2021 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, 教授 (80596841)
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Keywords | vertex operators / frobenius manifolds |
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| Outline of Annual Research Achievements |
Jointly with my student Chenghan Zha we computed the equivariant topological K-ring of the Milnor fiber of an invertible polynomial of chain type. The main application of our result is to describe the integral structure of the so-called Berglund--Hubsch dual singularity. Furthermore, in a joint work in progress with my other student Xiaokun Xia. We worked on the problem of computing the monodromy group of quantum cohomology. The problem is closely related to the constructions in my proposal. It will help us understand how the integrable system of differential equations changes under the blow up. Assuming that we know the monodromy group of the quantum cohomology of a given smooth projective variety, we determined the monodromy of the quantum cohomology of the blow up of the variety. Finally, in collaboration with Kyoji Saito, I completed the first draft of a book "Primitive forms and vertex operators". I explained very carefully the techniques used in my research papers and I extended many of my results to more general settings. In addition, I believe that our book will make the theory of primitive forms much more accessible to the general audience,because we did not just write an overview with summary of research results, but rather a self-contained text with complete proofs.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
One of the key objects in my proposal is a certain set of vertex operators defined in terms of the solutions of a certain system of Fuchsian differential equations defined by the so-called second structure connection. Thanks to Kyoji Saito, who suggested me to collaborate on a book, I had a good opportunity this year to write about the compatibility of analytic continuation and the monodromy representation. For each vertex operator, this property is automatically satisfied. However, when we take the product of two vertex operators, the compatibility becomes a very non-trivial problem. In our joint book with Saito I explained the background and I gave a careful proof of the compatibility property.This is a very important result for my future plans.
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| Strategy for Future Research Activity |
The next step in my project is to construct Hirota quadratic equations for the Gromov--Witten invariants of the Fano orbifold lines of type E and for the elliptic orbifold lines. Both problems are closely related to the theory of Kac-Wakimoto hierarchies of type E. The root systems of type E have an interesting interpretation in terms of the homology of del Pezzo surfaces. I am planning to use this description in my construction. I am also reconsidering the foundations of the theory of vertex algebras. The vertex operators mentioned above do not quite fit the current axioms. Victor Kac derived the definition of vertex algebras from the Wightman axioms in quantum field theory. The analytic properties somehow were not incorporated so I would like to re-examine Kac's derivation.
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| Causes of Carryover |
There was a scheduled conference for January 2022 which due to the COVID epidemic was cancelled again. The organizers told me that they are planning to have the conference in January 2023. Hopefully this time it will happen. I am planninh to use the rest of my grant to cover travel expenses.
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