A lattice version of the Atiyah-Singer index theorem

TL;DR

This paper develops a lattice-based formulation of the Atiyah-Singer index theorem, providing a $K$-theoretic index formula for operators on lattice approximations of manifolds, with applications to lattice gauge theory.

Contribution

It introduces a novel lattice version of the Atiyah-Singer index theorem and derives a $K$-theoretic formula applicable to lattice operators.

Findings

Established a lattice index theorem for operators on affine manifolds.

Derived a $K$-theoretic index formula for lattice operators.

Applied the theorem to Wilson-Dirac operators in gauge theory.

Abstract

We formulate and prove a lattice version of the Atiyah-Singer index theorem. The main theorem gives a $K$-theoretic formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds. We apply the main theorem to an index problem of Wilson-Dirac operators in lattice gauge theory.