2022 Fiscal Year Research-status Report
Integrability in Gromov--Witten theory
| Project/Area Number |
22K03265
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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) |
2022-04-01 – 2027-03-31
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| Keywords | Frobenius manifolds / quantum cohomology |
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| Outline of Annual Research Achievements |
I am writing a book in collaboration with K. Saito. We worked out very carefully the background from the theory of Frobenius manifolds that will be used in the current proposal. For example, we gave a self-contained proof of the so-called Painleve property of a semi-simple Frobenius manifold and we found a new formula for Saito's higher-residue pairing. My main progress is in proving a very important technical result which will be used in an essential way in the current proposal. Suppose that we have a semi-simple Frobenius manifold. Then we have a certain isomonodromic family of Fuchsian differential equations. The corresponding solutions can be viewed as generalization of the period integrals of analytic hypersurfaces. That is why we call them period vectors. We construct vertex operators whose coefficients are the period vectors. The product of two vertex operators involves a phase factor that can be represented by an integral along the path of a certain multivalued analytic 1-form called the phase form. We prove that for a given closed loop around the discriminant along which the two vertex operators are invariant, the corresponding periods of the phase form are integer multiples of 2\pi i. If we assume in addition that the Frobenius manifold has an integral structure, then our result implies that the vertex operators define a twisted representation of a certain lattice vertex algebra.
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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 technical result proved in our book with K. Saito is the main difficulty for proving that the generating function of Gromov--Witten invariants of a smooth projective variety satisfies Hirota quadratic equations. The next difficulty is in the construction of Hirota quadratic equation. As explained in my proposal, the idea is to use the qq-characters of Nekrasov. They are supposed to satisfy certain screening equations. I am currently investigating the physics literature on quiver gauged theories and the relevant mathematics literature on representation theory of quivers and moduli spaces of quiver sheaves. There is a lot of progress in this area, so I am optimistic about my project.
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| Strategy for Future Research Activity |
In this fiscal year the main goal will be to finish the book on vertex operators and primitive forms. I hope that I will be able also to give a mathematical proof of the statements made in the physics literature that the Necrasov qq-character satisfies the screening equations. I am also preparing to extend the scope of my project. Namely, my main goal in the proposal is to construct Hirota quadratic equations for the Gromov--Witten invariants of orbifold projetcive lines. However, I am also working very hard on the case when the target is P^2. In this case, one has to extend Nekrasov's construction to quiver with relations.
I am also investigating whether the vertex operators constructed from the periods of a semi-simple Frobenius manifold can be used to construct a quantum field theory in the sense of Wightman. Moreover, in physics the correlation functions are usually computed perturbatively as infinite sums of Feynman integrals. I would like to understand whether the correlation functions defined by the vertex operators of a semi-simple Frobenius manifold can be expressed also in terms of Feynman integrals.
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| Causes of Carryover |
The main reason for not using the budget for this fiscal year is that one of the places that I visited covered all my expenses. Also, at this point, I find it more beneficial to stay at IPMU and work instead of traveling.
I am planning to travel to Kyoto to collaborate with K. Saito. I am also interested in the theory of relative quantum cohomology introduced by Fenlong Yu and Hsian-Hua Tseng. I would like to invite one or maybe both of them to visit me.
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