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Representation Theoretic Study of Two Dimensional Quantum Field Theory

二维量子场论的表示理论研究

基本信息

批准号：
14204003
负责人：
TSUCHIYA Akihiro
金额：
$ 24.04万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

We studied the construction of the conformal field theories associated to Vertex Operator Algebra(V.O.A.) satisfying Zhu's C_2-finiteness condition.In order to construct the conformal field theories we defined universal enveloping algebra associated to V.O.A.satisfying Zhu's C_2-finiteness condition, and analyzed structure of Abelian categories constituted of modules of this algebra in more detail. These Abelian categories are Nothern and Artin, but not semi-simple.We gave an expression of Zhu algebra associated V.O.A.using enveloping algebra of V.O.A., and gave necessary and sufficient condition of the semi-simplicity of the Abelian categories using Zhu algebra.Using above results, we constructed coherent sheaves constituting of conformal blocks over moduli spaces of genus 0 N-pointed stable curves, and showed that they have D-module structures with regular singularities along the boundary of the moduli spaces.By analyzing of bi-module structure of enveloping algebra in further detail, we succeeded the construction of the Dx-module constituted conformal blocks over moduli spaces of genus g N-pointed stable curves, the formulation of the factorization properties of the coherent sheaves along the boundary of the moduli spaces. We are now preparing the paper.We are studying the so-called W(p) algebra satisfying Zhu's C_2-finiteness condition. These algebra are typical examples such that the Abelian categories are not semi-simple. We are now making detailed study of the Abelian categories, and Fusion product of conformal blocks, modular properties of genus 1 conformal blocks. Further more there are some nice relationships between representation theories of quantum groups at root of unity.

我们研究了与顶点算子代数(V.O.A.)有关的共形场论的构造。为了构造满足朱的C_2-有限条件的共形场论，我们定义了满足朱的C_2-有限条件的泛包络代数，并详细分析了该代数的模所构成的阿贝尔范畴的结构。这些Abelian范畴是Nothern和Artin，但不是半单的.我们利用V.O.A.的包络代数给出了伴随V.O.A.的朱代数的一个表达式，并给出了利用朱代数的Abelian范畴是半单的充要条件.利用上述结果，我们构造了亏格为0的N点稳定曲线的模空间上的共形块的凝聚层，并证明了它们在模空间的边界上具有正则奇异性的D-模结构.通过对包络代数的双模结构的进一步分析，我们得到了一些新的结果.我们成功地构造了亏格为g个N点稳定曲线的模空间上的Dx-模构成的共形块，得到了模空间边界上凝聚层的因式分解性质的公式。我们现在正在准备论文，我们正在研究满足朱的C_2-有限条件的所谓的W(P)代数。这些代数是阿贝尔范畴不是半单的典型例子。我们正在对Abel范畴、保形块的融合积、亏格1的保形块的模性质进行详细的研究。此外，在单位根量子群的表示理论之间也有一些很好的联系。