We use conformal supergravity techniques to study four-derivative corrections in four-dimensional gauged supergravity. We show that the four-derivative Lagrangian for the propagating degrees of freedom of the $$ \mathcal{N} $$ N = 2 gravity multiplet is determined by two real dimensionless constants. We demonstrate that all solutions of the two-derivative equations of motion in the supergravity theory also solve the four-derivative equations of motion. These results are then applied to explicitly calculate the regularized on-shell action for any asymptotically locally AdS4 solution of the two-derivative equations of motion. The four-derivative terms in the supergravity Lagrangian modify the entropy and other thermodynamic observables for the black hole solutions of the theory. We calculate these corrections explicitly and demonstrate that the quantum statistical relation holds for general stationary black holes in the presence of the four-derivative corrections. Employing an embedding of this supergravity model in M-theory we show how to use supersymmetric localization results in the holographically dual three-dimensional SCFT to determine the unknown coefficients in the four-derivative supergravity action. This in turn leads to new detailed results for the first subleading $$ {N}^{\frac{1}{2}} $$ N 1 2 correction to the large N partition function of a class of three-dimensional SCFTs on compact Euclidean manifolds. In addition, we calculate explicitly the first subleading correction to the Bekenstein-Hawking entropy of asymptotically AdS4 black holes in M-theory. We also discuss how to add matter multiplets to the supergravity theory in the presence of four-derivative terms and to generalize some of these results to six- and higher-derivative supergravity.

Abstract:The exact quantum entropy of BPS black holes can be evaluated using localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the QV operator). We compute the fluctuation determinant for vector multiplets and hyper multiplets around Q-invariant off-shell configurations in four-dimensional N = 2 supergravity with AdS 2 × S 2 boundary conditions, using the Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in N = 2 supergravity. We explain cancellations concerning 1 8 -BPS black holes in N = 8 supergravity that were observed in arXiv:1111.1161. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory.

We study four-derivative corrections to four-dimensional N ¼ 2 minimal gauged supergravity controlled by two real constants. The solutions of the equations of motion in the two-derivative theory are not modified by the higher-derivative corrections. We use this to derive a general formula for the regularized on-shell action for any asymptotically locally AdS 4 solution of the theory and show how the higher-derivative corrections affect black hole thermodynamic quantities in a universal way. We employ our results in the context of holography to derive explicit expressions for the subleading corrections in the large N expansion of supersymmetric partition functions on various compact manifolds for a large class of three-dimensional SCFTs.

We obtain holographic constraints on the microscopic degeneracies of black holes by computing the exact macroscopic quantum entropy using localization, including the effects of string worldsheet instantons in the supergravity effective action. For 1 4 -BPS black holes in type II string theory on K3 × T 2 , the constraints can be explicitly checked against expressions for the microscopic BPS counting functions that are known in terms of certain mock modular forms. We find that the effect of including the infinite sum over instantons in the holomorphic prepotential of the supergravity leads to a sum over Bessel functions with successively sub-leading arguments as in the Rademacher expansion of Jacobi forms -but begins to disagree with such a structure near an order where the mock modular nature becomes relevant. This leads to a systematic method to recover the polar terms of the microscopic degeneracies from the degeneracy of instantons (the Gromov-Witten invariants). We check explicitly that our formula agrees with the known microscopic answer for the first seven values of the magnetic charge invariant.

We analyze BPS black hole attractors in the conformal 4d gauged supergravity formalism and apply the technique known as supergravity localization in order to evaluate Sen's quantum entropy function [1] in the AdS 2 ×S 2 near-horizon geometry. Under certain assumptions, we reduce the exact expression of the functional integral to a finite-dimensional integral for a number of supersymmetric black holes in gauged supergravity with AdS asymptotics subject to a holographic description via a dual field theory. Examples include the asymptotically AdS 4 ×S 7 Cacciatori-Klemm black holes [2] in Mtheory and the asymptotically AdS 5 ×S 5 generalizations of Gutowski-Reall black holes [3] and Benini-Bobev black strings [4] in type IIB, as well as the recently constructed asymptotically AdS 4 ×S 6 solutions [5,6] in massive type IIA. Our results provide an important first step towards a gravitational counterpart to the exact evaluation of supersymmetric partition functions at finite N for the holographically dual field theories in these examples. arXiv:1803.05920v2 [hep-th]