Localization of unitary braid group representations

TL;DR

This paper investigates the localization of unitary braid group representations derived from R-matrices and fusion categories, demonstrating a specific case with the Jones representation at q=exp(πi/6) and proposing broader conjectures.

Contribution

It establishes the first explicit localization of a Jones braid group representation using a 9x9 R-matrix and conjectures about the localizability of a family of related theories.

Findings

Jones representation at q=exp(πi/6) can be localized by a 9x9 R-matrix

First explicit localization of a Jones braid group representation

Proposes conjectures on localizability of (SO(N),2) theories for odd prime N>1

Abstract

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary $R$-matrix and to a simple object in a unitary braided fusion category. Unitary $R$-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q=exp(\pi i/6) specialization of the unitary Jones representation of the braid groups can be localized by a unitary $9\times 9$ R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N),2) for an odd prime N>1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.