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 [1]. 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 Tσ(SO(n)∨) 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 \( \overline{A}\left({\mathcal{O}}_{\lambda}\right) \) 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 diagonal action of a \( {\mathbb{Z}}_2 \) group.
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Cabrera, S., Hanany, A. & Zhong, Z. Nilpotent orbits and the Coulomb branch of T σ(G) theories: special orthogonal vs orthogonal gauge group factors. J. High Energ. Phys. 2017, 79 (2017). https://doi.org/10.1007/JHEP11(2017)079
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DOI: https://doi.org/10.1007/JHEP11(2017)079