On the quantum entropy function in 4d gauged supergravity

TL;DR

This paper applies supergravity localization to evaluate the quantum entropy function of BPS black holes in 4d gauged supergravity, connecting gravitational calculations with dual field theory partition functions.

Contribution

It introduces a method to reduce the quantum entropy functional integral to a finite-dimensional integral for various AdS black holes in gauged supergravity.

Findings

Finite-dimensional integral expressions derived for specific black hole solutions.

Connection established between supergravity entropy calculations and dual field theory partition functions.

First steps towards a gravitational dual of exact supersymmetric partition functions at finite N.

Abstract

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 in the $\mathrm{AdS}_2\!\times\!\mathrm{S}^2$ 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 $\mathrm{AdS}_4\!\times\!\mathrm{S}^7$ Cacciatori-Klemm black holes in M-theory and the asymptotically $\mathrm{AdS}_5\!\times\!\mathrm{S}^5$ generalizations of Gutowski-Reall black holes and Benini-Bobev black strings in type IIB, as well as the recently constructed asymptotically $\mathrm{AdS}_4\!\times\!\mathrm{S}^6$ solutions 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.

Equations

$$d_{\mathrm{macro}}(p^I,q_I):={⟨\exp \left(4\pi q_I{\int }_0^{2\pi }{W}_{\tau }^I\mathrm{d}\tau \right)⟩}_{{\mathrm{AdS}}_2}^{\mathrm{finite}},$$

$$d_{\rm macro}(p^{I},q_{I})=\int_{-\infty}^{+\infty}\left(\prod_{I=0}^{n_{V}}\frac{{\rm d}\phi^{I}_{+}}{2\pi}\right)\delta(\xi_{I}\phi_{+}^{I}-2\pi)\,\exp\Bigl{[}-2\pi\bigl{(}p^{I}\mathcal{F}^{+}_{I}(\phi_{+})+q_{I}\phi_{+}^{I}\bigr{)}\Bigr{]}Z^{\rm reg}_{\mathrm{ind}}(\phi_{+})\,.$$

$$d_{\mathrm{micro}}({\mathfrak{p}}_a,{\mathfrak{q}}_a)={\int }_0^{2\pi }\left(\prod _a\frac{\mathrm{d}{\Delta }_a}{2\pi }\right)\delta (\sum _a{\Delta }_a-2\pi )Z({\mathfrak{p}}_a,{\Delta }_a)e^{-i{\sum }_a{\Delta }_a{\mathfrak{q}}_a},$$

$$\mathbb{W}=(e_{\mu }{}^a,{\psi }_{\mu }{}^i,b_{\mu },A_{\mu },{\mathcal{V}}_{\mu }{}^i{}_j,T_{ab},{\chi }^i,D),$$

$$\delta {\psi }_{\mu }{}^i=$$

$$\delta {\chi }^i=$$

$${\mathcal{D}}_{\mu }{ϵ}^i=({\partial }_{\mu }-\frac{1}{4}{\omega }_{\mu }{}^{ab}{\gamma }_{ab}+\frac{1}{2}{b}_{\mu }+\frac{1}{2}{A}_{\mu }{\gamma }^5){ϵ}^i+\frac{1}{2}{\mathcal{V}}_{\mu }{}^i{}_j{ϵ}^j,$$

$${\mathbb{V}}^I=(X_{+}^I,X_{-}^I,W_{\mu }^I,{\Omega }^{iI},Y^{ijI}),$$

$$\delta X_{\pm }^I=$$

$$\delta W_{\mu }^I=$$

$$\delta {\Omega }_{\pm }^{iI}=$$

$$\delta Y^{ijI}=$$

$$\hat{F}(W{)}_{\mu \nu }^I=2{\partial }_{[\mu }{W}_{\nu ]}^I+{\bar{\psi }}_{i[\mu }{\gamma }_{\nu ]}{\Omega }_{+}^{iI}-{\bar{\psi }}_{i[\mu }{\gamma }_{\nu ]}{\Omega }_{-}^{iI}+\mathrm{i}{X}_{-}^I{\bar{\psi }}_{\mu i}{\psi }_{\nu }{}^i{}_{+}-\mathrm{i}{X}_{+}^I{\bar{\psi }}_{\mu i}{\psi }_{\nu }{}^i{}_{-}.\text{vspace}4mm$$

