Classification of charge-conserving loop braid representations

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 $N$.

Classification of charge-conserving loop braid representations Institutional Repository Document