2023 Fiscal Year Annual Research Report
Differential geometry and integrable systems: exploiting new links
| Project/Area Number |
23H00083
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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) |
望月 拓郎 京都大学, 数理解析研究所, 教授 (10315971)
酒井 高司 東京都立大学, 理学研究科, 教授 (30381445)
中村 あかね 城西大学, 理学部, 准教授 (30782130)
金沢 篤 慶應義塾大学, 総合政策学部(藤沢), 准教授 (40784492)
大仁田 義裕 大阪公立大学, 数学研究所, 特別研究員 (90183764)
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| Project Period (FY) |
2023-04-01 – 2028-03-31
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| Keywords | Integrable systems / Quantum cohomology / tt* equations / Isomonodromy / Geometry |
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| Outline of Annual Research Achievements |
Progress was made by the Principal Investigator (Guest) on several aspects of the tt*-Toda equations. First, in the case of the A_n-type equations, comparisons were made between the tt*-Toda equations and the Toda equations. The difference of signs greatly affects the solvability of the equations, as well as the properties of monodromy data and asymptotic data. Second, the symplectic and Lie-theoretic structure of the monodromy data of the tt*-Toda equations was studied from various points of view. Joint research in these two areas was carried out with Chang-Shou Lin (NTU, Taiwan), Alexander Its (IUPUI, USA), Nan-Kuo-Ho (NTHU, Taiwan), and Ian McIntosh (York, UK). This was reported at several seminar talks. Progress was also made by the Co-Investigators in areas of mirror symmetry, integrable systems, and differential geometry related to this project.
Several research activities were partially supported. Guest and Ohnita were co-organisers of the 4th Taiwan-Japan Joint Conference on Differential Geometry at the NCTS, National Taiwan University, Taipei in November 2023. Guest co-organised a workshop in the series “Koriyama Geometry and Physics Days” at Nihon University (Koriyama) in March 2024. Visiting specialists from abroad included Omar Kidwai (University of Birmingham, UK), Eckhard Meinrenken (University of Toronto, Canada), Florent Schaffhauser (University of Heidelberg, Germany). Ohnita co-organised the 4th International Conference on Surfaces, Analysis, and Numerics in Differential Geometry in February 2024 at Kagawa Prefectural Hall (Takamatsu City, Kagawa).
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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
Progress by the Principal Investigator this year was mainly related to sub-project (a) on the tt*-Toda equations, and sub-project (b) on the Hitchin equations.
Regarding (a1), a thorough study of the monodromy data and asymptotic data of the A_2 Toda equations was made, in order to compare and contrast with the tt*-Toda equations, with a view to future generalizations. This resulted in a preprint "Connection formulae for the radial Toda equations I" (M. A. Guest, A. R. Its, M. Kosmakov, K. Miyahara, R. Odoi, arXiv:2309.16550). The preprint "The tt*-Toda equations of A_n type" (M. A. Guest, A. R. Its, and C.-S. Lin , arXiv:2302.04597) was also updated to its final form in October 2023. Regarding (a2), joint work of Guest and N.-K. Ho on the monodromy data for the A_n tt*-Toda equations continued, and an article on this is in preparation. Also related to subproject (a), Hosono and Kanazawa made further progress with their study of the BCOV equations for Calabi-Yau varieties. Guest and O. Kidwai initiated a new project with the goal of understanding the role of the 2D and 4D tt* equations in the context of BPS solitons and the theory of Dubrovin/Joyce structures.
Regarding (b2), Guest and I. McIntosh classified real forms of the Toda equations for general Lie groups whose solvability is expected to hold, as a generalization of the known results for the A_n tt*-Toda equations. An article on this is in preparation. Also related to subproject (b), Mochizuki and Q. Li continued their joint work on harmonic bundles over noncompact Riemann surfaces.
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| Strategy for Future Research Activity |
The focus for next year will be subprojects (a1), (a2), (b1), (b2). In addition to the task of generalizing results in the A_n case to the case of more general Lie groups, comparing tt*-Toda and Toda provides another promising direction. Although solvability of the Toda equation is more elementary than solvability of the tt*-Toda equation, the monodromy data and asymptotic data of the Toda equation presents interesting new phenomena.
Anticipated activities for next year include visits to Japan by I. McIntosh in order to continue the above work related to subproject (b2), and by N.K. Ho in order to continue the above work related to subproject (a2). Planned workshops include (1) a workshop on differential geometry and integrable systems at Waseda University, (2) a workshop in the series “Koriyama Geometry and Physics Days” at Nihon University (Koriyama). A research event co-organised by Guest, A. Its, and C.-S. Lin is also under consideration.
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