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$η$ invariant of massive Wilson Dirac operator and the index

Shoto Aoki, Hidenori Fukaya, Mikio Furuta, Shinichiroh Matsuo, Tetsuya Onogi, Satoshi Yamaguchi

TL;DR

This work reframes the lattice Dirac index problem in terms of $K$-theory, showing that the index defined by the overlap operator coincides with the $η$-invariant of the massive Wilson Dirac operator due to the suspension isomorphism between $K^0(\text{point})$ and $K^{1}(I,\partial I)$. It proves that for sufficiently small lattice spacing, the $η$-invariant of the Wilson operator matches the continuum Dirac index, thereby making the Ginsparg-Wilson relation and exact lattice chiral symmetry unnecessary for encoding gauge-field topology. The authors establish a concrete main theorem bounding the lattice spacing below which the lattice and continuum $K^{1}$-classes coincide, and discuss extensions to domains with boundaries, real Dirac operators, and curved backgrounds via KO-theory and APS-type indices. The results broaden the toolkit for lattice gauge theory topology beyond the overlap construction, highlighting the robustness of the $K$-theoretic viewpoint and spectral-flow centrality in nonperturbative topology.

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 $η$ 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 $η$-invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this $η$-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.

$η$ invariant of massive Wilson Dirac operator and the index

TL;DR

This work reframes the lattice Dirac index problem in terms of -theory, showing that the index defined by the overlap operator coincides with the -invariant of the massive Wilson Dirac operator due to the suspension isomorphism between and . It proves that for sufficiently small lattice spacing, the -invariant of the Wilson operator matches the continuum Dirac index, thereby making the Ginsparg-Wilson relation and exact lattice chiral symmetry unnecessary for encoding gauge-field topology. The authors establish a concrete main theorem bounding the lattice spacing below which the lattice and continuum -classes coincide, and discuss extensions to domains with boundaries, real Dirac operators, and curved backgrounds via KO-theory and APS-type indices. The results broaden the toolkit for lattice gauge theory topology beyond the overlap construction, highlighting the robustness of the -theoretic viewpoint and spectral-flow centrality in nonperturbative topology.

Abstract

We revisit the lattice index theorem in the perspective of -theory. The standard definition given by the overlap Dirac operator equals to the 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 -groups. Specifically, we identify the Wilson Dirac operator as an element of the group, which is characterized by the -invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this -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.
Paper Structure (6 sections, 20 equations, 2 figures)

This paper contains 6 sections, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $K$-theory
  3. Massless Dirac vs. massive Dirac
  4. Main theorem
  5. Comparison with the overlap Dirac index
  6. Summary

Figures (2)

  • Figure 1: Eigenvalue spectrum of the massive Dirac operator $H(m)$ as a function of $m$.
  • Figure 2: Eigenvalue spectrum of the massive Dirac operator $H(m)$ with chiral symmetry breaking effect, modeling the situation with a lattice Dirac operator.