A Companion Reader to Polchinski string theory, Study notes of String Theory

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A Companion Reader to Polchinski’s

String Theory

Stany M. Schrans

April 30, 2020

Contents

1.12 p 22: Eq. (1.3.32) Regularising

6.63 p 193: Eq. (6.6.8) The Four-Tachyon Closed String Amplitude and Factori- sation....................................... 316 6.64 p 193: Eq. (6.6.10) The Virasoro-Shapiro Amplitude............. 317 6.65 p 193: Eq. (6.6.12) The Regge Limit of the Virasoro-Shapiro Amplitude... 319 6.66 p 194: Eq. (6.6.13) The Hard Scattering Limit of the Virasoro-Shapiro Am- plitude....................................... 320 6.67 p 194: Eq. (6.6.14) The Amplitude for a Massless Closed String and Two Closed String Tachyons.............................. 321 6.68 p 194: Eq. (6.6.15) The Relation between the Coupling Constant of Tachy- onic and Massless Closed Strings........................ 323 6.69 p 194: Eq. (6.6.19) The Amplitude for Three Massless Closed Strings.... 325 6.70 p 195: Eq. (6.6.21) The Relation between I(x, y, z) and I(x, y)....... 326 6.71 p 195: Eq. (6.6.23) The Relation between Closed and Open String Four- Point Amplitudes................................. 327 6.72 p 195: Eq. (6.6.24-25) The OPE of Two Tachyon Vertex Operators and its Poles........................................ 327 6.73 p 198: Eq. (6.7.3) The One-Point Function from the Möbius Group..... 328 6.74 p 198: Eq. (6.7.4) The Two-Point Function from the Möbius Group..... 329 6.75 p 199: Eq. (6.7.5) The Two-Point Function Of Tensor Fields......... 330 6.76 p 199: Eq. (6.7.6) The Three-Point Function Of Tensor Fields........ 331 6.77 p 199: Eq. (6.7.7) The Four-Point Function Of Tensor Fields......... 333 6.78 p 200: Eq. (6.7.9) The Operator-State Mapping for the Two-Point Function. 337 6.79 p 200: Eq. (6.7.11) The Two-Point Function of Primary Fields with Zamolod- chikov’s Inner Product.............................. 338 6.80 p 200: Eq. (6.7.14) The Three-Point Function of Primary Fields as a Func- tion of the OPE Coefficients, I.......................... 339 6.81 p 200: Eq. (6.7.15) The Three-Point Function of Primary Fields as a Func- tion of the OPE Coefficients, II.......................... 339 6.82 p 201: Eq. (6.7.18) The Four-Point Function of Primary Fields as a Function of the OPE Coefficients.............................. 340 6.83 p 201: Eq. (6.7.19-22) The Four-Point Function from the Hilbert Space Expression, I................................... 341 6.84 p 202: Eq. (6.7.19-23) The Four-Point Function from the Campbell-Baker- Hausdorff Formula................................ 341 6.85 p 202: Eq. (6.7.24) The Four-Point Function from the Hilbert Space Expres- sion, II....................................... 343

