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Domain wall fermion and chiral gauge theories on the lattice with exact gauge invariance

We discuss how to construct anomaly-free chiral gauge theories on the lattice with exact gauge invariance in the framework of domain wall fermion. Chiral gauge coupling is realized by introducing a five-dimensional gauge field which interpolates between two different four-dimensional gauge fields at boundaries. The five-dimensional dependence is compensated by a local and gauge-invariant counter term. The cohomology problem to obtain the counter term is formulated in 5+1 dimensional space, using the Chern-Simons current induced from the five-dimensional Wilson fermion. We clarify the connection to the invariant construction based on the Ginsparg-Wilson relation using overlap Dirac operator. Formula for the measure and the effective action of Weyl fermions are obtained in terms of five-dimensional lattice quantities.


Introduction
Through the Ginsparg-Wilson relation [1] and the exact chiral symmetry based on it [2], Weyl fermion can naturally be introduced on the lattice. The chiral constraint imposed on the Weyl fermion is gauge-field dependent and by introducing the basis of Weyl fermion, the path integral can be set up [3,4]. In fact, it has been shown by Lüscher that the functional measure of the Weyl fermion can be constructed in anomaly-free abelian chiral gauge theories so that it satisfies the requirements of the smoothness, the locality and the gauge invariance [3,5,6,7]. The similar construction has also been argued for generic non-abelian chiral gauge theories, where to treat the exact cancellation of gauge anomaly, a local cohomology problem in 4 + 2-dimensions is formulated [4]. 1 This construction is generic and applies to any local lattice fermion theory with the Dirac operator satisfying the Ginsparg-Wilson relation. In the case using the overlap Dirac operator [14,15,16], the path integral formalism for the Weyl fermion reproduces the overlap formula for the chiral determinant by Narayanan and Neuberger [17]. 2 The above invariant construction of the functional measure provides the method to fix the phase factor of the chiral determinant in the overlap formalism in a gauge-invariant manner.
It was also suggested [4,28] that there is a close relation between the interpolation procedure in Lüscher's construction and the five-dimensional setup in Kaplan's domain wall fermion [29]. The purpose of this paper is to pursue this close connection and to show how to construct four-dimensional lattice chiral gauge theories with exact gauge invariance from the five-di- 1 The author refers the reader to [8,9] for recent review of this approach. In this approach, the exact cancellation of gauge anomalies in non-abelian chiral gauge theories has been shown in all orders of the expansion in lattice perturbation theory [10,11]. For SU(2) doublet, it has been shown that Witten's global anomaly is reproduced [12]. For SU(2)L× U(1)Y electroweak theory, the local cohomology problem in 4 + 2-dimensions has be solved in infinite volume lattice and the exact cancellation of gauge anomalies, including the mixed type, has been shown non-perturbatively [13]. 2 The author refers the reader to [18] for recent review of the overlap formalism. In the overlap formalism, reflecting chiral anomaly, the phase factor of the chiral determinant is not fixed in general and any reasonable choice of the phase factor should lead to the gauge anomaly for single Weyl fermion. The Wigner-Brillouin phase convention has been adopted for perturbative studies [19] and has also been tested numerically in a non-perturbative formulation of chiral gauge theories [20,21]. Geometrical treatment of the gauge anomaly in the overlap formalism has been discussed in detail in abelian theories [22] and non-abelian theories [23]. The SU(2) global anomaly has been examined in [24]. An adiabatic phase choice has been proposed in [25] and used in the construction of non-compact abelian chiral gauge theories. The overlap formalism in odd dimensions has been considered in [26,16,27]. mensional lattice framework of domain wall fermion.
For this purpose, we adopt the domain wall fermion in the vector-like formalism by Shamir [30]. But two different four-dimensional gauge fields are introduced at the boundaries and they are interpolated by a five-dimensional gauge field. This inevitably causes the five-dimensional dependence of the partition function of the domain wall fermion. To take account of this five-dimensional dependence, we formulate an integrability condition. It turns out that the dependence is governed by the lattice Chern-Simons term induced from the five-dimensional Wilson-Dirac fermion (with a negative mass) [31].
In order to compensate the five-dimensional dependence, we require a five-dimensional counter term. The counter term should be given by a smooth, local and gauge invariant functional of gauge field, in order to satisfy the requirement of the smoothness, the locality and the gauge invariance of the low energy effective action. We will argue that such local, gauge-invariant field can be obtained in anomaly-free chiral gauge theories, through the local cohomology problem in 5 + 1-dimensional space formulated with the lattice Chern-Simons current. Thus the reduction from the five-dimensional lattice to four-dimensional lattice is acheived in a local and gauge invariant manner.
The locality of the lattice Chern-Simons current is essential for the cohomological argument in 5 + 1-dimensional space and for this we require the so-called admissibility condition [32,5,3,4] extended to five-dimensional gauge fields (cf. [34]). With this condition, several properties of the Chern-Simons current are discussed. The earlier studies of the properties of the Chern-Simons current in the context of domain wall fermion can be found in [35,36].
Trying to formulate four-dimensional chiral gauge theories from the fivedimensional framework of domain wall fermion, our approach resembles to the wave-guide model [37,38] and the formalism proposed by Creutz et al. [39,40]. However, our approach is different from the wave-guide model in that we are considering the smooth (discrete, but smooth in lattice scale) interpolation in the fifth direction. The issue related to the disordered gauge degrees of freedom is taken account by the five-dimensional admissibility condition, which assures the existence of the chiral zero modes even with five-dimensional gauge fields (cf. [41,42,43]). Our approach is also different from the formalism by Creutz et al. in that we are isolating the chiral zero modes at one boundary as physical degrees of freedom, regarding the other boundary as reference.
This paper is organized as follows. In section 2 we formulate the domain wall fermion for chiral gauge theories with the interpolating five-dimensional gauge field. Then we derive the integrability condition for the partition function of the domain wall fermion and state a sufficient condition to obtain the five-dimensional counter term with the required properties. In section 3 we examine the properties of the lattice Chern-Simons current. In section 4 we argue how to reconstruct the counter term from the Chern-Simons current and formulate the local cohomology problem in 5+1 dimensional space. Section 5 is devoted to the discussions on the connection to the gauge-invariant construction based on the Ginsparg-Wilson relation.
2 Domain wall fermion for chiral gauge theories

