Project/Area Number  09440023 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
Algebra

Research Institution  KYUSHU UNIVERSITY 
Principal Investigator 
CHO Koji Kyushu University, Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (10197634)

CoInvestigator(Kenkyūbuntansha) 
NAKAYASHIKI Atsushi Kyushu University, Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (10237456)
渡辺 文彦 九州大学, 大学院・数理学研究科, 助手 (20274433)
SATO Eiichi Kyushu University, Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (10112278)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥5,400,000 (Direct Cost : ¥5,400,000)
Fiscal Year 1998 : ¥2,400,000 (Direct Cost : ¥2,400,000)
Fiscal Year 1997 : ¥3,000,000 (Direct Cost : ¥3,000,000)

Keywords  Integrable system / KZ equation / Quantum Group / integral solution / twisted (co) homology / Riemann relation / rational variety / Fano variety / 可積分場の量子論 / KZ方程式 / 量子群 / 積分解 / ツイストされた(コ)ホモロジー / リーマン関係式 / 有理多様体 / Fano 多様体 / K2方程式 / 量子論 / ツイストされた(2)ホモロシー / Fano多様体 / 形状因子 / qクニズニクザモロジコフ方程式 / レベル零 / 相関関数 / 頂点作用素 
Research Abstract 
We mainly study Geometric structure of two dimensional integrable quantumn field theory. It is expected that KZ equation at level zero should be a subsystem of GaussManinn system associated with some family of algebraic curves. In fact, it is true in the case of equations associated with S/N.In these cases, the Riemann relations of algebraic curves should play essential roles in order to study the structure of solvable models. We obtain some interesting results concerning with the Riemann relations of algebraic curves between the twisted homologies and cohomologies. It is also expected that the solutions of KZ equation at level zero can be expressed in terms of theta constants. If we deform these theta constants by introducing new parameters coming from the deformation of Jacobi varieties, we possibly find some relation between two dimensional integrable quantum field theory and the corresponding classical integrable systems. This study is closely related to modular forms, theta constants, Abel integrals and their classical relations. Though we cannot get any definite results yet, we get some results on Fano varieties with large dimensional rational varieties, which may have something to do with this field, and hope to contribute to these areas. As the next stage of our study, we must further investigate the structure of solvable models of two dimensional integrable quantum field theory on a basis of results of algebraic geometry such as ones we obtained.
