  Hypergravity is the theory in which the graviton, of spin-2, has a
supersymmetric partner of spin-5/2. There are "no-go" theorems that prevent
interactions in these higher spin theories. However, it appears that one can
circumvent them by bringing in an infinite tower of higher spin fields. With
this possibility in mind, we study herein the electric-magnetic duality
invariance of hypergravity. The analysis is carried out in detail for the free
theory of the spin-(2,5/2) multiplet, and it is indicated how it may be
extended to the infinite tower of higher spins. Interactions are not
considered. The procedure is the same that was employed recently for the
spin-(3/2,2) multiplet of supergravity. One introduces new potentials
("prepotentials") by solving the constraints of the Hamiltonian formulation. In
terms of the prepotentials, the action is written in a form in which its
electric-magnetic duality invariance is manifest. The prepotential action is
local, but the spacetime invariance is not manifest. Just as for the spin-2 and
spin-(3/2,2) cases, the gauge symmetries of the prepotential action take a form
similar to those of the free conformal theory of the same multiplet. The
automatic emergence of gauge conformal invariance out of demand of manifest
duality invariance, is yet another evidence of the subtle interplay between
duality invariance and spacetime symmetry. We also compare and contrast the
formulation with that of the analogous spin-(1,3/2) multiplet.
