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

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.

33 pages, 3 figures, 28 tables

References in corpus (5)

• Counting Gauge Invariants: the Plethystic Program
• Mirrors of 3d Sicilian theories
• T^σ_ρ(G) Theories and Their Hilbert Series
• Branes and the Kraft-Procesi Transition
• Algebraic properties of the monopole formula

Cited by in corpus (1)

• Magnetic Quivers, Higgs Branches, and 6d N=(1,0) Theories -- Orthogonal and Symplectic Gauge Groups