Crystal bases and three-dimensional $ \mathcal{N}=4 $ Coulomb branches

TL;DR

This paper establishes a correspondence between crystal bases and Coulomb branches in 3D $ abla=4$ gauge theories, providing new insights and explicit examples across various quiver types.

Contribution

It introduces a novel link between Kashiwara crystals and Coulomb branches, including derivations for non-simply laced cases and the effect of real masses.

Findings

Correspondence holds for all quiver types including affine.

Explicit examples demonstrating the equivalence.

Proposed link between infinite crystals and Hilbert spaces.

Abstract

We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional $ \mathcal{N} =4 $ gauge theories. The result holds for simply-laced, non-simply laced and affine quivers. Two equivalent derivations are given in the non-simply laced case, either by application of the axiomatic rules or by folding a simply-laced quiver. We also study the effect of turning on real masses and the ensuing simplification of the crystal. We present a multitude of explicit examples of the equivalence. Finally, we put forward a correspondence between infinite crystals and Hilbert spaces of theories with isolated vacua.