Interpolation with five-dimensional gauge field
Domain wall fermion, in its simpler vector-like formulation, is defined by the five-dimensional Wilson-Dirac fermion with a negative mass in a finite extent fifth dimension. (See Figure 1.) The four-dimensional lattice spacing a and the five-dimensional one a 5 are both set to unity. The fifth coordinate is denoted by t and takes integer values in the interval, t ∈ [−N + 1, N ]. In four dimensions, the lattice is assumed to have a finite volume L 4 and the periodic boundary condition is assumed for both fermion and gauge fields. Mass term is set to the negative value −m 0 where 0 < m 0 < 2.
This setup is equivalent to impose the Dirichlet boundary condition at the boundaries in the fifth dimension as In order to introduce chiral-asymmetric gauge interaction for the chiral zero modes at the two boundaries t = −N + 1 and N , the gauge field is assumed to be five-dimensional, where µ = 1, · · · , 5 and k = 1, · · · , 4. It is regarded to be interpolating a four-dimensional gauge field at t = −N + 1, say U 0 k (x), to another fourdimensional gauge field at t = N , say U 1 k (x). We assume that outside the finite interpolation region t ∈ [−∆, ∆](∆ < N ) the gauge field does not  Figure 2). ∆ should be chosen large enough in order to make sure that the interpolation is smooth enough. The precise condition for this will be discussed below. 0 r r r r r r r r r r r r r r r r r r The gauge fields at the boundaries, U 0 k (x) and U 1 k (x), are chosen so that their field strengths are small enough and satisfy the following bound where P kl (x) is the four-dimensional plaquette variable. This is the socalled admissibility condition [32,5,3,4]. It assures that the Hamiltonian defined through the transfer matrix of the five-dimensional Wilson-Dirac fermion, H = − ln T , has a finite gap and the overlap Dirac operator (the effective four-dimensional Dirac operator of the boundary chiral modes) is local within the exponentially suppressed tail [32,33,34]. This makes the limit that the size of the fifth dimension goes to infinity, N → ∞, welldefined. Furthermore, we also assume that the whole five-dimensional gauge field U µ (x, t), interpolating between U 0 k (x) and U 1 k (x), is smooth enough and satisfies the five-dimensional bound on the field strength as follows: This condition, as we will see below, assures that the Chern-Simons current induced from the five-dimensional Wilson-Dirac fermion is a local functional of the gauge field. The locality of the Chern-Simons current allows us the cohomological analyis of the gauge anomaly in the context of domain wall fermion. Since the difference of the four-dimensional gauge fields at the boundaries is estimated as with the five dimensional bound on the field strength, we can make the interpolation smooth enough by taking ∆ large enough. We also note a symmetry property of the five-dimensional Wilson-Dirac fermion. By the reflection of the gauge field in the fifth direction, the five-dimensional Wilson-Dirac operator transforms as follows: where P is the parity transformation operator in the fifth dimension: Therefore the interpolation in the reversed order implies the complex conjugation.

