Lattice realization of the axial U (1) non-invertible symmetry

In U (1) lattice gauge theory with compact U (1) variables, we construct the symmetry operator, i.e., the topological defect, for the axial U (1) non-invertible symmetry. This requires a lattice formulation of chiral gauge theory with an anomalous matter content and we employ the lattice formulation on the basis of the Ginsparg–Wilson relation. Then, the invariance of the symmetry operator under the gauge transformation of the gauge ﬁeld on the defect is realized, imitating the prescription by Karasik in continuum theory, by integrating the lattice Chern–Simons term on the defect over smooth lattice gauge transformations. The projection operator for allowed magnetic ﬂuxes on the defect then automatically emerges with lattice regularization. The resulting symmetry operator is manifestly gauge invariant under lattice gauge transformations. . .. . . . .. . . . .. . . .. . . . .. . . . .. . . . .. . . .. . . . .. . . . .. . . . .. . . ..

In the present paper, we make an attempt to implement the above argument of the axial U(1) noninvertible symmetry on a completely regularized framework of gauge theory, the lattice gauge theory. 1 It is well-known that one can construct a conserved axial U(1) current by adding the Chern-Simons form with an appropriate coefficient to the axial U(1) current.
However, since the Chern-Simons form is not gauge invariant, the conserved axial U(1) current cannot be regarded as a physical operator; see Ref. [36] for this classical issue.In terms of the exponential of the Noether charge defined on a 3D closed surface M 3 , which is called the symmetry operator associated with the defect M 3 , U α (M 3 ), where α is the axial U(1) rotation angle, the addition of the Chern-Simons form to the axial U(1) current induces the exponential of the Chern-Simons action on M 3 with a noninteger level.Corresponding to the conservation law of the modified axial U(1) current, U α (M 3 ) is topological in the sense that it is invariant under deformations of M 3 .However, since U α (M 3 ) is not gauge invariant, this way of symmetry restoration had been thought to be invalid.
The crucial observation of Refs.[1,2] is that, when the gauge group is U(1), the exponential of the Chern-Simons action with a fractional level can be made gauge invariant by coupling it to a topological field theory on M 3 .This is analogous to the gauge invariant low-energy description of the fractional quantum Hall effect [37].In this way, the modified symmetry operator, Ũα (M 3 ), which is still topological, can be physical.However, since Ũα (M 3 ) involves a functional integral for the topological field theory, Ũα (M 3 ) is not unitary; it is rather an infinite sum of unitary operators.Thus, the resulting symmetry is noninvertible because the Hermitian conjugate of Ũα (M 3 ) does not provide the inverse.
To realize the above axial U(1) noninvertible symmetry on the lattice, therefore, we have to do mainly two things: One is to formulate the lattice fermion which couples to the defect defined on the lattice through the γ 5 coupling.This is because the deformation of the defect induces the axial U(1) transformation on the fermion; this is nothing but the axial U(1) Ward-Takahashi identity.The other thing is to formulate the U(1) Chern-Simons action on the defect; we also have to implement the gauge invariance of the Chern-Simons action in a certain way.These two things are of course mutually related through the axial U(1) anomaly and both tasks are not straightforward.
For the first task, it is natural to regard the defect as a particular configuration of an external U(1) gauge field (for the details, see below).The lattice fermion couples to this U(1) lattice gauge field through the γ 5 coupling.This thus requires a formulation of chiral gauge theory on the lattice, which itself is known to be a very hard problem.Moreover, the matter content of our chiral gauge theory is anomalous, because the gauge anomaly in the chiral gauge theory corresponds to the axial U(1) anomaly in the target vector-like theory.For this, we employ the lattice formulation of the U(1) chiral gauge theory [38] on the basis of the Ginsparg-Wilson relation [39][40][41][42][43][44][45][46].Remarkably, in this lattice formulation, one can determine the structure of the axial U(1) anomaly with finite lattice spacings [47,49].
The second task, the Chern-Simons term on the lattice is also a quite nontrivial problem.
One possible approach is to use the Villain-type noncompact U(1) variable as in a recent work on the U(1) Chern-Simons action on the lattice [50].In the present paper, we consider a formulation on the basis of the compact U(1) variables with a possible generalization to non-Abelian gauge theories in mind.Then, it appears difficult to imitate the procedure in Refs.[1,2] directly, because it relies on a change of integration variables in terms of the gauge potential.Instead, we find that a formulation of Ref. [23], which is based on a "gauge average," is rather suitable for our lattice formulation.See also Ref. [24] for an analogous construction.Imitating the prescription of Ref. [23], we thus integrate the Chern-Simons term on the defect, which is defined so as to cancel the axial U(1) anomaly in our lattice formulation, over smooth lattice gauge transformations on the defect.Then, as per Ref. [23], the projection operator for allowed magnetic fluxes on the defect emerges.The resulting expression is manifestly invariant under lattice gauge transformations.In this way, we realize the axial U(1) noninvertible symmetry with lattice regularization.This paper is organized as follows: In Sect.2, we give a general setup of our lattice formulation.We introduce two lattice gauge fields, one is for the physical and dynamical U(1) gauge field and the other is for the external U(1) gauge field which defines the defect.Section 3 is a rather lengthy exposition of our construction of the lattice fermion with desired properties.We admit that our lattice formulation appears "too heavy" just in order to describe a simple system of a massless Dirac fermion coupled to the dynamical U(1) gauge field.
However, since the lattice fermion possesses the γ 5 coupling to the (albeit external) gauge field, we think that this formulation cannot be readily simplified.The most important result in Sect. 3 is Eq.(3.31), the axial U(1) Ward-Takahashi identity on the lattice, and, for the construction of the symmetry operator in Sect.4, it is sufficient to accept this relation; the purpose of the rest of Sect. 3 is to show the existence of a fermion integration measure which ensures Eq. (3.31).In Sect.4.1, on the basis of the formulation in Sect.3, we define the topological defect as a particular configuration of the external U(1) lattice gauge field.The Chern-Simons term on the defect emerges as the surface term of the volume integral of the axial U(1) anomaly.Then, in Sect.4.2, imitating the prescription of Ref. [23], we integrate the Chern-Simons term on the defect over gauge transformations on the defect.Here, to reproduce the picture assumed in Ref. [23], it is important to carry the integration only over smooth lattice gauge transformations. 2 Then, the projection operator for allowed magnetic fluxes on the defect emerges.We note that the resulting symmetry operator as the whole is manifestly invariant under lattice gauge transformations.Section 5 is devoted to Conclusion.In Appendix A, we elucidate the issue of the local counterterm which provides a desired form of the gauge anomaly; the information obtained in this appendix is utilized in the construction of our lattice fermion in Sect.3. In Appendix B, we provide the precise definition of the integration measure for the smooth gauge degrees of freedom.In Appendix C, 3 we give another way of constructing the symmetry operator for the rotation angle α = 2πp/N with even p, on the basis of a 3D Z N topological quantum field theory (TQFT), the level-N BF theory on the lattice.This construction resolves some unsatisfactory points in the construction in the main text.The construction is intrinsically 3D and can also be applied to homologically nontrivial closed 3-surfaces; the symmetry operator is manifestly invariant under 3D lattice gauge transformations.The computation of fusion rules of symmetry operators is also given.

