Geometrical approach to integrable and supersymmetric sigma models

One of the great successes of twentieth-century physics was the profound understanding of three of the four fundamental forces in a unified framework known as the Standard Model. The gravitational force, on the other hand, is well described only at longer distances by General Relativity while a satisfactory quantum mechanical description at short distances is still lacking. Such an obstacle begs for the unification of both descriptions, and ultimately all four forces, into a new theory of Quantum Gravity. At this moment, it is widely accepted that String Theory is the prime candidate to provide such a theory. In String Theory, elementary particles are little vibrating strings. A string propagating through the universe sweeps out a twodimensional surface called the worldsheet. The worldsheet dynamics is described by an intricate quantum field theory known as a sigma model. Quite remarkably, these tiny little strings are pretty demanding: the properties of the sigma model – such as its symmetries -- largely determine the geometry of the spacetime in which the string itself moves. We consider various aspects of sigma model symmetries with an eye on their applications in string theory. For instance, supersymmetric sigma models enable us to describe the known matter content of our universe. These particular sigma models constrain the spacetime in which the string is allowed to propagate even more severely. In this way, a deep relationship with the area of fundamental mathematics starts to unravel. In particular, we will be able to encode the full local geometry of the spacetimes associated with supersymmetric sigma models into a single potential. Additionally, we can consider sigma models with boundaries to study the propagation of open strings. Open strings end on higher-dimensional surfaces, known as D-branes, whose geometry is hard to identify in general. As a guiding principle, we will use integrable sigma models, which have a large hidden symmetry, to dictate their geometry. Hence, in many ways, we study the intense relationship between geometries and sigma model symmetries.