3d mirror symmetry for type D vertex functions

Establish the conjectured relation between the vertex functions of arbitrary Nakajima quiver varieties and the vertex functions of their 3d mirror dual varieties, including the type D quiver varieties with minuscule framings treated in the paper.

Background

The paper studies vertex functions of type D Nakajima quiver varieties and proves product formulas for zero-dimensional varieties with a single minuscule framing. These formulas are expected to provide the type D input needed for 3d mirror symmetry, analogous to formulas already used in previously established cases.

The authors note that a relation between vertex functions of arbitrary quiver varieties and those of their 3d mirror duals has been conjectured, while the existing proofs rely on formulas of the kind established in the paper. The general relation, and its application to the type D product formula, therefore remain unresolved.

References

Vertex functions for arbitrary quiver varieties are also conjectured to have a certain relation to the vertex functions of their 3d mirror dual varieties cite{aganagic2021elliptic, dinkins20203dflag, dinkins2022symplectic, smirnovzhou2022hypertoric}. All of the cases in which this has been proven use formulas like Theorem~\ref{thmintro1} as an ingredient. We hope that this will one day be the case for Theorem~\ref{thmintro1} as well.

— Vertex functions of type $D$ Nakajima quiver varieties  (2502.13937 - Dinkins et al., 19 Feb 2025) in Section 1, subsection “Consequences”