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On supertwistor geometry and integrability in super gauge theory [Elektronische Ressource] / von Martin Wolf

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Martin WolfOnSupertwistor GeometryAnd IntegrabilityIn Super Gauge TheoryInstitut fu¨r Theoretische PhysikFakult¨at fu¨r Mathematik und PhysikLeibniz Universit¨at HannoverCopyrightc 2006 by Martin Wolf. All rights reserved.ITP–UH–18/06Institut fu¨r Theoretische PhysikGottfried Wilhelm Leibniz Universit¨at HannoverAppelstraße 2, 30167 Hannover, Germanywolf@itp.uni-hannover.deTo My FamilyOnSupertwistor GeometryAnd Integrability In Super Gauge Theory¨ ¨Von der Fakultat fur Mathematik und Physik¨der Gottfried Wilhelm Leibniz Universitat HannoverZur Erlangung des GradesDoktor der NaturwissenschaftenDr. rer. nat.genehmigte DissertationvonDipl.-Phys. Martin Wolfgeboren am 19. Februar 1979 in Zwickau (Sachsen)Hannover 2006Tag der Promotion: 21. Juli 2006Betreuer: Prof. Dr. Olaf Lechtenfeld und Dr. Alexander D. PopovReferent: Prof. Dr. Olaf LechtenfeldKorreferent: Prof. Dr. Holger FrahmboreTaduriRt;araRerejioRworia;ipedohiR.welloRCelebriaowino.tiwtRarviThiAaRenyReSaAcknowledgmentsThis work could not have been carried out without the support of many people. It is thereforea pleasure to acknowledge their help.First of all, I would like to thank Olaf Lechtenfeld for allowing me to be a member of hisresearch group and for giving me the opportunity to do my Ph.D. studies here in Hannover.

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Published 01 January 2006
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Martin Wolf
On
Supertwistor Geometry
And Integrability
In Super Gauge Theory
Institut fu¨r Theoretische Physik
Fakult¨at fu¨r Mathematik und Physik
Leibniz Universit¨at HannoverCopyrightc 2006 by Martin Wolf. All rights reserved.
ITP–UH–18/06
Institut fu¨r Theoretische Physik
Gottfried Wilhelm Leibniz Universit¨at Hannover
Appelstraße 2, 30167 Hannover, Germany
wolf@itp.uni-hannover.deTo My FamilyOn
Supertwistor Geometry
And Integrability In Super Gauge Theory
¨ ¨Von der Fakultat fur Mathematik und Physik
¨der Gottfried Wilhelm Leibniz Universitat Hannover
Zur Erlangung des Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
von
Dipl.-Phys. Martin Wolf
geboren am 19. Februar 1979 in Zwickau (Sachsen)
Hannover 2006Tag der Promotion: 21. Juli 2006
Betreuer: Prof. Dr. Olaf Lechtenfeld und Dr. Alexander D. Popov
Referent: Prof. Dr. Olaf Lechtenfeld
Korreferent: Prof. Dr. Holger Frahmo
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iA
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araR
Celebri
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T
welloR
t;
a
bor
wino.
erejioR
iw
eTa

Rarvi
i
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eSa
Acknowledgments
This work could not have been carried out without the support of many people. It is therefore
a pleasure to acknowledge their help.
First of all, I would like to thank Olaf Lechtenfeld for allowing me to be a member of his
research group and for giving me the opportunity to do my Ph.D. studies here in Hannover. I
am grateful to him for his help with advice and expertise, and for numerous helpful discussions.
IamalsodeeplyindebtedtoAlexanderPopovwhowasalways abrilliantadvisor. Iwouldliketo
expressmygratitudetowardshimforpatientlyteachingmealotofmathematicsandphysics,for
ourcountless conversations on various topics which crossed the way, and foralways helping with
advice on any sort of problem. He has always encouraged my interest in mathematical physics
since the time I came to Hannover. Furthermore, I would like to thank Alexander Popov and
Christian S¨amann for very enjoyable collaborations.
I am also grateful to Holger Frahm for sparing his invaluable time reviewing the manuscript
of my thesis, and to Elmar Schrohe for kindlyagreeing to bethe head of the examination board.
In addition, I am thankful to the following people for their help and for numerous discussions:
Chong-SunChu,LouiseDolan,NorbertDragon,KlausHulek,AlexanderKling,NiallJ.MacKay,
Stefan Petersen, Ronen Plesser, Elmar Schrohe, Christian S¨amann, Richard Szabo, Sebastian
Uhlmann, Kirsten Vogeler and Robert Wimmer.