• 1 A First Look at Strings
  • 1.1 p 12: Eq. (1.2.15) The Variation of the Determinant of the Metric
  • 1.2 p 15: Eq. (1.2.32) The Change in the Curvature under a Weyl Rescaling
  • 1.3 p 15: Below Eq. (1.2.32) Invariance of χ under Weyl Rescaling
  • 1.4 p 16: The Variation of the Einstein-Hilbert Action
  • 1.5 p 16: Two-Dimensional Gravity has no Dynamics
  • 1.6 p 17: Below Eq. (1.3.7) Determining p+
  • 1.7 p 18: Eq. (1.3.9) Invariance of f dσ
  • 1.8 p 18: Below Eq. (1.3.9) Fixing the Gauge
  • 1.9 p 18: Eq. (1.3.10) Invariance of f dσ
  • 1.10 p 18: Eq. (1.3.11) The Lagrangian in the Light-Cone Gauge
  • 1.11 p 19: Eq. (1.3.13) The Open String Boundary Conditions - n n ∑
  • 1.13 p 24: Eq. (1.3.43) The Regge Slope for Open Strings
  • 1.14 p 29: Eq. (1.4.19) The Unoriented Strings
• 2 Conformal Field Theory
  • 2.1 p 33: Eq. (2.1.2) The Complex Coordinates
  • 2.2 p 33: Eq. (2.1.3) The Complex Derivatives
  • 2.3 p 33: Eq. (2.1.6) The Complex Metric
  • 2.4 p 33: Eq. (2.1.7) The Jacobian
  • 2.5 p 36: Eq. (2.1.23) The Equation of Motion as Operator Equation
  • 2.6 p 36: Eq. (2.1.24) ∂ ∂¯ ln |z|^2 = 2πδ^2 (z, ¯z)
  • 2.7 p 38: Eq. (2.2.4) A Taylor Expansion
  • 2.8 p 39: Eq. (2.2.5) and (2.2.8) Subtractions and Contractions
    • Motion 2.9 p 39: Below Eq. (2.2.6) Normal Ordered Products satisfy the Equation of
  • 2.10 p 39: Eq. (2.2.10) The Product of Normal Ordered Operators
  • 2.11 p 40: Eq. (2.2.11) Calculating an OPE
  • 2.12 p 41: Eq. (2.3.5) The Ward Identity
  • Transformation Law 2.13 p 42: Eq. (2.3.11) The OPE with the Conserved Current determines the
  • Translation 2.14 p 43: Eq. (2.3.14) Transformation of a Vertex Operator under a Space-Time
• 2.15 p 43: Eq. (2.3.15) The Energy-Momentum Tensor
• 2.16 p 43: Eq. (2.4.1) The Energy-Momentum Tensor is Traceless
  • and an Anti-holomorphic Part 2.17 p 43: Eq. (2.4.2) The Energy-Momentum Tensor Splits into a Holomorphic
• 2.18 p 44: Eq. (2.4.6) The OPE with the Energy-Momentum Tensor
• 2.19 p 44: Eq. (2.4.7) The Transformation of the Field Xμ
• 2.20 p 45: Fig 2.2. Examples of Conformal Transformations
• 2.21 p 46: Eq. (2.4.12) Conformal Transformation of an Operator A, I
• 2.22 p 46: Eq. (2.4.14) Conformal Transformation of an Operator A, II
• 2.23 p 46: Eq. (2.4.16) Conformal Transformation of a Primary Field
• 2.24 p 46: Eq. (2.4.17) Conformal Transformation of Typical Operators
  • Tensor 2.25 p 48: Eq. (2.4.23) Conformal Transformation of the Energy-Momentum
• 2.26 p 48: Eq. (2.4.27) The Schwarzian Derivative
• 2.27 p 49: Eq. (2.5.2) The Linear Dilaton Central Charge
• 2.28 p 49: Eq. (2.5.3) The Linear Dilaton Transformation
• 2.29 p 50: Eq. (2.5.4) The bc Action is Conformally Invariant
• 2.30 p 50: Eq. (2.5.11) The Ghost Energy-Momentum Tensor
• 2.31 p 51: Eq. (2.5.12) The Ghost Central Charge
• 2.32 p 51: Eq. (2.5.14) The Ghost Charge Current
• 2.33 p 51: Eq. (2.5.15) The Conformal Transformation of the Ghost Charge, I
• 2.34 p 51: Eq. (2.5.16) The Conformal Transformation of the Ghost Charge, II
• 2.35 p 51: Eq. (2.5.17) The Conformal Transformation of the Ghost Charge, III
• 2.36 p 52: Eq. (2.5.24) The Central Charge of the βγ System
• 2.37 p 53: Eq. (2.6.4) The Complex Coordinates
• 2.38 p 53: Eq. (2.6.7) The Fourier Expansion
• 2.39 p 54: Eq. (2.6.8) The Relation Between Lm and Tm
• 2.40 p 54: Eq. (2.6.9) The Relation Between Tzz and Tww
• 2.41 p 54: Eq. (2.6.10) The Hamiltonian
• 2.42 p 55: Fig 2.3 The Contracted Contour Integration
• 2.43 p 55: Eq. (2.6.14) Switching Between OPEs and Commutation Relations
• 2.44 p 56: Eq. (2.6.19) The Virasoro Algebra
