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Lectures Notes on String Theory
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Lucio Fontana Concetto Spaziale, These lecture notes are based upon a series of courses given at the master program ICFP. from 2018 by the author Comments and suggestions are welcome Some references that can. complement these notes are, Superstring Theory Green Schwarz Witten 1 2 the classic textbook from the eight. ies naturally outdated on certain aspects but still an unvaluable reference on many. topics including the Green Schwarz string and compactifications on special holonomy. String Theory Polchinski 3 4 the standard textbook with a very detailed derivation. of the Polyakov path integral and strong emphasis on conformal field theory methods. String Theory in a Nutshell Kiritsis 5 a concise presentation of string and super. string theory which moves quickly to rather advanced topics. String Theory and M Theory A Modern Introduction Becker Becker Schwarz 6. a good complement to the previous references with a broad introduction to modern. topics as AdS CFT and flux compactifications, A first course in String theory Zwiebach 7 an interesting and different approach. making little use of conformal field theory methods in favor of a less formal approach. Basic Concepts of String Theory Blumenhagen Lu st Theisen 8 As its name does. not suggest this book covers a lot of rather advanced topics about the worldsheet. aspects of string theory It is also rather appropriate for a math oriented reader. The lectures notes of David Tong http www damtp cam ac uk user tong string html. are rather enjoyable to read with a good balance between mathematical rigor and phys. ical intuition, The very lively online lectures of Shiraz Minwalla http theory tifr res in minwalla. Conventions, The space time metric is chosen to be of signature.
We work in units c 1, Latest update, April 28 2020. References, 1 M B Green J H Schwarz and E Witten Superstring theory Vol 1 Introduction. Cambridge Monographs on Mathematical Physics 1988, 2 M B Green J H Schwarz and E Witten Superstring theory Vol 2 loop amplitudes. anomalies and phenomenology Cambridge Uk Univ Pr 1987 596 P Cambridge. Monographs On Mathematical Physics 1988, 3 J Polchinski String theory Vol 1 An introduction to the bosonic string Cambridge. University Press 2007, 4 J Polchinski String theory Vol 2 Superstring theory and beyond Cambridge University.
Press 2007, 5 E Kiritsis String theory in a nutshell Princeton USA Univ Pr 2007 588 p 2007. 6 K Becker M Becker and J H Schwarz String theory and M theory A modern intro. duction Cambridge University Press 2006, 7 B Zwiebach A first course in string theory Cambridge University Press 2006. 8 R Blumenhagen D Lu st and S Theisen Basic concepts of string theory Theoretical. and Mathematical Physics Springer Heidelberg Germany 2013. 1 Introduction 5, 1 1 Gravity and quantum field theory 6. 1 2 String theory historical perspective 10, 2 Bosonic strings action and path integral 15. 2 1 Relativistic particle in the worldline formalism 16. 2 2 Relativistic strings 25, 2 3 Symmetries 30, 2 4 Polyakov path integral 39.
2 5 Open strings 45, 3 Conformal field theory 52, 3 1 The conformal group in diverse dimensions 54. 3 2 Radial quantization 55, 3 3 Conformal invariance and Ward identities 58. 3 4 Primary operators 62, 3 5 The Virasoro Algebra 66. 3 6 The Weyl anomaly 72, 3 7 Conformal field theory with boundaries 74. 4 Free conformal field theories 77, 4 1 Free scalar fields 78.
4 2 Free fermions 91, 4 3 The ghost CFT 101, 5 The string spectrum 107. 5 1 String theory and the dimension of space time 108. 5 2 BRST quantization 111, 5 3 The closed string spectrum 123. 5 4 Open string spectrum 128, 5 5 Physical degrees of freedom and light cone gauge 133. 5 6 General structure of the string spectrum 136, 6 String interactions 140. 6 1 The string S matrix 141, 6 2 Four tachyon tree level scattering 142.
