2020 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 |
The Principal Investigator focused on the Lie-theoretic structure of the Stokes data for the tt*-Toda equations, and, this year, especially on its role in theoretical physics. Relations were discovered between Stokes data and asymptotic data on the one hand and work on conformal field theories by Lerch-Vafa et al and Dorey et al. This resulted in 2 publications related to sub-projects 1 and 6. In sub-project 4, Co-Investigator Ohnita found a new proof of the classification of parallel Kaehler submanifolds of complex projective space. In sub-project 5 Co-Investigator Hosono continued his research on K3 manifolds and obtained new results in collaboration with H. Bong and S. T. Yau. The Principal Investigor and Co-Investigator Hosono studied applications of harmonic map theory to the quantum cohomology of K3 manifolds.
The Covid19 pandemic prevented face-to-face activities (including the invitation of foreign researchers) this year, but the organisation and implementation of online seminars and workshops was started. Two series of online lectures related to this project were given, one by Ian McIntosh (University of York, UK) on Riemann surface theory, and one by Takashi Otofuji (Nihon University) on representations of the Virasoro algebra. An online Japan-Taiwan workshop on geometric evolution equations was held. The 27th Osaka City University International Academic Symposium (postponed from 2020) was held online.
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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
Although the planned seminars and workshops were either postponed or held online, research activities continued at a satisfactory pace. Progress was made with sub-projects 1 and 6 (Lie theoretic properties of Stokes matrices, and their physical interpretation). Preparatory work was carried out with sub-project 3 (symplectic aspects of the tt*-Toda equations). Discussions were held regarding sub-project 2 with Co-Investigator Kobayashi and J. Dorfmeister (solutions of the sine-Gordon equation using loop group methods).
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| Strategy for Future Research Activity |
Plans of the Principal Investigator for 2021-22 include: (a) continuation of the collaboration between the Principal Investigator and A. Its (IUPUI, USA) and C.-S. Lin (NTU, Taiwan) on the classical differential equations aspects of the tt*-Toda equations in the case of SL(n,R), for arbitrary n (where new technical obstacles must be overcome); (b) continuation of the collaboration between the Principal Investigator and N.-K. Ho (NTHU, Taiwan) on symplectic aspects of the tt*-Toda equations.
Seminar and conference activities depend on the Covid19 pandemic situation. It is intended to support major activities organised jointly with Ohnita such as the 13th MSJ-SI "Differential Geometry and Integrable Systems" (postponed from 2020) and the RIMS Research Project "Differential Geometry and Integrable Systems - Mathematics of Symmetry, Stability and Moduli" (postponed from 2020). It is intended to support smaller-scale activities such as workshops held during the visits of foreign researchers such as V. Cortes (Hamburg), C.-S. Lin (Taipei), J. Dorfmeister (Munich).
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