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Geometry, Symmetry and Physics Seminar

Categorification of 2d cohomological Hall algebrasĀ 

Francesco Sala, IPMUĀ 

Location:  Serin Lab E372
Date & time: Thursday, 11 April 2019 at 1:00PM - 2:00PM

Abstract: Let \(\mathcal{M}\) 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 \(\mathcal{M}\) is an instance of two-dimensional cohomological Hall algebras. 

In the present talk, I will describe a full categorification of the cohomological Hall algebra of \(\mathcal{M}\). This is achieved by exhibiting a derived enhancement of \(\mathcal{M}\). 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 second part of the talk, I will focus on the case of curves and discuss some relations between the Betti, de Rham, and Dolbeaut categorified cohomological Hall algebras.

This is based on a paper with Mauro Porta ( arXiv:1903.07253). 

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