APS η-invariant, path integrals, and mock modularity
Atish Dabholkar (CNRS, UMR 7589, LPTHE, Paris, F-75005, France, Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, 34151, Italy, Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Énergies, LPTHE, Paris, F-75005, France); Diksha Jain (Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, Trieste, 34136, Italy, Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, 34151, Italy); Arnab Rudra (Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, 34151, Italy)
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.
