An introduction to Cohomological Hall algebras and their representations

Abstract

I am going to review for non-experts the notion of Cohomological Hall algebra (COHA) introduced in my joint paper with Maxim Kontsevich in 2010 (see arXiv:1006.2706).Restricting considerations to the case of COHA of quivers with potential I will recallsome structural results, like e.g. dimensional reduction of COHA from 3d to 2d. Then I plan to discuss a class of representations of COHA in the cohomology of the modulispaces of framed stable objects of a 3d Calabi-Yau category endowed with a stability structure, following my paper arXiv:1404.1606. Finally, if time permits, I will discuss some examples of representations of COHA and its double, including my recent joint work with Miroslav Rapcak, Yaping Yang and Gufang Zhao on the relation of COHA with affine Yangians and moduli spaces of Nekrasov spiked instantons (see arXiv:1810.10402). As will be explained in other talks at this conference this relation gives rise to a class of representations of the ``vertex algebra at the corner" (see Gaiotto and Rapcak, arXiv:1703.00982). Other classes of representations of the VOA at the corner are conjecturally related to the action of the double of spherical COHA on the cohomology of Hilbert schemes of non-reduced divisors in toric Calabi-Yau 3-folds.