Vertex functions for bow varieties and their Mirror Symmetry

In this paper, we study the vertex functions of finite type $A$ bow varieties. Vertex functions are $K$-theoretic analogs of $I$-functions, and 3d mirror symmetry predicts that the $q$-difference equations satisfied by the vertex functions of a variety and its 3d mirror dual are the same after a change of variable swapping the roles of the various parameters. Thus the vertex functions are related by a matrix of elliptic functions, which is expected to be the elliptic stable envelope of M. Aganagic and A. Okounkov. We prove all of these statements. The strategy of our proof is to reduce to the case of cotangent bundles of complete flag varieties, for which the $q$-difference equations can be explicitly identified with Macdonald difference equations. A key ingredient in this reduction, of independent interest, involves relating vertex functions of the cotangent bundle of a partial flag variety with those of a ``finer" flag variety. Our formula involves specializing certain Kähler parameters (also called Novikov parameters) to singularities of the vertex functions. In the $\hbar \to \infty$ limit, this statement is expected to degenerate to an analogous result about $I$-functions of flag varieties.