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Vertex functions for bow varieties and their Mirror Symmetry

Tommaso Maria Botta, Hunter Dinkins

TL;DR

The paper proves 3d mirror symmetry for vertex functions of finite type A bow varieties by showing that their $q$-difference equations match under a specific variable identification and by expressing the relation through elliptic stable envelopes. The authors reduce to the tractable case of cotangent bundles of complete flag varieties via a two-pronged brane-resolution strategy: D5 weight reductions (embedding Bow $X$ into $ ilde X$) and NS5 weight reductions (Lagrangian correspondences to refined NS5 resolutions). They establish compatibility of vertex functions with both D5 and NS5 resolutions, derive Macdonald-type difference equations in weight-one scenarios, and use analytic continuation and pole analysis to extend the results, culminating in a concrete mirror-map identity. The work connects curve-counting in bow varieties to Macdonald theory, elliptic cohomology, and fusion of $R$-matrices, with explicit examples (e.g., $X=T^*b P^1$) and broad implications for 3d mirror symmetry and related representation-theoretic structures. The results thus provide a robust enumerative realization of 3d mirror symmetry in $K$-theory and illuminate connections to quantum groups and stable envelopes.

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 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.

Vertex functions for bow varieties and their Mirror Symmetry

TL;DR

The paper proves 3d mirror symmetry for vertex functions of finite type A bow varieties by showing that their -difference equations match under a specific variable identification and by expressing the relation through elliptic stable envelopes. The authors reduce to the tractable case of cotangent bundles of complete flag varieties via a two-pronged brane-resolution strategy: D5 weight reductions (embedding Bow into ) and NS5 weight reductions (Lagrangian correspondences to refined NS5 resolutions). They establish compatibility of vertex functions with both D5 and NS5 resolutions, derive Macdonald-type difference equations in weight-one scenarios, and use analytic continuation and pole analysis to extend the results, culminating in a concrete mirror-map identity. The work connects curve-counting in bow varieties to Macdonald theory, elliptic cohomology, and fusion of -matrices, with explicit examples (e.g., ) and broad implications for 3d mirror symmetry and related representation-theoretic structures. The results thus provide a robust enumerative realization of 3d mirror symmetry in -theory and illuminate connections to quantum groups and stable envelopes.

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 Kähler parameters (also called Novikov parameters) to singularities of the vertex functions. In the limit, this statement is expected to degenerate to an analogous result about I-functions of flag varieties.

Paper Structure

This paper contains 63 sections, 64 theorems, 241 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation from 3d mirror symmetry
  3. Bow varieties
  4. Main theorem
  5. Strategy of the proof
  6. Reduction of D5 weights
  7. Reduction of NS5 weights
  8. Connection with quantum groups actions
  9. Extended worked example
  10. Outline of the paper
  11. Further directions
  12. Conventions and notation
  13. Acknowledgments
  14. Bow varieties
  15. Brane diagrams
  16. ...and 48 more sections

Key Result

Theorem 1.1

Let $X$ be a bow variety with dual $X^{!}$. For each $f \in X^{\mathop{\mathrm{\mathbb T}}\nolimits}$, we have where the identification of parameters $\kappa$ is defined by and $\mathop{\mathrm{w}}\nolimits(\cdot)$ is the weight of the brane.

Figures (4)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure A.1: Fixed points of $T^*\mathbb{P}^1$ (on top) and their dual (on the bottom).

Theorems & Definitions (111)

  • Theorem 1.1: Theorem \ref{['thm: mirsym']}
  • Theorem 1.2: Theorem \ref{['thm: d5 vertex']}
  • Theorem 1.3: Theorems \ref{['thm: ns5vertex']} and \ref{['thm: cosepNS5vertex']}
  • Remark 2.1
  • Theorem 2.2: rimanyi2020bow
  • Theorem 3.1: okounkov2020inductiveI
  • Remark 3.2
  • Proposition 3.3: BR
  • Theorem 3.4: BR
  • Proposition 3.5
  • ...and 101 more