顶点算子代数及其应用

The theory of vertex operator algebras has deep roots in both mathematics and physics. Vertex operators appeared in the early days of string theory and also representation theory of affine Kac-Moody algebras. It played a prominent role in the construction of the moonshine module by Frenkel, Lepowsky, and Meurman. The precise definition of a vertex operator algebra was given by Borcherds, who used them to prove the moonshine conjecture of Conway and Norton describing the unexpected connection between the representations of the Monster group and modular functions. This decisively established the importance of vertex algebras in representation theory and physics. ..We aim to study a family of vertex algebras associated to the modified regular representations of the Virasoro algebra. These vertex algebras are obtained by pairing up representations that have central charges adding up to 26. Having had a fairly good understanding of the vertex operations in the generic case, we intend to study the structure of these vertex algebras when the central charge parameter is rational. ..We also aim to study the application of vertex algebras to mirror symmetry through the chiral de Rham complex. The chiral de Rham complex on a smooth manifold is a sheaf of vertex superalgebras that locally are built from free bosons and free fermioins corresponding to the local coordinates and their partial derivatives on the cotangent bundle considered as a supermanifold. When the manifold is Calabi-Yau, the cohomology of its chiral de Rham complex is a vertex algebra with an N=2 superconformal structure. It is related to a certain limit of the physics models on the manifold considered in mirror symmetry. We aim to find an effective way of computing the cohomology of the chiral de Rham complex of the Grassmannian and CY complete intersections in it, furthermore to investigate their applications to mirror symmetry.

顶点算子代数出现在共形场论和对Moonshine猜想的研究中，Moonshine猜想把模形式和特殊有限单群的表示联系起来。在Frenkel,Lepowsky,Meurman构造的Moonshine模的基础上Borcherds给出了顶点算子代数的严格定义，并完全证明了Moonshine猜想，这确立了顶点算子代数在数学和物理中的重要性。本项目拟研究一类从Virasoro代数的两个对偶范畴中的表示构造而来的顶点算子代数，特别是当水平参数为有理数时这些顶点算子代数的内部结构。我们也希望研究chiral de Rham complex(cdRc)在镜像对称中的应用。一个卡拉比-丘流形上的cdRc同调是一个有N=2超对称保角不变结构的顶点算子代数，我们将研究如何有效地计算一些流形和它们的子流形（比如Grassmann流形）上的cdRc同调和这些计算在镜对称中的应用。

本项目主要研究顶点算子代数以及它们在表示论与数学物理中的应用的一些前沿问题。我们主要研究了用 Virasoro 代数的表示构造的一类顶点算子代数，顶点算子代数在镜像对称中的应用，Gorenstein Fano 环面簇中卡拉比－丘超曲面的镜像对称，以及齐次空间中卡拉比－丘超曲面的周期积分满足的 tautological 系统等方面的问题。我们的主要结果如下：证明了物理上计算的 Berglund-Hubsch Landau-Ginzburg orbifold 的椭圆亏格与一个顶点算子代数上某个算子的超迹相等，从而从数学上严格证明了镜像对称的 Berglund-Hubsch Landau-Ginzburg orbifold 的椭圆亏格满足预期的对偶性质；证明了关于复射影空间中卡拉比－丘超曲面的周期积分的超平面猜想，并在任意 Gorenstein Fano 环面簇中卡拉比－丘超曲面的周期积分的超平面猜想的证明上取得了重大进展。

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