Topological strings and large N phase transitions I: Nonchiral expansion of q-deformed Yang-Mills theory

TL;DR

This paper investigates the phase structure of q-deformed Yang-Mills theory on Riemann surfaces, revealing a phase transition at large N that connects topological string theory and black hole microstate counting.

Contribution

It provides an exact instanton expansion of q-deformed Yang-Mills on the sphere and identifies a phase transition related to topological string theory on the resolved conifold.

Findings

Exact instanton expansion matches Chern-Simons on Lens space

Identifies a phase transition at large N with topological string implications

Proposes a two-cut solution for strong coupling regime

Abstract

We examine the problem of counting bound states of BPS black holes on local Calabi-Yau threefolds which are fibrations over a Riemann surface by computing the partition function of q-deformed Yang-Mills theory on the Riemann surface. We study in detail the genus zero case and 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. The correspondence between two and three dimensional gauge theories is elucidated by an explicit mapping between two-dimensional Yang-Mills instantons and flat connections on the Lens space. In the large $N$ limit we find a peculiar phase structure in the model. At weak string coupling the theory reduces exactly to the trivial flat connection sector with instanton contributions exponentially suppressed, and the topological string partition function on the resolved conifold is reproduced in this regime. At a certain critical point all non-trivial vacua contribute, instantons are enhanced and the theory appears to undergo a phase transition into a strong coupling regime. We rederive these results by performing a saddle-point approximation to the exact partition function. We obtain a q-deformed version of the Douglas-Kazakov equation for two-dimensional Yang-Mills theory on the sphere, whose one-cut solution below the transition point reproduces the resolved conifold geometry. Above the critical point we propose a two-cut solution that should reproduce the chiral-antichiral dynamics found for black holes on the Calabi-Yau threefold and the Gross-Taylor string in the undeformed limit.