Furthermore, I would like to thank a number of wunderful people for having provided such
an enjoyable time here in Hannover. Therefore, many thanks to all of them: Hendrik Adorf,
Alexandra De Castro, Henning Fehrmann, Carsten Grabow, Matthias Ihl, Alexander Kling,
Michael Klawunn, Michael K¨ohn, Marco Krohn, Carsten Luckmann, Andreas Osterloh, Klaus
Osterloh, Guillaume Palacios, Leonardo Quevedo, Christian S¨amann, Sebastian Uhlmann and
CarstenvonZobeltitz. Inparticular,IhavetothankStefanPetersen,KirstenVogelerandRobert
Wimmer for a wonderful friendship. They made it an unforgetable time here in Hannover. In
addition, our countless discussions, their comments, suggestions and ideas were always a great
source of inspiration to me. Furthermore, I am also very grateful to Ingmar Glauche and Cindy
Rockstroh for always helping and supporting me. Thank you!
I want to express my deepest gratitude towards those this thesis has been dedicated to, i.e.,
to my parents Bettina and Hans-Peter Wolf and to my brother Henrik Wolf. I would not have
been in the position to carry out this work at all without their support and encouragement.
Finally, I would like to thank the Deutsche Forschungsgemeinschaft (DFG) for continued
support of my graduate work through the Graduiertenkolleg No. 282 entitled Quantenfeldtheo-
retische Methoden in der Teilchenphysik, Gravitation, Statistischen Physik und Quantenoptik.Zusammenfassung
In der vorliegenden Arbeit werden im Rahmen des Twistorzugangs verschiedene Aspekte der
Integrabilit¨at supersymmetrischer Eichtheorien betrachtet. Nach einer kurzen Darstellung
der Grundlagen der Twistortheorie im ersten Kapitel, untersuchen wir zun¨achst selbstduale
supersymmetrische Yang-Mills-Theorie (SYM). Insbesondere wird deren Twistorformulierung
erl¨autert und ausgehend von dieser, werden weitere selbstduale Modelle vorgestellt. Jene folgen
durch geeignete Reduktion aus selbstdualer SYM-Theorie und k¨onnen zum Teil im Rahmen
topologischer Feldtheorien verstanden werden. Fu¨r deren Twistorformulierung schlagen wir be-
stimmte gewichtete projektive Superr¨aume als Twistorra¨ume vor. Interessanterweise sind diese
Calabi-Yau-Supermannigfaltigkeiten, so daß es ebenfalls m¨oglich ist, geeignete Wirkungsprinzi-
pien zu formulieren. Im dritten Kapitel dieser Arbeit befassen wir uns mit der Twistorformu-
lierung eines supersymmetrischen Bogomolny-Modells in drei Raumzeitdimensionen. Die nicht-
supersymmetrische Variante dieses Modells beschreibt statische Yang-Mills-Higgs-Monopole
im Prasad-Sommerfield-Limes. Insbesondere wird eine supersymmetrische Erweiterung des
Minitwistorraumes betrachtet und, folgend aus einem der Grundprinzipien der Twistortheorie,
betten wir diesen in eine konkrete Doppelfaserung ein. Diese Methode hat den Vorteil der
M¨oglichkeit der Formulierung eines Chern-Simons a¨hnlichen Wirkungsfunktionals, welches fast
holomorphe Vektorbu¨ndel u¨ber dem entsprechenden Korrespondenzraum beschreibt. Letzterer
kann mit einer Cauchy-Riemann-Struktur ausgestattet werden. Weiterhin formulieren wir
holomorphe BF-Theorie auf dem Minisupertwistorraum und beweisen, daß die Modulr¨aume
dieser drei Theorien bijektiv zueinander sind. Im Anschluß werden bestimmte Deformationen
der komplexen Struktur des Minisupertwistorraumes betrachtet und ein daraus resultierendes
supersymmetrisches Bogomolny-Modell mit massiven Feldern konstruiert. Im vierten Kapitel
wird dann die Twistorformulierung von SYM-Theorien vorgestellt. Im letzten Kapitel vertiefen
wir die Untersuchung selbstdualer SYM-Theorien bezu¨glich ihrer (klassischen) Integrabilit¨at.
Insbesondere wird die Twistorkonstruktion unendlich dimensionaler Algebren nichtlokaler Sym-
metrien behandelt. Diese sind zum einen affine Erweiterungen globaler und zum anderen affine
Erweiterungen superkonformer Symmetrien. Im weiteren betrachten wir die Konstruktion von
self-dualer SYM-Hierarchien. Jene beschreiben unendlich viele Flu¨sse auf den entsprechenden
L¨osungsra¨umen, wobei die niedrigsten Raumzeittranslationen darstellen. Dies bedeutet, daß
eine gegebene L¨osung zu den Feldgleichungen in eine unendlich dimensionale Familie neuer
L¨osungen eingebettet werden kann. Weiterhin werden unendlich viele erhaltene nichtlokale
Str¨ome konstruiert.
