  We use supersymmetric localization to calculate correlation functions of
half-BPS local operators in 3d ${\cal N} = 4$ superconformal field theories
whose Lagrangian descriptions consist of vectormultiplets coupled to
hypermultiplets. The operators we primarily study are certain twisted linear
combinations of Higgs branch operators that can be inserted anywhere along a
given line. These operators are constructed from the hypermultiplet scalars.
They form a one-dimensional non-commutative operator algebra with topological
correlation functions. The 2- and 3-point functions of Higgs branch operators
in the full 3d ${\cal N}=4$ theory can be simply inferred from the 1d
topological algebra. After conformally mapping the 3d superconformal field
theory from flat space to a round three-sphere, we preform supersymmetric
localization using a supercharge that does not belong to any 3d ${\cal N} = 2$
subalgebra of the ${\cal N}=4$ algebra. The result is a simple model that can
be used to calculate correlation functions in the 1d topological algebra
mentioned above. This model is a 1d Gaussian theory coupled to a matrix model,
and it can be viewed as a gauge-fixed version of a topological gauged quantum
mechanics. Our results generalize to non-conformal theories on $S^3$ that
contain real mass and Fayet-Iliopolous parameters. We also provide partial
results in the 1d topological algebra associated with the Coulomb branch, where
we calculate correlation functions of local operators built from the
vectormultiplet scalars.
