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.

Appendices include the basic equations involved in coupling the population balance equation (PBE) with fluid flow, heat and mass transfer (Appendix A), the implementation of the conservative finite volume discretisation method (Appendix B), the derivation of the probability density function (PDF) transport equation (Appendix C) and the derivation of the stochastic field equation (Appendix D).

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.

The application of the theory and methodology presented in the previous chapters for formulating and solving the population balance equation (PBE), as well as its coupling with fluid flow and computational fluid dynamics (CFD), is here demonstrated via three case studies. The first case study is about synthesis of silica nanoparticles in a laminar flame. The second one involves soot formation in laminar and turbulent flames. The third one is about precipitation of barium sulphate crystals in a turbulent T-mixer flow. In each case, the deployment of the population balance methodology is presented in an educational manner, following the four main steps outlined in Chapter 1.

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.