A topological theory of unoriented SL(4) foams

TL;DR

This paper develops a topological framework for unoriented SL(4) foams, extending previous theories for SL(3), and explores their combinatorial evaluation and state spaces, revealing a connection to 4-colorings.

Contribution

It introduces a combinatorial evaluation method for SL(4) foams and analyzes their state spaces, advancing the understanding of higher-rank foam theories.

Findings

State space of any web is free over a localized ground ring.

The rank of the state space equals the number of 4-colorings.

Extension of foam evaluation to SL(4) case.

Abstract

Unoriented SL(3) foams are two-dimensional CW complexes with generic singularities embedded in 3- and 4-manifolds. They naturally come up in the Kronheimer-Mrowka SO(3) gauge theory for 3-orbifolds and, in the oriented case, in a categorification of the Kuperberg bracket quantum invariant. The present paper studies the more technically complicated case of SL(4) foams. Combinatorial evaluation of unoriented SL(4) foams is defined and state spaces for it are studied. In particular, over a suitably localized ground ring, the state space of any web is free of the rank given by the number of its 4-colorings.