An explicit construction of Wakimoto realizations of current algebras

TL;DR

This paper provides an explicit formula for Wakimoto realizations of current algebras associated with any parabolic subalgebra, extending previous work and confirming the expected stress-energy tensor form.

Contribution

It introduces a general explicit construction of Wakimoto realizations for current algebras using Hamiltonian reduction and quantization techniques.

Findings

Derived explicit Wakimoto realization formulas for general parabolic subalgebras.

Established the Poisson bracket realization using symplectic bosons.

Confirmed the quadratic form of the affine-Sugawara stress-energy tensor.

Abstract

It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra $\widehat{\cal G}_k$ can be associated with each parabolic subalgebra ${\cal P}=({\cal G}_0+{\cal G}_+)$ of the Lie algebra ${\cal G}$, where in the standard case ${\cal G}_0$ is the Cartan and ${\cal P}$ is the Borel subalgebra. In this letter we obtain an explicit formula for the Wakimoto realization in the general case. Using Hamiltonian reduction of the WZNW model, we first derive a Poisson bracket realization of the ${\cal G}$-valued current in terms of symplectic bosons belonging to ${\cal G}_+$ and a current belonging to ${\cal G}_0$. We then quantize the formula by determining the correct normal ordering. We also show that the affine-Sugawara stress-energy tensor takes the expected quadratic form in the constituents.