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

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2.6 Operator Algebra of Chiral Quasi-Primary Fields 29 • Applying the relation (2.45) to the Virasoro algebra (2.16) for values m = −1, 0, +1, we see that the chiral energy–momentum tensor is a quasiprimary field of conformal dimension (h, h) = (2, 0). In view of Eq. (2.25), this observation also explains the form of the Laurent expansion (2.42). • It is worth to note that the Schwarzian derivative S(w, z) vanishes for SL(2, C) transformations w = f (z) in agreement with the fact that T (z) is a quasi-primary field. 2.6 Operator Algebra of Chiral Quasi-Primary Fields The objects of interest in quantum field theories are n-point correlation functions which are usually computed in a perturbative approach via either canonical quantisation or the path integral method. In this section, we will see that the exact twoand three-point functions for certain fields in a conformal field theory are already determined by the symmetries. This will allow us to derive a general formula for the OPE among quasi-primary fields. 2.6.1 Conformal Ward Identity In quantum field theory, so-called Ward identities are the quantum manifestation of symmetries. We will now derive such an identity for the conformal symmetry of two-dimensional CFTs on general grounds. For primary fields φi, we calculate �� dz 2πi ɛ(z) T (z) φ1(w1, � w1) ...φN (wN , wN ) (2.46) = = N� � � φ1(w1, w1) ... � i=1 N� � � φ1(w1, w1) ... i=1 C(wi ) dz 2πi ɛ(z) T (z) φi(wi, � � wi) ...φN (wN , wN ) hi �ɛ(wi) + ɛ(wi) �wi � � φi(wi, wi) ...φN (wN , wN ) where we have applied the deformation of the contour integrals illustrated in Fig. 2.5 and used Eq. (2.37). Employing then the two relations shown in Eq. (2.38), we can write � dz 0 = 2πi ɛ(z) �� T (z) φ1(w1, w1) ...φN (wN , wN ) (2.47) − N� � i=1 hi 1 + �wi (z − wi) 2 z − wi �� φ1(w1, w1) ...φN (wN , wN ) � Since this must hold for all ɛ(z) of the form ɛ(z) =−z n+1 with n ∈ Z, the integrand must already vanish and we arrive at the Conformal Ward identity
30 2 Basics in Conformal Field Theory � dz w 1 0 w N w 2 w 3 Fig. 2.5 Deformation of contour integrals � � T (z) φ1(w1, w1) ...φN (wN , wN ) = N� � i=1 = = hi 1 + �wi (z − wi) 2 z − wi 2.6.2 Two- and Three-Point Functions 0 w 1 w N � dz � dz � dz w 2 �� . φ1(w1, w1) ...φN (wN , wN ) w 3 � dz (2.48) In this subsection, we will employ the global conformal SL(2, C)/Z2 symmetry to determine the two- and three-point function for chiral quasi-primary fields. The Two-Point Function We start with the two-point function of two chiral quasi-primary fields � φ1(z) φ2(w) � = g � z,w � . The invariance under translations f (z) = z + a generated by L−1 requires g to be of the form g(z,w) = g(z − w). The invariance under L0, i.e. rescalings of the form f (z) = λz, implies that � φ1(z) φ2(w) � → � λ h1 φ1(λz) λ h2 φ2(λw) � = λ h1+h2 g � λ(z − w) � != g � z − w � , from which we conclude d12 g � z − w � = (z − w) h1+h2 ,
Page 2 and 3: Lecture Notes in Physics Founding E
Page 4 and 5: R. Blumenhagen E. Plauschinn Introd
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 40 and 41: 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
Page 54 and 55: 2.9 Simple Examples of CFTs 45 S =
Page 56 and 57: 2.9 Simple Examples of CFTs 47 Let
Page 58 and 59: 2.9 Simple Examples of CFTs 49 c 1
Page 60 and 61: 2.9 Simple Examples of CFTs 51 The
Page 62 and 63: 2.9 Simple Examples of CFTs � L0
Page 64 and 65: 2.9 Simple Examples of CFTs 55 Howe
Page 66 and 67: 2.9 Simple Examples of CFTs 57 γ 0
Page 68 and 69: 2.9 Simple Examples of CFTs 59 Radi
Page 70 and 71: 2.9 Simple Examples of CFTs 61 whic
Page 72 and 73: 2.9 Simple Examples of CFTs 63 Comp
Page 74 and 75: 2.9 Simple Examples of CFTs 65 �
Page 80 and 81: 2.10 Highest Weight Representations
Page 86 and 87: 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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3.3 Highest Weight Representations
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3.3 Highest Weight Representations
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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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