HARMONIC SPACE AND QUATERNIONIC MANIFOLDS

We find a principle of harmonic analyticity underlying the quaternioni c (quaternion-Kahler) geometry and solve the differential constraints which define this geometry. To this end the original 4n-dimensional qu aternionic manifold is extended to a bi-harmonic space. The latter inc ludes additional harmonic coordinates associated with both the tangent local Sp(1) group and an extra rigid SU(2) group rotating the complex structures. Then the constraints can be rewritten as integrability co nditions for the existence of an analytic subspace in the bi-harmonic space and solved in terms of two unconstrained potentials on the analy tic subspace. Geometrically, the potentials have the meaning of vielbe ins associated with the harmonic coordinates. We also establish a one- to-one correspondence between the quaternionic spaces and off-shell N = 2 supersymmetric sigma-models coupled to N = 2 supergravity. The gen eral N = 2 sigma-model Lagrangian when written in the harmonic supersp ace is composed of the quaternionic potentials. Coordinates of the ana lytic subspace are identified with superfields describing N = 2 matter hypermultiplets and a compensating hypermultiplet of N = 2 supergravi ty. As an illustration we present the potentials for the symmetric qua ternionic spaces. 1994 Academic Press, Inc.