U (1) × U (1) ′ lattice gauge theory
We consider a lattice gauge theory defined on a finite hypercubic lattice of size L: The gauge group is U(1) × U(1) ′ . 5The first U(1) factor is for a physical and dynamical gauge field and the latter U(1) ′ is for an external nondynamical gauge field.Corresponding to these, we introduce two U(1) lattice gauge fields by link variables, where both obey periodic boundary conditions: gauge field which couples to fermion fields in a vector-like way.The external U(1) gauge field U(x, µ), on the other hand, couples to fermion fields in a chirally asymmetric, i.e. through γ 5 , way.See the next section.The sole role of the external gauge field U(x, µ) is to define the topological defect associated with the axial U(1) (noninvertible) symmetry.In this paper, we consider only a homologically trivial 3D defect M 3 on the dual lattice of Γ.As we will see below, U(x, µ) is given by a lattice analogue of the Poincaré dual of M 3 and we can assume that U(x, µ) is pure-gauge: where Λ(x + Lμ) = Λ(x).
Fermion fields, whose detailed description is deferred to the next section, are supposed to be residing on sites of Γ.The expectation value is then defined by the functional integral, where the partition function Z is determined by requiring 1 = 1.
The integral over the dynamical U(1) gauge field u(x, µ) is defined by the Haar measure at each link as usual.We adopt, however, a somewhat unconventional gauge action S G following Ref.[38], with g 0 being the bare coupling and, for a fixed number 0 < ǫ < π/3, (2.7) In this expression, the field strength of u(x, µ) is defined by from the plaquette variable, (2.9) The gauge action (2.6) is designed to impose the restriction, the so-called admissibility, on possible gauge field configurations: This is a gauge-invariant smoothness condition on the gauge field. 8In lattice gauge theory (with compact variables), the admissibility plays crucial roles to define topological quantities in a transparent way [52,53].For instance, in the present U(1) lattice gauge theory, the space of admissible gauge fields is given by a disjoint union of topological sectors, each of which is labeled by the magnetic fluxes [38], f µν (x + sμ + tν). (2.11) The fermion integration measure in Eq. (2.5) is thus defined for each magnetic flux sector separately.The factor w[m] in Eq. (2.5) parametrizes the relative weight and phase for each magnetic flux sector specified by m µν .
We note that if we parametrize the U(1) gauge field as then the field strength (2.8) is written as where z µν (x) ∈ Z.Note that z µν (x) is generally non-zero because of a possible mismatch between the logarithmic branch in Eq. (2.8) and that for the product of Eq. (2.12).From this, we have10 owing to Eqs. (2.13) and (2.10).Since the most left-hand side of this inequality is a sum of (six) integers, it must identically vanish, i.e. the integer field z ρσ (x) in Eq. (2.13) satisfies the Bianchi identity.This also implies that the field strength f µν (x) (2.13) itself satisfies the Bianchi identity, For the the external gauge field U(x, µ), on the other hand, it is pure-gauge (2.4) and the corresponding field strength identically vanishes: (2.17) We will use this fact frequently in what follows.

Fermion sector
The definition of the fermion integration in Eq. (2.5) requires an elaborate consideration as done in Ref. [38].This is because the fermion fields couples to the external gauge field U(x, µ) in a chirally asymmetric way and this requires a construction of chiral gauge theory on the lattice, a hard problem.Moreover, the U(1) ′ sector possesses the gauge anomaly to reproduce the axial U(1) anomaly in the target vector-like theory.Therefore, we have to define an "anomalous gauge theory" on the lattice. 11Having the results of Ref. [38], however, our task to define the fermion integration measure is not so quite hard, because the gauge field which causes the chiral coupling, U(x, µ), is not dynamical and moreover pure-gauge, Eq. (2.4).
Since ψ is Weyl, the coupling to U(1) ′ is chiral, whereas the coupling to U(1) is vector-like (as per the Dirac fermion in quantum electrodynamics).The forward and backward covariant difference operators on fermion fields are thus defined by where From these, the Wilson Dirac operator is given by In our construction, we assume a lattice Dirac operator which fulfills the Ginsparg-Wilson relation [39], with the chiral matrix γ 5 .The overlap Dirac operator [40,43] given by where λ(x) ∈ U(1) and Λ(x) ∈ U(1), then we have where

Chirality projection and the fermion action
Now, the Ginsparg-Wilson relation (3.5) enables a clear separation of chiralities of a lattice fermion.For this, we define the modified chiral matrix by [45,46], γ5 = γ 5 (1 − D). (3.9) Then, (γ 5 ) † = γ5 and Eq.(3.5) implies Thus, introducing chirality projection operators by we may define the left-handed Weyl fermion on the lattice as Then, the fermion action enjoys the form12 and chiralities are clearly separated in the lattice action (i.e., the action of the Dirac fermion is given by the sum of this and the corresponding right-handed one, S F = x∈Γ ψ(x)P − D P+ ψ(x)).This defines the fermion action S F in Eq. (2.5).
The fermion integration measure in Eq. (2.5), on the other hand, is defined by by employing (Grassmann-odd) expansion coefficients in where v j (x) and vk (x) are orthonormal basis vectors in the projected space (3.12), i.e. and Although the above provides a construction of a lattice Weyl fermion in the classical level, this is just a beginning of the construction for the full quantum theory [38].The point is that basis vectors v j depend on the gauge field through the dependence of P− on the gauge field.The latter dependence, however, does not uniquely determine the basis vectors; under a variation δ of the gauge field, from Eq. (3.16), P+ δv j = δ P− v j . (3.18) Therefore, P− δv j is not determined by Eq. (3.16) and there is wide ambiguity in the choice of basis vectors v j . 13One has to choose the basis vectors so that physical requirements such as the smoothness and locality of the fermion integration measure are fulfilled [38].
We note that, in terms of the expansion coefficients in Eq. (3.15), the action (3.13) is written as Corresponding to the fermion number anomaly in chiral gauge theory, the numbers of v j and vk can be generally different and if this is the case the matrix M is rectangular.When the numbers of v j and vk are the same, M is a square matrix and the partition function in the fermion sector is given by Also, in this case, the correlation functions of fermion fields are given by the Wick contractions by the fermion propagator S L (x, y), For instance, we have In what follows, we parametrize infinitesimal variations of the gauge fields as To parametrize variations of the basis vectors under these, we introduce the measure term and measure currents by [38], Then the variation of the partition function (3.20) is given by Particularly important variations are given by the gauge transformations (3.7), for which where we have set λ(x) = e iω(x) and Λ(x) = e iΩ(x) .Under these, by the gauge covariance of the Dirac operator, Eq. (3.8), where the representation matrices are introduced by Then the gauge anomaly (i.e.noninvariance of the fermion partition function (3.20) under the gauge transformation) is given by15 where (3.30)

