2016 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 geom etry, and integrable systems) were the main focus of our research. Motivated by quantum cohomology, we developed and ap plied methods to solve these equations.
The joint project with N.-K. Ho (National Tsinghua University, Taiwan) on the symplectic/differential geometry of the tt*-Toda equations was continued. The article "A Lie-theoretic description of the solution space of the tt*-Toda equations" was finished. In this article the convex set appearing in the joint articles with Its and Lin was described Lie-theoretically, using a framework of P. Boalch.
Several workshops and conferences related to this project were organised. The "1st Japan-Taiwan Conference on Differential Geometry" was held at Waseda University, 13-17 December 2016. The "String Theory Meeting in the Greater Tokyo Area" was held 28-29 November 2016 at Waseda University and 1-2 December 2016 at Tokyo Metropolitan University. The conference "Flat connections, Higgs bundles and Painleve equations" was held 1-5 May 2016 at National Taiwan University. A study meeting on the theme "Geometric quantization and related topics" was held in the framework of the Koriyama Geometry and Physics Days at Nihon University (Koriyama, Fukushima), 13-14 February 2017.
A series of lectures on Quasi-Hamiltonian Geometry was given by Eckhard Meinrenken (University of Toronto, Canada), 23-24 June 2017 at Waseda University.
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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) intri nsic approach via harmonic bundles and TERP structures. Progress was made with both themes.
Regarding (1), although the "generic case" of the article "Isomonodromy aspects of the tt* equations of Cecotti and Vafa III" with Its and Lin was completed in the previous year, the "non-generic case" presented some unexpected difficulties. These difficulties were resolved and the article was completed. The article with Ho "A Lie-theoretic description of the solution space of the tt*-Toda equations" was also finished. A second article with Ho "Kostant, Steinberg, and the Stokes matrices of the tt*-Toda equations" was started. The Lie-theoretic approach of these articles should facilitate applications (of previous work on the tt*-Toda equations) to symplectic/differential geometry in future.
Regarding (2), preparatory research was carried out in the framework of harmonic bundles. In particular, relations between the tt*-Toda equations and work of researchers such as P. Boalch, E. Meinrencken, T. Mochizuki on moduli spaces of flat connections were investigated.
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| Strategy for Future Research Activity |
In order to exploit our previous results on the tt*-Toda equations a more general and more powerful language is required. For this, the Lie-theoretic point of view, and the point of view of moduli spaces of flat connections, will be developed further. An international conference on this theme is planned for the academic year 2017-18. In addition, several experts in this area, and experts on its applications in symplectic/differential geometry, from Japan and abroad, will be invited to present their work.
In view of the origin of the tt* equations in physics, relations with (past and present) developments in physics should continue to provide new ideas. The work of Cecotti and Vafa in the 1990's on 2D supersymmetric field theories, and their most recent work on 4D supersymmetric field theories, are of great relevance to this project and will be investigated.
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