| 研究実績の概要 |
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.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
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.
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| 今後の研究の推進方策 |
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.
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