The Effective Action of $W_3$ Gravity to All Orders

TL;DR

This paper derives the all-order effective action for chiral $W_3$ gravity, showing it reduces to a constrained WZW model and analyzing renormalization constants consistent with previous results.

Contribution

It establishes a connection between the effective action of $W_3$ gravity and a constrained WZW model, extending previous one-loop calculations to all orders.

Findings

Effective action reduces to a constrained WZW model.

Renormalization constants are identified consistent with KPZ formulas.

Provides a framework for all-order calculations in $W_3$ gravity.

Abstract

The effective action for chiral $W_3$ gravity is studied. It is shown that the computation of the effective action can be reduced to that of a $SL(3,\re)$ Wess-Zumino-Witten theory. If one assumes that the effective action for the Wess-Zumino-Witten model is identical to the WZW action up to multiplicative renormalizations, then the effective action for $W_3$ gravity is, to all orders, given by a constrained WZW model. The multiplicative renormalization constants of the WZW model are discussed and it is analyzed which particular values of these constants are consistent with previous one-loop calculations, and which reproduce the KPZ formulas for gravity and their generalizations for $W_3$ gravity.