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Degenerations of CoHAs of 2-Calabi-Yau categories

Lucien Hennecart, Shivang Jindal

Abstract

By work of Davison and Meinhardt, the cohomological Hall algebra of a symmetric quiver with potential admits a geometrically defined filtration (the perverse filtration) whose associated graded is a supercommutative algebra. In the case of the triple quiver of a quiver with the canonical cubic potential, which corresponds to the preprojective algebra of the quiver via dimensional reduction, there is an additional filtration (the less perverse filtration), which is defined more generally for cohomological Hall algebras of suitably geometric 2-Calabi-Yau categories in work of Davison. In this paper, we show that the degenerations of the cohomological Hall algebras of preprojective algebras and more generally 2-Calabi-Yau categories with respect to the less perverse filtration is isomorphic to the enveloping algebra of the current Lie algebra of the BPS Lie algebra. This result applies in particular to CoHAs of local systems on Riemann surfaces and Higgs bundles on smooth projective curves. We extend this description to deformations of the cohomological Hall algebra obtained via torus actions on the arrows of the quiver and deformed canonical cubic potentials via the deformed dimensional reduction of Davison-Pădurariu. Last, we use our results to compare the less perverse filtration on CoHAs of preprojective algebras with the order filtration on the Maulik-Okounkov Yangian, via the comparison isomorphism of Botta-Davison and Schiffmann-Vasserot.

Degenerations of CoHAs of 2-Calabi-Yau categories

Abstract

By work of Davison and Meinhardt, the cohomological Hall algebra of a symmetric quiver with potential admits a geometrically defined filtration (the perverse filtration) whose associated graded is a supercommutative algebra. In the case of the triple quiver of a quiver with the canonical cubic potential, which corresponds to the preprojective algebra of the quiver via dimensional reduction, there is an additional filtration (the less perverse filtration), which is defined more generally for cohomological Hall algebras of suitably geometric 2-Calabi-Yau categories in work of Davison. In this paper, we show that the degenerations of the cohomological Hall algebras of preprojective algebras and more generally 2-Calabi-Yau categories with respect to the less perverse filtration is isomorphic to the enveloping algebra of the current Lie algebra of the BPS Lie algebra. This result applies in particular to CoHAs of local systems on Riemann surfaces and Higgs bundles on smooth projective curves. We extend this description to deformations of the cohomological Hall algebra obtained via torus actions on the arrows of the quiver and deformed canonical cubic potentials via the deformed dimensional reduction of Davison-Pădurariu. Last, we use our results to compare the less perverse filtration on CoHAs of preprojective algebras with the order filtration on the Maulik-Okounkov Yangian, via the comparison isomorphism of Botta-Davison and Schiffmann-Vasserot.
Paper Structure (42 sections, 47 theorems, 108 equations)

This paper contains 42 sections, 47 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Ackowledgements
  5. Conventions and Notations
  6. Cohomological Donaldson--Thomas theory of quivers with potential
  7. Constructible sheaves
  8. Associative and Lie algebras in categories of perverse sheaves
  9. Quivers with potentials
  10. Moduli stacks and moduli spaces
  11. Determinant line bundles
  12. Ext-quivers
  13. Cohomological Hall algebras
  14. BPS Lie algebras and associated graded with respect to the perverse filtration
  15. The cohomological Hall algebra of preprojective algebras
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $Q$ be a quiver and $\Pi_Q$ be the preprojective algebra of $Q$. We let $u$ be the first Chern class of a positive determinant line bundle over the stack $\mathfrak{M}_{\Pi_Q}$ of representations of $\Pi_Q$ (subsection:determinant_line_bundle_preproj). Then, there is an isomorphism of algebras where $\mathbf{U}$ denotes the universal enveloping algebra, $\mathfrak{n}_{\Pi_Q}^{+,\mathrm{BPS}}\

Theorems & Definitions (87)

  • Theorem 1.1: =\ref{['theorem:main_degeneration_affinized_BPS']}+\ref{['corollary:coha=envelopingalgebra']}
  • Theorem 1.2
  • Theorem 1.3
  • Example 1
  • Remark 1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • ...and 77 more