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In N=4 super-Yang-Mills theory at large N, large \lambda, and finitetemperature, the value of the Wilson-Maldacena loop wrapping the Euclidean timecircle (the Polyakov-Maldacena loop, or PML) is computed by the area of acertain minimal surface in the dual supergravity background. This prescriptioncan be used to calculate the free energy as a function of the PML (averagedover the spatial coordinates), by introducing into the bulk action a Lagrangemultiplier term that fixes the (average) area of the appropriate minimalsurface. This term, which can also be viewed as a chemical potential for thePML, contributes to the bulk stress tensor like a string stretching from thehorizon to the boundary (smeared over the angular directions). We find thecorresponding "hedgehog" black hole solutions numerically, within anSO(6)-preserving ansatz, and derive part of the free energy diagram for thePML. As a warm-up problem, we also find exact solutions for hedgehog blackholes in pure gravity, and derive the free energy and phase diagrams for thatsystem.

Hedgehog black holes and the Polyakov loop at strong coupling