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 $C{Y}_3$, i.e., BMPV black holes. We use supersymmetric localization in the framework of off-shell five dimensional $N=2$ supergravity coupled to $I=1,\dots ,N_V+1$ off-shell vector multiplets. The theory is governed at two-derivative level by the symmetric tensor ${\mathcal{C}}_{IJK}$ (the intersection numbers of the Calabi-Yau) and at four-derivative level by the gauge-gravitational Chern-Simons coupling $c_I$ (the second Chern class of the Calabi-Yau). The quantum entropy is an $N_V+2$-dimensional integral parameterised by one real parameter ${\phi }^I$ for each vector multiplet and an additional parameter ${\phi }^0$ for the gravity multiplet. The integrand consists of an action governed completely by ${\mathcal{C}}_{IJK}$ and $c_I$, 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 $f({\phi }^0)$. For $f=1$ our result agrees completely with the topological M-theory conjecture of Dijkgraaf, Gukov, Nietzke, 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.

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Cited by in corpus (4)

• Higher-derivative corrections to flavoured BPS black hole thermodynamics and holography
• Localization on $Ad{S}_3\times S^2$ I: the 4d/5d connection in off-shell Euclidean supergravity
• Black hole thermodynamics at 4 derivatives, natural variables and BPS limits
• Supersymmetric localization: $\mathcal{N}=(2,2)$ theories on S${}^2$ and AdS${}_2$