SELF-DUAL SUPERGRAVITIES

TL;DR

This paper develops a new formulation of N-extended supersymmetric self-dual supergravity equations using harmonic superspace, providing explicit methods to derive supervielbeins and superconnections, exemplified by a Taub-NUT-like solution.

Contribution

It introduces a harmonic superspace reformulation for self-dual supergravity, enabling explicit extraction of supergeometry from prepotentials, and demonstrates this with a concrete example.

Findings

Explicit method for decoding supervielbeins and superconnections.

Construction of an N=2 supersymmetric deformation of flat space.

Example solution analogous to Taub-NUT geometry.

Abstract

The N-extended supersymmetric self-dual Poincar\'e supergravity equations provide a natural local description of supermanifolds possessing hyperk\"ahler structure. These equations admit an economical formulation in chiral superspace. A reformulation in harmonic superspace encodes self-dual supervielbeins and superconnections in a graded skew-symmetric supermatrix superfield prepotential satisfying generalised Cauchy-Riemann conditions. A recipe is presented for extracting explicit self-dual supervielbeins and superconnections from such `analytic' prepotentials. We demonstrate the method by explicitly decoding a simple example of superfield prepotential, analogous to that corresponding to the Taub-NUT solution. The superspace we thus construct is an interesting $N=2$ supersymmetric deformation of flat space, having flat `body' and constant curvature `soul'.