The geometric categorical action conjecture for quantum toroidal algebras

Let $\widehat{\mathfrak{g}}$ be the affine Kac–Moody algebra and let the geometric categorical $\widehat{\mathfrak{g}}$ action on $\bigoplus_{n\in\mathbb N}D\operatorname{Coh}^{\mathbb C^\times}((\mathbb C^2{}^{[n]})^\Gamma)$ be the action from Theorem 3.1. A geometric categorical action conjecture asserts that this action extends to a 2-representation of the quantum toroidal algebra associated to $\widehat{\mathfrak{g}}$. The toroidal algebra is the affinization of the quantum affine Kac–Moody algebra; the conjecture was not proved in the cited construction, in part because a definition of 2-representations of toroidal algebras was not available.