2022 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 – 2024-03-31
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| Keywords | Frobenius manifolds / quantum cohomology |
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| Outline of Annual Research Achievements |
Every semi-simple Frobenius manifold can be viewed as a solution of a classical Riemann-Hilbert problem. The monodromy data is determined by a certain subset of a finite dimensional complex vector space equipped with a symmetric bilinear form. This subset has all the properties of a root system except that the bilinear form is not necessarily positive definite. The elements of this subset are called reflection vectors because the reflections with respect to the corresponding orthogonal hyperplanes generate the monodromy group of the Frobenius manifold. The problem is to classify the reflection vectors corresponding the semi-simple Frobenius manifolds that underly quantum cohomology. It is known that the blowup operation preserves semi-simplicity of quantum cohomology. Therefore, it is natural to investigate how does the reflection vectors change under the blow up operation. On the other hand, there is a very interesting conjecture that gives an explicit description of the reflections in terms of exceptional objects in the derived category. I have started a project in collaboration with my student in which the goal is to prove that if the conjecture holds for some manifold X, then it holds for the blowup of X at finitely many points. We did not complete the project yet but we made an interesting progress: we proved that certain exceptional objects supported on the exceptional divisor of the blowup are reflection vectors. We wrote a paper which is now available on the arXiv and it will be submitted to a journal soon.
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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
My expectation is that the smooth toric surfaces and the blowup operation give a complete classification of the smooth projective varieties that have semi-simple quantum cohomology. This should not be very hard to prove and I think that it is a very nice problem in algebraic geometry. I am very optimistic about finding a complete classification of the reflection vectors in the quantum cohomology of smooth surfaces and describing the corresponding monodromy groups. Having in mind the case of P^2, which I am currently working on, I expect to find many interesting relations to the theory of modular forms. I am still far from achieving the goals of my proposal, but nevertheless I found many interesting directions to investigate which might surpass the original goals of the proposal.
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| Strategy for Future Research Activity |
I am currently working very hard on the case when the target is P^2. There is a very interesting relation with modular forms which comes from the fact that the monodromy group of the quantum cohomology of P^2 is a modular group. It is known that the Gromov--Witten invariants can be reconstructed with the so-called local topological recursion. I was able to find a lift of the local recursion to a global one in which the spectral curve is the upper half-plane. The recursion kernel looks rather complicated but I still have my hopes that there is an elegant description in terms of the modular group. I am planning also to encourage my student to work out the case of Fano surfaces, or more generally, the blow up of P^2 at finitely many points.
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| Causes of Carryover |
I have invited Felong Yu to visit me but he could not make it. I guess traveling to Japan was still not that easy last fiscal year.
I am interested in the relative quantum cohomology introduced by Felong Yu and Hsian-Hua Tseng. I am planning to invite them to visit me. Also, if my student wants to go to workshops where he can present our work, I would like to support his trip.
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