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

Cabrera, Santiago  (Theoretical Physics, The Blackett Laboratory, Imperial College London, London, SW7 2AZ, United Kingdom) ; Hanany, Amihay (Theoretical Physics, The Blackett Laboratory, Imperial College London, London, SW7 2AZ, United Kingdom) ; Zhong, Zhenghao  (Theoretical Physics, The Blackett Laboratory, Imperial College London, London, SW7 2AZ, United Kingdom)

15 November 2017

Abstract: Coulomb branches of a set of 3 d N $\mathcal{N}$ = 4 supersymmetric gauge theories are closures of nilpotent orbits of the algebra so n $\mathfrak{so}(n)$ . 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 T σ (SO( n ) ∨ ) theories, utilizing the monopole formula . The results for all nilpotent orbits from so 3 $\mathfrak{so}(3)$ to so 10 $\mathfrak{so}(10)$ 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 A ¯ O λ $\overline{A}\left({\mathcal{O}}_{\lambda}\right)$ 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 ℤ 2 ${\mathbb{Z}}_2$ group.

Published in: JHEP 1711 (2017) 079