Conformal dimension measures the extent to which the Hausdorff dimension of a metric space can be lowered by quasisymmetric deformations. Introduced by Pansu in 1989, this concept has proved extremely fruitful in a diverse range of areas, including geometric function theory, conformal dynamics, and geometric group theory. This survey leads the reader from the definitions and basic theory through to active research applications in geometric function theory, Gromov hyperbolic geometry, and the dynamics of rational maps, amongst other areas. It reviews the theory of dimension in metric spaces and of deformations of metric spaces. It summarizes the basic tools for estimating conformal dimension and illustrates their application to concrete problems of independent interest. Numerous examples and proofs are provided. Working from basic definitions through to current research areas, this book can be used as a guide for graduate students interested in this field, or as a helpful survey for experts. Background needed for a potential reader of the book consists of a working knowledge of real and complex analysis on the level of first- and second-year graduate courses.
The theory of vertex operator algebras and their representations has been showing its power in the solution of concrete mathematical problems and in the understanding of conceptual but subtle mathematical and physical struc- tures of conformal field theories. Much of the recent progress has deep connec- tions with complex analysis and conformal geometry. Future developments, especially constructions and studies of higher-genus theories, will need a solid geometric theory of vertex operator algebras. Back in 1986, Manin already observed in Man) that the quantum theory of (super )strings existed (in some sense) in two entirely different mathematical fields. Under canonical quantization this theory appeared to a mathematician as the representation theories of the Heisenberg, Vir as oro and affine Kac- Moody algebras and their superextensions. Quantization with the help of the Polyakov path integral led on the other hand to the analytic theory of algebraic (super ) curves and their moduli spaces, to invariants of the type of the analytic curvature, and so on.He pointed out further that establishing direct mathematical connections between these two forms of a single theory was a big and important problem. On the one hand, the theory of vertex operator algebras and their repre- sentations unifies (and considerably extends) the representation theories of the Heisenberg, Virasoro and Kac-Moody algebras and their superextensions.
Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.
Conformal field theory is an elegant and powerful theory in the field of high energy physics and statistics. In fact, it can be said to be one of the greatest achievements in the development of this field. Presented in two dimensions, this book is designed for students who already have a basic knowledge of quantum mechanics, field theory and general relativity. The main idea used throughout the book is that conformal symmetry causes both classical and quantum integrability. Instead of concentrating on the numerous applications of the theory, the author puts forward a discussion of the general methods of conformal field theory as a physical theory. Hence the book provides in a self-contained way the necessary knowledge and “conformal” intuition which underline the various applications of conformal field theory. It is aimed to assist students and professionals in the study of the theory from its first principles and in applying the methods in their own research. The first of its kind, this book promises to give a detailed and comprehensive insight into the workings of conformal field theory.
The second edition of Non-Perturbative Methods in Two-Dimensional Quantum Field Theory is an extensively revised version, involving major changes and additions. Although much of the material is special to two dimensions, the techniques used should prove helpful also in the development of techniques applicable in higher dimensions. In particular, the last three chapters of the book will be of direct interest to researchers wanting to work in the field of conformal field theory and strings.This book is intended for students working for their PhD degree and post-doctoral researchers wishing to acquaint themselves with the non-perturbative aspects of quantum field theory.
Based on class-tested notes, this text offers an introduction to Conformal Field Theory with a special emphasis on computational techniques of relevance for String Theory. It introduces Conformal Field Theory at a basic level, Kac-Moody algebras, one-loop partition functions, Superconformal Field Theories, Gepner Models and Boundary Conformal Field Theory. Eventually, the concept of orientifold constructions is explained in detail for the example of the bosonic string. In providing many detailed CFT calculations, this book is ideal for students and scientists intending to become acquainted with CFT techniques relevant for string theory but also for students and non-specialists from related fields.
This book covers some recent advances in string theory and extra dimensions. Intended mainly for advanced graduate students in theoretical physics, it presents a rare combination of formal and phenomenological topics, based on the annual lectures given at the School of the Theoretical Advanced Study Institute (2001) a traditional event that brings together graduate students in high energy physics for an intensive course of advanced learning. The lecturers in the School are leaders in their fields.The first lecture, by E D'Hoker and D Freedman, is a systematic introduction to the gauge-gravity correspondence, focusing in particular on correlation functions in the conformal case. The second, by L Dolan, provides an introduction to perturbative string theory, including recent advances on backgrounds involving Ramond-Ramond fluxes. The third, by S Gubser, explains some of the basic facts about special holonomy and its uses in string theory and M-theory. The fourth, by J Hewett, surveys the TeV phenomenology of theories with large extra dimensions. The fifth, by G Kane, presents the case for supersymmetry at the weak scale and some of its likely experimental consequences. The sixth, by A Liddle, surveys recent developments in cosmology, particularly with regard to recent measurements of the CMB and constraints on inflation. The seventh, by B Ovrut, presents the basic features of heterotic M-theory, including constructions that contain the Standard Model. The eighth, by K Rajagopal, explains the recent advances in understanding QCD at low temperatures and high densities in terms of color superconductivity. The ninth, by M Sher, summarizes grand unified theories and baryogenesis, including discussions of supersymmetry breaking and the Standard Model Higgs mechanism. The tenth, by M Spiropulu, describes collider physics, from a survey of current and future machines to examples of data analyses relevant to theories beyond the Standard Model. The eleventh, by M Strassler, is an introduction to supersymmetric gauge theory, focusing on Wilsonian renormalization and analogies between three- and four-dimensional theories. The twelfth, by W Taylor and B Zwiebach, introduces string field theory and discusses recent advances in understanding open string tachyon condensation. The thirteenth, by D Waldram, discusses explicit model building in heterotic M-theory, emphasizing the role of the 8 gauge fields.The written presentation of these lectures is detailed yet straightforward, and they will be of use to both students and experienced researchers in high-energy theoretical physics for years to come.The proceedings have been selected for coverage in: Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings) CC Proceedings Engineering & Physical Sciences"
A twenty-fifth anniversary edition featuring a new preface, invaluable for graduate students and researchers in general relativity and elementary particle theory.
