1998 Fiscal Year Final Research Report Summary
Geomatric structure of Solvable Models
| Project/Area Number |
09440023
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KYUSHU UNIVERSITY |
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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)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAYASHIKI Atsushi Kyushu University, Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (10237456)
SATO Eiichi Kyushu University, Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (10112278)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Integrable system / KZ equation / Quantum Group / integral solution / twisted (co) homology / Riemann relation / rational variety / Fano variety |
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| 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 Gauss-Maninn 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.
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