Lectures on String Theory

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– 77 – 5. Geometry and topology of string world-sheet 5.1 From Lorentzian to Euclidean world-sheets String world-sheets and the target space both have Lorentzian signature. The connection to the theory of Riemann surfaces can be made by performing the Wick rotation of both world-sheet and the target space metrics. In particular, for the world-sheet time τ this means τ → −iτ. Thus, on the string-world sheet this Euclidean continuation corresponds to σ ± = τ ± σ → −i(τ ± iσ) . (5.1) A word of caution is needed here: One cannot really prove that the theory of the Lorentzian world-sheets is equivalent to the theory of the Euclidean ones. However, treating the world-sheets as Euclidean provides by itself a consistent theory of interacting strings. The Euclidean version can be then used to learn as much as possible about its Lorentzian counterpart. We start with considering a single closed string whose world-sheet has topology of a cylinder. Formula (5.1) suggests to introduce the complex coordinates w = τ + iσ , ¯w = τ − iσ . The Euclidean closed string covers only a finite interval of σ on the complex plane 0 ≤ σ ≤ 2π and, therefore, only a strip of the 2dim plane. w Fig. 5. Mapping the cylinder (τ, σ) to a strip (w, ¯w) on the complex w-plane. One can further map the strip to the whole complex plane by using the conformal map z = e w = e τ+iσ . Lines of constant τ are mapped into circles on the z plane and the operation of time translation τ → τ + a becomes the dilatation z → e a z . A procedure of identifying dilatations with the Hamiltonian and circles about the origin with equal-time surfaces is called sometimes radial quantization. Mapping
¡ ¡ £¢ ¢ – 78 – from the cylinder to the plane cannot change the physical content of the theory if the theory is conformally invariant, which is the case of string theory in critical dimension. We note that the plane is a non-compact manifold. However, one can compactify it by adding a point at infinity. The corresponding compact surface arising in this way is the Riemann sphere. A metric on a plane can be transformed to a metric on a sphere by a suitable choice of the conformal prefactor. For instance one can pick up the metric ds 2 = 4dzd¯z (1 + |z| 2 ) 2 The formula z = cot θ 2 eiφ defines a stereographic projection of the sphere onto the plane and under this projection the metric takes the form ds 2 = dθ 2 + sin 2 θdφ 2 , i.e. it is the standard round metric on a sphere. north pole (outgoing string) ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ §¥§¥§ ¨¥¨¥¨ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ z−plane south pole (incoming string) ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ §¥§¥§ ¨¥¨¥¨ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ §¥§¥§ ¨¥¨¥¨ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ §¥§¥§ ¨¥¨¥¨ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ §¥§¥§ ¨¥¨¥¨ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ £ ¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ ¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤¥¤ Fig. 6. Stereographic projection of a sphere onto z-plane. Asymptotic incoming and outgoing strings are mapped to the south and the north poles of the sphere respectively. Since cot π = 0 and cot(0) = ∞ the incoming and outgoing strings are mapped to 2 the south and the north poles of the sphere respectively. The example above can be generalized to general world-sheets corresponding to interacting strings. The crucial observation is that conformal invariance allows to consider compact world-sheets instead of surfaces with boundaries corresponding incoming and outgoing strings. The string boundaries are mapped to punctures on a compact Riemann surface. Under the Euclidean continuation the basic equation □X = 0 transforms into with a general solution ∂ z ∂¯z X = 0 X(z, ¯z) = X(z) + ¯X(¯z) , i.e. the left and right-moving excitation correspond now to analytic and anti-analytic fields on the complex z-plane.
