Supersymmetric Geometries in Type IIA Supergravity

Doctoral thesis, 2016

Supergravity theory is any supersymmetric theory with a local supersymmetry parameter. This thesis undertakes the study of type IIA supergravity, a supergravity theory in ten dimensions associated with type IIA string theory. In this framework we endeavour to classify all the classical geometries with minimal supersymmetry, where it should be noted that the constraints from minimal supersymmetry also apply to any solutions with enhanced supersymmetry. This thesis, together with the appended papers, provides a complete classification of all geometries in standard and massive type IIA supergravity, that preserve one supersymmetry. Such supergravity backgrounds locally admit one of four types of Killing spinors, with different isotropy groups. The Killing spinor equations have been solved for all four types, identifying the geometry of spacetime and examining the conditions on the fluxes. The picture that arises is that there are in fact three main cases, with isotropy groups Spin(7), SU(4) and G₂ ⋉ ℝ⁸, each with a special case, covariantly characterised by the vanishing of a certain spinor bilinear. In the Spin(7) case, this results in an enhancement of the isotropy group to Spin(7) ⋉ ℝ⁸. In the present work, I introduce the concepts and methods involved in making such a classification using Spinorial Geometry. The Spinorial Geometry method exploits the linearity of the Killing spinor equations and an explicit basis in the space of spinors, as well as a gauge choice, to produce a linear system of equations in the fluxes and the spin connection. The thesis describes the steps involved, and the simplification of the resulting linear system. The results are discussed and compared with results in other supergravity theories, focusing on type IIB supergravity and eleven-dimensional supergravity. This work has made heavy use of computer algebra, and a discussion of computer algebra is included.