2T R ,ℓ=1 annihilates the minimal 't Hooft lines in this theory.It also annihilates all 't Hooft lines T n j N/q j , n j ̸ = 0 Mod q.This is an alternative way to see the projective nature of this operator.

SU (N )/Z p
Next, we discuss SU (N )/Z p theory with matter with N -ality n, and Z p ⊆ Z q = Z gcd(N,n) .The partition function of this theory is given by the path integral in Eq. (3.2), now restricting the sum over the electric and magnetic twists (m, k) ∈ (N Z/p) 6 .
12 Similar to the discussion we had after Eq. (2.22), we can also consider the generators of the magnetic 1-form symmetry and argue that T N/q j inserts a magnetic flux N/q, as measured by the action of the magnetic 1-form symmetry on the state T N/q j |ψ⟩ phy .

The theory possess an electric Z
(1) q/p 1-form global symmetry.As before, T x is taken to be the generator of the electric Z (1) q symmetry.Then, the electric Z (1) q/p 1-form global symmetry is generated by T p x (as well as T p y and T p z ).The theory has q/p distinct Wilson's lines W e j p j , with e j = 0, 1, 2, .., q/p − 1.These lines are invariant under Z p , as they should be since Z p is gauged.The minimal admissible Wilson's line W p j carries one electric flux in units of pN/q.In the limiting case p = q, the line W p=q j coincides with the matter content and must be removed from the spectrum of line operators.Therefore, in this case, the theory does not possess a 1-form electric symmetry, as discussed in the previous section.
In addition, the theory has a magnetic Z m(1) p 1-form symmetry.If T j is the 't Hooft line with N -ality 1, then the minimal admissible 't Hooft line in the theory is T N/p j , which carries one magnetic flux in units of N/p.There are p distinct 't Hooft lines in the theory T n j N/p j , n j = 0, 1, .., p − 1, which are mutually local with Wilson's lines W e j p j .The Hamiltonian, Wilson's lines generators, and the 't Hooft lines generators of this theory can be simultaneously diagonalized.Therefore, the energies and eigenvalues of the set of Wilson and 't Hooft operators can be used to label the physical states of Hilbert space: (3.12) Next, we need to build a gauge invariant chiral symmetry operator.Our starting point, as usual, is the operator taken in the presence of the fractional magnetic fluxes m ∈ (N Z/p) 3 , which label the Hilbert space in Eq. (3.12).The operator T q/p x generates the electric Z (1) p 1-form symmetry, which is gauged.In other words, T q/p x implements the twists k ∈ (N Z/p) 3 .In analogy with SU (N )/Z q theories, we need to build gauge invariants of the chiral symmetry operator using the building block T q/p x U Z 2T R ,ℓ T −q/p x .To compute this block, we use the discussion around Eq. (2.33), taking the minimal twist N/p generated by T q/p x , to obtain and n x counts the magnetic fluxes in units of N/p.If ℓ N p 2 ̸ ∈ Z, there is a mixed anomaly between Z χ 2T R and the electric Z p symmetries in SU (N ) theory with matter, and we expect the chiral symmetry becomes noninvertible upon gauging Z (1)

