Lectures on String Theory

More documents

Recommendations

Info

– 113 – In the closed string case we have in addition the condition of level-matching which requires that for physical states M 2 L = M 2 R . Finally, we give expressions for the light-cone action S l.c = 1 ∫ ((Ẋi d 2 σ ) 2 − (X ′i ) 2 − 2i 8π ¯ψ ) i ρ a ∂ α ψ i and the Hamiltonian (7.1) H = (p i ) 2 + ∑ n>0 (α i −nα i n + ᾱ i −nᾱ i n) + ∑ r>0 r(b i −rb i r + ¯b i −r¯b i r) − 2a . Note that every sector, R and NS, has its own Hamiltonian. Let us now analyze the closed string spectrum. We first discuss the right-moving part which (up to a mass rescaling by 2) is equivalent to the spectrum of open fermionic string. • NS-sector. The ground state is the oscillator vacuum |0〉 with α ′ M 2 = −a. The first excited state is b i −1/2 |0〉 with α′ M 2 = 1 − a. This is a vector of SO(d − 2), 2 where the critical dimension d = 10. Since the little Lorentz group for massless states in d-dimensions is SO(d − 2) this state must be massless which gives the normal ordering constant to be a = 1 . At the next level one has the states 2 α−1|0〉 i and b i −1/2 bj −1/2 |0〉 with α′ M 2 = 1 . The number of these bosonic states 2 is 8 + 28 = 36 = 9×8 , they are comprise an antisymmetric representation of 2 SO(9), the little Lorentz group for massive states. • R-sector. The Ramond ground state is a spinor of SO(9, 1). The dimension of the Dirac spinor in d = 10 is 2 d 2 = 2 5 = 32, i.e. it has 32 complex or 64 real components. In ten dimensions it is possible to impose both Majorana and Weyl conditions 20 which reduce the number of independent components to 64 = 16. On shell the number of components is further reduced by two 2×2 because the Dirac equation Γ µ ∂ µ ψ relates half of the components to the other half (which satisfies the Klein-Gordon equation). The 8 remaining components can be viewed as the components of the Majorana-Weyl spinor of SO(8), the latter being the little Lorentz group for massless states in d = 10. Indeed, the spinor of SO(8) should have (2 8 2 complex components)/(Majorana × Weyl) = 32/4 = 8 20 In general, for the groups SO(p, q) the Majorana and Weyl conditions can be simultaneously imposed if and only if p − q = 0 mod 8. For Minkowski space, p = d − 1, q = −1, this gives d = 2 + 2n and for Euclidean space, p = d, q = 0, this gives d = 2n.
– 114 – real components. Thus, the Ramond ground state is massless. It turns out that the group SO(8) has three inequivalent representations of dimension 8: two of them are spinor representations and another is the vector one. Spinor representations are commonly denoted by 8 s and 8 c and the corresponding representation bases are depicted as |a〉 and |ȧ〉 . The vector representation is 8 v and the basis is |i〉. The first excited level consist of states α i −1|a〉 and b i −1|a〉 and their chiral partners with α ′ M 2 = 1. Once again, for d = 10, all the massive light-cone states can be uniquely assembled into representations of SO(9), the little Lorentz group for massive states. GSO projection It turns out that fermionic string with all the states we found in the R and NS sectors is inconsistent. This can seen, for instance, from the fact that the 1-loop amplitude is not modular invariant. In order to construct a consistent modular invariant theory one should truncate the string spectrum in a specific way. This truncation is known as the GSO (Gliozzi-Scherk-Olive) projection. It restores the modular invariance, removes from the theory the tachyon and, in addition, provides the space-time supersymmetry of the resulting string spectrum. Below we will use an inverse argument to motivate the GSO projection – we will show that it allows to achive a spectrum which exhibits space-time supersymmetry. Looking at the massless states in the Ramond sector we see that one has two SO(8) spinors 8 s and 8 c . On the other hand, the massless states of the NS sector comprise a vector 8 v . If we project one of the two spinors out then there will be match of NS bosonic (8) and R fermionic (also 8) degrees of freedom. These massless vector and the massless spinor is indeed a content of the N = 1 super Yang-Mills theory in ten dimensions. One has also to get rid of tachyon which is in the NS sector. This all can be achieved if one first defines an operator ∞∑ G = (−1) F , F = b i −rb i r − 1 r= 1 2 and then requires that all allowed states should have G = 1: G|Φ〉 = |Φ〉.
Page 1 and 2:
Preprint typeset in JHEP style - HY
Page 3 and 4:
- 2 - 6. Classical fermionic supers
Page 5 and 6:
- 4 - bosons and Ramond’s fermion
Page 7 and 8:
- 6 - ? Type I Type IIB E 8 x E 8 h
Page 9 and 10:
- 8 - 2. Relativistic particle Cons
Page 11 and 12:
- 10 - are called the first class c
Page 13 and 14:
- 12 - which is in complete agreeme
Page 15 and 16:
- 14 - The Nambu-Goto action ∫
Page 17 and 18:
- 16 - Exercise Show that the Hessi
Page 19 and 20:
- 18 - 3.2.3 Conformal gauge Consid
Page 21 and 22:
- 20 - which result into We see tha
Page 23 and 24:
- 22 - Analogously, {¯L m , X µ }
Page 25 and 26:
- 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 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
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: - 112 - and similar for ML 2 . For
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
show all

Lectures on String Theory

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?