Index Theorems and Soft Theorems

This thesis deals with Index theorems and Soft theorems for gravitini. In the first part, we derive the Atiyah-Patodi-Singer (APS) index theorem using supersymmetric quantum mechanics. We relate the APS η-invariant to the temperature dependence of the noncompact Witten index. It turns out that the temperature derivative of the Witten index depends solely on the asymptotic boundary of the noncompact target space. We also compute the elliptic genus of some noncompact superconformal field theories, namely N = (2, 2) cigar and N = (4, 4) TaubNUT. This elliptic genera is the completion of a mock Jacobi form. The holomorphic anomaly of this mock Jacobi form again depends on the boundary theory as in the case of the Witten index. We show that the APS index theorem can then be related to the completion of a mock Jacobi form via noncompact Witten index. In the second part, we derive the leading order soft theorem for multiple soft gravitini. We compute it in an arbitrary theory of supergravity with an arbitrary number of finite energy particles. Our results are valid at all orders in perturbation theory in more than three dimensions. We also comment on the infrared (IR) divergences in supergravity. It turns out that the leading order soft theorem is unaffected by the IR divergences