Quantum black hole entropy and the holomorphic prepotential of N=2 supergravity

TL;DR

This paper demonstrates that in N=2 supergravity, the quantum entropy of BPS black holes is determined solely by the holomorphic prepotential, as full-superspace integrals do not contribute, explaining the prepotential's exactness.

Contribution

Using localization techniques, the authors show that a broad class of full-superspace integrals do not affect BPS black hole entropy, supporting the prepotential's exclusive role.

Findings

Full-superspace integrals do not contribute to black hole entropy.

Prepotential terms capture the exact microscopic entropy.

Results extend to all derivative orders.

Abstract

Supersymmetric terms in the effective action of N=2 supergravity in four dimensions are generically classified into chiral-superspace integrals and full-superspace integrals. For a theory of N=2 vector multiplets coupled to supergravity, a special class of couplings is given by chiral-superspace integrals that are governed by a holomorphic prepotential function. The quantum entropy of BPS black holes in such theories depends on the prepotential according to a known integral formula. We show, using techniques of localization, that a large class of full-superspace integrals in the effective action of N=2 supergravity do not contribute to the quantum entropy of BPS black holes at any level in the derivative expansion. Our work extends similar results for semi-classical supersymmetric black hole entropy, and goes towards providing an explanation of why the prepotential terms capture the exact microscopic quantum black hole entropy.