A nonlocal Poisson bracket of the sine-Gordon model

TL;DR

The work demonstrates a nonabelian duality between the classical string on $\mathbb{R}\times S^2$ and the sine-Gordon model, yielding an affine-bundle relation between their phase spaces. By projecting the dual string's symplectic structure onto the sine-Gordon sector and restricting to an integrable foliation, it derives a nonlocal Poisson bracket on sine-Gordon that is compatible with the canonical bracket. This provides a concrete route to relate the quantization of the string sigma-model (under Virasoro constraints) to sine-Gordon quantization with a one-integration nonlocal bracket. The results extend the understanding of nonstandard Poisson structures in AdS/CFT contexts and connect dual string dynamics to sine-Gordon dynamics through a worldsheet-based construction.

Abstract

It is well known that the classical string on a two-sphere is more or less equivalent to the sine-Gordon model. We consider the nonabelian dual of the classical string on a two-sphere. We show that there is a projection map from the phase space of this model to the phase space of the sine-Gordon model. The corresponding Poisson structure of the sine-Gordon model is nonlocal with one integration.

Figures (2)

• Figure 1: The phase space of the classical string on an infinite line is an affine bundle over the phase space of the sine-Gordon model.
• Figure 2: The light cone