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


SIGMA 9 (2013), 053, 19 pages      arXiv:1305.3246      http://dx.doi.org/10.3842/SIGMA.2013.053

Parameterizing the Simplest Grassmann-Gaussian Relations for Pachner Move 3-3

Igor G. Korepanov and Nurlan M. Sadykov
Moscow State University of Instrument Engineering and Computer Sciences, 20 Stromynka Str., Moscow 107996, Russia

Received May 15, 2013, in final form August 08, 2013; Published online August 13, 2013

Abstract
We consider relations in Grassmann algebra corresponding to the four-dimensional Pachner move 3-3, assuming that there is just one Grassmann variable on each 3-face, and a 4-simplex weight is a Grassmann-Gaussian exponent depending on these variables on its five 3-faces. We show that there exists a large family of such relations; the problem is in finding their algebraic-topologically meaningful parameterization. We solve this problem in part, providing two nicely parameterized subfamilies of such relations. For the second of them, we further investigate the nature of some of its parameters: they turn out to correspond to an exotic analogue of middle homologies. In passing, we also provide the 2-4 Pachner move relation for this second case.

Key words: four-dimensional Pachner moves; Grassmann algebras; Clifford algebras; maximal isotropic Euclidean subspaces.

pdf (453 kb)   tex (25 kb)

References

  1. Berezin F.A., The method of second quantization, Pure and Applied Physics, Vol. 24, Academic Press, New York, 1966.
  2. Berezin F.A., Introduction to superanalysis, Mathematical Physics and Applied Mathematics, Vol. 9, D. Reidel Publishing Co., Dordrecht, 1987.
  3. Björk J.E., Rings of differential operators, North-Holland Mathematical Library, Vol. 21, North-Holland Publishing Co., Amsterdam, 1979.
  4. Cartier P., Démonstration "automatique" d'identités et fonctions hypergéométriques (d'après D. Zeilberger), Séminaire Bourbaki, Vol. 1991/92, Astérisque 206 (1992), Exp. No. 746, 3, 41-91.
  5. Chevalley C., The algebraic theory of spinors and Clifford algebras, Collected works, Vol. 2, Springer-Verlag, Berlin, 1997.
  6. Korepanov I.G., Geometric torsions and an Atiyah-style topological field theory, Theoret. and Math. Phys. 158 (2009), 344-354, arXiv:0806.2514.
  7. Korepanov I.G., Relations in Grassmann algebra corresponding to three- and four-dimensional Pachner moves, SIGMA 7 (2011), 117, 23 pages, arXiv:1105.0782.
  8. Korepanov I.G., Deformation of a 3→3 Pachner move relation capturing exotic second homologies, arXiv:1201.4762.
  9. Korepanov I.G., Special 2-cocycles and 3-3 Pachner move relations in Grassmann algebra, arXiv:1301.5581.
  10. Korepanov I.G., Sadykov N.M., Pentagon relations in direct sums and Grassmann algebras, SIGMA 9 (2013), 030, 16 pages, arXiv:1212.4462.
  11. Lickorish W.B.R., Simplicial moves on complexes and manifolds, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., Vol. 2, Geom. Topol. Publ., Coventry, 1999, 299-320, math.GT/9911256.
  12. Pachner U., P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), 129-145.
  13. Petkovsek M., Wilf H.S., Zeilberger D., A=B, A K Peters Ltd., Wellesley, MA, 1996.

Previous article  Next article   Contents of Volume 9 (2013)