Joost Hoogeveen University of Amsterdam. Faculty of Science Pure

180 Pages·2010·1.68 MB·English

AUP-Hoogeveen:AUP/Buijn 31-05-2010 15:42 Pagina 1
UvA Dissertation J Fundamentals of the
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H Pure Spinor Formalism
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Faculty of Science e
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Joost Hoogeveen
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Joost Hoogeveen (1984) studied physics and mathematics, initially at the University l
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of Amsterdam. In 2004 he continued these studies at the University of Cambridge.
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In 2005 he started his PhD research at the Institute of Theoretical Physics of the h
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University of Amsterdam. P
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This thesis presents recent developments within the pure spinor formalism, which o
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has simplified amplitude computations in perturbative string theory, especially when m
a
spacetime fermions are involved. Firstly the worldsheet action of both the minimal li
s
m
and the non-minimal pure spinor formalism is derived from first principles, i.e. from
an action with two dimensional diffeomorphism and Weyl invariance. Secondly the
decoupling of unphysical states in the minimal pure spinor formalism is proved.
9 789056 296414
Fundamentals of the Pure
Spinor Formalism
This workhas been accomplishedatthe Institute for TheoreticalPhysics(ITFA) of
theUniversityofAmsterdam(UvA)andisfinanciallysupportedbyaSpinozagrant
of the Netherlands Organisationfor Scientific Research (NWO).
Lay out: Joost Hoogeveen, typeset using LATEX
Cover design: Ren´e Staelenberg, Amsterdam
Cover illustration: Wayne Ruston
ISBN 978 90 5629 641 4
NUR 910
c J. Hoogeveen / Vossiuspers UvA - Amsterdam University Press, 2010
(cid:13)
All rights reserved. Without limiting the rights under copyright reserved above, no
partofthisbookmaybereproduced,storedinorintroducedintoaretrievalsystem,
or transmitted, in any formor by any means (electronic, mechanical,photocopying,
recordingor otherwise)without the written permission of both the copyrightowner
and the author of the book.
Fundamentals of the Pure
Spinor Formalism
Academisch Proefschrift
ter verkrijging van de graad van doctor
aan de Universiteit van Amsterdam
op gezag van de Rector Magnificus
prof. dr. D. C. van den Boom
ten overstaanvan een door het college voor promoties ingestelde
commissie, in het openbaar te verdedigen in de
Agnietenkapel
op vrijdag 9 juli 2010, te 12:00 uur
door
Joost Hoogeveen
geboren te Leidschendam
Promotiecommissie
Promotor
prof. dr. R. H. Dijkgraaf
Co-Promotor
dr. K. Skenderis
Overige leden
prof. dr. N. J. Berkovits
prof. dr. J. de Boer
prof. dr. P. van Nieuwenhuizen
dr. M. M. Taylor
prof. dr. E. P. Verlinde
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Publications
This thesis is based on the following publications:
1. Joost Hoogeveen and Kostas Skenderis,
BRST quantization of the pure spinor superstring,
JHEP 0711 (2007) 081 [arXiv:0710.2598].
2. Joost Hoogeveen and Kostas Skenderis,
Decoupling of unphysical states in the minimal pure spinor formalism I,
JHEP 1001 (2010) 041 [arXiv:0906.3368].
3. Nathan Berkovits, Joost Hoogeveen, Kostas Skenderis,
Decoupling of unphysical states in the minimal pure spinor formalism II,
JHEP 0909 (2009) 035 [arXiv:0906.3371].
