Super-Rényi Entropy & Wilson Loops for N=4 SYM and their Gravity Duals

TL;DR

This work computes the supersymmetric Rényi entropy for ${\cal N}=4$ SYM across a spherical entangling surface by localizing the theory on a four-dimensional ellipsoid, then matches the universal logarithmic piece to a gravity dual described by a hyperbolically sliced BPS black hole in five-dimensional ${\cal N}=4^+$ gauged supergravity. It extends the analysis to Wilson-loop insertions, showing that the full ten-dimensional IIB uplift is necessary to capture the dual Wilson-loop observable and yields exact agreement with the field theory in the large-$N$, large-$\lambda$ limit. The gravity calculation reproduces the field-theory universal part and the replica-index dependence, with the Euclidean action and holographic Wilson loop scaling as $I_n=\frac{(n+1)^2}{4n}I_1$ and $\ln W_n=\frac{n+1}{2}\sqrt{\lambda}$, respectively. The results demonstrate a precise AdS/CFT correspondence for these refined entanglement diagnostics and highlight the necessity of 10D geometry for certain observables, offering insights into universal structures across dimensions and potential links to ${\cal N}=2^*$ deformations. The work thus provides a robust framework for exploring supersymmetric entanglement in holography and prompts further study of bulk representations of boundary deformation independence and Wilson-line observables.

Abstract

We compute the supersymmetric Rényi entropies across a spherical entanglement surface in N=4 SU(N) SYM theory using localization on the four-dimensional ellipsoid. We extract the leading result at large N and λ, and match its universal part to a gravity calculation involving a hyperbolically sliced supersymmetric black hole solution of N=4+ SU(2) X U(1) gauged supergravity in five dimensions. We repeat the analysis in the presence of a Wilson loop insertion and find again a perfect match with the dual string theory. Understanding the Wilson loop operator requires knowledge of the full ten-dimensional IIB supergravity solution which we elaborate upon.