Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices

TL;DR

This paper explores the geometric structure of Kac-Moody affine Grassmannian slices, demonstrating how fundamental monopole operators facilitate embeddings across all symmetric Kac-Moody types, extending known finite type results.

Contribution

It constructs embeddings of slices compatible with fundamental monopole operators for all symmetric Kac-Moody types, generalizing finite type geometric insights.

Findings

Embeddings are compatible with FMOs in finite types.

Constructed embeddings for all symmetric Kac-Moody types.

Embeddings respect Poisson structures under certain conditions.

Abstract

Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definition in finite ADE types. The Coulomb branch construction has good algebraic properties, but its geometry is hard to understand in general. In finite types, an essential geometric feature is that slices embed into one another. We show that these embeddings are compatible with the fundamental monopole operators (FMOs), remarkable regular functions arising from the Coulomb branch construction. Beyond finite type these embeddings were not known, and our second result is to construct them for all symmetric Kac-Moody types. We show that these embeddings respect Poisson structures under a mild "goodness" hypothesis. These results give an affirmative answer to a question posed by Finkelberg in his 2018 ICM address and demonstrate the utility of FMOs in studying the geometry of Kac-Moody affine Grassmannian slices, even in finite types.