2000 Fiscal Year Final Research Report Summary
Integrable systems with infinite degrees of freedom
| Project/Area Number |
09304002
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| Research Category |
Grant-in-Aid for Scientific Research (A).
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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 | KYOTO UNIVERSITY |
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Principal Investigator |
UENO Kenji Kyoto Univ.Graduate School of Sci., Professor, 大学院・理学研究科, 教授 (40011655)
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| Co-Investigator(Kenkyū-buntansha) |
SAITO Masahiko Kobe Univ.Graduate School of Sci., Professor, 大学院・理学研究科, 教授 (80183044)
SHIMIZU Yuji Int.Chrisitan Univ.Dept.of Gen.Culture., Assoc.Professor, 教養学部, 準教授 (80187468)
MARUYAMA Masaki Kyoto Uriv.Research Inst.Math.Sci., Professor, 大学院・理学研究科, 教授 (50025459)
NAMIKAWA Yukihiko Nagoya Univ.Graduate School of Polymath., Professor, 大学院・多元数理科学研究科, 教授 (20022676)
FUJIWARA Kazuhiro Nagoya Univ. Graduate School of Polymath., Assoc.Professo, 大学院・多元数理科学研究科, 講師 (00229064)
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| Project Period (FY) |
1997 – 2000
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| Keywords | conformal filed theory / moduli space / modular functor / degeneration of curves / Heisenberg algebra / KZB equation / Painleve equation / Painleve equation |
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| Research Abstract |
In the present research we studied mainly conformal filed theory and string theory related to geometry of moduli spaces. We obtained the following main results.
1. Construction of modular functor :
Taking the tensor product of non-abelian conformal field theory and a fractional power of abelian conformal field theory we constructed modularfunctor. This implies that we can construct new invariants of threefolds associated with complex simple Lie algebras.
2. Reconstruction of abelian conformal field theory and study of relationship with degeneration of curves and conformal blocks :
Using Heisenberg algebra and vertex operator algebra we reconstruct abelian conformal field theory. It is a similar construction of non-abelian conformal field theory. This clarifies relationship between degeneration of curves and abelian conformal blocks.
3. Study of KZB equation :
The differential equations describing projectively flat connection of conformal blocks over the moduli space of curves of genus greater than or equal to one is called KZB equation. In the present study we gave new simple description of KZB equation and studied its properties.
4. Study of the moduli spaces of abelian surfaces and K3 surfaces :
Katsura and van der Geer gave stratification of the moduli spaces of abelian surfaces and K3 surfaces using the Artin-Mazur formal groups. They gave explicit description of cycle classes of the loci corresponding to supersingular surfaces.
5. The spaces of initial conditions of Painleve equations :
Saito and his group gave new method of classification of Painleve equations by using the fact the spaces of initial conditions are rational surfaces.
6. Study of superstring theory :
Eguchi and his group studied Landau-Ginzburg models and found a new description of isolated singularities of type E.Saito and his group studied mirror symmetry of rational elliptic surfaces.
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