W-Gravity

TL;DR

This paper explores the geometric structure of higher spin gauge theories, focusing on W-infinity gravity, and analyzes its gauge group, actions, and connections to twistor methods in two-dimensional cases.

Contribution

It introduces a higher spin geometric framework for W-infinity gravity, detailing gauge groups, action construction, and twistor analysis for specific dimensions.

Findings

W-infinity gravity involves symplectic diffeomorphisms as gauge group.

Actions can be constructed only in 1 and 2 dimensions.

Twistor methods relate to self-dual geometries and Legendre transforms.

Abstract

The geometric structure of theories with gauge fields of spins two and higher should involve a higher spin generalisation of Riemannian geometry. Such geometries are discussed and the case of $W_\infty$-gravity is analysed in detail. While the gauge group for gravity in $d$ dimensions is the diffeomorphism group of the space-time, the gauge group for a certain $W$-gravity theory (which is $W_\infty$-gravity in the case $d=2$) is the group of symplectic diffeomorphisms of the cotangent bundle of the space-time. Gauge transformations for $W$-gravity gauge fields are given by requiring the invariance of a generalised line element. Densities exist and can be constructed from the line element (generalising $\sqrt { \det g_{\mu \nu}}$) only if $d=1$ or $d=2$, so that only for $d=1,2$ can actions be constructed. These two cases and the corresponding $W$-gravity actions are considered in detail. In $d=2$, the gauge group is effectively only a subgroup of the symplectic diffeomorphism group. Some of the constraints that arise for $d=2$ are similar to equations arising in the study of self-dual four-dimensional geometries and can be analysed using twistor methods, allowing contact to be made with other formulations of $W$-gravity. While the twistor transform for self-dual spaces with one Killing vector reduces to a Legendre transform, that for two Killing vectors gives a generalisation of the Legendre transform.