Spinorial geometry and Killing spinor equations of 6-D supergravity

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)·Sp(1)⋉H(1), U(1)·Sp(1)⋉H(2), Sp(1)⋉H(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)⋉H 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. On study leave from the Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK.