Equivalence conjecture for the two categorifications of the basic toroidal representation

Let $\widehat{\widehat{\mathfrak{g}}}$ denote the double-affine extension associated to the affine Kac–Moody algebra $\widehat{\mathfrak{g}}$. Consider the 2-representations of $\widehat{\widehat{\mathfrak{g}}}$ arising from Theorem and from the geometric categorical action conjecture. An equivalence conjecture asserts that there is an equivalence between these two 2-representations. The two constructions use derived categories of coherent sheaves on closely related Nakajima quiver varieties, realized with different stability conditions; the source presents their equivalence as a further conjectural relationship.