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Capturing the Atiyah-Patodi-Singer index from the lattice

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

TL;DR

This work constructs a rigorous lattice realization of the Atiyah-Patodi-Singer index for Dirac operators on manifolds with boundary by coupling a lattice Wilson Dirac discretization to a domain-wall mass term and interpreting the APS index as a spectral flow. A finite-element interpolator links lattice and continuum operators, enabling a precise comparison and a proof that, for sufficiently small lattice spacing $a$, the lattice-domain-wall construction reproduces the continuum APS index on a flat torus. The authors develop a comprehensive $K$-theory framework for unbounded and bounded Dirac-type operators, establishing the equivalence of $K^{\mathrm{Riesz}}$ and $K^{\mathrm{bounded}}$ groups and a robust notion of spectral flow that captures both integer and mod-2 (KO) invariants. The result provides the first mathematically rigorous lattice formulation of the APS index, with potential extensions to symmetric and real structures, and offers a practical lattice surrogate for bulk-boundary topological phenomena in lattice gauge theories and related areas.

Abstract

Using the Wilson Dirac operator in lattice gauge theory with a domain-wall mass term, we construct a discretization of the Atiyah-Patodi-Singer index for domains with compact boundary in a flat torus. We prove that, for sufficiently small lattice spacings, this discretization correctly captures the continuum Atiyah-Patodi-Singer index.

Capturing the Atiyah-Patodi-Singer index from the lattice

TL;DR

This work constructs a rigorous lattice realization of the Atiyah-Patodi-Singer index for Dirac operators on manifolds with boundary by coupling a lattice Wilson Dirac discretization to a domain-wall mass term and interpreting the APS index as a spectral flow. A finite-element interpolator links lattice and continuum operators, enabling a precise comparison and a proof that, for sufficiently small lattice spacing , the lattice-domain-wall construction reproduces the continuum APS index on a flat torus. The authors develop a comprehensive -theory framework for unbounded and bounded Dirac-type operators, establishing the equivalence of and groups and a robust notion of spectral flow that captures both integer and mod-2 (KO) invariants. The result provides the first mathematically rigorous lattice formulation of the APS index, with potential extensions to symmetric and real structures, and offers a practical lattice surrogate for bulk-boundary topological phenomena in lattice gauge theories and related areas.

Abstract

Using the Wilson Dirac operator in lattice gauge theory with a domain-wall mass term, we construct a discretization of the Atiyah-Patodi-Singer index for domains with compact boundary in a flat torus. We prove that, for sufficiently small lattice spacings, this discretization correctly captures the continuum Atiyah-Patodi-Singer index.
Paper Structure (28 sections, 21 theorems, 151 equations, 4 figures)

This paper contains 28 sections, 21 theorems, 151 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Wilson Dirac operators on a lattice
  3. Dirac operators in continuum space
  4. Wilson Dirac operators
  5. From lattice to continuum
  6. Finite element interpolator
  7. Properties
  8. Domain-wall Dirac operators and the Atiyah-Patodi-Singer index
  9. Dirac operators on manifolds with boundaries and the APS index
  10. Domain-wall fermion Dirac operators
  11. The domain-wall Dirac operators and the APS index
  12. Domain-wall Dirac operators on a flat torus
  13. K-groups and unbounded selfadjoint operators
  14. Definitions of Kpq-cocycles
  15. Set-theoretic techniques
  16. ...and 13 more sections

Key Result

Theorem 3

holds, where $\mathrm{sf}$ denotes the spectral flow defined later in Section sec:sf.

Figures (4)

  • Figure 1: The function $\rho_a^{(1)}(t)$. The function $\bar{\rho}_a(t)$ is its restriction to $t\in [0,1]$ where the two end points are identified.
  • Figure 2: The function $\xi_{\lambda_0}(\lambda)$ normalized by $\lambda_0$.
  • Figure 3: The function $T_{\mathbb{R} \to [-1/2,1/2]}(\lambda)$.
  • Figure 4: The staple-shaped parameter region in the $t$-$s$ plane where we prove that the lattice-continuum combined Dirac operator $D^{\rm cmb}(m,t,s)$ is invertible.

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Remark 9
  • Remark 10
  • ...and 41 more