arXiv 2001.080482020-01-22math.RThep-thmath.QA

顶点代数$\mathcal R^{(p)}$和$\mathcal V^{(p)}$

The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$

Drazen Adamovic, Thomas Creutzig, Naoki Genra, Jinwei Yang

发表信息：: Commun. Math. Phys. 383, 1207-1241 (2021)
DOI：: 10.1007/s00220-021-03950-1

摘要：

在[2]中引入的顶点代数$V^{(p)}$和$R^{(p)}$是著名的对数CFT三元组代数的非常有趣的亲戚。代数$V^{(p)}$（分别为$R^{(p)}$）是单仿射顶点代数$L_k(\mathfrak{sl}_2)$（分别为$L_k(\mathfrak{sl}_2)$乘以海森堡代数）的大扩展，在$k=-2+1/p$层上对正整数$p$。在本文中，我们得到了这些代数的结构结果，并证明了来自表示论和物理学的各种推论。我们证明了SU（2）作为$V^{(p)}$上的自同构，并将$V^{(p)}$分解为$L_k(\mathfrak{sl}_2)$ -模，将$R^{(p)}$分解为$L_k(\mathfrak{gl}_2)$ -模.对$V^{(p)}$的分解表明，$V^{(p)}$是角顶点代数在S-对偶下的大水平极限.我们还证明了$V^{(p)}$的量子哈密顿约化是在[12]中引入的对数偶线代数$A^{(p)}$，而$R^{(p)}$的约化得到[39]中的$B^{(p)}$ -代数.相反，我们实现了$V^{(p)}$和$R^{(p)}$从$A^{(p)}$和$B^{(p)}$通过一个过程，值得被称为逆量子哈密顿约化。作为推论，我们得到了$k=-2+1/p$层上的普通$L_k(\mathfrak{sl}_2)$ -模范畴$KL_{k}$是一个刚性顶点张量范畴，等价于范畴Rep $(SU(2))$的一个扭曲.这最终完成了$L_k(\mathfrak{sl}_2)$在所有级别$k$上的刚性编织张量范畴结构。我们还建立了某些顶点算子代数扩张的唯一性结果，并利用这个结果证明了$R^{(p)}$和$B^{(p)}$在边界容许水平上都是$A$型非主W-代数.同样的唯一性结果也表明$R^{(p)}$和$B^{(p)}$是$(A_1, D_{2p})$和$(A_1, A_{2p-3})$型的Argyres-Douglas理论的手征代数.

英文摘要：

The vertex algebras $V^{(p)}$ and $R^{(p)}$ introduced in [2] are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{(p)}$ (respectively, $R^{(p)}$) is a large extension of the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ (respectively, $L_k(\mathfrak{sl}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $V^{(p)}$ and we decompose $V^{(p)}$ as an $L_k(\mathfrak{sl}_2)$-module and $R^{(p)}$ as an $L_k(\mathfrak{gl}_2)$-module. The decomposition of $V^{(p)}$ shows that $V^{(p)}$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $V^{(p)}$ is the logarithmic doublet algebra $A^{(p)}$ introduced in [12], while the reduction of $R^{(p)}$ yields the $B^{(p)}$-algebra of [39]. Conversely, we realize $V^{(p)}$ and $R^{(p)}$ from $A^{(p)}$ and $B^{(p)}$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $KL_{k}$ of ordinary $L_k(\mathfrak{sl}_2)$-modules at level $k=-2+1/p$ is a rigid vertex tensor category equivalent to a twist of the category Rep$(SU(2))$. This finally completes rigid braided tensor category structures for $L_k(\mathfrak{sl}_2)$ at all levels $k$. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $R^{(p)}$ and $B^{(p)}$ are certain non-principal W-algebras of type $A$ at boundary admissible levels. The same uniqueness result also shows that $R^{(p)}$ and $B^{(p)}$ are the chiral algebras of Argyres-Douglas theories of type $(A_1, D_{2p})$ and $(A_1, A_{2p-3})$.

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英文摘要：

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