Infinite Family of Integrable Sigma Models Using Auxiliary Fields

TL;DR

The paper introduces an infinite-class of two-dimensional sigma models built by coupling the principal chiral model (PCM) to an auxiliary field, parameterized by a single-variable function $E(\nu)$ with $\nu = \mathrm{tr}(v_+ v_+ )\mathrm{tr}(v_- v_- )$. A Lax representation with $\mathfrak{L}_\pm = \frac{j_\pm \pm z\, \mathfrak{J}_\pm}{1 - z^2}$ and a non-ultralocal Maillet Poisson bracket (with twist function $\varphi(z) = \frac{z^2 - 1}{z^2}$) show that all such theories are classically integrable, including PCM deformations by functions of the stress tensor. The contraction of the stress tensor yields flows in $E(\nu)$; this yields deformations such as $T\overline{T}$, root-$T\overline{T}$, and linear $E(\nu)$, all preserving integrability. The work establishes a unifying auxiliary-field framework for generating PCM-like integrable deformations and hints at broader applications in holography and dualities.

Abstract

We introduce a class of $2d$ sigma models which are parameterized by a function of one variable. In addition to the physical field $g$, these models include an auxiliary field $v_α$ which mediates interactions in a prescribed way. We prove that every theory in this family is classically integrable, in that it possesses an infinite set of conserved charges in involution, which can be constructed from a Lax representation for the equations of motion. This class includes the principal chiral model (PCM) and all deformations of the PCM by functions of the energy-momentum tensor.