Twistor geometry, supersymmetric field theories in supertring theory c samann

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arXiv:hep-th/0603098 v1 13 Mar 2006 Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory Von der Fakult¨at f¨ur Mathematik und Physik der Universit¨at Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Christian S¨amann 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 m¨ussen wissen, 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, Strin g Theory ITP-UH-26/05 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 Superr¨aumen entstehen auf nat¨urliche Wei- se bei Typ II Superstringtheorie in einem nichttrivialen Graviphoton-Hintergrund, und solchen Deformationen wurde in den letzten zwei Jahren viel Beachtung geschenkt. Zu- n¨achst konzentrieren wir uns auf die Definition d er nichtantikommutativen Deformation von N = 4 super Yang-Mills-Theorie. Da es f¨ur die Wirkung dieser Theorie keine Super- raumformulierung gibt, weichen wir statt dessen auf die ¨aquivalenten constraint equations aus. W¨ahrend der 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 F¨allen halbieren s ie 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 erm¨oglicht es, bestimmte chirale Ringe zu definieren und ergibt supersymmetrische Ward-Takahashi-Identit¨aten, welche von gew¨ohnlichen super- symmetrischen Feldtheorien bekannt sind. Wenn man Seibergs naturalness argument, welches die Symmetrien von Niederenergie-Wirkungen betrifft, auch im nichtantikom- mutativen Fall zustimmt, so erh¨alt man Nichtrenormierungstheoreme selbst f¨ur nichtan- tikommutative Feldtheorien. Im zweiten und umfassenderen Teil dieser Arbeit untersuchen wir detailliert geome- trische Aspekte von Supertwistorr¨aumen, die gleichzeitig Calabi-Yau-Supermannigfal- tigkeiten sind und dadurch als target space f¨ur topologische Stringtheorien geeignet sind. Zun¨achst stellen wir die Geometrie des bekanntesten Beispiels f¨ur einen solchen Super- twistorraum, P 3|4 , vor und f¨uhren die Penrose-Ward-Transformation, die bestimmte holomorphe Vektorb¨undel ¨uber dem Supertwistorraum mit L¨osungen zu den N = 4 supersymmetrischen selbstdualen Yang-Mills-Gleichungen verbindet, explizit aus. An- schließend diskutieren wir mehrere dimensionale Reduktionen des Supertwistorraumes P 3|4 und die implizierten Ver¨anderungen an der Penrose-Ward-Transformation. Fermionische dimensionale Reduktionen bringen uns dazu, exotische Supermannig- faltigkeiten, d.h. Supermannigfaltigkeiten mit zus¨atzlichen (bosonischen) nilpotenten Di- mensionen, zu studieren. Einige dieser R¨aume k¨onnen als target space f¨ur topologische Strings dienen und zumindest bez¨uglich des Satzes von Yau f¨ugen diese sich gut in das Bild der Calabi-Yau-Supermannigfaltigkeiten ein. Bosonische dimensionale Redu ktionen ergeben die Bogomolny-Gleichungen sowie Ma- trixmodelle, die in Zu s amm en hang mit den ADHM- und Nahm-Gleichungen stehen. (Tats¨achlich betrachten w ir die Supererweiterungen dieser Gleichungen.) Indem wir bes- timmte Terme zu der Wirkung dieser Matrixmodelle hinzuf¨ugen, k¨onnen wir eine kom- plette ¨ Aquivalenz zu den ADHM- und Nahm-Gleichungen erreichen. Schließlich kann die nat¨urliche Interpretation dieser zwei Arten von BPS-Gleichungen als spezielle D- Branekonfigurationen in Typ IIB Superstringtheorie vollst¨andig auf die Seite der topo- logischen Stringtheorie ¨ubertragen werden. Dies f¨uhrt zu einer Korrespondenz zwischen topologischen und physikalischen D-Branesystemen und er¨offnet die interessante Perspek- tive, Resultate von beiden Seiten auf die jeweils andere ¨ubertragen zu k¨onnen. Abstract There are two major topics within string theory to which the results presented in this thesis are related: non-anticommutative field theory on the one hand and twistor str ing 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 definition 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 defined on the superspace 4|16  , which are equivalent to the N = 4 super Yang-Mills equations. In deriving the deformed field 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 field 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 define certain chiral rings and it yields su- persymmetric Ward-Takahashi-identities, well-known from ordinary supersymmetric field theories. If one agrees with Seiberg’s naturalness arguments concerning symmetries of low-energy effective actions also in the non-anticommutative situation, one even arrives at non-renormalization theorems for non-anticommutative field 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 first present the geometry of the most prominent example of s uch a supertwistor sp ace, P 3|4 , 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 P 3|4 and the implied modifications 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 fit nicely into the picture of Calabi-Yau supermanifolds. Bosonic dimensional reductions yield the Bogomolny equations describing static mo- nopole configurations 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 bund les and sheaves . . . . . . . . . . . . . . . . . . . . . . . . 31 II.2.1 Vector bu ndles . . . . . . . . . . . . . . . . . . . . . . . . . 31 II.2.2 Sheaves and line bundles . . . . . . . . . . . . . . . . . . . 35 II.2.3 Dolbeault and ˇ Cech cohomology . . . . . . . . . . . . . . . 36 II.2.4 Integrable distributions and Cauchy-Riemann structures . 39 II.3 Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 II.3.1 Definition 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 . . . . . . . . . 