2018 Fiscal Year Research-status Report
Riemann-Hilbert problem for Gromov-Witten invariants
| Project/Area Number |
17K05193
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MILANOV Todor 東京大学, カブリ数物連携宇宙研究機構, 准教授 (80596841)
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Keywords | Gromov-Witten invariants / quantum cohomology / integrable hierarchies |
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| Outline of Annual Research Achievements |
The main problem in my proposal can be approached in two different ways -- via the representation theory of vertex algebras or via integrable systems. My work in FY2018 was concentrated on the second approach. In collaboration with Jipeng Chang we found an integrable hierarchy that governs the Gromov--Witten (GW) theory of Fano orbifold lines of type D. The starting point of our project was an earlier work by myself, Hsian-Hua Tseng, and Yefeng Shen, in which we proved a partial result. Namely, we found that a certain Kac--Wakimoto (KW) hierarchy governs the stationary GW invariants. Our goal with Jipeng was to find an extension of the KW hierarchy that governs all GW invariants not only the stationary ones. Our project can be divided into two parts. First, we found the extension of the KW hierarchy in terms of Hirota bilinear equations (HBEs). I had already developed some methods (in collaboration with various people) to construct HBEs in GW theory so this part went rather smoothly. The second part of our project was based on Shiota's work on the 2-component BKP hierarchy. We proved that the HBEs yield an integrable hierarchy of Lax equations involving 3 Lax operators. There are still some details left to work out, but the main difficulties are resolved.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
I was expecting that the project described above will be finished by now, i.e., the papers available on the arXiv and submitted to a journal. However, there were many technical difficulties in the second part of our project. We had to work with differential-difference operators on several variables which made the proofs quite involved and time consuming.
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| Strategy for Future Research Activity |
I think that the second part of our work with Jipeng is very interesting from the point of view of the theory of integrable hierarchies. I am planning to continue my collaboration with him. Our next goal will be to find the Lax equations of all Kac--Wakimoto hierarchies of type ABCD. I guess we will start with types A and D, since by now we have quite good intuition for these cases. I am planning also to return to the first approach in my proposal based on vertex algebras. There was a recent paper by Taro Kimura and Valeri Pestun on quiver W-algebras, which I studied in great details. I already worked out some examples and I think that their results are relevant for my project.
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| Causes of Carryover |
Because we have efficiently saved funds, I will use it to buy a book.
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