This graduate textbook covers the basic formalism of supergravity, as well as its modern applications, suitable for a focused first course. Assuming a working knowledge of quantum field theory, Part I gives basic formalism, including on- and off-shell supergravity, the covariant formulation, superspace and coset formulations, coupling to matter, higher dimensions, and extended supersymmetry. A wide range of modern applications are introduced in Part II, including string theoretical (T- and U-duality, anti-de Sitter/conformal field theory (Ads/CFT), susy and sugra on the worldsheet, and superembeddings), gravitational (p-brane solutions and their susy, attractor mechanism, and Witten’s positive energy theorem), and phenomenological (inflation in supergravity, supergravity no-go theorems, string theory constructions at low energies, and minimal supergravity and its susy breaking). The broader emphasis on applications than competing texts gives Ph.D. students the tools they need to do research that uses supergravity and benefits researchers already working in areas related to supergravity.

To define irreducible representations, spinors with dotted and undotted indices are defined. Irreducible representations of susy are defined, first in the massless case, then in the massive case, with and without central charges. The R-symmetry of the algebras is defined. The Lagrangians for the d = 4 multiplets, N = 1 chiral, N = 1 vector, their coupling and the N = 2 models, and finally the unique N = 4 model, are described.

From the unique N = 1 11-dimensional supergravity, all the other supergravities in lower dimensions are thought to be obtained. To obtain the seven-dimensional gauged supergravity, we first describe a first-order formulation of 11-dimensional supergravity. Then, we describe a nonlinear ansatz, leading to a consistent truncation. The concepts of consistent truncation and nonlinear ansatze are described, and the linearized ansatz on S4 and the spherical harmonics on S4 are reviewed. Relatedly, we describe the Lagrangians and transformation rules for the (maximal) N = 8 d = 4, N = 8 d = 5, and N = 4 d = 7 gauged supergravities, the massive type IIA 10-dimensional supergravity, type IIB 10-dimensional supergravity, and some general properties of gaugings, in particular the non-compact ISO(7) gauging. We end with modified supergravities: the one with SO(1, 3) × SU(8) local Lorentz covariance in 11 dimensions, the generalization known as exceptional field theory, and the geometric approach to supergravity, in particular d'Auria and Fré’s 11-dimensional supergravity with OSp(1|32) invariance.

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.