p . The corresponding gauge invariant operator of the Zχ
2T R symmetry is then given by the summations This noninvertible operator generalizes (3.6) to any Z p ⊆ Z q , and it projects onto sectors with finer topological charges than the sectors admissible by (3.6).This means there exist sectors where ŨZ 2T R ,ℓ act invertibly for all ℓ = 1, 2, .., T R if and only if with similar conditions in the y and z directions.As special cases, we may first set p = q to readily cover (3.6).Also, setting p = 1, the operator ŨZ 2T R ,ℓ becomes invertible, as can be easily seen from the second line in (3.15).Notice that ŨZ 2T R ,ℓ does not act on Wilson's lines in this theory, as the noninvertible operator is built from (T j ) qp j /p and its inverse; thus, one can push a Wilson line through ŨZ 2T R ,ℓ without being affected 13 .We can write this observation as = W e j p j ŨZ 2T R ,ℓ , e j = 0, 1, 2, .., q/p − 1 , j = x, y, z . (3.17) This is very different from the action of ŨZ 2T R ,ℓ on 't Hooft lines, as we discussed before.
The procedure employed to construct the noninvertible operator ŨZ 2T R ,ℓ contains an additional layer of underlying physics.It is essential to keep in mind that this operator is constructed in SU (N )/Z p theory, where its creation involved a sum over magnetic m ∈ (N Z/p) 3 and electric k ∈ (N Z/p) 3 twists.These twists do not encompass the entire range of permissible twists that can be applied.Recall that the theory encompasses a global Z (1) q/p symmetry, which affords us the opportunity to introduce the electric twists 13 Although we do not give the explicit form of T j , it can be thought of as an exponential of an integral of the chromoelectric field over a 2-dimensional submanifold; see [45].A Wilson line would acquire a phase as we push it past T q/p j (we use [A a j (x, t), E b k (y, t)] = iδ jk δ(x − y)δ ab , where a, b are the color indices, along with the Baker-Campbell-Hausdorff formula).It also acquires the negative of the same phase as it is pushed past T −q/p j .Therefore, the phases cancel out, and hence, the result in Eq. (3.17).
k ∈ (pN Z/q) 3 .Moreover, we can turn on magnetic twists m ∈ (pN Z/q) 3 , compatible with the cocycle condition 14 .This broader scope of twists provides a richer set of possibilities within the theory.We recall that T p x is the generator of Z q/p symmetry that implements the twists k x ∈ pN Z/q.Then, one can write the partition function of SU (N )/Z p theory in these background twists as and we used Eq.(3.12) along with T Next, consider the commutation relation between T p x and ŨZ 2T R ,ℓ , the latter operator is being in the background of the magnetic twist m ∈ (pN Z/q) 3 .Using the discussion and procedure around Eq. (2.33), we obtain The failure of the commutation between T p x and ŨZ 2T R ,ℓ by the phase e ∈ Z , signals a mixed anomaly between the noninvertible Zχ 2T R chiral symmetry and the electric Z (1) q/p 1-form global symmetry.This anomaly means that certain sectors in Hilbert space exhibit degeneracy.Let us analyze this situation more closely.We assume there exists a sector with n x , n y , n z that satisfies Eq. (3.16), and thus, in this sector, the symmetry operator ŨZ 2T R ,ℓ acts invertibly for all elements ℓ = 1, 2, .., T R .Now, ŨZ 2T R ,ℓ , being a global symmetry operator, commutes with the Hamiltonian: Likewise, since Z q/p is a global symmetry, its generators T p j commute with the Hamiltonian: This commutation relation, along with Eq. (2.25), implies that T p j acts on physical states in Hilbert space as (the label l = (l x , l y , l z ) emphasizes that such states satisfy condition (3.16), such that ŨZ 2T R ,ℓ acts invertibly on such states.Also, we suppressed the detailed dependence on the different quantum numbers to reduce clutter) and that the states are labeled by their energies as well as e j = 1, 2, ..., q/p distinct labels; these are the eigenvalues (fluxes) of the Z q/p symmetry operator.The algebra defined by the commutation relations Eqs.(3.20, 3.21), along with the mixed anomaly represented as Eq.(3.19), under the assumption of a nontrivial phase, furnishes a finitedimensional space with a minimum dimension of q 2 /gcd(n x p 2 N, q 2 ) (we take n x = n y = n z ).The Hilbert space of physical states, which are labeled by q/p distinct fluxes, sit in q 2 /gcd(n x p 2 N, q 2 ) orbits, and a rotation by ŨZ 2T R ,ℓ=1 links a state with a flux e j to a state with a flux e j+gcd(n j p 2 N,q 2 )/(qp) as: Using the commutation relation (3.20), one immediately sees that the states |E, e j ⟩ l and |E, e j + gcd(n j p 2 N, q 2 )/(qp)⟩ l have the same energy 15 .
In the following subsections, we apply our formalism to examples of theories with fermions in specific representations.The SU (4n + 2)/Z 2 gauge theory with a 2-index anti-symmetric Dirac fermion (N -ality 2) has a Z χ 8n chiral symmetry.The SU (4n+2) theory possesses an electric Z

