It has been a long-standing problem to construct a gauge-invariant regularization for a chiral gauge theory. Grabowska and Kaplan proposed a formulation of the chiral gauge theory on the lattice [1], which is developed based on the idea of the domain-wall fermion proposed by Kaplan [2]. In the formulation of the domain-wall fermions an extra dimension is introduced, and the left-handed fermion is localized on one domain wall and the right-handed fermion is localized on the other domain wall. In this formulation, the left- and right-handed fermions are coupled with the same gauge field because the gauge field is constant along the extra dimension. Thus this formulation is vector-like. On the other hand, in the Grabowska–Kaplan formulation the gauge field along the extra dimension is given by the gradient flow [3–6], (1.1) where is the solution of the flow equation of Eq. (1.1), and and are the covariant derivative and the field strength constructed from , respectively. In other words, the gauge field is modified along the extra dimension by the gradient flow from the gauge field on one domain wall to on the other domain wall. This means that the left- and right-handed fermions are coupled with the gauge field differently. Thus their formulation is expected to be a non-perturbative formulation of a chiral gauge theory. Grabowska and Kaplan also formulated a four-dimensional effective theory from the formulation explained above and obtained the chiral overlap operator [7], which obeys the Ginsparg–Wilson relation [8]. This operator only depends on the gauge fields and , and in the tree-level continuum limit, the left-handed fermion is only coupled with and the right-handed fermion is only coupled with . For recent works related to Refs. [1,7], see Refs. [9–13].

The effective action of the -dimensional domain-wall fermion with the gauge field evolved by the gradient flow is composed of three parts: the effective action of the -dimensional left-handed fermion, the effective action of the -dimensional right-handed fermion, and the Chern–Simons term which is induced by the heavy modes in the bulk. Since the gradient flow assures the gauge invariance of the theory, the effective action of the -dimensional domain-wall fermion itself is gauge invariant. However, the effective actions of the left- and right-handed fermions are not gauge invariant because of gauge anomalies. In other words, the Chern–Simons term plays the role of cancelling out the gauge variation from the effective actions of the boundary modes.

If the formulation is free from gauge anomalies, the Chern–Simons term vanishes. Moreover, as shown in the two-dimensional gauge theory in Refs. [1,7], the gauge field is expected to be evolved into a pure gauge so that the right-handed fermion on the other domain wall does not interact with the physical degrees of freedom of the gauge field. Thus we expect that the -dimensional domain-wall fermion with the gauge field evolved by the gradient flow results in a -dimensional effective theory in which only the left-handed fermion couples to the physical degrees of freedom of the gauge field.

In the lattice theory, we expect the same structure of the effective action in the continuum limit. The effective action constructed from the chiral overlap operator is composed of three parts: the functional of the gauge field , the functional of the gauge field , and the cross terms of the gauge fields and . This effective action is gauge invariant under the simultaneous gauge transformation of and . In the case of four-dimensional effective theories, the cross terms of the gauge fields and were calculated in the continuum limit and it was confirmed that the parity-odd part of the gauge variation of the functional of the gauge field coincides with the gauge anomaly known in the continuum theory [12].

In order to confirm the correspondence of the structures of the effective actions between the formulation of the domain-wall fermion and the chiral overlap operator, we generalize this result; i.e., we calculate the gauge variation of the functional of the gauge field for an arbitrary even-dimensional effective action of the chiral overlap operator in the continuum limit, and explicitly check that the parity-odd part indeed coincides with the gauge anomaly in the continuum theory in the case of two, four, and six dimensions.

In Ref. [7], the chiral overlap operator is defined through the transfer matrix which depends on the flow time due to the -dependence of the gauge field. They consider the simplification that the gauge field is constant in the half of the interval near the boundary and is in the remaining region. is the length of the extra dimension and in the large- limit the effective theory for the boundary modes is obtained. In this case, the chiral overlap operator in arbitrary even dimensions is expressed as follows: (2.1)

Here, is the sign function, (2.2) of the Hermitian Wilson Dirac operator, (2.3) where is the lattice spacing, and and are free parameters. is also defined by replacing the gauge field with , which is obtained from the original gauge field according to the gradient flow equation, Eq. (1.1). Here we consider Euclidean arbitrary even dimensions . Gamma matrices satisfy the following equations: (2.4)

The Greek letters, , run from to . Therefore, (2.5) follow from Eq. (2.4). and are the forward and backward lattice covariant derivatives, respectively, which are defined as (2.6) (2.7)

The generators of the gauge group satisfy the following equations: (2.8)

Here the covariant derivative is defined as , where and is real. Thus, by defining the link variable as (2.9) where denotes the path-ordered product and is the unit vector in the direction , we obtain (2.10) (2.11) in the continuum limit . In Ref. [12], the fermion one-loop effective action defined by (2.12) is studied and the following expression is obtained: (2.13) where and denotes the trace over the spinor and gauge indices. and are the infinitesimal variations which only act on and , respectively: (2.14) (2.15)

Here, we treat the gauge fields and independently, and the infinitesimal variations and are independent.¹ Equation (2.13) is decomposed into the parity-odd and parity-even parts. The former is written as (2.16) (2.17) and the latter is written as (2.18) (2.19)

By expressing the infinitesimal gauge transformation as (2.20) (2.21) and using the equations (2.22) (2.23) we obtain the following equation which is related to the gauge anomaly: (2.24)

The parity-even part can be removed by local counterterms. As discussed in Ref. [12] for , the parity-even part contains a mass term of the gauge field even if the anomaly-free condition is satisfied, and this part should be subtracted by local counterterms.