Schlagworte: Supertwistorgeometrie, Supersymmetrische Eichtheorien, Integrabilit¨atAbstract
In this thesis, we report on different aspects of integrability in supersymmetric gauge theories.
Our main tool of investigation is supertwistor geometry. In the first chapter, we briefly review
the basics of twistor geometry. Afterwards, we discuss self-dual super Yang-Mills (SYM) theory
and some of its relatives. In particular, a detailed twistor description of self-dual SYM theory
is presented. Furthermore, we introduce certain self-dual models which are, in fact, obtainable
from self-dual SYM theory by a suitable reduction. Some of them can be interpreted within the
context of topological field theories. To provide a twistor description of these models, we pro-
pose weighted projective superspaces as twistor spaces. These spaces turn out to be Calabi-Yau
supermanifolds. Therefore, it is possible to write down appropriate action principles, as well. In
chapterthree, wethendealwiththetwistorformulation ofacertain supersymmetricBogomolny
model in three space-time dimensions. The nonsupersymmetric version of this model describes
static Yang-Mills-Higgs monopolesinthePrasad-Sommerfieldlimit. Inparticular, weconsidera
supersymmetric extension of mini-twistor space. This space is in turn a part of a certain double
fibration. It is then possible to formulate a Chern-Simons type theory on the correspondence
spaceofthisfibration. Asweexplain,thistheorydescribespartiallyholomorphicvectorbundles.
It should be noticed that the correspondence space can be equipped with a Cauchy-Riemann
structure. Moreover, we formulate holomorphic BF theory on mini-supertwistor space. We then
prove that the moduli spaces of all three theories are bijective. In addition, complex struc-
ture deformations on mini-supertwistor space are investigated eventually resulting in a twistor
correspondence involving a supersymmetric Bogomolny model with massive fields. In chapter
four, we review the twistor formulation of non-self-dual SYM theories. Theremaining chapter is
devoted to a more detailed investigation of (classical) integrability of self-dual SYM theories. In
particular, we explain the twistor construction of infinite-dimensional algebras of hidden sym-
metries. Our discussion is exemplified by deriving affine extensions of internal and space-time
symmetries. Furthermore, we construct self-dual SYM hierarchies and their truncated versions.
These hierarchies describean infinitenumberof flows on the respective solution space. Thelow-
est level flows are space-time translations. The existence of such hierarchies allows us to embed
a given solution to the equations of motion of self-dual SYM theory into an infinite-parameter
family of new solutions. The dependence of the self-dual SYM fields on the additional moduli
can berecovered bysolvingtheequations ofthehierarchy. We in addition deriveinfinitelymany
nonlocal conservation laws.
Keywords: Supertwistor Geometry, Supersymmetric Gauge Theories, IntegrabilityMotivation and introduction
Twistor geometry
In 1967, Penrose [187] introduced the foundations of twistor geometry. The corner stone
of twistor geometry is the substitution of space-time as background for physical pro-
cesses by some new background manifold – called twistor space, and furthermore the
reinterpretation of physical theories on this new space. As originally proposed, twistor
space, or rather the projectivization thereof, which is associated with complexified four-
3dimensional Minkowski space, is the complex projective space P . Geometrically, this
space parametrizes all isotropic two-planes in complexified Minkowski space. Upon this
correspondence, it is possible toreinterpret solutions to zero rest mass free field equations
on Minkowski space in terms of certain cohomology groups on twistor space [187]–[190].
Forinstance, ifU issomeopensubset incomplexified Minkowski spaceandifZ denotes±h
the sheaf of solutions to the helicity±h zero rest mass field equations, then there is an
1 ′ 0isomorphism between the two cohomology groupsH (U ,O ′(∓2h−2)) andH (U,Z ),U ±h
′where U is an appropriate region on twistor space related to U. Here,O ′(∓2h−2) isU
′the sheaf of sections of a certain holomorphic line bundle (over U ) having first Chern
class∓2h−2. The map between representatives of these cohomology groups has been
termed Penrose transform. Putting it differently, any solution to zero rest mass free field
equations can be represented by certain holomorphic “functions” on twistor space, which
are “free” of differential constraints. For a detailed discussion of this correspondence in
ˇterms of Cech cohomology, we refer the reader to [90]. Let us mention in passing that,
as was shown by Eastwood in [91], it is also possible to generalize this description to the
case of massive free fields.
Besides giving insights into the geometric nature of solutions to linear field equa-
tions, the ideas of twistor geometry turned out to be extremely powerful for studying
various nonlinear classical field equations. Twistor methods have successfully been ap-
plied to subclasses of the Einstein’s equations of General Relativity and of the (super)
Yang-Mills equations, to name just the most prominent ones. Indeed, it is possible
to associate with any self-dual oriented Riemannian four-dimensional manifold X, that
is, a manifold with self-dual Weyl tensor, a complex three-dimensional twistor space
C