2023 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 | Hall algebras / COHAs / Categorification / t-structures / Stability conditions / Calabi-Yau completions / Yangians / Quantum groups |
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| Outline of Annual Research Achievements |
During FY2023, the preprint [arXiv:2207.08926] underwent a significant revision. The revision includes the construction of the categorified, K-theoretical, and cohomological Hall algebra associated to a t-structure on a finite type dg-category, satisfying certain natural conditions. This new framework yields a multitude of novel examples. For instance, we can now construct categorified, K-theoretical, and cohomological Hall algebras associated to the underlying t-structure of a stability condition on a Kuznetsov component (of a smooth cubic 4fold, etc.).
Furthermore, Diaconescu, Porta, Schiffmann, Vasserot, and I are in the final stages of a project concerning the cohomological Hall algebra of coherent sheaves on a surface supported set-theoretically on a fixed curve. We are establishing a PBW-type theorem for such algebras. Additionally, when we restrict ourselves to minimal resolutions of a type ADE singularity, we can establish a connection between the corresponding COHA and the affine Yangian of the same type.
Finally, I started a project with Luis Alvarez-Consul about moduli spaces of objects on a 2-Calabi-Yau completion of a category of framed quiver representations with valued in a fixed dg-category. When the quiver is A_2 and the target category is the category of coherent sheaves on a curve, the corresponding category admits a torsion pair whose torsion object are Higgs sheaves on the curve. We anticipate that the moduli space of stable torsion-free objects of this category will serve as the curve analog of Nakajima quiver varieties.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I have established a framework enabling the construction of categorified Hall algebras associated with t-structures arising from stability conditions on 2CY categories, advancing the part about categorified Yangians.
One of the anticipated outcomes of the nearly completed project with Diaconescu, Porta, Schiffmann, and Vasserot is a description of the Dolbeault COHA of the projective line in terms of the Yangian of affine type A_1.
During a visit to Kavli IPMU in March 2024, I initiated a project with Diaconescu, Porta, and Yu Zhao aimed at studing Hecke operators associated to modifications along divisors. This is an initial step toward understanding the Dolbeault COHA of a curve of positive genus.
Finally, I have found the right candidate for the curve analog of Nakajima quiver varieties.
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| Strategy for Future Research Activity |
During FY 2024, I am at completing the revision of [arXiv:2207.08926] and the project with Diaconescu, Porta, Schiffmann, and Vasserot.
I plan to complete the project with Diaconescu, Porta, and Yu Zhao about Hecke operators associated to modifications along effective divisors. More precisely, our operators act on the K-theory of certain moduli stacks of vectors bundles on a smooth projective surface satisfying certain conditions. When the divisor is an affine ADE configuration of (-2)-rational curves, we obtain a geometric realization of the corresponding quantum toroidal algebra. I will compute relations between these operators when the divisor is a smooth projective curve of positive genus.
Finally, I will study the moduli stacks and spaces arising from the project with Alvarez-Consul.
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| Causes of Carryover |
During FY2023, I used the grant only to support a visit to Kavli IPMU during March 2024. I will use the unused amount to support a visit to Kavli IPMU during February and March 2025 and to fund an international conference that will take place at Kavli IPMU during the first week of March 2025.
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