Quantum black hole entropy and the holomorphic prepotential of $ \mathcal{N}=2 $ supergravity

Abstract

Abstract Supersymmetric terms in the effective action of $ \mathcal{N}=2 $ supergravity in four dimensions are generically classified into chiral-superspace integrals and full-superspace integrals. For a theory of $ \mathcal{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 $ \mathcal{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.