Construction of the fermion integration measure
Appropriately modifying the proofs of the theorems in Ref. [38], we can show the following statements: Assuming the admissibility (2.10) with a sufficiently small ǫ < 1/30, there exists a fermion integration measure which depends smoothly on gauge fields.The variation of the measure under the change of the dynamical gauge field u(x, µ) depends locally on u(x, µ) but nonlocally on the external gauge field U(x, µ).The variation of the measure under the change of the external gauge field U(x, µ) depends nonlocally on the dynamical gauge field u(x, µ).These nonlocal dependences are worrisome, but we do not think they are problematic, because U(x, µ) is not dynamical and is not the subject of the functional integral.The fermion measure possesses the gauge transformation property under the infinitesimal gauge transformations (3.26), as where: (3.32) In the expectation value (3.31), gauge transformations on the fermion fields are given by Equation (3.31) shows that the first U(1) has no gauge anomaly whereas the second U(1) ′ possesses the gauge anomaly with a definite structure.The anomaly relation (3.31) is the most fundamental relation for our construction of the symmetry operator in the next section.
The proof of the above statements proceeds as follows.
First, the above statements follow if there exist currents j µ (x) and J µ (x) which fulfill the following conditions: (i) The currents depend smoothly on gauge fields and j µ (x) depends locally on the dynamical gauge field u(x, µ).(ii) The functional satisfies the "integrability condition,"17 (iii) The currents satisfies the "anomalous conservation laws," This part of the proof corresponds to the proof of Theorem 5.1 of Ref. [38] (Sect.10).The idea is that [38] the measure term L η in Eq. (3.24) can be regarded as a connection of a principal U(1) bundle over the configuration space of admissible gauge fields, which is denoted by U in what follows.The curvature 2-form of the bundle is given by the quantity on the right-hand side of Eq. (3.35).The global existence of the currents j µ (x) and J µ (x) on U and the corresponding functional L η which satisfies Eq. (3.35) then implies that this U(1) bundle is trivial.Writing a measure term defined from a certain set of basis vectors v j by there exists a fermion integration measure with the above properties if and only if, for any closed curve in U, the Wilson lines where To show this, one has to compute the above Wilson lines for all nontrivial loops in U.
In Ref. [38], it is shown that U in U(1) lattice gauge theory has the topology of a multidimensional torus Wilson loops in real space and the others correspond to U(1) gauge transformations).Thus, it suffices to consider Wilson lines wrapping each S 1 .In our present system, as far as the pure-gauge external field (2.4) without integration over Λ(x) with nontrivial windings (this is the situation we are interested in) is considered, no new S 1 in the U(1) ′ gauge theory emerges. 18Therefore, we may invoke the results of Sect. 10 of Ref. [38] without change.
Next, we have to show the existence of the currents in Eq. (3.34) with required properties.As per Theorem 5.3 of Ref. [38], we first show the existence of currents with similar properties in an infinite volume lattice, Z 4 .That is, we show that there exist currents j ⋆ µ (x) and J ⋆ µ (x) on Z 4 which fulfill the following conditions: (i) Both currents depend smoothly on gauge fields and j ⋆ µ (x) depends locally on u(x, µ).(ii) The corresponding functional,19 satisfies the integrability condition, The currents satisfy the anomalous conservation laws, where Note that all these expressions refer to an infinite volume.
In an infinite volume, one can construct the required currents rather explicitly.The point is that, in infinite volume, one can parametrize admissible gauge fields by vector fields as [47], This is shown in the following way [47,48].We recall Eq. (2.13) which holds for admissible gauge fields on infinite volume as well as on Γ.Then, we can solve on Z 4 by taking the complete axial gauge referring to a certain point o in Z 4 .Then, a µ (x) is given by The important point here is that the definition of a µ (x) refers to the point o and it depends on the choice of o.However, it can be seen that different choice of o corresponds to a different choice of the gauge and the dependence on o disappears in gauge invariant quantities.This fact plays a crucial role in the construction in Ref. [38].
Now, using the parametrizations in Eq. (3.44), from the topological invariance of A(x) and A ′ (x), 20 an ingenious cohomological argument with finite lattice spacings shows [47], where γ is a constant 21 and the currents k µ (x) and K µ (x) are gauge invariant.See also Ref. [49].Here, we have used the charge assignment in Eq. (3.1) to determine possible anomalous terms of the form ∼ ǫ µνρσ f µν f ρσ .Generally speaking, we may also have anomalous terms of the forms, A(x) ∼ ǫ µνρσ f µν F ρσ and A ′ (x) ∼ ǫ µνρσ F µν F ρσ .However, since F µν (x) = 0 in our problem as noted in Eq. (2.17), we do not write these possible anomalies in Eq. (3.47). 20Taking a local variation of (γ 5 ) 2 = 1 in Eq. (3.10), we have γ5 Tr[(γ 5 ) 2 γ 5 tδ η D] = 0 follows; similarly, we have x∈Z 4 δ η A ′ (x) = 0.These are the topological invariance of A(x) and A ′ (x) meant in the text. 21We will later argue that this constant is given by Eq. (3.32).
