Mécanique Quantique Matricielle et la Théorie des Cordes à Deux Dimensions dans des Fonds Non-triviaux

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Domaine: Physique
La théorie des cordes est le candidat le plus promettant pour la théorie unissant toutes les interactions en incluant la gravitation. Elle a la dynamique très compliquée. C'est pourquoi c'est utile d'étudier ses simplifications. Une de celles-ci est la théorie des cordes non-critiques qui peut être définie dans les dimensions inférieures. Le cas particulièrement intéressant est la théorie des cordes à deux dimensions. D'une part elle a la structure très riche et d'autre part elle est résoluble exactement. La solution complète de la théorie des cordes à deux dimensions dans le fond le plus simple du dilaton linéaire a été obtenue en utilisant sa représentation comme la mécanique quantique matricielle. Ce modèle de matrices fournit une technique très puissante et découvre l'intégrabilité cachée dans la formulation habituelle de CFT. Cette thèse prolonge la formulation de la théorie des cordes à deux dimensions par des modèles de matrices dans des fonds non-triviaux. Nous montrons comment les perturbations changeants le fond sont incorporés à la mécanique quantique matricielle. Les perturbations sont intégrables et dirigées par la hiérarchie de Toda. Cette intégrabilité est utilisée pour extraire l'information divers sur le système perturbé: les fonctions des corrélations, le comportement thermodynamique, la structure de l'espace-temps. Les résultats concernant ces et autres questions, comme des effets non-perturbatifs dans la théorie des cordes non-critiques, sont présentés dans cette thèse.

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Service de Physique Theorique { C.E.A.-Saclay
THESE DE DOCTORAT DE L’UNIVERSITE PARIS XI
Specialite : Physique Theorique
presentee par
Serguei ALEXANDROV
pour obtenir le grade de
Docteur de l’Universite Paris XI
Mecanique Quantique Matricielle et
la Theorie des Cordes a Deux Dimensions
dans des Fonds Non-triviaux
Soutenue le 23 septembre 2003 devant le jury compose de :
M. BREZIN Edouard, president du jury,
M. KAZAKOV Vladimir, directeur de these,
M. KOSTOV Ivan, de these,
M. NEKRASOV Nikita, rapporteur,
M. STAUDACHER Matthias, rapp
M. ZUBER Jean-Bernard.
tel-00003512, version 1 - 9 Oct 2003tel-00003512, version 1 - 9 Oct 2003Acknowledgements
This work was done at the Service de Physique Theorique du centre d’etudes de Saclay. I
would like to thank the laboratory for the excellent conditions which allowed to accomplish
my work. Also I am grateful to CEA for the nancial support during these three years.
Equally, my gratitude is directed to the Laboratoire de Physique Theorique de l’Ecole Nor-
male Superieure where I also had the possibility to work all this time. I am thankful to all
members of these two labs for the nice stimulating atmosphere.
Especially, I would like to thank my scienti c advisors, Volodya Kazakov and Ivan Kostov
who opened a new domain of theoretical physics for me. Their creativity and deep knowledge
were decisive for the success of our work. Besides, their care in all problems helped me much
during these years of life in France.
I am grateful to all scientists with whom I had discussions and who shared their ideas
with me. In particular, let me express my gratitude to Constantin Bachas, Alexey Boyarsky,
Edouard Brezin, Philippe Di Francesco, David Kutasov, Marcus Marino,~ Andrey Marshakov,
Yuri Novozhilov, Volker Schomerus, Didina Serban, Alexander Sorin, Cumrum Vafa, Pavel
Wiegmann, Anton Zabrodin, Alexey Zamolodchikov, Jean-Bernard Zuber and, especially, to
Dmitri Vassilevich. He was my rst advisor in Saint-Petersburg and I am indebted to him
for my rst steps in physics as well as for a fruitful collaboration after that.
Also I am grateful to the Physical Laboratory of Harvard University and to the Max{
Planck Institute of Potsdam University for the kind hospitality during the time I visited
there.
