$\eta$ invariant of massive Wilson Dirac operator and the index

TL;DR

This paper explores the mathematical relationship between the $ ext{Wilson}$ Dirac operator's $ ext{eta}$ invariant and the index theorem, revealing deep $K$-theoretic insights and implications for lattice gauge theory topology.

Contribution

It identifies the Wilson Dirac operator as a $K^1$-group element linked to the $ ext{eta}$-invariant and shows this invariant equals the continuum Dirac index at small lattice spacings, challenging the necessity of Ginsparg-Wilson relations.

Findings

$ ext{eta}$-invariant equals the continuum index at small lattice spacings

Wilson Dirac operator characterized by $K$-theoretic suspension isomorphism

Ginsparg-Wilson relation not essential for gauge topology understanding

Abstract

We revisit the lattice index theorem in the perspective of $K$-theory. The standard definition given by the overlap Dirac operator equals to the $\eta$ invariant of the Wilson Dirac operator with a negative mass. This equality is not coincidental but reflects a mathematically profound significance known as the suspension isomorphism of $K$-groups. Specifically, we identify the Wilson Dirac operator as an element of the $K^1$ group, which is characterized by the $\eta$-invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this $\eta$-invariant equals to the index of the continuum Dirac operator. Our results indicate that the Ginsparg-Wilson relation and the associated exact chiral symmetry are not essential for understanding gauge field topology in lattice gauge theory.