Tensor categories and the mathematics of rational and logarithmic conformal field theory

TL;DR

This paper reviews the mathematical framework of braided and modular tensor categories derived from vertex operator algebras, crucial for understanding rational and logarithmic conformal field theories, including recent advances in their construction and properties.

Contribution

It provides a comprehensive review of the construction of tensor categories from vertex operator algebras, highlighting new results on their structure and applications to conformal field theories.

Findings

Established operator product expansion for intertwining operators.

Constructed braided tensor categories for rational and logarithmic CFTs.

Proved the rigidity of modular tensor categories for WZNW models.

Abstract

We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess-Zumino-Novikov-Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet \mathcal{W}-algebras as an example of the applications of our general construction of the braided tensor category structure.