Deconstruction hierarchies and locality diagrams of conformal models

The relationship between locality graphs and deconstruction hierarchies of conformal models is explained, leading to computationally effective procedures for determining the latter, and the relevant notions are illustrated with several examples.

Escape from supercooling with or without bubbles: gravitational wave signatures

Quasi-conformal models are an appealing scenario that can offer naturally a strongly supercooled phase transition and a period of thermal inflation in the early Universe. A crucial aspect for the viability of these models is how the Universe escapes from the supercooled state. One possibility is that thermal inflation phase ends by nucleation and percolation of true vacuum bubbles. This route is not, however, always efficient. In such case another escape mechanism, based on the growth of quantum fluctuations of the scalar field that eventually destabilize the false vacuum, becomes relevant. We study both of these cases in detail in a simple yet representative model. We determine the duration of the thermal inflation, the curvature power spectrum generated for the scales that exit horizon during the thermal inflation, and the stochastic gravitational wave background from the phase transition. We show that these gravitational waves provide an observable signal from the thermal inflation in almost the entire parameter space of interest. Furthermore, the shape of the gravitational wave spectrum can be used to ascertain how the Universe escaped from supercooling.

The Arena of Conformal Models

Chapter 14 discusses how the identification of a class of universality is one of the central questions needing an answer for those in the field of statistical physics. This chapter discusses in detail the class of universality of several models, and provides examples that include the Ising model, the tricritical Ising model and its structure constants, the Yang–Lee model and the 3-state Potts model. This chapter also covers the study of the statistical models of geometric type (as, for instance, those that describe the self-avoiding walks) and their formulation in terms of conformal minimal models, including conformal models with O(n) symmetry.

Conformal Field Theories with Extended Symmetries

The conformal transformations may be part of a larger group of symmetry. Chapter 13 discusses several of the extensions of conformal field theory, including supersymmetry, Z N transformations and current algebras. It also covers superconformal models, the Neveu–Schwarz and Ramond sectors, irreducible representations and minimal models, additional symmetry, the supersymmetric Landau–Ginzburg theory, parafermion models, the relation to lattice models, Kac–Moody algebras, Virasoro operators, the Sugawara Formula, maximal weights and conformal models as cosets. The appendix provides for the interested reader a self-contained discussion on the Lie algebras, include the dual Coxeter numbers, properties of weight vectors and roots/simple roots.

Minimal Conformal Models

Chapter 11 discusses the so-called minimal conformal models, all of which are characterized by a finite number of representations. It goes on to demonstrate how all correlation functions of these models satisfy linear differential equations. It shows how their explicit solutions are given by using the Coulomb gas method. It also explains how their exact partition functions can be obtained by enforcing the modular invariance of the theory. The chapter also covers null vectors, the Kac determinant, unitary representations, operator product expansion, fusion rules, Verlinde algebra, screening operators, structure constants, the Landau–Ginzburg formulation, modular invariance, and Torus geometry. The appendix covers hypergeometric functions.