Cohomological Hall algebra of a curve

2019 Fiscal Year Final Research Report

• Project/Area Number: 18K13402
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: The University of Tokyo
• Principal Investigator: Sala Francesco 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (60800555)
• Project Period (FY): 2018-04-01 – 2020-03-31
• Keywords: Quantum groups / Hall algebras / moduli space / moduli stack / Betti shape / De Rham shape / Dolbeault shape / Gauge theory

Outline of Final Research Achievements

The project fits into the realm of geometric representation theory, which stems from synergies between algebra and geometry. The goal of the project was the discovery of new algebraic structures (quantum groups) arising from the study of (moduli) spaces in geometry. We have introduced new quantum groups naturally associated to curves (opposite to those known in the literature, associated to quivers). Our approach to their definition has been geometric, based on the theory of Hall algebras and their refined versions (cohomological, K-theoretical, etc). Specifically, we have defined three algebras: the Betti, the de Rham, and the Dolbeault algebras of a curve. They represent new symmetries arising from the geometry of the corresponding moduli spaces, which play a preeminent role in geometry (e.g., in non-abelian Hodge theory) and in physics (e.g., in gauge theory). Thus, they unlock a new striking connection between geometry, algebra, and physics, which needs to be investigated further.

Free Research Field

Algebraic geometry, representation theory

Academic Significance and Societal Importance of the Research Achievements

この研究によってベッチ代数、ドラーム代数、ドルボー代数というまったく新しいタイプの量子群が導入されました。これらはさらに研究する価値があります。これらの代数は幾何学的に導入されましたが、もとになった空間は数え上げ幾何や幾何学的ラングランズプログラムなど数学のいろいろな分野で研究されて来ましたので、それぞれのモジュライ空間の形(トポロジー)に新たな観点を提供します。さらに、これらの代数はアルダイ=ガイオット=立川予想などの、量子物理と幾何学との間の未解決予想の更なる理解に重要であろうと思われます。