Topology of SU ( N ) lattice gauge theories coupled with Z N 2 -form gauge ﬁelds

: We extend the deﬁnition of L¨uscher’s lattice topological charge to the case of 4d SU ( N ) gauge ﬁelds coupled with Z N 2-form gauge ﬁelds. This result is achieved while maintaining the locality, the SU ( N ) gauge invariance, and Z N 1-form gauge invariance, and we ﬁnd that the manifest 1-form gauge invariance plays the central role in our construction. This result gives the lattice regularized derivation of the mixed ’t Hooft anomaly in pure SU ( N ) Yang–Mills theory between its Z N 1-form symmetry and the θ periodicity.

C Bound on ε for the admissibility condition 17 1 Introduction Gauge fields in the continuum description enjoy topological classifications, and those in different topological sectors cannot be continuously deformed from one another.This fact allows us to introduce a new parameter, called the θ angle, in quantum Yang-Mills theories [1][2][3], and the presence of topological sectors makes the dynamics of Yang-Mills theories quite rich and highly nontrivial.However, we should note that it is nontrivial if these considerations based on topology are robust under quantum fluctuations.Quantum field theories (QFTs) are subject to ultraviolet divergences, and we need to introduce some regularization to control it.Lattice regularization achieves it in a gauge invariant manner, but it becomes unclear if the notion of continuous gauge fields survives under this process.
Lüscher addressed this issue for SU (2) gauge theories and pointed out the presence of topological sectors on the lattice by introducing the admissibility condition [4] (see also Ref. [5]).There, the topological charge on the lattice is explicitly defined in a local and gauge-invariant way, and one can extend the construction to simple and simply-connected gauge groups such as SU (N ) straightforwardly.In this paper, we consider its extension for non-simply-connected gauge groups, in particular SU (N )/Z N .SU (N )/Z N gauge fields can be thought of the SU (N ) gauge theories coupled to Z N 2-form gauge fields.Treating the Z N 2-form gauge fields as the background field, it is equivalent to the SU (N ) gauge theories in the 't Hooft twisted boundary condition [6].van Baal computed the topological charge for smooth gauge fields in this setup, and it turns out that the topological charge has a fractional shift in the unit of 1/N [7].These observations now acquire renewed interest from the viewpoint of generalized symmetries in QFTs.Pure SU (N ) Yang-Mills theory enjoys the Z N 1-form symmetry as the global symmetry [8], and the Z N 2-form gauge field obtains a natural interpretation as the gauge field coupled to the 1-form symmetry [9,10].Studying the response of background gauge fields provides a concise and systematic way to extract nontrivial consequences out of global symmetries, and the fractional shift of the topological charge turns out to have a huge impact on our understanding of the Yang-Mills vacua [11].
In this paper, we construct the topological charge of lattice SU (N )/Z N gauge fields that maintains the locality, SU (N ) gauge invariance, and Z N 1-form gauge invariance.Introducing the admissibility condition for SU (N ) gauge fields coupled to flat Z N 2-form gauge fields, we show that the SU (N )-valued transition functions can be locally defined from the lattice SU (N )/Z N gauge fields.We show that those transition functions satisfy the cocycle condition up to the center elements specified locally by the 2-form gauge fields.Thus, the principal SU (N )/Z N bundle is obtained from the admissible lattice gauge fields, and we can compute their topological charge as its second Chern class.We note that all of this procedure is completely parallel to Lüscher's one in Ref. [4].It turns out that we can keep the manifest 1-form gauge invariance at every stage of our construction, which is the essential ingredient to circumvent various complications potentially caused by the presence of higher-form gauge fields.Using this lattice topological charge, we define the lattice Yang-Mills theory with the θ angle coupled to the background Z N 2-form gauge fields, and we obtain the anomalous relation for the Yang-Mills partition function on the lattice.Our result justifies the observation of Ref. [11] that had been obtained in the continuum analysis assuming smoothness of gauge fields, so we rigorously realize the mixed 't Hooft anomaly between the Z N 1-form symmetry and the θ periodicity with the lattice regularization.