6 3 One loop partition function 149, 7 Introduction to superstring theories 160. 7 1 Two dimensional local supersymmetry 161, 7 2 Superconformal symmetry 169. 7 3 BRST quantization 176, 7 4 Type II superstring theories 183. 7 5 One loop vacuum amplitudes 189, Introduction, Introduction. In Novembrer 1994 Joe Polchinski published on the ArXiv repository a preliminary ver. sion of his celebrated textbook on String theory based on lectures given at Les Houches. under the title What is string theory 1 If he were asked the same question today the. answer would probably be rather different as the field has evolved since in various directions. some of them completely unexpected at the time, One may try to figure out what string theory is about by looking at the program of Strings.
2017 the last of a series of annual international conferences about string theory that have. taken place at least since 1989 all over the world Among the talks less than half were about. string theory proper i e the theory you will read about in these notes while the others. pertained to a wide range of topics such as field theory amplitudes dualities in field theory. theoretical condensed matter or general relativity. The actual answer to the question raised by Joe Polchinski What is string theory may. be answered at different levels, litteral the quantum theory of one dimensional relativistic objects that interact by. joining and splitting, historical before 1974 a candidate theory of strong interactions after that date a. quantum theory of gravity, practical a non perturbative quantum unified theory of fundamental interactions whose. degrees of freedom in certain perturbative regime are given by relativistic strings. sociological a subset of theoretical physics topics studied by people that define them. selves as doing research in string theory, In these notes we will provide the construction of a consistent first quantized theory of. interacting quantum closed strings We will show that such theory automatically includes a. perturbative theory of quantum gravity We will introduce also open strings that incorpo. rate gauge interactions and give rise to the concept of D branes that plays a prominent role. in the AdS CFT correspondence, Along the way we will introduce some concepts and techniques that are as useful in other.
areas of theoretical physics as they are in string theory for instance conformal field theories. BRST quantization of gauge theories or supersymmetry. 1 1 Gravity and quantum field theory, String theory has been investigated by a significant part of the high energy theory community. for more than forty years as it provides a compelling answer and maybe the answer to. the following outstanding question what is the quantum theory of gravity. A successful theory of quantum gravity from the theoretical physics viewpoint should at. least satisfy the following properties, Introduction. 1 the theory should reproduce general relativity in an appropriate classical low energy. 2 the theory should be renormalizable or better UV finite in order to have predictive. 3 it should satisfy the basic requirements for any quantum theory such as unitarity. 4 it should explain the origin of black hole entropy possibly the only current prediction. for quantum gravity, 5 last but not least the physical predictions should be compatible with experiments in. particular with the Standard Model of particle physics astrophysical and cosmological. observations, According to our current understanding string theory passes successfully the first four. tests Whether string theory reproduces accurately at energies accessible to experiments the. known physics of fundamental particles and interaction is still unclear given that such physics. occurs in a regime of the theory that is beyond our current analytical control to draw an. analogy one cannot reproduce analytically the physics of condensed matter systems from the. microscopic quantum mechanical description in terms of atoms At least it is clear that the. main ingredients are there chiral fermions non Abelian gauge interactions and Higgs like. The problem with quantizing general relativity, The classical theory of relativistic gravity in four space time dimensions or Einstein theory.
