Overlap lattice fermion in a gravitational field

We construct a lattice Dirac operator of the overlap-type which describes the propagation of the Dirac fermion in an external gravitational field. The local Lorentz symmetry is manifestly realized as a lattice gauge symmetry, while the general coordinate invariance is expected to be restored only in the continuum limit. Our doubler-free Dirac operator obeys the conventional Ginsparg-Wilson relation and possesses a γ5 hermiticity with respect to the inner product which is suggested from the general coordinate invariance. The lattice index theorem in the presence of a gravitational field holds and the classical continuum limit of the index density reproduces the Dirac genus. Reduction to a single Majorana fermion is possible for 8k + 2 and 8k + 4 dimensions, but not for 8k dimensions as being consistent with the existence of the global gravitational/gauge anomalies in the latter dimensions. Other Lorentz representations, such as the spinor-vector and the bispinor representations, can also be treated. Matter fields with a definite chirality (with respect to the lattice-modified chiral matrix) is briefly considered.