On the generalized Miura transformation

TL;DR

This paper generalizes the Miura transformation for $ ext{W}_N$ algebras, providing new realizations in terms of simple current algebras, and demonstrates quantization and extensions of these algebraic structures.

Contribution

It introduces a generalized Miura transformation for classical $ ext{W}$ algebras via Hamiltonian reduction, leading to new realizations in terms of non-abelian current algebras.

Findings

Realizations of $ ext{W}_N$ in terms of simple current algebras

Quantization of the $ ext{W}_3^2$ realization

Extension of $ ext{W}_N$ realizations using $ ext{W}_{N-1}$ currents and free bosons

Abstract

By generalizing the Miura transformation for $\Ww_N$ to other classical $\Ww$ algebras obtained by hamiltonian reduction, we find realisations of these algebras in terms of relatively simple non-abelian current algebras, e.g. $\widehat{sl(2)}\times \widehat{u(1)}^N$, generalizing the free field realisation of $\Ww_N$. As an example, we present the $\widehat{sl(2)}\times \widehat{u(1)}$ realisation of $\Ww_3^2$, which we also quantize. By a specific example, we also show how the realisation of $\Ww_N$ with the currents of $\Ww_{N-1}$ and a free boson can be generalized to certain classes of ``extended'' $\Ww_N$ algebras.