The Gravity Dual of Supersymmetric Renyi Entropy

TL;DR

The paper identifies a gravity dual for supersymmetric Renyi entropies of 3d $N=2$ SCFTs by mapping the branched $S^3$ to $S^1\times H^2$ and employing a charged topological $AdS_4$ black hole within 4d $N=2$ gauged supergravity, preserving half of the supersymmetries. Using localization results for $Z_n$ (with squashing $b=\sqrt{n}$) and large-$N$ saddle-point techniques, it derives a universal relation $S_n^{\text{susy}}= -\frac{3n+1}{4n} F_1$ for a broad class of theories, with ABJM giving explicit $N^{3/2}$ scaling. The holographic analysis computes the on-shell action and shows that the gravity results reproduce the field theory large-$N$ expressions for both the SUSY Renyi entropy and the Wilson-loop contributions, including a Wilson-loop independent term $S_{W,n}=\frac{\pi}{2}\sqrt{2\lambda}$. The work thus strengthens the AdS/CFT correspondence for supersymmetric Renyi entropies and clarifies the role of Wilson loops in holographic Renyi calculations, providing explicit cross-checks between field theory localization and gravitational on-shell actions.

Abstract

Supersymmetric Renyi entropies are defined for three-dimensional N=2 superconformal field theories on a branched covering of a three-sphere by using the localized partition functions. Under a conformal transformation, the branched covering is mapped to S^1 x H^2, whose gravity dual is the charged topological AdS_4 black hole. The black hole can be embedded into four-dimensional N=2 gauged supergravity where the mass and charge are related so that it preserves half of the supersymmetries. We compute the supersymmetric Renyi entropies with and without a certain type of Wilson loop operators in the gravity theory. We find they agree with those of the dual field theories in the large-N limit.

Figures (2)

• Figure 1: The conformal map from the branched covering of a three-sphere \ref{['ConicalS3']} to $S^1 \times H^2$\ref{['Hyp']}. The orange circle at $\theta=\pi/2$ in the left figure is mapped to a circle (also shown in orange color) along $\tau$ direction at $u=0$ of the hyperbolic space $H^2$ in the right figure.
• Figure 2: The gravity dual of a Wilson loop in a fundamental representation is given by the fundamental string in AdS$_4$ spacetime. In our case, it has two ends at the UV boundary $r=r_\infty$ and the horizon $r=r_H$. We suppress the $\tau$ direction in this figure.