Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator

TL;DR

This paper reformulates the Atiyah-Patodi-Singer index theorem in a physics-friendly manner for domain-wall fermions, clarifying its connection to physical boundary conditions and anomalies in topological materials.

Contribution

It provides a new, more accessible derivation of the APS index theorem tailored for physicists using domain-wall fermions in flat space.

Findings

APS index embedded in domain-wall fermion determinant

Clarifies the relation between APS boundary conditions and physical boundaries

Links the index to anomaly descent equations

Abstract

Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.