Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory

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 string theory on the other hand. Non-anticommutative deformations of superspaces arise naturally in type II superstring theory in a non-trivial graviphoton background and they have received much attention 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 R4|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 supercharges). 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 supersymmetric 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 such a supertwistor space, CP 3|4, and make explicit the Penrose-Ward transform which relates certain holomorphic vector bundles over the supertwistor space to solutions to theN = 4 supersymmetric self-dual Yang-Mills equations. Subsequently, we discuss several dimensional reductions of the supertwistor space CP 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 monopole configurations as well as matrix models related to the ADHMand the Nahm equations. (In fact, we describe the superextensions of these equations.) By adding certain 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 systems and opens interesting perspectives for carrying over results from either sides to the respective other one.