Quantum geometric Langlands conjectures and New development in W-algebras

2022 Fiscal Year Final Research Report

• Project/Area Number: 21K20317
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: Genra Naoki 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (00909202)
• Project Period (FY): 2021-08-30 – 2023-03-31
• Keywords: 頂点代数 / アファインLie代数 / 量子幾何学的Langlands予想 / スクリーニング作用素

Outline of Final Research Achievements

We have obtained several results on representation theories of affine Lie algebras and W-algebras. In particular, the joint work with Drazen Adamovic and Thomas Creutzig on the invese Hamiltonian reduction of Bereshadosky-Polyakov algebras to affine sl3 algebras implies certain relationships between representation theories of affine Lie algebras and W-algebras. Based on the relation, with Thibault Juillard we showed that W-algebras are obtained by Hamiltonian reductions of suitable W-algebras, which suggests conjectual relationship between quantum geometric Langlands and W-algebras.

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

W代数の研究は一部の特別なクラスを除いて進展途中であるにも関わらず，理論物理の発展により様々な予想が生まれ，量子幾何学的Langlands予想のような数学的にも重要な様々な問題と結びついている．本研究結果はW代数の表現論とアファインLie代数や量子幾何学的Langlands予想との関係性をより深く具体的に明らかにするものであり，今後の数理物理全体において重要な意義を持つものだと考えられる．