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

Abstract

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 $$ {\mathcal{C}}_{IJK} $$ C IJK (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 $$ {\mathcal{C}}_{IJK} $$ C IJK 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.