Integrability condition for domain wall fermion
Now we consider the partition function of the domain wall fermion, and examine its dependence on the path of the interpolation. Let us denote by c 1 the original choice of the path of the interpolation, which is represented by the five-dimensional gauge field U µ (x, t). In order to examine the dependence on the path, we introduce another path, say c 2 and consider the difference of the logarithm of the partition function: Let us assume first that the five-dimensional gauge field representing the path c 2 can be deformed to that representing the path c 1 , while satisfying the constraint on the five-dimensional plaquette variables Eq. (2.5). Whether this is always possible for any two paths depends on the topological structure of the space of the admissible gauge fields in consideration. Since the two paths can be regareded to form a loop in the space of the gauge fields (Fig. 3), the above condition is equivalent to whether any loops in the space of the gauge fields can be contractible to the point, or not. The case with the non-contractible loops will be discussed later. We then parameterize the smooth deformation of the path by the parameter By differentiating the partition function with respect to the parameter s and integrating back, we obtain an expression for the difference of the partition function, Dir.
where η a µ (z, s)T a = ∂ s U s µ (z) U s µ (z) −1 and J a µ (z)| Dir. is given by Since the variation of the gauge field η a µ (z, s) is restricted in the interpolation region t ∈ [−∆ + 1, ∆], the summation over t in the above expression is restricted in the same finite region.
Because of this fact, as will be shown in detail in appendix D, the difference Eq. (2.12) is well-defined and finite in the limit N → ∞, as long as the gauge fields at the boundaries, U 0 k and U 1 k , are topologically trivial and do not cause any accidental zero modes of the low energy effective Dirac operator. In particular, the inverse five-dimensional Wilson-Dirac operator in Eq. (2.13), satisfying the Dirichlet boundary condition, can be replaced by the inverse five-dimensional Wilson-Dirac operator defined in the infinite extent of the fifth dimension, where t, t ′ ∈ [−∆ + 1, ∆]. This implies that the path-dependence of the partition function of the domain wall fermion actually does not depend on the specific choice of the Dirichlet boundary condition, which supports the chiral zero modes at the boundaries. In order to make this point clear and to formulate an integrability condition for the domain wall fermion, we introduce the five-dimensional Wilson-Dirac fermion defined in the interval t ∈ [−N + 1, 3N ] in the fifth dimension and with the anti-periodic boundary condition. The five-dimensional gauge field which couples to this Wilson-Dirac fermion is assumed to form a loop in the space of the gauge field so that it goes along the path c 1 (or c 2 ) and comes back along a certain path c 0 (Fig. 4). Then we can infer that, in the limit N → ∞, Dir.
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 4: Five-dimensional gauge field representing the loop In the r.h.s. of this identity, we may choose any paths for c 0 . By choosing c 2 and c 1 , respectively, we can further obtain two identities as follows: (2.17) By averaging these two expressions and using the property of the fivedimensional Wilson-Dirac operator under the reflection t → −t + 1, we finally obtain This identity may be regarded as the integrability condition for the domain wall fermion. Several comments about this result are in order. First of all, in the r.h.s. of this identity, the partition functions are defined with the anti-periodic boundary condition and it is clear that there is no contribution from the low-lying chiral modes. They are defined well for any topological sector of U 0 k and U 1 k . Secondly, the first and second terms in the r.h.s. are turned out to be real by the reflection property and they depend on the single paths c 2 and c 1 , respectively. These terms then can be used to subtract the bulk contribution in the partition function of the domain wall fermion as Thirdly, an important observation is that the imaginary part of the r.h.s. is the complex phase induced from the five-dimensional Wilson-Dirac fermion with a negative mass, which has been known to reproduce the Chern-Simons term in the classical continuum limit [31]. We denote this lattice Chern-Simons term associated with the loop c 2 + (−c 1 ) as Q (2.20)

A sufficient condition for the path-independence of the domain wall fermion
Now we discuss the requirement to obtain the partition function of the domain wall fermion which does not depend on the path of the interpolation and which is gauge invariant. A sufficient condition for this can be stated as follows: It is possible to construct a local counter term which makes the partition function of the domain wall fermion independent of the path of the interpolation in a gauge-invariant manner, if the lattice Chern-Simons term in the r.h.s. of the integrability condition Eq.(2.18) can be expressed in the form where q 5w (z) is a smooth, local, and gauge-invariant functional of the fivedimensional gauge field, for all possible loops in the space of gauge fields.
We first assume that this is the case and discuss the consequences of this condition. In the next sections, we will discuss how to obtain the local and gauge-invariant functional q 5w (z).

Local counter term
The immediate consequence of the localization property of q 5w (z) is that Q can be decomposed into two parts C c 2 5w and C (−c 1 ) 5w = −C c 1 5w , which are associated with the paths c 1 and c 2 , respectively, and do not depend on other paths. Namely, Q Then from the integrability condition Eq. (2.18) we can infer that in the (2.25) This holds for any two paths. Therefore the (subtracted) partition function of the domain wall fermion plus the local counter term does not depend on the path of the interpolation and is determined uniquely by the fourdimensional gauge fields, U 0 k and U 1 k . Thus the reduction from the fivedimensional lattice to the four-dimensional lattice is achieved.
We will see in section 5 that this result holds for all topological sectors, through the more direct calculation of the partition function of the domain wall fermion.

Gauge invariance
Let us examine the gauge transformation property of the (subtracted) partition function of the domain wall fermion with the local counter term under the gauge transformation for U 1 k , Since the partition function does not depend on the interpolation between g U 1 and U 0 , one may choose any interpolation path between them. In this case, it turns out to be convenient to choose the interpolation as follows: we first interpolate the gauge function as and then apply it as a five-dimensional gauge transformation to the gauge field representing the original interpolation between U 1 k and U 0 k . (2.28) In Eq. (2.25), the partition functions of the five-dimensional Wilson-Dirac fermions with both boundary conditions are invariant under such five-dimensional gauge transformation. Then the gauge transformation is given solely by the gauge variation of the counter term associated with the path c 1 .
Thus the question of the gauge invariance of the system reduces to the question of the gauge invariance of the counter term associated with the path c 1 . When q 5w (z) is a gauge invariant local field, it is gauge invariant.

Properties of lattice Chern-Simons current
In this section, we discuss properties of the Chern-Simons current which is obtained from the lattice Chern-Simons term by the variation with respect to gauge field. We will argue that the lattice Chern-Simons current is a smooth and local functional of the gauge field.

