Matrix Quantum Mechanics and 2-D ...

Matrix Quantum Mechanics and 2-D String Theory [thesis] - S. Alexandrov, Angielskie techniczne

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Service de Physique Theorique – C.E.A.-Saclay
UNIVERSITE PARIS XI
PhD Thesis
Matrix Quantum Mechanics and
Two-dimensional String Theory
in Non-trivial Backgrounds
Sergei Alexandrov
Abstract
String theory is the most promising candidate for the theory unifying all interactions
including gravity. It has an extremely di
cult dynamics. Therefore, it is useful to study
some its simplications. One of them is non-critical string theory which can be dened in
low dimensions. A particular interesting case is 2D string theory. On the one hand, it has a
very rich structure and, on the other hand, it is solvable. A complete solution of 2D string
theory in the simplest linear dilaton background was obtained using its representation as
Matrix Quantum Mechanics. This matrix model provides a very powerful technique and
reveals the integrability hidden in the usual CFT formulation.
This thesis extends the matrix model description of 2D string theory to non-trivial back-
grounds. We show how perturbations changing the background are incorporated into Matrix
Quantum Mechanics. The perturbations are integrable and governed by Toda Lattice hier-
archy. This integrability is used to extract various information about the perturbed system:
correlation functions, thermodynamical behaviour, structure of the target space. The results
concerning these and some other issues, like non-perturbative eects in non-critical string
theory, are presented in the thesis.
Acknowledgements
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 scientic advisers, 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.
Contents
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
String 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 σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2
Weyl invariance and eective 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
Compactication, 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 . . . . . . . . . . . . . . . . . . . 27
6.3
FZZ conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
II Matrix models
31
1
Matrix models in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2
Matrix models and random surfaces . . . . . . . . . . . . . . . . . . . . . . . 33
2.1
Denition of one-matrix model . . . . . . . . . . . . . . . . . . . . . 33
2.2
Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3
Discretized surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4
Topological expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5
Continuum and double scaling limits . . . . . . . . . . . . . . . . . . 38
v
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