Classical Symmetries of Some Two-Dimensional Models

TL;DR

This paper investigates the classical hidden symmetries of two-dimensional principal chiral and symmetric space models, revealing algebraic structures like Kac--Moody and Virasoro-like generators, with implications for string theory dualities.

Contribution

It provides a detailed analysis of the algebraic structure of hidden symmetries in classical two-dimensional models, emphasizing the doubled current algebra and boundary conditions.

Findings

Hidden symmetries include Kac--Moody and Virasoro-like algebras.

Doubled current algebra on a line segment offers a better interpretation.

Symmetry stress tensor is singular at the segment ends.

Abstract

It is well-known that principal chiral models and symmetric space models in two-dimensional Minkowski space have an infinite-dimensional algebra of hidden symmetries. Because of the relevance of symmetric space models to duality symmetries in string theory, the hidden symmetries of these models are explored in some detail. The string theory application requires including coupling to gravity, supersymmetrization, and quantum effects. However, as a first step, this paper only considers classical bosonic theories in flat space-time. Even though the algebra of hidden symmetries of principal chiral models is confirmed to include a Kac--Moody algebra (or a current algebra on a circle), it is argued that a better interpretation is provided by a doubled current algebra on a semi-circle (or line segment). Neither the circle nor the semi-circle bears any apparent relationship to the physical space. For symmetric space models the line segment viewpoint is shown to be essential, and special boundary conditions need to be imposed at the ends. The algebra of hidden symmetries also includes Virasoro-like generators. For both principal chiral models and symmetric space models, the hidden symmetry stress tensor is singular at the ends of the line segment.