Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces

Abstract

For any free oriented Borel-Moore homology theory $A$, we construct an associative product on the $A$-theory of the stack of Higgs torsion sheaves over a projective curve $C$. We show that the resulting algebra $A{\mathbf{Ha}}_C^0$ admits a natural shuffle presentation, and prove it is faithful when $A$ is replaced with usual Borel-Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose $A$-theory admits an $A{\mathbf{Ha}}_C^0$-action. These triples can be interpreted as certain sheaves on $\mathbb{P}(T^{\ast }C)$. In particular, we obtain an action of $A{\mathbf{Ha}}_C^0$ on the cohomology of Hilbert schemes of points on $T^{\ast }C$.

arXiv comment51 pages, completely rewritten section 7

References in corpus (11)

• Algebraic cobordism
• The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of ${\mathbb{A}}^2$
• On the Hall algebra of an elliptic curve, I
• Lectures on Nakajima's Quiver Varieties
• The cohomological Hall algebra of a preprojective algebra
• Homological algebra of twisted quiver bundles
• Two-dimensional categorified Hall algebras
• Invariants of moduli spaces of stable sheaves on ruled surfaces
• On the $K$-theoretic Hall algebra of a surface
• Moduli of framed sheaves on projective surfaces
• W-algebras associated to surfaces

Cited by in corpus (10)

• Quiver Yangian from Crystal Melting
• Two-dimensional categorified Hall algebras
• On the $K$-theoretic Hall algebra of a surface
• W-algebras associated to surfaces
• BPS Lie algebras and the less perverse filtration on the preprojective CoHA
• Purity and 2-Calabi-Yau categories
• Generators for categorical Hall algebras of surfaces
• Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and W-algebras
• The Wheel Conditions and K-theoretic Hall Algebras
• KLR and Schur algebras for curves and semi-cuspidal representations