Supersymmetric indices factorize

The extent to which quantum mechanical features of black holes can be understood from the Euclidean gravity path integral has recently received significant attention. In this paper, we examine this question for the calculation of the supersymmetric index. For concreteness, we focus on the case of charged black holes in asymptotically flat four-dimensional $\mathcal{N}=2$ ungauged supergravity. We show that the gravity path integral with supersymmetric boundary conditions has an infinite family of Kerr-Newman classical saddles with different angular velocities. We argue that fermionic zero-mode fluctuations are present around each of these solutions making their contribution vanish, except for a single saddle that is BPS and gives the expected value of the index. We then turn to non-perturbative corrections involving spacetime wormholes and show that a fermionic zero mode is present in all these geometries, making their contribution vanish once again. This mechanism works for both single- and multi-boundary path integrals. In particular, only disconnected geometries without wormholes contribute to the gravitational path integral which computes the index, and the factorization puzzle that plagues the black hole partition function is resolved for the supersymmetric index. Finally, we classify all other single-centered geometries that yield non-perturbative contributions to the gravitational index of each boundary.