String Theory - University of Cambridge

1 Preprint typeset in JHEP style - HYPER VERSIONJ anuary 2009 String TheoryUniversity of Cambridge Part III Mathematical TriposDr David TongDepartment of Applied Mathematics and Theoretical Physics,Centre for Mathematical Sciences,Wilberforce Road, Cambridge , CB3 OWA, 1 Recommended Books and Resources J. Polchinski, String TheoryThis two volume work is the standard introduction to the subject. Our lectures willmore or less follow the path laid down in volume one covering the bosonic String . Thebook contains explanations and descriptions of many details that have been deliberately(and, I suspect, at times inadvertently) swept under a very large rug in these two covers the superstring. M. Green, J. Schwarz and E. Witten,Superstring TheoryAnother two volume set. It is now over 20 years old and takes a slightly old-fashionedroute through the subject, with no explicit mention of conformal field Theory .

2 How-ever, it does contain much good material and the explanations are uniformly one is most relevant for these lectures. B. Zwiebach,A First Course in String TheoryThis book grew out of a course given to undergraduates who had no previous exposureto general relativity or quantum field Theory . It has wonderful pedagogical discussionsof the basics of lightcone quantization. More surprisingly, it also has some very cleardescriptions of several advanced topics, even though it misses out all the bits in between. P. Di Francesco, P. Mathieu and D. S en echal,Conformal Field TheoryThis big yellow book is affectionately known as the yellow pages. It s a great wayto learn conformal field Theory . At first glance, it comes across as slightly dauntingbecause it s big. (And yellow). But you soon realise that it s big because it starts atthe beginning and provides detailed explanations at every step.

3 The material necessaryfor this course can be found in chapters 5 and References: String Theory and M- Theory by Becker, Becker and Schwarzand String Theory in a Nutshell (it s a big nutshell) by Kiritsis both deal with thebosonic String fairly quickly, but include more advanced topics that may be of book D-Branes by Johnson has lively and clear discussions about the many joysof D-branes. Links to several excellent online resources, including video lectures byShiraz Minwalla, are listed on the course Gravity31. The Relativistic Relativistic Point Nambu-Goto of the Nambu-Goto of Polyakov of the Polyakov a Constraints Revisited262. The Quantum Lightning Look at Covariant String First Excited Excited Invariance Nod to the Superstring483. Open Strings and Ground Excited States: A World of Light55 1 Excited States and Regge Nod to the Dynamics: The Dirac Branes: A World of Glue594.

4 Introducing Conformal Field Holomorphy of Conformal Stress-Energy Example: The Free Scalar Product Example: The Free Scalar Aside: No Goldstone Bosons in Two Stress-Energy Tensor and Primary Central is for Weyl is for has a Virasoro Virasoro of the Virasoro of State-Operator Simple Favourite Example: The Free Scalar Comments on Conformal Field Theories with Boundaries1055. The Polyakov Path Integral and Path Faddeev-Popov Method109 2 Faddeev-Popov Ghost Critical Dimension of String Usual Nod to the Aside: Non-Critical and Vertex Example: Closed Strings in Flat Example: Open Strings in Flat General CFTs1246. String to Compute? Over String Amplitudes at Tree Gauge Symmetry: SL(2,C) Virasoro-Shapiro to String Veneziano Tension of Moduli Space of the One-Loop Partition the String Partition is String Theory Finite?

5 Perturbation Theory ? : Games with Integrals and Gamma Functions1547. Low Energy Effective s Beta Strings and Low-Energy Effective Action167 3 Frame and Einstein to Einstein s Once More to the Simple String Away from the Critical Elephant in the Room: The Revisited: Background Gauge Beta Born-Infeld DBI to Closed String Yang-Mills in Type II Superstring Theories1958. Compactification and View from around the View from the Gauge Big Circles are the Same as Small Path Integral Derivation of for Open for 4 AcknowledgementsThese lectures are aimed at beginning graduate students. They assume a workingknowledge of quantum field Theory and general relativity. The lectures were given overone semester and are based broadly on Volume one of the book by Joe Polchinski. Iinherited the course from Michael Green whose notes were extremely useful.

