Representation Theoretic Study of Two Dimensional Quantum Field Theory

一、与顶点算子代数相关的共形场论80年代中期，Borcherds、Frenkel等人利用共形场论中的算子积展开，创立了手性顶点算子代数理论，并将其应用于有限群论中的Monster研究。但是除了Virasoro代数的最小级数和仿射李代数的积分表示外，还没有研究与顶点算子代数相关的共形场论。在与大阪大学的Kiyokazu nagatomu的合作中，我定义了与手性顶点代数相关的通用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，提出了共形块sp ^1理论，给出了沿模空间边界的共形块的有限维数、KZ连接的定义和族分解性质。这些结果于2001年11月在加州大学洛杉矶分校的会议上公布。我们就这个题目写了一篇论文。我们发展了有理椭圆曲面的周期理论，作为1994年Seiberg - Witten提出的N=2超级杨米尔斯理论的推广。在P^1上沿光纤上具有1阶极点的亚纯2型模型- Weil晶格的周期积分可以看作质量参数。我们证明了该周期图的单性可以用E_8^<(1)>型椭圆Weyl群来描述。并由Kyoji Saito证明了该理论与E_8^<(1)>简单椭圆奇点的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.