In a series of papers, Nekrasov introduces moduli spaces of “crossed instantons,” certain quiver representations modeling instantons on unions of 2-planes in 4-space. The geometry of these spaces encodes information about moduli of sheaves on surfaces and produces deformations of characters of quantum affine algebras. In work with Martijn Kool and Woonam Lim, we study Nekrasov’s moduli spaces from the perspective of algebraic geometry. I’ll explain how invariants of moduli spaces of crossed instantons can be defined using a construction from sheaf-counting on Calabi-Yai 4-folds. I’ll also state a speculative description of these spaces as moduli spaces of framed sheaves on a projective 4-fold.