Spencer cohomology, supersymmetry and the structure of Killing superalgebras

Abstract

We review and expand upon work from the last decade on the application of Spencer cohomology to the study of supersymmetric bosonic backgrounds of supergravity. The central observation of this project is that the symmetry superalgebras of such backgrounds, known as Killing superalgebras, are filtered subdeformations of the Poincaré superalgebra. Such deformations of Z-graded Lie superalgebras are governed by Spencer cohomology, thus the structure of Killing superalgebras can be determined from the Spencer cohomology of (graded subalgebras of) the Poincaré superalgebra. Moreover, the cohomology calculation often allows one to write down the Killing spinor equation and determine much of the structure of supersymmetric supergravity backgrounds, and in some cases it allows one to generalise the notion of supersymmetric geometry.

This thesis contributes to the existing literature on Spencer cohomology and supersymmetry both in terms of general theoretical results and in terms of explicit calculations and examples. We consider the algebraic structure of Killing superalgebras, with particular focus on the highly supersymmetric case, generalising to arbitrary dimension and amount of supersymmetry a number of results previously only proven for 11-dimensional supergravity. We characterise possible obstructions to ``integrating'' infinitesimal filtered deformations ((2,2)-Spencer cohomology classes) of the Poincaré superalgebra to full deformations. We also give a geometric description of backgrounds with gauged R-symmetry, clarifying and improving upon previous treatments in the literature, and generalise the aforementioned results to Killing superalgebras of such backgrounds. Some worked examples in 2 dimensions are presented. We determine the (2,2)-Spencer cohomology of the N-extended Poincaré superalgebra in 5 and 6 spacetime dimensions with gauged R-symmetry. We also show that this cohomology is trivial in Type IIA. We then consider the structure of filtered deformations and maximally supersymmetric backgrounds in the minimal 5-dimensional case with gauged R-symmetry, showing that obstructions to integrating infinitesimal deformations exist and arise in both algebraic and geometric guises. We also classify some sub-classes of the maximally supersymmetric backgrounds and their superalgebras in this case.