Difference equations associated with quantum toroidal algebra and hyper-Kaehler quotient

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03274
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Nagoya University
• Principal Investigator: Kanno Hiroaki 名古屋大学, 多元数理科学研究科, 教授 (90211870)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 量子トロイダル代数 / 量子可積分系 / 超対称ゲージ理論

Outline of Final Research Achievements

We have derived difference equations of KZ type for the correlation functions or　the trace of the products of intertwining operators of the quantum toroidal　algebra of type gl_1. We also investigated the structure of solutions and　clarified the relation to the Nekrasov partition function of elliptic type. We have investigated the non-stationary difference equation for the K-theoretic　Nekrasov partition function with a surface defect, which we can regard as a　conformal block of the deformed Virasoro algebra. We have proved that by gauge
transformation the equation is transformed to a quantized version of discrete　Painreve VI equation. The underlying moduli space is the affine Laumon space,　which is only Kaehler quotient. In this sense this equation belongs to a new class of difference equations which was not expected in the beginning of the project.

Free Research Field

数理物理学

Academic Significance and Societal Importance of the Research Achievements

gl_1 型の量子トロイダル代数の絡作用素に対する KZ 型の差分方程式の導出法について，かなり一般的な処方箋を確立することができた．これは，インスタントン分配関数が満たす量子 KZ 方程式との関係を明らかにする上で技術的に重要になると期待される．そのためには，今後，遮蔽作用素の役割を明らかにしていく必要がある．アフィン Laumon 空間が背後にある非定常差分方程式は，研究の開始当初には想定されていなかった新しいクラスの差分方程式である．量子トロイダル代数との関係を明らかにすることことは今後の課題である．