follows in the absence of matter from the Einstein Hilbert EH action in four space time. dimensions that takes the form, Seh 2 d4 x det g R g 2 1 1. where R is the Ricci scalar associated with the space time manifold M endowed with a met. ric g and the cosmological constant that has no a priori reason to vanish The coupling. constant of the theory is related to the Newton s constant through 8 G by dimen. sional analysis it has dimension of length Its inverse defines the Planck mass MPl 1019 GeV. Quantizing general relativity raises a number of deep conceptual issues that can be raised. even before attempting to make any explicit computation Some of them are. Because of diffeormorphism invariance there are no local observables in general relativ. A path integral formulation of quantum gravity should include by definition a sum. over space time geometries Which geometries should be considered Should we specify. boundary conditions, Introduction, A Hamiltonian formulation of quantum gravity would require a foliation of space time. in terms of space like hypersurfaces Generically such foliation does not exist. Classical dynamics of general relativity predicts the formation of event horizons shield. ing regions of space time from the exterior This challenges the unitarity of the theory. through the black hole information paradox, Quantum gravity with a positive cosmological constant which seems to be relevant to de. scribe the Universe raises a number of additional conceptual issues that will be ignored. in the rest of the lectures We will mainly focus on theories with a vanishing cosmologi. cal constant the case of negative cosmological constant will be discussed in the AdS CFT. Perturbative QFT for gravity, One may try to ignore these conceptual problems and build a quantum field theory of gravity. in the usual way i e by defining propagators vertices Feynman rules etc from the. non linear EH action eqn 1 1 2, With vanishing cosmological constant one considers fluctuations of the metric around a.
reference Minkowski space time metric, Linearizing the equations of motion that follows from 1 1 in the absence of sources we. h 2 h h 0 1 3, where we have defined the trace reversed tensor h h 21 h This theory possesses. a gauge invariance that comes from the diffeomorphism invariance of the full theory The. equations of motion are invariant under, One can choose to work in a Lorentz gauge defined by h 0 in which case the field. equations 1 3 amounts to a wave equation for each component h 0. The solutions of these equations are naturally plane waves h x h0 exp ik x and. the Lorentz gauge condition means that they are transverse Finally the residual gauge. invariance that remains in the Lorentz gauge corresponding to vector fields satisfying the. wave equation 0 can be fixed by choosing the longitudinal gauge h 0 0 As a result. the gravitational waves have two independent transverse polarizations The corresponding. quantum theory is a theory of free gravitons that are massless bosons of helicity two. The interactions between gravitons are added by expanding the EH action around the. background 1 2 in powers of h In pure gravity one obtains three graviton and four. graviton vertices that have a rather complicated form For instance the four graviton vertex. Introduction, looks roughly like, G 1 1 4 4 k1 k4 2 k1 k2 1 1 4 4 k 1 3 k 2 3 1 2 1 2 1 5. Using these vertices one can define Feynman rules for the quantum field theory of gravitons. and compute loop diagrams like the one below, Figure 1 1 Graviton scattering.
As in most quantum field theories such loops integrals diverge when the internal momenta. propagating in the loop become large and should be regularized By dimensional analysis. the regularized loop diagrams will be weighted by positive powers of uv MPl where uv. is the ultraviolet cutoff, In renormalizable QFTs as quantum chromodynamics such high energy or ultraviolet. divergences can be absorbed into redefinitions of the couplings and fields of the theory which. leads to theories with predictive power In contrast this cannot be done for general relativity. for the simple reason that the coupling constant is dimensionfull it has the dimension of. length Therefore the divergences cannot be absorbed by redefining fields and couplings in. the original two derivative action rather higher derivative terms should be included to do so. General relativity is thus a prominent example of non renormalizable quantum field theory. Still it doesn t mean that such a theory is meaningless in the Wilsonian sense it can describe. the low energy dynamics well below the Planck scale MPl of an ultraviolet theory of. quantum gravity that is not explicitly known However as in any non renormalizable theory. this effective action has little predictive power as higher loop divergences need to be absorbed. in extra couplings that were not present in the original action 1 but become less important. as the energy becomes lower As we shall see string theory solves the problem in a rather. remarkable way by removing all the ultraviolet divergences of the theory. Strictly speaking the one loop divergence of pure GR can be absorbed by field redefinition This not the. case when matter is present and from two loops onwards for pure gravity. A rst course in String theory Zwiebach 7 an interesting and di erent approach making little use of conformal eld theory methods in favor of a less formal approach Basic Concepts of String Theory Blumenhagen L ust Theisen 8 As its name does not suggest this book covers a lot of rather advanced topics about the worldsheet

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