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.

arXiv comment33 pages, 3 figures, 28 tables

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Cited by in corpus (15)

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• Magnetic Quivers from Brane Webs with O5 Planes
• Branes and the Kraft-Procesi transition: classical case
• Magnetic Lattices for Orthosymplectic Quivers
• Folding Orthosymplectic Quivers
• Quiver origami: discrete gauging and folding
• Discrete quotients of 3-dimensional N = 4 Coulomb branches via the cycle index
• Magnetic quivers and line defects -- On a duality between 3d N=4 unitary and orthosymplectic quivers
• Orthosymplectic Implosions
• Manifestly Covariant Worldline Actions from Coadjoint Orbits. Part I: Generalities and Vectorial Descriptions
• Minimally Unbalanced Quivers
• Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional N=4 theories
• Atomic Higgsings of 6D SCFTs
• Crystal bases and three-dimensional $\mathcal{N}=4$ Coulomb branches