Blackhole / string transition for the small Schwarzschild blackhole
of AdS(5) x S**5 and critical unitary matrix models
(2006)Abstract
n this paper we discuss the blackhole-string transition of the small
Schwarzschild blackhole of $AdS_5 S^5$ using the AdS/CFT correspondence
at finite temperature. The finite temperature gauge theory effective
action, at weak and strong coupling, can be expressed entirely
in terms of constant Polyakov lines which are $SU (N)$ matrices.
In showing this we have taken into account that there are no Nambu-Goldstone
modes associated with the fact that the 10 dimensional blackhole
solution sits at a point in $S^5$. We show that the phase of the
gauge theory in which the eigenvalue spectrum has a gap corresponds
to supergravity saddle points in the bulk theory. We identify the
third order $N = ınfty$ phase transition with the blackhole-string
transition. This singularity can be resolved using a double scaling
limit in the transition region where the large N expansion is organized
in terms of powers of $N^-2/3$. The $N = ınfty$ transition now
becomes a smooth crossover in terms of a renormalized string coupling
constant, reflecting the physics of large but finite N. Multiply
wound Polyakov lines condense in the crossover region. We also discuss
the implications of our results for the resolution of the singularity
of the Lorenztian section of the small Schwarzschild blackhole.