Examples
2 one-form symmetry.In [33], the authors argued that upon gauging Z (1) 2 , the odd rotations of Z χ 8n become non-invertible.We can show this is the case on T 3 using our construction.Setting N = 4n + 2 in (3.6), we obtain 15 It is helpful to give a numerical example.Take N = 1000, q = 500, and p = 20.Such numbers are contrived and do not necessarily correspond to any realistic theory.Condition (3.16) is satisfied if we take n x = 2.Then, the phase in the anomaly Eq. (3.19) is e −i2π/5 , implying a 5-fold degeneracy.The theory has an electric Z For ℓ odd, ŨZ 8n ,ℓ projects onto untwisted gauge sectors and becomes non-invertible.
The SU (4n)/Z 2 theory with a 2-index anti-symmetric Dirac fermion has a Z χ 8n−4 chiral symmetry.The cocycle conditions, say in the x-direction, must satisfy (see Eq. (2.9)) Therefore we must have n yz ∈ 2nZ.There is no mixed anomaly between Z χ 8n−4 and the electric Z 2 symmetries in the SU (4n) theory since the anomaly phase nyz 2 ∈ nZ is trivial.Thus, the full chiral symmetry Z χ 8n−4 is invertible.This is also in agreement with [33].This theory has a Z χ 6 chiral symmetry.What is special about this theory is that its bilinear fermion operator vanishes identically because of Fermi statistics.Moreover, the SU (6) theory exhibits a mixed anomaly between its electric Z (1) 3 1-form center and chiral symmetries [46,47].Assuming confinement, then the chiral symmetry must be broken in the infrared.Yet, this breaking has to be accomplished via higher-order condensate.Using (3.6), we find that the operator corresponding to a chiral transformation in SU (6)/Z 3 theory is Hence, for ℓ ∈ {1, 2, 4, 5}, the operator ŨZ 6 ,ℓ projects onto untwisted gauge sectors, and the chiral symmetry operator becomes noninvertible.

2-index SU (6) chiral gauge theory
Our next example is a chiral gauge theory.This is SU (6) YM theory with a single left-handed Weyl fermion ψ in the 2-index symmetric representation and 5 flavors of left-handed Weyl fermions χ in the complex conjugate 2-index anti-symmetric representation.The fermion budget ensures the theory is free from gauge anomalies.The theory encompasses continuous global symmetry SU (5) χ × U (1) A , where SU (5) χ acts on χ.The charges of ψ and χ under U (1) A are q ψ = −5 , q χ = 2.The theory is also endowed with a Z χ 4 chiral symmetry, which is taken to act on χ with a unit charge.It can be checked that this is a genuine symmetry since neither Z 4 nor a subgroup of it can be absorbed in rotations in the centers of SU (6) × SU (5) χ .It turns out, see [48] for details (also see [49]), that we must divide the global symmetry by Z 3 × Z 5 to remove redundancies.Putting everything together and remembering that the theory possesses an electric Z (1) 2 1-form center symmetry (since all fermions have N -ality n = 2), we write the faithful global group as: 2 . (3.27) This theory has an anomaly between its Z 2 center symmetry and Z χ 4 chiral symmetry.To see the anomaly, we recall that we can turn on the magnetic and electric twists (m, k) ∈ (3Z) 6 .This gives the topological charge Q ∈ Z/2.Thus, under a chiral transformation, the partition function acquires a phase where N χ = 5 is the number of the χ flavors and T χ = 4 is the Dynkin index of χ.Therefore, we expect that Z χ 4 becomes noninvertible in the SU (6)/Z 2 chiral theory.Using (3.6), the noninvertible operator corresponding to a chiral transformation in SU (6)/Z 2 theory is Hence, for ℓ ∈ {1, 3}, the operator ŨZ 4 ,ℓ projects onto untwisted gauge sectors, and the chiral symmetry operator becomes noninvertible.
4 SU (N ) × U (1)/Z p , Z p ⊆ Z N theories, noninvertible symmetries, and their anomalies In this section, we also gauge the U (1) baryon number symmetry.Thus, we are discussing SU (N )×U (1) gauge theory with a Dirac fermion in a representation R, N -ality n, and U (1) charge +1.This theory, as we discussed in Section 2, is endowed with an invertible Z χ 2gcd(T R ,d R ) chiral symmetry as well as an electric Z N center symmetry acting on its Wilson's lines; see Eqs. (2.17).However, in [32], it was shown that SU (N )×U (1) theories also have noninvertible Zχ 2T R chiral symmetry.In the following, we first review the construction of the noninvertible Zχ 2T R operator in SU (N ) × U (1) theories, and next, we discuss this operator in SU (N ) × U (1)/Z p , Z p ⊆ Z N , theories.

