2013 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 | 可積分系 / 幾何学 / 量子コホモロジー |
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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.
In previous work, we found all (global) solutions the tt*-Toda equations by p.d.e. methods, and then computed the isomonodromic Stokes data corresponding to these solutions. In this period we used the Riemann-Hilbert method to give an alternative proof of the existence of solutions, in an article with A. Its and C.-S. Lin "Isomonodromy aspects of the tt* equations of Cecotti and Vafa II. Riemann-Hilbert problem" (arXiv:1312.4825).
In order to discuss the above work with other specialists, and in order to prepare for future applications to geometry, a number of workshops were held: Isomonodromic deformations and related topics (November 2013, Waseda University); Symplectic geometry of moduli spaces of connections (February 2014, Waseda University); Koriyama Geometry and Physics Days (February 2014, Nihon University, Koriyama); 6-th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics (March 2014, Taiwan National University); Moduli spaces of flat connections on surfaces and related topics (November 2014, Waseda University). Visits to Tokyo of C. Hertling, N.-K. Ho, C.-S. Lin were also supported. Travel expenses of collaborators 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), a preliminary version of a joint article with Its and Lin (arXiv:1312.4825) was finished, and progress was made on a further joint article with the same coauthors.
Regarding (2), joint research with Hertling was continued. This will describe in detail the vector bundle approach to the sinh-Gordon equation, the simplest case of the tt*-Toda equations.
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| Strategy for Future Research Activity |
Work will continue to strengthen further the above results by applying them to more general cases of the tt*-Toda equations, and then to apply them to other equations in differential geometry.
We intend to support further activities such as workshops and conferences in order to make progress with these goals.
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