标题： 共形嵌入的插值类别 显示英文标题

标题： Interpolation categories for Conformal Embeddings

摘要： 在本文中，我们给出了来自共形嵌入$\mathcal{V}(\mathfrak{sl}_N,N) \subset \mathcal{V}(\mathfrak{so}_{N^2-1},1)$的模的范畴的图示描述。 这个构造的一个小变体（在道德上对应于$\mathfrak{gl}_N$级别$N$到$\mathfrak{o}_{N^2-1}$级别$1$的共形嵌入）具有统一的生成元和关系，这些关系是$q = e^{2 \pi i/4N}$的有理函数，这使我们能够构建一个新的连续族张量范畴，在非整数级别之间进行插值，从而在这些范畴之间进行插值。 这是继 Zhengwei Liu 的插值范畴$\mathcal{V}(\mathfrak{sl}_N, N\pm 2) \subset \mathcal{V}(\mathfrak{sl}_{N(N\pm 1)/2},1)$之后，此类插值范畴的第二个例子，他使用自己的分类 Yang-Baxter 平面代数构造了这些插值范畴。 我们的方法不同于刘的方法，我们构建了一个双色辫子理论，其中一条线来自$X$定义表示的图像，即$\mathfrak{sl}_N$的定义表示，另一条线来自局部模态范畴中的一个可逆对象$g$，以及一个来自映射$X \otimes X^* \rightarrow g$的三价顶点。我们预计我们方法的小变化将为每一个无限离散共形嵌入族产生插值范畴。 显示英文摘要

摘要： 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 $X \otimes X^* \rightarrow g$. We anticipate small variations on our approach will yield interpolation categories for every infinite discrete family of conformal embeddings.