APS $η$-invariant, path integrals, and mock modularity

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.

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