SU (N ) × U (1)
Our starting point is the SU (N )×U (1) theory and its where Q 5 is the conserved chiral charge defined in Eq. (2.43) in the background of the fractional n x,y,z and integer N x,y,z magnetic fluxes in the x, y, z directions.We remind that we can turn on fractional fluxes in Z N irrespective of the N -ality of the matter content since we use U (1) transition functions to impose the cocycle condition; see Eq. (2.36).No nontrivial electric twists are applied at this stage, i.e., we take k ∈ (N Z) 3 , since our nonabelian gauge group is SU (N ) rather than SU (N )/Z p .The operator U Z 2T R ,ℓ is invariant under SU (N ).To see that, we apply a large SU (N ) gauge transformation, recalling Eq. (2.24) and setting k ∈ (N Z) 3 , which immediately gives the change in the nonabelian winding number by Q ∈ Z.In addition, U Z 2T R ,ℓ must be invariant under U (1) gauge symmetry.The photon gauge field a i transforms under U (1) gauge symmetry as a j (x + êk L k ) = a j − ∂ k ξ(x), and ξ(x) is a periodic gauge function: Applying a large U (1) gauge transformation to Q 5 , we find (see [32] for the derivation) where p x,y,z are arbitrary integers corresponding to the U (1) gauge transformation.Eq. (4.1) shows that the operator U Z 2T R ,ℓ fails to be gauge invariant under U (1) gauge symmetry.To remedy this problem, we follow the procedure of the previous section and define a new operator ŨZ 2T R ,ℓ by summing arbitrary copies of the gauge-transformed The operator ŨZ 2T R ,ℓ implements the chiral transformation of the now-noninvertible Zχ 2T R symmetry, as it acts projectively by selecting certain nonvanishing sectors in Hilbert space labeled by the integers l x,y,z , such that for ℓ = 1 we must have with identical expressions for l y and l z .Condition (4.3) ensures that all the symmetry elements ℓ = 1, 2, .., T R act invertibly on the same admissible sector.To explicitly see the projective nature of ŨZ 2T R ,ℓ on states in Hilbert space, we use the partition function of the SU (N ) × U (1) theory given by Eq. (2.40) (we set the electric flux background k=0 and, as usual, we use n to label a specific fractional magnetic flux background: n = (n x , n y , n z )) to compute ⟨ ŨZ 2T R ,ℓ ⟩: 16 We immediately see from the Kronecker deltas in Eq. (4.2) that only those sectors with N satisfying Eq. ( 4.3) are selected.
Turning off the fractional magnetic flux background (i.e., setting n = 0), the operator ŨZ 2T R ,ℓ becomes invertible for ℓ ∈ T R Z/gcd(T R , d R ).We recognize that we have just recovered the invertible Z χ 2gcd(T R ,d R ) subgroup of Zχ 2T R .Furthermore, setting n = 0, the operator ŨZ 2T R ,ℓ=1 destructs all Hilbert space sectors characterized with integral magnetic fluxes N / ∈ T R Z 3 /gcd(T R , d R ).This noninvertible nature of the chiral operator should have been anticipated.When we start with the SU(N) theory with matter, we find an 't Hooft anomaly between its invertible Z χ 2T R chiral symmetry and U (1) baryon symmetry.This anomaly is valued in Z T R /gcd(T R ,d R ) .Upon gauging U (1), this anomaly becomes of the ABJ type, and the chiral symmetry becomes noninvertible.Now, If we take the Euclidean version of our theory in the infinite volume limit and apply a π/2 rotation to ŨZ 2T R ,ℓ=1 , the operator becomes a defect.Alternatively, we may also use the half-gauging procedure to construct this defect, which was done in [32].Inserting this defect at some position will generally create a domain wall (since it enforces a chiral transformation) dressed with a TQFT that accounts for the noninvertible nature of the defect.It will be interesting to analyze what happens to the domain walls when we turn on an external magnetic field with flux N / ∈ T R Z 3 /gcd(T R , d R ).
SU (N )×U (1) gauge theory has an electric Z N 1-form global center symmetry, and the immediate exercise would be checking whether there is a mixed anomaly between the center and the noninvertible chiral symmetries.To this end, we turn on both electric and magnetic twists 17 (m, k) ∈ Z 6 , giving rise to nonabelian fractional topological charge Q SU (N ) ∈ Z/N as well as abelian topological charge Q u = n N 2 ; see Eq. (2.15). 16Recall from our earlier analysis that the theory is endowed with electric Z  m (1) symmetries, and the states of the theory are labeled by the eigenstates of these symmetries, e and N , respectively. 17Notice that these electric twists k ∈ Z 3 are e that label the physical states in Hilbert space: |E, e, N ⟩ n .In principle, k j should be in Z Mod N , but, as usual, we drop the modding as this does not affect the cocycle conditions.