We note that this study is a non-Abelian extension of the previous study about U (1) lattice gauge theory [12], while the actual construction in this paper is based on a different idea from that of Ref. [12].Still, applying this construction to the U (1) gauge group should reproduce the result of Ref. [12] due to the uniqueness of the principal bundle with given data.
2 Lattice SU (N ) gauge theories coupled with Z N 2-form gauge fields Approximating the closed Euclidean spacetime M as a lattice, we introduce the SU (N )valued link variables U ∈ SU (N ) for each link and the Z N -valued plaquette variables e 2πi N Bp for each plaquette p.For the purpose of numerical simulations, one usually takes M = T 4 as the Euclidean spacetime and approximates it as the hypercubic lattice Λ L = (Z/LZ) 4 since the hypercubic symmetry restricts possible UV divergences and simplifies renormalization procedures.In this section, let us consider more generic cases for the discussion of general properties, although we shall also restrict to the hypercubic case later for the construction of a lattice topological charge.Following the nomenclature in Ref. [4], the 0-cells are referred to as sites, the 1-cells are links, the 2-cells are plaquettes, the 3-cells are faces, and the 4-cells are just cells.
When and describe the same link with the opposite orientation, = − , we require that U − = U −1 .Similarly, B −p = −B p mod N .Along a plaquette p, we define the SU (N )-valued plaquette variable by the path-ordered product of link variables, where ∈ ∂p refers to the link in the positive relative orientation for the plaquette p.We require that the Z N plaquette variables satisfy the flatness condition, for each face f .We define the Wilson plaquette action as where β is the coupling constant for the lattice Yang-Mills theory.This Wilson action is invariant under the SU (N ) gauge transformation: For the link connecting two sites x and y, which we denote as = x → y , the SU (N ) transformation is given by with g x ∈ SU (N ).Let us denote n as the initial and final point of the closed loop for p, then U p → g −1 n U p g n under the SU (N ) gauge transformation so that tr(U p ) is invariant.The Wilson action with Z N plaquette fields B p is invariant also under the Z N 1-form gauge transformation, defined by where λ ∈ Z N and (dλ) p = ∈∂p λ .We note that the flatness condition (2.2), or (dB) f = 0 mod N , is invariant under the Z N 1-form gauge transformations.When M has no torsion, we can think of B p as a closed surface on the dual lattice using the Poincaré duality, and the Z N 1-form gauge invariance requires the invariance under its continuous deformation and the addition/subtraction of contractible closed surfaces.
In order to introduce the notion of topological sectors for the lattice gauge fields, the admissibility condition [4] plays the key role.We say that the SU (N ) gauge field {U } is admissible, if and only if, for a given 0 < ε 2, all the plaquettes U p satisfy Here, • refers the matrix norm.After diagonalization e − 2πi N Bp U p = V e diag(iθ1 ,...,iθ We denote the set of admissible gauge fields as and A ε [B p ] is SU (N ) gauge invariant and Z N 1-form gauge covariant.The functional integral is performed over A ε [B p ] so that the partition function is given as 1 where DU is the product of the SU (N ) Haar measure for all the link variables.We note that ε shall be determined independently of the lattice size, coupling constant β, and the plaquette gauge fields B p .It will be discussed in Appendix C and we find ε 0.074 is sufficiently small.

