I will talk about cohomological Hall algebras (COHA) of toric Calabi-Yau 3-folds and their representations. Let $X$ be a smooth local toric Calabi-Yau 3-fold. On the cohomology of the moduli spaces of certain sheaves (e.g. torsion free sheaves, or sheaves supported on a toric divisor) on X, there is an action of the COHA of Kontsevich and Soibelman via “raising operators”. I will discuss the “double” of the COHA that acts on the cohomology of the moduli space by adding the “lowering operators”. We associate a root system to $X$. The double COHA is expected to be the shifted Yangian of this root system. We also give a prediction for the shift in terms of an intersection pairing. We provide evidence of the aforementioned expectation in various examples.

This is based on my joint work with M. Rapcak, Y. Soibelman, and G. Zhao.

This is a part of the "GRT at home" seminar series, grt-home.org