Dai-Freed theorem and topological phases of matter

TL;DR

This paper provides a physics-based derivation of Dai-Freed theorems related to the eta-invariant, connecting anomalies and topological phases of matter through the analysis of massive fermions and their ground state properties.

Contribution

It offers a novel physics derivation of mathematical theorems linking the eta-invariant to topological phases using fermion ground state wave functions.

Findings

Ground state wave function takes values in the determinant line bundle.

Nontrivial Berry phases characterize low energy topological phases.

The derivation connects anomalies with topological phases via fermion analysis.

Abstract

We describe a physics derivation of theorems due to Dai and Freed about the Atiyah-Patodi-Singer eta-invariant which is important for anomalies and topological phases of matter. This is done by studying a massive fermion. The key role is played by the wave function of the ground state in the Hilbert space of the fermion in the large mass limit. The ground state takes values in the determinant line bundle and has nontrivial Berry phases which characterize the low energy topological phases.