An explicit construction of Wakimoto realizations of current algebras

Abstract

It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra ${\hat{\mathcal{G}}}_k$ can be associated with each parabolic subalgebra $\mathcal{P}=({\mathcal{G}}_0+{\mathcal{G}}_{+})$ of the Lie algebra $\mathcal{G}$, where in the standard case ${\mathcal{G}}_0$ is the Cartan and $\mathcal{P}$ 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 $\mathcal{G}$-valued current in terms of symplectic bosons belonging to ${\mathcal{G}}_{+}$ and a current belonging to ${\mathcal{G}}_0$. 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.

13 pages, LaTeX; a typo corrected in (5.5-6), refs and a remark added

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