The general form of supersymmetric solutions of N=(1,0) U(1) and SU(2) gauged supergravities in six dimensions

TL;DR

This paper characterizes all supersymmetric solutions in six-dimensional N=(1,0) U(1) and SU(2) gauged supergravities, revealing their geometric structures, constructing new solutions, and analyzing supersymmetry enhancements and Penrose limits.

Contribution

It provides the necessary and sufficient conditions for supersymmetric configurations, classifies the underlying structures, and constructs explicit new solutions in both theories.

Findings

Supersymmetric solutions are associated with an $SU(2)\ltimes \mathbb{R}^4$ structure.

In U(1) theory, the base space is Kähler with the gauge field as the Ricci form.

In SU(2) theory, gauge fields relate to the base's spin bundle curvatures.

Abstract

We obtain necessary and sufficient conditions for a supersymmetric field configuration in the N=(1,0) U(1) or SU(2) gauged supergravities in six dimensions, and impose the field equations on this general ansatz. It is found that any supersymmetric solution is associated to an $SU(2)\ltimes \mathbb{R}^4$ structure. The structure is characterized by a null Killing vector which induces a natural 2+4 split of the six dimensional spacetime. A suitable combination of the field equations implies that the scalar curvature of the four dimensional Riemannian part, referred to as the base, obeys a second order differential equation. Bosonic fluxes introduce torsion terms that deform the $SU(2)\ltimes\mathbb{R}^4$ structure away from a covariantly constant one. The most general structure can be classified in terms of its intrinsic torsion. For a large class of solutions the gauge field strengths admit a simple geometrical interpretation: in the U(1) theory the base is K\"{a}hler, and the gauge field strength is the Ricci form; in the SU(2) theory, the gauge field strengths are identified with the curvatures of the left hand spin bundle of the base. We employ our general ansatz to construct new supersymmetric solutions; we show that the U(1) theory admits a symmetric Cahen-Wallach$_4\times S^2$ solution together with a compactifying pp-wave. The SU(2) theory admits a black string, whose near horizon limit is $AdS_3\times S_3$. We also obtain the Yang-Mills analogue of the Salam-Sezgin solution of the U(1) theory, namely $R^{1,2}\times S^3$, where the $S^3$ is supported by a sphaleron. Finally we obtain the additional constraints implied by enhanced supersymmetry, and discuss Penrose limits in the theories.