WGravity
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 spacetime, 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 spacetime. 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 selfdual fourdimensional geometries and can be analysed using twistor methods, allowing contact to be made with other formulations of $W$gravity. While the twistor transform for selfdual spaces with one Killing vector reduces to a Legendre transform, that for two Killing vectors gives a generalisation of the Legendre transform.
 Publication:

arXiv eprints
 Pub Date:
 November 1992
 arXiv:
 arXiv:hepth/9211113
 Bibcode:
 1992hep.th...11113H
 Keywords:

 High Energy Physics  Theory
 EPrint:
 49 pages, QMW926