2019 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 | matrix models / integrable hierarchies |
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| Outline of Annual Research Achievements |
In the first part of the fiscal year I completed my project with Jipeng Cheng that started in FY2018. We were able to construct an extension of the Kac--Wakimoto hierarchy of type D that govern the Gromov--Witten invariants of Fano orbifold lines of type D. Our paper is available on the arXiv and submitted to a journal. In the second part of the fiscal year my main focus was on constructing a matrix model, similar to the Kontsevich matrix model, for simple singularities of type D. This is a joint work with Alexander Alexandrov. The total descendent potential of a simple singularity of type D can be identified with a tau-function of the 2-component BKP hierarchy. This identification allows us to interpret the total descendent potential in terms of correlation functions for a certain set of neutral fermions. Our goal is to prove that the correlation functions are asymptotic expansions of certain matrix integrals. We are almost ready, except that again there is a technical difficulty. We have to prove that the two-point correlators, also known as propagators, are asymptotic expansions of oscillatory integrals whose amplitudes have singularities. Our work can be related also to the so called Fan-Jarvis-Ruan-Witten invariants for simple singularities of type D. However, in order to do this one has to solve an interesting problem -- find the image of the Milnor lattice under the period map. I was able to solve this problem in collaboration with my student Chenghan Zha. Our paper is available on the arXiv and submitted to a journal.
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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
I was able to resolve the technical difficulty from my project with Jipeng Cheng. The project that I started with Alexander Alexandrov is something that I wanted to do for a long time. Organizing Gromov--Witten invariants into a set of correlation functions is an approach to Gromov--Witten theory that is not explored that much. I think that it is a very promising direction. We were able to make an interesting progress.
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| Strategy for Future Research Activity |
The main problem in my research proposal is still faraway from being solved in the most general case. So far I understand how the problem can be solved in the settings of simple singularities and to some extend in the settings of Gromov--Witten theory of 1 -dimensional orbifolds. I was able to make a progress for the case of Gromov--Witten theory of P^2. However, I came across a problem that requires new ideas. Namely, I have to understand how to quantize holomorphic functions. In the cases that I handled so far I had to quantize polynomials. This is straightforward -- we replace each variable with an operator series. For holomorphic functions, however the quantization is much more subtle. This is what I am trying to understand right now. I am studying very carefully the standard model in physics. I think that there are many ideas there that are still not explored by mathematicians.
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| Causes of Carryover |
I went to conferences that covered my travel expenses. I am planning to use the extra funds to buy books and support my research collaborator to go to workshops.
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