• 2.45 p 56: Eq. (2.6.24) The Transformation of Primary Fields
• 2.46 p 56: Eq. (2.6.25) The Open String Boundary
• 2.47 p 58: Eq. (2.7.2) The Single Valuedness of Xμ
• 2.48 p 58: Eq. (2.7.3) The Space-Time Momentum
• 2.49 p 58: Eq. (2.7.4) Integrating ∂Xμ
  • 2.50 p 59: Eq. (2.7.7) Normal Ordering for L
  • 2.51 p 59: Eq. (2.7.9) aX =
  • 2.52 p 60: Eq. (2.7.11) The Creation-Annihilation Normal Ordering
  • 2.53 p 60: aX from the Normal Ordering
  • 2.54 p 61: Eq. (2.7.15) The Virasoro Generators for the Linear Dilaton CFT
  • 2.55 p 61: Eq. (2.7.17) The bc Ghost Commutators
  • 2.56 p 61: Eq. (2.7.18) The bc Vacuum States
  • 2.57 p 61: Eq. (2.7.19) The bc Virasoro Generators
  • 2.58 p 61: Eq. (2.7.21) The bc Normal Ordering Constant ag
  • 2.59 p 62: ag from Normal Ordering
  • 2.60 p 62: Eq. (2.7.22) The Ghost Number Operator
  • 2.61 p 62: Eq. (2.7.23) The Ghost Number of the Ghost Fields
  • 2.62 p 62: Eq. (2.7.24) The Ghost Number of the Vacuum
  • 2.63 p 63: Eq. (2.8.1) From the Semi-Infinite Cylinder to the Unit Disk
  • 2.64 p 63: The State-Operator Isomorphism in 2d-CFTs
  • 2.65 p 63: Eq (2.8.2) The Unit Operator and the Ground State
  • 2.66 p 64: Eq (2.8.4) The Isomorphism for General States
    • ing at the Origin, I 2.67 p 64: Eq (2.8.6) The Isomorphism for General States with an Operator Act-
    • ing at the Origin, II 2.68 p 65: Eq (2.8.7) The Isomorphism for General States with an Operator Act-
  • 2.69 p 65: Eq (2.8.10) The Ghost Operators Acting on the Ground State
  • 2.70 p 65: Eq (2.8.11) The Ground State and the Ghost Ground State
  • 2.71 p 65: The Ghost Number of the Ground State
  • 2.72 p 65: Eq (2.8.16) The Complex Coordinates for the Open String
  • 2.73 p 66: Eq (2.8.17) The State-Operator Mapping: from Operator to State
  • 2.74 p 67: Eq (2.8.18) The State-Operator Mapping: from State to Operator
    • State 2.75 p 67-68: The State-Operator Mapping for the Scalar Field Xμ: The Ground
    • State for the Operator ∂kXμ 2.76 p 68: Eq (2.8.28) The State-Operator Mapping for the Scalar Field Xμ: The
  • 2.77 p 70: Eq (2.9.3) The OPE of Three Operators
  • 2.78 p 72: Eq (2.9.14) Non-Highest Weight States in Unitary CFTs
  • 2.79 p 72: Eq (2.9.15) hO = 0 Operators
  • 2.80 p 73: The Normal Ordering Constants from the State-Operator Mapping
• 3 The Polyakov Path Integral
  • 3.1 p 79: Fig 3.4 Open String Processes
  • 3.2 p 82: Eq (3.2.3b) The Weyl Invariance of the Euler Number
  • 3.3 p 83: Eq (3.2.7) String Coupling Constants
  • Tensor in 2D 3.4 p 85: Eq (3.3.6) The Relations Between the Ricci Scalar and the Riemann
• 3.5 p 85: Eq (3.3.8) The Residual Conformal Symmetry after Gauge Fixing
• 3.6 p 87: Footnote 2 The Gauge Invariance of the Delta Function
• 3.7 p 88: Eq (3.3.16) The Infinitesimal Transformation of the Metric
• 3.8 p 88: Eq (3.3.18) The Faddeev-Popov Determinant
• 3.9 p 89: Eq (3.3.21) The Faddeev-Popov Action
• 3.10 p 89: Eq (3.3.24) The Faddeev-Popov Action in the Conformal Gauge
• 3.11 p 90-91: The Anomaly of a Global Scale Symmetry
• 3.12 p 92: Eq (3.4.6) Weyl Invariance of an Expectation Value
• 3.13 p 92: Eq (3.4.8) The General Form of the Weyl Anomaly
  • dinates, I 3.14 p 92: Eq (3.4.9) The General Form of the Weyl Anomaly in Complex Coor-
  • dinates, II 3.15 p 92: Eq (3.4.10) The General Form of the Weyl Anomaly in Complex Coor-
  • dinates, III 3.16 p 93: Eq (3.4.11) The General Form of the Weyl Anomaly in Complex Coor-
  • dinates, I 3.17 p 93: Eq (3.4.12) The Actual Form of the Weyl Anomaly in Complex Coor-
  • dinates, II 3.18 p 93: Eq (3.4.15) The Actual Form of the Weyl Anomaly in Complex Coor-
