Continuum and Lattice Overlap for Chiral Fermions on the Torus

The overlap formulation is applied to calculate the chiral determinant on a two-dimensional torus with twisted boundary conditions. We evaluate first the continuum overlap, which is convergent and well-defined, and yields the correct string theory result for both the real and imaginary parts of the effective action. We then show that the lattice version of the overlap gives the continuum overlap results in the limit when the lattice spacing tends to zero, and that the subleading terms in that limit are irrelevant.