Conformal Dimension
Conformal Dimension

Author: John M. Mackay

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 162

ISBN-13: 0821852299

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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.

Assouad Dimension and Fractal Geometry
Assouad Dimension and Fractal Geometry

Author: Jonathan M. Fraser

Publisher: Cambridge University Press

Published: 2020-10-29

Total Pages: 287

ISBN-13: 1108478654

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The first thorough treatment of the Assouad dimension in fractal geometry, with applications to many fields within pure mathematics.

Two-Dimensional Conformal Geometry and Vertex Operator Algebras
Two-Dimensional Conformal Geometry and Vertex Operator Algebras

Author: Yi-Zhi Huang

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 289

ISBN-13: 1461242762

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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.

Conformal Field Theory
Conformal Field Theory

Author: Philippe Francesco

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 908

ISBN-13: 1461222567

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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.

Non-perturbative Methods in 2 Dimensional Quantum Field Theory
Non-perturbative Methods in 2 Dimensional Quantum Field Theory

Author: Elcio Abdalla

Publisher: World Scientific

Published: 2001

Total Pages: 834

ISBN-13: 9812810153

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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. Contents: Free Fields; The Thirring Model; Determinants and Heat Kernels; Self-Interacting Fermionic Models; Nonlinear a Models: Classical Aspects; Nonlinear a Models OCo Quantum Aspects; Exact S-Matrices of 2D Models; The Wess-Zumino-Witten Theory; QED 2: Operator Approach; Quantum Chromodynamics; QED 2: Functional Approach; The Finite Temperature Schwinger Model; Non-Abelian Chiral Gauge Theories; Chiral Quantum Electrodynamics; Conformally Invariant Field Theory; Conformal Field Theory with Internal Symmetry; 2D Gravity and String-Related Topics. Readership: Graduate students and researchers in high energy and quantum physics."

Geometry and Analysis of Metric Spaces via Weighted Partitions
Geometry and Analysis of Metric Spaces via Weighted Partitions

Author: Jun Kigami

Publisher: Springer Nature

Published: 2020-11-16

Total Pages: 164

ISBN-13: 3030541541

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The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.

Infinite Dimensional Lie Algebras And Groups
Infinite Dimensional Lie Algebras And Groups

Author: Victor G Kac

Publisher: World Scientific

Published: 1989-07-01

Total Pages: 642

ISBN-13: 9814663174

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Contents:Integrable Representation of Kac-Moody Algebras: Results and Open Problems (V Chari & A Pressley)Existence of Certain Components in the Tensor Product of Two Integrable Highest Weight Modules for Kac-Moody Algebras (SKumar)Frobenius Action on the B-Cohomology (O Mathieu)Certain Rank Two Subsystems of Kac-Moody Root Systems (J Morita)Lie Groups Associated to Kac-Moody Lie Algebras: An Analytic Approach (E Rodriguez-Carrington)Almost Split-K-Forms of Kac-Moody Algebras (G Rousseau)Global Representations of the Diffeomorphism Groups of the Circle (F Bien)Path Space Realization of the Basic Representation of An(1) (E Date et al)Boson-Fermion Correspondence Over (C De Concini et al)Classification of Modular Invariant Representations of Affine Algebras (V G Kac & M Wakimoto)Standard Monomial Theory for SL2 (V Lakshmibai & C S Seshadri)Some Results on Modular Invariant Representations (S Lu)Current Algebras in 3+1 Space-Time Dimensions (J Mickelson)Standard Representations of An(1) (M Primc)Representations of the Algebra Uq(sI(2)), q-Orthogonal Polynomials and Invariants of Links (A N Kirillov & N Yu Reshetikhin)Infinite Super Grassmannians and Super Plücker Equations (M J Bergvelt)Drinfeld-Sokolov Hierarchies and t-Functions (H J Imbens)Super Boson-Fermion Correspondence of Type B (V G Kac & J W van de Leur)Prym Varieties and Soliton Equations (T Shiota)Polynomial Solutions of the BKP Hierarchy and Projective Representations of Symmetric Groups (Y You)Toward Generalized Macdonald's Identities (D Bernard)Conformal Theories with Non-Linearly Extended Virasoro Symmetries and Lie Algebra Classification (A Bilal & J-LGervais)Extended Conformal Algebras from Kac-Moody Algebras (P Bouwknegt)Meromorphic Conformal Field Theory (P Goddard)Local Extensions of the U(1) Current Algebra and Their Positive Energy Representations (R R Paunov & I T Todorov)Conformal Field Theory on Moduli Family of Stable Curves with Gauge Symmetries (A Tsuchiya & Y Yamada) Readership: Mathematicians and mathematical physicists

Introduction to Conformal Field Theory
Introduction to Conformal Field Theory

Author: Ralph Blumenhagen

Publisher: Springer Science & Business Media

Published: 2009-07-31

Total Pages: 270

ISBN-13: 3642004490

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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.

Supersymmetric Gauge Field Theory and String Theory
Supersymmetric Gauge Field Theory and String Theory

Author: D. Bailin

Publisher: CRC Press

Published: 1994-01-01

Total Pages: 672

ISBN-13: 9781420050820

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Designed as a sequel to the authors' Introduction to Gauge Field Theory, Supersymmetric Gauge Field Theory and String Theory introduces first-year graduate students to supersymmetric theories, including supergravity and superstring theories. Starting with the necessary background in quantum field theory, the book covers the three key topics of high-energy physics. The emphasis is on practical calculations rather than abstract generalities or phenomenological results. Where possible, the authors show how to calculate, connecting the theoretical with the phenomenological. While the field continues to advance and grow, this book addresses the basic theory at the core and will likely remain relevant even if more advanced ideas change.

Gauge/Gravity Duality
Gauge/Gravity Duality

Author: Martin Ammon

Publisher: Cambridge University Press

Published: 2015-04-09

Total Pages: 549

ISBN-13: 1316239721

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Gauge/gravity duality creates new links between quantum theory and gravity. It has led to new concepts in mathematics and physics, and provides new tools to solve problems in many areas of theoretical physics. This book is the first textbook on this important topic, enabling graduate students and researchers in string theory and particle, nuclear and condensed matter physics to get acquainted with the subject. Focusing on the fundamental aspects as well as on the applications, this textbook guides readers through a thorough explanation of the central concepts of gauge/gravity duality. For the AdS/CFT correspondence, it explains in detail how string theory provides the conjectured map. Generalisations to less symmetric cases of gauge/gravity duality and their applications are then presented, in particular to finite temperature and density, hydrodynamics, QCD-like theories, the quark-gluon plasma and condensed matter systems. The textbook features a large number of exercises, with solutions available online at www.cambridge.org/9781107010345.