Nilpotent orbits and the Coulomb branch of $T^\sigma (G)$ theories: special orthogonal vs orthogonal gauge group factors

TL;DR

This paper investigates the Coulomb branches of 3d $ ext{N}=4$ supersymmetric gauge theories related to nilpotent orbits of $ ext{so}(n)$, exploring the impact of orthogonal versus special orthogonal gauge group factors through Hilbert series computations.

Contribution

It introduces a novel analysis connecting the choice of $SO/O(N)$ gauge group factors with Lusztig's Canonical Quotient, and develops a new method for projecting magnetic lattices using $ ext{Z}_2$ actions.

Findings

Computed Hilbert series for nilpotent orbits from $ ext{so}(3)$ to $ ext{so}(10)$.

Discovered a relationship between $SO/O(N)$ gauge factors and Lusztig's Canonical Quotient.

Proposed a new projection method for magnetic lattices via $ ext{Z}_2$ group actions.

Abstract

Coulomb branches of a set of $3d\ \mathcal{N}=4$ supersymmetric gauge theories are closures of nilpotent orbits of the algebra $\mathfrak{so}(n)$. From the point of view of string theory, these quantum field theories can be understood as effective gauge theories describing the low energy dynamics of a brane configuration with the presence of orientifold planes. The presence of the orientifold planes raises the question to whether the orthogonal factors of a the gauge group are indeed orthogonal $O(N)$ or special orthogonal $SO(N)$. In order to investigate this problem, we compute the Hilbert series for the Coulomb branch of $T^\sigma(SO(n)^\vee)$ theories, utilizing the monopole formula. The results for all nilpotent orbits from $\mathfrak {so} (3)$ to $\mathfrak{so}(10)$ which are special and normal are presented. A new relationship between the choice of $SO/O(N)$ factors in the gauge group and the Lusztig's Canonical Quotient of the corresponding nilpotent orbit is observed. We also provide a new way of projecting several magnetic lattices of different $SO(N)$ gauge group factors by the simultaneous action of a $\mathbb Z_2$ group.