BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks

TL;DR

This work develops a refined cohomological Hall algebra (CoHA) framework for suitably geometric 2-dimensional categories, culminating in a sheaf-theoretic BPS algebra that sits inside the absolute CoHA. Under total negativity, the BPS algebra is freely generated by intersection complexes of coarse moduli spaces, and a PBW-type theorem expresses the CoHA as a symmetric product of a BPS Lie data, yielding a cohomological integrality result. The authors connect these constructions to nonabelian Hodge theory for stacks, establishing χ-independence results and a stacky nonabelian Hodge correspondence between Dolbeault and Betti sides at the level of Borel–Moore homology. They verify the Bozec–Schiffmann positivity conjecture for totally negative quivers and provide a framework that unifies representations of preprojective algebras, Higgs bundles, local systems, and coherent sheaves on symplectic surfaces within a single refined CoHA/BPS paradigm.

Abstract

We define and study a sheaf-theoretic cohomological Hall algebra for suitably geometric Abelian categories $\mathcal{A}$ of homological dimension at most two, and a sheaf-theoretic BPS algebra under the conditions that $\mathcal{A}$ is 2-Calabi-Yau and has a good moduli space. We show that the BPS algebra for the preprojective algebra $Π_Q$ of a totally negative quiver is the free algebra generated by the intersection cohomology of the closure of the locus parameterising simple $Π_Q$-modules in the coarse moduli space. We define and study the BPS Lie algebra of arbitrary 2-Calabi-Yau categories $\mathcal{A}$ for which the Euler form is negative on all pairs of non-zero objects, which recovers the BPS algebra as its universal enveloping algebra for such "totally negative" 2CY categories. We show that for totally negative 2CY categories the BPS algebra is freely generated by intersection complexes of certain coarse moduli spaces as above, and the Borel-Moore homology of the stack of objects in such $\mathcal{A}$ satisfies a Yangian-type PBW theorem for the BPS Lie algebra. In this way we prove the cohomological integrality theorem for these categories. We use our results to prove that for $C$ a smooth projective curve, and for $r$ and $d$ not necessarily coprime, there is a nonabelian Hodge isomorphism between the Borel-Moore homologies of the stack of rank $r$ and degree $d$ Higgs bundles, and the appropriate stack of twisted representations of the fundamental group of $C$. In addition we prove the Bozec-Schiffmann positivity conjecture for totally negative quivers; we prove that their polynomials counting cuspidal functions in the constructible Hall algebra for $Q$ have positive coefficients, strengthening the positivity theorem for the Kac polynomials of such quivers.

Theorems & Definitions (180)

• Theorem 1.1: §\ref{['subsubsection:ThecohaproductKV']}
• Theorem 1.2: =Theorem \ref{['theorem:FreeAlgPreProj']}
• Corollary 1.3
• Theorem 1.4: =Theorem \ref{['theorem:FreeAlg2CY']}
• Theorem 1.5: see §\ref{['subsection:PBW-2CY']}
• Corollary 1.6
• Theorem 1.7: =Theorem \ref{['theorem:NAHT-body']}
• Remark 2.1
• Proposition 2.2
• proof