Introduction to Conformal Field Theory: With Applications to String ...

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R. Blumenhagen E. Plauschinn Introduction to Conformal Field Theory With Applications to String Theory 123
Ralph Blumenhagen Max-Planck-Institut für Physik Werner-Heisenberg-Institut Föhringer Ring 6 80805 München Germany Erik Plauschinn Max-Planck-Institut für Physik Werner-Heisenberg-Institut Föhringer Ring 6 80805 München Germany Blumenhagen, R., Plauschinn, E., Introduction to Conformal Field Theory: With Applications to String Theory, Lect. Notes Phys. 779 (Springer, Berlin Heidelberg 2009), DOI 10.1007/ 978-3-642-00450-6 ISBN 978-3-642-00449-0 e-ISBN 978-3-642-00450-6 DOI 10.1007/978-3-642-00450-6 Springer Dordrecht Heidelberg London New York Lecture Notes in Physics ISSN 0075-8450 e-ISSN 1616-6361 Library of Congress Control Number: 2009921991 c○ Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Integra Software Service Pvt. Ltd. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Page 2 and 3: Lecture Notes in Physics Founding E
Page 6 and 7: Preface These lecture notes are bas
Page 8 and 9: Contents 1 Introduction ...........
Page 10 and 11: Contents xi Concluding Remarks ....
Page 12 and 13: 2 1 Introduction phase transition i
Page 14 and 15: Chapter 2 Basics in Conformal Field
Page 16 and 17: 2.1 The Conformal Group 7 ηρσ
Page 18 and 19: 2.1 The Conformal Group 9 • The c
Page 20 and 21: 2.1 The Conformal Group 11 transfor
Page 22 and 23: 2.1 The Conformal Group z 13 ′ =
Page 24 and 25: 2.1 The Conformal Group 15 z ↦→
Page 26 and 27: 2.2 Primary Fields 17 • Since p(m
Page 28 and 29: 2.3 The Energy-Momentum Tensor 19 2
Page 30 and 31: 2.4 Radial Quantisation 21 directio
Page 32 and 33: 2.5 The Operator Product Expansion
Page 38 and 39: 2.6 Operator Algebra of Chiral Quas
Page 46 and 47: 2.7 Normal Ordered Products 37 �
Page 48 and 49: 2.7 Normal Ordered Products 39 The
Page 50 and 51: 2.8 The CFT Hilbert Space 41 togeth
Page 52 and 53: 2.8 The CFT Hilbert Space 43 for n
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2.9 Simple Examples of CFTs 45 S =
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2.9 Simple Examples of CFTs 47 Let
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2.9 Simple Examples of CFTs 49 c 1
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2.9 Simple Examples of CFTs 51 The
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2.9 Simple Examples of CFTs � L0
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2.9 Simple Examples of CFTs 55 Howe
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2.9 Simple Examples of CFTs 57 γ 0
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2.9 Simple Examples of CFTs 59 Radi
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2.9 Simple Examples of CFTs 61 whic
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2.9 Simple Examples of CFTs 63 Comp
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2.9 Simple Examples of CFTs 65 �
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2.10 Highest Weight Representations
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2.11 Correlation Functions and Fusi
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2.11 Correlation Functions and Fusi
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2.12 Non-Holomorphic OPE and Crossi
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2.12 Non-Holomorphic OPE and Crossi
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2.13 Fusing and Braiding Matrices 8
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Chapter 3 Symmetries of Conformal F
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3.2 The Sugawara Construction 89 wh
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3.2 The Sugawara Construction 91 c
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3.4 The �so(N)1 Current Algebra 9
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3.5 The Knizhnik-Zamolodchikov Equa
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3.5 The Knizhnik-Zamolodchikov Equa
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3.6 Coset Construction 103 Quotient
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3.6 Coset Construction 105 Hilbert
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3.7 W Algebras 107 where the first
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3.7 W Algebras 109 However, one sti
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Further Reading 111 N (N (LL)L) CWW
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114 4 Conformal Field Theory on the
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Chapter 5 Supersymmetric Conformal
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5.1 N = 1 Superconformal Models 171
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5.3 Chiral Ring 181 primaries, the
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5.3 Chiral Ring 183 always two comp
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5.4 Spectral Flow 185 � + ′ −
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5.5 Coset Construction for the N =
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5.5 Coset Construction for the N =
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5.6 Gepner Models 191 in moduli spa
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5.6 Gepner Models 193 S �so(10)1
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5.6 Gepner Models 195 Except the tw
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5.6 Gepner Models 197 Q � state
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5.6 Gepner Models 199 Before perfor
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5.7 Massless Modes of Gepner Models
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Further Reading 203 (3, −3, 0) (2
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Chapter 6 Boundary Conformal Field
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6.1 The Free Boson with Boundaries
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6.2 Boundary States for the Free Bo
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6.3 Boundary States for RCFTs 225
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6.3 Boundary States for RCFTs 227 T
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6.4 CFTs on Non-Orientable Surfaces
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6.5 Crosscap States for the Free Bo
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6.6 Crosscap States for RCFTs 245 w
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6.6 Crosscap States for RCFTs 247
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6.7 The Orientifold of the Bosonic
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Concluding Remarks Let us conclude
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Index A A-D-E classification, 148 a
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Index 263 Klein bottle partition fu
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Index 265 simple current, 161 summa
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