Two-dimensional categorified Hall algebras

  • Mauro Porta

    Université de Strasbourg, France
  • Francesco Sala

    Università di Pisa, Italy; The University of Tokyo, Japan
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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 of complexes of sheaves with bounded coherent cohomology on a derived moduli stack . In the surface case, is a suitable derived enhancement of the moduli stack 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 , the moduli stack of vector bundles with flat connections on , and the moduli stack of finite-dimensional local systems on , 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 , new K-theoretical Hall algebras, and by passing to , 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.

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Mauro Porta, Francesco Sala, Two-dimensional categorified Hall algebras. J. Eur. Math. Soc. 25 (2023), no. 3, pp. 1113–1205

DOI 10.4171/JEMS/1303