Interpolation categories for Conformal Embeddings

Abstract

In this paper we give a diagrammatic description of the categories of modules coming from the conformal embeddings $\mathcal{V}({\mathfrak{sl}}_N,N)\subset \mathcal{V}({\mathfrak{so}}_{N^2-1},1)$. A small variant on this construction (morally corresponding to a conformal embedding of ${\mathfrak{gl}}_N$ level $N$ into ${\mathfrak{o}}_{N^2-1}$ level $1$) has uniform generators and relations which are rational functions in $q=e^{2\pi i/4N}$, which allows us to construct a new continuous family of tensor categories at non-integer level which interpolate between these categories. This is the second example of such an interpolation category for families of conformal embeddings after Zhengwei Liu's interpolation categories $\mathcal{V}({\mathfrak{sl}}_N,N\pm 2)\subset \mathcal{V}({\mathfrak{sl}}_{N(N\pm 1)/2},1)$ which he constructed using his classification Yang-Baxter planar algebras. Our approach is different from Liu's, we build a two-color skein theory, with one strand coming from $X$ the image of defining representation of ${\mathfrak{sl}}_N$ and the other strand coming from an invertible object $g$ in the category of local modules, and a trivalent vertex coming from a map $X\otimes X^{\ast }\to g$. We anticipate small variations on our approach will yield interpolation categories for every infinite discrete family of conformal embeddings.

29 pages

References in corpus (12)

• Braided tensor categories and extensions of vertex operator algebras
• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra
• Operator algebras in rigid C*-tensor categories
• Skein theory for the D_{2n} planar algebras
• Constructing the extended Haagerup planar algebra
• Tensor envelopes of regular categories
• Skein theory for SU(n)-quantum invariants
• Level-rank duality via tensor categories
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Knot polynomial identities and quantum group coincidences
• Examples of subfactors from conformal field theory
• The quantum isomeric supercategory