$$\mathbb{H}=(A_i{}^{\alpha },{\zeta }^{\alpha }).$$

$$\delta A_i{}^{\alpha }=$$

$$\delta {\zeta }_{\pm }^{\alpha }=$$

$$D_{\mu }{A}_i{}^{\alpha }=({\partial }_{\mu }-b_{\mu })A_i{}^{\alpha }+\frac{1}{2}{\mathcal{V}}_{\mu i}{}^j{A}_j{}^{\alpha }-{\xi }_I{W}_{\mu }^I{t}^{\alpha }{}_{\beta }{A}_i{}^{\beta }+\mathrm{fermions},$$

$$\big{[}\delta_{\mathrm{Q}}(\epsilon_{1}),\,\delta_{\mathrm{Q}}(\epsilon_{2})\big{]}=\delta_{\mathrm{cgct}}(v^{\mu})+\delta_{\mathrm{M}}(\varepsilon_{ab})+\delta_{\mathrm{S}}(\hat{\eta}^{i})+\delta_{\mathrm{K}}(\Lambda_{\mathrm{K}}{\!}^{a})+\delta_{\mathrm{gauge}}(\theta^{I})\,,$$

$$v^{\mu }=$$

$${\epsilon }_{ab}=$$

$${\hat{\eta }}^i=$$

$${\Lambda }_{\mathrm{K}}{}^a=$$

$${\theta }^I=4\mathrm{i}{\overline{ϵ}}_{2i-}{ϵ}_1{}^i{}_{-}{X}_{+}^I-4\mathrm{i}{\overline{ϵ}}_{2i+}{ϵ}_1{}^i{}_{+}{X}_{-}^I.$$

$$\displaystyle\big{[}\delta_{\mathrm{Q}}(\epsilon),\,\delta_{\mathrm{S}}(\eta)\big{]}=$$

$$+{\delta }_{\mathrm{SU}(2)}(2\mathrm{i}{\overline{ϵ}}_j{\gamma }^5{\eta }^i-{\delta }_j{}^i\mathrm{i}{\overline{ϵ}}_k{\gamma }^5{\eta }^k),$$

$$\displaystyle\big{[}\delta_{\mathrm{S}}(\eta_{1}),\,\delta_{\mathrm{S}}(\eta_{2})\big{]}=$$

$$ds^{2}=v_{1}\bigl{(}\sinh^{2}\eta\,\mathrm{d}\tau^{2}+\mathrm{d}\eta^{2}\bigr{)}+v_{2}\bigl{(}\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\varphi^{2}\bigr{)}\,,$$

$$e^1=\sqrt{v_1}\sinh \eta \mathrm{d}\tau ,e^2=\sqrt{v_1}\mathrm{d}\eta ,e^3=\sqrt{v_2}\mathrm{d}\theta ,e^4=\sqrt{v_2}\sin \theta \mathrm{d}\phi .$$

$$ds^{2}=v_{1}\Bigl{(}(r^{2}-1)\,\mathrm{d}\tau^{2}+\frac{\mathrm{d}r^{2}}{r^{2}-1}\Bigr{)}+v_{2}\bigl{(}\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\varphi^{2}\bigr{)}\,.$$

$$T_{12}^{\mp }=-w_{\mp },T_{34}^{\mp }=\pm w_{\mp }.$$

$$v_1^{-1}=-\frac{1}{4}{w}_{-}{w}_{+}.$$

$$v_1^{-1}=-\frac{1}{4}{w}_{-}$$