Coherent sheaves on surfaces, COHAs and deformed W_1+∞-algebras

2023/11/22 by Anton Mellit, Mellit, Anton, Alexandre Minets +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2311.13415

openalex publication_date 2023/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface S with pure cohomology, deriving an explicit presentation by generators and relations. When S has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric W_1+∞-algebra associated to the ring H^*(S,ℚ). We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on S and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of 1-dimensional sheaves on a K3 surface in an ample class. The double COHA is shown to contain Nakajima's Heisenberg algebra, as well as a copy of the Virasoro algebra.

Citations

• Derived projectivizations of complexes
• Cohomological Hall algebras, their categorification, and their representations via torsion pairs
• BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

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• Nilpotent cohomological Hall algebras of surfaces
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