| 研究課題/領域番号 |
26800003
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| 研究機関 | 東京大学 |
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研究代表者 |
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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| 研究期間 (年度) |
2014-04-01 – 2017-03-31
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| キーワード | Gromov-Witten invariants / Eynard-Orantin recursion / period integrals / Frobenius structures / vertex operators |
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| 研究実績の概要 |
In my FY2014 report, I have announced that I found a global Eynard―Orantin recursion for simple singularities. Once we have a global spectral curve, we can recall a construction of Bouchard and Eynard, which allows us to express the original recursion via a contour integral of a certain meromorphic differential 1-form. My idea was to degenerate the spectral curve, which would allow me to obtain a differential operators annihilating the total descendant potential of the simple singularity. Since, we know that the total descendant potential is a highest weight vector of a W-algebra, the main conjecture is that using the topological recursion we can obtain states in the W-algebra. However the problem of degenerating the recursion deserves a separate study and it will take more time than I was expecting. Nevertheless, I managed to overcome the difficulties in the case of a singularity of type A. The key idea was to use the language of Vertex Operator Algebra (VOA) and to replace the Bouchard―Eynard recursion with a set of differential operators representing states in the Hesenberg VOA associated with the Milnor lattice of the singularity. My construction is very convenient to use if we want to degenerate the spectral curve, so in the case of singularity of type A, I was able to obtain all the results that I was hoping for. In particular, I managed to find that the elementary symmetric polynomials are states in the W-algebra of type A.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
I completely solved the main problem in my proposal in the case of a simple singularity of type A. My original expectation was that I will solve the problem in the case of all simple singularities. Although, I have succeeded only for type A, the remaining two cases are only technically more involved. The main point is that the idea in my proposal is correct and that the current achievement is sufficient to think about more complicated cases. Namely, the quantum cohomology of an orbifold projective line with two orbifold points, provides an example in which the set of vanishing cycles is an affine root system of type A. Also, for a boundary singularity of type B, the set of vanishing and half vanishing cycles is a root system of type B, which can be viewed as a folding of a root system of type A. The underlying Frobenius manifold has a higher-genus theory governed by an integrable hierarchy in both cases. However, the W-constraints and the Lie algebra interpretation of the integrability are unknown. I think that my current progress allows me to start working on new problems and my methods look very promising. That is why I would say that I progressed well in the project of my proposal. Also, the fact that I am still missing the cases of D and E singularity simply means that the problem that I proposed is more interesting than originally expected.
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| 今後の研究の推進方策 |
I would like to complete the case of D and E singularities. I have to options to proceed, and I am planning to investigate both of them. The first one, is to find the limit of the Eynard―Orantin recursion when the spectral curve degenerates, i.e., what is the Eynard―Orantin recursion for a singular spectral curve? The 2nd approach is to use again VOA approach for the Bouchard―Eynard recursion. The root system of type D and E are more complicated, but I believe that something similar to what I did in the A-case should work. I have started to work also on the case of a simple boundary singularity. This case is particularly interesting for the specialists in integrable systems. We have semi-simple Frobenius structures too, but if we apply Dubrovin―Zhang's integrable hierarchy construction, we obtain a hierarchy, which is different from the natural candidate, i.e., a Drinfeld―Sokolov hierarchy. Givental's higher genus reconstruction also yields a total descendant potential, which is not a highest weight vector for the W-algebra associated to a classical root system. The idea that I have applied in the case of type A singularity can be used also for the boundary singularity of type B. I am very optimistic that, the Eynard―Orantin recursion can help us to understand the symmetries of the higher genus theory for a boundary singularity.
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| 次年度使用額が生じた理由 |
The reason for the left over amount is that for some of my trips, the host institutions covered all my expenses.
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| 次年度使用額の使用計画 |
In FY2016, I have invited Valentin Tonita to visit me in May, and I am going to use this grant to pay his airfare (120,000). I am planning to attend the LMS Durham Research Symposium "Geometric and Algebraic Aspects of Integrability", July 25 - August 4. I am planning to use the grant to pay for my travel expenses (184,059).
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