| Project/Area Number |
22K03265
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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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) |
2022-04-01 – 2027-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2026: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2025: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2024: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2023: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2022: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
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| Keywords | K-theory / Gromov-Witten / Fock space / Frobenius manifolds / quantum cohomology / vertex operators / Gromov-Witten invariants |
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| Outline of Research at the Start |
Using methods from algebra and complex geometry, I am planning to construct systems of differential equations similar to the so-called KdV equation. The latter governs the motion of a wave in shallow waters. These differential equations have applications to geometry and possibly to string theory.
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| Outline of Annual Research Achievements |
I worked on two projects in FY2023 related to the current grant proposal. First, I wanted to find rigorous mathematical proofs for several statements in physics (the work of Kimura and Pestun) about the partition function of a 5d quiver gauge theory. I was able to work out the details in the most elementary case when the quiver is of type A_1. Using the Lefschetz trace formula, I was able to compute the partition function which essentially coincides with the physics prediction but not quite. In particular, the physics statements about the relation between Nekrasov's qq-characters and W-algebras, even in this elemenatry case, are still hard to establish. My second project was in K-theoretic Gromov-Witten (KGW) theory. I found a closed formula for the genus-0 permutation-equivariant Gromov-Witten invariants of the point. My formula generalizes a well known formula by Y.P. Lee in the ordinary KGW theory. Moreover, I was able to find an interesting application of the K-theoretic Fock space of Weiqiang Wang to premutation equivariant KGW theory. Namely, there is a natural pushforward map which realizes the GW invariants as vectors in the Fock space. This new point of view allows me to extend one of my old results with Valentin Tonita to permutation-equivariant KGW, i.e., the genus-0 KGW invariants are governed by an integrable hierarchy of hydrodynamic type.
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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
The work of Pestun and Kimura contains many interesting statements which however are hard to prove mathematically. The case of the A_1 quiver is definitely the starting point. Here one has to compute torus characters of representations defined via sheaf cohomology on the moduli spaces of torsion free sheaves on P^2. I am quite satisfied with the fact that I was able to understand very well the geometry of the moduli space and to perform actual computations. I got to the main point, i.e., investigating the relation between W-algebras and Nekrasov qq-characters. The Nekrasov quiver gauge theory provides a (q_1,q_2)-deformation of the W-algebras that I am interested in. On the other hand, K-theoretic Gromov-Witten (KGW) theory also has a very interesting relation to q-deformations. Namely, the mirror symmetry for KGW invariants involves the Jackson integral and q-difference equations. I do not claim that the gauge theory has something to do with KGW theory but I would like to understand the q-deformations coming from KGW theory too. I made many attempts before to understand Givental's permutation-equivariant KGW theory. I finally was able to find a point of view (based on the K-theoretic Fock space) which helps me to go through the basics of Givental's permutation-equivariant KGW theory. Here the main problem is to find an efficient way to compute the higher-genus KGW invariants of the point. I was able to understand genus-0 and I think that I am in very good shape to pursue the higher-genus case.
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| Strategy for Future Research Activity |
I am planning to continue my investigation for both the quiver gauge theory of the A_1 quiver and the K-theoretic Gromov-Witten theory (KGW) of the point. For the quiver gauge theory, I have to understand the significance of two more results. One of them is the so called compactness result of Nekrasov. At least Nekrasov made an informal statement to me that his compactness result implies that his qq-characters are solutions to the screening equations. The other result concerns a limit in the quiver gauge theory which in the physics literature is known as the topological string limit. Apparently, the quiver gauge theory has an application in computing some formal expressions that appear in the localization formulas for the Gromov-Witten invariants of toric Calabi-Yau manifolds. For the other project, in KGW theory, my next goal is to test the ideas of the topological recursion. In the cohomological case, the Gromov-Witten invariants of the point can be computed via oscillatory integrals where the holomorphic forms involved in the construction of the integrals are defined via the topological recursion. In KGW theory, I am expecting a similar phenomenon but we will need to use q-oscillatory integrals instead. I am planning to test first the genus-0 case because I already have explicit formulas. I would like also to work out a similar formula for the KGW theory of the moduli spaces of r-spin structures. This would give me a q-deformation of the oscillatory integrals of A_r-1 singularity.
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