In string theory one considers the dynamics of one-dimensional objects(strings), which move through (curved) space. In analogy to the fact that the trajectory of point particles is the shortest distance between two points, one requires that the ara that a string sweeps out is minimal.
This kind of problem is well-known in mathematics, in particular in differential geometry:
A minimal surface in a (curved) space is a surface, that minimizes the surface area. A typical example would be a soap film.
In the mathematical formulation of this problem one applies the calculus of variation: One studies the so-called volume functional, which maps each surface in space to a real number. Making use of the methods from geometric analysis one can find those surfaces, which minimize the surface area. In mathematics one calls this „critical point“ of the functional. The volume functional has the advantage, that it is always positive.
All variational problems that appear in theoretical physics are formulated in the mathematical language of differential geometry. However, their structure is usually more complicated.
On the other hand, there are interesting interactions between geometry and physics:
For example, the so-called Thirring-model describes the interaction of four elementary particles in physics,
in differential geometry it describes surfaces of constant curvature in three-dimensional space.
This project focusses on the study of geometric and analytic properties of various variational problems arising in string theory.
On the one hand it will be investigated under which conditions there exist critical points for the functionals arising in string theory and what their properties are. In contrast to the volume functional the functionals from string theory are not always bounded from below leading to severe mathematical difficulties.
Thus, it is required to develop new mathematical methods to ensure the existence of critical points.
In the second part of the project the focus is put on the equations, that govern the dynamics of a superstring
moving through some curved target space. Formally, these equations are nonlinear wave equations.
Linear wave equations describe the unperturbed expansion of waves, like for example in the case of sound waves. If we add a nonlinear term to the wave equations, then their solutions can develop singularities:
In the case of sound waves this would correspond to the occurrence of supersonic waves.
From a mathematical point of view the nonlinearities arising in the equations for the superstring are manageable. Thus, one can expect that their solutions can both have global solutions without singularities as well as solutions that develop singularities.
Another aspect that shall be investigated is the influence of the geometry of the target space on the qualitative behavior of the solution.