Two-dimensional categorified Hall algebras

Abstract

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category ${\mathsf{Coh}}^{\mathsf{b}}(\mathbb{R}\mathsf{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathsf{M}$. In the surface case, $\mathbb{R}\mathsf{M}$ is a suitable derived enhancement of the moduli stack $\mathsf{M}$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve $X$, the moduli stack of vector bundles with flat connections on $X$, and the moduli stack of finite-dimensional local systems on $X$, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in the other two cases our construction yields, by passing to ${\mathsf{K}}_0$, new K-theoretical Hall algebras, and by passing to ${\mathsf{H}}_{\ast }^{\mathsf{BM}}$, new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve.

v4: 71 pages; Final version. v3: 69 pages; introduction expanded; proofs in Sections 2 and 3 strengthened; new appendix about ind quasi-compact stacks and their correspondences included. The main results are unchanged

References in corpus (4)

• Higher Topos Theory
• Theorie homotopique des DG-categories
• Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces
• On the $K$-theoretic Hall algebra of a surface

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