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arXiv · 2503.23921

$K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators

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

We show that the Wilson Dirac operator in lattice gauge theory can be identified as a mathematical object in $K$-theory and that its associated spectral flow is equal to the index. In comparison to the standard lattice Dirac operator index, our formulation does not require the Ginsparg-Wilson relation and has broader applicability to systems with boundaries and to the mod-two version of the indices in general dimensions. We numerically verify that the $K$ and $KO$ group formulas reproduce the known index theorems in continuum theory. We examine the Atiyah-Singer index on a flat two-dimensional torus and, for the first time, demonstrate that the Atiyah-Patodi-Singer index with nontrivial curved boundaries, as well as the mod-two versions, can be computed on a lattice.

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Shoto Aoki, Hidenori Fukaya, Mikio Furuta, Shinichiroh Matsuo, Tetsuya Onogi, Satoshi Yamaguchi. 2025-03-31. $K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators. https://doi.org/10.1093/ptep%2Fptaf087

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