2022 Fiscal Year Final Research Report
Integrable hierarchies related to Gromov-Witten invariants
| Project/Area Number |
18K03350
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Kindai University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | グロモフ-ウィッテン不変量 / 可積分階層 / Dubrovin-Zhang理論 / Givental理論 / 格子KP階層 / 戸田階層 / フルヴィッツ数 / ホッジ積分 |
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| Outline of Final Research Achievements |
The Gromov-Witten invariants are a rich source of studies on integrable hierarchies. Major progress therein has been achieved by the Dubrovin-Zhang theory and the Givental theory. The present research is focused on the cases that are related to the lattice P and Toda hierarchies and various reductions thereof. To be more precise, we have considered the Hurwitz numbers and the Gromov-Witten invariants of the Riemann sphere and the Hodge integrals on the moduli space of stable curves, and found that the Volterra-type hierarchies, the equivariant Toda hierarchy, the lattice Gelfand-Dickey hierarchy and the generalized ILW hierarchy show up as the integrable structures of these geometric objects. Moreover, these integrable hierarchies urn out to possess many novel features.
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| Free Research Field |
代数解析，数理物理
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は代数解析的な可積分系研究の一環である．その要となるのはτ函数の概念であり，無限次元グラスマン多様体，無限次元リー群とその表現，自由フェルミ場とそのフォック空間などを駆使してτ函数の構造や性質を記述する．グロモフ-ウィッテン不変量に関するDubrovin-Zhang理論やGivental理論もτ函数の概念を共有しているが，方法論的には代数解析的方法とかなり異質である．本研究はDubrovin-Zhang理論やGivental理論をヒントにして代数解析的な可積分階層の理論の拡張を試みたことに学術的意義がある．この試みはまだ道半ばであり，今後も継続して行く価値がある．
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