Canonical structure of the non-linear σ-model in a polynomial formulation

Summary

Abstract

We study the canonical formulation of the SU(N) non-linear σ-model in a polynomial, first-order representation. The fundamental variables in this description are a non-Abelian vector field L_µ and a non-Abelian antisymmetric tensor field θ_µν, which constrains L_µ to be a ‘pure gauge’ (F_µν(L) = 0) field. The second-class constraints that appear as a consequence of the first-order nature of the Lagrangian are solved, and the corresponding reduced phase-space variables explicitly found. We also treat the first-class constraints due to the gaugein variance under transformations of the antisymmetric tensor field, constructing the corresponding most general gauge-invariant functionals, which are used to describe the classical dynamics of the physical degrees of freedom. We present these results in detail in 1 + 1, 2 + 1 and 3 + 1 dimensions, mentioning some properties of the d + 1-dimensional case. We show that there is a kind of duality between this description of the non-linear σ-model and the massless Yang-Mills theory. The duality is further extended to more general first-class systems.

Introduction

One of the distinctive properties of the non-linear σ-model [1], is that its dynamical variables belong to a non-linear manifold [2], thus realising the corresponding symmetry group in a non-linear fashion [3]. Whence either the Lagrangian becomes non-polynomial in terms of unconstrained variables, or it becomes polynomial but in variables which satisfy a non-linear constraint. It is often convenient to work in a polynomial or ‘linearized’ representation of the model, where the symmetry is linearly realised.