Vertex functions for bow varieties and their Mirror Symmetry

TL;DR

This paper proves the predicted equivalence of vertex functions for bow varieties and their 3d mirror duals, revealing deep connections between K-theoretic invariants, elliptic functions, and mirror symmetry in algebraic geometry.

Contribution

It establishes the equality of q-difference equations for vertex functions under 3d mirror symmetry and relates them via elliptic stable envelopes, extending understanding of mirror symmetry in bow varieties.

Findings

Vertex functions satisfy identical q-difference equations after parameter change.

The relation involves an elliptic matrix related to elliptic stable envelopes.

Reduction to flag varieties allows explicit identification with Macdonald difference equations.

Abstract

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 K\"ahler 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.