2014 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.
Continuing our work with A. Its and C.-S. Lin, a joint article "Isomonodromy aspects of the tt* equations of Cecotti and Vafa III. Asymptotics and Iwasawa factorization" was prepared during this period. In this article we exploit a relation with loop groups, thus unifying all our methods used so far.
The space of (local) solutions of the sinh-Gordon equation was studied using meromorphic connections on vector bundles in an article with C. Hertling "The Painleve III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles" (arXiv:1501.04812).
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: Koriyama Geometry and Physics Days (October 2014, Nihon University, Koriyama); Workshop on Geometry and Nonlinear PDE (January 2015, Waseda University); 7th OCAMI-TIMS Joint International Workshop on Differential Geometry and Geometric Analysis (March 2015, Osaka City University). Visits to Tokyo of S.-C. Chang, Y.-I. Lee, and junior researchers from Taiwan 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), progress was made on a joint article with A. Its and C.-S. Lin. These results will establish a new loop group approach to the tt*-Toda equations.
Regarding (2), a preliminary version of a joint article with Hertling was finished (arXiv:1501.04812). This describes in detail the vector bundle approach to the sinh-Gordon equation, the simplest case of the tt*-Toda equations.
The combined results of both themes form the basis of a systematic method which should be applicable to problems in differential geometry.
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| Strategy for Future Research Activity |
One direction for the remaining period will be to apply the vector bundle approach (theme 2) to other situations beyond the sinh-Gordon equation. The results of theme 1 should make this possible.
A more important (but more difficult) goal is to understand and formulate the results sufficiently deeply in order to apply them to other equations arising in differential geometry. This will involve (a) exploiting (and understanding the limitations of) the theory of loop groups, and (b) identifying suitable targets for the application of our methods. The theory of harmonic maps is such a target, and more specifically the theory of harmonic bundles (or Higgs bundles) over Riemann surfaces. Recent developments in the physics of the four-dimensional tt* equations offer further challenges for our methods.
We intend to support activities such as workshops and conferences in order to make progress with these goals. In particular, an international conference on differential geometry and integrable systems in November 2015 is planned, which will allow us to present our systematic approach to the differential geometry community. A workshop on the Hitchin equations is planned for December 2015.
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