New Special Operators in $W$-Gravity Theories

TL;DR

This paper discovers new special operators in W_3-gravity theories, connecting them to minimal models and filling gaps in understanding the relationship between W-gravity and 2D gravity coupled to the Ising model.

Contribution

It introduces new physical operators with non-trivial ghost sectors in W_3-gravity and formulates a selection rule for correlator calculations, extending to W_N-gravity and minimal models.

Findings

Identified new operators as Liouville dressings of Ising model energy operators.

Constructed non-ghost operators representing minimal model operators.

Derived the canonical spectrum of W_N-gravity for all N.

Abstract

We find new special physical operators of $W_3 -$gravity having non trivial ghost sectors. Some of these operators may be viewed as the liouville dressings of the energy operator of the Ising model coupled to {\it 2d~gravity} and this fills in a gap in the connection between pure $W_3 -$gravity and Ising model coupled to 2d gravity found in our previous work. We formulate a selection rule required for the calculation of correlators in $W -$gravity theories. Using this rule, we construct the non ghost part of the new operators of $W_N -$gravity and find that they represent the $(N , N+1)$ minimal model operators from both inside and outside the minimal table. Along the way we obtain the canonical spectrum of $W_N -$gravity for all $N$ .