Strings in plane wave backgrounds [Elektronische Ressource] / von Ari Pankiewicz

Ari Pankiewicz

Sujets

• Physik
• Astronomie

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2003
• Nombre de lectures: 17
• Langue: English

Extrait

Strings in plane wave backgrounds
D I S S E R T A T I O N
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
im Fach Physik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultat¨ I
Humboldt-Universit¨at zu Berlin
von
Dipl.-Phys. Ari Pankiewicz
geboren am 02.11.1974 in Heidelberg
Pr¨asident der Humboldt-Universit¨at zu Berlin:
Prof. Dr. Jurg¨ en Mlynek
Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at I:
Prof. Dr. Michael Linscheid
Gutachter:
1. Prof. Dr. Stefan Theisen
2. Prof. Dr. Albrecht Klemm
3. Dr. habil. Jan Plefka
eingereicht am: 03. April 2003
Tag der mundlic¨ hen Prufung:¨ 13. Juni 2003Abstract
The interplay between string and gauge theory has led to many new
insights in recent years. The most prominent example is the AdS/CFT
correspondence, a duality between string theory on Anti-de Sitter (AdS)
spaces and conformal gauge theories defined on their boundary. The study
of string theory on plane wave backgrounds, which are connected to AdS by
the Penrose limit, opens up the possibility of testing this duality beyond the
low-energy supergravity approximation. Generalized plane wave geometries
are interesting in themselves, as they provide a large class of exact classical
space-time backgrounds for string theory.
Inthisthesisaspectsofstringtheoryonplanewavebackgroundsarestud-
ied, with an emphasis on the connection to gauge theory. String interactions
in the plane wave space-time with maximal supersymmetry are investigated
intheframeworkoflight-conestringfieldtheory. Intheprocess,manyresults
that had been found for the case of flat Minkowski space-time are general-
ized to the more complex plane wave background. The leading non-planar
corrections to the anomalous dimensions of gauge theory operators dual to
string states are recovered within light-cone string field theory.
Keywords: String theory, AdS/CFT correspondence, Penrose limit and
pp-wave background, Light-cone string field theoryZusammenfassung
Das Wechselspiel zwischen String- und Eichtheorien hat in den letzten
Jahren zu vielen neuen Einsichten gefuhrt. Das herausragendste Beispiel ist¨
diesogenannteAdS/CFTKorrespondenz,eineDualitat¨ zwischenStringtheo-
rienaufAnti-deSitter-Raumen(AdS)undkonformenEichtheorienaufderen¨
Rand. Die Untersuchung von Stringtheorie auf ebenfrontigen Gravitations-
wellen, die sich im sogenannten Penrose-Limes aus AdS-Raumzeiten gewin-
nen lassen, erlaubt es, diese Dualitat¨ ub¨ er die niederenergetische Supergravi-
tationsnaherung hinausgehend zu uberprufen. Verallgemeinerte ebenfrontige¨ ¨ ¨
Gravitationswellen sind auch fur¨ sich gesehen interessant, da sie eine grosse
KlassevonRaumzeitenbilden,dieexakteklassischeL¨osungenderStringtheo-
rie sind.
IndieserArbeitwerdenAspektederStringtheorieaufebenfrontigenGra-
vitationswellen untersucht. Besonderes Interesse gilt dabei der Verbindung
dieser Stringtheorien zu Eichtheorien. Wechselwirkungen von Strings in der-
jenigenGravitationswellen-RaumzeitmitmaximalerSupersymmetriewerden
im Rahmen der Lichtkegel-Stringfeldtheorie behandelt. Viele Ergebnisse, die
fur den Fall der flachen Minkowski-Raumzeit bekannt sind, werden dabei¨
vollstandig¨ aufdiekomplizierterenebenfrontigenGravitationswellenverallge-
meinert.Diefuhrenden¨ nicht-planarenKorrekturenzudenanomalenDimen-
sionen von Operatoren in der Eichtheorie, die eine duale Beschreibung von
Stringzust¨andenliefern,werdeninnerhalbderLichtkegel-Stringfeldtheoriere-
produziert.
Schlagwor¨ ter: Stringtheorie, AdS/CFT Korrespondenz, Penrose-Limes
und ebenfrontige Gravitationswellen, Lichtkegel-StringfeldtheorieContents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Strings on the plane wave from gauge theory 5
2.1 pp-waves in supergravity . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Penrose-Guv¨ en limit . . . . . . . . . . . . . . . . . . . . . 10
2.3 The BMN correspondence . . . . . . . . . . . . . . . . . . . . 15
3 Extensions of the BMN duality 26
3.1 Various approaches . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Strings on orbifolded plane waves from quiver gauge theory . . 30
3.2.1 IIB superstring on plane wave orbifold . . . . . . . . . 30
3.2.2 Operator analysis inN = 2 quiver gauge theory . . . . 33
3.3 Further directions . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 D-branes on the plane wave . . . . . . . . . . . . . . . 37
3.3.2 Strings on pp-waves and interacting field theories . . . 41
