Classification of charge-conserving loop braid representations

TL;DR

This paper classifies charge-conserving loop braid representations as monoidal functors into a category of charge-conserving matrices, revealing a bijection with pairs of plane partitions of total degree N.

Contribution

It provides a complete classification and construction of all N-charge-conserving loop braid representations, linking them to pairs of plane partitions.

Findings

Representations are classified by pairs of plane partitions of total degree N.

All such representations can be explicitly constructed.

The classification establishes a bijection with a combinatorial set of plane partitions.

Abstract

Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{Mat}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that representations fall into varieties indexed by a set in bijection with the set of pairs of plane partitions of total degree $N$.