Translational anomaly in chiral gauge theories on a torus and the overlap formalism

We point out that a fermion determinant of a chiral gauge theory on a 2D to rus has a phase ambiguity proportional to the Polyakov loops along the boun daries, which can be reproduced by the overlap formalism. We show that the requirement on the fermion determinant that a singularity in the gauge fiel d can be absorbed by a change of the boundary condition for the fermions, i s not compatible with translational invariance in general. As a consequence , the gauge anomaly for singular gauge transformations discovered by Naraya nan-Neuberger actually exists in any 2D U(1) chiral gauge theory unless the theory is vector-like. We argue that the gauge anomaly is peculiar to the overlap formalism with the Wigner-Brillouin phase choice and that it is not necessarily a property required in the continuum. We also generalize our r esults to any even dimension.