With these preparations, we define the measure term in infinite volume by where and In Eq. (3.49), the dependence of gauge fields on the parameter s is introduced using the parametrization in Eq. (3.44) as The smooth dependence of L ⋆ η (3.48) on gauge fields is obvious from the well-definedness of the overlap Dirac operator for admissible gauge fields with ǫ < 1/30 [53].As per Sect.6 of Ref. [38], the integrability condition (3.41) can be shown straightforwardly for the sum of expressions in Eqs. and (3.50) being proportional to γ is peculiar to our "anomalous gauge theory" and it is a variation of a counterterm, −i∆ = (8/3)γ x∈Z 4 ǫ µνρσ A µ (x)a ν (x + μ)f ρσ (x + μ + ν).This counterterm makes the vector current in the target theory conserving and the coefficient of the ǫ µνρσ f µν f ρσ term in the axial anomaly the naively expected one.See Appendix A for a detailed exposition on this point.
An important difference of our system from that of Ref. [38] is the gauge noninvariance of the currents, j ⋆ µ (x) and J ⋆ µ (x).Under the gauge transformations, a µ (x) → a µ (x) + ∂ µ ω(x) and A µ (x) → A µ (x) + ∂ µ Ω(x), in a similar way to deriving Eq. (3.53), we find This shows that the current j ⋆ µ (x) in Eq. (3.40) is not invariant under the U(1) ′ gauge transformation and the current J ⋆ µ (x) in Eq. (3.40) is not invariant under the U(1) gauge transformation.These gauge noninvariances have an important consequence on the locality of the currents, because the relation between the vector fields a µ (x) and A µ (x) in Eq. (3.44), from which the measure term (3.48) is constructed, and link variables u(x, µ) and U(x, µ) is local only up to the U(1) × U(1) ′ gauge transformations [47].A closer look at Eq. (3.54) shows that J ⋆non-inv µ (x), which couples to the external gauge field U(x, µ), does not locally depend on the dynamical gauge field u(x, µ) and j ⋆non-inv µ (x), which couples to the dynamical gauge field u(x, µ), does not locally depend on the external gauge field U(x, µ).However, since the gauge field U(x, µ) is not dynamical, we do not think these nonlocalities are problematic.
We have to then show the existence of the currents j µ (x) and J µ (x) on the finite volume lattice Γ, which possess the properties (i)-(iii) listed around Eq. (3.34).This is accomplished [38] by starting from the currents j ⋆ µ (x) and J ⋆ µ (x) in infinite volume and invoking the locality of the overlap Dirac operator D, which is guaranteed for ǫ < 1/30 [53].First, setting we can show the existence of j inv µ (x) and and This is done by starting from the currents j ⋆inv µ (x) and J ⋆inv µ (x) in infinite volume given by Eq. (3.49) and repeating the argument in Sect.11 of Ref. [38].The measure term L ⋆inv η (3.49) fulfills the same prerequisites as the measure term L ⋆ η of Ref. [38].In particular, since the currents j ⋆inv µ (x) and J ⋆inv µ (x) are gauge invariant as Eq.(3.54) shows, j ⋆inv µ (x) and J ⋆inv µ (x) locally depend on gauge fields.Thus, we can literally apply the argument of Ref. [38] to show the existence of j inv µ (x) and J inv µ (x) on Γ.The currents j inv µ (x) and J inv µ (x) obtained above, however, are not quite the same as the desired currents j µ (x) and J µ (x).A comparison of Eqs.(3.57) and (3.36) shows that J inv µ (x) does not have the desired anomaly.To remedy this point, we define In this expression, the gauge potentials a µ (x) and A µ (x) are given by first extending the gauge fields on Γ to Z 4 by periodic copies and then apply the construction as Eq.(3.44). 22Then, since the currents j non-inv µ (x) and J non-inv µ (x) are not gauge invariant, these currents generally do not depend locally on gauge fields.As in infinite volume, J non-inv µ (x), which couples to the external gauge field U(x, µ), does not locally depend on the dynamical gauge field u(x, µ) and j non-inv µ (x), which couples to the dynamical gauge field u(x, µ), does not locally depend on the external gauge field U(x, µ).We do not think, however, these nonlocalities are harmful because U(x, µ) is an external field and is not dynamical.
Finally, we define the full measure currents by The addition of j non-inv µ (x) and J non-inv µ (x) does not influences the integrability (3.56), because L non-inv η (3.58) is δ η of a certain combination.The desired anomaly of J µ (x) (3.36) is reproduced by J non-inv µ (x) as in Eq. (3.53).This completes our argument on the existence of the fermion integration measure. 23t the end of this section, we argue that the value of the coefficient γ in the above expressions is given by Eq. assuming that u n is normalized, where n ± are the numbers of zero modes, Hu n = 0, with positive and negative chiralities, Γ 5 = γ 5 = ±1, respectively.We note that the left-hand side of Eq. (3.60) can also be written as where, in the second equality, we have used [47], with k µ (x) being a gauge invariant current on Z 4 . 24The gauge field in this expression is given by periodic copies of that on Γ.Since k µ (x) is gauge invariant, it is periodic on Γ and thus x∈Γ ∂ * µ k µ (x) = 0.In the last equality, we have used Eq.(7.14) of Ref. [38]; the integers m µν are the magnetic fluxes defined by Eq. (2.11).The equality of Eqs.(3.60) and (3.61), n + − n − = 4π 2 γǫ µνρσ m µν m ρσ , is the lattice index theorem in Abelian gauge theory.Since the minimal value of ǫ µνρσ m µν m ρσ is 8, the coefficient (3.32) is consistent with this index theorem.From these facts, we believe that the value in Eq. (3.32) is the only possible one for a physically sensible lattice Dirac operator. 25 Symmetry operator and the topological defect for the axial U (1) noninvertible symmetry

