$T\overline{T}$-deformation of $q$-Yang-Mills theory

Abstract

We derive the $T\overline{T}$-perturbed version of two-dimensional $q$-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the $T\overline{T}$-deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large $N$ factorization into chiral and anti-chiral sectors. For the $U(N)$ gauge theory on the sphere, we show that the large $N$ phase transition persists, and that it is of third order and induced by instantons. The effect of the $T\overline{T}$-deformation is to decrease the critical value of the 't Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for $(q,t)$-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large $N$ limit of Yang-Mills theory, showing that the $T\overline{T}$-deformation decreases the contribution of the Boltzmann entropy.

Figures (12)

• Figure 1: The disk (left), the cylinder (center) and the pair of pants (right), homeomorphic to the complex plane with respectively zero, one or two holes.
• Figure 2: Obtaining a torus from elementary pieces. On the left, the gluing is an integration over boundary conditions (in blue).
• Figure 3: A sphere with three Wilson loops is cut into three disks plus a remaining pair of pants.
• Figure 4: Non-relativistic fermions: ground state (left) and two excited states (center, right). The two excited states are excitations over the positive Fermi surface. In the center, a fermion occupying a positive energy level jumps above the positive Fermi surface: this will correspond to a chiral state at large $N$. On the right, a fermion occupying a negative energy level jumps above the positive Fermi surface: this will be exponentially suppressed at large $N$.
• Figure 5: The critical curve of $q$-Yang-Mills theory, in terms of the parameter $A= \lambda/p$ as a function of $p$. The horizontal asymptote (dashed) is the DK critical point $A_{\text{cr}}= \pi^2$. The vertical asymptote (red) is the point $p=2$. This plot is inspired by Arsiwalla.