2018 Fiscal Year Annual Research Report
The tt* equations: a bridge between the differential geometry of moduli spaces and classical isomonodromy theory
| Project/Area Number |
18H03668
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| Research Institution | Waseda University |
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Principal Investigator |
Guest Martin 早稲田大学, 理工学術院, 教授 (10295470)
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| Co-Investigator(Kenkyū-buntansha) |
細野 忍 学習院大学, 理学部, 教授 (60212198)
大仁田 義裕 大阪市立大学, 大学院理学研究科, 教授 (90183764)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | Integrable systems / Geometry / Quantum cohomology / tt＊ equations / Isomonodromy |
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| Outline of Annual Research Achievements |
Progress was made by the Principal Investigator (Guest) on various aspects of the tt*-Toda equations and their relation with geometry. This was reported at several seminars in Europe. Progress made by the Co-Investigators (Hosono, Ohnita) was as follows. Hosono worked on mirror symmetry of a certain family of K3 surfaces with Bong Lian, S.-T. Yau and H. Takagi. An explicit K3 analogue of the elliptic lambda function was computed in terms of period integrals and genus two theta functions. Ohnita collaborated with Naoyuki Koike, Makiko Sumi Tanaka and Takashi Sakai on differential geometric aspects of this research project. He obtained a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein- Kaehler C-spaces.
Several research activities were partially supported. Guest and Ohnita organised a UK-Japan Winter School on Variational Problems in Geometry and Mathematical Physics in January 2019 at the University of Leeds, UK. Discussions there with F. Burstall, P. Dorey, I. McIntosh and J. Wood were productive. Guest organised lectures at Waseda in February 2019 by Murad Alim on tt* geometry. Hosono visited B. Lian and S.-T. Yau at Harvard for intensive discussions. He also studied certain pairs of Calabi-Yau manifolds, called Fourier-Mukai partners. He invited Michal Kapustka (Jagiellonian University, Krakow, Poland) to Gakushuin University for discussions on this topic. Ohnita organised an international workshop "Geometry of Submanifolds and Integrable Systems" at Osaka City University in March 2019.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
This project investigates classical approaches (using methods of p.d.e. and isomonodromy theory) and modern approaches (geometric, Lie-theoretic) to solving the tt*-Toda equations, and studies the geometrical and/or physical meaning of special solutions. In particular 3 sub-projects were planned in the initial period of the project.
Sub-projects 1 and 3 were initiated by the Principal Investigator and N.-K. Ho during the first year. The symplectic structure of the space of local solutions of the tt*-Toda equations was identified from two points of view: first the asymptotic data of the solutions, then the monodromy data of the solutions. This was carried out explicitly in the case of any complex Lie group, but only for part of the space of local solutions. It was shown (in joint work with Waseda PhD student Ryosuke Odoi) that the transformation between asymptotic data and monodromy data respects the symplectic structures. Sub-project 2 was also initiated by Guest with J. Dorfmeister. Here the tt*-Toda equations were considered from the viewpoint of loop group theory. In preparation for sub-project 6, Guest clarified a relation between the DPW theory of harmonic maps and earlier work of C.-S. Lin and collaborators.
Hosono completed the preprint "K3 surfaces from configurations of six lines in P^2 and mirror symmetry II - lambda_K3 functions" as joint work with B. Lian and S.-T. Yau (arXiv:1903.09373). Ohnita investigated properties and structures of R-spaces canonically embedded in Kaehler C-spaces from the viewpoint of symplectic geometry and Kaehler geometry.
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| Strategy for Future Research Activity |
Sub-project 1 (Lie-theoretic aspects of the monodromy data) and sub-project 3 (symplectic aspects of the monodromy data) will be continued in joint work with N.-K. Ho. An article is in preparation on this topic. Sub-project 2 (loop group aspects of the tt*-Toda equations) will be continued in collaboration with J. Dorfmeister. Joint work with A. Its and C.-S. Lin on the tau functions of the tt*-Toda equations will be initiated. The W-algebra associated to the Toda equations will be investigated in joint work with T. Otofuji. This is related to sub-projects 3 and 6 (and to previous work of C.-S. Lin and his collaborators).
Activities planned for this year include: (1) lectures by Yvette Kosmann-Schwarzbach (Ecole Polytechnique) and Nan-Kuo Ho (National Tsing-Hua University) on "Poisson geometry, moduli spaces, and applications", to be held 24-26 June 2019 at Waseda University; (2) 2nd Taiwan-Japan Joint Conference on Differential Geometry, to be held 1-5 November 2019 at the NCTS, National Taiwan University; (3) a workshop in the series “Koriyama Geometry and Physics Days” at Nihon University (Koriyama); (4) Osaka City University International Academic Symposium 2019, to be held in March 2020 at Osaka City University.
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