A matrix solution to pentagon equation with anticommuting variables

We construct a solution to pentagon equation with anticommuting variables living on two-dimensional faces of tetrahedra. In this solution, matrix coordinates are ascribed to tetrahedron vertices. As matrix multiplication is noncommutative, this provides a “more quantum” topological field theory than in our previous works.

💡 Research Summary

The paper presents a novel solution to the pentagon equation, a fundamental algebraic identity that underlies the invariance of three‑dimensional topological quantum field theories (TQFTs) under Pachner moves. While earlier constructions employed commuting scalar variables attached to the faces or edges of tetrahedra, the authors introduce two key innovations: (1) matrix coordinates are assigned to the vertices of each tetrahedron, and (2) anticommuting (Grassmann) variables are placed on the two‑dimensional faces.

By endowing each vertex with an (n\times n) complex matrix, the connectivity of a tetrahedron is encoded in a product of matrices. Because matrix multiplication is non‑commutative, the order in which vertices are traversed matters, thereby capturing the subtle ordering information that scalar models cannot represent. This non‑commutativity is essential when one performs a 3‑2 Pachner move, which replaces three tetrahedra sharing an edge with two tetrahedra sharing a face. The move reshuffles the vertex ordering, and the matrix formalism naturally reflects the resulting change in the algebraic expression.

The anticommuting variables (\xi_{ijk}) attached to each triangular face satisfy (\xi_i\xi_j = -\xi_j\xi_i) and (\xi_i^2=0). They are integrated using Berezin integration, which extracts a scalar amplitude from the mixed matrix‑Grassmann polynomial. The authors construct a combined object (\Phi(A,B,C,D,E;\xi)) that is a polynomial in the matrix entries of the five vertex matrices (A,\dots,E) and the Grassmann variables of the ten faces of a 4‑simplex. The pentagon relation is then expressed as an equality of Berezin integrals before and after the Pachner move:

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