Integrable Generalized Principal Chiral Models

TL;DR

This paper investigates a special class of 2D sigma models on group manifolds, deriving conditions for their integrability, providing explicit solutions, and analyzing their behavior under renormalization group flow.

Contribution

It introduces algebraic conditions for integrability in generalized principal chiral models and provides explicit solutions and analysis of their RG flow.

Findings

Derived algebraic conditions for integrability.

Constructed explicit Lax pairs for solutions.

Analyzed RG flow, showing integrability along the flow.

Abstract

We study 2D non-linear sigma models on a group manifold with a special form of the metric. We address the question of integrability for this special class of sigma models. We derive two algebraic conditions for the metric on the group manifold. Each solution of these conditions defines an integrable model. Although the algebraic system is overdetermined in general, we give two examples of solutions. We find the Lax field for these models and calculate their Poisson brackets. We also obtain the renormalization group (RG) equations, to first order, for the generic model. We solve the RG equations for the examples we have and show that they are integrable along the RG flow.