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

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.

In response to the referees' helpful comments, further discussion of several issues is added, including in particular rigidity. Much of the paper is devoted to explaining what has been mathematically proved and where people can find these proofs. Several references added. 27 pages. Invited review article for publication in a special issue of J. Phys. A on logarithmic conformal field theory

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• Symplectic fermions and a quasi-Hopf algebra structure on ${\bar{U}}_{i}sl(2)$
• On the tensor structure of modules for compact orbifold vertex operator algebras
• Gluing vertex algebras
• Twisted modules and $G$-equivariantization in logarithmic conformal field theory
• Direct limit completions of vertex tensor categories
• Unitarity of the modular tensor categories associated to unitary vertex operator algebras, I
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• Integral forms for tensor powers of the Virasoro vertex operator algebra $L(\frac{1}{2},0)$ and their modules
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
• Unitarity of the modular tensor categories associated to unitary vertex operator algebras, II
• Intertwining operators among modules for affine Lie algebra and lattice vertex operator algebras which respect integral forms
• A general mirror equivalence theorem for coset vertex operator algebras
• Vertex algebraic intertwining operators among generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$
• Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras
• $K$-theory of AF-algebras from braided C*-tensor categories
• The Drinfeld-Kohno theorem for the superalgebra $gl(1|1)$
• Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
• Regular vertex operator subalgebras and compressions of intertwining operators
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras