APS η-invariant, path integrals, and mock modularity

Abstract

Abstract We show that the Atiyah-Patodi-Singer η-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 η-invariant to a Callias index and compute it using localization of a supersymmetric path integral. We show that the η-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.

Keywords

• Anomalies in Field and String Theories
• Differential and Algebraic Geometry
• Scattering Amplitudes
• Sigma Models