2020 Fiscal Year Annual Research Report
Ktheoretic enumerative invariants and qdifference equations
Project/Area Number 
19F19802

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, 准教授 (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

Keywords  quantum cohomology / qdifference equations 
Outline of Annual Research Achievements 
The Ktheoretic Jfunction of a projective manifold X is a function on r+1 variables q, Q_1,...,Q_r defined through genus0 Ktheoretic GromovWitten invariants of X. Here q is an auxiliary parameter, while Q_1,...,Q_r, known also as Novikov variables or Kahler parameters, are coordinates on the space of Kahler forms on X. We were able to prove confluence of the Ktheoretic Jfunction for an important class of varieties known as Fano varieties. Namely, we proved that after rescaling the Novikov variables appropriately, the Ktheoretic Jfunction has a limit q>1 which coincides with the cohomological Jfunction.
Givental has constructed a solution for the quantum Ktheoretic differential equations in terms of certain oscillatory integrals. In particular, since the Jfunction is also a solution to the quantum Ktheoretic differential equations, there is an interesting problem of comparing the oscillatory integral and the Jfunction. We studied in great details the example of Fano toric manifolds for which r=2. We were able to express Givental's oscillatory integral in terms of the Ktheoretic Jfunction. In order to do this, we had to deal with various subtle estimates related to the asymptotic growth of the qexponential and the qGamma function. Our results should be helpful when analyzing other toric manifolds.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We were able to solve the main problem in our proposal for the case of all Fano manifolds, which is a rather large class of manifolds. Moreover, since our argument does not rely on finding explicit formulas for the Jfunction, it is possible that it can be generalized to include all smooth projective varieties.

Strategy for Future Research Activity 
We are currently working on the text of our paper. Our future plan is to extend our current argument to nonFano targets. We will do this in two different ways. First, we would like to consider the case of all toric manifolds with Picard number 2, that is, r=2. We already understand the Fano case, so we will try to generalize our computation. In cohomological GromovWitten theory Iritani found an important integral structure in quantum cohomology. We would like to understand if a similar structure is available in Ktheoretic GromovWitten theory. Second, we would like to generalize our proof of the confluence of a Fano manifold. Currently, our argument can be split into several steps and only one of them uses the Fano condition.
