From spinors to forms: results on g-structures in supergravity and on topological field theories

This thesis is divided in two parts, that can be read separately even if both use the possibility of replacing spinors with differential forms in theories with supersymmetry. The first part explores some recent results that have been obtained by applying the G-structure approach to type II supergravities. Using generalized complex geometry it is possible to reformulate the conditions for unbroken supersymmetry in type II supergravity in terms of differential forms. We use this result to find a classification for AdS7 and AdS6 solutions in type II supergravity. Concerning AdS7 solutions we find that in type IIB no solutions can be found, whereas in massive type IIA many new AdS7×M3 solutions are at disposal with the topology of the internal manifold M3 given by a three-sphere. We develop a classification for such solutions. Concerning AdS6 solutions, very few AdS6×M4 supersymmetric solutions are known in literature: one in massive IIA, and two IIB solutions dual to it. The IIA solution is known to be unique. We obtain a classification for IIB supergravity, by reducing the problem to two PDEs on a two-dimensional space Σ. The four-dimensional space M4 is then given by a fibration of S2 over Σ. We also explore other two contexts in which the G-structure approach has revealed its usefulness: first of all we derive the conditions for unbroken supersymmetry for a Mink2 (2,0) vacuum, arising from type II supergravity on a compact eight-dimensional manifold M8. When M8 enjoys SU(4)×SU(4) structure the resulting system is elegantly rewritten in terms of generalized complex geometry. Finally we rewrite the equations for ten-dimensional supersymmetry in a way formally identical to an analogous system in N = 2 gauged supergravity; this provides a way to look for lifts of BPS solutions without having to reduce the ten-dimensional action. The second part is devoted to study some aspects of two different Chern-Simons like theories: holomorphic Chern-Simons theory on a six-dimensional Calabi-Yau space and three-dimensional supersymmetric theories involving vector multiplets (both with Yang-Mills and Chern-Simons terms in the action). Concerning holomorphic Chern-Simons theory, we construct an action that couples the gauge field to off-shell gravitational backgrounds, comprising the complex structure and the (3,0)-form of the target space. Gauge invariance of this off-shell action is achieved by enlarging the field space to include an appropriate system of Lagrange multipliers, ghost and ghost-for-ghost fields. From this reformulation it is possible to uncover a twisted supersymmetric algebra for this model that strongly constrains the anti-holomorphic dependence of physical correlators. Concerning three-dimensional theories, we will develop a new way of computing the exact partition function of supersymmetric three-dimensional gauge theories, involving vector supermultiplet only. Our approach will reduce the problem of computing the exact partition function to the problem of solving an anomalous Ward identity. To obtain such a result we will describe the coupling of three-dimensional topological gauge theories to background topological gravity. The Seifert condition for manifolds supporting global supersymmetry is elegantly deduced from the topological gravity BRST transformations. We will show how the geometrical moduli that affect the partition function can be characterized cohomologically. In the Seifert context Chern-Simons topological (framing) anomaly is BRST trivial and we will compute explicitly the corresponding local Wess-Zumino functional. As an application, we obtain the dependence on the Seifert moduli of the partition function of three-dimensional supersymmetric gauge theory on the squashed sphere by solving the anomalous topological Ward identities, in a regularization independent way and without the need of evaluating any functional determinant.