It was nice to work in the friendly atmosphere created by Paolo Ribeca and Thomas
Quella at Saclay and Nicolas Couchoud, Yacine Dolivet, Pierre Henry-Laborder, Dan Israel
and Louis Paulot at ENS with whom I shared the o ce.
Finally, I am thankful to Edouard Brezin and Jean-Bernard Zuber who accepted to be
the members of my jury and to Nikita Nekrasov and Matthias Staudacher, who agreed to
be my reviewers, to read the thesis and helped me to improve it by their corrections.
tel-00003512, version 1 - 9 Oct 2003tel-00003512, version 1 - 9 Oct 2003Contents
Introduction 1
I String theory 5
1 Strings, elds and quantization . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 A little bit of history . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 String action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 theory as two-dimensional gravity . . . . . . . . . . . . . . . . 8
1.4 Weyl invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Critical string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Critical bosonic strings . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Branes, dualities and M-theory . . . . . . . . . . . . . . . . . . . . . 13
3 Low-energy limit and string backgrounds . . . . . . . . . . . . . . . . . . . . 16
3.1 General -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Weyl invariance and e ectiv e action . . . . . . . . . . . . . . . . . . . 16
3.3 Linear dilaton background . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Inclusion of tachyon . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Non-critical string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Two-dimensional string theory . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.1 Tachyon in two-dimensions . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Discrete states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Compacti cation, winding modes and T-duality . . . . . . . . . . . . 23
6 2D string theory in non-trivial backgrounds . . . . . . . . . . . . . . . . . . 25
6.1 Curved backgrounds: Black hole . . . . . . . . . . . . . . . . . . . . . 25
6.2 Tachyon and winding condensation . . . . . . . . . . . . . . . . . . . 26
II Matrix models 29
1 Matrix models in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 models and random surfaces . . . . . . . . . . . . . . . . . . . . . . . 31
2.1 De nition of one-matrix model . . . . . . . . . . . . . . . . . . . . . 31
2.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Discretized surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Topological expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Continuum and double scaling limits . . . . . . . . . . . . . . . . . . 36
3 One-matrix model: saddle point approach . . . . . . . . . . . . . . . . . . . 38
v
tel-00003512, version 1 - 9 Oct 2003CONTENTS
3.1 Reduction to eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Saddle point equation . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 One cut solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Critical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 General solution and complex curve . . . . . . . . . . . . . . . . . . . 42
4 Two-matrix model: method of orthogonal polynomials . . . . . . . . . . . . 44
4.1 Reduction to eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Critical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Complex curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.6 Free fermion representation . . . . . . . . . . . . . . . . . . . . . . . 49
5 Toda lattice hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1 Integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Lax formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Free fermion and boson representations . . . . . . . . . . . . . . . . . 55
5.4 Hirota equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.5 String equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.6 Dispersionless limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.7 2MM as -function of Toda hierarchy . . . . . . . . . . . . . . . . . . 61
IIIMatrix Quantum Mechanics 63
1 De nition of the model and its interpretation . . . . . . . . . . . . . . . . . 63
2 Singlet sector and free fermions . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1 Hamiltonian analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Reduction to the singlet sector . . . . . . . . . . . . . . . . . . . . . . 66
2.3 Solution in the planar limit . . . . . . . . . . . . . . . . . . . . . . . 67
2.4 Double scaling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3 Das{Jevicki collective eld theory . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1 E ectiv e action for the collective eld . . . . . . . . . . . . . . . . . . 72
3.2 Identi cation with the linear dilaton background . . . . . . . . . . . . 74
3.3 Vertex operators and correlation functions . . . . . . . . . . . . . . . 77
3.4 Discrete states and chiral ring . . . . . . . . . . . . . . . . . . . . . . 79
