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

Abstract

Let $X$ be a Nakajima quiver variety and $X^{\prime }$ its $3d$-mirror. We consider the action of the Picard torus $\mathsf{K}=\mathrm{Pic}(X)\otimes {\mathbb{C}}^{\times }$ on $X^{\prime }$. Assuming that $(X^{\prime }{)}^{\mathsf{K}}$ 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.