A BRST Analysis of $W$-symmetries

TL;DR

This paper conducts a classical BRST analysis of $w_N$-algebras, revealing a nested subalgebra structure and providing a quantum BRST operator, with implications for $W$-string theories and minimal models.

Contribution

It introduces a new basis for $w_N$-algebras that makes their nested subalgebra structure explicit and derives a corresponding nested BRST charge, extending to quantum cases.

Findings

Nested subalgebra structure of $w_N$-algebras revealed.

Quantum BRST operator for $W_4$-algebra constructed.

Connections to minimal models discussed.

Abstract

We perform a classical BRST analysis of the symmetries corresponding to a generic $w_N$-algebra. An essential feature of our method is that we write the $w_N$-algebra in a special basis such that the algebra manifestly has a ``nested'' set of subalgebras $v_N^N \subset v_N^{N-1} \subset \dots \subset v_N^2 \equiv w_N$ where the subalgebra $v_N^i\ (i=2, \dots ,N)$ consists of generators of spin $s=\{i,i+1,\dots ,N\}$, respectively. In the new basis the BRST charge can be written as a ``nested'' sum of $N-1$ nilpotent BRST charges. In view of potential applications to (critical and/or non-critical) $W$-string theories we discuss the quantum extension of our results. In particular, we present the quantum BRST-operator for the $W_4$-algebra in the new basis. For both critical and non-critical $W$-strings we apply our results to discuss the relation with minimal models.