Solutions to generalized Yang-Baxter equations via ribbon fusion categories

TL;DR

This paper explores solutions to generalized Yang-Baxter equations using ribbon fusion categories, providing explicit examples from quantum theories like Ising, $SO(N)_2$, and $SU(3)_3$, and connecting to braid group representations.

Contribution

It introduces a novel approach to constructing solutions to generalized Yang-Baxter equations via objects in ribbon fusion categories, extending known theories.

Findings

Constructed braid group representations from ribbon fusion categories.

Provided explicit examples from Ising, $SO(N)_2$, and $SU(3)_3$ theories.

Described solutions related to Jones-Kauffman theory at roots of unity.

Abstract

Inspired by quantum information theory, we look for representations of the braid groups $B_n$ on $V^{\otimes (n+m-2)}$ for some fixed vector space $V$ such that each braid generator $\sigma_i, i=1,...,n-1,$ acts on $m$ consecutive tensor factors from $i$ through $i+m-1$. The braid relation for $m=2$ is essentially the Yang-Baxter equation, and the cases for $m>2$ are called generalized Yang-Baxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case $m=3$. Examples are given from the Ising theory (or the closely related $SU(2)_2$), $SO(N)_2$ for $N$ odd, and $SU(3)_3$. The solution from the Jones-Kauffman theory at a $6^{th}$ root of unity, which is closely related to $SO(3)_2$ or $SU(2)_4$, is explicitly described in the end.