Loop Grassmannian of quivers and Compactified Coulomb branch of quiver gauge theory with no framing

TL;DR

This paper demonstrates that the loop Grassmannian for symmetric matrices and the compactified Coulomb branch for quiver gauge theories are equivalent when the matrix corresponds to the quiver, unifying two generalizations of Zastava.

Contribution

It establishes the isomorphism between the loop Grassmannian of quivers and the compactified Coulomb branch, linking geometric structures in representation theory and gauge theory.

Findings

The loop Grassmannian generalizes Zastava for symmetric matrices.

The compactified Coulomb branch generalizes Zastava for quiver gauge theories.

These two structures coincide for the associated matrix of a quiver.

Abstract

Mirkovi\'c introduced the notion of loop Grassmannian for symmetric integer matrix $\kappa$. It is a two-step limit of the local projective space $Z_{\kappa}^{\alpha}$, which generalizes the usual Zastava for a simply laced group $G$. The usual loop Grassmannian of $G$ is recovered when the matrix $\kappa$ is the Cartan matrix of $G$. On the other hand, Braverman, Finkelberg, and Nakajima showed that the Compactified Coulomb branch $\mathbf{M}_{Q}^{\alpha}$ for the quiver gauge theory with no framing also generalizes the usual Zastava. We show that in the case when $\kappa$ is the associated matrix of the quiver $Q$, these two generalizations of Zastava coincide, i.e $\mathbf{M}_{Q}^{\alpha}\cong Z_{\kappa(Q)}^{\alpha}$.