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arXiv · 2407.17708

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

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

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Shoto Aoki, Hidenori Fukaya, Mikio Furuta, Shinichiroh Matsuo, Tetsuya Onogi, Satoshi Yamaguchi. 2024-07-25. The index of lattice Dirac operators and $K$-theory. https://arxiv.org/abs/2407.17708

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