Lectures on String Theory - stringworld.ru

1 Preprint typeset in JHEP style - HYPER VERSIONL ectures on String TheoryGleb ArutyunovaaInstitute for Theoretical Physics and Spinoza Institute, Utrecht University3508 TD Utrecht, The NetherlandsAbstract:The course covers the basic concepts of modern String Theory . Thisincludes covariant and light-cone quantisation of bosonic and fermionic strings,geometry and topology of String world-sheets, vertex operators and String scatteringamplitudes, world-sheet and space-time supersymmetries, elements of conformalfield Theory , Green-Schwarz superstrings, strings in curved backgrounds, low-energyeffective actions, D-brane Introduction to the remarks22. Relativistic particle83. Classical relativistic bosonic Polyakov of String in the first order of motion. Classical Virasoro of the equations of e symmetry.

2 String in physical order structure of the light-cone symmetry374. Quantization of bosonic on canonical constraints in quantum operators. Tachyon scattering in the physical symmetry and critical quantization665. Geometry and topology of String Lorentzian to Euclidean space84 1 6. Classical fermionic in General action and its gauge and in the superconformal algebra1077. Quantum fermionic quantization and superstring spectrum111A. Dynamical systems of classical mechanics119B. OPE and conformal blocks123C. Useful formulae127D. Riemann normal coordinates128E. Exercises1301. Introduction to the Historical remarksString Theory arose at the end of sixties in an attempt to describe the Theory ofstrong interactions. In 1969 Veneziano found a beautiful formula for the scatteringamplitude of four particles.

3 This amplitude comprised many features that physicistsexpected to be found in the Theory of strong interactions. It was realized very soon byNambu and Susskind that the underlying dynamical object from which the Venezianoformula can be derived is arelativistic String . The fundamental difference of stringsfrom the Theory of point particles is that strings are extended, one-dimensional,objects; when String moves through space and time it sweeps two-dimensional surfacecalled the String world-sheet . The strings can be of two types: open with topologyof an interval and closed with topology of a investigations revealed however severe difficulties to treat the stringas the Theory of strong interactions. These difficulties of a critical dimension . of a massless spin two particle which is absent in the hadronic world.

4 2 If one tries to constructthe quantum mechanicsof relativistic strings one findsthat mathematically consistent Theory exists if and only if the dimension of space-time where String propagates is 26. The number 26 was named the critical dimen-sion . On the one hand, it was pretty remarkable and unexpected to find an exampleof physical Theory which puts restrictions on space-time where it is defined. On theother hand, it was certainly not clear why a Theory that shared at least some qualita-tive features with hadronic physics should exist in 26 dimensions only. A subsequentdiscovery of QCD (Quantum Chromo Dynamics) as the most appropriate candidateto describe the Theory of strong interactions led to a considerable loss of interest tostring 1974, Scherk and Schwarz came up with a proposal to completely alter theview on String Theory .

5 They suggested to consider the massless spin two particleabsent in the hadronic world as thegraviton the quantum of the gravitationalinteraction. Indeed, this particle neatly fits the properties of the graviton stringtheory predicts that this particle interacts according to the standard laws of GeneralRelativity. Gravitational interactions have a natural scale, called the Planck mass,which is around 1019 GeV. This is a huge number in comparison with characteristicenergies of hadronic physics, 100 200 MeV. Thus, according to their view, stringtheory could provide the unifying description of all the particles and matter forces,including gravity and it operates on a new fundamental if one accepts that quantum mechanics of relativistic strings can be definedin the unusual number 26 of the space-time dimensions, another problem arises.

6 Suchstring does not contain fermionic degrees of freedom and it predicts the existence ofa particle with the negative mass squared:m2<0. Such a particle,tachyon, is asource of instability and its existence indicates that either the Theory is ill-definedor it is formulated around a wrong ground state, or as physicists say, around a wrong vacuum. Critical dimension, tachyon and absence of fermions were thepuzzling features the String Theory had to status of String Theory changed again with the discovery of universe is made of two fundamental types of particles: bosons and constitute all the matter and bosons mediate interactions of the matterparticles. Supersymmetry is a new type of symmetry between bosons and fermions(Wess and Zumino 1974). Many physicists hope today that supersymmetry couldprovide an underlying principle for unification of all first success in incorporating supersymmetry in String Theory was achievedin 1971 by Ramond, who constructed a String analogue of the Dirac equation (thespinning String ).

7 Shortly afterwards, Neveu and Schwarz constructed a new bosonicstring Theory . They realized that the two constructions were different facets of asingle Theory - an interacting superstring Theory containing Neveu and Schwarz s 3 bosons and Ramond s fermions. The supersymmetry of the two-dimensional stringworld sheet was recognized by Gervais and Sakita in 1971. This was advent of theNSR (Neveu-Ramond-Schwarz) 1972 Schwarz demonstrated the consistency of the superstring Theory in 10 di-mensions. Instead of 26 found for purely bosonic String , the critical dimension for theNSR String appears to be 10. In 1977 Gliozzi, Scherk and Olive realized that furtherconditions should be imposed on the spectrum (the GSO projection mechanism) ofthe NSR String which lead to both the so-calledspace-time supersymmetry(to com-pare with the world-sheet supersymmetry mentioned above) and to the removal oftachyon.

8 Thus, superstring Theory has at least two advantages in comparison withbosonic strings: critical dimension 10<26 and the absence of tachyon. It also turnedout that the GSO projection can be imposed in two different ways which lead to twodifferent types of superstrings, called the Type IIA and Type theories have a natural particle limit, when the length of String this limit superstrings give rise to the low-energy effective theories, known assupergravities. These theories can be defined in a way completely independent ofstring Theory : they can be thought of as supersymmetric generalizations of the pureEinstein gravity. As is known, attempts to quantize gravity in the standard frame-work of quantum mechanics fail because gravity is a non-renormalizable Theory (thereare infinitely many divergent graphs with any number of external legs and with anarbitrarily high index of divergence, cf.)

9 The course on Quantum Field Theory ). Su-persymmetric theories tend to be less divergent than non-supersymmetric ones whichgave initially a hope that supersymmetry could cure the nonrenormalizable infinitiesof the quantum gravity. It seems that supergravities themselves are still not capableto solve the divergency problem1. Quite remarkably, there is a strong evidence thatthe divergency problem of quantum (super)gravities is resolved by String Theory . Resolution of the four-fermi interaction. At high energies the weak force ismediated by a heavy instance, it is unknown if the so-calledN= 8 supergravity is finite or not. 4 To get a better feeling why it happens it is convenient to envoke an analogywith the Theory of weak interactions. Trying to describe the decay of the neutron bythe Fermi-type Lagrangian containing the quartic, point-like, interaction vertex onefinds irremovable ultraviolet divergencies at higher loop orders.

10 Again, the reasonis that the corresponding Theory of Fermi-interactions in non-renormalizable. Thesolution of this problem lies in a fact that at higher energies (more than 100 GeV),the pointlike vertex gets resolved and the interaction is mediated by a heavy W-boson. In the new Theory one has qubic vertices and this ultimately makes thetheory similar phenomenon occurs in String Theory . Expanding the Einstein-Hilbert action gRone gets more and more complicated point-like vertices whichrender the Theory non-renormalizable, analogously to the old Fermi Theory . In oppo-site, in String Theory all these vertices get dissolved by the exchange of the massivestring states. String states form an infinite tower of particles of arbitrarily high massand spin and all of them participate in the interaction process.