• 3.19 p 93: Eq (3.4.16a) The Ricci Scalar in the Conformal Gauge
• 3.20 p 93: Eq (3.4.16b) The Laplacian in the Conformal Gauge
• 3.21 p 93: Eq (3.4.17) The Weyl Variation of Z[g]
• 3.22 p 93: Eq (3.4.18) Z[g] in the Conformal Gauge
• 3.23 p 94: Eq (3.4.19) Z[g] for an Arbitrary Metric
• 3.24 p 94: Eq (3.4.21) The Second Way to Calculate the Variation of Z[g], I
• 3.25 p 94: Eq (3.4.22) The Second Way to Calculate the Variation of Z[g], II
• 3.26 p 95: Theories with a Quantum Anomaly
• 3.27 p 95: Eq (3.4.26) The Energy-Momentum Tensor of the Cosmological Term
• 3.28 p 96: Eq (3.4.27) The Most General Form δW ln Z[g] with Boundary Terms
• 3.29 p 96: Eq (3.4.29) The Weyl Transformation of the Counterterms
• 3.30 p 96: Eq (3.4.30) The Wess-Zumino Consistency Condition
• 3.31 p 97: Eq (3.4.31) The Central Charge is Constant
• 3.32 p 98: Fig 3.8 Scattering of Closed Strings
• 3.33 p 100: Compact Connected Topologies
• 3.34 p 102: Eq (3.6.3) The Normalisation of the First Excited States
• 3.35 p 102: Eq (3.6.4) The On-Shell Condition for the First Excited States
• 3.36 p 103: Eq (3.6.7) The Weyl Transformation of a Renormalised Operator
  • Polyakov String 3.37 p 103: Eq (3.6.8) The Weyl Transformation for the Tachyon Vertex for the
  • 3.38 p 103: Eq (3.6.11) The Weyl Transformation of the Geodesic Distance, I
  • 3.39 p 105: Eq (3.6.15) The Weyl Transformation of the Geodesic Distance, II
    • tor for the Polyakov String 3.40 p 105: Eq (3.6.16) The Weyl Transformation for the Massless Vertex Opera-
  • 3.41 p 105: Eq (3.6.18) Linking ∇^2 Xμ with kμR
    • erator, I 3.42 p 106: Eq (3.6.20) The Independent Parameters of the Massless Vertex Op-
    • erator, II 3.43 p 106: Eq (3.6.21) The Independent Parameters of the Massless Vertex Op-
    • erator, III 3.44 p 106: Eq (3.6.22) The Independent Parameters of the Massless Vertex Op-
  • 3.45 p 108: Eq (3.7.5) The Graviton from the Background Field
    • Tensor 3.46 p 109: Eq (3.7.7) The Spacetime Gauge Invariance of the Antisymmetric
  • 3.47 p 109: Eq (3.7.7) The Spacetime Gauge Invariance of the Three-Tensor Hωμν
    • Transformation 3.48 p 110: The Most General Classical Action Invariant under a Rigid Weyl
  • 3.49 p 110: Eq (3.7.11) The Linear Approximation of the Non-linear Sigma Model
  • 3.50 p 111: Eq (3.7.13) The β Functions to First Order
  • 3.51 p 111: Eq (3.7.14) The β Functions with two Spacetime Derivatives
  • 3.52 p 113: Eq (3.7.19) The β Function for the Linear Dilaton Model
  • 3.53 p 114: Eq (3.7.20) The Effective Spacetime Action
  • 3.54 p 114: Eq (3.7.23) The Ricci Scalar after a Weyl Transformation
  • 3.55 p 114: Eq (3.7.25) The Space Time Action with Einstein Metric
    • Beltrami Equations and all that Stuff 3.56 Appendix: Almost Complex Structures, Holomorphic Normal Coordinates,
• 4 The String Spectrum
  • 4.1 p 122: Eq (4.1.8) Spurious States, I
  • 4.2 p 122: Eq (4.1.9) Spurious States, II
  • 4.3 p 123: Eq (4.1.11) The Physical Hilbert Space, I: the Tachyon State
  • 4.4 p 123: Eq (4.1.16) The L 0 Condition for the Level One State
  • 4.5 p 123: Eq (4.1.17) The Lm≥ 1 Condition for the Level One State
  • 4.6 p 124: Eq (4.1.18) The Spurious Level One State
  • 4.7 p 124: Eq (4.1.18) The Level One States for Different Values of A
  • 4.8 p 124: Eq (4.1.18) The Level Two States
  • 4.9 p 126: Eq (4.2.6) The BRST Invariance of the Quantum Action
  • 4.10 p 127: Ghost Number Conservation
  • 4.11 p 127: Eq (4.2.7) δB(bAF A) = i(S 2 + S 3 )
  • 4.12 p 127: Eq (4.2.8) A Change in the Gauge-Fixing Condition
  • 4.13 p 128: Eq (4.2.13) The BRST Charge is Nilpotent
