Black-holes, topological strings and large N phase transitions

TL;DR

This paper investigates the microstate counting of BPS black holes on specific Calabi-Yau manifolds using q-deformed Yang-Mills theory, revealing a phase transition at large N between weak and strong coupling regimes.

Contribution

It establishes a connection between black hole microstate counting, topological strings, and large N phase transitions through exact partition function computations.

Findings

Exact instanton expansion of the gauge theory at finite N.

Identification of a third-order phase transition at large N.

Reproduction of the topological string partition function in the weak coupling phase.

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 is of third order.