On supertwistor geometry and integrability in super gauge theory

In this thesis, we report on different aspects of integrability in supersymmetric gauge theories. The main tool of investigation is twistor geometry. In trying to be self-contained, we first present a brief review about the basics of twistor geometry. We then focus on the twistor description of various gauge theories in four and three space-time dimensions. These include self-dual supersymmetric Yang-Mills (SYM) theories and relatives, non-self-dual SYM theories and supersymmetric Bogomolny models. Furthermore, we present a detailed investigation of integrability of self-dual SYM theories. In particular, the twistor construction of infinite-dimensional algebras of hidden symmetries is given and exemplified by deriving affine extensions of internal and space-time symmetries. In addition, we derive self-dual SYM hierarchies within the twistor framework. These hierarchies describe an infinite number of flows on the respective solution space, where the lowest level flows are space-time translations. We also derive infinitely many nonlocal conservation laws.