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 4n-dimensional quaternionic manifold is extended to a biharmonic 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. † On leave from the Laboratory of Theoretical Physics, JINR, Dubna, Head Post Office,

Quaternionic Taub-NUT from the harmonic space approach

“…In [4], a generalization of this approach to the quaternionic-Kähler (QK) manifolds was given. These manifolds generalize the HK ones in that the extra SU(2) which transforms complex structures becomes an essential part of the holonomy group.…”

“…These manifolds generalize the HK ones in that the extra SU(2) which transforms complex structures becomes an essential part of the holonomy group. It was shown in [4], that the QK geometry constraints can be also solved in terms of some unconstrained potential L +4 living on the analytic subspace parametrized by SU(2) harmonics and half of the original coordinates. The specificity of the QK case is the presence of a non-zero constant Sp(1) curvature on all steps of the way from L +4 to the related metric.…”

“…The specificity of the QK case is the presence of a non-zero constant Sp(1) curvature on all steps of the way from L +4 to the related metric. It is interesting to consider some examples in order to see in detail how the machinery proposed in [4] works. Only the simplest case of the homogeneous QK manifold Sp(n + 1)/Sp(1) × Sp(n) (corresponding to L +4 = 0) was considered in ref.…”

“…Only the simplest case of the homogeneous QK manifold Sp(n + 1)/Sp(1) × Sp(n) (corresponding to L +4 = 0) was considered in ref. [4]. The aim of this paper is to demonstrate the power of the harmonic geometric approach on the example of less trivial QK metric, a quaternionic generalization of the well-known four-dimensional Taub-NUT (TN) metric [5].…”

“…We first recall some salient features of the construction of [4]. One starts with a 4n-dimensional Riemann manifold parametrized by local coordinates {x µm }, µ = 1, 2, ..., 2n; m = 1, 2, and uses a vielbein formalism.…”