| 研究実績の概要 |
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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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
3: やや遅れている
理由
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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