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Communications in Mathematical Physics

, Volume 156, Issue 2, pp 245–275 | Cite as

W-Geometry

  • C. M. Hull
Article

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 ofW-gravity is analysed in detail. While the gauge group for gravity ind dimensions is the diffeomorphism group of the space-time, the gauge group for a certainW-gravity theory (which isW-gravity in the cased=2) is the group of symplectic diffeomorphisms of the cotangent bundle of the space-time. Gauge transformations forW-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 ifd=1 ord=2, so that only ford=1,2 can actions be constructed. These two cases and the correspondingW-gravity actions are considered in detail. Ind=2, the gauge group is effectively only a subgroup of the symplectic diffeomorphism group. Some of the constraints that arise ford=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 ofW-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.

Keywords

Gauge Group High Spin Gauge Transformation Geometric Structure Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • C. M. Hull
    • 1
  1. 1.Physics DepartmentQueen Mary and Westfield CollegeLondonUK

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