Chern-Simons current
Let us consider the smooth deformation of the five-dimensional gauge field U µ (z) representing the loop in the Chern-Simons term. Under the deformation, the variation of the lattice Chern-Simons term is given by We refer the current J a µ (z) as Chern-Simons current. Since the Chern-Simons current is defined by the variation of the Chern-Simons term which is the gauge-invariant functional of the five-dimensional gauge field, it is gauge-covariant, integrable and conserved: 3

Locality of the Chern-Simons current
Next we argue that the Chern-Simons current is a local functional of the five-dimensional gauge field, as long as the constraint on the five-dimensional plaquette variables Eq. (2.5) is full-filled. This fact follows from the following consideration. With the bound Eq. (2.5), we infer that the five-dimensional Wilson-Dirac operator is bounded from below by a positive constant [32,34], Given the positive lower and upper bounds for the five-dimensional Wilson-Dirac operator,α it follows that the inverse five-dimensional Wilson-Dirac operator in the infinite lattice decays exponentially at large distance in the five dimensions [32,34]: 3 ∂µ and ∂ * µ denote the forward and backward difference operators, respectively, while Dµ and D * µ denote the covariant counterparts, Similar exponential bound can be established for the differentiation with respect to the gauge field.
The Chern-Simons current in consideration is defined in the finite volume lattice. It can be expressed in terms of the inverse five-dimensional Wilson-Dirac operator in the infinite lattice as where n · L = ( k n k )L + n 5 (4N ). Then we see that the dependence of the Chern-Simons current on the gauge field is exponentially suppressed at large distance in the leading contribution with n = 0, while the remaining dependences on the gauge field are at most of order O(exp(−θ L/2), exp(−θ 2N )) and exponentially small. In this sense, the Chern-Simons current in the finite volume lattice can be regarded as a local functional of gauge field.

The Chern-Simons current and gauge anomaly
If we consider the variation of the Chern-Simons term under the gauge transformation, we obtain (3.14) The flow of the Chern-Simons current should be responsible for the gauge anomaly associated with the chiral zero mode at the boundaries [44]. In fact, it has been shown by Golterman, Jansen and Kaplan [35] that the asymptotic value of the fifth component of the Chern-Simons current reproduces the known result of the gauge anomaly in the classical continuum limit:

Reconstruction of the Chern-Simons term from the Chern-Simons current
In this section, we discuss how to obtain the local and gauge invariant field of the Chern-Simons term. Using the Chern-Simons current obtained in the previous section, we introduce a local topological field on 5 + 1-dimensional space and formulate a local cohomology problem. We will see that the trivial solution of the cohomology problem leads to the local field with the required properties.

Contractible loops
Let us first assume that a loop l in the space of the gauge fields can be contractible to a point. Namely, we assume that the five-dimensional gauge field U µ (z) representing the loop l can be deformed to the uniform gauge field U µ (z) = U 0 k (x) (for all t), while satisfying the constraint on the fivedimensional plaquette variables Eq. (2.5). It depends on the topological structure of the space of the admissible gauge fields in consideration, whether all possible loops are contractible or not. The case with the non-contractible loops will be discussed later.
Let us parameterize the smooth deformation of the loop l by the param- The trivial interpolation (the point) at s = 0 is denoted by l 0 . By differentiating the Chern-Simons term with respect to the parameter s and then integrating back, we obtain an expression for the Chern-Simons term as Here we took account of the fact that for the point l 0 represented by the uniform gauge field U 0 µ (x), the Chern-Simons term vanishes identically, ImTrLn (D 5w − m 0 )| AP (l 0 ) = 0, because of the reflection property of the five-dimensional Wilson-Dirac operator discussed in section 2.
The Chern-Simons term so reconstructed is now given by the sum of the local field in the curly bracket, because the Chern-Simons current is a local functional of the gauge field. We note, however, that the local field is not gauge invariant. In fact, under an infinitesimal gauge transformation, , the variation of the local field is given by The clue to obtain the gauge invariant local field is to note that the field in Eq. (4.2) may be corrected by the total divergence of a certain local current K µ (z) without affecting the Chern-Simons term: As we can see from Eq. (4.4), the question whether J a µ (z) would satisfy the above relation and how to find the local current K µ (z) defines a local cohomology problem.
This cohomology problem can be reformulated as a local cohomology problem in higher dimensions [4]. In the next subsection, we formulate the cohomology problem in five-dimensinal lattice plus one dimensional continuum space.

Cohomology problem in 5 + 1 dimensional space
In order to reformulate the local cohomology problem in five-dimensinal lattice plus one dimensional continuum space, let us introduce a gauge field on the 5 + 1-dimensional space as where A 6 = T a A a 6 and its gauge transformation property is specified as (4.6) Accordingly, the covariant derivative in the continuum dimension can be defined by We now introduce a gauge invariant and local field in the 5+1 dimensional space by This local field is topological. Namely, the summation of the field over the 5 + 1-dimensional space is invariant under the local variation of the 5 + 1-dimensional gauge field: (4.10) In fact, denoting δU µ U −1 µ = T a ζ a µ and using Eq. (3.6), it follows from the second expression of the topological field in Eq. (4.9) that (4.11) Now let us assume that this topological field is cohomologically trivial, that is, it can be written in the form q(z, s) = ∂ * µ k µ (z, s) + ∂ s k 6 (z, s), (4.12) where (k µ (z, s), k 6 (z, s)) is a local current which is gauge invariant under the 5 + 1-dimensional gauge transformation. Then we can see that k 6 (z, s) provides the desired local, gauge invariant field with the required properties. In fact, from the 5 + 1-dimensional gauge invariance and the fact that q(z, s) is a linear functional of D s U µ U µ −1 , we infer that k 6 (z, s) cannot depend on A 6 (z, s) and its s-dependence comes from that of the link variable U s µ . Then we can seť , (4.14) assuming z k 6 (z, s = 0) = 0. From the 5 + 1-dimensional gauge invariance of k µ (z, s) and the fact that q(z, s) is a linear functional of D s U µ U µ −1 , we also infer that k µ (z, s) must be a linear functional of D s U µ U µ −1 written as This in turn implies that Thus Eq. (4.4) is indeed satisfied.
In this manner, we can reconstruct the Chern-Simons term with the required properties for all contractible loops, through the cohomology problem with the topological field Eq. (4.8).

