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

We present a formula for the quantum entropy of supersymmetric five-dimensional spinning black holes in M-theory compactified on $CY_3$, 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 $\varphi^I$ for each vector multiplet and an additional parameter $\varphi^0$ 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 $f(\varphi^0)$. 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.