Generalization of lattice Dirac operator index

TL;DR

This paper develops a comprehensive lattice formulation for various Dirac operator indices using K-theory, extending applicability to manifolds with boundary, curved boundaries, and gravitational backgrounds, with mathematical proofs and numerical evidence.

Contribution

It introduces a unified lattice framework for Dirac indices applicable to manifolds with boundaries and gravitational effects, surpassing previous flat-torus limitations.

Findings

Mathematical proof of the formulation

Numerical evidence supporting the approach

Extension of the mod-2 index to various dimensions

Abstract

We provide a comprehensive lattice formulation of various types of the Dirac operator indices, employing $K$-theory to classify the Wilson Dirac operator via its spectral flow. In contrast to the index of the overlap Dirac operator defined through the Ginsparg-Wilson relation, which is restricted to flat tori in even dimensions, our formulation offers several key advantages: 1) It can be applied straightforwardly to the Atiyah-Patodi-Singer index for manifolds with boundary. 2) The boundary can be curved, allowing for the inclusion of gravitational background effects. 3) The mod-2 index in both even and odd dimensions can be defined as a natural extension of the same formulation. In this talk, we present the mathematical proof and provide numerical evidence supporting the formulation.