Representation Theoretic Study of Two Dimensional Quantum Field Theory

在80年代中期，S、博尔切尔德、弗伦克尔等人利用共形场论中的算符乘积展开创立了手征顶点算符代数理论，并将其应用于研究有限群理论中的怪物。但除了Virasoro代数的极小级数和仿射Lie代数的积分表示的情况外，还没有研究与顶点算符代数相关的共形场论。我与大阪大学的Kiyokazu Nagtopo合作，定义了与手征顶点代数相关的泛包络代数和零模代数，并重新阐述了顶点算符代数的表示理论。在正则性条件下，我们发展了共形块P^1的理论，并证明了沿模空间边界的共形块的有限维、KZ联络的定义和族分解性质.这些结果在2001年11月的UCLA会议上公布.我们发展了有理椭圆曲面的周期理论，作为Seiberg-Witten于1994年提出的N=2超杨Mills理论的推广。在P^1的∞处沿纤维上具有1阶极点的亚纯2形式的Model-Weil晶格的周期积分可视为质量参数。证明了该周期映射的单调性可以用E_8^&lt；(1)&gt；型椭圆Weyl群来刻画。并由Kyoji Saito证明了该理论与E_8^&lt；(1)&gt；简单椭圆奇点变形理论之间的关系。

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.