Representation Theoretic Study of Two Dimensional Quantum Field Theory

I. 80年代中期，Borcherds，Frenkel，等人利用共形场论中的算子乘积展开建立了手征顶点算子代数理论，并将其应用于研究群理论中的Monster，但除了Virasoro代数的极小级数和仿射李代数。在与大坂大学的长友清和的合作中，我定义了与手征顶点代数相关的泛包络代数和零模代数，并重新表述了顶点算子代数的表示理论。在正则性条件下，我们发展了共形块P^1的理论，证明了模空间边界上共形块的有限维性、KZ联络的定义以及共形块沿着的族因子分解性质，这些结果已在2001年11月的UCLA会议上公布.我们写了一篇关于这个题目的论文。拓扑场论及其带质量参数的变形我们发展了有理椭圆曲面的周期理论，作为Seiberg -维滕在1994年发展的N=2超杨米尔斯理论的推广。Mordel-Weil格沿着亚纯2-形式的周期积分可以看作是质量参数，其中亚纯2-形式在纤维上的极点为1阶，在P ^1上的极点为∞。证明了这种周期映射的单值性可用E_8^<（1）>型椭圆Weyl群来描述。并指出了该理论与Kyoji Saito的E_8^<（1）>简单椭圆奇点的形变理论之间的关系。

I. Conformed field theory associated with vertex operator algebraIn the middle of 80's, Borcherds, Frenkel, etc. founded the theory of chiral vertex operator algebra using the operator product expansion in the conformal field theory and apply it to study Monster in finity group theory.But there has not studied conformal field theory associated with vertex operator algebra except case of minimal series of Virasoro algebra and integral representation of Affine Lie algebra.In the collaboration with Kiyokazu Nagatomo at Osaka University, I defined the universal enveloping algebra and zero mod algebra associated with chiral vertex algebra, and reformulated the representation theory of vertex operator algebra. And under the regularity condition we developed the theory the conformed blocksP^1 and showed finite dimensionality, definition of KZ connection and family factorization properties of conformal blocks along the boundary of Moduli spaces.These results was announsed at the meeting at UCLA in November 2001. We wrote a paper on this subject.II. Topological field theory and their deformation with mass parameterWe developed the period theory of rational elliptic surfaces as extension of N=2 super Yang Mills theory developed by Seiberg - Witten in 1994. The period integral of the Mordel - Weil lattice along the meromorphic 2-form which has order 1 pole on the fiber at ∞ on P^1, can be regarded as mass parameters. We showed that monodromy of this period map can be described by E_8^<(1)> type elliptic Weyl group. And we showed the relation ship between this theory and deformation theory of E_8^<(1)> simply elliptic singularity by Kyoji Saito.