Light Cone $W_n$ Geometry and its Symmetries and Projective Field Theory

TL;DR

This paper develops a geometric framework for light cone $W_n$ gravity using generalized projective structures, linking symmetries to gauge transformations of flat ${SL}(n,{f C})$ bundles, and explores quantum aspects and anomalies.

Contribution

It provides a novel geometric formulation of classical light cone $W_n$ geometry and connects $W_n$ symmetries to gauge transformations of flat ${SL}(n,{f C})$ bundles, extending previous results for specific cases.

Findings

Generalized Beltrami differentials parametrize projective structures.

$W_n$ symmetries correspond to gauge transformations of flat ${SL}(n,{f C})$ bundles.

Quantum $W_n$ gravity can be formulated as an induced gauge theory with analyzed anomalies.

Abstract

I show that the generalized Beltrami differentials and projective connections which appear naturally in induced light cone $W_n$ gravity are geometrical fields parametrizing in one-to-one fashion generalized projective structures on a fixed base Riemann surface. I also show that $W_n$ symmetries are nothing but gauge transformations of the flat ${SL}(n,{\bf C})$ vector bundles canonically associated to the generalized projective structures. This provides an original formulation of classical light cone $W_n$ geometry. From the knowledge of the symmetries, the full BRS algebra is derived. Inspired by the results of recent literature, I argue that quantum $W_n$ gravity may be formulated as an induced gauge theory of generalized projective connections. This leads to projective field theory. The possible anomalies arising at the quantum level are analyzed by solving Wess-Zumino consistency conditions. The implications for induced covariant $W_n$ gravity are briefly discussed. The results presented, valid for arbitrary $n$, reproduce those obtained for $n=2,3$ by different methods.