Lectures on String Theory

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– 91 – where a, b, c, d ∈ Z and ad − bc = 1. The action on the modular parameter τ is in the form of the fractional-linear transformation τ → τ ′ = aτ + b cτ + d . One can check that these transformations preserve the area of the parallelogram. Since two SL(2, Z)-matrices ( ) ( ) a b a b + − c d c d act on τ in the same way, the modular group of the torus is PSL(2, Z) = SL(2, Z)/Z 2 . i 8 ¥¡¥¡¥ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ £¡£¡£¡£¡£¡£¡£¡£ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ Fixed points of S and ST ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ £¡£¡£¡£¡£¡£¡£¡£ £¡£¡£¡£¡£¡£¡£¡£ £¡£¡£¡£¡£¡£¡£¡£ £¡£¡£¡£¡£¡£¡£¡£ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¦¡¦¡¦ ¥¡¥¡¥ ¦¡¦¡¦ ¥¡¥¡¥ ¨¡¨ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ §¡§ £¡£¡£¡£¡£¡£¡£¡£ ¡¡¡¡¡¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¦¡¦¡¦ §¡§ ¨¡¨ −1 −1/2 0 1/2 1 Fig. 8. Fundamental domain M g=1 of the Teichimüller space describing the moduli space of conformally non-equivalent tori. The moduli space of conformally non-equivalent tori is then the quotient of the Teichmüller space of the modular group M g=1 = Teichmüller space modular group . One usually uses the following generators of the modular group T : τ → τ + 1 , S : τ → − 1 τ . Any element of SL(2, Z) is a composition of a certain number of S and T generators: SST ST T T ST....SST
– 92 – Any point in the upper-half plane is related by a modular transformation to a point in the so-called fundamental domain F ≡ M g=1 of the modular group. It can be chosen as { M g=1 = − 1 2 ≤ Reτ ≤ 0, |τ|2 ≥ 1 ∪ 0 < Reτ < 1 } 2 , |τ|2 > 1 . The modular group does not act freely on the upper-half plane because some of the modular transformations have fixed points. The point τ = i is the fixed point of S: τ → − 1 2πi and τ = e τ 3 = − 1 + i √ 3 is the fixed point of ST . The existence of the 2 2 fixed points implies that M g=1 is not a smooth manifold, rather it has singularities of the orbifold type. Since it does not matter which fundamental region to choose in order to integrate over in the one-loop path integral the integrand the corresponding string amplitude should be invariant under modular transformations. The requirement of the modular invariance is one of the most important principles of string theory. In particular, it leads to strong restrictions on the possible gauge groups for the heterotic string. 6. Classical fermionic superstring Introduction of the world-sheet fermionic degrees of freedom requires understanding of how spinors on curved manifolds (world-sheets) are defined. The discussion in the next paragraph is general and can be applied to a manifold of an arbitrary dimension d. 6.1 Spinors in General Relativity It is not straightforward to introduce spinors in General Relativity. If we have a tensor field T i 1...i p j 1 ...j q of rank (p, q) on a manifold M then under general coordinate transformations of the coordinates x i on M: x i → x ′i (x j ), this field transforms as follows T ′k 1...k p l 1 ...l q (x ′ ) = ∂x′k 1 ∂x · · · ∂x′k p i 1 ∂x i p ∂x j 1 ∂x ′l 1 · · · ∂xj q ∂x ′l q T i 1...i p j 1 ...j q (x) . Here tensor indices are acted with the matrices ∂x′i which form a group GL(d, R). ∂x j This is a group of all invertible real d × d matrices. This group does not have spinor representations.
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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
Page 23 and 24:
- 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
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- 30 - 2. Varying w.r.t. γ τσ le
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- 32 - The physical Hamiltonian and
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- 34 - Orbits of the Virasoro algeb
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- 36 - Let us outline the computati
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- 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 and 78: - 76 - One can check that these obj
Page 79 and 80: ¡ ¡ £¢ ¢ - 78 - from the cylin
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: - 90 - gluing the opposite sides to
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
Page 129 and 130: - 128 - D. Riemann normal coordinat
Page 131 and 132: - 130 - E. Exercises Exercise 1. Sh
Page 133 and 134: - 132 - Exercise 10. • Show that
Page 135 and 136: - 134 - Exercise 16. Prove that tha
Page 137 and 138: - 136 - where A(m) is a function of
Page 139 and 140: - 138 - Exercise 38. Compute the th
Page 141 and 142: - 140 - as Here Exercise 50. Under
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