Interpolation categories for Conformal Embeddings
Cain Edie-Michell, Noah Snyder
TL;DR
This work develops a two-color diagrammatic framework to interpolate between quantum subgroup categories arising from a conformal embedding, focusing on V( rak{sl}_N,N) V( rak{so}_{N^2-1},1). By introducing SE_N at q = e^{2 π i/(4N)} with a q-Braid and a determinant-based Pair relation, the authors relate the semisimplified Cauchy completion Ab(SE_N) to Rep(U_q( rak{sl}_N))_A, where A is an étale algebra encoding the embedding. They construct a dominant functor Phi from SE_N to the A-module category, prove faithfulness and fullness, and show Ab(SE_N) Rep(U_q( rak{sl}_N))_A, thereby identifying SE_N as a presentation of the quantum subgroup category. The interpolating category E is then built as a generic-q extension, with End_{ E_q}(+^n) isomorphic to the even subalgebras of Hecke-Clifford algebras, and a basis for Hom spaces established via Deligne interpolation, yielding a nontrivial continuous family of tensor categories that interpolate between conformal embedding categories. The approach promises analogous interpolations for other infinite families of embeddings and connects to isomeric Lie superalgebras through endomorphism algebras. Overall, the paper advances diagrammatic and Deligne-style interpolation techniques in the study of tensor categories associated with 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 $q = e^{2 πi/4N}$, 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 $X \otimes X^* \rightarrow g$. We anticipate small variations on our approach will yield interpolation categories for every infinite discrete family of conformal embeddings.