2017 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 / period integrals / Frobenius structures / vertex operators |
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| Outline of Annual Research Achievements |
In FY 2017 I have worked on three different projects. First, I finished a project that started in FY2016 and whose goal was to classify the Frobenius manifolds that correspond to an Eynard-Orantin recursion. Moreover, using K. Saito’s theory of primitive forms I have developed a systematic way to construct Frobenius structures on the Hurwitz spaces.The second project was about the period map for the quantum cohomology of the projective plane P2. The quantum cohomology of a smooth projective variety is equipped with a very interesting Fuchsian connection, called the second structure connection. It can be thought of as a generalization of the classical hypergeometric equation. The fundamental solution of the second structure connection is a multi-valued analytic function. It defines an analytic map from the universal cover of an appropriate configuration space to a vector space. I called this map the period map because in all cases when the variety is known to have a mirror model in the sense of Givental, the fundamental solution can be described in terms of classical period integrals. I determined the image of the period map and found analytic functions that provide an inverse to it in the case when the variety is P2. In particular, my result yields an identification of the quantum cohomology of P2 with a quotient of an open Stein domain by a discrete group.The last project was started towards the end of FY2016. The goal is to construct extension of a certain class of Kac-Wakimoto hierarchies that govern the Gromov-Witten invariants of the Fano orbifold lines.
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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
Since I managed to solve an interesting problem outlined in my proposal, I would classify the status of my work.
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| Strategy for Future Research Activity |
My main focus in FY2018 will be on the construction of an extension of the Kac-Wakimoto hierarchies that govern the Gromov-Witten invariants of the Fano orbifold lines. The latter are classified via the Dynkin diagrams of type ADE. The extension in the case of type A Fano orbifold lines is already known. The corresponding extended hierarchies are known as the extended bi-graded Toda hierarchies. Recently in collaboration with my visitor Jipeng Cheng we were able to understand the relation between the principal Kac-Wakimoto hierarchy of type D and the Shiota Grassmanian. I think that the Shiota Grassmanian will provide a convenient set up to construct the extension of the Kac-Wakimoto hierarchy corresponding to the Fano orbifold lines of type D.I am planning also to continue my work on the quantum cohomology of P2. Since I know the inverse of the period map, I can obtain the W-constraints for the genus-0 Gromov-Witten invariants of P2. These are certain system of Hamilton-Jacobi equations. Furthermore, I would like to solve the main problem of my proposal in the case of P2. One possibility is to quantize the Hamilton Jacobi equations using ideas from the theory of vertex algebras. The knowledge of the inverse of the period map allows us also to use the topological recursion of Eynard and Orantin.
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| Causes of Carryover |
The reason to carry over some amount from FY2017 to FY2018 is that I was invited to co-organize a conference in UC Berkeley in FY2018. I decided to decrease my spending in FY2017 so that I have enough funds to make my trip to Berkeley. I am planning to use the amount for FY2018 to go to 2 conferences: "Quantum cohomology and mirror symmetry", May 12-14 UC Berkeley and another one with yet unknown title in Aug 28-30 in Novosibirsk. I am also planning to visit Sofia University for 3 weeks Aug 15-Sep 7.
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