Ktheoretic enumerative invariants and qdifference equations
Project/Area Number 
19F19802

Research Category 
GrantinAid for JSPS Fellows

Allocation Type  Singleyear Grants 
Section  外国 
Review Section 
Basic Section 11010:Algebrarelated

Research Institution  The University of Tokyo 
Host Researcher 
MILANOV Todor 東京大学, カブリ数物連携宇宙研究機構, 教授 (80596841)

Foreign Research Fellow 
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

Project Period (FY) 
20191108 – 20220331

Project Status 
Completed (Fiscal Year 2021)

Budget Amount *help 
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2021: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2020: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2019: ¥700,000 (Direct Cost: ¥700,000)

Keywords  quantum Ktheory / qdifference equations / quantum cohomology 
Outline of Research at the Start 
GromovWitten theory was discovered by physicists via the so called topological string theory. It has many interesting applications to both geometry and differential equations. The Ktheoretic version of GromovWitten theory is much more involved, but nevertheless, there was a recent progress by Givental, which in principle clarifies the relation between Ktheoretic and cohomological GromovWitten theory. We are planning to investigate the applications of Ktheoretic GromovWitten invariants to differential equations, such as, confluence of difference equations and integrable hierarchies.

Outline of Annual Research Achievements 
We got two interesting results. The first one is related to the problem of confluence in the theory of qdifference equations. Namely, we proved that the small Ktheoretic Jfunctions of a smooth projectve variety with nonnegative first Chern class has a limit as q>1 and this limit coincides with the small cohomological Jfunction. Here, nonnegative 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 nonnegative number. The limit is taken after rescaling each Novikov variable in the Ktheoretic Jfunction by an appropriate power of q1.Moreover, we expect that our argument can be generalized so one can prove the confluence of the big Jfunction and the confluence of the quantum qdifference 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 Jfunction of a Fano toric manifold of Picard rank 2 with a certain qoscillatory integral. The latter was introduced by Givental in order to provide a solution of the quantum qdifference equations and it can be viewed as a first step towards constructing or fomrulating mirror symmetry in quantum Ktheory.

Research Progress Status 
令和3年度が最終年度であるため、記入しない。

Strategy for Future Research Activity 
令和3年度が最終年度であるため、記入しない。

Report
(3 results)
Research Products
(4 results)