We give the proof of Witten’s positive energy theorem in general relativity and make the connection to supergravity.

The vielbein–spin connection formulation of general relativity is described, and this being the one that appears in supergravity. Anti-de Sitter space, as a Lorentzian version of Lobachevsky space, is described. It is a symmetric space solution for the case of a cosmological constant. Black holes, as objects with event horizons and singularities at the center, are described.

We describe what the susy invariance of a solution means and how to calculate the mass of solutions of supergravity. The supersymmetry of various solutions is considered, and it is shown that these correspond to fundamental objects (states) in string theory. These states are then shown to be classified via the susy algebra. Finally, intersecting brane solutions are considered in the same analysis.

After obtaining the transformation rules and constraints from rigid superspace described as a coset, we define the covariant formulation of four-dimensional YM in rigid superspace and solve the constraints and Bianchi identities, and relate this formulation to the prepotential formalism. Then, we describe the coset approach to three-dimensional supergravity (as a generalization of the covariant YM formalism). Finally, we describe the general super-geometric approach to supergravity.

After reviewing YM superfields in rigid superspace, we defined them in curved superspace. We define invariant measures for the superspace actions, and finally describe supergravity actions. Then, we discuss couplings of supergravity with matter, describing things first in superspace and then in components.

We consider compactification of low-energy string theory, mostly in the supergravity regime and mostly for the heterotic case, and we discuss the conditions for obtaining N = 1 in four dimensions. We review topology issues, in particular the relation of spinors with holonomies, Kahler and Calabi–Yau manifolds, cohomology, homology, and their relation to mass spectra in four dimensions. We explain the moduli space of Calabi–Yau space, the Kahler moduli, and complex structure moduli. We then consider new features of the type IIB and heterotic E8 × E8 models.

Supersymmetry is defined in superspace, via superfields. Superspace actions are described for the chiral and vector multiplets, and the N = 2 superspace and actions are also described. Perturbative susy breaking is defined, via the Witten index, and in particular the tree-level susy breaking.

The general theory of coset manifolds (coset formalism) is defined. The notion of parallel transport and general relativity on the coset manifold are explained. In particular, one has a notion of H-covariant Lie derivatives. Finally, rigid superspace is obtained as a particular type of coset manifold, using this formalism.

We define the notion of spherical harmonics, as a generalization from the two-sphere case. We use coset theory to define them, and then we describe examples of spherical harmonics. The KK decomposition is defined, and then the particular cases of groups spaces and spheres are considered for the spherical harmonics.

We describe various solution-generating techniques (dualities and transformations). We start with abelian T-duality, generalized to nonabelian T-duality, and then TsT transformations, O(d,d) transformations, and null Melvin twists.

We start by describing the particle action in the first-order and second-order formalism. This is then generalized to the bosonic string, for which we discuss actions and equations of motion, constraints, quantization, and oscillators, and we add background fields. The particle is generalized to the particle, and from that, we find we can generalize the bosonic string to the GS superstring, the NS-R (spinning) string, and the Berkovits superstring, using pure spinors.

We consider supersymmetric AdS/CFT gravity dual pairs and their deformations. First, we consider supersymmetric and integrable deformations: the beta deformation of N = 4 SYM and the gamma deformation, a three-parameter generalization. Then, we consider the eta and lambda deformations of the string worldsheet in AdS5 × S5. Then, the Yang–Baxter deformation, and the generalized supergravity equations.

For extremal black holes, we have the attractor mechanism, originally defined in the context of N = 2 supergravity. This is then interpreted and described in the Sen’s entropy function formalism. The attractor mechanism exists also in five-dimensional gauged supergravity, and by embedding it in string theory, we can relate it to holography and the AdS/CFT correspondence.

After an introduction to general relativity and supersymmetry, the formalism of supergravity is defined, on-shell, off-shell, and in superspace, using coset theory and local superspace. Higher dimensions, extended susy, and KK reduction are also defined. Then, various applications are described: dualities and solution-generating techniques, solutions and their susy algebra, gravity duals and deformations, supergravity on the string worldsheet and superembeddings, cosmological inflation, no-go theorems and Witten’s positive energy theorem, compactification of low-energy string theory and toward embedding the Standard Model using supergravity, susy breaking and minimal supergravity.

We examine cosmological inflation in supergravity. We start with N = 1 supergravity with a single chiral superfield, then consider D-term inflation, with the example of the FI model. We examine possible field redefinitions. Supergravity models with slow-roll conditions satisfied are found. A special embedding of any inflationary model into supergravity is defined. The “alpha attractors” defined by Kallosh and Linde in N = 1 supergravity are defined.