Integrability and braided tensor categories

TL;DR

This paper introduces a topological approach using braided tensor categories to construct conserved currents in integrable models, generalizing quantum-group methods and providing a new perspective on solving the Yang-Baxter equation.

Contribution

It develops a topological framework for integrability, deriving conditions for Boltzmann weights to admit conserved currents, and introduces a linear method for Baxterising solutions.

Findings

Derived a simple constraint on Boltzmann weights for conserved currents.

Provided a topological method to construct solutions to the Yang-Baxter equation.

Presented examples including a potential new integrable model.

Abstract

Many integrable statistical mechanical models possess a fractional-spin conserved current. Such currents have been constructed by utilising quantum-group algebras and ideas from "discrete holomorphicity". I find them naturally and much more generally using a braided tensor category, a topological structure arising in knot invariants, anyons and conformal field theory. I derive a simple constraint on the Boltzmann weights admitting a conserved current, generalising one found using quantum-group algebras. The resulting trigonometric weights are typically those of a critical integrable lattice model, so the method here gives a linear way of "Baxterising", i.e. building a solution of the Yang-Baxter equation out of topological data. It also illuminates why many models do not admit a solution. I discuss many examples in geometric and local models, including (perhaps) a new solution.