Two-dimensional categorified Hall algebras

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 ∞-category Cohᵇ(ℝℳ) of complexes of sheaves with bounded coherent cohomology on a derived moduli stack ℝℳ. In the surface case, ℝℳ is a suitable derived enhancement of the moduli stack 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 K₀ , new K-theoretical Hall algebras, and by passing to H∗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.