Representation Theoretic Study of Two Dimensional Quantum Field Theory

点击翻译按钮获取中文摘要

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.