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.

arXiv comment44 pages, 5 figues

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Cited by in corpus (7)

• General anomaly matching by Goldstone bosons
• Duality and Mock Modularity
• Quantum modularity of partial theta series with periodic coefficients
• Atiyah-Patodi-Singer index theorem from axial anomaly
• Worldline approach for spinor fields in manifolds with boundaries
• Three Avatars of Mock Modularity
• Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase