Spinors on manifolds with boundary: APS index theorems with torsion

TL;DR

This paper extends the Atiyah-Patodi-Singer index theorem to manifolds with boundary and torsion, analyzing boundary corrections and bulk contributions, and verifies results through multiple methods including supersymmetric quantum mechanics.

Contribution

It introduces a new approach to index theorems with torsion, modifies boundary and bulk terms, and confirms results via duality and explicit examples.

Findings

Modified boundary Chern-Simons correction and eta invariant with torsion

Agreement with heat kernel and Pauli-Villars techniques

Explicit computation of indices for Taub-NUT and dual manifolds

Abstract

Index theorems for the Dirac operator allow one to study spinors on manifolds with boundary and torsion. We analyse the modifications of the boundary Chern-Simons correction and APS eta invariant in the presence of torsion. The bulk contribution must also be modified and is computed using a supersymmetric quantum mechanics representation. Here we find agreement with existing results which employed heat kernel and Pauli-Villars techniques. Nonetheless, this computation also provides a stringent check of the Feynman rules of de Boer et al. for the computation of quantum mechanical path integrals. Our results can be verified via a duality relation between manifolds admitting a Killing-Yano tensor and manifolds with torsion. As an explicit example, we compute the indices of Taub-NUT and its dual constructed using this method and find agreement for any finite radius to the boundary. We also suggest a resolution to the problematic appearance of the Nieh-Yan invariant multiplied by the regulator mass^2 in computations of the chiral gravitational anomaly coupled to torsion.