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Aspects of Twistor Geometry and Supersymmetric Field Theories Within Superstring Theory

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Aspects of Twistor Geometry and Supersymmetric Field Theories Within Superstring Theory Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory Von der FakultÄatfÄurMathematik und Physik der UniversitÄatHannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Christian SÄamann geboren am 23. April 1977 in Fulda For there is nothing hidden, except that it should be made known; neither was anything made secret, but that it should come to light. Mark 4,22 \Wir mÄussenwissen, wir werden wissen." David Hilbert To those who taught me Betreuer: Prof. Dr. Olaf Lechtenfeld und Dr. Alexander D. Popov Referent: Prof. Dr. Olaf Lechtenfeld Korreferent: Prof. Dr. Holger Frahm Tag der Promotion: 30.01.2006 Schlagworte: Nichtantikommutative Feldtheorie, Twistorgeometrie, Stringtheorie Keywords: Non-Anticommutative Field Theory, Twistor Geometry, String Theory Zusammenfassung Die Resultate, die in dieser Arbeit vorgestellt werden, lassen sich im Wesentlichen zwei Forschungsrichtungen in der Stringtheorie zuordnen: Nichtantikommutative Feldtheorie sowie Twistorstringtheorie. Nichtantikommutative Deformationen von SuperrÄaumenentstehen auf natÄurliche Wei- se bei Typ II Superstringtheorie in einem nichttrivialen Graviphoton-Hintergrund, und solchen Deformationen wurde in den letzten zwei Jahren viel Beachtung geschenkt. Zu- nÄachst konzentrieren wir uns auf die De¯nition der nichtantikommutativen Deformation von N = 4 super Yang-Mills-Theorie. Da es fÄurdie Wirkung dieser Theorie keine Super- raumformulierung gibt, weichen wir statt dessen auf die Äaquivalenten constraint equations aus. WÄahrendder Herleitung der deformierten Feldgleichungen schlagen wir ein nichtan- tikommutatives Analogon zu der Seiberg-Witten-Abbildung vor. Eine nachteilige Eigenschaft nichantikommutativer Deformationen ist, dass sie Super- symmetrie teilweise brechen (in den einfachsten FÄallenhalbieren sie die Zahl der erhal- tenen Superladungen). Wir stellen in dieser Arbeit eine sog. Drinfeld-Twist-Technik vor, mit deren Hilfe man supersymmetrische Feldtheorien derart reformulieren kann, dass die gebrochenen Supersymmetrien wieder manifest werden, wenn auch in einem getwisteten Sinn. Diese Reformulierung ermÄoglicht es, bestimmte chirale Ringe zu de¯nieren und ergibt supersymmetrische Ward-Takahashi-IdentitÄaten,welche von gewÄohnlichen super- symmetrischen Feldtheorien bekannt sind. Wenn man Seibergs naturalness argument, welches die Symmetrien von Niederenergie-Wirkungen betri®t, auch im nichtantikom- mutativen Fall zustimmt, so erhÄaltman Nichtrenormierungstheoreme selbst fÄurnichtan- tikommutative Feldtheorien. Im zweiten und umfassenderen Teil dieser Arbeit untersuchen wir detailliert geome- trische Aspekte von SupertwistorrÄaumen,die gleichzeitig Calabi-Yau-Supermannigfal- tigkeiten sind und dadurch als target space fÄurtopologische Stringtheorien geeignet sind. ZunÄachst stellen wir die Geometrie des bekanntesten Beispiels fÄureinen solchen Super- twistorraum, CP 3j4, vor und fÄuhrendie Penrose-Ward-Transformation, die bestimmte holomorphe VektorbÄundelÄuber dem Supertwistorraum mit LÄosungenzu den N = 4 supersymmetrischen selbstdualen Yang-Mills-Gleichungen verbindet, explizit aus. An- schlie¼end diskutieren wir mehrere dimensionale Reduktionen des Supertwistorraumes CP 3j4 und die implizierten VerÄanderungenan der Penrose-Ward-Transformation. Fermionische dimensionale Reduktionen bringen uns dazu, exotische Supermannig- faltigkeiten, d.h. Supermannigfaltigkeiten mit zusÄatzlichen (bosonischen) nilpotenten Di- mensionen, zu studieren. Einige dieser RÄaumekÄonnenals target space fÄurtopologische Strings dienen und zumindest bezÄuglich des Satzes von Yau fÄugendiese sich gut in das Bild der Calabi-Yau-Supermannigfaltigkeiten ein. Bosonische dimensionale Reduktionen ergeben die Bogomolny-Gleichungen sowie Ma- trixmodelle, die in Zusammenhang mit den ADHM- und Nahm-Gleichungen stehen. (TatsÄachlich betrachten wir die Supererweiterungen dieser Gleichungen.) Indem wir bes- timmte Terme zu der Wirkung dieser Matrixmodelle hinzufÄugen,kÄonnenwir eine kom- plette AquivalenzÄ zu den ADHM- und Nahm-Gleichungen erreichen. Schlie¼lich kann die natÄurliche Interpretation dieser zwei Arten von BPS-Gleichungen als spezielle D- Branekon¯gurationen in Typ IIB Superstringtheorie vollstÄandigauf die Seite der topo- logischen Stringtheorie Äubertragen werden. Dies fÄuhrtzu einer Korrespondenz zwischen topologischen und physikalischen D-Branesystemen und erÄo®netdie interessante Perspek- tive, Resultate von beiden Seiten auf die jeweils andere Äubertragen zu kÄonnen. Abstract There are two major topics within string theory to which the results presented in this thesis are related: non-anticommutative ¯eld theory on the one hand and twistor string theory on the other hand. Non-anticommutative deformations of superspaces arise naturally in type II super- string theory in a non-trivial graviphoton background and they have received much at- tention over the last two years. First, we focus on the de¯nition of a non-anticommutative deformation of N = 4 super Yang-Mills theory. Since there is no superspace formulation of the action of this theory, we have to resort to a set of constraint equations de¯ned on R4j16 the superspace ~ , which are equivalent to the N = 4 super Yang-Mills equations. In deriving the deformed ¯eld equations, we propose a non-anticommutative analogue of the Seiberg-Witten map. A mischievous property of non-anticommutative deformations is that they partially break supersymmetry (in the simplest case, they halve the number of preserved super- charges). In this thesis, we present a so-called Drinfeld twisting technique, which allows for a reformulation of supersymmetric ¯eld theories on non-anticommutative superspaces in such a way that the broken supersymmetries become manifest even though in some sense twisted. This reformulation enables us to de¯ne certain chiral rings and it yields su- persymmetric Ward-Takahashi-identities, well-known from ordinary supersymmetric ¯eld theories. If one agrees with Seiberg's naturalness arguments concerning symmetries of low-energy e®ective actions also in the non-anticommutative situation, one even arrives at non-renormalization theorems for non-anticommutative ¯eld theories. In the second and major part of this thesis, we study in detail geometric aspects of supertwistor spaces which are simultaneously Calabi-Yau supermanifolds and which are thus suited as target spaces for topological string theories. We ¯rst present the geometry of the most prominent example of such a supertwistor space, CP 3j4, and make explicit the Penrose-Ward transform which relates certain holomorphic vector bundles over the supertwistor space to solutions to the N = 4 supersymmetric self-dual Yang-Mills equations. Subsequently, we discuss several dimensional reductions of the supertwistor space CP 3j4 and the implied modi¯cations to the Penrose-Ward transform. Fermionic dimensional reductions lead us to study exotic supermanifolds, which are supermanifolds with additional even (bosonic) nilpotent dimensions. Certain such spaces can be used as target spaces for topological strings, and at least with respect to Yau's theorem, they ¯t nicely into the picture of Calabi-Yau supermanifolds. Bosonic dimensional reductions yield the Bogomolny equations describing static mo- nopole con¯gurations as well as matrix models related to the ADHM- and the Nahm equations. (In fact, we describe the superextensions of these equations.) By adding cer- tain terms to the action of these matrix models, we can render them completely equivalent to the ADHM and the Nahm equations. Eventually, the natural interpretation of these two kinds of BPS equations by certain systems of D-branes within type IIB superstring theory can completely be carried over to the topological string side via a Penrose-Ward transform. This leads to a correspondence between topological and physical D-brane sys- tems and opens interesting perspectives for carrying over results from either sides to the respective other one. Contents Chapter I. Introduction 15 I.1 High-energy physics and string theory . 15 I.2 Epistemological remarks .......................... 19 I.3 Outline ................................... 21 I.4 Publications ................................ 23 Chapter II. Complex Geometry 25 II.1 Complex manifolds ............................. 25 II.1.1 Manifolds ............................ 25 II.1.2 Complex structures ...................... 27 II.1.3 Hermitian structures ..................... 28 II.2 Vector bundles and sheaves ........................ 31 II.2.1 Vector bundles ......................... 31 II.2.2 Sheaves and line bundles ................... 35 II.2.3 Dolbeault and Cech· cohomology . 36 II.2.4 Integrable distributions and Cauchy-Riemann structures . 39 II.3 Calabi-Yau manifolds ........................... 41 II.3.1 De¯nition and Yau's theorem . 41 II.3.2 Calabi-Yau 3-folds ....................... 43 II.3.3 The conifold .......................... 44 II.4 Deformation theory ............................ 46 II.4.1 Deformation of compact complex manifolds . 47 II.4.2 Relative deformation theory . 48 Chapter III. Supergeometry 49 III.1 Supersymmetry ............................... 49 III.1.1 The supersymmetry algebra . 50 III.1.2 Representations of the supersymmetry algebra . 51 III.2 Supermanifolds ............................... 52 III.2.1 Supergeneralities ........................ 53 III.2.2 Gra¼mann variables ...................... 54 III.2.3 Superspaces .......................... 56 III.2.4
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