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We present a gauge-invariant and non-perturbative construction of the Glashow-Weinberg-Salam model on the lattice, based on the lattice Dirac operator satisfying the Ginsparg-Wilson relation. Our construction covers all SU(2) topological sectors with vanishing U(1) magnetic flux and would be usable for a description of the baryon number non-conservation. In infinite volume, it provides a gauge-invariant regularization of the electroweak theory to all orders of perturbation theory. First we formulate the reconstruction theorem which asserts that if there exists a set of local currents satisfying cetain properties, it is possible to reconstruct the fermion measure which depends smoothly on the gauge fields and fulfills the fundamental requirements such as locality,gauge-invariance and lattice symmetries. Then we give a closed formula of the local currents required for the reconstruction theorem. JHEP05(2008)095suggested by 't Hooft's anomaly matching condition [5] and so on. Unfortunately, very little is known so far about the actual behavior of chiral gauge theories beyond perturbation theory. It is desirable to develop a formulation to study the non-perturbative aspect of chiral gauge theories.Despite the well-known problem of the species doubling [6 -9], lattice gauge theory can now provide a framework for non-perturbative formulation of chiral gauge theories. The clue to this development is the construction of local and gauge-covariant lattice Dirac operators satisfying the Ginsparg-Wilson relation [10 -15]. By this relation, it is possible to realize an exact chiral symmetry on the lattice [16], without the species doubling problem. It is also possible to introduce Weyl fermions on the lattice and this opens the possibility to formulate anomaly-free chiral lattice gauge theories [17 -29]. In the case of U(1) chiral gauge theories, Lüscher [18] proved rigorously that it is possible to construct the fermion path-integral measure which depends smoothly on the gauge field and fulfills the fundamental requirements such as locality, gauge-invariance and lattice symmetries. Although it is believed that a chiral gauge theory is a difficult case for numerical simulations because the effective action induced by Weyl fermions has a non-zero imaginary part, it would be still interesting and even useful to develop a formulation of chiral lattice gauge theories by which one can work out fermionic observables numerically as the functions of link field with exact gauge invariance. 1 In this article, we construct the SU(2) × U(1) chiral gauge theory of the Glashow-Weinberg-Salam model on the lattice, keeping the exact gauge invariance. As in the case of U(1) theories, we first formulate the reconstruction theorem which asserts that if there exists a set of local currents satisfying cetain properties, it is possible to reconstruct the chiral fermion measure which depends smoothly on the gauge field and fulfills the fundamental requirements such as locality, 2 gauge-invariance and lattice symmetries. 3 We then give a closed expr...

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