Transition functions from lattice gauge fields
In this section, we construct the transition function for admissible lattice gauge fields coupled to the Z N 2-form gauge fields B p extending the seminal work by Lüscher [4].We work on the 4-torus M = T 4 and its hypercubic lattice discretization Λ L = (Z/LZ) 4 , i.e., each direction is cut into L pieces.As illustrated in Fig. 1, we define some variables on the hypercubic structure as follows: The unit cell is denoted as for n ∈ Λ L , and we would like to define the transition function on each face where μ is the unit vector along the µth direction.
For the link = n → n + μ , we write the link variable as U (n, µ) = U .For the plaquette including n, n + μ, n + ν, and n + μ + ν, we have the SU (N ) plaquette variable and and the admissibility condition (2.6) requires 1 − Ũµν (n) < ε for all the plaquettes.
At this moment, ṽn,µ (x) is defined only at the corner of f (n, µ), and we later interpolate it.

Complete axial gauge and transition functions at the corner
We first define the complete axial gauge for each cell c(n) and introduce the transition function at the corner of the cells as the connection formula of the corresponding link variables.Let x ∈ c(n) be a corner, i.e., x = n + µ z µ μ with z µ ∈ {0, 1}, and we define the standard parallel transporter from n to x as2 We then set, for two adjacent corners x and y of c(n), This can be thought of the link variable in the complete axial gauge of c(n) for = x → y , since all the link variables along the standard parallel transporter are set to be 1.When the link = x → y is shared by two cells c(n) and c(n − μ), the link variables u n xy and u n−μ xy take different values due to the different choice of the gauge between these cells, and we introduce the transition functions at the corner v n,µ (x) to make the connection between them: The explicit formula for v n,µ (x) is given by when x is at the corner of f (n, µ) = c(n)∩c(n− μ).For later purpose, we denote the corner of f (n, µ) as s i (i = 0, 1, . . ., 7) as illustrated in Fig. 2, and we introduce the coordinate on f (n, µ) as x = n + y α α + y β β + y γ γ with α, β, γ ∈ {1, 2, 3, 4} \ {µ} and α < β < γ.
We introduce the coordinate for the face f (n, µ) as x = n + y α α + y β β + y γ γ with 0 ≤ y α,β,γ ≤ 1 and α < β < γ.We label the corners of f (n, µ) (i.e., y α,β,γ ∈ {0, 1}) as s i (i = 0, 1, . . ., 7) as shown in this figure .The link in the complete axial gauge u m xy is the product of link variables along a closed loop that starts and ends at the site m.Therefore, we can make them invariant under the 1-form gauge transformation by multiplying appropriate factors of e − 2πi N Bµν (n) .Due to the flatness condition (2.2), there is a unique local way to achieve the Z N 1-form gauge invariance, and such 1-form invariant u m xy are denoted with the tilde, ũm xy .This construction is illustrated in Fig. 3 by taking an example of ũm=n− , where s 7 and s 2 denote corners of f (n, 3), and it shows that ũn− 3 (3.9) The list of the concrete expressions for ũm xy is given in Appendix B.1.We further want to define ṽn,µ (x) for a corner x of f (n, µ) such that ũn−μ xy = ṽn,µ (x)ũ n xy ṽn,µ (y where x and y are the corners of f (n, µ).Also, we impose that ṽn,µ (x) for the corner x behaves covariantly under the 1-form gauge transformation, ṽn,µ (x) → e 2πi N λµ(n−μ) ṽn,µ (x).(3.11)This is again achieved by multiplication of appropriate factors of e 2πi N Bµν (n) , and the concrete expressions are given in Appendix B.2.We can then confirm Eq. (3.10) by using the flatness condition, (dB) f = 0 mod N .From these, we observe the cocycle condition at the corner (3.12) We can check it using the relation n) and the formulas given in Appendix B.2.This is the desired cocycle condition in the SU (N )/Z N gauge theory and actually is expected from the transformation property under the 1-form gauge transformation alone: From the cocycle condition for transition functions without tildes, the right-hand side must be an element of Z N .This Z N element should transform identically as the left-hand side does under the Z N 1-form gauge transformation (3.11).Then, the unique possibility is e 2πi N Bµν (n−μ−ν) 1.

Fractional topological charge on the lattice and its applications
We now obtain the SU (N )/Z N bundle on T 4 characterized by transition functions ṽnµ (x) between the cells c(n) and c(n − μ).The topological charge, or the second Chern class, is given by [4,7] Q where Q top introduces the topological classification of admissible lattice gauge fields A ε [B p ], and gauge fields in different sectors cannot be continuously deformed with each other within A ε [B p ].We note that this topological charge is manifestly SU (N ) and Z N 1-form gauge invariant.Moreover, since ṽn,µ (x) is constructed out of the plaquettes Ũp in the adjacent cells, the topological charge density is defined in the local way on the lattice.

