2023 Fiscal Year Annual Research 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 – 2024-03-31
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| Keywords | vertex operators / Fuchsian singularities / Frobenius manifolds |
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| Outline of Annual Research Achievements |
Given a semi-simple Frobenius manifold, we define the notion of periods by generalizing the period map of Kyoji Saito in singularity theory. The periods are multivalued analytic functions on P^1 satisfying a system of ordinary differential equations that has only Fuchsian singularities. Following Givental we construct a set of vertex operators and we ask the question if their products satisfy locality in the sense of the Wightman axioms in quantum field theory, i.e., the vertex operators commute up to analytic continuation. The main achievement of my proposal is that I was able to find a connection formula for the product formulas. Namely, if we take the product of two vertex operators, then it is easy to express it near each singularity of the Fuchsian system in terms of Givental's R-matrix and S-matrix. The fundamental question is the following: given two different singularities, let us analytically continue the product from one to the other. How does the product changes? I am currently writing a book in collaboration with Kyoji Saito in which we solved this problem. In FY2023 I spent some time on improving the exposition of our book. I also made progress on the applications of the connection formula to quantum field theory and integrable systems. In collaboration with Bojko Bakalov we found a method to extend the Kac-Wakimoto hierarchies of ADE type. In particular, this completes my project on integrable hierarchies and Fano orbifold lines from 10 years ago in which the extension of the corresponding Kac-Wakimoto hierarchy was left unknown.
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