  The Ginsparg-Wilson relation, if written in a suitable form, can be used as a
condition for lattice Dirac operators of massless fermions also in odd
dimensions. The fermion action with such a Dirac operator is invariant under a
generalized parity transformation, which reduces to the ordinary parity
transformation in the (naive) continuum limit. The fermion measure, however,
transforms non-trivially under the generalized parity transformation, and hence
the parity anomaly arises solely from the fermion measure. The analogy to the
lattice construction of chiral gauge theories in even dimensions is clarified
by considering a dimensional reduction. We also propose a natural definition of
a lattice Chern-Simons term, which is consistent with odd dimensional
Ginsparg-Wilson fermions.

High Energy Physics - Lattice