Establish the conjectured 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 three-dimensional mirror-dual varieties, including the type D vertex functions covered by the product formula.

Background

The paper proves product formulas for vertex functions of zero-dimensional type D Nakajima quiver varieties with a single minuscule framing. These formulas are presented as ingredients for broader three-dimensional mirror symmetry statements relating vertex functions of dual quiver varieties.

The authors note that the corresponding relation is conjectural for arbitrary quiver varieties and explicitly indicate that it remains to be established for the type D product formula proved in the paper.

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 . 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”