N = 4 Super-Yang-Mills on Conic Space as Hologram of STU Topological Black Hole

TL;DR

$N=4$ SYM on a conic sphere $S^4_q$ is analyzed using heat-kernel and localization to extract free energy and supersymmetric Renyi entropy; the universal, coupling-protected $q$-scaling is demonstrated both at weak coupling and in the large-$N$ strong coupling limit. The branched sphere is connected to holography via a conformal map to $S^1\times H^3$, with the gravity dual given by five-dimensional STU topological black holes (TBH$_5$). Localization on a resolved branched sphere yields a matrix model whose leading large-$N$ behavior reproduces the $q$-dependence found in the free field analysis, confirming a gravity dual description. The TBH$_5$/qSCFT$_4$ correspondence is established by matching supersymmetric Rényi entropy and free energy across single, double, and triple R-charge configurations, validating the duality and suggesting a broader universality for supersymmetric Rényi observables.

Abstract

We construct four-dimensional N=4 super-Yang-Mills theories on a conic sphere with various background R-symmetry gauge fields. We study free energy and supersymmetric Renyi entropy using heat kernel method as well as localization technique. We find that the universal contribution to the partition function in the free field limit is the same as that in the strong coupling limit, which implies that it may be protected by supersymmetry. Based on the fact that, the conic sphere can be conformally mapped to $S^1\times H^3$ and the R-symmetry background fields can be supported by the R-charges of black hole, we propose that the holographic dual of these theories are five-dimensional, supersymmetric STU topological black holes. We demonstrate perfect agreement between N=4 super-Yang-Mills theories in the planar limit and the STU topological black holes.