2021 Fiscal Year Research-status Report
Yangians and Cohomological Hall algebras of curves
| Project/Area Number |
21K03197
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Sala Francesco 東京大学, カブリ数物連携宇宙研究機構, 客員准科学研究員 (60800555)
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| Project Period (FY) |
2021-04-01 – 2026-03-31
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| Keywords | Yangians / Quantum Groups / Hall Algebras / Higgs bundles / Flat bundles / stable pairs |
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| Outline of Annual Research Achievements |
During the FY2021, I have focused on different aspects of the theory of 2-dimensional cohomological Hall algebras of curves. In [arXiv:2111.00249], Negut, Schiffmann, and I characterize explicitly the classical spherical Hall algebra of a smooth projective curve providing a realization of it as a quantum loop group (answering a long-standing question). Since the classical spherical Hall algebra of a genus g curve is related to the spherical de Rham K-theoretical Hall algebra of a curve of the same genus, our result provides an explicit description of the latter as a quantum loop group.
Diaconescu, Porta and I are finishing up the construction of a right module for the Dolbeault cohomological Hall algebra of a curve, and its categorification. The right module is defined geometrically by using cyclic Higgs bundles on the curve. By the spectral correspondence, cyclic Higgs bundles on a curve correspond to Pandharipande-Thomas stable pairs on the cotangent of the curve. Our construction applies also in the surface case: indeed, we can define a right module of the cohomological and categorified Hall algebras of a K3 surface via moduli spaces of stable pairs.
Diaconescu, Porta, Schiffmann, Vasserot, and I are completing a project about a description of the cohomological Hall algebra of a minimal resolution of a type ADE singularity as a new positive part of the Maulik-Okounkov affine Yangian of the same type. In type A_2, our work shall provide a description of the Dolbeault cohomological Hall algebra of the projective line in terms of the affine Yangian of gl(2).
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
Because of the COVID19 pandemic, I have had not the chance to meet in person my current collaborators nor to visit Kavli IPMU for a research visit. The lack of in-person communications has slightly delayed the conclusions of the project with Diaconescu and Porta, about representations of the Dolbeault cohomological and categorified Hall algebras of curves via cyclic Higgs bundles and stable pairs, and of the project with Diaconescu, Porta, Schiffmann, and Vasserot about the Dolbeault cohomological Hall algebra of the projective line (and, more generally, the cohomological Hall algebra of type ADE minimal resolutions). The lack of in-person interactions with permanent researchers at IPMU (such as Nakajima, Toda, and Kapranov) has somehow slightly delayed these projects.
Now, we are catching up, thanks to the easing of the pandemic restrictions: in particular, we plan to meet during FY2022 to complete these projects. Moreover, I plan to visit Kavli IPMU for at least a month, in order to have enough time for fruitful interactions with Nakajima, Toda, Kapranov, and the other researchers there.
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| Strategy for Future Research Activity |
During FY2022, I plan to finish the project about representations via cyclic Higgs bundles and stable pairs. Moduli spaces of cyclic Higgs bundles are candidates of curve analog of Nakajima quiver varieties, indeed they are symplectic and (for genus greater or equal than two) they have a proper map to an affine space (the Hitchin fibration). On the other hand, they are quasi-smooth derived schemes and not smooth varieties, hence one has to investigate a possible generalization of Maulik-Okounkov formalism from symplectic varieties to quasi-smooth symplectic derived stacks. Moreover, I plan to describe the relation between moduli spaces of cyclic Higgs bundles and moduli spaces of Minet's stable Higgs triples, which provide another approach to the construction of representations for the Dolbeault cohomological Hall algebra of a curve.
In addition, I plan to finish the project about the explicit characterization of the Dolbeault cohomological Hall algebra of the projective line (and, more generally, the cohomological Hall algebra of type ADE minimal resolutions) via Maulik-Okounkov Yangians. The realization of this new positive half of the Maulik-Okounkov Yangian can be seen as a first step in providing an equivalent realization of the Maulik-Okounkov Yangian for affine ADE quivers, which could show a direction into the understanding of the Alday-Gaiotto-Tachikawa conjectures for gauge theories on minimal resolutions of type ADE singularities.
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| Causes of Carryover |
Due the COVID19 pandemic, I was unable to use the funds to visit Kavli IPMU. I will use the funds to fully support a visit during the FY2022.
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Research Products
(8 results)
