The integrality conjecture and the cohomology of preprojective stacks

Abstract

We study the Borel-Moore homology of stacks of representations of preprojective algebras ${\Pi }_Q$, via the study of the DT theory of the undeformed 3-Calabi-Yau completion ${\Pi }_Q[x]$. Via a result on the supports of the BPS sheaves for ${\Pi }_Q[x]$-mod, we prove purity of the BPS cohomology for the stack of ${\Pi }_Q[x]$-modules, and define BPS sheaves for stacks of ${\Pi }_Q$-modules. These are mixed Hodge modules on the coarse moduli space of ${\Pi }_Q$-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure, and thus that the Borel-Moore homology of stacks of ${\Pi }_Q$-modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of ${\Pi }_Q$-modules. Among these and other applications, we use our results to prove positivity of a number of "restricted" Kac polynomials, determine the critical cohomology of ${\mathrm{Hilb}}_n({\mathbb{A}}^3)$, and the Borel-Moore homology of genus one character stacks, as well as various applications to the cohomological Hall algebras associated to Borel-Moore homology of stacks of preprojective algebras, including the PBW theorem, and torsion-freeness.

v4: 60 pages, greatly improved presentation, improved some results and proofs. Rewritten introduction. Added new section on (non)torsion-freeness and (non)commutativity

References in corpus (3)

• Motivic Donaldson-Thomas invariants and McKay correspondence
• On two cohomological Hall algebras
• Donaldson-Thomas theory for categories of homological dimension one with potential

Cited by in corpus (14)

• Two-dimensional categorified Hall algebras
• On cohomological Hall algebras of quivers : Yangians
• K-theoretic Hall algebras for quivers with potential
• The local motivic DT/PT correspondence
• Dimensional reduction in cohomological Donaldson-Thomas theory
• BPS Lie algebras and the less perverse filtration on the preprojective CoHA
• Gopakumar-Vafa invariants and wall-crossing
• Purity and 2-Calabi-Yau categories
• Parallels between moduli of quiver representations and vector bundles over curves
• Theta-stratifications, Theta-reductive stacks, and applications
• Deformed dimensional reduction
• Enumerating coloured partitions in 2 and 3 dimensions
• McKay correspondence, cohomological Hall algebras and categorification
• Cohomology of non-generic character stacks