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.

The population balance is introduced as an approach for modelling problems involving a population of particles with a distribution of one or more properties. Numerous applications are identified. The general methodology of applying the population balance in four basic steps is introduced. Basic concepts such as distributions, choice of distributed variables, kinetic and transport processes and the coupling of the population balance with fluid dynamics, are also introduced.

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.

Models for the kinetic and transport processes in the population balance equation (PBE) are discussed. Since the range of population balance problems is very wide, the focus is on the general forms of models and their incorporation into the PBE. More specific examples are drawn from the fields of aerosols and crystallisation that feature in the case studies of Chapter 6. The determination of model parameters from experimental data, or the inverse problem, is also discussed.

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