Representation theory of vertex operator algobras and contormal tield theory

顶点算子算法的表示论与共形领域理论

• 批准号: 14540025
• 负责人: NAGATOMO Kiyokazu
• 金额: $ 2.3万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540025/
• 关键词: conformal field theory; algebraic geometry; representation theory; vertex operator; vertex algebra; mathematical physics; 数理物理; 共形ブロック; モジュラー変換; リーマン面

We have formulated conformal field theory over genus one compact curve by establishing notions of sheaves of coinvariants and sheaves of conformal blocks, and have studied their mathematical structures. One of our tools to analyze these objects is the deformation theory of elliptic curve including the degenerate case. In fact we have used the structure theory of the space of meromorphic functions over elliptic curves with possible singularities only at the origin. The first step of our research is a canonical construction of connections defined on sheaves of coinvariants. These connections describe the dependence of sheaves on the deformation parameter of elliptic curves. Then Weierstrass' zeta function naturally appears as one of key ingredients in this construction. One of the most important problems is the construction of horizontal sections of sheaves of conformal blocks. We also have used the deformation theory to solve the differential equations and we have found that any solutions can be obtained from linear functional over an algebra which is associated to a vertex operator algebra. Consequently we are able to express any horizontal section in terms of traces of the finite-dimensional algebra. More generally there is a one to one correspondence between the space of horizontal sections and. the space of linear functional over the algebra.

通过建立共形块层和共形不变层的概念，建立了亏格为一紧曲线上的共形场论，并研究了它们的数学结构。我们分析这些对象的工具之一是椭圆曲线的变形理论，包括退化的情况。实际上，我们已经使用了椭圆曲线上的亚纯函数空间的结构理论，它只在原点处具有可能的奇点。我们的研究的第一步是一个规范的建设上定义的联系层的coinvariants。这些关系描述了层对椭圆曲线变形参数的依赖性。然后维尔斯特拉斯的zeta函数自然出现在这个建设的关键成分之一。其中最重要的问题之一是共形块体的水平剖面的构造。我们还利用变形理论来求解微分方程，我们发现任何解都可以从与顶点算子代数相关联的代数上的线性泛函中得到。因此，我们能够用有限维代数的迹来表示任何水平截面。更一般地，在水平部分的空间和水平部分的空间之间存在一一对应关系。代数上的线性泛函空间。

项目成果

Toshiyuki Abe, Kiyokazu Nagatomo: "Finiteness of conformal blocks on the projective line"Fields Institute Communications. 39. 1-12 (2003)

Toshiyuki Abe、Kiyokazu Nagatomo：“射影线上共形块的有限性”菲尔兹通信研究所。

Hiroyuki Yamane: "A central extension of Uqsl(2|2)^<(1)> and R-matrices with a new parameter"J.Math.Phys.. 40. 5450-5455 (2003)

Hiroyuki Yamane：“Uqsl(2|2)^<(1)> 和带有新参数的 R 矩阵的中心扩展”J.Math.Phys.. 40. 5450-5455 (2003)

F.Huang, X.Shi, Akitaka Matsumura: "On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary"Osaka Journal of Mathematics. (to appear in).

F.Huang、X.Shi、Akitaka Matsumura：“关于具有自由边界的可压缩纳维-斯托克斯方程的接触不连续稳定性”大阪数学杂志。

Yousuke Ohyama: "Isomonodromy Deformations and Twister theory"ContemMath.. 309. 185-193 (2002)

Yousuke Ohyama：“等单性变形和扭曲理论”ContemMath.. 309. 185-193 (2002)

Toshiyuki Abe: "Finiteness of conformal blocks on the projective line"Fields Institute Communications. 39. 1-12 (2003)

Toshiyuki Abe：“射影线上共形块的有限性”菲尔兹通信研究所。