论文标题
带有大中心的顶点代数和kazhdan-lusztig通信
Vertex algebras with big center and a Kazhdan-Lusztig Correspondence
论文作者
论文摘要
某些可变形的顶点代数的家族以变形参数的极限获取一个大中心，类似于临界水平的仿射谎言代数。然后，顶点代数及其表示类别成为这个大中心定义的品种的捆绑包。零纤维成为顶点代数，其他纤维变成了该顶点代数上的扭曲模块。 我们在一类顶点代数$ a^{（p）} [\ mathfrak {g}，κ] $中探索这些想法和猜想的对应关系这些代数以量子几何兰兰兹内核的名称引入，并在4维量子场理论中进行了解释。在限制$κ\至\ infty $的情况下，他们获得了一个大型的中央subergebra，该子代理在$ \ mathfrak {g} $ - 连接的空间上识别出功能环。零纤维预计将是Feigin-Tipunin代数$ W_P（\ Mathfrak {G}）$，其代表类别有望与对数Kazhdan Lusztig猜想相当于小量子组。 我们的严格结果集中在$（\ Mathfrak {g}，1）$和$（\ Mathfrak {SL} _2,2）$的情况下。在本文的第一部分中，我们可能具有独立的兴趣，我们讨论了Aggine Lie代数和三胞胎代数$ W_2（\ Mathfrak {sl} _2 _2）$的扭曲模块（包括单一元素）的扭曲模块（曲折），并介绍了一种扭曲的现场实现方法，以获取扭曲的模块以获取扭曲的模量$（g）$（g f inth pp）$（g f \ g。我们还将发现与大中心与各自的量子组相匹配。在本文的第二部分中，我们将结果与顶点代数束匹配，我们从$ a^{（p）} [\ mathfrak {g}，κ] $的极限进行计算，在我们的情况下，该$具有GKO coset resp。 n = 4个超符号代数。
Certain deformable families of vertex algebras acquire at a limit of the deformation parameter a large center, similar to affine Lie algebras at critical level. Then the vertex algebra and its representation category become a bundle over the variety defined by this large center. The zero-fibre becomes a vertex algebra, the other fibres become twisted modules over this vertex algebra. We explore these ideas and a conjectural correspondence in a class of vertex algebras $A^{(p)}[\mathfrak{g},κ]$ associated to a choice of a finite-dimensional semisimple Lie algebra and an integer $(\mathfrak{g},p)$ and a level $κ$, and some related algebras. These algebras were introduced under the name quantum geometric Langlands kernels and have an interpretation in 4-dimensional quantum field theory. In the limit $κ\to \infty$, they acquire a large central subalgebra identified with the ring of functions on the space of $\mathfrak{g}$-connections. The zero fibre is expected to be the Feigin-Tipunin algebra $W_p(\mathfrak{g})$, whose category of representations is expected to be equivalent to the small quantum group by the logarithmic Kazhdan Lusztig conjecture. Our rigorous results focus on the cases $(\mathfrak{g},1)$ and $(\mathfrak{sl}_2,2)$. In the first part of the paper, which might be of independent interest, we discuss the twisted modules (including twists by unipotent elements) of the affine Lie algebra and of the triplet algebra $W_2(\mathfrak{sl}_2)$, and introduce a method of twisted free field realization to obtain twisted modules for arbitrary $(\mathfrak{g},p)$. We also match our findings to the respective quantum group with big center. In the second part of the paper we match the results to the vertex algebra bundle we compute from the limit of $A^{(p)}[\mathfrak{g},κ]$, which in our cases have realizations as GKO coset resp. N=4 superconformal algebras.
