Characters of tangent spaces at torus fixed points and $3d$-mirror symmetry

TL;DR

This paper explores the relationship between tangent space characters at torus fixed points in Nakajima quiver varieties and their 3d-mirror counterparts, linking geometric invariants with enumerative vertex functions.

Contribution

It proposes a new formula connecting the K-character of tangent spaces at fixed points with enumerative invariants, advancing understanding of 3d-mirror symmetry in quiver varieties.

Findings

Derived a formula for tangent space characters at fixed points

Linked geometric invariants with enumerative vertex functions

Provided insights into 3d-mirror symmetry mechanisms

Abstract

Let $X$ be a Nakajima quiver variety and $X'$ its $3d$-mirror. We consider the action of the Picard torus $\mathsf{K}=\mathrm{Pic}(X)\otimes \mathbb{C}^{\times}$ on $X'$. Assuming that $(X')^{\mathsf{K}}$ is finite, we propose a formula for the $\mathsf{K}$-character of the tangent spaces at the fixed points in terms of certain enumerative invariants of $X$ known as vertex functions.