  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.