46 II.4.2 Relative deformation theory . . . . . . . . . . . . . . . . . 47 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 Graßmann variables . . . . . . . . . . . . . . . . . . . . . . 54 III.2.3 Superspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 56 III.2.4 Supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . 58 III.2.5 Calabi-Yau supermanifolds and Yau’s theorem . . . . . . . 59 III.3 Exotic supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 III.3.1 Partially formal supermanifolds . . . . . . . . . . . . . . . 60 III.3.2 Thick complex manifolds . . . . . . . . . . . . . . . . . . . 61 III.3.3 Fattened complex manifolds . . . . . . . . . . . . . . . . . 63 III.3.4 Exotic Calabi-Yau supermanifolds and Yau’s theorem . . . 64 III.4 Spinors in arbitrary dimensions . . . . . . . . . . . . . . . . . . . . . . 66 III.4.1 Spin groups and Clifford algebras . . . . . . . . . . . . . . 66 III.4.2 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 [...]... breaking mechanisms it can come with The variety of such imaginable breaking mechanisms remains, however, a serious problem 9 It is doubtful that these critics would accept the exception of twistor string theory, which led to new ways of calculating certain gauge theory amplitudes 10 Contrary to the logical positivism, Popper attributes some meaning to such theories in the process of developing new theories. .. certain aspects of conformal field theories which will prove useful in what follows The aspects of string theory entering into this thesis are introduced in the following chapter We give a short review on string theory basics and superstring theories before elaborating on topological string theories One of the latter, the topological B-model, will receive much attention later due to its intimate connection... The connecting link in this correspondence is a partially holomorphic Chern-Simons theory on a Cauchy-Riemann supermanifold which is a real one-dimensional fibration over the mini-supertwistor space While the supertwistor spaces examined so far naturally yield Penrose-Ward transforms for certain self-dual subsectors of super Yang-Mills theories, the superambitwistor space L5|6 introduced in the following... calculating the locus in M , where C has rank less than i We will present an example in paragraph §28 For more details, see e.g [113] §21 Chern character Let us also briefly introduce the characteristic classes called Chern characters, which play an important rˆle in the Atiyah-Singer index theorem We o will need them for instanton configurations, in which the number of instantons is given by an integral... degeneracy loci Chern classes essentially make statements about the degeneracy of sets of sections of vector bundles via a Gauß-Bonnet formula More precisely, given a vector bundle E of rank e over M , the ce+1−i th Chern class is Poincar´-dual to the degeneracy cycle of i generic global sections This degeneracy locus e is obtained by arranging the i generic sections in an e × i-dimensional matrix C and calculating... discussion Calabi-Yau manifolds are complex manifolds which have a trivial first Chern class They are Ricci-flat and come with a holomorphic volume element The latter property allows to define a Chern-Simons action on these spaces, which will play a crucial rˆle o throughout this thesis Calabi-Yau manifolds naturally emerge in string theory as candidates for internal compactification spaces In particular,... ˇ Note that C 0 (U, S) will denote the set of Cech 0-cochains taking values in the sheaf S 7 A special case of locally ringed spaces are the better-known schemes 36 Complex Geometry §28 Holomorphic line bundles A holomorphic line bundle is a holomorphic vector bundle of rank 1 Over the Riemann sphere P 1 ∼ S 2 , these line bundles can be = completely characterized by an integer d ∈ , cf §14 Given... approaches for describing such manifolds and introduce an integration operation on a certain class of them, the so-called thickened and fattened complex manifolds We furthermore examine the validity of Yau’s theorem for such exotic Calabi-Yau supermanifolds, and we find, after introducing the necessary tools, that the results fit nicely into the picture of ordinary Calabi-Yau supermanifolds which was... example is certainly the AdS/CFT correspondence [187] These dualities provide a dictionary between certain pairs of string theories and gauge theories, which allows to perform field theoretic calculations in the mathematically often more powerful framework of string theory The recently proposed twistor string theory [296] gives rise to a second important example of such a duality It has been in its context... underlying structure of spacetime and can be nicely incorporated into the quantum field theoretic framework, there is a strong hint that this extension is a first step towards combining quantum field theory with gravity: As stated above, we naturally obtain a theory describing gravity by promoting supersymmetry to a local symmetry Besides being in some cases the low-energy limit of certain string theories, . as candi- dates for internal compactification spaces. In particular, topological strings of B-type – a subsector of the superstrings in type IIB superstring. devoted to studying certain aspects of this twistor string theory, let us present this theory in more detail. Twistor strin g theory was introduced in 2003 by