Relations in Grassmann Algebra Corresponding to Three- and Four-Dimensional Pachner Moves

Abstract

New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally - using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewise-linear manifold with triangulated boundary, and present example calculations confirming its nontriviality.

References in corpus (5)

• The Ponzano-Regge model
• A Euclidean Geometric Invariant of Framed (Un)Knots in Manifolds
• A simple topological quantum field theory for manifolds with triangulated boundary
• Two deformations of a fermionic solution to pentagon equation
• A matrix solution to pentagon equation with anticommuting variables

Cited by in corpus (6)

• Simplex and Polygon Equations
• Matrix factorizations and pentagon maps
• Pentagon Relations in Direct Sums and Grassmann Algebras
• Parameterizing the Simplest Grassmann-Gaussian Relations for Pachner Move 3-3
• Multiplicative expression for the coefficient in fermionic 3-3 relation
• Free fermions on a piecewise linear four-manifold. II: Pachner moves