Lectures on String Theory

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– 3 – If one tries to construct the quantum mechanics of relativistic strings one finds that mathematically consistent theory exists if and only if the dimension of spacetime where string propagates is 26. The number 26 was named the “critical dimension”. On the one hand, it was pretty remarkable and unexpected to find an example of physical theory which puts restrictions on space-time where it is defined. On the other hand, it was certainly not clear why a theory that shared at least some qualitative features with hadronic physics should exist in 26 dimensions only. A subsequent discovery of QCD (Quantum Chromo Dynamics) as the most appropriate candidate to describe the theory of strong interactions led to a considerable loss of interest to string theory. In 1974, Scherk and Schwarz came up with a proposal to completely alter the view on string theory. They suggested to consider the massless spin two particle absent in the hadronic world as the graviton – the quantum of the gravitational interaction. Indeed, this particle neatly fits the properties of the graviton – string theory predicts that this particle interacts according to the standard laws of General Relativity. Gravitational interactions have a natural scale, called the Planck mass, which is around 10 19 GeV. This is a huge number in comparison with characteristic energies of hadronic physics, 100 − 200 MeV. Thus, according to their view, string theory could provide the unifying description of all the particles and matter forces, including gravity and it operates on a new fundamental scale. Even if one accepts that quantum mechanics of relativistic strings can be defined in the unusual number 26 of the space-time dimensions, another problem arises. Such string does not contain fermionic degrees of freedom and it predicts the existence of a particle with the negative mass squared: m 2 < 0. Such a particle, tachyon, is a source of instability and its existence indicates that either the theory is ill-defined or it is formulated around a “wrong” ground state, or as physicists say, around a “wrong” vacuum. Critical dimension, tachyon and absence of fermions were the puzzling features the string theory had to face. The status of string theory changed again with the discovery of supersymmetry. All universe is made of two fundamental types of particles: bosons and fermions. Fermions constitute all the matter and bosons mediate interactions of the matter particles. Supersymmetry is a new type of symmetry between bosons and fermions (Wess and Zumino 1974). Many physicists hope today that supersymmetry could provide an underlying principle for unification of all interactions. The first success in incorporating supersymmetry in string theory was achieved in 1971 by Ramond, who constructed a string analogue of the Dirac equation (the spinning string). Shortly afterwards, Neveu and Schwarz constructed a new bosonic string theory. They realized that the two constructions were different facets of a single theory - an interacting superstring theory containing Neveu and Schwarz’s
– 4 – bosons and Ramond’s fermions. The supersymmetry of the two-dimensional string world sheet was recognized by Gervais and Sakita in 1971. This was advent of the NSR (Neveu-Ramond-Schwarz) superstring. In 1972 Schwarz demonstrated the consistency of the superstring theory in 10 dimensions. Instead of 26 found for purely bosonic string, the critical dimension for the NSR string appears to be 10. In 1977 Gliozzi, Scherk and Olive realized that further conditions should be imposed on the spectrum (the GSO projection mechanism) of the NSR string which lead to both the so-called space-time supersymmetry (to compare with the world-sheet supersymmetry mentioned above) and to the removal of tachyon. Thus, superstring theory has at least two advantages in comparison with bosonic strings: critical dimension 10 < 26 and the absence of tachyon. It also turned out that the GSO projection can be imposed in two different ways which lead to two different types of superstrings, called the Type IIA and Type IIB. String theories have a natural particle limit, when the length of string vanishes. In this limit superstrings give rise to the low-energy effective theories, known as supergravities. These theories can be defined in a way completely independent of string theory: they can be thought of as supersymmetric generalizations of the pure Einstein gravity. As is known, attempts to quantize gravity in the standard framework of quantum mechanics fail because gravity is a non-renormalizable theory (there are infinitely many divergent graphs with any number of external legs and with an arbitrarily high index of divergence, cf. the course on Quantum Field Theory). Supersymmetric theories tend to be less divergent than non-supersymmetric ones which gave initially a hope that supersymmetry could cure the nonrenormalizable infinities of the quantum gravity. It seems that supergravities themselves are still not capable to solve the divergency problem 1 . Quite remarkably, there is a strong evidence that the divergency problem of quantum (super)gravities is resolved by string theory. ¢¡¢ ¡ ¢¡¢ ¡ ¢¡¢ ¡ Resolution of the four-fermi interaction. At high energies the weak force is mediated by a heavy boson. 1 For instance, it is unknown if the so-called N = 8 supergravity is finite or not.
Page 1 and 2: Preprint typeset in JHEP style - HY
Page 3: - 2 - 6. Classical fermionic supers
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 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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Lectures on String Theory

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