The ansatz for non-contractible loops
When the topological field Eq. (4.8) is shown to be cohomologically trivial, the remaining issue is to show the condition Eq. (2.21) for all possible noncontractible loops. One may try k 6 (z, s) as the anzatz for the Chern-Simons term.Q 5w = z k 6 (z, s = 1). (4.19) The question is then to show exp(iQ 5w ) = exp(iQ 5w ) (4.20) for all non-contractible loops. This problem requires first to figure out the topological structure of the space of the gauge fields in consideration, which is constrained by the admissibility condition. So far, the topological structure of the space of the admissible gauge field is known only for abelian gauge theories [3]. In this case, it is indeed possible to show that the topological field Eq.

Some results in the infinite four-dimensional volume
In the infinite four-dimensional volume, the cohomology problem in 5+1dimensional space defined in the previous subsections can be solved in certain cases. (cf. [5,7,10,11]) In this subsection, we describe how to construct the local counter terms for the theories in the infinite four-dimensional volume.

Abelian chiral gauge theories
In the abelian gauge theories, the lattice Chern-Simons current is a gaugeinvariant conserved current, which is a local functional of the gauge fields: With these two conditions we can directly apply the cohomological method using the Poincaré lemma on the lattice [5,7] to J µ (z), to obtain where χ νµ (z) is a gauge-invariant and local anti-symmetric tensor field and From the lattice symmetries, we infer that α µ and β µνρ vanish identically. Through explicit calculations in the weak coupling expansion, we can also verify that α µ and β µνρ vanish identically and γ = − 1 32π 2 [28]. From this result, we can reconstruct the lattice Chern-Simons term as 4 Here Aµ(z) is the vector potential which represents the original admissible (five-dim.) link variable Uµ(z) and the field strength as follows [5]: This vector potential itself is not a local functional of the original link variable Uµ(z), but its local, gauge invariant functional becomes a local functional of Uµ(z). Then, by rescaling the gauge field as A µ → e α A µ in each Weyl fermion contributions, we can see that the first term of the lattice Chern-Simons term vanishes identically. As to the second term, we may add the following total divergence term without affecting the lattice Chern-Simons term: Then we obtain the local expression of the lattice Chern-Simons term (the local counter term) which is manifestly gauge invariant:

Non-abelian chiral gauge theories in lattice perturbation theory
In the anomaly-free non-abelian chiral gauge theories, it is possible to show that the topological field Eq. (4.8) is cohomologically trivial to any orders of the lattice perturbation theory, as shown in the approach based on the Ginsparg-Wilson relation [10,11]. In particular, one can construct the local field of the Chern-Simons term directly from the lattice Chern-Simons current, to any orders of the perturbation theory.
In the lattice perturbation theory, non-abelian lattice gauge fields are treated in the expansion of the gauge coupling constantant, (4.31) Accordingly, the local field of the lattice Chern-Simons term may be assumed to have the expansion in the gauge coupling constant: Let us assume that the local fields q 5w (z) are constructed to the order l = n and consider how to construct the local field of the order l = n + 1.
Since the Chern-Simons term should produce the Chern-Simons current under the local variation of the gauge field, Then we may expand the variation of the Chern-Simons current which is subtracted by the local fields up to the order l = n: From these conditions, we can directly apply the cohomological method for the abelian gauge theories [5,7] toJ where χ a(n) νµ (z) is a local field which is invariant under the linearized gauge transformation and the global gauge transformation. When and only when n = 2, a cohomologically non-trivial term can appear aš which vanishes identically by the anomaly-free condition From this result, we can obtain the local field of order l = n + 1, which reproducesJ a(n) µ (z) under the local variation of the gauge field, aš Here we have used the fact that χ a(n) νµ (z) has the following structure in the vector potentials: The above fieldq (z) which is gauge invariant under the full non-abelian gauge transformation, while its leading term in the expansion of the gauge coupling constant remains to coincide withq (n+1) 5w (z). For this purpose, we follow the method adopted in [11]. Namely, this step can be achieved by the replacement of the field strength aš and by the replacement of the vector potentials in Eq. (4.44) as 46) where W (z, z k ) is defined as the ordered product of the link variables from z k to z along the shortest path that first goes in direction 1, then direction 2, and so on. Since it coincides with the original vector potential up to the linearized gauge transformation in the expansion of the gauge coupling constant asÂ where ω(z, z k ) is the oriented line sum of the gauge potential from z k to z along the same path as defined for W (z, z k ), the leading term of q (n+1) 5w (z) so defined actually coincides withq (n+1) 5w (z).

