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__Francesco Sala__ (IPMU Tokyo)

Categorification of 2d cohomological Hall algebras

Tuesday 17th Sep, 14:00, J11
*Abstract:* Let M denote the moduli stack of either coherent sheaves on a smooth projective surface or Higgs sheaves on a smooth projective curve X. The convolution algebra structure on the Borel-Moore homology of M is an instance of two-dimensional cohomological Hall algebras. These examples were defined by Kapranov-Vasserot and by Schiffmann and me, respectively. In the present talk, I will describe a full categorification of the cohomological Hall algebra of M. This is achieved by exhibiting a derived enhancement of M. Furthermore, this method applies also to several other moduli stacks, such as the moduli stack of vector bundles with flat connections on X and the moduli stack of finite-dimensional representations of the fundamental group of X. In the curve case, we call the corresponding categorified algebras the Betti, de Rham, and Dolbeaut categorified Hall algebras of the curve X, respectively. In the second part of the talk, I will discuss some relations between these categorified Hall algebras. This is based on a joint work with Mauro Porta.