Chiral symmetry and Atiyah-Patodi-Singer index theorem for staggered fermions

TL;DR

This paper explores the Atiyah-Patodi-Singer index theorem for staggered fermions, specifically Kähler-Dirac fermions, revealing two types of chiral symmetry with different boundary conditions and implications for lattice gauge theories.

Contribution

It formulates APS index theorems for two chiral symmetries of Kähler-Dirac fermions on manifolds with boundary, clarifying their behavior and boundary conditions.

Findings

Two notions of chiral symmetry with mixed anomalies identified.

APS index theorems derived for each symmetry on manifolds with boundary.

A fundamental difference in boundary conditions for the two symmetries established.

Abstract

We consider the Atiyah-Patodi-Singer (APS) index theorem corresponding to the chiral symmetry of a continuum formulation of staggered fermions called K\"ahler-Dirac fermions, which have been recently investigated as an ingredient in lattice constructions of chiral gauge theories. We point out that there are two notions of chiral symmetry for K\"ahler-Dirac fermions, both having a mixed perturbative anomaly with gravity leading to index theorems on closed manifolds. By formulating these theories on a manifold with boundary, we find the APS index theorems corresponding to each of these symmetries, necessary for a complete picture of anomaly inflow, using a recently discovered physics-motivated proof. We comment on a fundamental difference between the nature of these two symmetries by showing that a sensible local, symmetric boundary condition only exists for one of the two symmetries. This sheds light on how these symmetries behave under lattice discretization, and in particular on their use for recent symmetric-mass generation proposals.