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- W2921084750 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." @default.
- W2921084750 created "2019-03-22" @default.
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- W2921084750 date "2019-04-03" @default.
- W2921084750 modified "2023-09-27" @default.
- W2921084750 title "Categorification of two-dimensional cohomological Hall algebras" @default.
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