Hedgehog black holes and the Polyakov loop at strong coupling

Understand

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.