Abstract
A parameterization of Grassmann-algebraic relations corresponding to the Pachner move 3–3 is proposed. In these relations, each 4-simplex is assigned a Grassmann weight depending on five anticommuting variables associated with its 3-faces. The weights are chosen to have the “simplest” form—a Grassmann–Gaussian exponent or its analogue (satisfying a similar system of differential equations). Our parameterization works for a Zariski open set of such relations, looks relevant from the algebraic-topological viewpoint, and reveals intriguing nonlinear relations between objects associated with simplices of different dimensions.
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References
Berezin, F.A.: The Method of Second Quantization (in Russian). Nauka, Moscow (1965) [English transl.: Academic Press, New York (1966)]
Berezin, F.A.: Introduction to algebra and analysis with anticommuting variables. In: Palamodov, V.P. (ed.) Moscow State University Press, Moscow (1983) [Expanded transl. into English: Kirillov, A.A. (ed.) Introduction to Superanalysis. D. Reidel, Dordrecht (1987) (Mathematical Physics and Applied Mathematics, vol. 9)]
Chevalley C.: The Algebraic Theory of Spinors and Clifford Algebras (Collected Works, vol. 2), pp. 215. Springer, New York (1997)
Korepanov, I.G.: Geometric torsions and invariants of manifolds with a triangulated boundary. Theor. Math. Phys. 158(1), 82–95 (2009). arXiv:0803.0123
Korepanov, I.G.: Geometric torsions and an Atiyah-style topological field theory. Theor. Math. Phys. 158(3), 344–354 (2009). arXiv:0806.2514
Korepanov, I.G.: Special 2-cocycles and 3–3 Pachner move relations in Grassmann algebra. arXiv:1301.5581
Korepanov, I.G., Sadykov, N.M.: Parameterizing the simplest Grassmann–Gaussian relations for Pachner move 3–3. SIGMA 9, 053 (2013). arXiv:1305.3246
Lickorish, W.B.R.: Simplicial moves on complexes and manifolds. Geom. Topol. Monogr. 2, 299–320 (1999). arXiv:math/9911256
Pachner U.: PL homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Combin. 12, 129–145 (1991)
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Korepanov, I.G. Two-Cocycles Give a Full Nonlinear Parameterization of the Simplest 3–3 Relation. Lett Math Phys 104, 1235–1261 (2014). https://doi.org/10.1007/s11005-014-0707-0
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DOI: https://doi.org/10.1007/s11005-014-0707-0