Supersymmetric indices factorize

TL;DR

This work provides a concrete gravitational framework for computing supersymmetric indices of black holes from Euclidean path integrals. By analyzing the near-horizon AdS$_2$ region and performing a careful one-loop/zero-mode analysis in ${ m N}=4$ JT supergravity, the authors show that only BPS saddles contribute to the index, while non-BPS saddles are killed by gravitino zero modes, and spacetime wormholes do not affect the index under supersymmetric boundary conditions. They demonstrate that the index factorizes across multiple boundaries and classify potential nonperturbative contributions from defects, with supersymmetric defects yielding controlled, finite corrections and the Hawking–Horowitz–Ross solution contributing nothing. The results reinforce the view that certain protected gravitational observables are immune to ensemble averaging and topology-changing ambiguities, while offering a precise, calculable bridge to the dual quantum-mechanical picture in highly supersymmetric settings.

Abstract

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.

Figures (13)

• Figure 1: The gravitational path integral defined for asymptotically flat or AdS geometry with supersymmetric boundary conditions, i.e. with periodic bosons ($B$) and fermions ($F$). We will first clarify how to explicitly impose such boundary conditions. Then, our goal is to find the geometries that contribute to the gravitational path integral with the above boundary conditions.
• Figure 2: The index with multiple asymptotically-flat boundaries is computed by integrating over geometries and fields with index boundary conditions. We will identify which geometries involving topology change near the horizon appear in the gravitational path integral with multiple boundaries, and determine whether they contribute.
• Figure 3: Cartoon emphasizing the factorization property in gravity with index boundary conditions (on the green boundaries) and non-supersymmetric boundary conditions on other boundaries (represented in blue). The red jagged lines on the right denote the separation between the asymptotic and near-horizon regions. In this paper, we shall prove that, at least in the near-horizon region, the supersymmetric boundaries always factorize while connected geometries with non-supersymmetric boundary conditions could in principle contribute to the gravitational path integral. Consequently, the red blob in the right figure represents a sum over all allowed near-horizon geometries.
• Figure 4: Summary of boundary conditions and the singular $SU(2)$ gauge field when working in the Boyer-Lindquist coordinates.
• Figure 5: Summary of boundary conditions in corotating coordinates. The $SU(2)$ gauge field is now smooth.