Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 091, 7 pages      arXiv:0708.3788      https://doi.org/10.3842/SIGMA.2007.091
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Dimensional Reduction of Conformal Tensors and Einstein-Weyl Spaces

Roman Jackiw
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received August 29, 2007, in final form September 16, 2007; Published online September 21, 2007

Abstract
Conformal Weyl and Cotton tensors are dimensionally reduced by a Kaluza-Klein procedure. Explicit formulas are given for reducing from four and three dimensions to three and two dimensions, respectively. When the higher dimensional conformal tensor vanishes because the space is conformallly flat, the lower-dimensional Kaluza-Klein functions satisfy equations that coincide with the Einstein-Weyl equations in three dimensions and kink equations in two dimensions.

Key words: conformal tensors; dimensional reductions; Kaluza-Klein.

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References

  1. Bergmann P.G., Introduction to the theory of relativity, Prentice Hall, New York, 1942.
  2. Grumiller D., Jackiw R., Kaluza-Klein reduction of conformally flat spaces, Intern. J. Modern Phys. D 15 (2006), 2075-2094, math-ph/0609025.
  3. Guralnik G., Iorio A., Jackiw R., Pi S.Y., Dimensionally reduced gravitational Chern-Simons term and its kink, Ann. Physics 308 (2003), 222-236, hep-th/0305117.
  4. Grumiller D., Kummer W., The classical solutions of the dimensionally reduced gravitational Chern-Simons theory, Ann. Physics 308 (2003), 211-221, hep-th/0306036.
  5. Cacciatori S., Calderelli M., Klemm D., Mansi D., More on BPS solutions of N = 2, D = 4 gauged supergravity, JHEP 2004 (2004), no. 7, 061, 31 pages, hep-th/0406238.
    Bergamin L., Grumiller D., Iorio A., Nuñez C., Chemistry of Chern-Simons supergravity: reduction to a BPS kink, oxidation to M-theory and thermodynamical aspects, JHEP 2004 (2004), no. 11, 021, 40 pages, hep-th/0409273.
    Sahoo B., Sen A., BTZ black hole with Chern-Simons and higher derivative terms, JHEP 2006 (2006), no. 7, 008, 11 pages, hep-th/0601228.
    Sahoo B., Sen A., a¢-correction to external dyonic black holes in heterofic string theory, JHEP 2007 (2007), no. 1, 010, 18 pages, hep-th/0608182.
  6. Eastwood M., Private communication.
  7. Jones P., Tod K., Minitwistor spaces and Einstein-Weyl spaces, Classical Quantum Gravity 2 (1985), 565-577.
    Tod K., Compact 3-dimensional Einstein-Weyl structures, J. Lond. Math. Soc. (2) 42 (1992), 341-351.
    Eastwood M., Tod K., Local constraints on Einstein-Weyl geometries, J. Reine Angew. Math. 491 (1997), 183-198.

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