In this section, we evaluate the parity-odd part following Ref. [14], in which the axial anomaly defined by the overlap operator is calculated in the continuum limit for arbitrary even dimensions. By expanding the parity-odd part of Eq. (2.24) in powers of , we obtain the following equations: (3.1)

Since , we only need to calculate terms with in the trace of (3.2)

From Eqs. (2.2) and (2.3), it is clear that and contain one . Thus, the s appear an odd number of times in all of the terms of Eq. (3.2), and these terms are reduced to the form that contains the factor , which is zero if . Therefore, we only need to take account of the terms in which the s appear at least times. Note that a diagonal element of the kernel of an operator on the lattice is calculated from (3.3) where (3.4)

From Eqs. (2.2) and (2.3), we obtain (3.5) where the ellipsis denotes the terms which do not contribute to Eq. (3.2) in the continuum limit for the reason explained later. The expressions that appear in Eq. (3.5) are defined as follows: (3.6) (3.7) (3.8) (3.9) where (3.10) (3.11) and denotes the field strength of the gauge field . is defined as the coefficient of in the power series of ; that is, , and the explicit form of is given as follows: (3.12)

Since we have if , we obtain the following expressions: (3.13) (3.14) and (3.15)

The number of s for each of the terms in Eqs. (3.13), (3.14), and (3.15) is less than or equal to , except for the terms which contain s. On the other hand, the maximum number of s for the terms that contain s is . Therefore, in the terms in Eq. (3.2) where s appear times, s appear at most times. However, the number of s can be reduced as we explain below.

From the equation, (3.16) we obtain the following equations: (3.17) (3.18)

Here, means that is omitted. From Eqs. (3.17) and (3.18), the terms in Eq. (3.2) are reduced to a form in which the number of s is less than or equal to . In addition, since the maximum numbers of s in and are odd, and the terms in Eq. (3.2) are the products of an odd number of them, the maximum number of s in the terms in Eq. (3.2) is odd, that is, not but . Therefore, we only need to consider the terms in Eq. (3.2) in the continuum limit.

Let be the number of s in each of the terms and the number of s without s in s in each of the terms. Then each of the terms in Eq. (3.2) can be classified into one of six cases.

Therefore, the terms classified into cases (i) and (ii) only contribute to Eq. (3.2) in the continuum limit. Note that the terms which contain the factors of ellipses in Eqs. (3.13), (3.14), and (3.15) do not contribute to Eq. (3.2) in the continuum limit, because they correspond to the cases (iii-a), (iii-b), (iv-a), and (iv-b).

Now, we can write down the terms which contribute to Eq. (3.2) in the continuum limit. Here we define the following functions: (3.21) (3.22) and (3.23) where . Then, we obtain (3.24) and (3.25)

Equations (3.24) and (3.25) can be simplified further by evaluating the trace over the spinor index and the integral with respect to the variable . In general, by defining (3.26) (3.27) where is independent of and valued in the Lie algebra of the gauge group with subscripts running from to , we have the following equations: (3.28) where denotes the trace over the spinor only, and (3.29) denotes the maximum integer which is less than or equal to . For the derivation of Eq. (3.29), see Ref. [14]. In the last equality of Eq. (3.28) we used by assuming . Therefore, using Eqs. (3.24), (3.25), and (3.28), we finally obtain the following expression: (3.30) where (3.31) and (3.32)

Here, is defined by Eqs. (2.20) and (2.21), and we used the differential form. That is, , , and . We can obtain the explicit form in the specific dimensions as follows:

It turns out from the above results that the gauge anomalies in two, four, and six dimensions in the continuum limit obtained here are equivalent to those known in the continuum theory up to total derivatives (for a review of the gauge anomaly, see Ref. [15]).

In this paper, we generalized the result in Ref. [12], in which the gauge anomaly of the four-dimensional effective theory is calculated; i.e., we performed the explicit calculation of the arbitrary even-dimensional gauge anomaly with the chiral overlap operator in the continuum limit. The resultant expressions in two, four, and six dimensions are found to be equivalent to those known in the continuum theory up to total derivatives. If the gauge field is evolved by the gradient flow, the total effective action is gauge invariant, and the anomalies are cancelled by the cross terms of the gauge fields and . This means that the parity-odd part of the cross terms corresponds to the Chern–Simons term. Thus the parity-odd part of the cross terms vanishes if the anomaly cancellation condition is satisfied.