Symplectic leaves for generalized affine Grassmannian slices

TL;DR

This paper proves the smoothness of a dense open subset of generalized affine Grassmannian slices, confirming a conjecture and revealing their symplectic leaf structure, with implications for Coulomb branch studies.

Contribution

It establishes the smoothness of the dense open subset of generalized affine Grassmannian slices, confirming a conjecture and describing their symplectic leaf decomposition.

Findings

The dense open subset is smooth.

The variety decomposes into symplectic leaves.

The results hold over arbitrary rings, including complex numbers.

Abstract

The generalized affine Grassmannian slices $\overline{\mathcal{W}}_\mu^\lambda$ are algebraic varieties introduced by Braverman, Finkelberg, and Nakajima in their study of Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories. We prove a conjecture of theirs by showing that the dense open subset $\mathcal{W}_\mu^\lambda \subseteq \overline{\mathcal{W}}_\mu^\lambda$ is smooth. An explicit decomposition of $\overline{\mathcal{W}}_\mu^\lambda$ into symplectic leaves follows as a corollary. Our argument works over an arbitrary ring and in particular implies that the complex points $\mathcal{W}_\mu^\lambda(\mathbb{C})$ are a smooth holomorphic symplectic manifold.