2021 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 / Quantum cohomology / tt* equations / Isomonodromy |
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| Outline of Annual Research Achievements |
Progress was made by Principal Investigator in collaboration with A. Its (IUPUI, USA) and C.-S. Lin (NTU, Taiwan) on describing all global solutions of the tt*-Toda equations in the case of SL(n,R), for arbitrary n. A comprehensive article on this topic is in preparation. Progress was also made by Principal Investigator and Co-Investigator Otofuji on the role of representations of the Virasoro algebra in the theory of the tt* equations. This resulted in the first of planned series of publications on this topic. Co-Investigator Hosono continued his research on K3 manifolds and obtained further new results. The Principal Investigator and Co-Investigator Hosono initiated a collaboration with A. Kanazawa on the role of the tt* equations in the theory of K3 manifolds from the point of view of harmonic map theory.
An online conference on "Special Geometry and Integrable Systems" (postponed from 2020) was held with foreign speakers including V. Cortes (Hamburg), C. Hertling (Mannheim), Ian Strachan (Glasgow), A. Swann (Aarhus), as well as Co-Investigators Hosono and Iritani. Guest and Co-Investigator Ohnita co-organised the online 3rd Japan-Taiwan Joint Conference on Differential Geometry. An online workshop was held on "Toda equations, parabolic Higgs bundles, and related topics". The first part of the 13th MSJ-SI "Differential Geometry and Integrable Systems" (postponed from 2020) was held in hybrid format at the end of the year.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The research of the Principal Investigator up to this point has mainly contributed to sub-projects 1 and 6. Preparations have been made with sub-project 3, and new ideas have developed regarding the physics and representation theory aspects of sub-project 6. Sub-project 5 was started concretely through online meetings between the Principal Investigator, Co-Investigator Hosono and A. Kanazawa.
In addition to individual research activities and online activities, it was possible to restart some face to face activities with domestic experts. Seminars at Waseda University on topics related to this project were given by Akane Nakamura (Josai University), Toru Kajigaya (Tokyo University of Science), and Shimpei Kobayashi (Hokkaido University). A workshop (restricted to a small group of participants because of the Covid19 pandemic) was held at Nihon University (Koriyama) in the series "Koriyama Geometry and Physics Days", focusing on field theories and representation theory.
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| Strategy for Future Research Activity |
Plans for 2022-23 include the continuation of the above projects, together with a new project on the asymptotics of arbitrary local solutions of the tt*-Toda equations (initially in the case of SL(4,R)) is planned in collaboration with A. Its (IUPUI, USA) and former Waseda students Ryosuke Odoi and Kenta Miyahara. This will provide useful input from classical o.d.e. theory for sub-project 3 (symplectic aspects of the tt*-Toda equations), in collaboration with N.-K. Ho (NTHU, Taiwan).
It is intended to support the continuation of major activities organised jointly with Co-Investigator Ohnita such as the 13th MSJ-SI at Osaka City University and the RIMS Research Project at Kyoto University. In both activities a resumption of face-to-face participation (including by international speakers) is expected to be possible. Expected visitors for workshops and seminars at Waseda University include K. Wendland (Dublin), C. Meneses (Kiel), E. Heintze (Augsburg).
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