Introduction of the topological defect
We first construct a defect which corresponds to the axial U(1) (anomalous) symmetry on the lattice.Since this is a 0-form symmetry acting on fermion fields, we consider a codimension 1, i.e. 3D closed surface M 3 on the dual lattice of Γ (see Fig. 1).In this paper, we assume that M 3 is homologically trivial, i.e. there exists a 4D region V 4 on Γ such that M 3 is given by the boundary of V 4 , M 3 = ∂(V 4 ).We also assume that M 3 is endowed with a natural orientation from inside to outside.
For a given defect M 3 , we then associate a particular configuration of the external U(1) lattice gauge field U(x, µ) according to the rule, where α ∈ [0, 2π) is a fixed real number.One can see that, for any M 3 with properties assumed above, U(x, µ) (4.1) defines a pure-gauge U(1) gauge field as per Eq.(2.4). 26This Fig. 1 A 2D slice of an example of the 3D defect M 3 (the broken line) and the interior of the defect, V 4 (the shaded area).The external gauge field U(x, µ) on the links indicated by thick ticks acquire nontrivial phases e ±iα/2 according to the rule in Eq. (4.1).
is expected because Eq. (4.1) defines a lattice analogue of the Poincaré dual of a homologically trivial surface M 3 .Now, we want to consider a deformation of the defect.We see that the deformation can be realized by a gauge transformation on the external gauge field U(x, µ).Suppose M 3 is a defect that surrounds a particular single site y ∈ Γ as per Fig. 2; we take the orientation of M 3 such that the site y is inside M 3 .Then, it is easy to see that the U(1) ′ gauge transformation in Eq. (3.7) with eliminates the defect, i.e. all U(x, µ) becomes unity.From this observation, it is clear that one can freely deform a general defect M 3 by repeatedly applying this kind of gauge transformations site by site.
The functional integral over the fermion fields that we have defined in the previous section is, however, not invariant under the U(1) ′ gauge transformation (4.2); we must have an anomaly.The finite gauge transformation (4.2) can be constructed by accumulating Λ(x) is independent of the chosen path and periodic on Γ.Since Λ(x)U (x, µ)Λ(x + μ) −1 = 1, Λ(x) gives the gauge transformation in Eq. (2.4).infinitesimal gauge transformations parametrized by (see Eq. (3.26)) Then, Eq. (3.31) yields where, for the fermion fields inside the expectation value (see Eq. (3.33)), where we have noted T αβ = −δ αβ .Integrating Eq. (4.4) with respect to α, we thus have where, on the right-hand side, fermion fields within the operator O α are given by 27 ψ(x) α = e −iα/2δxy ψ(x), ψ(x) α = ψ(x)e iα/2δxy .(4.7) In Eq. (4.6), the left-hand side is the expectation value with the defect M 3 surrounding a single site y (Fig. 2).The right-hand side is the expectation value without the defect.By 27 Since we introduced ψ as a left-handed Weyl fermion, in terms of the Dirac fermion, these transformations correspond to ψ(x) α = e i(α/2)γ5 ψ(x) and ψ(x) α = ψ(x)e i(α/2)γ5 .The reason of the factor 1/2 in the rotation angle is that the axial rotation with α = 2π, ψ(x) 2π = −ψ(x) and ψ(x) 2π = − ψ(x), corresponds to a vectorial gauge transformation and thus is regarded as the identity transformation.repeatedly using the relation (4.6) site by site, therefore, for an arbitrary defect M 3 = ∂V 4 , we have where, on the right-hand side, (4.9) Equation (4.8) is nothing but the anomalous chiral Ward-Takahashi identity.That is, we may deform away the defect M 3 .This induces the chiral rotation of the angle α/2 on fermion fields on the sites swept by the deformation.The deformation, however, also leaves the effect of the axial anomaly.That is, the defect is not quite topological for this anomalous symmetry.
We may rewrite Eq. (4.8) as and define the symmetry operator U α ( M 3 ) in terms of the functional integral.U α ( M 3 ) is topological because the most right-hand side of Eq. (4.10) is independent of the precise form of the defect M 3 .We note that the combination in Eq. (4.10) is invariant under the physical U(1) lattice gauge transformations; the physical U(1) does not have the anomaly as Eq.(3.31) shows and the field strength f µν (x) is also invariant under U(1).
We now note from Eq. (2.13) that the topological density appearing in Eq. (4.10) can be written as28