Connection to the lattice theory of Weyl fermion based on the Ginsparg-Wilson relation
In this section, we discuss the connection of the chiral domain wall fermion discussed so far to the gauge-invariant construction of chiral gauge theories based on the Ginsparg-Wilson relation [3,4]. We will establish an identity relating the partition function of the domain wall fermion (subtracted and with the local counter term) and the partition function of the Weyl fermion defined with the overlap Dirac operator satisfying the Ginsparg-Wilson relation.

Weyl fermion defined through the overlap Dirac operator
The lattice Dirac fermion theory defined with the Dirac operator which satisfies the Ginsparg-Wilson relation Based on this exact chiral symmetry, the lattice Weyl fermion can be defined by imposing the constraint with the chiral operatorsγ 5 and γ 5 , By introducing the orthnormal basis for the Weyl fermion, {v i (x)|i = 1, 2, · · ·} and {v k (x)|k = 1, 2, · · ·} 5 aŝ the functional measure of the Weyl fermion can be set up and the pathintegral formula of the partition function can be defined.
The choice of the basis {v i (x)|i = 1, 2, · · ·} may be different by a unitary which dependents on the gauge field in general becauseγ 5 does so. Then the measure is changed by the phase factor Accordingly, the partition function is changed by In the gauge-invariant construction of the chiral gauge theories based on the Ginsparg-Wilson relation [3,4], a general method to fix the phase of the functional measure (and the partition function) has been described so that it satisfies the requirements of the smoothness, the locality and the gauge invariance. Our target in this paper is the Weyl fermion theory which is defined with the overlap Dirac operator satisfying the Ginsparg-Wilson relation [14,15]. The overlap Dirac operator, D, is given by the explicit formula [14] Here H is chosen as the hermitian operator obtained through the transfer matrix of the five-dimensional Wilson-Dirac fermion. 6 H = − ln T. (5.12) 6 Boriçi has pointed out that the transfer matrix can be expressed by a simpler fourdimensional hermitian matrix as which leads to the same spectrum asymmetry operator, H/ √ H 2 = H/ √ H 2 [45].
In this caseγ 5 is given by the spectrum asymmetry of H, 13) and the chiral basis, {v i (x)|i = 1, 2, · · ·}, can be chosen as the eigenvectors of H belonging to the negative eigenvalues (up to the global phase choice). The partition function resulted from the path-integral formula Eq. (5.7), reproduces the overlap formula of the chiral determinant [17]: Here the phase of the chiral basis can be chosen following the gauge-invariant method of [3,4] in anomaly-free chiral gauge theories.

Gauge-invariant partition function of Weyl fermions from the domain wall fermion
We now argue the connection of the (subtracted) partition function of the domain wall fermion with the local counter term, to the partition function of the Weyl fermion given by Eq. (5.14). 7

Partition functions of the 5-dim. Wilson fermions
The partition functions of the five-dimensional Wilson fermions with the Dirichlet boundary condition and with the anti-periodic boundary condition 7 It may be worth while to recall the situation in the vector-like case. In the vectorlike theories [30], the connection between the domain wall fermion and the overlap Dirac operator is directly seen in the fact that the determinant of domain wall fermion (subject to Dirichlet boundary condition in the fifth dimension) is factorized into four-dimensional part for the low-lying massless mode and five-dimensional part for the remaining massive modes [46,47]: In the l.h.s., N is the size of the fifth dimension and the contribution of the massive modes is factorized in the determinant of the five-dimensional Wilson fermion subject to the anti-periodic boundary condition in the fifth dimension [47].
can be expressed explicitly in terms of the transfer matrix, as shown in the appendix B. The results are given as follows: where T t is the transfer matrix with U k (x, t), which explicit form is given in appendix A, and N t = det (P L + P R B t ). The subscript 0 and 1 denote the quantities with U 0 k (x) and U 1 k (x), respectively. U 5 (x, t) appears in between the product of the transfer matrices so that the five-dimensional gauge covariance is maintained. Note that for abelian gauge group, the extra phase factor, which consists of the product of U 5 (x, t) −1 , appears.
In these formula, the dominant term in T (N −∆) 0 in the limit N → ∞ is estimated as follows: (5.19) where the projection operator P 0 is introduced by . With these results, we can infer that in the limit N → ∞ (5.23)

Factorization of the partition function of the domain wall fermion
Now we introduce the chiral basis {v 0 i (x)} associated with the gauge fields U 0 k (x) as {v 0 i (x)} may be chosen as the eigenfunctions of H 0 = − ln T 0 belonging to the negative eigenvalues. Similarly, we introduce the chiral basis {v 1 i } associated with the gauge fields U 1 k (x). We also introduce the chiral basis for the anti-field {v k } as v k (x)P L =v k (x) (k = 1, 2, · · ·). (5.25) In terms of these chiral bases, the formula Eq. (5.23) of the (subtracted) partition function of the domain wall fermion with the local counter term can be rewritten further as (5.27) The first factor in the r.h.s. of Eq. (5.26) is nothing but the overlap formula, which gives the partition function of the right-handed Weyl fermion at t = N coupled to the gauge field U 1 k (x), Similarly, the third factor in the r.h.s. of Eq. (5.26) reproduces the partition function of the left-handed Weyl fermion at t = −N + 1, which couples to the gauge field U 0 k (x). On the other hand, the second factor in the r.h.s. of Eq. (5.26), which is the phase factor defined by Eq. (5.27), comes from the interpolation between the gauge fields U 0 k (x) and U 1 k (x). 8 It depends on the choices of the chiral bases {v 0 i }, {v 1 i } and the path c 1 , but the path-dependence is to be compensated by the local counter term, e −iC 5w (c 1 ) . (5.30) From these results, it is quite natural to choose the chiral basis of the Weyl fermion coupled to the gauge field U 1 k as follows: With this choice, the (subtracted) partition function of the domain wall fermion with the local counter term is factorized into two chiral determinants: (5.32) The path-independence and the gauge invariance of det (v k , Dv j ) are obvious from this identity. 9

