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It has long been conjectured that the large $N$ deconfinement phasetransition of $\mathcal{N}=4$ ${\rm SU}(N)$ super-Yang-Mills corresponds viaAdS/CFT to the Hawking-Page transition in which black holes dominate thethermal ensemble, and quantitative evidence of this has come through the recentmatching of the superconformal index of ${1\over 16}$-BPS states to thesupersymmetric black hole entropy. We introduce the half-BPS Gukov-Wittensurface defect as a probe of the superconformal index, which also serves as anorder parameter for the deconfinement transition. This can be studied directlyin field theory as a modification of the usual unitary matrix model or in thedual description as a D3-brane probe in the background of a (complex)supersymmetric black hole. Using a saddle point approximation, we determine ourdefect index in the large $N$ limit as a simple function of the chemicalpotentials and show independently that it is reproduced by the renormalizedaction of the brane in the black hole background. Along the way, we alsocomment on the Cardy limit and the thermodynamics of the D3-brane in thegeneralized ensemble. The defect index sharply distinguishes between theconfining and the deconfining phases of the gauge theory and thus is asupersymmetric non-perturbative order parameter for these large $N$ phasetransitions which deserves further investigation. Finally, our work provides anexample where the properties of a black hole coupled to an external system canbe analyzed precisely.

Probing Supersymmetric Black Holes with Surface Defects