# 3d mirror symmetry of the cotangent bundle of the full flag variety

@article{Dinkins20203dMS, title={3d mirror symmetry of the cotangent bundle of the full flag variety}, author={Hunter Dinkins}, journal={Letters in Mathematical Physics}, year={2020}, volume={112} }

Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of q-difference equations holomorphic in variables z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document…

## 8 Citations

### Euler characteristic of stable envelopes

- MathematicsSelecta Mathematica
- 2022

In this paper we prove a formula relating the equivariant Euler characteristic of K-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag…

### Exotic Quantum Difference Equations and Integral Solutions

- Mathematics
- 2022

Hunter Dinkins: One of the fundamental objects in the K -theoretic enumerative geometry of Nakajima quiver varieties is known as the the capping operator. It is uniquely determined as the fundamental…

### Elliptic Stable Envelopes of Affine Type A Quiver Varieties

- MathematicsInternational Mathematics Research Notices
- 2022

We generalize Smirnov’s formula for the elliptic stable envelopes of the Hilbert scheme of points in $\mathbb {C}^2$ to the case of affine type $A$ Nakajima quiver varieties constructed with…

### Quantum difference equation for Nakajima varieties

- MathematicsInventiones mathematicae
- 2022

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the…

### A G ] 1 3 Ju l 2 02 1 VIRTUAL COULOMB BRANCH AND VERTEX FUNCTION

- Mathematics
- 2021

We introduce a variation of the K-theoretic quantized Coulomb branch constructed by Braverman–Finkelberg–Nakajima, by application of a new virtual intersection theory. In the abelian case, we define…

### Virtual Coulomb branch and vertex functions

- Mathematics
- 2021

We introduce a variant of the $K$-theoretic quantized Coulomb branch constructed by Braverman--Finkelberg--Nakajima, by application of a new virtual intersection theory. In the abelian case, we…

### Three-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

- Mathematics
- 2021

We consider a pair of quiver varieties $(X;X^{\prime})$ related by 3D mirror symmetry, where $X =T^*{Gr}(k,n)$ is the cotangent bundle of the Grassmannian of $k$-planes of $n$-dimensional space. We…

### Quantum differential and difference equations for $\mathrm{Hilb}^{n}(\mathbb{C}^2)$

- Mathematics
- 2021

We consider the quantum difference equation of the Hilbert scheme of points in C2. This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and…

## References

SHOWING 1-10 OF 29 REFERENCES

### Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2019

Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}=…

### Elliptic and K-theoretic stable envelopes and Newton polytopes

- MathematicsSelecta Mathematica
- 2019

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $$K$$K-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We…

### Pursuing Quantum Difference Equations II: 3D mirror symmetry

- MathematicsInternational Mathematics Research Notices
- 2022

Let $\textsf {X}$ and $\textsf {X}^{!}$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $K$-theoretic limit of the elliptic duality interface is an equivariant…

### 3d mirror symmetry and quantum K-theory of hypertoric varieties

- MathematicsAdvances in Mathematics
- 2021

### A-type Quiver Varieties and ADHM Moduli Spaces

- MathematicsCommunications in Mathematical Physics
- 2021

We study quantum geometry of Nakajima quiver varieties of two different types—framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection…

### Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, II

- Mathematics
- 2015

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch,…

### Symplectic Duality of $T^*Gr(k,n)$

- Mathematics
- 2020

In this paper, we explore a consequence of symplectic duality (also known as 3d mirror symmetry) in the setting of enumerative geometry. The theory of quasimaps allows one to associate hypergeometric…

### Quantum difference equation for Nakajima varieties

- MathematicsInventiones mathematicae
- 2022

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the…

### Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, I

- Mathematics
- 2016

Consider the 3-dimensional N = 4 supersymmetric gauge theory associated with a compact Lie group G and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact…

### Lectures on K-theoretic computations in enumerative geometry

- Mathematics
- 2015

These are notes from my lectures on quantum K-theory of Nakajima quiver varieties and K-theoretic Donaldson-Thomas theory of threefolds given at Columbia and Park City Mathematics Institute. They…