  • the Point Particle 4.14 p 129: Eq (4.2.20) The Structure Constants for the BRST Transformation of
• 4.15 p 129: Eq (4.2.22) The BRST Transformation for the Point Particle
• 4.16 p 129: Eq (4.2.23) The BRST Action for the Point Particle
  • the Point Particle 4.17 p 130: Eq (4.2.25) The BRST Transformation of the Gauge Fixed Action for
• 4.18 p 130: Eq (4.2.26) The Canonical Commutation Relations for the Point Particle
• 4.19 p 131: Eq (4.3.1) The BRST Transformation for the Bosonic String
• 4.20 p 131: Nilpotency of the BRST Transformation for the Bosonic String
• 4.21 p 131: Eq (4.3.3) The BRST Current for the Bosonic String
• 4.22 p 132: Eq (4.3.4) OPEs with the BRST Current
• 4.23 p 132: Eq (4.3.6) The Anticommutator {QB, bm}
• 4.24 p 132: Eq (4.3.7) The Mode Expansion of the BRST Operator
• 4.25 p 132: Eq (4.3.7) The BRST Normal Ordering Constant
  • Charge 4.26 p 132: Eq (4.3.10) The jB(z)jB(w) OPE and the Nilpotency of the BRST
• 4.27 p 133: Eq (4.3.11) The BRST Current as a Primary Field
• 4.28 p 133: Eq (4.3.15) The Algebra Satisfied by the Constraints
• 4.29 p 133: Eq (4.3.16) The Nilpotency of the General BRST Charge
• 4.30 p 134: Eq (4.3.17) The Hermitian Conjugate of the Ghost Modes
  • States 4.31 p 134: Eq (4.3.18) The Ghost Insertions for the Inner Product of the Ground
• 4.32 p 134: QB takes Hˆ into itself
• 4.33 p 134: The Need for a New Inner Product on Hˆ
• 4.34 p 135: Eq (4.3.23) The Level Zero Mass Shell Condition
• 4.35 p 135: Eq (4.3.24) The Level Zero Physical State Condition
• 4.36 p 135: Eq (4.3.25) The Level One Mass Shell Condition
• 4.37 p 135: Eq (4.3.26) The Level One Negative Norm States
• 4.38 p 135: Eq (4.3.27) The Level One Physical State Condition
• 4.39 p 139: Eq (4.4.7) The Commutation Relations of the Light-Cone Oscillators
• 4.40 p 139: Eq (4.4.10) The Splitting of the BRST Operator
• 4.41 p 139: Eq (4.4.11) The Ghost Number of the BRST Operator
• 4.42 p 139: Eq (4.4.13) The Simplified BRST Operator Q
• 4.43 p 140: Eq (4.4.13) The Operator S
• 4.44 p 140: The Cohomology of Q
• 4.45 p 140: The Cohomology of QB
• 4.46 p 141: Eq (4.4.23) The BRST Operator Acting a a Hilbert Space State
• 5 The String S -Matrix
  • 5.1 p 147: Eq (5.1.9-11) The Torus as a Parallelogram
  • 5.2 p 148: Eq (5.1.12) The Transformations S and T
  • 5.3 p 148: Eq (5.1.13) P SL(2, Z ) Group
  • 5.4 p 148: Eq (5.1.14) P SL(2, Z ) Transforming the Metric
  • 5.5 p 148: Eq (5.1.15) The Fundamental Region of P SL(2, Z )
  • 5.6 p 151: Eq (5.2.4) The Diff×Weyl Transformation of the Metric, I
  • 5.7 p 151: Eq (5.2.5) The Diff×Weyl Transformation of the Metric, II
  • 5.8 p 151: Eq (5.2.7) The Conformal Killing Equation
    • mal Gauge 5.9 p 152: Eq (5.2.8) The Moduli and Conformal Killing Vectors in the Confor-
    • positive Euler number 5.10 p 152: Eq (5.2.10) No CKVs for negative Euler number and no moduli for
  • 5.11 p 155: Eq (5.3.2) The Gauge-Fixed Measure
  • 5.12 p 155: Eq (5.3.5) The Variation of the Metric Including the Moduli
  • 5.13 p 156: Eq (5.3.6) Inverse Faddeev-Popov Determinant
  • 5.14 p 156: Eq (5.3.8) The Faddeev-Popov Ghosts
  • 5.15 p 156: Eq (5.3.9) The S-Matrix for the Bosonic String
  • 5.16 p 157: Eq (5.3.14) P 1 CJ is an Eigenfunction of P 1 P 1 T
  • 5.17 p 158: Eq (5.3.15) The Relation Between The B and C Eigenfunctions
    • Ghost Eigenfunctions, I 5.18 p 158: Eq (5.3.16) The Faddeev-Popov Determinant as a Function of the
    • Ghost Eigenfunctions, II 5.19 p 158: Eq (5.3.17) The Faddeev-Popov Determinant as a Function of the
    • Ghost Eigenfunctions, II| 5.20 p 158: Eq (5.3.18) The Faddeev-Popov Determinant as a Function of the