Examples
4.3.1 SU (4k + 2) × U (1)/Z p with 2-index antisymmetric fermions SU (4k + 2) × U (1) theory with a single 2-index anti-symmetric Dirac fermion was considered in [32].Here, we study this theory when we gauge a Z p ⊆ Z N subgroup of the center.Numerical scans reveal that condition (4.10) is always satisfied for specific values of n x and N x .Also, the anomaly (4.11) is trivial unless both p and l x are odd; then, the anomaly is valued in Z 2 .The Hilbert space is spanned by the physical states and the anomaly means that the states live in two orbits such that |E, pe, n/p + N ⟩ m , |E, p(e + gcd(N, p 2 l)/p), n/p + N ⟩ m , |E, p(e + 2gcd(N, p 2 l)/p), n/p + N ⟩ m , etc. have the same energy (we take n x = n y = n z ).

The Standard Model
The methods presented in this paper provide a systematic means to find noninvertible symmetries in any given gauge theory.As an important application, we employ our approach to search for noninvertible symmetries in the nongravitational sector of the Standard Model (SM).SM is based on su(3)×su(2)×u(1) Lie algebra.Yet, the faithful gauge group, i.e., the global structure of the group, is to be uncovered.The matter content and charges under the gauge and global symmetries are displayed in Table 1, and all fermions are taken to be left-handed Weyls.The anomalies associated with the U (1) B and U (1) L symmetries are given by: U Thus, we see that U (1) B−L symmetry is anomaly-free symmetry (we neglect gravity in this context).Under a U (1) B+L rotation, the path integral picks up an ABJ phase exp (iα where N f is the number of families, c 2 (F ) is the second Chern class for SU (2) and c 2 (f ) is the second Chern class for U (1).The ABJ anomaly breaks the U (1) B+L down to a Z B+L gcd(2,36)N f = Z B+L 2N f symmetry.Notice that SU (3) does not play a role in the ABJ anomaly.
Matter content and charges of SM: q L and l L are the quark and lepton doublets, ẽR , ũR , dR are the electron and up and down quarks singlets, while h is the Higgs doublet.Notice that we take the hyper U (1) charges to be integers, while the matter content has the standard charges under the baryon number U (1) B and lepton number U (1) L symmetries.
The matter content is consistent with the existence of an electric Z 6 1-form global symmetry [50,51].The cocycle conditions satisfied by SM on T 4 with a gauged Z (1) 6 are given by [51]: ) . Ω (i) , i = 2, 3, and ω (1) are the transition functions of the gauge bundles, n µν are the 't Hooft twists, and the superscript/subscript (i) = (3), ( 2), (1) denote the condition for the SU (3), SU (2), U (1) gauge groups respectively.The electric Z j , and the U (1) center t j , such that the full Z (1) 6 symmetry generator is given by T The anomalous U (1) B+L current conservation law is given by where K µ SU (2) is the SU (2) topological current.The corresponding unbroekn Z B+L 2N f symmetry operator on T 3 is given by: where the conserved charge Q 5 is given by (here we turn on a Z 6 magnetic twist) Therefore, U Z 2N f ,ℓ is U (1) gauge invariant, as required.Further, we examine U Z 2N f ,ℓ after gauging the electric Z 6 1-form center by sandwiching U Z 2N f ,ℓ between its generators (this is a generalization of Eq. (4.5)): T (3)  x T (2)  x t x U Z 2N f ,ℓ T (3)  x T (2)  x t We used Eq.(2.33), setting k x = m x = 1, to find the first exponent.The second exponent is found by applying Eq. (2.44) and using n = 1, N = 6.Here, n x , n x , and N x are the SU (2) and SU (3) fractional twists and U (1) integral magnetic flux, respectively.This analysis shows that SM does not possess noninvertible symmetries in its nongravitational sectors.Our findings are consistent with [27].

