# 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 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.
Document type :
Preprints, Working Papers, ...
44 pages, LATEX, JHU-TIPAC-920023, ENSLAPP-L-405-92, MPI-Ph/92-85. 1992
Domain :

https://hal.archives-ouvertes.fr/hal-00473312
Contributor : Oleg Ogievetsky <>
Submitted on : Thursday, April 15, 2010 - 10:22:57 AM
Last modification on : Thursday, April 15, 2010 - 10:22:57 AM

### Citation

A. Galperin, E. Ivanov, O. Ogievetsky. Harmonic space and quaternionic manifolds. 44 pages, LATEX, JHU-TIPAC-920023, ENSLAPP-L-405-92, MPI-Ph/92-85. 1992. <hal-00473312>

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