4 Compact target space and winding modes in MQM . . . . . . . . . . . . . . 82
4.1 Circle embedding and duality . . . . . . . . . . . . . . . . . . . . . . 82
4.2 MQM in arbitrary representation: Hamiltonian analysis . . . . . . . . 86
4.3 in partition function . . . . . . . . . 88
4.4 Non-trivial SU(N) representations and windings . . . . . . . . . . . . 90
IV Winding perturbations of MQM 93
1 Introduction of winding modes . . . . . . . . . . . . . . . . . . . . . . . . . . 93
1.1 The role of the twisted partition function . . . . . . . . . . . . . . . . 93
1.2 Vortex couplings in MQM . . . . . . . . . . . . . . . . . . . . . . . . 95
1.3 The partition function as -function of Toda hierarchy . . . . . . . . 96
2 Matrix model of a black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
vi
tel-00003512, version 1 - 9 Oct 2003CONTENTS
2.1 Black hole background from windings . . . . . . . . . . . . . . . . . . 99
2.2 Results for the free energy . . . . . . . . . . . . . . . . . . . . . . . . 100
2.3 Thermodynamical issues . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Correlators of windings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
V Tachyon perturbations of MQM 107
1 Tachyon p as pro les of Fermi sea . . . . . . . . . . . . . . . . . 107
1.1 Tachyons in the light-cone representation . . . . . . . . . . . . . . . . 108
1.2 Toda description of tachyon perturbations . . . . . . . . . . . . . . . 110
1.3 Dispersionless limit and interpretation of the Lax formalism . . . . . 113
1.4 Exact solution of the Sine{Liouville theory . . . . . . . . . . . . . . . 113
2 Thermodynamics of tachyon perturbations . . . . . . . . . . . . . . . . . . . 115
2.1 MQM partition function as -function . . . . . . . . . . . . . . . . . 115
2.2 Integration over the Fermi sea: free energy and energy . . . . . . . . 116
2.3 Thermodynamical interpretation . . . . . . . . . . . . . . . . . . . . 117
3 Backgrounds of 2D string theory . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.1 Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.2 Global properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
VI MQM and Normal Matrix Model 125
1 Normal matrix model and its applications . . . . . . . . . . . . . . . . . . . 125
1.1 De nition of the model . . . . . . . . . . . . . . . . . . . . . . . . . . 125
1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2 Dual formulation of compacti ed MQM . . . . . . . . . . . . . . . . . . . . . 130
2.1 Tachyon perturbations of as Normal Matrix Model . . . . . . . 130
2.2 Geometrical description in the classical limit and duality . . . . . . . 131
VIINon-perturbative e ects in matrix models and D-branes 135
Conclusion 137
1 Results of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2 Unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
ARTICLES
I Correlators in 2D string theory with vortex condensation 143
II Time-dependent backgrounds of 2D string theory 165
IIIThermodynamics of 2D string theory 193
IV 2D String Theory as Normal Matrix Model 209
V Backgrounds of 2D string theory from matrix model 231
VI Non-perturbative e ects in matrix models and D-branes 255
vii
tel-00003512, version 1 - 9 Oct 2003CONTENTS
References 281
viii
tel-00003512, version 1 - 9 Oct 2003Introduction
This thesis is devoted to application of the matrix model approach to non-critical string
theory.
More than fteen years have passed since matrix models were rst applied to string theory.
Although they have not helped to solve critical string and superstring theory, they have
taught us many things about low-dimensional bosonic string theories. Matrix models have
provided so powerful technique that a lot of results which were obtained in this framework
are still inaccessible using the usual continuum approach. On the other hand, those results
that were reproduced turned out to be in the excellent agreement with the results obtained
by eld theoretical methods.
One of the main subjects of interest in the early years of the matrix model approach
was the c = 1 non-critical string theory which is equivalent to the two-dimensional critical
string theory in the linear dilaton background. This background is the simplest one for the
low-dimensional theories. It is at and the dilaton eld appearing in the low-energy target
space description is just proportional to one of the spacetime coordinates.
In the framework of the matrix approach this string theory is described in terms of Matrix
Quantum Mechanics (MQM). Already ten years ago MQM gave a complete solution of the
2D string theory. For example, the exact S-matrix of scattering processes was found and
many correlation functions were explicitly calculated.
However, the linear dilaton background is only one of the possible backgrounds of 2D
string theory. There are many other backgrounds including ones with a non-vanishing cur-
vature which contain a dilatonic black hole. It was a puzzle during long time how to describe
such backgrounds in terms of matrices. And only recently some progress was made in this
direction.
