Table Of ContentLecture Notes
ni Physics
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346
.D ts)LL
Theisen S.
Lectures on String Theory
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Authors
Dieter Lfist
Stefan Theisen
CERN, CH-1211 Geneva 23, Switzerland
ISBN 3-540-51882-7 Springer-Verlag Berlin Heidelberg New York
ISBN 0-38?-51882-7 Springer-Verlag New York Berlin Heidelberg
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Preface
These Lectures on String Theory are an extended version of
lectures we gave at the Max-Planck-Institut riif Physik und Astro-
physik in Munich in fall and winter 1987/88. They were meant to
be an introduction to the subject and this si also the intention of
these notes.
We have not attempted to give a complete tsil of references. In-
stead, at the end of each chapter we give a short tsil of the original
papers and some reviews we found helpful and were familiar with.
An indispensable general reference si the book by Green, Schwarz
and Witten: Superstring Theory, 2 Vols., Cambridge University
Press, 1987. tI also contains a more complete set of references,
in particular for our chapters ,2 ,3 5 and 7 - .9 Early reviews are
collected in Dual Resonance Theory, ed. M. Jacob, Physics Reports
Reprints Vol. ,I North Holland, 1974.
We would like to thank our collaborators S. Ferrara, I.-G. Koh,
.J Lauer, W. Lerche, B. Schellekens and G. Zoupanos, from whom
we have learned much of the subject presented here. We also ac-
knowledge helpful discussions with many other people and thank
in particular L. Alvarez-Gaum6, W. Buchmiiller, L. Castellani, P.
Forg£cs, F. Gieres, G. Horowitz, P. Howe, L. Ibafiez, T. Jacobsen, .J
Kubo, H. Nicolai, H.-P. Nilles, R. Rohm, M. Schmidt, .J Schnittger,
.J Schwarz, A. Shapere, M. Srednicki, .~i Stora, N. Warner, P. West,
A. Wipf and R. Woodard. We thank P. Breitenlohner for supply-
ing his powerful TEX macro package and .J Paxon for her help in
preparing the manuscript.
,NREC Geneva, 1989 Dieter Liist
Stefan Theisen
Table of Contents
1. Introduction ................................................ 1
2. The Classical Bosonic String .............................. 5
2.1 The relativistic particle .................................. 5
2.2 The Nambu-Goto action ................................. 9
2.3 The Polyakov action and its symmetries .................. 11
2.4 Oscillator expansions ..................................... 22
2.5 Examples of classical string solutions ...................... 27
3. The Quantized Bosonic String ............................. 31
3.1 Canonical quantization of the bosonic string .............. 31
3.2 Light cone quantization of the bosonic string .............. 38
3.3 Spectrum of the bosonic string ............................ 40
3.4 Covariant path integral quantization ...................... 47
4. Introduction to Conformal Field Theory ................. 57
4.1 General introduction ..................................... 57
4.2 Application to string theory .............................. 77
5. Reparametrization Ghosts and BRST Quantization ..... 85
5.1 The ghost system as a conformal field theory .............. 85
5.2 BRST quantization ....................................... 88
6. Global Aspects of String Perturbation Theory
and Riemann Surfaces ..................................... 98
7. The Classical Closed Fermionic String .................... 128
7.1 Superstring action and its symmetries ..................... 129
7.2 Superconformal gauge ..................................... 134
8. The Quantized Closed Fermionic String .................. 147
8.1 Canonical quantization .................................... 147
8.2 Light cone quantization ................................... 152
8.3 Path integral quantization ................................ 159
9. Spin Structures and Superstring Partition Function ..... 163
10. Toroidal Compactification of the Closed Bosonic String -
10-Dimensional Heterotic String .......................... 177
10.1 Toroidal compactification of the closed bosonic string ...... 178
10.2 The heterotic string ....................................... 193
11. Conformal Field Theory II: Lattices and Kac-Moody
Algebras ..................................................... 202
11.1 Kac-Moody algebras ...................................... 202
11.2 Lattices and Lie algebras .................................. 207
11.3 Frenkel-Kac-Segal construction ............................ 220
11.4 Fermionic construction of the current algebra -
Bosonization .............................................. 225
11.5 Unitary representations and characters of Kac-Moody
algebras .................................................. 228
12. Conformal Field Theory III: Superconformal Field
Theory ....................................................... 237
Vl
13. Bosonization of the Fermionic String - Covariant Lattices 258
13.1 First order systems ....................................... 261
13.2 Covariant vertex operators, BRST and picture changing ... 270
13.3 The covariant lattice ...................................... 278
14. Heterotic Strings in Ten and Four Dimensions ........... 288
14.1 Ten-dimensional heterotic strings .......................... 288
14.2 Four-dimensional heterotic strings in the covariant lattice
approach ................................................. 293
14.3 General aspects of four-dimensional heterotic string
theories .................................................. 308
15. Low Energy Field Theory .................................. 328
VII
Chapter 1
Introduction
String theory is currently one of the main activities among high energy
theorists. It began at the end of the 1960's as an attempt to explain the
spectrum of hadrons and their interactions. It was however discarded as a
theory of strong interactions, a development which was supported by the
rapid success of quantum chromodynamics. One problem was the existence
of a critical dimension, which is 26 for the bosonic string and 10 for the
fermionic string. Another obstacle in the interpretation of string theory as
the theory of strong interactions was the existence of a massless spin two
particle which is not present in the hadronic world.
