2019 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 
Givental has proposed an oscillatory integral that solves the quantum Ktheoretic differential equations of any Fano toric manifold. On the other hand, the Ktheoretic Jfunction provides also a solution for the quantum Ktheoretic differential equations in terms of a Taylor power series in the Novikov variables. Finding the relation between the oscillatory integral and the Jfunction amounts to finding an appropriate integration cycle and finding the Taylor series expansion of the corresponding oscillatory integral. We were able to solve this problem in the case of the projective space. The Taylor series expansion can be obtained by using a Melin transform with respect to the Novikov variables, which turns the oscillatory integral into a product of qgamma functions. Taking the inverse Melin transform yields a formula for the oscillatory integral in terms of qversion of a MelinBarns integral. We worked out various estimates involving the qexponential and the qgamma function which should be useful in the general case too. Our formula generalizes a formula found by Iritani in the case of cohomological GromovWitten theory of Fano toric orbifolds. In particular, we found a qgamma integral structure in the quantum Ktheoretic ring that should play an important role in mirror symmetry.

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 work out an important example that will serve as a model for more complicated cases. Also our computation gives a hint of how confluence should work in general. Having in mind that our project started in the 2nd half of FY2019 we believe that the current progress is satisfactory.

Strategy for Future Research Activity 
We are planning to work out a more complicated example. Namely, we would like to consider the case of a compact Fano toric manifold of Picard rank 2. The case of a toric manifold of Picard rank 2 is very important, because it is simple enough and at the same time, after some small modifications it could allow us to test confluence for nonFano targets too. Our result from this fiscal year suggests that the Ktheoretic Jfunction can be expressed in terms of Givental's oscillatory integrals. It is very easy to see that the latter, after rescaling the Novikov variables appropriately has a limit as q>1. Therefore, we have a very natural conjecture about the confluence of the Ktheoretic Jfunction. We would like to prove this conjecture by using Givental and Tonita's work on reconstructing genus0 Ktheoretic GromovWitten invariants from cohomological ones.
