The braided Doplicher-Roberts program and the Finkelberg-Kazhdan-Lusztig equivalence: A historical perspective, recent progress, and future directions

TL;DR

This paper offers a historical overview and recent progress on the Finkelberg-Kazhdan-Lusztig equivalence, focusing on fiber functors, algebraic structures, and applications to conformal field theory.

Contribution

It introduces a new fiber functor approach to the equivalence theorem, elucidating algebraic structures and suggesting future research directions.

Findings

Constructed a fiber functor explaining algebraic structures

Analyzed structural properties of braided fusion categories

Proposed future research directions

Abstract

Our recent approach to the Finkelberg-Kazhdan-Lusztig equivalence theorem centers on the construction of a fiber functor associated with the categories in the equivalence theorem, which in turn explains the underlying algebraic and analytic structure of the corresponding weak Hopf algebra in a new sense. We provide a non-technical and historical overview of the core arguments behind our proof, discuss these structural properties, and its applications to rigidity and unitarizability of braided fusion categories arising from conformal field theory. We conclude proposing some natural directions for future research.