  It has long been conjectured that the large $N$ deconfinement phase
transition of $\mathcal{N}=4$ ${\rm SU}(N)$ super-Yang-Mills corresponds via
AdS/CFT to the Hawking-Page transition in which black holes dominate the
thermal ensemble, and quantitative evidence of this has come through the recent
matching of the superconformal index of ${1\over 16}$-BPS states to the
supersymmetric black hole entropy. We introduce the half-BPS Gukov-Witten
surface defect as a probe of the superconformal index, which also serves as an
order parameter for the deconfinement transition. This can be studied directly
in field theory as a modification of the usual unitary matrix model or in the
dual description as a D3-brane probe in the background of a (complex)
supersymmetric black hole. Using a saddle point approximation, we determine our
defect index in the large $N$ limit as a simple function of the chemical
potentials and show independently that it is reproduced by the renormalized
action of the brane in the black hole background. Along the way, we also
comment on the Cardy limit and the thermodynamics of the D3-brane in the
generalized ensemble. The defect index sharply distinguishes between the
confining and the deconfining phases of the gauge theory and thus is a
supersymmetric non-perturbative order parameter for these large $N$ phase
transitions which deserves further investigation. Finally, our work provides an
example where the properties of a black hole coupled to an external system can
be analyzed precisely.

High Energy Physics - Theory