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}\mathcal{M}) of complexes of sheaves with bounded coherent cohomology on a derived moduli stack \mathbb{R}\mathcal{M} . In the surface case, \mathbb{R}\mathcal{M} is a suitable derived enhancement of the moduli stack \mathcal{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.

Topics & Concepts