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.