.11)
The first term on the right-hand side is a total difference on the lattice and the combination in the square brackets would be regarded as a lattice counterpart of the Chern-Simons 3form.In fact, since z µν (x) in the last term are integers, the change of Eq. (4.11) under an infinitesimal variation of the lattice gauge field u(x, µ) depends only on the variation of u(x, µ) on a 3D closed surface corresponding to M 3 ; we denote this closed 3-surface on the original lattice as M 3 .See Fig. 3.In what follows, we call link variables u(x, µ) with A quite important characterization of the Chern-Simons action is its gauge noninvariance.The Chern-Simons action on a closed manifold can be regraded gauge invariant only when the level is an integer.With this fact and the above idea on the lattice Chern-Simons term in mind, in what follows, we consider the invariance of Eq. (4.10) under the gauge transformation on the boundary variables only.As we will analyze below, Eq. (4.10) is not invariant such a transformation.We regard this noninvariance as a good indication because the Chern-Simons action in continuum is not gauge invariant in general.How to make the expression invariant also under the gauge transformation on the boundary variables only is the subject of the next subsection.

Gauge average and the projection operator for magnetic fluxes on the defect
The symmetry operator U α ( M 3 ) in Eq. (4.10) is topological.However, since Eq.(4.10) is not invariant under gauge transformations on the boundary variables only as we will see below, U α ( M 3 ), when seeing as an object depending only on the boundary variables, is not gauge invariant.In this sense, U α ( M 3 ) may be regarded as an unphysical operator.
To make the symmetry operator also invariant under gauge transformations on the boundary variables only, we imitate the prescription by Karasik [23] in continuum and integrate U α ( M 3 ) over gauge transformations on the boundary variables only. 29That is, we consider the average over gauge transformations: where D[λ] is the integration over gauge transformations on the boundary variables only; the precise definition of the measure is given in Appendix B. 30 The symbol [ ] λ indicates the gauge transformation on the boundary variables only corresponding to the gauge average of the Chern-Simons action on the defect in the continuum [23].
Suppose that, in the plaquette, only u(x + μ, ν) is the boundary variable, while others are bulk variables.See Fig. 4.Then, we consider the gauge transformation acting only on the boundary variable, where we have set To realize the picture of Ref. [23], however, it turns out that we have to impose a certain smoothness condition on possible gauge transformations λ to be integrated in Eq. (4.12). x Fig. 4 A plaquette extending between M 3 and V 4 .Only the gauge field u(x + μ, ν) on the link x + μ → x + μ + ν is the boundary variable which is subject to the gauge transformation in Eq. (4.18).The plaquette with the replacement x → x + ρ + σ also contributes to Eq. (4.22).
That is, setting, where l ν (x + μ) ∈ Z, we require that the gauge transformation function φ(x) varies sufficiently smoothly on the lattice, satisfying sup (4.17) The transformation (4.14) then induces the change of the field strength, Note that the right-hand side is within the desired branch −π < f µν (x) + ∂ ν φ(x + μ) + 2πl ν (x + μ) ≤ π because of Eqs.(2.10) and (4.17). 31Since Eq. (4.17) implies 31 Although in Eq. ( 4.18) we have considered the case in which only one link variable of a plaquette is belonging to the boundary variables, there exist cases in which two, three, and four link variables of a plaquette are belonging to the boundary variables.The bound ǫ ′ < π/6 is designed to include these possibilities.and l µ (x) are integers, the integer field l µ (x) is flat (i.e., rotation-free): The motivation for the gauge average in Eq. (4.12) is to make the lattice analogue of the Chern-Simons term invariant under gauge transformations of boundary variables only.The invariance is archived, however, in a restricted sense.Under the gauge transformation λ ′ on boundary variables only, From this, one might think that λ ′ can be absorbed by the redefinition λ → λ(λ ′ ) −1 and the symmetry operator is made invariant.However, since our consideration below is based on the flatness (4.20), the change of the gauge transformation λ → λλ ′ in Eq. (4.21) should not influence the flatness (4.20).Also, our measure D x) , then the integer field l µ (x) defined in Eq. (4.16) does not change under λ → λλ ′ .We thus require this sufficient smoothness for λ ′ in Eq. (4.21).The gauge invariance of our lattice Chern-Simons term on the defect as a function of boundary variables is restricted in this sense.We expect that this point is not physically problematic because in continuum limit only differentiable gauge transformations are relevant; in the cutoff scale, these satisfy |∂ µ φ ′ (x)| ≪ 1. 32  Let us now fix a direction µ and assume that M 3 is perpendicular to μ and intersects a link x → x + μ as Fig. 4. In what follows, when necessary, we consider the case M 3 = T 3 as an explicit example, although we think that general M 3 can be treated by appropriate modifications of expressions.Then, under the gauge transformation on the boundary variables, the topological density ǫ µνρσ f µν (x)f ρσ (x + μ + ν) (4.11) with x + μ ∈ M 3 = T 3 changes by (here µ is fixed and not summed over; the 3-surface M 3 is extending in the directions of ν, ρ, and σ) 32 Since if we restore the lattice spacing, this reads where we have used the Bianchi identity (2.16).In deriving this, we have noted that f αβ (x) is gauge invariant if α and β do not contain µ and f µα (x) with x + μ ∈ M 3 receives a nontrivial gauge transformation.Then, under the sum over x + μ ∈ M 3 , the last line vanishes.
At this point, we note that the flatness (4.20) allows us to introduce a "scalar potential" ϕ(x) for the integer vector field l ν (x) as (here, sites o, x and y all are belonging to M 3 ), This number is independent of the site x in M 3 again because of the flatness (4.20).From the definition (4.24), one sees that k ν is the number of how many times e −iφ(x) e +iφ (x+ν)   winds around U(1) as x goes along the ν direction.The smoothness (4.19) is required to make this winding number well-defined on the lattice.Therefore, if we integrate over possible smooth gauge transformations in Eq. ( 4.12), one has a summation over integers k ν for each ν.Because of Eq. (4.17), for finite ℓ, the winding number is bounded by Finally, using the representation (4.23) in Eq. (4.22), we have where we have used the Bianchi identity (2.16) and the fact that ϕ(x) acquires the number k ν after a single return around the ν direction.From this, we see that the integration over smooth gauge transformations in Eq. (4.12) produces a factor, where M ν 2 is a closed surface such that M 3 = S 1 × M ν 2 with S 1 being the direction of ν.The precise form of M ν 2 does not matter because of the Bianchi identity (2.16).That is, a directional sum of f µν (x) over a surface of a 3D cube vanishes and one can freely deform M ν 2 as far as it can be done by the adding/subtracting of a cube.As noted above, with the integration measure in Appendix B, the integration over smooth gauge transformations becomes the sum over the winding number k ν .Introducing the function δ ℓ (x) by we know Therefore, the gauge average gives rise to, for each ν, since γ = −1/(32π 2 ) (recall Eq. (3.32)).This implies that, when α/(2π) is an irrational number, i.e., any magnetic fluxes along M ν 2 are not allowed, while, when α/(2π) is a rational number, p/N, where p and N are co-prime integers, This precisely corresponds to the constraint on the magnetic flux in the continuum obtained in Ref. [23]: These observations show that, for ℓ → ∞, we can write the symmetry operator defined by Eq. (4.12) as Ũα ( where P α ( M 3 ) is a projection operator on the subspace of allowed magnetic fluxes in M 3 .
From this expression, it is obvious that Ũα ( M 3 ) represents a noninvertible symmetry because it contains a projection operator.Gathering Eqs.(2.5), (4.10), and (4.30), in terms of the functional integral, the lattice realization of the axial U(1) noninvertible symmetry is thus given by where µ is the direction normal to M 3 = ∂(V 4 ); ν is the direction of S 1 in the decomposition The functional integral over the fermion, O M 3 F , is defined by the formulation in Sect. 3 with the external gauge field (4.1),where α = 2πp/N.The integrand in Eq. (4.35) is manifestly invariant under the physical U(1) lattice gauge transformations and thus is well-defined.Also, the defect M 3 is topological especially because Although our argument on the lattice correctly reproduces the constraint (4.33) in continuum theory, something intriguing is happening here: For α = 2πp/N, the quantity in the square brackets of Eq. (4.12) would correspond to the Chern-Simons action −4ip/(16πN) M 3 ada.Under the gauge transformation, a → a + dφ, this action naively changes by −ip/(4πN) M 3 d(φda).Then, setting M 3 = S 1 × M 2 and considering the winding k of φ around S 1 , the change under the gauge transformation would be −ipk/(2N) M 2 da.If this were correct, the sum over k would produce a constraint 1/(2π) M 2 da = 2NZ and this does not coincide with Eq. (4.33) by a factor 2. In Appendix A of Ref. [23], the necessity of the other factor 2 under the gauge transformation is elucidated in detail.Here, it is interesting that our lattice regularization provides this factor 2 automatically; the point is that Eq. (4.22) is obtained from the gauge transformation on the boundary variables which acts both of field strengths in ǫ µνρσ f µν (x)f ρσ (x + μ + ν).

