Axial Anomaly in Lattice Abelian Gauge Theory in Arbitrary Dimensions

Axial anomaly of lattice abelian gauge theory in hyper-cubic regular lattice in arbitrary even dimensions is investigated by applying the method of exterior differential calculus. The topological invariance, gauge invariance and locality of the axial anomaly determine the explicit form of the topological part. The anomaly is obtained up to a multiplicative constant for finite lattice spacing and can be interpreted as the Chern character of the abelian lattice gauge theory. ∗On leave of absence from Institute of Theoretical Physics, Academia Sinica, P.O.Box 2735, Beijing 100080, China Recent discovery of lattice Dirac operator [1, 2, 3] satisfying Ginsperg-Wilson (GW) relation [4] and implementing exact chiral symmetry [5] have opened up new possibility of understanding nonperturbative behaviors of chiral gauge theories on the lattice. The axial anomaly arises as the nontrivial Jacobian factor of the fermion measure [6] under the chiral transformations and is related to the index of the Dirac operator [2, 5]. Perturbative evaluation of the axial anomaly in the continuum limit was carried out in ref. [7] by using the overlap Dirac operator [3] and the axial anomaly of the contimuum theories were reproduced. See also refs. [8, 9]. Such explicit analysis becomes rather involved by its own right. However, it is plausable that the axial anomaly on the lattice is also related to some topological object as in continuum theory and its structure can be determined by invoking the method of differential geometry on the discrete lattice. In fact it was argued in ref. [10] that the topological invariance of the index of the GW Dirac operator and the gauge invariance almost fix the form of the axial anomaly, and the topologcal part of the anomaly is obtained up to a multiplicative constant for finite lattice spacing in four dimensions. In this note we investigate the axial anomaly of lattice abelian gauge theory in euclidean hyper-cubic regular lattice in arbitrary even dimensions by applying the method of exterior differential calculus. We find that the topological invariance, gauge invariance and locality of the axial anomaly also determine the explicit form of the topological part in arbitrary dimensions. The axial anomaly is a natural extension of the result obtained in ref. [10] and has characteristic structure of products of field strengths contracted with the Levi-Civita symbol but each argument of the field strengths is shifted so that the axial anomaly acquires topological nature. We argue that such shifts in the arguments can be naturally understood within the framework of noncommutative differential calculus [11] and the axial anomaly is indeed the Chern character of abelian gauge theory on the discrete lattice. Let us begin with some basic definitions in noncommutative differential calculus [11] on the hyper-cubic regular lattice Z of unit lattice spacing. See also ref. [10]. We denote the generators of exterior differential algebra by dxμ (μ = 1, · · · , D). They satisfy dxμdxν = −dxνdxμ , f(x)dxμ = dxμf(x− μ̂) , (1) where f(x) is an arbitrary function on Z. In noncommutative differential calculus dxμ does not commute with the coordinates xμ as in ordinary differential calculus. Instead, it generates a shift of the coordinates in the direction indicated by μ̂. Differential k-forms on Z can be defined by f = 1 k! fμ1···μk(x)dxμ1 · · ·dxμk , (2) where fμ1···μk(x) is completely antisymmetric in μ1, · · · , μk. The vector space of k-forms on Z is denoted by Ωk. 2 On the discrete lattice one can introduce two kind of difference schemes, the forward and backward difference operators ∂μ and ∂ ∗ μ defined by ∂μf(x) = f(x+ μ̂)− f(x) , ∂ ∗ μf(x) = f(x)− f(x− μ̂). (3) Exterior differential operator d : Ωk → Ωk+1 on forms can be defined by the forward difference operator as df = 1 k! ∂μfμ1···μk(x)dxμdxμ1 · · ·dxμk . (4) Since two successive differences commute ∂μ∂ν = ∂ν∂μ, the exetrior differential operator satisfies nilpotency relation d = 0. This enables us to define closed forms and exact forms as in ordinary exterior calculus. The remarkable property of noncommutative differential calculus is that the Leibniz rule for the ordinary exterior differential calculus holds true because of the second property in (1). Let f and g be kand l-forms, then one easily finds d(f(x)g(x)) = df(x)g(x) + (−1)f(x)dg(x) . (5) We shall use another kind of exterior differential operator, the divergence operator d : Ωk → Ωk−1, defined by df = 1 (k − 1)! ∂ μfμμ2···μk(x)dxμ2 · · ·dxμk . (6) It also satisfies nilpotency d = 0. The nilpotency of the exterior difference operator naturally leads to an analog of Poincaré lemma. We quote it here from ref. [10]: Lemma 1 Let f be a closed k-form on Z with compact support and ∑ x f(x) = 0 for k = D, then there exists a (k − 1)-form g such that f = dg. Since d is also nilpotent, one can state Poincaré lemma in the following form: Lemma 2 Let f be a k-form on Z with compact support satisfying df = 0 and ∑ x f(x) = 0 for k = 0, then there exists a (k + 1)-form g such that f = dg. Let us introduce another copy of the lattice Z and the generators of exterior algebra dyμ (μ = 1, · · · , D) satisfying dxμdyν = −dyνdxμ , dyμdyν = −dyνdyμ , f(x, y)dxμ = dxμf(x− μ̂, y) , f(x, y)dyμ = dyμf(x, y − μ̂) , (7)