Bow varieties, stable envelopes, and their 3d mirror symmetry.

Mirror symmetry for 3d N=4SUSY QFTs has recently received much attention in geometry and representation theory. Theories within this class give rise to interesting moduli spaces of vacua, whose most relevant components are called the Higgs and Coulomb branches. Nakajima initiated themathematical study of Higgs branches in the 90s; since then, their geometry has been pivotal in diverse areas of enumerative geometry and geometric representation theory. On the other hand, a mathematically precise definition of the Coulomb branch has onlyrecently been proposed, and its study has started. 

Physically, 3d mirror symmetryis understood as a duality for pairs of theories whose Higgs and Coulomb branches are interchanged. Mathematically, it descends to a number of statements relating invariants attached to the dual sides. One of its key predictions is the identification of dualpairs of elliptic stable envelopes, which are certain topological classes intimately related to elliptic quantum groups.