In this thesis we try to develop the matrix model description of 2D string theory in non-
trivial backgrounds. Our research covers several possibilities to deform the initial simple
target space. In particular, we analyze winding and tachyon perturbations. We show how
they are incorporated into Matrix Quantum Mechanics and study the result of their inclusion.
A remarkable feature of these perturbations is that they are exactly solvable. The reason
is that the perturbed theory is described by Toda Lattice integrable hierarchy. This is the
result obtained entirely within the matrix model framework. So far this integrability has
not been observed in the continuum approach. On the other hand, in MQM it appears quite
naturally being a generalization of the KP integrable structure of the c < 1 models. In
this thesis we extensively use the Toda description because it allows to obtain many exact
results.
We tried to make the thesis selfconsistent. Therefore, we give a long introduction into
the subject. We begin by brie y reviewing the main concepts of string theory. We introduce
1
tel-00003512, version 1 - 9 Oct 2003Introduction
the Polyakov action for a bosonic string, the notion of the Weyl invariance and the anomaly
associated with it. We show how the critical string theory emerges and explain how it is
generalized to superstring theory avoiding to write explicit formulae. We mention also the
modern view on superstrings which includes D-branes and dualities. After that we discuss
the low-energy limit of bosonic string theories and possible string backgrounds. A special
attention is paid to the linear dilaton background which appears in the discussion of non-
critical strings. Finally, we present in detail 2D string theory both in the linear dilaton and
perturbed backgrounds. We elucidate its degrees of freedom and how they can be used to
p the theory.
The next chapter is an introduction to matrix models. We explain what the matrix models
are and how they are related to various physical problems and to string theory, in particular.
The relation is established through the sum over discretized surfaces and such important
notions as the 1=N expansion and the double scaling limit are introduced. Then we consider
the two simplest examples, the one- and the two-matrix model. They are used to present two
of the several known methods to solve matrix models. First, the one-matrix model is solved
in the large N-limit by the saddle point approach. Second, it is shown how to obtain the
solution of the two-matrix model by the technique of orthogonal polynomials which works,
in contrast to the rst method, to all orders in perturbation theory. We nish this chapter
giving an introduction to Toda hierarchy. The emphasis is done on its Lax formalism. Since
the Toda integrable structure is the main tool of this thesis, the presentation is detailed and
may look too technical. But this will be compensated by the power of this approach.
The third chapter deals with a particular matrix model | Matrix Quantum Mechanics.
We show how it incorporates all features of 2D string theory. In particular, we identify
the tachyon modes with collective excitations of the singlet sector of MQM and the wind-
ing modes of the compacti ed string theory with degrees of freedom propagating in the
non-trivial representations of the SU(N) global symmetry of MQM. We explain the free
fermionictation of the singlet sector and present its explicit solution both in the
non-compacti ed and compacti ed cases. Its target space interpretation is elucidated with
the help of the Das{Jevicki collective eld theory.
Starting from the forth chapter, we turn to 2D string theory in non-trivial backgrounds
and try to describe it in terms of perturbations of Matrix Quantum Mechanics. First, the
winding perturbations of the compacti ed string theory are incorporated into the matrix
framework. We review the work of Kazakov, Kostov and Kutasov where this was rst
done. In particular, we identify the perturbed partition function with a -function of Toda
hierarchy showing that the introduced perturbations are integrable. The simplest case of
the windings of the minimal charge is interpreted as a matrix model for the string theory
in the black hole background. For this case we present explicit results for the free energy.
Relying on these description, we explain our rst work in this domain devoted to calculation
of winding correlators in the theory with the simplest winding perturbation. This work is
little bit technical. Therefore, we concentrate mainly on the conceptual issues.
The next chapter is about tachyon perturbations of 2D string theory in the MQM frame-
work. It consists from three parts representing our three works. In the rst one, we show
how the tachyon perturbations should be introduced. Similarly to the case of windings, we
nd that the p are integrable. In the quasiclassical limit we interpret them in
terms of the time-dependent Fermi sea of fermions of the singlet sector. The second work
2
tel-00003512, version 1 - 9 Oct 2003

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