  Exact chiral symmetry at finite lattice spacing would preclude the axial
anomaly. In order to describe a continuum quantum field theory of Dirac
fermions, lattice actions with purported exact chiral symmetry must break the
flavor-singlet axial symmetry. We demonstrate that this is indeed the case by
using a minimally doubled fermion action. For simplicity we consider the
Abelian axial anomaly in two dimensions. At finite lattice spacing and with
gauge interactions, the axial anomaly arises from non-conservation of the
flavor-singlet current. Similar non-conservation also leads to the axial
anomaly in the case of the naive lattice action. For minimally doubled actions,
however, fine tuning of the action and axial current is necessary to arrive at
the anomaly. Conservation of the flavor non-singlet vector current additionally
requires the current to be fine tuned. Finally we determine that the chiral
projection of a minimally doubled fermion action can be used to arrive at a
lattice theory with an undoubled Dirac fermion possessing the correct anomaly
in the continuum limit.