Fractional shift of the topological charge and mixed 't Hooft anomaly
By performing the 1-form gauge transformations and redefinition of variables, we find that the SU (N )/Z N bundle constructed above is equivalent to the 't Hooft twisted boundary condition with the 't Hooft flux (see, e.g., Appendix A of Ref. [13] for details) As shown by van Baal in Ref. [7], we then find that and thus the lattice topological charge is shifted by a fractional value in the unit of 1/N .In the above expression, the fractional part may look to be written using the global data, but it can be written in the local way by using the cohomological operations: where the Pontryagin square is defined by For the definition of (higher-)cup products on the hypercubic lattice, see Ref. [14].The correction by the 1-cup product is introduced so that the Z N 1-form gauge invariance is achieved at the cochain level.We can then confirm the equivalence between Eqs. (4.4) and (4.5) by performing the gauge transformation so that (dB) f = 0 in Z and the Poincaré dual of B p always has transverse intersections.Then, 1 2 P 2 (B p ) in Eq. (4.5) counts the number of intersection points in mod N and it gives the fractional part of Eq. (4.4).
As we have defined the local topological charge, we can introduce it as the θ term for the Boltzmann weight of the path integral, and we define the partition function as In order for the well-definedness of the lattice topological charge, we have to restrict the domain for the path integral to the admissible gauge fields A ε [B p ] with some ε 0.074.Using Eq. (4.5), we find the following equality for the lattice Yang-Mills partition function: In Ref. [11] (see also Refs.[15][16][17][18][19][20]), this is interpreted as the 't Hooft anomaly for the Z N 1-form symmetry and the θ periodicity of 4d SU (N ) pure Yang-Mills theory, and one can derive various nontrivial consequences on the Yang-Mills vacua from this relation.
In previous studies, its derivation is based on the classical analysis of the path-integral measure for smooth gauge field configurations, and it is assumed that the path integral over quantum gauge fields does not spoil the relation for classical action.Here, we have given the complete proof of the anomalous relation (4.8) using the lattice regularization that maintains the locality and the gauge invariance, and thus it is now shown to be a rigorous relation for the quantum Yang-Mills theory.

Classical continuum limit of the topological charge
Although we obtained the lattice local expression for the topological charge density (4.2), it is somewhat unrealistic to use this expression for the practical computations due to the complicated construction of transition functions ṽn,µ (x).Therefore, it is desirable to find a simpler expression for the lattice topological charge.Due to the quantization of the topological charge, we do not even have to perform its exact calculation as long as we can control the approximation.
Here, we take the classical continuum approximation for computing the lattice topological charge and make its connection to the naive continuum formula.Since Q top is invariant under the continuous deformation of {U } as long as the admissibility (2.6) is satisfied, one can use the following approximate expression with smeared lattice gauge fields when one ensures (or monitors) the preservation of the admissibility in the smearing procedures such as the gradient flow [21][22][23][24][25].
To take the classical continuum limit, we formally introduce the lattice constant a and take the a → 0 limit with fixed torus size aL.When B p = 0 everywhere, we can put the assumption that all the link variables are close to the identity and then the gauge fields are introduced as their phase factor.When B p is present, however, this assumption has a self-contradiction as some link variables necessarily have O(1) deviations from 1, so we need more careful treatment about the notion of continuum gauge field.In contrast, we can safely assume the continuity about the 1-form gauge-invariant plaquettes.We introduce the 1-form gauge invariant field strength Fµν by Ũµν (n) = exp ia 2 Fµν (an) 1 + ia 2 Fµν (an).(4.9) For the discussion of continuity, we would like to compare Fµν at two different lattice points n and m.When m ∈ c(n), we use the standard parallel transporter w n (m) to bring back Fµν (am) to the point n, and we say that the gauge field is continuous if Under this assumption of the continuity, we obtain Then, we substitute this expression to the definition (4.2), and the second term on the righthand-side gives the leading contribution.The rest of calculations is completely identical to the one for SU (N ) theories [4], and we can find the result by quoting them as q(n) = a 4 1 32π 2 ε µνρσ tr Fµν (an) Fρσ (an) + O(a 5 ).(4.12) The continuum expression for the topological charge is reproduced at the leading order under the presence of generic B p .

