Counting of degrees of freedom of fields up to spin 2 is described, both on-shell and off-shell. Then, the supergravity models, based on matching Bose and Fermi degrees of freedom, are inferred. The N = 1 on-shell supergravity model is defined, and the first-order and second-order formalisms, as well as the better 1.5-order formalism, in which the susy invariance of the action is shown.

We described actions and equations of motion for general supergravities, out of which we derive extremal and black p-brane solutions. We find electric p-brane solutions and magnetic p-brane solutions and show the duality between them, then we generalize to black p-branes and Dp-branes in 10 dimensions, and then we discuss fundamental string and NS5-brane solutions in 10 dimensions. Tseytlin’s harmonic function rule for writing intersecting brane solutions is explained.

By adding WZ terms to the superstring actions, we find actions with kappa symmetry. Similarly, for super-p-branes, we can describe actions, and find a brane scan, related to the existence of a kappa symmetry. In curved superspace, the supergravity equations of motion in 11 dimensions are obtained from the condition of kappa symmetry. The superembedding formalism starts with the superembedding conditions. For the case of the particle, we describe it and give example. For the superstring, we sketch how it is done.

We define the Maldacena–Núñez no-go theorem for supergravity compactifications and show that it implies that there are no de Sitter or Minkowski compactifications, both in massless and in massive supergravity. The case of no Randall–Sundrum solutions in d = 5 gauged supergravity is treated separately. The swampland conjecture for string theory compactifications is based on some “sporadic” results, and there is a more general no-go theorem, but there are loopholes.

Supersymmetry is defined as a Bose–Fermi symmetry. Spinors are defined in general dimensions. The Wess–Zumino model is defined first in two dimensions on-shell, where the invariance of the action is proven using Majorana spinor identities. The susy algebra is defined, and using Fierz identities, one proves the closure of the algebra and resulting off-shell susy. Then, the four-dimensional free off-shell Wess–Zumino model is defined as a simple generalization.

N = 2 supergravity in four dimensions is defined, and the related special geometry is defined. First one starts with the rigid susy case, then special geometry is defined, and then the subset of very-special geometry and associated duality symmetries are defined. The general properties of other, more general supergravity theories (with more susy or in higher dimensions) are described. The unique N = 1 11-dimensional supergravity theory is described. We end with some comments on off-shell and superspace models in the more general cases.

We first define the notion of Kaluza–Klein (KK) compactification, the three types of KK metrics one can define, and then we consider fields with (Lorentz) spin in the KK theory. The original KK theory, for compactification on S1 from five dimensions to four dimensions, is described, and we end with general properties of KK reductions.

The N = 1 four-dimensional supergravity is described in superspace, in the super-geometric approach. We discuss the invariances, gauge choices, and the fields, then the superspace constraints, and then solve the constraints and the Bianchi identities.

We consider Minimal Supergravity, starting with the masses and parameters of MSSM, followed by the supergravity extension. Then, susy breaking is treated, in particular in the mechanisms for the MSSM and MinSugra cases. The MinSugra case, with its gravity mediation mechanism, is described in detail, and the Polonyi model is given as an example.

We describe the N = 1 four-dimensional off-shell supergravity, the susy transformation rules and action, and then the susy algebra closure.

The three-dimensional N = 1 off-shell supergravity action is described, with its symmetries explained, and the susy transformation rules and invariance of the action. The closure of the susy algebra is discussed.

We describe the AdS/CFT correspondence, obtained from string theory in certain backgrounds, in a decoupling limit. We also consider the M theory cases of gravity duals and then the gravity duals of N extremal Dp-branes. In the Penrose limit of supergravity solutions, we obtain pp waves. The Penrose limit can also be taken on the isometry group and algebra.

We define U-duality as being generated by T-dualities and S-dualities together. We show how this leads to the unification of string theory states (and their corresponding supergravity solutions) under M theory. The string duality web is then defined. Finally, we show how U-duality is obtained from M theory.

We start by reviewing the Standard Model, its spectrum, symmetries, representations, and Lagrangian. Then, we consider the Grand Unified Theories extensions, in particular the SU(5) GUT, the SO(10) GUT, and other (bigger) groups. Then, we consider the Minimal Supersymmetric Standard Model (MSSM) and Minimal Supergravity, and new low-energy string (supergravity) constructions.

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.