Cohomological Hall algebra of a curve
Project/Area Number 
18K13402

Research Category 
GrantinAid for EarlyCareer Scientists

Allocation Type  Multiyear Fund 
Review Section 
Basic Section 11020:Geometryrelated

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) 
20180401 – 20200331

Project Status 
Granted (Fiscal Year 2018)

Budget Amount *help 
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)

Keywords  Hall algebras / Betti moduli stack / de Rham moduli stack / Dolbeaut moduli stack / Deligne moduli stack / categorification / Betti / De Rham / Dolbeaut / Deligne moduli stacks 
Outline of Annual Research Achievements 
The first two goals of my project for the FY2018 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.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Denote by COHA^{Dol}_X the cohomological Hall algebra of the Dolbeaut moduli stack of a fixed smooth projective complex curve X.
The third goal of FY2018 was about the relation between COHA^{Dol}_X when X is the projective line and the MaulikOkounkov Yangian of gl(2). At the moment, I am working on two independent projects in order to achieve this goal. Both of them consists of realizing a representation of COHA^{Dol}_X and, in the case of the projective line, comparing such a representation with those of the MaulikOkounkov Yangian of gl(2). As a tool for such a comparison, I plan to use some results of Feigin and collaborators on the representation theory of MaulikOkounkov Yangian of gl(2) (and quantum toroidal algebra of gl(2)) and some construction of Gaberdiel and collaborators on the physics side.
The first project, which I am carrying with Porta, is about a representation of COHA^{Dol}_X on the cohomology of moduli spaces of prolonged Higgs sheaves on X (which were defined by Katzarkov, Orlov, and Pantev); while the second project, which is together with D.E. Diaconescu and Porta, is about a representation of COHA^{Dol}_X on the cohomology of moduli spaces of cyclic Higgs sheaves on X. (The two moduli spaces I am considering should be related by some wallcrossing in a suitable space of stability conditions  we plan to clarify this point in our projects.)

Strategy for Future Research Activity 
Denote by COHA^{Dol}_X the cohomological Hall algebra of the Dolbeaut moduli stack of a fixed smooth projective complex curve X.
(a) Goal: study CoHA^{Dol}_X when the curve X is the complex projective line. I plan to finish the two projects I was mentioning in "Reasons", to establish a relation between COHA^{Dol}_X when X is the projective line and the MaulikOkounkov Yangian of gl(2).
(b) Goal: show that COHA^{Dol}_X is a deformation of a universal enveloping algebra of a (graded) Lie algebra. The goal is to construct a filtration of COHA^{Dol}_X such that the corresponding graded object is commutative. In particular, I expect that such a graded object is a universal enveloping algebra of a certain Lie algebra. Such a Lie algebra should have graded dimension given by the Kac polynomial of the curve. The construction of the filtration should be a consequence of the construction of a relative version of COHA^{Dol}_X with respect to the Hitchin fibration of the Dolbeaut moduli stack.

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