  • 5.21 p 158: Eq (5.3.19) The Weyl Anomaly of the Ghost Current
  • 5.22 p 159: Eq (5.3.20) The Riemann-Roch Theorem, I
  • 5.23 p 159: Eq (5.3.21) The Riemann-Roch Theorem, II
  • 5.24 p 160: Eq (5.4.3) Weyl Invariance of the b Insertions
  • 5.25 p 160: Eq (5.4.4) The Diffeomorphism Invariance of the S-matrix
  • 5.26 p 161: Eq (5.4.5) The BRST Variation of a Vertex Operator
  • 5.27 p 161: Eq (5.4.6) The BRST variation of the b-Ghost Insertion
    • ential 5.28 p 162: Eq (5.4.8) The b-Ghost Insertion as a Function of the Beltrami Differ-
  • 5.29 p 162: Eq (5.4.10) The Metric under a Change of Moduli
  • 5.30 p 162: Eq (5.4.11) The Infinitesimal Version of the Beltrami Equations
  • 5.31 p 162: Eq (5.4.12) The b-Insertion in terms of the Transition Functions, I
    • Moduli 5.32 p 162: Eq (5.4.14) The Change in Transition Functions under a Change of
  • 5.33 p 163: Eq (5.4.15) The b-Insertion in terms of the Transition Functions, II
  • 5.34 p 164: Eq (5.4.18) Simplifying the b-Ghost Insertions
  • 5.35 Appendix: The Schwinger Dyson Equations in a QFT
• 6 Tree-Level Amplitudes
  • 6.1 p 166: The Two-sphere S
  • 6.2 p 167: Eqs. (6.4.5a,b)The CKVs on S
  • 6.3 p 168: The Two-Disk D
  • 6.4 p 168: The Two-dimensional Projective Plane RP
    • Fields 6.5 p 169: Eq. (6.2.3) The Functional Integral in Terms of a Complete Set of
  • 6.6 p 169: Eq. (6.2.5) The Zero Mode Normalisation
  • 6.7 p 170: Eq. (6.2.6) The Functional Integral as a Determinant
  • 6.8 p 170: Eq. (6.2.8) Green’s Function PDE
  • 6.9 p 170: Eq. (6.2.9) Green’s Function on the S
    • the Renormalised Green’s Function 6.10 p 171: Eq. (6.2.13) From the Zero Mode to Momentum Conservation and
  • 6.11 p 171: Eq. (6.2.16) The Renormalised Green’s Function
  • 6.12 p 171: Eq. (6.2.17) The Tachyon amplitude on S 2 : Final Result
    • Operators 6.13 p 172: Eq. (6.2.18) Amplitudes for Higher Order Vertex
    • tion Values 6.14 p 172: Eqs. (6.2.21-6.2.23) How Holomorphicity can Determine Expecta-
  • 6.15 p 173: Eq. (6.2.25) The Expectation with one Level One Vertex Operator
  • 6.16 p 173: Eq. (6.2.26) Momentum Conservation in the Expectation Value
  • 6.17 p 174: Eq. (6.2.28) Expanding around z → z
    • the Holomorphicity Condition 6.18 p 174: Eq. (6.2.31) The Expectation Value of Vertex Operators on S 2 from
  • 6.19 p 174: Eq. (6.2.32) The Green’s Function on the Two-Disk D
  • 6.20 p 175: Eq. (6.2.33) The Tachyon Vertex Amplitude in the Two-Disk D
  • 6.21 p 175: Eq. (6.2.34) Boundary Normal Ordering
  • 6.22 p 176: Eq. (6.2.38) The Green’s Function on the Projective Plane RP
  • 6.23 p 176: Eq. (6.3.1) The Simplest Ghost Non-Vanishing Expectation Value
  • 6.24 p 177: Eq. (6.3.5) The Multi-Ghost Field Amplitude
    • Ghost Insertions 6.25 p 177: Eq. (6.3.6) The Holomorphic Derivation for the Need for Three c-
    • Amplitude 6.26 p 177: Eq. (6.3.8) The Alternative Expression for the Multi-Ghost Field
  • 6.27 p 179: Eq. (6.4.1) The Three Tachyon Open String Amplitude, I
  • 6.28 p 179: Eq. (6.4.2) The Three Tachyon Open String Amplitude, II
  • 6.29 p 179: Eq. (6.4.3-6.4.4) The Three Tachyon Open String Amplitude, III
  • 6.30 p 179: Eq. (6.4.5) The Four Tachyon Open String Amplitude, I
• 6.31 p 180: Eq. (6.4.7) The Mandelstam Variables
• 6.32 p 180: Eq. (6.4.8) The Four Tachyon Open String Amplitude, II
  • stam Variables 6.33 p 180: Eq. (6.4.9) The Four Tachyon Open String Amplitude with Mandel-
  • Tachyon State 6.34 p 181: Eq. (6.4.11) The Divergence of the Amplitude at the Intermediate
