Hedgehog black holes and the Polyakov loop at strong coupling

TL;DR

This paper explores the computation of the Polyakov-Maldacena loop in N=4 super-Yang-Mills theory at strong coupling using holographic duality, introducing a new approach with hedgehog black hole solutions to analyze free energy and phase structure.

Contribution

It introduces a novel numerical method to find hedgehog black hole solutions in supergravity, linking minimal surface areas to free energy calculations at strong coupling.

Findings

Derived new black hole solutions with a Lagrange multiplier term.

Mapped the free energy and phase diagram for the Polyakov-Maldacena loop.

Provided exact solutions for hedgehog black holes in pure gravity.

Abstract

In N=4 super-Yang-Mills theory at large N, large \lambda, and finite temperature, the value of the Wilson-Maldacena loop wrapping the Euclidean time circle (the Polyakov-Maldacena loop, or PML) is computed by the area of a certain minimal surface in the dual supergravity background. This prescription can be used to calculate the free energy as a function of the PML (averaged over the spatial coordinates), by introducing into the bulk action a Lagrange multiplier term that fixes the (average) area of the appropriate minimal surface. This term, which can also be viewed as a chemical potential for the PML, contributes to the bulk stress tensor like a string stretching from the horizon to the boundary (smeared over the angular directions). We find the corresponding "hedgehog" black hole solutions numerically, within an SO(6)-preserving ansatz, and derive part of the free energy diagram for the PML. As a warm-up problem, we also find exact solutions for hedgehog black holes in pure gravity, and derive the free energy and phase diagrams for that system.