The two-dimensional Nϭ2 Wess-Zumino model is constructed on the lattice through Nicolai mapping with a Ginsparg-Wilson fermion. The Nicolai mapping requires a certain would-be surface term in the bosonic action which ensures the vacuum energy cancellation even on the lattice, but inevitably breaks chiral symmetry. With the Ginsparg-Wilson fermion, the holomorphic structure of the would-be surface term is maintained, leaving a discrete subgroup of the exact chiral symmetry intact for a monomial scalar potential. Through this feature both the boson and fermion can be kept massless on the lattice without any fine-tuning.

We consider the cohomological classification of the 4+2-dimensional topological field, which is proposed by Lüscher, for SU(2) L × U(1) Y electroweak theory. The dependence on the admissible abelian gauge field of U(1) Y is determined through topological argument, with SU(2) L gauge field fixed as background. We then show the exact cancellation of the local gauge anomaly of the mixed type SU(2) L 2 × U(1) Y at finite lattice spacing, as well as U(1) Y 3 , using the pseudo reality of SU(2) L and the anomaly cancellation conditions in the electroweak theory given in terms of the hyper-charges of U(1)1 The topological aspect of the non-abelian anomaly for Weyl fermions defined based on the overlap formalism / the Ginsparg-Wilson relation has been examined by D.H. Adams in close relation to the argument of L. Alvarez-Gaumé and P. Ginsparg in the continuum theory [13,14]. The global SU(2) anomaly has been examined by H. Neuberger and O. Bär and I. Campos [15,16] in detail. A lattice implementation of the η-invariant and its relation to the effective action for chiral Ginsparg-Wilson fermions has been examined by T. Aoyama and Y.K. in [17]. Non-compact formulation of abelian chiral gauge theories has been considered recently by Neuberger [18].

In the gauge-invariant construction of abelian chiral gauge theories on the lattice based on the Ginsparg-Wilson relation, the gauge anomaly is topological and its cohomologically trivial part plays the role of the local counter term. We give a prescription to solve the local cohomology problem within a finite lattice by reformulating the Poincaré lemma so that it holds true on the finite lattice up to exponentially small corrections. We then argue that the path-integral measure of Weyl fermions can be constructed directly from the quantities defined on the finite lattice.

Based on the Ginsparg-Wilson relation, a gauge invariant formulation of electroweak SU(2)×U(1) gauge theory on the lattice is considered. If the hypercharge gauge coupling is turned off in the vacuum sector of the U(1) gauge fields, the theory consists of four left-handed SU(2) doublets and it is possible, as in vector-like theories, to make the fermion measure defined globally in all topological sectors of SU(2). We then try to incorporate U(1) gauge field, following Lüscher's reconstruction theorem. The global integrability condition is proved for "gauge loops" in the space of the U(1) gauge fields with arbitrary SU(2) gauge field fixed in the background. For "non-gauge loops", however, the proof is given so far only for the classical SU(2) instanton backgrounds.