In 1974 Scherk and Schwarz suggested to turn the existence of the mass-
less spin two particle into an advantage for string theory by interpreting this
particle as the graviton, the field quantum of gravitation. This implies that
the string tension has to be related to the characteristic mass scale of gravity,
namely the Planck mass Mp - ~/--~/G ~ 1019GeV. They also recognized
that at low energies this graviton interacts according to the covariance laws
of general relativity. In this way string theory could, at least in princi-
ple, achieve a unification of gravitation with all the other interactions in a
quantized theory.
At that time it was only known how to incorporate non-abelian gauge
symmetries in open string theories. Moreover, any open string theory with
local interactions which consist of splitting and joining of strings automat-
ically also contains closed strings with the massless spin two state in its
spectrum. These theories are however plagued by gravitational and gauge
anomalies which were beheved to be fatal.
The renewed interest in string theory started again in 1984 when Green
and Schwarz showed that the open superstring si anomaly free fi and only fi
the gauge group si SO(32). In addition they found that the ten-dimensional
supersymmetric Einstein-Yang-Mills field theory si anomaly free for the
gauge group SO(32) and also for the phenomenologically more interesting
group 8 E x E8, which is however excluded in the open string theory. This
puzzle was resolved soon after by Gross, Harvey, Martinec and Rohm with
the formulation of the heterotic string. It is a theory of closed strings only
and represents the most economical way of incorporating both gravitational
and gauge interactions. The allowed gauge symmetries 8 E x 8 E or SO(32)
arise in a way different from the open string due to the incorporation of the
so-called Kac-Moody algebras, which are infinite-dimensional extensions of
ordinary Lie algebras.
On the other hand, the heterotic string is formulated as a ten-dimen-
sional theory and obviously fails to reproduce an important experimen-
tally estabhshed fact, namely that we hve in four-dimensional (almost) flat
Minkowski space-time.
The first approach to obtain four-dimensional string theories was along
the old ideas of Kaluza and Klein, namely to consider compactifications
of the ten-dimensional heterotic string theory. In this case, four string
coordinates are uncompactified, whereas the remaining six are curled up and
describe a tiny compact space whose size is of the order of the Planck length.
The internal space can however not be arbitrarily chosen; the requirement
of preserving conformal invariance puts severe constraints on it. Analyzing
these constraints it turns out that the internal six-dimensional space must
have vanishing Ricci-curvature. Examples are tori or the so-called Calabi-
Yau manifolds.
More recently, it was discovered how one can construct (heterotic) string
theories directly in four dimensions without ever referring to any compacti-
fication scheme. This opened a wide range of possibilities to obtain consis-
tent string theories in four dimensions. In the most general four-dimensional
string theories the part which refers to the extra dimensions (above four),
which is needed because of conformal invariance, is replaced by a general
conformal field theory. This internal conformal field theory has to obey
some additional consistency requirements (like modular invariance, as we
will discuss in some detail), but does however not need to admit an inter-
pretation as a compact six-dimensional space. Unfortunately there exists a
huge number of consistent internal conformal field theories, destroying the
once celebrated uniqueness of string theory. In addition there exists so far
no compelling principle which determines the number of space-time dimen-
sions to be four. All dimensions below ten seem to be on an equal footing.
However, the uniqueness in string theory could still be true in the sense that
all different models are just different ground states, i.e. different classical
solutions of an unique second quantized string theory. Then one specific
string vacuum with a specific (hopefully correct) choice of gauge group and
number (hopefully four) of flat space-time dimensions could be singled out
by an underlying dynamical principle. At the moment, this is however
wishful thinking and all ideas in this direction must be considered as pure
speculations. Nevertheless, many of the four-dimensional heterotic string
models exhibit promising aspects for phenomenology. However, within the
context of string theory the word phenomenology should not be taken too
seriously. At present one should only expect an explanation of generic fea-
tures of the observed world, such as the presence of chiral fermions, the
number of generations etc.
These lecture notes are intended to provide some of the tools which are
necessary for the construction of four-dimensional (heterotic) string theo-
ries. Our main emphasis is on the relation to conformal field theory. One of
the constructions of four-dimensional heterotic strings, namely the covari-
ant lattice construction, will be discussed in detail. For an outline of the
topics covered we refer to the table of contents. The selection we made was
dictated by limitations of space and time and by our preferences.