4 String interactions in the plane wave background 44
4.1 Review of free string theory on the plane wave . . . . . . . . . 47
4.2 Principles of light-cone string field theory . . . . . . . . . . . . 51
4.3 The kinematical part of the vertex . . . . . . . . . . . . . . . 55
4.4 The completeO(g ) superstring vertex . . . . . . . . . . . . . 60s
4.4.1 The bosonic constituents of the prefactors . . . . . . . 61
4.4.2 The fermionic constituents of the prefactors . . . . . . 63
4.4.3 The dynamical generators at orderO(g ) . . . . . . . . 64s
4.5 Functional expressions . . . . . . . . . . . . . . . . . . . . . . 68
4.6 Anomalous dimension from string theory . . . . . . . . . . . . 71
iii4.6.1 Contribution of one-loop diagrams . . . . . . . . . . . 73
4.6.2 Contr of contact terms . . . . . . . . . . . . . . 75
5 Summary and outlook 76
A The kinematical part of the vertex 79
A.1 The Delta-functional . . . . . . . . . . . . . . . . . . . . . . . 79
A.2 Structure of the bosonic Neumann matrices . . . . . . . . . . 81
A.3 The kinematical constraints atO(g ) . . . . . . . . . . . . . . 82s
A.3.1 The bosonic part . . . . . . . . . . . . . . . . . . . . . 82
A.3.2 The fermionic part . . . . . . . . . . . . . . . . . . . . 84
A.4 Neumann matrices at leading order . . . . . . . . . . . . . . . 85
B The dynamical constraints 88
B.1 More detailed computations . . . . . . . . . . . . . . . . . . . 88
B.2 Proof of the dynamical constraints . . . . . . . . . . . . . . . 90
eB.3 {Q,Q} at orderO(g ) . . . . . . . . . . . . . . . . . . . . . . 92s
Bibliography 95
ivChapter 1
Introduction
1.1 Motivation
The intimate connection between string and gauge theories has been one of
thedominantthemesintheoreticalhighenergyphysicsoverthelastyears. A
famousexampleistheequivalence(duality)ofstringtheoryonAnti-deSitter
spaces with conformal field theories, the AdS/CFT correspondence [1, 2, 3],
see [4] for a review.
Severalargumentssupporttheexpectationofadualitybetweenstringand
gauge theories or, even more generally, gravitational and non-gravitational
theories. For example, a qualitative one comes from the fact that QCD, the
SU(3)gaugetheoryofstronginteractions,confineschromoelectricfluxtoflux
tubes – the QCD string – at low energies. After all, this is how string theory
was originally discovered. A quantitative argument is ’t Hooft’s analysis
of the large N limit of SU(N) gauge theories [5]. ’t Hooft showed that
2for large N and fixed ’t Hooft coupling λ = g N, the Feynman diagramYM
expansioncanberearrangedaccordingtothegenusg oftheRiemannsurface
which the diagram can be drawn on and every amplitude can be written in
P∞ 2−2g 2an expansion of the form N f (λ), i.e. 1/N is the effective genusgg=0
counting parameter. This is like the perturbation series of a string theory,
where the string coupling g is identified with 1/N and λ corresponds tos
the loop-counting parameter of the string non-linear σ-model. This a very
general argument for the largeN duality between gauge theories and certain
string theories, but it does not give an answer to what kind of string theory
one should look for.
Further hints come from the study of black holes. The simplest exam-
12
ple is the Schwarzschild solution of general relativity depending on a single
parameter, the mass M of the black hole. They have a horizon and are
black classically, everything crossing the horizon is inevitably pulled into the
black hole singularity. However, semi-classical analysis shows that due to
quantum processes black holes start to emit Hawking radiation: the emis-
sion spectrum is roughly that of a blackbody with temperature T ∼ 1/M;
the deviation of the pure blackbody spectrum is encoded in the so called
‘greybody factor’. As radiating systems black holes are expected to obey the
laws of thermodynamics. If one defines the black hole entropy, as first pro-
1 2posed by Bekenstein and Hawking byS = A∼M ,A the area of the black
4
hole horizon, these laws are in fact satisfied. A quantum theory of gravity
should e.g. provide the framework for a microscopic derivation of the black
hole entropy via a counting of states and predict its greybody factor. As the
Bekenstein-Hawking entropy involves the area instead of the volume, as is
the case for statistical mechanics and local quantum field theories, one may
wonder if one can find a holographic description in terms of local quantum
field theories ‘living’ on the horizon, such that S ∼ A. More generally,QFT
the holographic principle [6, 7] asserts that the number of degrees of freedom
of quantum gravity on some manifold scales as the ar