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

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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-02147943
Contributor : Inspire Hep <>
Submitted on : Wednesday, June 5, 2019 - 10:40:09 AM
Last modification on : Wednesday, June 12, 2019 - 10:49:10 AM

### Citation

Atish Dabholkar, Diksha Jain, Arnab Rudra. APS $\eta$-invariant, path integrals, and mock modularity. 2019. ⟨hal-02147943⟩

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