Integrability and Braided Tensor Categories
MATHEMATICAL PICTURE LANGUAGE
Paul Fendley - All Souls College, Oxford University
Many integrable critical classical statistical mechanical models and the corresponding quantum spin chains possess a fractional-spin conserved current. Such currents have been constructed by utilizing quantum-group algebras, fermionic and parafermionic operators, and ideas from "discrete holomorphicity''. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory. I derive simple constraints on the Boltzmann weights necessary and sufficient for such a current to exist, generalizing those found using quantum-group algebras. I find many solutions, in both geometric and local models. In all cases, the resulting weights are those of an integrable lattice model, giving a linear construction for "Baxterising'', i.e. building a solution of the Yang-Baxter equation out of topological data.