Tachyons, Boundary Interactions and Genus Expansion in String Theory [thesis]

148 Pages·2003·0.887 MB·English

by  M. Laidlaw

Tachyons, Boundary Interactions, and the Genus
Expansion in String Theory
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Mark Colin Andrew Laidlaw
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B.Sc., The University of Victoria, 1998
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e M.Sc., The University of British Columbia, 2000
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4 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

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DOCTOR OF PHILOSOPHY
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The Faculty of Graduate Studies
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THE UNIVERSITY OF BRITISH COLUMBIA
28 July, 2003
c Mark Colin Andrew Laidlaw, 2003
(cid:13)
In presenting this thesis in partial fulfilment of the requirements for an
advanced degree at the University of British Columbia, I agree that the
Library shall make it freely available for reference and study. I further agree
thatpermissionforextensive copying ofthisthesisforscholarlypurposesmay
be granted by the head of my department or by his or her representatives.
It is understood that copying or publication of this thesis for financial gain
shall not be allowed without my written permission.
(Signature)
Department of Physics and Astronomy
The University Of British Columbia
Vancouver, Canada
Date
Abstract ii
Abstract
This thesis examines the interaction of both bosonic- and superstrings with
various backgrounds with a view to understanding the interplay between
tachyon condensation and world-sheet conformal invariance, and to under-
standing the d-branes that overlap with closed string modes. We briefly
review the development of both background independent string field the-
ory and cubic string field theory, as these provide insight into the problem of
tachyon condensation. We then develop the boundary state and show that in
backgrounds of interest to tachyon condensation the conformal invariance of
the string world-sheet is broken, which suggests a generalized boundary state
obtained by integrating over the conformal group of the disk. We find that
this prescription reproduces particle emission amplitudes calculated from the
string sigma model for both on- and off-shell boundary interactions. The
boundary state appears as a coherent superposition of closed string states,
and using this a method for calculating amplitudes beyond tree level is de-
veloped. The interaction of closed strings with other backgrounds is also
discussed. An extension of the boundary state to encode fields other than
a gauge or tachyon field is described. A modification of the boundary state
which encodes the time dependence of tachyon condensation is reviewed, and
an examination of spherically symmetric tachyon condensation in the 1/D
expansion is presented.
Contents iii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 Background Independent String Field Theory . . . . . . . . . 23
2.1.1 Bosonic String Case . . . . . . . . . . . . . . . . . . . 23
2.1.2 Superstring Case . . . . . . . . . . . . . . . . . . . . . 28
2.2 Cubic String Field Theory . . . . . . . . . . . . . . . . . . . . 31
3 Boundary States . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 The Bosonic Boundary State . . . . . . . . . . . . . . . . . . . 41
3.2.1 Conformal Transformation of Bosonic Boundary State . 44
3.2.2 Boundary State Single Particle Emission . . . . . . . . 46
Contents iv
3.2.3 Boundary State Multiple Particle Emission . . . . . . . 49
3.2.4 Bosonic Sigma Model . . . . . . . . . . . . . . . . . . . 52
3.2.5 Conformal Transformation in the Sigma Model . . . . . 54
3.2.6 Sigma Model Single Particle Emission . . . . . . . . . . 55
3.2.7 Sigma Model Multiple Particle Emission . . . . . . . . 58
3.2.8 Bosonic Boundary State Summary . . . . . . . . . . . 62
3.3 Bosonic Amplitudes in the Euler Number Expansion . . . . . 63
3.3.1 χ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 χ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.3 χ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 73
−
3.3.4 χ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 82
−
3.3.5 χ = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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3.3.6 Beyond the Born-Infeld Action . . . . . . . . . . . . . 87
3.4 Fermionic Boundary State . . . . . . . . . . . . . . . . . . . . 88
3.4.1 Particle Emission from Fermionic Boundary State . . . 90
3.4.2 Particle Emission in the Superstring Sigma Model . . . 91
3.4.3 Euler Number Expansion for Fermions . . . . . . . . . 94
3.5 Ghosts and Antighosts . . . . . . . . . . . . . . . . . . . . . . 94
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Generalized Boundary Interactions . . . . . . . . . . . . . . . 97
4.1 Additional Boundary State Fields . . . . . . . . . . . . . . . . 98
4.2 Time Dependent Tachyons . . . . . . . . . . . . . . . . . . . . 102
4.3 Spherically Symmetric Tachyon Condensation . . . . . . . . . 107
5 Conclusions and Future Directions . . . . . . . . . . . . . . . . 112
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Contents v
A Properties of the Conformal Transformation Matrices . . . 126
(a,b)
A.1 Bosonic Matrix M . . . . . . . . . . . . . . . . . . . . . . 126
mn
(a,b)
A.2 Fermionic Matrix N . . . . . . . . . . . . . . . . . . . . . 130
rm
B Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.1 Bosonic Tree level . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.2 Fermionic Tree level . . . . . . . . . . . . . . . . . . . . . . . 136
List of Tables vi
List of Tables
2.1 Level truncation in string field theory . . . . . . . . . . . . . . 35
3.1 Orientable Surfaces χ = 1 . . . . . . . . . . . . . . . . . . . 80
−
3.2 Non-Orientable Surfaces χ = 1 . . . . . . . . . . . . . . . . . 81
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List of Figures vii
List of Figures
1.1 Open string theory interaction . . . . . . . . . . . . . . . . . . 7
1.2 String Theory Modulus Space . . . . . . . . . . . . . . . . . . 15
2.1 Tachyon Condensation . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Cubic String Field theory integration . . . . . . . . . . . . . . 37
3.1 A schematic of the disk tadpole (a) and the emission of one particle by the boundary state
3.2 The Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Surfaces with χ = 1 . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Orientable surfaces with χ = 0 . . . . . . . . . . . . . . . . . . 65
3.5 Non-orientable surfaces with χ = 0 . . . . . . . . . . . . . . . 66
3.6 Orientable surfaces with χ = 1 . . . . . . . . . . . . . . . . . 74
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3.7 Non-orientable surfaces with χ = 1 . . . . . . . . . . . . . . 75
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3.8 Orientable surfaces with χ = 2 . . . . . . . . . . . . . . . . . 82
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3.9 Non-orientable surfaces with χ = 2 . . . . . . . . . . . . . . 83
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Preface viii
Preface
This work investigates a number of aspects of the interplay between the
conformal invariance in string theory and interaction terms confined to the
boundaries of the string world-sheet. A brief synopsis of some of the theoret-
ical basis for the work is presented in chapter 2, while chapters 3 and 4 have
sections of extensive overlap with, respectively, [4, 69, 70] and [54], works on
which the author collaborated.
Acknowledgements ix
Acknowledgements
This thesis would not have been possible without the advice and tutelage of
my supervisor, Professor Semenoff, and the members of my PhD. committee,
Professors McKenna, Rozali, Schleich, and Zhitnitsky. I am very grateful to
them, and also my fellow graduate students, who have answered more than
their share of questions, bothconceptual andtechnical, while I was preparing
this. Finally, I am exceptionally grateful for the unwavering support and
encouragement of my wife, Susan, to whom I dedicate this work.