6 I alsobenefited enormously from the insightful and entertaining video lectures by m grateful to Anirban Basu, Niklas Beisert, Joe Bhaseen, Diego Correa, Nick Dorey,Michael Green, Anshuman Maharana, Malcolm Perry and Martin Schnabl for discus-sions and help with various aspects of these notes. I m also grateful to the students,especially Carlos Guedes, for their excellent questions and superhuman typo-spottingabilities. Finally, my thanks to Alex Considine for infinite patience and understandingover the weeks these notes were written. I am supported by the Royal Society. 5 0. IntroductionString Theory is an ambitious project. It purports to be an all-encompassing theoryof the universe, unifying the forces of Nature, including gravity, in a single quantummechanical premise of String Theory is that, at the fundamental level, matter does not consistof point-particles but rather of tiny loops of String .

7 From this slightly absurd beginning,the laws of physics emerge. General relativity, electromagnetism and Yang-Mills gaugetheories all appear in a surprising fashion. However, they come with baggage. Stringtheory gives rise to a host of other ingredients, most strikingly extra spatial dimensionsof the universe beyond the three that we have observed. The purpose of this course isto understand these statements in lectures differ from most other courses that you will take in a physics Theory is speculative science. There is no experimental evidence that stringtheory is the correct description of our world and scant hope that hard evidence willarise in the near future. Moreover, String Theory is very much a work in progress andcertain aspects of the Theory are far from understood. Unresolved issues abound andit seems likely that the final formulation has yet to be written. For these reasons, I llbegin this introduction by suggesting some answers to the question: Why study stringtheory?

8 Reason 1. String Theory is a Theory of quantum gravityString Theory unifies Einstein s Theory of general relativity with quantum , it does so in a manner that retains the explicit connection with both quantumtheory and the low-energy description of quantum gravity contains many puzzles, both technical and conceptual. Whatdoes spacetime look like at the shortest distance scales? How can we understandphysics if the causal structure fluctuates quantum mechanically? Is the big bang truelythe beginning of time? Do singularities that arise in black holes really signify the endof time? What is the microscopic origin of black hole entropy and what is it tellingus? What is the resolution to the information paradox? Some of these issues will bereviewed later in this or not String Theory is the true description of reality, it offers a frameworkin which one can begin to explore these issues.

9 For some questions, String theoryhas given very impressive and compelling answers. For others, String Theory has beenalmost silent. 1 Reason 2. String Theory may bethetheory of quantum gravityWith broad brush, String Theory looks like an extremely good candidate to describe thereal world. At low-energies it naturally gives rise to general relativity, gauge theories,scalar fields and chiral fermions. In other words, it contains all the ingredients thatmake up our universe. It also gives the only presently credible explanation for the valueof the cosmological constant although, in fairness, I should add that the explanation isso distasteful to some that the community is rather amusingly split between whetherthis is a good thing or a bad thing. Moreover, String Theory incorporates several ideaswhich do not yet have experimental evidence but which are considered to be likelycandidates for physics beyond the standard model.

10 Prime examples are supersymmetryand , while the broad brush picture looks good, the finer details have yet tobe painted. String Theory does not provide unique predictions for low-energy physicsbut instead offers a bewildering array of possibilities, mostly dependent on what ishidden in those extra dimensions. Partly, this problem is inherent to any Theory ofquantum gravity: as we ll review shortly, it s a long way down from the Planck scaleto the domestic energy scales explored at the LHC. Using quantum gravity to extractpredictions for particle physics is akin to using QCD to extract predictions for howcoffee makers work. But the mere fact that it s hard is little comfort if we re lookingfor convincing evidence that String Theory describes the world in which we String Theory cannot at present offer falsifiable predictions, it has nonethelessinspired new and imaginative proposals for solving outstanding problems in particlephysics and cosmology.