Phase transitions in q-deformed 2d Yang-Mills theory and topological strings

TL;DR

The paper analyzes large $N$ phase structure of $U(N)$ $q$-deformed 2d YM on the sphere, revealing a phase diagram with a line of critical points $A_*(p)$ for $p>2$ that smoothly connects to the Douglas–Kazakov transition, while no transition occurs for $p\le 2$. It shows the transition is triggered by instantons through a detailed instanton analysis yielding a function $\gamma(A,p)$ that vanishes at $A_*(p)$, and it constructs the strongly coupled two-cut solution for the large-area phase. The results imply that, on certain Calabi–Yau backgrounds, nonperturbative topological string theory may exhibit new phase transitions at small radius and offer a mechanism to smooth large $N$ transitions in gauge theory. The work also links the small-area phase to a CS/Stieltjes-Wigert matrix model and discusses implications and open problems for OSV-type holography and topological strings.

Abstract

We analyze large N phase transitions for U(N) q-deformed two-dimensional Yang-Mills theory on the sphere. We determine the phase diagram of the model and we show that, for small values of the deformation parameter, the theory exhibits a phase transition which is smoothly connected to the Douglas-Kazakov phase transition. For large values of the deformation parameter the phase transition is absent. By explicitly computing the one-instanton suppression factor in the weakly coupled phase, we also show that the transition is triggered by instanton effects. Finally, we present the solution of the model in the strongly coupled phase. Our analysis suggests that, on certain backgrounds, nonperturbative topological string theory has new phase transitions at small radius. From the point of view of gauge theory, it suggests a mechanism to smooth out large N phase transitions.

Figures (5)

• Figure 1: This figure shows the density $\rho(h)$ before and after the Douglas-Kazakov transition. The solution for $A\ge \pi^2$ can be interpreted as a two-cut solution of an auxiliary matrix model.
• Figure 2: This figure shows the deformation of the contour needed to compute the resolvent in (\ref{['solwo']}). We pick a residue at $z=p$, and we have to encircle the singularity at the origin as well as the branch cut of the logarithm, which on the left hand side is represented by the dashed lines.
• Figure 3: This figure represents the phase diagram of $q$-deformed 2d YM theory. The horizontal axis represents the parameter $p$, while the vertical axis represents $A$. The curve shown in the figure is the critical line (\ref{['apcrit']}), which separates the phases of small and large area. The horizontal dashed line, which is the asymptote of the curve as $p\rightarrow \infty$, represents the $A=\pi^2$ critical point of Douglas and Kazakov.
• Figure 4: This figure shows the function $\gamma(A,p)$ appearing in the one-instanton suppression factor, plotted as a function of $A$, and for the values $p=2.1, 3, \infty$, from top to bottom. For each $p$ it is a decreasing function of the area and vanishes at the critical value $A_*(p)$.
• Figure 5: This figure shows the deformation of the contour needed to compute the resolvent in the two-cut solution. We have to encircle the singularity at the origin, and the two branch cuts denoted by thick lines on the left.