Harmonic Space and Quaternionic Manifolds

Abstract

We find a principle of harmonic analyticity underlying the quaternionic (quaternion-Kähler) geometry and solve the differential constraints which define this geometry. To this end the original 4 n-dimensional quaternionic manifold is extended to a bi-harmonic space. The latter includes 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 conditions for the existence of an analytic subspace in the bi-harmonic space and solved in terms of two unconstrained potentials on the analytic subspace. Geometrically, the potentials have the meaning of vielbeins 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 general N = 2 sigma-model Lagrangian when written in the harmonic superspace is composed of the quaternionic potentials. Coordinates of the analytic subspace are identified with superfields describing N = 2 matter hypermultiplets and a compensating hypermultiplet of N = 2 supergravity. As an illustration we present the potentials for the symmetric quaternionic spaces.