LAUSR

A bstractCoulomb branches of a set of 3dN$$ \mathcal{N} $$ = 4 supersymmetric gauge theories are closures of nilpotent orbits of the algebra son$$ \mathfrak{so}(n) $$. 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 so3$$ \mathfrak{so}(3) $$ to so10$$ \mathfrak{so}(10) $$ 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 QuotientA¯Oλ$$ \overline{A}\left({\mathcal{O}}_{\lambda}\right) $$ 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 ℤ2$$ {\mathbb{Z}}_2 $$ group.