Coupling gauge theories to axions and noninvertible symmetries
In this section, we introduce axions into the game, taking T 4 to be larger than any scale in the theory.To be specific, we take SU (N )/Z p or SU (N ) × U (1)/Z p gauge theories of the previous sections and follow the setup of [52] by adding a complex scalar Φ that is neutral under the gauge groups but couples to the fermions.Thus, we add the following terms to the Lagrangian: L ⊃ |∂ µ Φ| 2 + V (Φ) + yΦ ψψ + h.c., where ψ, ψ are two left-handed Weyl fermions in representations R and its complex conjugate R, respectively, and y is a Yukawa coupling.The potential of the complex field is , where λ is O(1) dimensionless parameter.We take the scalar field v.e.v.v ≫ Λ, where Λ the strong scale of the gauge sector.We shall pretend that we did not know about the noninvertible symmetries or how to construct them, and let us see if we can reproduce them in the IR.
Let us first consider the SU (N ) gauge theory before gauging U (1) and the electric Z p symmetry.Under Z χ 2T R and U (1) baryon number, the different fields transform as and notice that the axion is inert under the Z F 2 fermion number subgroup of Z χ 2T R .
Next, we consider SU (N )/Z p or SU (N ) × U (1)/Z p gauge theories with axions.Flowing to an energy scale below v, the radial degree of freedom ρ freezes in, i.e., we set ρ = v, and the fermions acquire a mass ∼ yv and decouple.What remains is the light degree of freedom, the axion φ.However, the axion should reproduce all the UV anomalies.Thus, we can write the following IR effective Lagrangian of φ: (5.4) Variation of L φ w.r.t φ produces the anomalous current conservation law: where j µ (φ) = v 2 ∂ µ φ.This is exactly the anomalous current conservation law we had previously, now written down for the axion current.Therefore, everything we said in the previous sections applies here.In particular, we can define an operator of the Z χ 2T R symmetry as: where the dots denote the contribution from the U (1) gauge field (see Eq. (2.43)).We used a calligraphic letter for the operator to emphasize that it is constructed in the IR.Yet, all the anomalies and failure of invariance under gauge symmetries that lead to the noninvertibility of the UV operators apply here as well.Thus, similar to what we did before, we can construct the noninvertible operator ŨZ 2T R ,ℓ , which implements the noninvertible symmetry Zχ 2T R in the IR.Such operators shall project onto magnetic sectors and also exhibit mixed anomalies with the global 1-form electric center symmetry, exactly as we discussed previously.
It was pointed out in [53] that SU (N )/Z p theories with axions have noninvertible symmetries.However, our construction shows that such a conclusion is not general and depends on the UV completion.Consider two theories SU (4k)/Z 2 and SU (4k + 2)/Z 2 with a Dirac fermion in the 2-index antisymmetric representation and coupled to a complex scalar field Φ as above.As we flow to the IR, we can construct the operators corresponding to the chiral symmetries.We discussed in Section 3.3.1 that SU (4k)/Z 2 theory does not exhibit an anomaly between its chiral symmetry and the 1-form symmetry of the corresponding SU (4k) theory, and hence, the chiral symmetry operator is invertible.Therefore, an axion domain wall (DW), implemented by the action of ŨZ 8k−4 ,ℓ , will not be dressed with TQFT degrees of freedom.On the contrary, SU (4k + 2)/Z 2 exhibits an anomaly between its chiral symmetry and the 1-form center of the corresponding SU (4k+2) theory, and thus, the minimal chiral symmetry operator ŨZ 8k ,ℓ=1 is noninvertible.The axion DW implemented by ŨZ 8k ,ℓ=1 must be dressed with a fractional quantum Hall TQFT.
We may also consider axions in SU (N ) × U (1)/Z p theory of Section 4. Everything we said there is transcendent to the IR axion domain walls.In particular, for p = 1, the operator ŨZ 2T R ,ℓ=1 destructs the Hilbert space sectors characterized by vanishing fractional n = 0 and integral magnetic fluxes N / ∈ T R Z 3 /gcd(T R , d R ).It will be interesting to examine what happens to the axion domain walls of this theory as we place them in such an external magnetic field.

( 1 )
N/p global symmetry.The anomaly implies that certain sectors of the theory, designated by certain Z (1) N/p electric fluxes, exhibit multi-fold degeneracy.
that acts on Wilson lines.
in the 1-2 and 3-4 planes, and similar expressions in the rest of the planes.Substituting into Q u = T 4 f ∧f 8π 2 , we obtain the fractional U (1) topological charge.

3. 3 . 1
SU (4n + 2)/Z 2 and SU (4n)/Z 2 with a Dirac fermion in the 2-index anti-symmetric representation form symmetry, and thus, 25 distinct flux states.These states set in 5 different orbits such that the states labeled with e 1 , e 6 , e 11 , e 16 , e 21 have the same energy, and the states e 2 ,e 7 ,...,e 22 , have the same energy, etc.