Conclusion
In this paper, we made an attempt to realize the axial U(1) noninvertible symmetry of Refs.[1,2] in the framework of lattice gauge theory.The structures of the axial U (1) anomaly and the associated Chern-Simons term with finite lattice spacings are controlled by the lattice formulation of chiral gauge theory in Ref. [38] based on the Ginsparg-Wilson relation with appropriate modifications for our anomalous gauge theory.Imitating the prescription of Ref. [23], the symmetry operator/topological defect is constructed by integrating the boundary variables along the defect over smooth lattice gauge transformations.The projection operator for allowed magnetic fluxes on the defect, with correct values, then emerges.
The resulting expression as the whole is manifestly invariant under the physical lattice gauge transformations.
Our lattice formulation provides a firm basis for the axial U(1) noninvertible symmetry in the U(1) gauge theory and awaits possible applications; the computation of the condensate and the fusion rule naturally comes to mind. 33Also, a generalization to the non-Abelian gauge theory is an intriguing problem.We hope to return to these problems in the near future.
for our chiral gauge theory with an anomalous matter content.This analysis also gives the important information of what sort of counterterm should be included in our lattice formulation in the main text.We consider the left-handed Weyl fermion, The gauge group is U(1) × U(1) ′ and the charge assignment is the same as that in the main text.Hence, we define the Dirac operator by where representation matrices are We define the effective action Γ , that is a functional of a µ and A µ , in the following way [63].First, we note where we have introduced gauge currents, lµ (x) and Lµ (x).
The representation (A5) of the effective action is, however, yet formal because the gauge currents contain UV divergences.We hence define the gauge currents lµ (x) and Lµ (x) by the prescription [64], with the UV cutoff Λ.Note that here the Dirac operator / D (A2) does not contain the chiral projection operator and the chirality projection is implemented by the insertion of P + in the trace.This is a possible definition of the gauge current of a Weyl fermion and this prescription is known as the covariant gauge current [64].Correspondingly, the way to define the effective action through Eqs.(A5) and (A6) is known as the covariant regularization [63].In Ref. [65], it is shown that the lattice formulation of Ref. [38], at least in infinite volume, can be understood in terms of the covariant regularization.In our present context, therefore, it is natural to study the above prescription and resulting anomalies.We can also define another sort of gauge current, the consistent gauge current [66,67], as the functional derivative of the effective action Γ with respect to the gauge potential.From Eq. (A5), one has [63] In a similar way, we have Thus, the last two terms in Eqs.(A7) and (A8) provide the difference between the consistent current and the covariant current (this difference is known as the Bardeen-Zumino current [62]).
Similarly, we find: 34 Plugging these into Eqs.(A7) and (A8) and noting 1 0 ds s 2 = 1/3, we have the relation between the consistent and covariant gauge currents, On the other hand, the covariant gauge currents (A6) give rise to the so-called covariant gauge anomaly [64]: The anomaly of the consistent gauge currents (A11), the so-called consistent gauge anomaly [66,67], is therefore given by 34 Λ → ∞ is appropriately understood in what follows.
From Eq. (A13), we observe that in the present prescription (i.e., in the covariant regularization), (i) ℓ µ (x), which corresponds to the vector current in the target theory, does not conserve; and (ii) the coefficient of ǫ µνρσ f µν f ρσ in ∂ µ L µ (x), which corresponds to the axial vector anomaly in the target theory, is 1/3 of the naively expected one in Eq. (A12); if one computes the divergence of the axial vector current while respecting the vectorial gauge invariance, one has the coefficient of ǫ µνρσ f µν f ρσ in ∂ µ Lµ (x) of Eq. (A12).These two facts are actually related to each other.One can make ℓ µ (x) conserving by adding an appropriate counterterm to the effective action.The required counterterm is This provides the additional contribution to the consistent gauge anomalies and This is the desired form of the consistent gauge anomalies in our context.That is, the vector current conserves and the γǫ µνρσ f µν f ρσ term in the axial anomaly has the naively expected coefficient.We want to realize this structure also in the lattice formulation in the main text.Equation (A14) indicates what sort of counterterm should be added in the effective action in our lattice formulation.In the measure term (3.48), the term being proportional to γ is the variation of a lattice counterpart of the counterterm (A14).This term renders the coefficient of the ǫ µνρσ f µν f ρσ term in the axial anomaly the naively expected one in Eq. (3.53).

B Gauge integration measure
In this Appendix, we give a precise definition of the integration measure for the smooth gauge degrees of freedom assumed in the main text, e.g., in Eqs.(4.12) and (4.27).
We define the measure by where C Symmetry operator in terms of a Z N TQFT In the construction of the symmetry operator in the main text, some points remain unsatisfactory.One is that the construction of the symmetry operator is limited to cases of homologically trivial 3-dimensional (3D) closed surfaces, because the construction refers to an auxiliary 4-dimensional (4D) volume on the lattice.This is unsatisfactory from the perspective of the construction of the symmetry operator for generic cases and possible applications.The another point is that the invariance of the symmetry operator under the gauge transformation along the 3D surface is limited to sufficiently smooth lattice gauge transformations.These two points are mutually related and have the same root.In this Appendix, we provide an intrinsically 3D construction of the symmetry operator by employing a 3D Z N TQFT, the level-N BF theory, whose properties have been well-understood [6,8,15,[69][70][71].
In particular, we give the construction of the symmetry operator for generic 3D closed surfaces.The 3D expression is moreover manifestly invariant under arbitrary 3D lattice gauge transformations.As an application of our construction of the symmetry operator, we give the evaluation of fusion rules of symmetry operators within our lattice regularized framework.
In what follows, we set the rotation angle as by integers p and N. We first assume that p and N are coprime and later relax this restriction.Our formulation on the basis of the level-N BF theory is possible only when p is an even integer p = 2Z.First, using Eq.(4.11), Eq. (4.10) is written as In terms of the cochain and the cup product on the hypercubic lattice [50,72], 36 this can also be written as where a is a 1-cochain in Γ, a ∈ C 1 (Γ; R), z is a Z 2-cocycle in Γ (because of Eq. (2.15), δz = 0, where δ is the coboundary operator), z ∈ Z 2 (Γ; Z), and f is a 2-cocycle, f = δa + 2πz ∈ Z 2 (Γ; R).
In the anomalous Ward-Takahashi identity (C3), the first exponential on the left-hand side has the form of a 3D sum and this part can be generalized to arbitrary, i.e. not necessarily homologically trivial, closed 3D surfaces.The problem is the second exponential because it refers to a 4D volume V 4 .

