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

Abstract

Coulomb branches of a set of 3dN = 4 supersymmetric gauge theories are closures of nilpotent orbits of the algebra 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 so(3) to 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) 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 group.