A Mathematical Introduction to Conformal Field Theory
by Schottenloher, MartinEdition: 2nd
Format: Hardcover
Pub. Date: 2008-12-04
Publisher(s): Textstream
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Summary
"The first part of this book gives a detailed, self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in two dimensions. The second part surveys some more advanced topics of conformal field theory, such as the representation theory of the Virasoro algebra, conformal symmetry within string theory, an axiomatic approach to Euclidean conformally covariant quantum field theory and a mathematical interpretation of the Verlinde formula in the context of moduli spaces of holomorphic vector bundles on a Riemann surface." "The substantially revised and enlarged second edition makes in particular the second part of the book more self-contained and tutorial, with many more examples given. Furthermore, two new chapters on Wightman's axioms for quantum field theory and vertex algebras broaden the survey of advanced topics. An outlook making the connection with most recent developments has also been added."--BOOK JACKET.
Table of Contents
| Introduction | p. 1 |
| References | p. 3 |
| Mathematical Preliminaries | |
| Conformal Transformations and Conformal Killing Fields | p. 7 |
| Semi-Riemannian Manifolds | p. 7 |
| Conformal Transformations | p. 9 |
| Conformal Killing Fields | p. 13 |
| Classification of Conformal Transformations | p. 15 |
| Case 1: n = p + q > 2 | p. 15 |
| Case 2: Euclidean Plane (p = 2, q = 0) | p. 18 |
| Case 3: Minkowski Plane (p = q = 1) | p. 19 |
| The Conformal Group | p. 23 |
| Conformal Compactification of R[superscript p,q] | p. 23 |
| The Conformal Group of R[superscript p,q] for p + q > 2 | p. 28 |
| The Conformal Group of R[superscript 2,0] | p. 31 |
| In What Sense Is the Conformal Group Infinite Dimensional? | p. 33 |
| The Conformal Group of R[superscript 1,1] | p. 35 |
| References | p. 38 |
| Central Extensions of Groups | p. 39 |
| Central Extensions | p. 39 |
| Quantization of Symmetries | p. 44 |
| Equivalence of Central Extensions | p. 56 |
| References | p. 61 |
| Central Extensions of Lie Algebras and Bargmann's Theorem | p. 63 |
| Central Extensions and Equivalence | p. 63 |
| Bargmann's Theorem | p. 69 |
| References | p. 73 |
| The Virasoro Algebra | p. 75 |
| Witt Algebra and Infinitesimal Conformal Transformations of the Minkowski Plane | p. 75 |
| Witt Algebra and Infinitesimal Conformal Transformations of the Euclidean Plane | p. 77 |
| The Virasoro Algebra as a Central Extension of the Witt Algebra | p. 79 |
| Does There Exist a Complex Virasoro Group? | p. 82 |
| References | p. 84 |
| First Steps Toward Conformal Field Theory | |
| Representation Theory of the Virasoro Algebra | p. 91 |
| Unitary and Highest-Weight Representations | p. 91 |
| Verma Modules | p. 92 |
| The Kac Determinant | p. 95 |
| Indecomposability and Irreducibility of Representations | p. 99 |
| Projective Representations of Diff[subscript +] (S) | p. 100 |
| References | p. 102 |
| String Theory as a Conformal Field Theory | p. 103 |
| Classical Action Functionals and Equations of Motion for Strings | p. 103 |
| Canonical Quantization | p. 111 |
| Fock Space Representation of the Virasoro Algebra | p. 115 |
| Quantization of Strings | p. 119 |
| References | p. 120 |
| Axioms of Relativistic Quantum Field Theory | p. 121 |
| Distributions | p. 122 |
| Field Operators | p. 129 |
| Wightman Axioms | p. 131 |
| Wightman Distributions and Reconstruction | p. 137 |
| Analytic Continuation and Wick Rotation | p. 142 |
| Euclidean Formulation | p. 148 |
| Conformal Covariance | p. 149 |
| References | p. 151 |
| Foundations of Two-Dimensional Conformal Quantum Field Theory | p. 153 |
| Axioms for Two-Dimensional Euclidean Quantum Field Theory | p. 153 |
| Conformal Fields and the Energy-Momentum Tensor | p. 159 |
| Primary Fields, Operator Product Expansion, and Fusion | p. 163 |
| Other Approaches to Axiomatization | p. 168 |
| References | p. 169 |
| Vertex Algebras | p. 171 |
| Formal Distributions | p. 172 |
| Locality and Normal Ordering | p. 177 |
| Fields and Locality | p. 181 |
| The Concept of a Vertex Algebra | p. 185 |
| Conformal Vertex Algebras | p. 192 |
| Associativity of the Operator Product Expansion | p. 199 |
| Induced Representations | p. 209 |
| References | p. 212 |
| Mathematical Aspects of the Verlinde Formula | p. 213 |
| The Moduli Space of Representations and Theta Functions | p. 213 |
| The Verlinde Formula | p. 219 |
| Fusion Rules for Surfaces with Marked Points | p. 221 |
| Combinatorics on Fusion Rings: Verlinde Algebra | p. 228 |
| References | p. 232 |
| Some Further Developments | p. 235 |
| References | p. 236 |
| References | p. 239 |
| Index | p. 245 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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