Generalized and quasi-localizations of braid group representations

TL;DR

This paper develops a theory of localizations for braid group representations in braided fusion categories and Yang-Baxter operators, exploring conditions for their uniform modeling and introducing new conjectures.

Contribution

It introduces a framework for localizing braid group representations in monoidal categories, including cases with non-integral dimensions, and formulates a conjecture relating localization existence to object dimensions.

Findings

Localization can exist for objects with non-integral dimension.

Proved special cases of the conjecture relating object dimension to localization.

Established a connection between localizations and square-root of integer dimensions.

Abstract

We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to Yang-Baxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act "locally". Although related to the notion of (quasi-)fiber functors for fusion categories, remarkably, such localizations can exist for representations associated with objects of non-integral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the square-root of an integer and prove several key special cases of the conjecture.