Harmonic space and quaternionic manifolds

TL;DR

This paper develops a harmonic space approach to quaternionic geometry, solving differential constraints by extending the manifold to a bi-harmonic space with additional harmonic coordinates, and establishes a link with off-shell N=2 supersymmetric sigma-models.

Contribution

It introduces a harmonic analyticity principle for quaternionic manifolds, solves the defining constraints via unconstrained potentials, and connects quaternionic geometry with N=2 supersymmetric sigma-models.

Findings

Reformulation of quaternionic constraints as integrability conditions.

Solution of constraints using two unconstrained potentials.

Establishment of a correspondence between quaternionic spaces and N=2 supersymmetric sigma-models.

Abstract

We find a principle of harmonic analyticity underlying the quaternionic (quaternion-K\"ahler) geometry and solve the differential constraints which define this geometry. To this end the original $4n$-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.