Spinorial geometry and Killing spinor equations of 6-D supergravity

TL;DR

This paper classifies solutions to Killing spinor equations in 6D supergravity, revealing geometric structures and symmetries associated with different supersymmetry-preserving isotropy groups.

Contribution

It provides a complete solution to the Killing spinor equations in 6D (1,0)-supergravity with various multiplets, detailing the geometric and algebraic structures for all supersymmetry cases.

Findings

Spacetime admits parallel null 1-forms with torsion in non-compact cases.

Associated vector fields are Killing and related to spacetime geometry.

Certain cases lead to principal bundle structures with Lorentzian Lie group fibers.

Abstract

We solve the Killing spinor equations of 6-dimensional (1,0)-supergravity coupled to any number of tensor, vector and scalar multiplets in all cases. The isotropy groups of Killing spinors are $Sp(1)\cdot Sp(1)\ltimes \bH (1)$, $U(1)\cdot Sp(1)\ltimes \bH (2)$, $Sp(1)\ltimes \bH (3,4)$, $Sp(1) (2)$, $U(1) (4)$ and $\{1\} (8)$, where in parenthesis is the number of supersymmetries preserved in each case. If the isotropy group is non-compact, the spacetime admits a parallel null 1-form with respect to a connection with torsion the 3-form field strength of the gravitational multiplet. The associated vector field is Killing and the 3-form is determined in terms of the geometry of spacetime. The $Sp(1)\ltimes \bH$ case admits a descendant solution preserving 3 out of 4 supersymmetries due to the hyperini Killing spinor equation. If the isotropy group is compact, the spacetime admits a natural frame constructed from 1-form spinor bi-linears. In the $Sp(1)$ and U(1) cases, the spacetime admits 3 and 4 parallel 1-forms with respect to the connection with torsion, respectively. The associated vector fields are Killing and under some additional restrictions the spacetime is a principal bundle with fibre a Lorentzian Lie group. The conditions imposed by the Killing spinor equations on all other fields are also determined.