Overlap lattice fermion in a gravitational field

TL;DR

This paper develops a lattice Dirac operator of overlap type that incorporates gravitational fields, maintaining local Lorentz symmetry and reproducing continuum topological properties, thus advancing lattice formulations of fermions in curved spacetime.

Contribution

It introduces a doubler-free overlap lattice Dirac operator that respects local Lorentz symmetry and reproduces continuum index theorems in gravitational backgrounds.

Findings

The lattice index theorem holds with gravitational fields.

The continuum limit reproduces the Dirac genus.

Reduction to Majorana fermions depends on spacetime dimension.

Abstract

We construct a lattice Dirac operator of overlap type that describes the propagation of a Dirac fermion in an external gravitational field. The local Lorentz symmetry is manifestly realized as a lattice gauge symmetry, while it is believed that the general coordinate invariance is restored only in the continuum limit. Our doubler-free Dirac operator satisfies the conventional Ginsparg-Wilson relation and possesses gamma_5 hermiticity with respect to the inner product, which is suggested by 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, which is consistent with the existence of the global gravitational/gauge anomalies in 8k dimensions. Other Lorentz representations, such as the spinor-vector and the bi-spinor representations, can also be treated. Matter fields with a definite chirality (with respect to the lattice-modified chiral matrix) are briefly considered.