Interpolation categories for Conformal Embeddings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Edie-Michell, Cain Snyder, Noah |
| author_facet | Edie-Michell, Cain Snyder, Noah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interpolation categories for Conformal Embeddings Edie-Michell, Cain Snyder, Noah Quantum Algebra Operator Algebras Representation Theory 17B37, 18M15, 18M20, 18M30, 46L37 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. |
| title | Interpolation categories for Conformal Embeddings |
| topic | Quantum Algebra Operator Algebras Representation Theory 17B37, 18M15, 18M20, 18M30, 46L37 |
| url | https://arxiv.org/abs/2503.13641 |