| 研究課題/領域番号 |
19F19802
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| 研究種目 |
特別研究員奨励費
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| 配分区分 | 補助金 |
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| 応募区分 | 外国 |
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| 審査区分 |
小区分11010:代数学関連
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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, 教授 (80596841)
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| 研究分担者 |
ROQUEFEUIL ALEXIS 東京大学, カブリ数物連携宇宙研究機構, 外国人特別研究員
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| 研究期間 (年度) |
2019-11-08 – 2022-03-31
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| 研究課題ステータス |
完了 (2021年度)
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| 配分額 *注記 |
2,100千円 (直接経費: 2,100千円)
2021年度: 600千円 (直接経費: 600千円)
2020年度: 800千円 (直接経費: 800千円)
2019年度: 700千円 (直接経費: 700千円)
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| キーワード | quantum K-theory / q-difference equations / quantum cohomology |
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| 研究開始時の研究の概要 |
Gromov-Witten theory was discovered by physicists via the so called topological string theory. It has many interesting applications to both geometry and differential equations. The K-theoretic version of Gromov-Witten theory is much more involved, but nevertheless, there was a recent progress by Givental, which in principle clarifies the relation between K-theoretic and cohomological Gromov-Witten theory. We are planning to investigate the applications of K-theoretic Gromov-Witten invariants to differential equations, such as, confluence of difference equations and integrable hierarchies.
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| 研究実績の概要 |
We got two interesting results. The first one is related to the problem of confluence in the theory of q-difference equations. Namely, we proved that the small K-theoretic J-functions of a smooth projectve variety with non-negative first Chern class has a limit as q->1 and this limit coincides with the small cohomological J-function. Here, non-negative first Chern class means that the natural pairing of the 1st Chern class of the tangent bundle and the homology class of an irreducible curve is a non-negative number. The limit is taken after rescaling each Novikov variable in the K-theoretic J-function by an appropriate power of q-1.Moreover, we expect that our argument can be generalized so one can prove the confluence of the big J-function and the confluence of the quantum q-difference equations. It is also expected that the positivity condition of the 1st Chern class is redundant but removing this condition seems to be a challenging problem. Our second result is in the settings of toric geometry. We were able to identify explicitly the small J-function of a Fano toric manifold of Picard rank 2 with a certain q-oscillatory integral. The latter was introduced by Givental in order to provide a solution of the quantum q-difference equations and it can be viewed as a first step towards constructing or fomrulating mirror symmetry in quantum K-theory.
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| 現在までの達成度 (段落) |
令和3年度が最終年度であるため、記入しない。
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| 今後の研究の推進方策 |
令和3年度が最終年度であるため、記入しない。
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