### 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 γ_{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.

Original language | English |
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Pages (from-to) | 197-215 |

Number of pages | 19 |

Journal | Progress of Theoretical Physics |

Volume | 116 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)

### Cite this

*Progress of Theoretical Physics*,

*116*(1), 197-215. https://doi.org/10.1143/PTP.116.197