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Journal of High Energy Physics

, 2017:79 | Cite as

Nilpotent orbits and the Coulomb branch of T σ(G) theories: special orthogonal vs orthogonal gauge group factors

  • Santiago Cabrera
  • Amihay Hanany
  • Zhenghao Zhong
Open Access
Regular Article - Theoretical Physics

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.

Keywords

Field Theories in Lower Dimensions Gauge Symmetry Supersymmetric Gauge Theory Brane Dynamics in Gauge Theories 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Theoretical PhysicsThe Blackett Laboratory, Imperial College LondonLondonUnited Kingdom

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