Canonical Structure of the Non-Linear Sigma-Model in a Polynomial Formulation

Abstract

We study the canonical structure of the $SU(N)$ non-linear Sigma-model in a polynomial, first-order representation. The fundamental variables in this description are a non-Abelian vector field L_mu and a non-Abelian antisymmetric tensor field theta_{mu nu}, which constrains L_{mu} to be a `pure gauge' (F_{mu nu}(L) = 0) field. The second-class constraints that appear as a consequence of the first-order nature of the Lagrangian are solved, and the reduced phase-space variables explicitly found. We also treat the first-class constraints due to the gauge-invariance under transformations of the antisymmetric tensor field, constructing the corresponding most general gauge-invariant functionals, which are used to describe the dynamics of the physical degrees of freedom. We present these results 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. This duality is further extended to more general first-class systems.