Higher-Derivative Supergravity, AdS$_4$ Holography, and Black Holes

Abstract

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 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 AdS 4 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 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 AdS 4 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.

The Supersymmetry of Higher-Derivative Supergravity in AdS$_4$ Holography

“…In fact, we show that all the supersymmetric solutions to the four-derivative theory are the supersymmetric solutions of the twoderivative theory. Starting from this, we find a formula for their on-shell action including the four-derivative corrections, extending the two-derivative case of [5], and proving a conjecture in [4]. One feature of this formula is that it suggests a localization theorem is at play: every supersymmetric solution admits a Killing vector ξ, and the on-shell action is expressed in terms of contributions from the fixed point sets of ξ, whether isolated (nuts) or two-dimensional (bolts).…”

“…The action has been obtained in [3,4] starting from the Weyl multiplet of four-dimensional off-shell N = 2 conformal supergravity coupled to one vector multiplet and one hypermultiplet (the latter two being compensator multiplets). Then to begin to reduce to Poincaré supergravity, a number of the superconformal symmetries are gauge-fixed.…”

“…(5) The fact that the action ( 1) is invariant under the supersymmetry transformations of two-derivative minimal supergravity is enforced by construction: each term is independently invariant under the supersymmetry transformations, and the "two-derivative ansatz" takes the first term to the two-derivative action (as the name suggests). Whilst the method just outlined leads to a supersymmetric action that includes four-derivative corrections to S 2∂ and starts from the most general conformal supergravity action consistent with physical assumptions [4], it would be interesting to investigate whether this is the most general form while remaining in Poincaré supergravity and imposing a modified supersymmetry transformation. This is beyond the scope of this note, but we plan to return to this question in the future [6].…”

“…g and E 4∂ A (see Appendix A) [4] . Therefore, any two-derivative solution is also a four-derivative solution.…”

“…It is generically difficult to write down the higherderivative corrections to a supergravity theory. However, for the theory of interest here, a four-derivative action has been recently suggested [3,4]. We are going to take this as our starting point, and we are going to focus on the solutions to this theory that preserve supersymmetry.…”