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

SIGMA 9 (2013), 053, 19 pages      arXiv:1305.3246      https://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.

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