The index of lattice Dirac operators and $K$-theory

TL;DR

This paper establishes a mathematical equivalence between the index of continuum Dirac operators on a torus and the eta invariant of lattice Dirac operators, using K-theory without requiring modified chiral symmetry.

Contribution

It introduces a K-theoretic framework to relate continuum and lattice Dirac operators, proving their indices are equal at finite lattice spacing without modified chiral symmetry.

Findings

Proves the index equality between continuum and lattice Dirac operators.

Shows the Wilson Dirac operator's eta invariant matches the continuum index.

Demonstrates the K-theoretic classification of Dirac operators at finite lattice spacing.

Abstract

We mathematically show an equality between the index of a Dirac operator on a flat continuum torus and the $\eta$ invariant of a lattice Dirac operator known as the Wilson Dirac operator with a negative mass when the lattice spacing is sufficiently small. Unlike the standard approach, our formulation using $K$-theory does not require modified chiral symmetry on the lattice. We prove that a one-parameter family of continuum massive Dirac operators and the corresponding Wilson Dirac operators belong to the same equivalence class of the $K^1$ group at a finite lattice spacing. Their indices, which are evaluated by the spectral flow or equivalently by the $\eta$ invariant at a finite mass, are proved to be equal.