The functional measure of the Weyl fermion from the domain wall fermion
The choice of the chiral basis in Eq. (5.31) indeed leads to the functional measure of the Weyl fermion which is independent of the path of the interpolation. To see this, we choose another path, say c 2 , then we get another basis,ṽ These two bases are related by the unitary transformation which determinant turns out to be det Q = e −iφ(c 2 ) e iC 5w (c 2 ) · e iφ(c 1 ) e −iC 5w (c 1 ) .

(5.35)
But it is not difficult to see that the phase factors along the paths c 1 and c 2 multiply to give The complex phase part of this identity can be regarded as the lattice counter part of the relation between the η-invariant and the effective action for the chiral fermions [48,49,50,51,28]. Note also that this result is a generalization of Eq. (5.16) to the case of chiral gauge theories.
, (5.36) and this identity can be regarded as the integrability condition for the phase factor exp(iφ(c i )).
Since we are assuming that the lattice Chern-Simons term can be decomposed into the two parts C c 1 5w and C −c 2 5w locally and gauge-invariantly, the determinant of the unitary transformation, det Q, turns out to be unity. This holds for any two paths. Therefore the measure does not depend on the path of the interpolation and is determined uniquely. We should note that from the integrability condition Eq. (5.36) and the expression for the (subtracted) partition function Eq. (5.26), we can directly infer that Eq. (2.25) holds and the (subtracted) partition function with the local counter term is independent on the path of the interpolation for all topological sectors.

The connection to the gauge-invariant construction based on the Ginsparg-Wilson relation
Let us now look closely at the connection to the gauge-invariant construction by Lüscher [3,4]. We note first that our result Eq. (5.32) should be compared with the equation Eq. (7.1) in [4] 10 : where Q 1 is the evolution operator of the chiral projector (5.41) Then we note the correspondence as (in the gauge U 5 (z) = 1), which implies that in the domain wall fermion the evolution of the basis of the Weyl fermion is realized by the successive multiplications of the transfer matrices, like time-development along the fifth dimension. 11 For the loop in the space of the gauge fields, this leads to the correspondence of the integrability conditions: The above correspondences become exact in the continuum limit of the fifth dimension, which should be taken as a 5 → 0 with ∆a 5 fixed. In taking the continuum limit, one may smooth the interpolation further by replacing T t with T Nt t and take the limit N t → ∞ first as an intermediate step, to get Then, using P t = Q t P 0 Q t −1 , we can see that this becomes identical to from the topological field in 4 + 2-dimensional space. W corresponds to the local counter term exp(iC 5w (c 1 )) in the domain wall fermion, where the local field k 6 (x, t) is obtained from the topological field in 5 + 1-dimensional space: (5.45) In this respect, it is possible to see that the topological field in 5 + 1dimensional space reduces to the topological field in 4 + 2-dimensional space introduced by Lüscher [4], in the continuum limit. For this, we recall that the 5 + 1-dimensional topological field is defined from the local variation of the Chern-Simons term. From Eq. (5.44) and Eq. (5.36) the Chern-Simons term can be expressed as follows, in the same intermediate step as above: In this equation, one may express each P t using the chiral basis as P t =  (5.47) and evaluate the variation of the Chern-Simons term, to obtain (5.48) In the limit ∆s → 0 and a 5 → 0, this variation reduces to iTrP [∂ s P, ∂ 5 P ] a 5 ∆s. (5.49) In the generic gauge U 5 (z) = 1, this result reads iTr {P [∂ s P, D 5 P ] + (∂ s A 5 ) P } a 5 ∆s, (5.50) and if we replace ∂ s to the covariant derivative D s , it exactly coincides with the 4 + 2-dimensional topological field [4].
Finally, we note that the gauge anamaly obtained from the asymptotic value of the Chern-Simons current in the domain wall fermion is related to the gauge anomaly expressed by the overlap Dirac operator D [52]: To see this, we infer from the locality property of the Chern-Simons current that the fifth component at becomes independent of the interpolation in the limit N → ∞, depending only on the asymptotic value of the gauge field at t = N , U 1 k (x). Then using the formula of the overlap Dirac operator expressed in terms of the inverse five-dimensional Wilson-Dirac operator [52,34] 12 given by we obtain Eqs. (5.51) and (5.52). The gauge anomalies Eqs. (5.51) and (5.52) can be evaluated in the classical continuum limit as in [53,28] (see also [54,55,56,57]) and the earlier calculation [35] is reproduced.

