Black hole microstate counting from the gravitational path integral

TL;DR

The paper shows that supersymmetric localization of the gravitational path integral for 1/8-BPS black holes in N=8 supergravity reproduces the exact string-theory index, including non-perturbative orbifold saddles. It provides a complete computation of the bulk one-loop determinants for all N=2 supermultiplets arising in the N=8 theory, along with careful treatment of boundary zero-modes governed by the N=4 super-Schwarzian, and incorporates topological/Kloosterman phases from orbifolds. By summing over AdS_2×S^2 saddles and their Z_c orbifolds, it exactly matches the Hardy–Ramanujan–Rademacher expansion coefficients C(Δ) with K_c(Δ) and the modified Bessel factors, yielding W_micro(Δ) as the gravitational index and degeneracy in an energy window. The results illuminate the role of boundary dynamics, non-perturbative geometries, and duality-invariant measures, and offer a concrete bridge between macroscopic quantum entropy and microscopic black hole microstates, with broader implications for the structure of observables in quantum gravity.

Abstract

Reproducing the integer count of black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In this paper, we show that, by using supersymmetric localization, the gravitational path integral for $\frac{1}8$-BPS black holes in $\mathcal{N}=8$ supergravity reproduces the index obtained in the string theory construction of such black holes, including all non-perturbatively suppressed geometries. A more refined argument then shows that, not only the black hole index, but also the total number of black hole microstates within an energy window above extremality that is polynomially suppressed in the charges, also matches this string theory index. To achieve such a match we compute the one-loop determinant arising in the localization calculation for all $\mathcal{N}=2$ supergravity supermultiplets in the $\mathcal{N}=8$ gravity supermultiplet. Furthermore, we carefully account for the contribution of boundary zero-modes, that can be seen as arising from the zero-temperature limit of the $\mathcal{N}=4$ super-Schwarzian, and show that performing the exact path integral over such modes provides a critical contribution needed for the match to be achieved. A discussion about the importance of such zero-modes in the wider context of all extremal black holes is presented in a companion paper.

Figures (1)

• Figure 1: Left: Indices of $1/8$-BPS black holes estimated to four different levels of precison: (i) the dotted black curve represents the roughest Bekenstein-Hawking estimate given by $\exp(S_0)$; (ii) the dashed red curve represents the degeneracy corrected at one-loop around the leading saddle; (iii) the solid blue curve is the estimate from considering the entire asymptotic perturbation theory series around the leading AdS$_2 \times S^2$ localizing saddle, neglecting non-perturbative corrections; (iv) the dots represent the exact index as obtained from string theory. Right: The difference between the string theory index and the entire contribution of the leading AdS$_2 \times S^2$ localizing saddle. Note that while the plot on the left shows the various degeneracies on a logarithmic scale, the plot on the right does not. This emphasizes the great accuracy of even the leading localization result before other non-perturbative corrections (the orbifold geometries) are included.