论文标题
$ \ mathfrak {sl} _2 $ -type Tensor类别的Virasoro代数为Central Charge $ 25 $和申请
An $\mathfrak{sl}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
论文作者
论文摘要
令$ \ Mathcal {o} _ {25} $为Virasoro Lie代数的有限长度模块的顶点代数编织张量张量张量，以中央费用为25 $，其组成因子是可减轻的Verma模块的不可修复的代数。我们表明，$ \ Mathcal {O} _ {25} $是刚性的，其简单对象会产生一个半密度张量子类别类别，该类别的编织张量等同于Abelian $ 3 $ -COCYCLE twister the Of-Mathfrak $ \ Mathfrak {Sl sl} $ -Mmodules。我们还表明，这个$ \ mathfrak {sl} _2 $ -type子类别是编织反向反向的张量，等效于Virasoro Algebra的类似类别中央电荷$ 1 $。作为一个应用程序，我们构建了一个简单的共形顶点代数，其中包含virasoro顶点操作员中央电荷的代数$ 25 $作为$ psl_2（\ mathbb {c}）$ - orbifold。我们还利用结果来研究Arakawa的手性通用中心机代数为$ sl_2 $，$ -1 $，表明它具有相当于$ \ MATHRM {REP} \，PSL_2（\ MATHBB {C}）的对称张量类别的表示类别。该代数是中央费用$ 1 $和25美元的Virasoro Vertex操作员代数的张量产品的扩展，类似于先前由I. Frenkel-Styrkas和I. Frenkel-m构建的Virasoro代数的定期代表。朱。
Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.