Conclusion
The introduction of chirally asymmetric gauge-couplings to the chiral zero modes of domain wall fermion, as the original proposal by Kaplan, inevitably makes the system five-dimensional. We have shown that the five-dimensional dependence can be compensated by the local and gauge-invariant counter term in anomaly-free chiral gauge theories. The chiral structure of the dimensionally reduced low energy effective action of the chiral zero modes can be understood again by the Ginsparg-Wilson relation. In fact, it provides a concrete example of the gaugeinvariant construction of the chiral gauge theories based on the Ginsparg-Wilson relation, where the continuous interpolation in the space of the gauge fields is partly replaced by the discrete step-wise interpolation. Hope is that such discrete treatment of the interpolation of the gauge fields would be useful for a practical implementation of the gauge-invariant lattice chiral gauge theories.
We note that in the gauge-invariant construction with the domain wall fermion, the Ginsparg-Wilson relation is not used explicitly, in sharp contrast to the invariant construction by Lüscher based on the Ginsparg-Wilson relation. This is because one of the points of the invariant construction is the formulation of the integrability condition in the space of gauge fields and the idea behind it is generic. The local cohomology problem follows from the integrability condition, as long as the requirement of locality is fulfilled.
In this respect, we also note that our construction is applicable to any domain wall fermion theory defined with a proper five-dimensional Dirac operator of the structure D 5 (−m 0 ) = (D 4 (−m 0 ) + 1) δ ts − P L δ t+1,s − P R δ t,s+1 , (6.1) Such a five-dimensional fermion theory can lead to a certain four-dimensional lattice Dirac operator satisfying the Ginsparg-Wilson relation [59,34,45] 13 . In this sense, our construction partly shares the general applicability with the gauge-invariant construction based on the Ginsparg-Wilson relation. 14 In the original proposal by Kaplan [29], the dynamical treatment of the five-dimensional gauge field was also intended. The question of this ambitious attempt is still open.
where C and B are two by two matrices in the spinor space which define the four-dimensional Wilson-Dirac operator D w as Explicitly, they are given as follows: For the gauge field satisfying the admissibility condition Eq. (2.4), the Hamiltonian defined through the transfer matrix has a finite gap [30,32,33].
forward to construct the five-dimensional overlap Dirac operator [16] from such a generic five-dimensional Dirac operator.

B Evaluation of the partition functions of 5-dim. Wilson fermions
In this appendix, we describe the calculation of the functional determinant of the five-dimensional Wilson-Dirac fermion, in the cases with the antiperiodic boundary condition and Dirichlet boundary condition in the fifth dimension. Here we follow the method given by Neuberger in [46], with a slight extension to include the fifth component of the five-dimensional gauge field. More generic method has been given by Lüscher [58,59]. Let us consider the five-dimensional Wilson-Dirac operator where t, s ∈ [−T + 1, T ] for Dirichlet boundary condition and t, s ∈ [−T + 1, 3T ] for anti-periodic boundary condition. We denote the size of the fifth dimension as N in both cases. (N is an even integer.) In the chiral basis of the gamma matrices, W is written explicitly in the matrix form as follows: where X = 0, Y = 0 for Dirichlet boundary condition and X = U 5,N , Y = U −1 5,N for anti-periodic boundary condition. In order to make W almost lower tridiagonal, we first exchanges the right-and left-handed component columns for each t. Then we move the leftmost column to the place of the rightmost column.
We then introduces the following abbreviations for the blocked matrix elements.
Using these, W assumes the following form, In order to take account of the boundary element, β Y , we assume the following factorization and consider the recursion equations for the elements V t : These equations can easily be solved to get Then the determinant of W is evaluated as follows: Here we have omitted the sign factor given by (−1) q(N +1) where q = 2N c L 4 and N c is the dimension of the gauge group representation, because it turns out to be unity. The products −α −1 t β t , α −1 N α X and β −1 1 β Y are evaluated as Collecting these results, we finally obtain C The partition functions in the limit N → ∞ In this appendix, we evaluate the partition function of the five-dimensional Wilson-Dirac fermions in the limit N → ∞ and derive Eqs. (5.21) and (5.22). Here we describe the case with the anti-periodic boundary in some detail. The case with the Dirichlet boundary condition can be evaluated in the same manner. As shown in the appendix B, the partition function of the five-dimensional Wilson-Dirac fermion with the anti-periodic boundary condition is given by where Therefore, the r.h.s. of Eq. (C.2) reduces to the following expression: In the last expression the determinant is taken about the indices i and j of basis v 0 i . Moreover, the term T 2(N −∆) 1 is also evaluated as (C.9) Accordingly, the dominant contribution of the determinant in Eq.(C.7) is evaluated as

D The inverse five-dimensional Wilson-Dirac operator
In this appendix, we discuss the relation between the inverse five-dimensional Wilson-Dirac operator defined with the Dirichlet boundary condition and that defined in the infinite extent of the fifth dimension, and show that the replacement of Eq. (2.15) is allowed in the limit N → ∞.
For this purpose, we first note that the Dirichlet boundary condition can be implemented by including the surface term in the infinite volume. Namely, if we consider the five-dimensional Dirac fermion defined in the infinite extent of fifth dimension, but with the couplings between the lattice sites (−N, −N + 1) and between the lattice sites (N, N + a 5 ) omitted, then the field in the interval [−N + 1, N ] does not have any coupling to those outside the region and it is nothing but the field defined with the Dirichlet boundary condition imposed at t = −N and t = N + 1. 0 q q q q q q q q q q q q q q q q q q q q q q q q −N + 1 N U 0 k (x) U 1 k (x) Figure 5: Implementation of Dirichlet B.C. by surface interaction We denote the lattice Dirac operator for this fermion by D ∨ 5w , which can be expressed with the surface interaction as follows: Then it follows immediately that 1