  We point out that a fermion determinant of a chiral gauge theory on a 2D
torus has a phase ambiguity proportional to the Polyakov loops along the
boundaries, which can be reproduced by the overlap formalism. We show that the
requirement on the fermion determinant that a singularity in the gauge field
can be absorbed by a change of the boundary condition for the fermions, is not
compatible with translational invariance in general. As a consequence, the
gauge anomaly for singular gauge transformations discovered by
Narayanan-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
results to any even dimension.
