KPZ Analysis for $W_3$ Gravity

TL;DR

This paper explores the BRST quantization of $W_3$ gravity, connecting it to the quantum Drinfeld--Sokolov reduction and deriving the KPZ formula, with implications for Toda actions and recent constructions.

Contribution

It establishes a link between BRST quantization of $W_3$ gravity and the quantum Drinfeld--Sokolov reduction, deriving the KPZ formula within this framework.

Findings

BRST charge interpretation via Drinfeld--Sokolov reduction

Derivation of KPZ formula for $W_3$ gravity

Equivalence of BRST charge to recent constructions in conformal gauge

Abstract

Starting from the covariant action for $W_3$ gravity, we discuss the BRST quantization of $W_3$ gravity. Taking the chiral gauge the BRST charge has a natural interpretation in terms of the quantum Drinfeld--Sokolov reduction for $Sl(3,\re)$. Nilpotency of this charge leads to the KPZ formula for $W_3$. In the conformal gauge, where the covariant action reduces to a Toda action, the BRST charge is equivalent to the one recently constructed by Bershadsky et al.