Characterizing Invariants for Local Extensions of Current Algebras

TL;DR

This paper investigates invariants of local extensions in current algebras within quantum field theory, comparing algebraic and analytical methods to characterize and classify these extensions.

Contribution

It introduces and compares two methods for computing invariants of local extensions, linking braid group representations with operator algebra techniques.

Findings

Both methods effectively compute characteristic invariant ratios.

The approaches are applicable to classifying local extensions.

The study bridges conformal field theory and operator algebra techniques.

Abstract

Pairs $\aa \subset \bb$ of local quantum field theories are studied, where $\aa$ is a chiral conformal \qft and $\bb$ is a local extension, either chiral or two-dimensional. The local correlation functions of fields from $\bb$ have an expansion with respect to $\aa$ into \cfb s, which are non-local in general. Two methods of computing characteristic invariant ratios of structure constants in these expansions are compared: $(a)$ by constructing the monodromy \rep of the braid group in the space of solutions of the Knizhnik-Zamolodchikov differential equation, and $(b)$ by an analysis of the local subfactors associated with the extension with methods from operator algebra (Jones theory) and algebraic quantum field theory. Both approaches apply also to the reverse problem: the characterization and (in principle) classification of local extensions of a given theory.