# APS $\eta$-invariant, path integrals, and mock modularity

## TL;DR

This paper connects the Atiyah-Patodi-Singer eta-invariant to supersymmetric path integrals and mock modular forms, providing new proofs and insights into index theory and quantum modularity.

## Contribution

It introduces a novel relation between the eta-invariant, scattering theory, and mock modular forms, with a new proof of the APS theorem via localization of path integrals.

## Key findings

- Relates eta-invariant to temperature-dependent Witten index.
- Provides a new proof of the APS theorem using scattering theory.
- Connects eta-invariant of elliptic genus to quantum modular forms.

## Abstract

We show that the Atiyah-Patodi-Singer $\eta$-invariant can be related to the temperature dependent Witten index of a noncompact theory and give a new proof of the APS theorem using scattering theory. We relate the $\eta$-invariant to a Callias index and compute it using localization of a supersymmetric path integral. We show that the $\eta$-invariant for the elliptic genus of a finite cigar is related to quantum modular forms obtained from the completion of a mock Jacobi form which we compute from the noncompact path integral.