Wakimoto realizations of current algebras: an explicit construction

TL;DR

This paper explicitly constructs generalized Wakimoto realizations of affine current algebras associated with simple Lie algebras, using Hamiltonian reduction and quantum corrections, and explores their properties and applications.

Contribution

It provides an explicit construction of Wakimoto realizations from a novel viewpoint, including formulas at the classical and quantum levels, and details on quantum coordinate transformations and screening charges.

Findings

Derived explicit Wakimoto current formulas at Poisson bracket level.

Quantized the classical formulas with normal ordering and quantum corrections.

Constructed quantum coordinate transformations and identified screening charges.

Abstract

A generalized Wakimoto realization of $\widehat{\cal G}_K$ can be associated with each parabolic subalgebra ${\cal P}=({\cal G}_0 +{\cal G}_+)$ of a simple Lie algebra ${\cal G}$ according to an earlier proposal by Feigin and Frenkel. In this paper the proposal is made explicit by developing the construction of Wakimoto realizations from a simple but unconventional viewpoint. An explicit formula is derived for the Wakimoto current first at the Poisson bracket level by Hamiltonian symmetry reduction of the WZNW model. The quantization is then performed by normal ordering the classical formula and determining the required quantum correction for it to generate $\widehat{\cal G}_K$ by means of commutators. The affine-Sugawara stress-energy tensor is verified to have the expected quadratic form in the constituents, which are symplectic bosons belonging to ${\cal G}_+$ and a current belonging to ${\cal G}_0$. The quantization requires a choice of special polynomial coordinates on the big cell of the flag manifold $P\backslash G$. The effect of this choice is investigated in detail by constructing quantum coordinate transformations. Finally, the explicit form of the screening charges for each generalized Wakimoto realization is determined, and some applications are briefly discussed.