Ing. Electronico

January 2009

Preprint typeset in JHEP style - HYPER VERSION

String Theory
University of Cambridge Part III Mathematical Tripos

Dr David Tong
Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences,
Wilberforce Road,
Cambridge, CB3 OWA, UK
http://www.damtp.cam.ac.uk/user/tong/string.html
d.tong@damtp.cam.ac.uk

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Recommended Books andResources
• J. Polchinski, String Theory
This two volume work is the standard introduction to the subject. Our lectures will
more or less follow the path laid down in volume one covering the bosonic string. The
book 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 lectures.
Volumetwo covers the superstring.
• M. Green, J. Schwarz and E. Witten, Superstring Theory
Another two volume set. It is now over 20 years old and takes a slightly old-fashioned
route through the subject, with no explicit mention of conformal field theory. However, it does contain much good material and the explanations are uniformly excellent.
Volume one is most relevant for these lectures.
• B.Zwiebach, A First Course in String Theory
This book grew out of a course given to undergraduates who had no previous exposure
to general relativity or quantum field theory. It has wonderful pedagogical discussions
of the basics of lightcone quantization. More surprisingly, it also has some very clear
descriptions of several advanced topics, even though it misses out all the bits in between.
• P.Di Francesco, P. Mathieu and D. S´n´chal, Conformal Field Theory
ee
This big yellow book is affectionately known as the yellow pages. It’s a great way
to learn conformal field theory. At first glance, it comes across as slightly daunting
because it’s big. (And yellow). But you soon realise that it’s big because it starts at
the beginning and provides detailed explanations at every step. Thematerial necessary
for this course can be found in chapters 5 and 6.

Further References: “String Theory and M-Theory” by Becker, Becker and Schwarz
and “String Theory in a Nutshell” (it’s a big nutshell) by Kiritsis both deal with the
bosonic string fairly quickly, but include more advanced topics that may be of interest.
The book “D-Branes” by Johnson has lively and clear discussions about themany joys
of D-branes. Links to several excellent online resources, including video lectures by
Shiraz Minwalla, are listed on the course webpage.

Contents
0. Introduction
0.1 Quantum Gravity

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1. The Relativistic String
1.1 The Relativistic Point Particle
1.1.1 Quantization
1.1.2 Ein Einbein
1.2 The Nambu-Goto Action
1.2.1 Symmetries of the Nambu-Goto Action
1.2.2 Equationsof Motion
1.3 The Polyakov Action
1.3.1 Symmetries of the Polyakov Action
1.3.2 Fixing a Gauge
1.4 Mode Expansions
1.4.1 The Constraints Revisited

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2. The Quantum String
2.1 A Lightning Look at Covariant Quantization
2.1.1 Ghosts
2.1.2 Constraints
2.2 Lightcone Quantization
2.2.1 Lightcone Gauge
2.2.2 Quantization
2.3 The StringSpectrum
2.3.1 The Tachyon
2.3.2 The First Excited States
2.3.3 Higher Excited States
2.4 Lorentz Invariance Revisited
2.5 A Nod to the Superstring

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3. Open Strings and D-Branes
3.1 Quantization
3.1.1 The Ground State
3.1.2 First Excited States: A World of Light

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3.1.3 Higher Excited States and Regge Trajectories3.1.4 Another Nod to the Superstring
Brane Dynamics: The Dirac Action
Multiple Branes: A World of Glue

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4. Introducing Conformal Field Theory
4.0.1 Euclidean Space
4.0.2 The Holomorphy of Conformal Transformations
4.1 Classical Aspects
4.1.1 The Stress-Energy Tensor
4.1.2 Noether Currents
4.1.3 An Example: The Free Scalar Field
4.2 Quantum Aspects
4.2.1 Operator...