Quantum entropy of BMPV black holes and the topological M-theory conjecture

Abstract

We present a formula for the quantum entropy of supersymmetric five-dimensional spinning black holes in M-theory compactified on CY3, i.e., BMPV black holes. We use supersymmetric localization in the framework of off-shell five dimensional N = 2 supergravity coupled to I = 1, . . . , NV + 1 off-shell vector multiplets. The theory is governed at two-derivative level by the symmetric tensor CIJK (the intersection numbers of the Calabi-Yau) and at four-derivative level by the gauge-gravitational Chern-Simons coupling cI (the second Chern class of the Calabi-Yau). The quantum entropy is an NV +2-dimensional integral parameterised by one real parameter φI for each vector multiplet and an additional parameter φ0 for the gravity multiplet. The integrand consists of an action governed completely by CIJK and cI, and a one-loop determinant. Consistency with the on-shell logarithmic corrections to the entropy, the symmetries of the very special geometry of the moduli space, and an assumption of analyticity constrains the one-loop determinant up to a scale-independent function g(φ0). For g = 1 our result agrees completely with the topological M-theory conjecture of Dijkgraaf, Gukov, Neitzke, and Vafa for static black holes at two derivative level, and provides a natural extension to higher derivative corrections. For rotating BMPV black holes, our result differs from the DGNV conjecture at the level of the first quantum corrections.