Heriot-Watt Mathematics Report Series
HWM05-30, 13 Dec 2005
Black Holes, topological strings and large N phase transitions
N Caporaso, M Cirafici, L Griguolo, S Pasquetti, D Seminara and R J Szabo
Abstract
The counting of microstates of BPS black-holes on local Calabi-Yau
of the form ${\mathcal O}(p-2)\oplus{\mathcal
O}(-p)~\longrightarrow~S^2$ is explored by computing the
partition function of $q$-deformed Yang-Mills theory on $S^2$.
We obtain, at finite $N$, the instanton
expansion of the gauge theory. It can be written exactly as the
partition function for $U(N)$ Chern-Simons gauge theory on a Lens
space, summed over all non-trivial vacua, plus a tower of
non-perturbative instanton contributions. In the large $N$ limit we find a
peculiar phase structure in the model. At weak string coupling the
theory reduces to the trivial sector and the
topological string partition function on the resolved conifold is
reproduced in this regime. At a certain critical point, instantons
are enhanced and the theory undergoes a phase transition into a strong
coupling regime. The
transition from the strong coupling phase to the weak coupling phase
appears to be of third order.
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