2015 Fiscal Year Annual Research Report
Systematic development and application of methods in differential geometry and integrable systems motivated by quantum cohomology
| Project/Area Number |
25247005
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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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| Project Period (FY) |
2013-10-21 – 2018-03-31
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| Keywords | Integrable systems / Geometry / Quantum cohomology |
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| Outline of Annual Research Achievements |
The tt*-Toda equations (certain differential equations which play an important role in supersymmetry, differential geometry, and integrable systems) were the main focus of our research. Motivated by quantum cohomology, we developed and applied methods to solve these equations.
The main part of our third and final joint article in the series with A. Its (IUPUI, USA) and C.-S. Lin (National Taiwan University) "Isomonodromy aspects of the tt* equations of Cecotti and Vafa III. Iwasawa factorization and asymptotics" was finished during this period.
Progress was made with a related joint project with N.-K. Ho (National Tsing-Hua University, Taiwan). The purpose of this project is to study the space of solutions of the tt*-equations from the point of view of symplectic geometry.
The TIMS-OCAMI-WASEDA International Workshop on Painleve Equations and Related Topics was held at National Taiwan University, 10-13 May 2015, in connection with this project. A study meeting on the theme "Painleve equations, integrable systems and moduli spaces" was held in the framework of Koriyama Geometry and Physics Days at Nihon University (Koriyama, Fukushima), 6-8 February 2016. The OCAMI-KOBE-WASEDA Joint International Workshop on Differential Geometry and Integrable Systems was held at Osaka City University and Kobe University, 13-17 February 2016.
Visits to Tokyo of Nan-Kuo Ho (National Tsing-Hua University), Yuan-Pin Lee (University of Utah), Kaoru Ono (Kyoto University) were also supported.
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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
Two main themes were proposed: (1) extension and interpretation of previous results on the tt*-Toda equations, (2) intrinsic approach via harmonic bundles and TERP structures. Progress was made with both themes.
Regarding (1), loop group methods were applied to the description of solutions of the tt*-Toda equations, thus completing the main part of the joint article "Isomonodromy aspects of the tt* equations of Cecotti and Vafa III. Iwasawa factorization and asymptotics" with A. Its (IUPUI, USA) and C.-S. Lin (National Taiwan University). This prepared the ground for a systematic Lie-theoretic description of he monodromy data which parametrizes such solutions. It also prepared the ground for a joint project with N.-K. Ho (National Tsing-Hua University, Taiwan), to study the space of solutions of the tt*-Toda equations from the point of view of symplectic geometry.
Regarding (2), the above loop group methods will form the starting of a second joint article with C. Hertling (Mannheim University, Germany) on the intrinsic approach to the sinh-Gordon equation, and (if possible) beyond the sinh-Gordon equation.
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| Strategy for Future Research Activity |
The main priority, following the completion of the series of joint articles with A. Its and C.-S. Lin on the monodromy data and asymptotic data of the tt*-Toda equation, will be to exploit these results in differential geometry.
For this it will be important to exchange ideas with researchers in differential geometry. The following activities are planned in his direction: (1) lecture series by J. Dorfmeister (Technical University of Munich); (2) 1st Japan-Taiwan Conference on Differential Geometry & 8th OCAMI-TIMS Joint International Workshop on Differential Geometry and Geometric Analysis, 13-17 December 2016, Waseda University (co-organized with collaborator Y. Ohnita); (3) UK-Japan Winter School on "Singularities, symmetries and submanifolds", University College London, UK, 4-7 January 2017 (co-organized with collaborators Y. Maeda, Y. Ohnita).
A new (but related) direction is motivated by recent results in mathematical physics. The work of Gaiotto-Moore-Neitzke has used the Hitchin moduli space as a crucial example, and their ideas have led to work of Mazzeo-Swoboda-Weiss-Witt on the "ends" of this moduli space. For rank 2 bundles, these ends are described by the solutions of the sinh-Gordon equation which we have studied. For rank n bundles, it can be expected that solutions of the tt*-Toda equation will play a similar role. A 4-day workshop on String Theory and Mathematics at Waseda University and Tokyo Metropolitan University is planned for November 2016 (co-organized with collaborator S. Ketov).
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[Presentation] Painleve III, old and new2015
Author(s)
Guest Martin
Organizer
TIMS-OCAMI-WASEDA International workshop on Painleve equations and related topics
Place of Presentation
National Taiwan University (Taipei)
Year and Date
2015-05-10
Int'l Joint Research / Invited
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