Hall algebras and gauge theory on a surface
Project/Area Number 
17H06598

Research Category 
GrantinAid for Research Activity Startup

Allocation Type  Singleyear Grants 
Research Field 
Algebra

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) 
20170825 – 20190331

Project Status 
Completed (Fiscal Year 2017)

Budget Amount *help 
¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)

Keywords  Hall algebras / Higgs bundles / Dolbeaut moduli stack / de Rham moduli stack / Betti moduli stack / categorification / Higgs sheaves 
Outline of Annual Research Achievements 
My research achivevements consisted of the construction and characterization of the twodimensional cohomological Hall algebras of a fixed smooth projective complex curve X. In the paper arXiv:1801.03482, together with O. Schiffmann, I have defined the cohomological Hall algebra associated with the Dolbeaut moduli stack of X (that is, the moduli stack parameterizing Higgs sheaves on X). We have characterized such an algebra describing, for example, a set of generators of it. Recall that the de Rham moduli stack is the stack parameterizing vector bundles with flat connections on X, while the Betti moduli stack is the stack parameterizing finitedimensional representations of the fundamental group of X. In the paper arXiv:1903.07253, together with M. Porta, I constructed cohomological Hall algebras for the de Rham and Betti moduli stacks of X, respectively. This result is a consequence of a more general construction of convolution algebra structures on the bounded derived category of coherent sheaves on the Dolbeaut, de Rham and Betti moduli stacks (this gives rise to categorified Hall algebras). In the Dolbeaut case, the resulting categorified Hall algebra indeed categorifies the algebra constructed with Schiffmann. In addition, I have established some relations between these 3 categorified Hall algebras, which can be interpreted as Hall algebra versions of the RiemannHilbert correspondence, in the de Rham & Betti case, and of the nonabelian Hodge correspondence, in the Dolbeaut & de Rham case.

Research Progress Status 
平成29年度が最終年度であるため、記入しない。

Strategy for Future Research Activity 
平成29年度が最終年度であるため、記入しない。

Report
(1 results)
Research Products
(15 results)


[Presentation] Continuum quantum groups2019
Author(s)
Francesco Sala
Organizer
XXI Congresso dell'Unione Matematica Italiana, Sezione "Teoria di Lie". September 2  6, 2019, University of Pavia, Italy.
Related Report
Int'l Joint Research / Invited












