Current Algebra and Integrability of Principal Chiral Model on the World-sheet with General Metric

TL;DR

This paper investigates the classical current algebra and integrability of the principal chiral model on a two-dimensional world-sheet with a general metric, establishing metric-independent Poisson brackets and Lax connection properties.

Contribution

It develops the Hamiltonian formalism for the model, derives the Poisson brackets for currents and Lax connection, and proves the existence of a metric-independent Lax pair.

Findings

Poisson brackets between currents are derived

Lax connection's Poisson bracket is metric-independent

Existence of Lax connection is established

Abstract

We study the classical current algebra for principial chiral model defined on two dimensional world-sheet with general metric. We develop the Hamiltonian formalism and determine the form of the Poisson brackets between currents. Then we determine the Poisson bracket for Lax connection and we show that this Possion bracket does not depend on the world-sheet metric. We also study the Nambu-Gotto form of this model. We prove an existence of the Lax connection and determine their Poisson bracket.