Wightman fields for two-dimensional conformal field theories with pointed representation category

TL;DR

This paper constructs explicit Wightman fields for two-dimensional conformal field theories with pointed braided tensor categories, providing new insights into their Hilbert space structure and local extensions.

Contribution

It introduces a method to explicitly realize Wightman fields and local extensions for theories with pointed automorphism categories, connecting to the Longo-Rehren construction.

Findings

Explicit Hilbert space structure for these theories

Construction of primary Wightman fields

Examples including the U(1)-current

Abstract

Two-dimensional full conformal field theories have been studied in various mathematical frameworks, from algebraic, operator-algebraic to categorical. In this work, we focus our attention on theories with chiral components having pointed braided tensor representation subcategories, namely having automorphisms whose equivalence classes necessarily form an abelian group. For such theories, we exhibit the explicit Hilbert space structure and construct primary fields as Wightman fields for the two-dimensional full theory. Given a finite collection of chiral components with automorphism categories with trivial total braiding, we also construct a local extension of their tensor product as a chiral component. We clarify the relations with the Longo-Rehren construction, and illustrate these results with concrete examples including the U(1)-current.