Quasimap critical cohomology, Coulomb branches, and quantum groups

This article presents a systematic study of the critical cohomology of ${\text{QM}}^{\xi }({\text{bbP}}^1,X)$, the moduli space of based quasimaps from ${\text{bbP}}^1$ to a Nakajima quiver variety $X$. We develop two derived equivalent presentations of this moduli space as a global critical locus. Accordingly, we obtain two equivalent explicit realizations of the canonical DT sheaf of the moduli space. Using each critical presentation, we describe canonical actions of shifted Yangians and quantized Coulomb branches on the quasimap critical cohomologies. We produce, by geometric means, a canonical morphism from an appropriate Hall algebra to the Coulomb branch, and show that the two actions we describe are compatible via this morphism under an additional hypothesis. We conjecture the compatibility holds generally, and reduce the claim to a sheaf-theoretic statement that may be approached via Joyce conjecture.