  • sation 6.35 p 182: Eq. (6.4.14) The Four-Tachyon Open String Amplitude and Factori-
• 6.36 p 182: Eq. (6.4.17) The Pole of I(s, t) at α′s =
• 6.37 p 183: Eq. (6.4.17) The Pole of the Amplitude α′s = 0 is Actually not There
• 6.38 p 183: Eq. (6.4.22) Relating the Beta and Gamma Functions
• 6.39 p 183: Eq. (6.4.23) The Veneziano Amplitude
• 6.40 p 184: Eq. (6.4.27) The Center of Mass Frame Kinematics
• 6.41 p 183: Eq. (6.4.28) The Regge Behaviour of the Veneziano Amplitude
  • tude, I 6.42 p 183: Eq. (6.4.29) The Hard Scattering Behaviour of the Veneziano Ampli-
  • tude, II 6.43 p 183: Eq. (6.4.30) The Hard Scattering Behaviour of the Veneziano Ampli-
• 6.44 p 185: The Hermiticity of the Chan-Paton Factors
• 6.45 p 185: Eq. (6.5.4) The Trace of Chan-Paton Factors
• 6.46 p 186: Eq. (6.5.6) The Four Tachyon Amplitude with Chan-Paton Factors
• 6.47 p 186: Eq. (6.5.7-8) The Four Tachyon Amplitude and Unitarity
• 6.48 p 187: Eq. (6.5.9) Traces and the Completeness Relation
• 6.49 p 187: Eq. (6.5.10) One Gauge Boson and two Tachyons, I
• 6.50 p 187: Eq. (6.5.11) One Gauge Boson and two Tachyons, II
• 6.51 p 187: Eq. (6.5.12) One Gauge Boson and two Tachyons: Final Result
• 6.52 p 188: Eq. (6.5.15) The Three Gauge Boson Amplitude
• 6.53 p 188: Eq. (6.5.16) The Yang-Mills Effective Field Theory
  • time Symmetry 6.54 p 189: Eq. (6.5.18) From a Global Worldsheet Symmetry to a Local Space-
• 6.55 p 190: Eq. (6.5.21) Worldsheet Parity for the Open String
• 6.56 p 190: Eq. (6.5.23) Unoriented Open Strings with Chan-Paton factors
  • String, I 6.57 p 191: Eq. (6.5.26) The Orientation Reversing Symmetries of the Oriented
  • String, II 6.58 p 191: Eq. (6.5.27) The Orientation Reversing Symmetries of the Oriented
  • String, III 6.59 p 191: Eq. (6.5.31), The Orientation Reversing Symmetries of the Oriented
• 6.60 p 192: Eq. (6.6.2) The Three Tachyon Tree Amplitude for Closed Strings
• 6.61 p 193: Eq. (6.6.4) The Four Tachyon Tree Amplitude for Closed Strings
• 6.62 p 193: Eq. (6.6.7) The Pole at α′s = −
• 1.1 Mathematica code for 2d-gravity
• 2.1 Conformal transformation examples, I
• 2.2 Conformal transformation examples, II
• 2.3 Deforming Contours
• 2.4 From the semi-infinite cylinder to the unit disk
• 2.5 From the semi-infinite strip to the upper-half unit disk
• 2.6 From Operator to State
• 2.7 From State to Operator
• 2.8 Radius of Convergence for Three Operators
• 2.9 Conformal Bootstrap
• 3.1 Open string processes
• 3.2 String coupling constants
• 3.3 Mathematica code for the relationship between R and Rabcd in 2D
• 3.4 Mathematica code and result for R with a linearised metric
• 3.5 From the semi-infinite cylinder to the unit disk
• 3.6 Closed string scattering amplitude
• 3.7 2D compact connected surfaces
• 3.8 Möbius strip
• 3.9 Klein bottle
• 4.1 The degenerate ghost vacuum
• 5.1 Modular transformations of the torus
• 5.2 The fundamental region of the modular group
• 5.3 Divergence theorem on the toruss
• 5.4 Contour integration encircling two patches
• 6.1 Stereographic projection for S
• 6.2 The two-disk D 2 from the two-sphere S 2 , I
• 6.3 The two-disk D 2 from the two-sphere S 2 , II
• 6.4 The two-disk D 2 as the upper half complex plane H
• 6.5 The projective plane RP 2 from the two-sphere S
• 6.6 Mathematica code for multi-ghost expectation value
  • plane 6.7 Mapping the three open string tachyon amplitude to the upper half complex
• 6.8 Kinematics for the Mandelstam Variables
• 6.9 Center of mass frame kinematics
• 6.10 Open string Chan Paton factors
• 6.11 Mathematica code for the three-point function
• 6.12 Mathematica code for the four-point function

