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

TL;DR

This work develops a general Killing-spinor-based framework for six-dimensional chiral N=(1,0) U(1) and SU(2) gauged supergravities, revealing that supersymmetric configurations align with an $SU(2) times R^4$ structure and a null Killing vector that enforces a 2+4 base split. The authors derive necessary and sufficient conditions for supersymmetry, express the fluxes and field strengths in terms of base data, and show how the intrinsic torsion of the base controls the geometry and the deformations induced by fluxes. They construct a variety of explicit solutions, including non-twisting U(1) dyonic strings, Cahen–Wallach$_4 imes S^2$ backgrounds, SU(2) black strings with AdS$_3 imes S^3$ near-horizon limits, and Yang–Mills analogues of Salam–Sezgin backgrounds, and they analyze enhanced supersymmetry and Penrose limits that connect gauged and ungauged theories. The results provide geometric insight into supersymmetric solutions via four-dimensional base geometry, with potential applications to holography and string-realizations of these six-dimensional vacua. The methodology demonstrates that, despite complexity, a G-structure approach yields tractable reductions to four-dimensional geometry and yields a rich set of physically interesting backgrounds, including nonabelian generalizations of known Salam–Sezgin-type solutions and their pp-wave limits.

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 Kähler, 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.