The gravitational path integral for $ N=4$ BPS black holes from black hole microstate counting

TL;DR

This paper refines the quantum entropy function for 4D N=4 BPS black holes by summing over Euclidean backgrounds, including orbifolds, and explores a gravitational interpretation involving 2D Euclidean wormholes.

Contribution

It improves the existing formulation of the quantum entropy function using supersymmetric localization and introduces a gravitational interpretation with Euclidean wormholes.

Findings

Sum over orbifolded Euclidean backgrounds in the entropy calculation

Explicit form of gauge potentials supporting these backgrounds

Semi-classical gravitational interpretation involving 2D Euclidean wormholes

Abstract

We use the exact degeneracy formula of single-centred $\frac14$ BPS dyonic black holes with unit torsion in 4D $N=4$ toroidally compactified heterotic string theory to improve on the existing formulation of the corresponding quantum entropy function obtained using supersymmetric localization. The result takes the form of a sum over Euclidean backgrounds including orbifolds of the Euclidean $AdS_2 \times S^2$ attractor geometry. Using an $N=2$ formalism, we determine the explicit form of the Abelian gauge potentials supporting these backgrounds. We further show how a rewriting of the degeneracy formula is amenable, at a semi-classical level, to a gravitational interpretation involving 2D Euclidean wormholes. This alternative picture is useful to elucidate different aspects of the gravitational path integral capturing the microstate degeneracies. We also comment on the relation between the associated 1D holographic models.