Lectures on String Theory

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– 115 – α ′ mass 2 rep of SO(8) little (−1) F (−1) ¯F rep of little group group (NS, NS) − sector −2 |0〉 L × |0〉 R SO(9) −1 −1 1 0 b i −1/2|0〉 L × b j −1/2 } {{ } |0〉 R SO(8) +1 +1 1 + 28 + 35 v } {{ } 8 v 8 v (R, R) − sector +1 |a〉 }{{} L × |b〉 }{{} R +1 1 + 28 + 35 s 8 v 8 v −1 |ȧ〉 }{{} L × |ḃ〉 R }{{} −1 1 + 28 + 35 c 8 c 8 c 0 SO(8) |ȧ〉 }{{} L 8 c × |b〉 }{{} R 8 s −1 +1 8 v + 56 v |a〉 }{{} L × |ḃ〉 R }{{} +1 −1 8 v + 56 v 8 s 8 c (R, NS) − sector × +1 +1 8 c + 56 c × −1 +1 8 s + 56 s +1 +1 |a〉 }{{} L b i −1/2|0〉 } {{ R } 8 s 8 v 0 SO(8) |ȧ〉 }{{} L b i −1/2|0〉 } {{ R } 8 c 8 v (NS, R) − sector ¯bi −1/2 |a〉 L } {{ } × |a〉 }{{} R +1 8 c + 56 c 8 v 8 s 0 SO(8) ¯bi −1/2 |a〉 L } {{ } 8 v × |ȧ〉 }{{} R 8 c −1 8 s + 56 s Tab. 6. The lowest levels of the closed fermionic string spectrum.
– 116 – we get Since a general state in the NS sector has the form |Φ〉 = α i 1 −n1 · · · α i N −nN b j 1 −r1 · · · b j M −rM |0〉 G|Φ〉 = (−1) M−1 |Φ〉. Thus, all states with M even are projected out, in particular, the tachyon. This removes the tachyon and all the states with half-integer α ′ M 2 . Indeed, α ′ M 2 = r M ∑ i=1 r i − 1 2 }{{} a and for M even the sum of half-integers is always an integer α ′ M 2 is half-integer number. In the Ramond sector the operator G is defined as follows G = (−1) F = b 1 0 · · · b 8 0(−1) P ∞ n=1 bi −n bi n . The transversal zero modes b i 0 form the Clifford algebra {b i 0, b j 0} = δ ij . Thus, these operators can be represented by SO(8) γ-matrices Γ i which have size 16 by 16. These matrices act on the 16-dimensional Majorana (i.e. real) spinor whose components can be thought to combine two Weyl projections, which are precisely |a〉 and |ȧ〉: ( ) |a〉8s |ψ〉 = . |ȧ〉 8c In the Majorana representation these matrices Γ i can be taken in the block-diagonal form as ( ) 0 γ Γ i i = (γ i ) t . 0 The fact that Γ i obey the standard Glifford algebra {Γ i , Γ j } = 2δ ij implies that 8×8 real matrices γ i satisfy the following algebra γ i (γ j ) t + γ j (γ i ) t = 2δ ij . If we introduce the standard Pauli matrices σ 1 = 0 1 1 0 , σ 2 = 0 −i 1 0 , σ i 0 3 = 0 −1 then the matrices γ i can be defined as γ 1 = −iσ 2 ⊗ σ 2 ⊗ σ 2 , γ 2 = iI ⊗ σ 1 ⊗ σ 2 , γ 3 = iI ⊗ σ 3 ⊗ σ 2 , γ 4 = iσ 1 ⊗ σ 2 ⊗ I , γ 5 = iσ 3 ⊗ σ 2 ⊗ I , γ 6 = iσ 2 ⊗ I ⊗ σ 1 , γ 7 = iσ 2 ⊗ I ⊗ σ 3 , γ 8 = I ⊗ I ⊗ I .
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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
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- 26 - i.e. the total mass momentum
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- 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− ,
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- 40 - One can correctly anticipate
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- 42 - Using α µ q α µ m+n−q
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- 44 - Here N is the number operato
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- 46 - string spectrum and makes it
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- 48 - Thus, for τ > τ ′ one ob
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where the expression on the r.h.s.
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- 52 - Plugging everything together
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- 54 - where we assume that τ 1 >
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- 56 - It also follows from the sca
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- 58 - 4.2 Quantization in the phys
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- 60 - Here by arrow we indicated t
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- 62 - Thus, our commutator takes t
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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 and 92: - 90 - gluing the opposite sides to
Page 93 and 94: - 92 - Any point in the upper-half
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Page 97 and 98: - 96 - The two-dimensional Dirac ma
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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
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Page 113 and 114: - 112 - and similar for ML 2 . For
Page 115: - 114 - real components. Thus, the
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,
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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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Lectures on String Theory

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