APS index theorem, Path integral and mock-modularity

APS index theorem is an extension of Atiyah-Singer index theorem for cases of the manifolds with boundary. We will present an introduction to the index theorem in general and then an introduction to the APS index theorem. And after that, we will present a physicist\'s derivation of APS index theorem. We will also show that the Atiyah-Patodi-Singer η-invariant can be related to the temperature-dependent Witten index of a noncompact theory. In the end, we will present some observations on the relations between mock-modular forms and APS eta invariant. Arxiv: 1905.05107