The Relation between Quantum W algebras and Lie algebras

TL;DR

This paper establishes a systematic, algorithmic approach to understanding quantum W algebras via affine Lie algebra cohomologies, providing new methods for their construction and realization.

Contribution

It introduces a formalism that quantizes the Drinfeld-Sokolov reduction, enabling explicit calculation and embedding of W algebras into affine Lie algebra structures.

Findings

Large set of quantum W algebras can be derived as affine Lie algebra cohomologies.

Provides an algorithmic method for calculating W algebra generators and operator product expansions.

Offers a general approach for constructing free field realizations and Fock resolutions for W algebras.

Abstract

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary $sl_2$ embeddings we show that a large set $\cal W$ of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set $\cal W$ contains many known $W$ algebras such as $W_N$ and $W_3^{(2)}$. Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any $W$ algebra in $\cal W$ can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore {\em any} realization of this semisimple affine Lie algebra leads to a realization of the $W$ algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in $\cal W$. Some examples are explicitly worked out.