Lax Pair Formulation of the W-gravity Theories in two Dimensions

TL;DR

This paper extends the Lax pair formulation to two-dimensional $w_N$ gravity theories, providing a systematic approach for arbitrary $w_N$ cases and analyzing their constraints, Ward identities, and peculiarities.

Contribution

It introduces a general prescription for formulating $w_N$ gravity theories using Lax pairs, expanding beyond $w_2$ and $w_3$ to arbitrary $w_N$ cases.

Findings

Lax pair formulation applied to $w_2$, $w_3$, and $w_4$ gravities.

Systematic expression of constraints and cosmological constants in zero-curvature conditions.

Analysis of gauge parameter restrictions in presence of constraints.

Abstract

The Lax pair formulation of the two dimensional induced gravity in the light-cone gauge is extended to the more general $w_N$ theories. After presenting the $w_2$ and $w_3$ gravities, we give a general prescription for an arbitrary $w_N$ case. This is further illustrated with the $w_4$ gravity to point out some peculiarities. The constraints and the possible presence of the cosmological constants are systematically exhibited in the zero-curvature condition, which also yields the relevant Ward identities. The restrictions on the gauge parameters in presence of the constraints are also pointed out and are contrasted with those of the ordinary 2d-gravity.