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Preprint typeset in JHEP style - HY
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- 2 - 6. Classical fermionic supers
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- 4 - bosons and Ramond’s fermion
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- 6 - ? Type I Type IIB E 8 x E 8 h
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- 8 - 2. Relativistic particle Cons
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- 10 - are called the first class c
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- 12 - which is in complete agreeme
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- 14 - The Nambu-Goto action ∫
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- 16 - Exercise Show that the Hessi
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- 18 - 3.2.3 Conformal gauge Consid
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- 20 - which result into We see tha
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- 22 - Analogously, {¯L m , X µ }
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- 24 - 3.3.1 Solutions of the equat
Page 27 and 28: - 26 - i.e. the total mass momentum
Page 29 and 30: - 28 - It follows from the Schwarz
Page 31 and 32: - 30 - 2. Varying w.r.t. γ τσ le
Page 33 and 34: - 32 - The physical Hamiltonian and
Page 35 and 36: - 34 - Orbits of the Virasoro algeb
Page 37 and 38: - 36 - Let us outline the computati
Page 39 and 40: - 38 - Thus, we arrive at {l i− ,
Page 41 and 42: - 40 - One can correctly anticipate
Page 43 and 44: - 42 - Using α µ q α µ m+n−q
Page 45 and 46: - 44 - Here N is the number operato
Page 47 and 48: - 46 - string spectrum and makes it
Page 49 and 50: - 48 - Thus, for τ > τ ′ one ob
Page 51 and 52: where the expression on the r.h.s.
Page 53 and 54: - 52 - Plugging everything together
Page 55 and 56: - 54 - where we assume that τ 1 >
Page 57 and 58: - 56 - It also follows from the sca
Page 59 and 60: - 58 - 4.2 Quantization in the phys
Page 61 and 62: - 60 - Here by arrow we indicated t
Page 63 and 64: - 62 - Thus, our commutator takes t
Page 65 and 66: - 64 - level α ′ mass 2 rep of S
Page 67 and 68: - 66 - representations of the trans
Page 69 and 70: - 68 - Let us illustrate this proce
Page 71 and 72: - 70 - Here c α is called a ghost
Page 73 and 74: - 72 - The ghosts and anti-ghosts a
Page 75 and 76: - 74 - The expression in the bracke
Page 77: - 76 - One can check that these obj
Page 81 and 82: - 80 - coordinate chart ¡ ¢¡¢¡
Page 83 and 84: - 82 - The last formula can be unde
Page 85 and 86: - 84 - i.e the conformal factor φ
Page 87 and 88: - 86 - The last term here reflects
Page 89 and 90: - 88 - i.e. it is not normalizable.
Page 91 and 92: - 90 - gluing the opposite sides to
Page 93 and 94: - 92 - Any point in the upper-half
Page 95 and 96: - 94 - d(d−1) 2 local Lorentz tra
Page 97 and 98: - 96 - The two-dimensional Dirac ma
Page 99 and 100: - 98 - The “flat” and “curved
Page 101 and 102: - 100 - Here D α ɛ = ∂ α ɛ
Page 103 and 104: - 102 - By using reparametrizations
Page 105 and 106: - 104 - Consider first the commutat
Page 107 and 108: - 106 - Now we note that in additio
Page 109 and 110: - 108 - One can also check that the
Page 111 and 112: - 110 - The oscillator ground state
Page 113 and 114: - 112 - and similar for ML 2 . For
Page 115 and 116: - 114 - real components. Thus, the
Page 117 and 118: - 116 - we get Since a general stat
Page 119 and 120: - 118 - that of Type IIB supergravi
Page 121 and 122: - 120 - In this form the Hamiltonia
Page 123 and 124: - 122 - group” was introduced by
Page 125 and 126: - 124 - where the operator Q k (x,
Page 127 and 128: - 126 - where we have substituted
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- 128 - D. Riemann normal coordinat
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- 130 - E. Exercises Exercise 1. Sh
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- 132 - Exercise 10. • Show that
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- 134 - Exercise 16. Prove that tha
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- 136 - where A(m) is a function of
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- 138 - Exercise 38. Compute the th
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- 140 - as Here Exercise 50. Under
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Lectures on String Theory

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