C.1 Symmetry operator in terms of the BF theory on the lattice
To express that second exponential in Eq. (C3) solely in terms of 3D notions, we introduce the level-N BF theory defined on a 3-dimensional closed surface (i.e.3-cycle) M 3 in Γ.We define the action on a cubic lattice by where b ∈ C 1 ( M 3 ; Z N ) and c ∈ C 1 (M 3 ; Z N ); z is a background 2-cocycle z ∈ Z 2 (M 3 ; Z), δz = 0.In the lattice action (C4), we put the Z N 1-cochain b on the dual lattice, whose sites are given by assuming that M 3 extends in µ, ν, ρ directions.The first term of the action (C4) has the structure depicted in Fig. C1; note that this is not the cup product.The partition function of the BF theory is then defined by where Now, taking the sum over b in Eq. (C6), for p even, we have where we have introduced the delta functional: Equation (C8) immediately shows that, if there exists a 2D closed surface M 2 ⊂ M 3 such that the magnetic flux is nonzero modulo N, i.e.Let us first set z = 0 in Eq. (C8).Then, letting numbers of the sites, links, faces (plaquettes), and cubes of M 3 , be s, l, f , and c, respectively, the partition function is evaluated this with Eq. (C3), we arrive at the conclusion that the symmetry operator on an arbitrary 3-cycle M 3 is given by using the BF theory as If the deformation of the defect M 3 → M ′ 3 can be realized as above, from Eq. (C20), where we have noted that the anomalous Ward-Takahashi identity (C3) applied to the present small deformation implies, O . This shows that the symmetry operator is actually topological.
Under the 0-form U(1) lattice gauge transformation, the gauge potential in Eq. (2.12) changes as where l ∈ C 1 (Γ, Z).Since the field strength (2.13) is gauge invariant, the gauge transformation of z is given by z → z + δl.
From Eq. (C8), we find that Z M 3 [z| M 3 ] changes under the gauge transformation as Since O M 3 F in Eq. (C21) is gauge invariant for a gauge-invariant operator O as we have demonstrated in the main text, the symmetry operator defined by Eq. (C21) is manifestly invariant under arbitrary 3D lattice gauge transformations.Some remarks are in order.First, the continuum BF theory corresponding to Eq. (C8) would be given by This system gives rise to the 't Hooft anomaly under the background 1-form gauge transformation, z → z + dl, This anomaly is reproduced by a 4D symmetry-protected topological (SPT) action, assuming that M 3 = ∂(V 4 ).From this anomaly inflow, one may expect the relation (C20), for which we provided a precise proof on the lattice.Another point is the case in which the integers p and N are not co-prime and p = kp ′ and N = kN ′ with the greatest common divisor k.Since the chiral rotation angle (C1) depends only on the ratio, α = 2πp/N = 2πp ′ /N ′ , we may expect a certain relation between Z where we have repeated the calculation in Eq. (C11).This relation, however, is not satisfactory in that it depends on the way of discretization of M 3 through the number s − l.We may evade this point by generalizing the definition (C6) to where k is the greatest common divisor of p and N.Then, we obtain a simple equality, Z (p,N )

C.2 Fusion rules
As an application of our representation (C21), we evaluate fusion rules of two symmetry operators sharing a common 3-cycle M 3 .From Eq. (C8), we have (here we assume that gcd(p 1 + p 2 , N) = 1) where, in the last line, we have noted that Z (p 1 ,N ) M 3 [0] is given by Z M 3 [0] in Eq. (C11).In the second equality, we have noted that if z = 0 mod N ∈ H 2 (M 3 ; Z N ), the left-hand side identically vanishes and the right-hand side does too; the equality therefore holds.Otherwise, if z = 0 mod N ∈ H 2 (M 3 ; Z N ), there exists ν ∈ C 1 (M 3 ; Z N ) such that z = δν mod N and then A more general fusion can be obtained by considering (here we assume that gcd(p 1 , N 1 ) = gcd(p 2 , N 2 ) = 1) This identically vanishes unless z = 0 mod N 1 in Z 2 (M 3 ; Z N 1 ) and z = 0 mod N 2 in Z 2 (M 3 ; Z N 2 ).Thus, let us assume these are matched; this implies that z = 0 mod N in Z 2 (M 3 ; Z N ), where N = lcm(N 1 , N 2 ).Then, there exists ν ∈ C 1 (M 3 ; Z N ) such that z = δν mod N. Using this in Eq. (C38), after the shift of variables, c 1 → c 1 + ν and c 2 → c 2 + ν, we have On the other hand, setting β 1 = N/N 1 and β 2 = N/N 2 for N = lcm(N 1 , N 2 ), and assuming that gcd(β 1 p 1 + β 2 p 2 , N) = 1, This identically vanishes unless z = 0 mod N in Z 2 (M 3 ; Z N ).Assuming the latter, we have

6 )
is an explicit example of such a lattice Dirac operator.From the above expressions, one sees the γ 5 -hermiticity, D † = γ 5 Dγ 5 .The gauge covariance of the lattice difference operators (3.2) is inherited by D. Under the U(1) × U(1) ′ gauge transformation,

e +iα/ 2 e +iα/ 2 e −iα/ 2 e −iα/ 2 y M 3 Fig. 2 A
Fig. 2 A 2D slice of a defect M 3 (the broken line) surrounding a single site y.The phases e ±α/2 on links are the values of U(x, µ) assigned according to the rule in Eq. (4.1).

Fig. 3 A
Fig. 3 A 2D slice of the defect M 3 (the broken line) and the interior of the defect, V 4 (the shaded area).The corresponding 3-surface M 3 (the bold line) defines the boundary variables.
x) = y→y+ρ ∈a path connecting o and x in M 3 l ρ (y), (4.23)where we have assumed a certain fixed point o ∈ M 3 .ϕ(x) is invariant under any deformation of the path because of the flatness (4.20) and then the first relation obviously holds.ϕ(x) is, however, not single-valued in M 3 because of a possibility that the path winds nontrivial cycles of M 3 .When M 3 = T 3 with the size ℓ, the path can wind one of three cycles of T 3 .Under a single winding to the ν direction, ϕ(x) acquires an additional integer

)
Fig.C1The structure of the product in the first term of the lattice action (C4).The 1-cochain b is put on the dual link (the blue line) and the 1-cochain c is put on the original lattice (the black lines); 2-cochain δc − z is therefore put on the original plaquette (the square).
) then there is no configuration of c such that δc = z mod N. In this case, therefore, the partition function identically vanishes, Z M 3 [z] = 0. Later, we substitute z → z| M 3 , where z| M 3 denotes the 2-cocycle z ∈ Z 2 (Γ, Z) in Eq. (C3) restricted on M 3 .If this z| M 3 satisfies Eq. (C10) (this is equivalent to [1/(2π)] face∈M 2 f = 0 mod N), we cannot construct the symmetry operator.The existence of the symmetry operator, therefore, requires that the magnetic fluxes in Eq.(2.11)  are quantized in unit of N.