Originally the notion of Cohomological Hall algebra (COHA for short) was introduced in my joint paper with Maxim Kontsevich (arXiv: 1006.2706) for the purposes of motivic Donaldson-Thomas theory of 3-dimensional Calabi-Yau categories. A physicist can think informally of COHA as of quantized enveloping algebra of the Lie algebra of single-particle closed BPS states. From this point of view there should be a class of representations of COHA associated with spaces of open BPS states. Mathematically we are talking about representations of COHAs in the cohomology of the moduli spaces of stable framed objects of the category (e.g. stable framed coherent sheaves on a Calabi-Yau 3-fold).

I am going to discuss recent results which relate this class of representations to the (generalization of the ) AGT conjecture. In particular, a version of COHA of the category of torsion sheaves on the standard complex 3-dimensional vector space acts on the cohomology of the moduli space of Nekrasov spiked instantons. This action can be upgraded to the action on this space of the “vertex algebra at the corner” of Gaiotto and Rapcak. The underlying geometry is the one of non-reduced toric divisors in toric Calabi-Yau 3-folds. The above-mentioned result is yet another incarnation of the idea that quantum algebras which initially might look as “2-dimensional objects” (e.g. quiver Yangians) should be upgraded to those in dimension 3.