^{3}

We investigate the property of the effective action with the chiral overlap operator, which was derived by Grabowska and Kaplan. They proposed a lattice formulation of four-dimensional chiral gauge theory, which is derived from their domain-wall formulation. In this formulation, an extra dimension is introduced and the gauge field along the extra dimension is evolved by the gradient flow. The chiral overlap operator satisfies the Ginsparg–Wilson relation and only depends on the gauge fields on the two boundaries. We start from the arbitrary even-dimensional chiral overlap operator. We treat the gauge fields on the two boundaries independently, and derive the general expression to calculate the gauge anomaly with the chiral overlap operator in the continuum limit. As a result, we show that the gauge anomalies with the chiral overlap operator in two, four, and six dimensions in the continuum limit are equivalent to those known in the continuum theory up to total derivatives.

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 [

The effective action of the

If the formulation is free from gauge anomalies, the Chern–Simons term vanishes. Moreover, as shown in the two-dimensional

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

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

In Ref. [

Here,

The Greek letters,

The generators

Here the covariant derivative is defined as

Here, we treat the gauge fields ^{1}

By expressing the infinitesimal gauge transformation as

The parity-even part can be removed by local counterterms. As discussed in Ref. [

In this section, we evaluate the parity-odd part following Ref. [

Since

From Eqs. (

From Eqs. (

Since we have

The number of

From the equation,

Here,

Let

(i)

The terms classified into case (i) are the products of

Since

(ii)

The terms classified into case (ii) are the products of

The second terms of Eqs. (

(iii-a)

(iii-b)

(iv-a)

(iv-b)

Therefore, the terms classified into cases (i) and (ii) only contribute to Eq. (

Now, we can write down the terms which contribute to Eq. (

Equations (

Here,

(1)

Thus we obtain

(2)

Thus we obtain

This result is derived in Ref. [

(3)

Thus we obtain

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. [

In this paper, we generalized the result in Ref. [

The author would like to thank Takeo Moroi and Natsumi Nagata for helpful discussion and advice.

Open Access funding: SCOAP

We also assume that