K-theoretic enumerative invariants and q-difference equations

2019 Fiscal Year Annual Research Report

• Project/Area Number: 19F19802
• Research Institution: The University of Tokyo
• 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)
• Co-Investigator(Kenkyū-buntansha): 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): 2019-11-08 – 2022-03-31
• Keywords: quantum cohomology / q-difference equations

Outline of Annual Research Achievements

Givental has proposed an oscillatory integral that solves the quantum K-theoretic differential equations of any Fano toric manifold. On the other hand, the K-theoretic J-function provides also a solution for the quantum K-theoretic differential equations in terms of a Taylor power series in the Novikov variables. Finding the relation between the oscillatory integral and the J-function 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 q-gamma functions. Taking the inverse Melin transform yields a formula for the oscillatory integral in terms of q-version of a Melin--Barns integral. We worked out various estimates involving the q-exponential and the q-gamma function which should be useful in the general case too. Our formula generalizes a formula found by Iritani in the case of cohomological Gromov--Witten theory of Fano toric orbifolds. In particular, we found a q-gamma integral structure in the quantum K-theoretic ring that should play an important role in mirror symmetry.

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 non-Fano targets too. Our result from this fiscal year suggests that the K-theoretic J-function 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 K-theoretic J-function. We would like to prove this conjecture by using Givental and Tonita's work on reconstructing genus-0 K-theoretic Gromov--Witten invariants from cohomological ones.