List of Tables

3.1 2D compact connected surfaces......................... 119 3.2 Weyl transformation of the massless vertex operator, I............ 134 3.3 Weyl transformation of the massless vertex operator, II............ 154

Chapter 1

A First Look at Strings

1.1 p 12: Eq. (1.2.15) The Variation of the Determinant of the Metric

Use

ln det M = tr ln M [1.1]

to write

γ−^1 δγ = δ ln γ = δtr ln γ = tr δ ln γ = tr γ−^1 δγ = γabδγba [1.2]

We have used the fact that

γ−^1

ab =^ γ

ab. So δγ = γγabδγab. The second equation is

obtained by using γabγbc = δac from which it follows that δγabγbc + γabδγbc = 0.

1.2 p 15: Eq. (1.2.32) The Change in the Curvature under a Weyl Rescaling

This is a formula that will come back several times, and it is quite rare to see it worked out in detail, so it is useful to do this here. We wish to show that under a local Weyl rescaling gab → g ab′ = e^2 ω(σ)gab the Ricci scalar satisfies

(g′)^1 /^2 R′^ = g^1 /^2 (R − 2 ∇^2 ω) [1.3]

We have gone to Euclidean space and called the worldsheet metric g in stead of γ, just to save us some typing. One way to show this to write the Ricci scalar out in terms of the Riemann curvature, write that one out in terms of the connections and those in terms of the metric. We then transform the metric, make sure we don’t get dizzy from all the terms, indices, and different contractions and hope this all works out. The other way is to be smart about it and ignore all terms we don’t need, focussing on only what we do need.

Let us first recall some basic facts. The Ricci scalar is given by

R = gabRab = gabRcacb = gab^

∂cΓcba − ∂bΓcca + ΓccdΓdba − ΓcbdΓdca

[1.4]

We have used the definition of the Riemann curvature

Rabcd = ∂cΓadb − ∂dΓacb + ΓaceΓedb − ΓadeΓecb [1.5]

The connection is given by

Γabc =

gad^ (∂bgcd + ∂cgbd − ∂dgbc) [1.6]

When we replace the metric by g′ ab = e^2 ω(σ)gab we have g′ab^ = e−^2 ω(σ)gab^ and the connec- tion becomes

Γ′bca =

g′ad^

∂bg′ cd + ∂cg′ bd − ∂dg bc′

e−^2 ωgad^

[

∂b

e^2 ωgcd

• ∂c

e^2 ωgbd

− ∂d

e^2 ωgbc

)]

gad^ (∂bgcd + ∂cgbd − ∂dgbc) + gad^ (gcd∂bω + gbd∂cω − gbc∂dω) = Γabc + ∆abc [1.7]

where

∆abc = gad^ (gcd∂bω + gbd∂cω − gbc∂dω) [1.8]

Let us now think, before we blindly start calculating. The Ricci scalar contains connections and their derivatives and these in turn contain derivatives of the Weyl factor ω. The √g on both sides just makes sure that the e^2 ω^ is overall cancelled. So, R′^ is an expression that will contain terms without ω’s and terms with ∂nω, ∂nω∂mω or ∂n∂mω. The terms without any ω obviously combine to give g^1 /^2 R again, so it is the terms containing ω’s that should carry our attention. Now R is a scalar under diffeomorphism, as all its indices are nicely contracted. There- fore we should be able to write everything in terms of covariant derivatives of ω. A mo- ment’s thought reveals that there are only two possible combinations with at most a second order derivative, viz. ∇ω · ∇ω and ∇^2 ω. We should therefore be able to write

g′^1 /^2 R′^ = g^1 /^2

R + a∇ω · ∇ω + b∇^2 ω

[1.9]

for some a and b that may depend on the metric and its derivatives, but not on ω. Let us now think about how we can fix these coefficients. We will do this for a general dimension D as we will need that formula later as well, and set D = 2 at the end.