K-theoretic enumerative invariants and q-difference equations

2020 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

The K-theoretic J-function of a projective manifold X is a function on r+1 variables q, Q_1,...,Q_r defined through genus-0 K-theoretic Gromov-Witten 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 K-theoretic J-function for an important class of varieties known as Fano varieties. Namely, we proved that after rescaling the Novikov variables appropriately, the K-theoretic J-function has a limit q-->1 which coincides with the cohomological J-function.

Givental has constructed a solution for the quantum K-theoretic differential equations in terms of certain oscillatory integrals. In particular, since the J-function is also a solution to the quantum K-theoretic differential equations, there is an interesting problem of comparing the oscillatory integral and the J-function. 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 K-theoretic J-function. In order to do this, we had to deal with various subtle estimates related to the asymptotic growth of the q-exponential and the q-Gamma function. Our results should be helpful when analyzing other toric manifolds.

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 J-function, 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 non-Fano 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 Gromov-Witten theory Iritani found an important integral structure in quantum cohomology. We would like to understand if a similar structure is available in K-theoretic Gromov--Witten 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.