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W geometry

Nov, 1992
48 pages
Published in:
  • Commun.Math.Phys. 156 (1993) 245-275
e-Print:
DOI:
Report number:
  • QMW-92-6,
  • QMW-92-06
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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∞W_\infty-gravity is analysed in detail. While the gauge group for gravity in dd dimensions is the diffeomorphism group of the space-time, the gauge group for a certain WW-gravity theory (which is W∞W_\infty-gravity in the case d=2d=2) is the group of symplectic diffeomorphisms of the cotangent bundle of the space-time. Gauge transformations for WW-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 det⁡gμν\sqrt { \det g_{\mu \nu}}) only if d=1d=1 or d=2d=2, so that only for d=1,2d=1,2 can actions be constructed. These two cases and the corresponding WW-gravity actions are considered in detail. In d=2d=2, the gauge group is effectively only a subgroup of the symplectic diffeomorphism group. Some of the constraints that arise for d=2d=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 WW-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.
Note:
  • 49 pages, QMW-92-6
  • gravitation
  • gauge field theory: W(infinity)
  • spin: high
  • high: spin
  • transformation: gauge
  • transformation: diffeomorphism
  • constraint
  • twistor
  • differential geometry
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