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

TL;DR

This paper introduces new algebraic relations involving Grassmann variables that correspond to Pachner moves in 3D and 4D, leading to a novel invariant for 3D manifolds, discovered through computer algebra.

Contribution

It presents the first known relations for Grassmann algebra related to Pachner moves in 3D and 4D, and constructs a new invariant for 3D manifolds based on these relations.

Findings

New algebraic relations for Grassmann variables linked to Pachner moves.

An invariant of 3D manifolds with nontrivial examples.

Relations cannot be fully explained as deformations of torsion-based relations.

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.