Summary and outlooks
We have constructed the topological charge on the lattice for SU (N ) gauge fields coupled with Z N 2-form gauge fields.Introducing the admissibility condition for the SU (N ) link variables with generic flat B p , we obtain a manifestly local, SU (N ) and Z N 1-form gaugeinvariant expression for the topological charge density.As an application of this result, we provide the rigorous derivation of the mixed 't Hooft anomaly between the Z N 1form symmetry and the θ periodicity.We also give the classical continuum limit of the topological charge density, as the exact expression is so complicated for the practical use.We introduce the gauge-invariant criterion for the validity of continuum approximation, so one can use it with smeared gauge field if one ensures that the admissibility is kept intact under the smearing process such as the gradient flow.
In this work, we have treated B p as a flat background gauge field, but we can perform its path integral to obtain the SU (N )/Z N Yang-Mills theory.Here, we can add the discrete theta angle as the local counter term, so there are N distinct SU (N )/Z N Yang-Mills theory as clarified in Ref. [26].In the continuum analysis, it is suggested that they have dyonic line operators as gauge-invariant line operators, and their electric charge is specified by the choice of the discrete theta parameter.In the lattice formulation, we need to violate the flatness condition for B p along the dyonic line.It would be an interesting future study to uncover various topological phenomena, such as the Witten effect [27], on the lattice.As we have defined the admissible gauge fields only for SU (N ) link variables coupled with the flat 2-form gauge fields in this paper, we would need further generalizations for studying those phenomena.
It would also be interesting to extend this work to the case with fundamental matter fields.In such cases, the flavor symmetry and the gauge redundancy has the common center, and we may have a 't Hooft anomaly between the projective nature of the flavor symmetry and the θ periodicity [28][29][30].To obtain such an anomaly, the background 2form gauge field plays an important role even though the 1-form symmetry is not present, so we expect that our result is useful to give the fully lattice regularized derivation of those 't Hooft anomalies.
In order to get some insight, let us discuss what kinds of elements in SU (N ) are excluded from the definition of U y .We can immediately see that the nontrivial center elements, such as e which is a nontrivial center element for a subgroup SU (2) ⊂ SU (N ).Especially when N = 2, this is the element treated as the exceptional configuration in Ref. [4].Let V ∈ SU (N ).Since the set of eigenvalues are not affected by the conjugate operation, V XV −1 ∈ D if X ∈ D. This shows that, for U ∈ exp(iD) ⊂ SU (N ), This is the key property of U y for the discussions in the main text.

B Explicit formulas
In this appendix, we give the concrete expressions for the 1-form gauge invariant links in the complete axial gauge, ũn xy , and the 1-form covariant transition function, ṽn,µ (x), at the corner of f (n, µ).For µ = 3.

1 Introduction 1 2 2 3 3 . 1 7 4 11 5and outlooks 12 A
Lattice SU (N ) gauge theories coupled with Z N 2-form gauge fields Transition functions from lattice gauge fields 4 Complete axial gauge and transition functions at the corner 5 3.2 Continuum interpolation of transition functions and cocycle conditions Fractional topological charge on the lattice and its applications 10 4.1 Fractional shift of the topological charge and mixed 't Hooft anomaly 10 4.2 Classical continuum limit of the topological charge Summary Fractional power of special unitary matrices 13 B Explicit formulas 14 B.1 Formulas for ũn xy 14 B.2 Formulas for ṽn,µ (x) 16

Figure 3 .
Figure 3. Illustration for the construction of u m xy (left) and its 1-form invariant counterpart ũm xy (right).We take an example with m = n − 3, x = s 7 = n + 2 + 4, and y = s 2 = n + 2, so we draw two cubes for faces f (n, 3) and f (n − 3, 3).(Left) Each building block of u n− 3 s7s2 = w n− 3(s 7 )U (s 7 , −4)w n− 3(s 2 ) −1 is shown in different colors, in which the links in the 3 direction are drawn with curved lines.(Right) As u m xy forms a closed loop starting and ending at the site m, we attach the Z N plaquette fields e 2πi N Bµν (n) when the loop surrounds the corresponding plaquettes.