Quadratic heptagon cohomology

TL;DR

This paper introduces a new quadratic cohomology theory for the heptagon relation, an algebraic structure related to 5-dimensional Pachner moves, revealing nontrivial cocycles in specific dimensions.

Contribution

It proposes the quadratic cohomology for the heptagon relation and provides explicit nontrivial cocycles in dimensions 4 and 5.

Findings

Quadratic cohomology is nontrivial for the heptagon relation.

Explicit cocycles are found in dimensions 4 and 5.

Heptagon cohomology shows richer structure compared to pentagon and higher analogues.

Abstract

A cohomology theory is proposed for the recently discovered heptagon relation -- an algebraic imitation of a 5-dimensional Pachner move 4--3. In particular, `quadratic cohomology' is introduced, and it is shown that it is quite nontrivial, and even more so if compare heptagon with either its higher analogues, such as enneagon or hendecagon, or its lower analogue, pentagon. Explicit expressions for the nontrivial quadratic heptagon cocycles are found in dimensions 4 and 5.