The index theorem of lattice Wilson--Dirac operators via higher index theory

TL;DR

This paper proves an index theorem linking lattice Wilson--Dirac operators to twisted Dirac operators on a torus, using higher index theory of almost flat vector bundles, advancing mathematical understanding of lattice gauge theories.

Contribution

It introduces a novel proof of the index theorem for lattice Wilson--Dirac operators utilizing higher index theory, connecting lattice and continuum formulations.

Findings

Established the index theorem for lattice Wilson--Dirac operators.

Connected lattice operators with twisted Dirac operators on a torus.

Applied higher index theory to lattice gauge theory context.

Abstract

We give a proof of the index theorem of lattice Wilson--Dirac operators, which states that the index of a twisted Dirac operator on the standard torus is described in terms of the corresponding lattice Wilson--Dirac operator. Our proof is based on the higher index theory of almost flat vector bundles.