Sphere quantization of Higgs and Coulomb branches and Analytic Symplectic Duality

We employ the protected sphere correlation functions of three-dimensional Super Conformal Field Theories with eight supercharges in order to define a quantization of their Higgs and Coulomb branches of vacua as real phase spaces. We also employ hemisphere correlation functions to define a quantization of certain real loci in the Higgs and Coulomb branches. Localization formulae and dualities applied to these quantizations result in a body of predictions about unitary representations of certain algebras, which may perhaps be understood as an "analytic" form of the symplectic duality program. In particular, the protected correlation functions in the class of theories denoted as T[G] are naturally related to the theory of unitary representations of complex or real semi-simple Lie groups.