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Large $N$ phase transition in $T \overline{T}$-deformed $2d$ Yang-Mills theory on the sphere

Leonardo Santilli, Miguel Tierz

TL;DR

This work analyzes the $T\overline{T}$-deformed $U(N)$ Yang--Mills theory on the sphere at large $N$. By recasting the deformed theory in a matrix-model framework, the authors perform a perturbative analysis in the deformation parameter $\tau$ and a nonperturbative two-cut construction for the strong-coupling phase, revealing that the Douglas--Kazakov phase transition persists for a range of $\tau$ with a lowered critical area $A_{cr}(\tau)$ and retains its third-order character. Instanton physics is shown to continue driving the transition, with the deformation enhancing instanton effects and lowering the critical area; in particular, the single-monopole sector yields a deformed exponential suppression controlled by $b_{\infty}$, consistent with a modified onset of the transition. These results illuminate how TTbar deformations modify nonperturbative phase structure in solvable gauge theories, potentially informing holographic and integrable perspectives on deformations of quantum field theories.

Abstract

We study the partition function of a $T \overline{T}$-deformed version of Yang-Mills theory on the two-sphere. We show that the Douglas-Kazakov phase transition persists for a range of values of the deformation parameter, and that the critical area is lowered. The transition is of third order and also induced by instantons, whose contributions we characterize.

Large $N$ phase transition in $T \overline{T}$-deformed $2d$ Yang-Mills theory on the sphere

TL;DR

This work analyzes the -deformed Yang--Mills theory on the sphere at large . By recasting the deformed theory in a matrix-model framework, the authors perform a perturbative analysis in the deformation parameter and a nonperturbative two-cut construction for the strong-coupling phase, revealing that the Douglas--Kazakov phase transition persists for a range of with a lowered critical area and retains its third-order character. Instanton physics is shown to continue driving the transition, with the deformation enhancing instanton effects and lowering the critical area; in particular, the single-monopole sector yields a deformed exponential suppression controlled by , consistent with a modified onset of the transition. These results illuminate how TTbar deformations modify nonperturbative phase structure in solvable gauge theories, potentially informing holographic and integrable perspectives on deformations of quantum field theories.

Abstract

We study the partition function of a -deformed version of Yang-Mills theory on the two-sphere. We show that the Douglas-Kazakov phase transition persists for a range of values of the deformation parameter, and that the critical area is lowered. The transition is of third order and also induced by instantons, whose contributions we characterize.

Paper Structure

This paper contains 11 sections, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Two-dimensional Yang--Mills theory and its $T\overline{T}$-deformation
  3. Large $N$ limit of the $T \overline{T}$-deformed theory
  4. Perturbative solution
  5. Strong coupling phase
  6. Third order phase transition
  7. Instanton analysis
  8. Instantons in the undeformed theory
  9. Instantons in the $T \overline{T}$-deformed theory
  10. Outlook
  11. Instanton sectors and Poisson resummation

Figures (3)

  • Figure 1: Convergence of the sequence $\left\{ b_{k}\right\} _{k}$, for $\tau =0.1$ (left) and $\tau =0.5$ (right).
  • Figure 2: On the left: a comparison of the function $\gamma \left( \frac{x}{\pi^2} \right)$ for the deformed (blue) and undeformed (orange) case. On the right: a zoom on the tail of the function $\frac{1}{x} \gamma \left( \frac{x}{\pi^2} \right)$ for the deformed (blue) and undeformed (orange) case. The plots are at $\tau = 0.1$.
  • Figure 3: On the left: a comparison of the function $\gamma \left( \frac{x}{\pi^2} \right)$ for the deformed (blue) and undeformed (orange) case. On the right: a zoom on the tail of the function $\frac{1}{x} \gamma \left( \frac{x}{\pi^2} \right)$ for the deformed (blue) and undeformed (orange) case. The plots are at $\tau = 0.5$.