We present theories of N = 2 hypermultiplets in four spacetime dimensions
that are invariant under rigid or local superconformal symmetries.
The target spaces of theories with rigid superconformal invariance
are (4n)-dimensional special hyper-K¨ahler manifolds. Such
manifolds can be described as cones over tri-Sasakian metrics and
are locally the product of a flat four-dimensional space and a quaternionic
manifold. The latter manifolds appear in the coupling of hypermultiplets
to N = 2 supergravity. We employ local sections of
an Sp(n) × Sp(1) bundle in the formulation of the Lagrangian and
transformation rules, thus allowing for arbitrary coordinatizations of
the hyper-K¨ahler and quaternionic manifolds
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