Kac - Moody algebra is not hidden symmetry of chiral models

A detailed examination of the infinite dimensional loop algebra of hidden symmetry transformations of the Principal Chiral Model reveals it to have a structure differing from a standard centreless Kac-Moody algebra. A new infinite dimensional abelian symmetry algebra is shown to preserve a symplectic form on the space of solutions. 1. We have recently obtained [1] a novel free-field parametrisation of classical solutions of the Principal Chiral Model (PCM). This afforded a very efficient way of determining and classifying the symmetries of the model from natural transformations on the free-field data. In the process we unveiled the precise nature of the ‘loop’ algebra of the celebrated ‘hidden symmetries’ of the PCM ∂−(g ∂+g) + ∂+(g ∂−g) = 0 , (1) where g is a mapping from two-dimensional Minkowski space to U(N). This symmetry algebra has been wrongly identified in the literature as a centreless Kac-Moody algebra. The first purpose of this letter is to correct the record and to highlight the nature of these remarkable symmetries which have recently attracted renewed attention [2]. We also discuss some new commuting symmetries of the PCM discovered in [1], which preserve a natural symplectic structure on the space of solutions of the PCM. 2. The algebraic structure of the hidden symmetry transformations associated with the infinite set of non-local conserved currents [3, 4] of the PCM was first discussed by Dolan [5], who determined the algebra [J r , J b s ] = ∑ c f c J c r+s , r, s ≥ 0 , (2) where f c are structure constants of the Lie algebra of U(N) in a basis {T }; [T , T ] = ∑ c f ab c T . By using, for the infinite set, a compact generating function form of transformation, g 7→ g ( I − Y (x, λ)TY (x, λ) ) , (3) Dolan’s inductive arguments were streamlined in [6], allowing a direct verification of the closure of the commutation relations (2) on the fields g. Here T is a constant infinitesimal antihermitian matrix and Y (x, λ) satisfies the PCM Lax-pair [3, 7],