论文标题
顶点代数$ \ MATHCAL R^{（P）} $和$ \ MATHCAL V^{（P）} $
The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$
论文作者
论文摘要
在[2]中引入的顶点代数$ v^{（p）} $和$ r^{（p）} $是对数CFT的著名三胞胎代数的非常有趣的亲戚。代数$ v^{（p）} $（分别分别，$ r^{（p）} $）是简单的仿射顶点algebra $ l_k（\ mathfrak {sl} _2 _2）$的很大扩展对于正整数$ p $。在本文中，我们得出了这些代数的结构结果，并证明了来自代表理论和物理学的各种猜想。 我们表明，su（2）在$ v^{（p）} $上充当自动形态，我们将$ v^{（p）} $分解为$ l_k（\ mathfrak {sl} _2 _2 _2）$ - 模块和$ r^{（p）} $作为$ l_k（$ l_k（$ l_k（p） $ v^{（p）} $的分解表明，$ v^{（p）} $是在s偶尔的上下文中出现的角顶点代数的大级别限制。我们还表明，$ v^{（p）} $的量子减少是[12]中引入的对数Doublet代数$ a^{（p）} $，而$ r^{（p）} $减少$ b^{（p）} $ - algebra of [39]。相反，我们从$ a^{（p）} $和$ a^{（p）} $和$ b^{（p）} $的过程中意识到$ v^{（p）} $和$ r^{（p）} $，通过一个值得称为倒数量子的汉密尔顿减少的过程。作为推论，我们获得了普通$ l_k的类别$ kl_ {k} $（\ mathfrak {sl} _2）$ - 级别$ k = -2+1/p $的模块是一个刚性的顶点tensor类别，等效于类别rep rep $（su（2））$。最终，在所有级别$ k $上完成了$ l_k（\ mathfrak {sl} _2）$的刚性编织张量类别结构。 我们还建立了某些顶点运算符代数扩展的唯一性结果，并使用此结果证明$ r^{（p）} $和$ b^{（p）} $都是某些非主要的$ a $ a $ a $的非主要w- algebras。相同的唯一性结果还表明，$ r^{（p）} $和$ b^{（p）} $是类型$（a_1，d_ {2p}）$（a_1，d_ {2p}）$和$（a_1，a_1，a_1，a_ {2p-3}）的手性代数。
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 $(A_1, A_{2p-3})$.