v
vi
Contents
Preface xi
1 String theory 1
1.1 Motivations for supersymmetric string theory . . . . . . . . . . . . . 1
1.2 Bosonic string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 BRST quantisation of Polyakovaction . . . . . . . . . . . . . 14
1.2.2 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.3 Curved backgrounds . . . . . . . . . . . . . . . . . . . . . . . 28
1.3 RNS formalism and its limitations . . . . . . . . . . . . . . . . . . . 31
1.3.1 Tree-level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.2 Higher genus and limitations of RNS . . . . . . . . . . . . . . 41
2 Pure spinor formalism 45
2.1 Minimal pure spinor formalism . . . . . . . . . . . . . . . . . . . . . 46
2.1.1 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.1.2 Tree-level prescription . . . . . . . . . . . . . . . . . . . . . . 49
2.1.3 One and higher-loop prescription . . . . . . . . . . . . . . . . 52
2.1.4 Decoupling of Q exact states and PCO positions . . . . . . 55
S
2.2 Non-minimal pure spinor formalism. . . . . . . . . . . . . . . . . . . 56
2.3 Results from the pure spinor formalism . . . . . . . . . . . . . . . . 59
3 Basic techniques 63
3.1 Invariant tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Clifford algebra and pure spinors . . . . . . . . . . . . . . . . . . . . 65
3.2.1 The SU(N) subgroup of SO(2N) . . . . . . . . . . . . . . . . 66
3.2.2 Charge conservation and tensor products . . . . . . . . . . . 70
3.2.3 Dynkin labels and gamma matrix traceless tensors . . . . . . 71
3.2.4 Explicit expression for gamma matrices . . . . . . . . . . . . 72
3.2.5 Pure spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Pure spinor Lorentz generators . . . . . . . . . . . . . . . . . . . . . 73
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Contents
3.3.1 Lorentz current OPE’s with unconstrained spinors . . . . . . 74
3.3.2 Gauge fixing w invariance . . . . . . . . . . . . . . . . . . . 75
α
3.3.3 Currents containing pure spinors . . . . . . . . . . . . . . . . 77
3.4 Lorentz invariant measures . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Measure for the zero modes of λ . . . . . . . . . . . . . . . . 80
3.4.2 Measure for the zero modes of Nmn . . . . . . . . . . . . . . 80
3.5 Gamma matrix traceless projectors . . . . . . . . . . . . . . . . . . . 81
3.5.1 Arbitrary rank . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6 Chain of operators for b ghost . . . . . . . . . . . . . . . . . . . . . . 83
4 BRST quantisation of the pure spinor superstring 85
4.1 Coupling to 2d gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.1 Topological gravity and Q invariance . . . . . . . . . . . . . 88
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4.2 Adding vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 BRST quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4 Pure spinor measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.1 Minimal formulation . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.2 Non-minimal formulation . . . . . . . . . . . . . . . . . . . . 99
4.5 Ghost contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.6 Replacing worldsheet fields by zero modes . . . . . . . . . . . . . . . 102
5 Decoupling of unphysical states 105
5.1 Tree level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.1 No C integration . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1.2 Including C integration . . . . . . . . . . . . . . . . . . . . . 109
5.1.3 Global issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 One loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.1 No B integration . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.2 Including B integration . . . . . . . . . . . . . . . . . . . . . 122
5.2.3 Non-vanishing of the trace of (ǫTR) . . . . . . . . . . . . . . 124
5.3 No-go theorem for Q closed, Lorentz invariant PCOs . . . . . . . . 126
S
5.4 Proof of decoupling of unphysical states . . . . . . . . . . . . . . . . 130
5.4.1 Tree-level amplitudes. . . . . . . . . . . . . . . . . . . . . . . 130
5.4.2 Higher-loop amplitudes . . . . . . . . . . . . . . . . . . . . . 133
5.5 Origin of the problems . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5.1 Minimal formalism . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5.2 Non-minimal formalism . . . . . . . . . . . . . . . . . . . . . 138
5.5.3 Toy example . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5.4 Singular gauge and possible resolution . . . . . . . . . . . . . 141
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Contents
6 Discussion and conclusion 143
A Detailed computations of I 147
k
A.1 Coefficients in λ integrals . . . . . . . . . . . . . . . . . . . . . . . . 147
A.1.1 Proof of equations (A.5) and (A.6) . . . . . . . . . . . . . . . 148
A.2 Computing the I ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
k
Bibliography 155
Samenvatting 161
Acknowledgments 167
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