An action variable of the sine-Gordon model

TL;DR

The article identifies and constructs an action variable for the sine-Gordon model by exploiting a hidden U(1)_L symmetry generated through Backlund transformations, which act by shifting the phases of breather excitations. This symmetry can be understood as the nonlinear analogue of decomposing a free field into positive- and negative-frequency components and is shown to emerge from logarithms of Backlund transformations, yielding a local expansion in the null-surface limit via improved currents. The construction mirrors the string-theoretic picture where a length-like quantity is produced by an infinite linear combination of Pohlmeyer charges, thereby linking sine-Gordon dynamics to the integrable structure of classical strings in AdS5×S5 and extending to the O(N) sigma-model. Together, these results illuminate hidden symmetries in integrable field theories and provide a concrete, local realization of action variables within the sine-Gordon framework and its stringy origins.

Abstract

It was conjectured that the classical bosonic string in AdS times a sphere has a special action variable which corresponds to the length of the operator on the field theory side. We discuss the analogous action variable in the sine-Gordon model. We explain the relation between this action variable and the Backlund transformations and show that the corresponding hidden symmetry acts on breathers by shifting their phase. It can be considered a nonlinear analogue of splitting the solution of the free field equations into the positive- and negative-frequency part.

Figures (2)

• Figure 1: The Bäcklund transformation $B_{\mu}$ can be expanded in $1/\mu$ near $\mu=\infty$ and $\tilde{B}_{\tilde{\mu}}$ can be expanded in $\tilde{\mu}$ near $\tilde{\mu}=0$. We analytically continue $B$ and $\tilde{B}$ to $\mu=\tilde{\mu}=1$
• Figure 2: The relation between the hidden symmetry $U(1)_L$ and the Bäcklund transformations. We pick a point $O$ on a "Liouville torus" (for a finite-gap solution, this is an actual finite-dimensional torus). The solid curve $OXY$ represents the 1-parameter family of points $B_{\tilde{\mu}^{-1}}\tilde{B}_{\tilde{\mu}}^{-1}O$ parametrized by $\tilde{\mu}\in [0,1]$. When $\tilde{\mu}=1$ the interval $OY$ (where $Y=B_{1}\tilde{B}_{1}^{-1}O$) is the b-cycle of the torus. The symmetry $U(1)_L$ acts by shifts along this cycle. The arrow $OX$ represents the 1-parameter family of points $\exp\left[t\log(B_{\tilde{\mu}^{-1}}\tilde{B}_{\tilde{\mu}}^{-1}) \right]O$ with $t\in [0,1]$. When $\tilde{\mu}\to 1$ the 1-parameter group $\exp\left[t\log(B_1\tilde{B}_1^{-1})\right]$ with $t\in [0,2]$ is $U(1)_L$.