2020 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 – 2022-03-31
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| Keywords | matrix model / FJRW invariants |
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| Outline of Annual Research Achievements |
I completed the project that started in FY2019 about constructing a matrix model for simple singularities of type D. Namely, in a joint work with Alexander Alexandrov we proved that the total descendent potential of a simple singularity of type D, after changing the variables according to the so-called Miwa parametrization can be expressed as a matrix integral similar to the Kontsevich matrix model. Let me recall also that in a joint work with my student Chenghan Zha we found explicit formulas for the period map image of the Milnor lattice in the Milnor ring. Using this result, we were also able to identify the total descendent potential with the generating function of FJRW-invariants of the Berglund-Hubsch dual singularity. We also proved that the principal Kac-Wakimoto hierarchy of type D has a unique tau-function satisfying the so-called string equation. Therefore, we obtained an explicit identification between the following 3 seemingly different formal functions: tau-function of the Kac--Wakimoto hierarchy, total descendent potential of the simple singularity of type D, and the generating function of FJRW invariants. Our paper is available on the arXiv and submitted to a journal. Our matrix model, unlike the Konstevich matrix model, is a two matrix model. There are various interesting questions that one has to answer in order to understand better the place of our result in the general picture. For example, possible relation to the Chekhov-Eynard-Orantin recursion, applications to W-constriants, etc.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
There was a technical problem to resolve in our project with A. Alexandrov that took us more time than expected. Also, I started working on a book about the applications of K. Saito's theory of primitive forms to integrable hierarchies. I am planning to explain in great details the techniques that I have developed so far for constructing integrable hierarchies in the form of Hirota Quadratic Equations. Currently, these techniques are spread in various research papers. Since, I am planning to use them in the current project, I found it necessary to explain my theory in a more pedagogical and self-contained manner.
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| Strategy for Future Research Activity |
I am planning to continue my project in two different directions. The first one is similar to my work with Jipeng Cheng. Namely, I would like to find a formalism describing the flows of the Kac--Wakimoto hierarchy of type E and use it to find the extended Kac--Wakimoto hierarchies relevant for the Gromov--Witten theory of Fano orbifold lines of type E and the elliptic orbifold lines. The type E-case is rather challenging because the basic representation does not have a fermionic realization. The 2nd direction is related to formulating the higher genus reconstruction for P^2 in terms of Checkhov--Eynard--Orantin topological recursion. Solving this problem should give me a method to construct W-constriants and Hirota Quadratic Equations.
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| Causes of Carryover |
I am planning to go to a conference in UK in January 2022. The conference was postponed twice due to the pandemic. I am also planning to make trips to Kyoto to discuss my joint book project with K. Saito or to invite him to visit me.
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