Communications in Mathematical Physics

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


  • C. M. Hull


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.


Gauge Group High Spin Gauge Transformation Geometric Structure Quantum Computing 
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© Springer-Verlag 1993

Authors and Affiliations

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

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