Geometric Variational Problems from String Theory

In string theory one studies the dynamics of one-dimensional objects, called strings, which propagate through a curved space. In analogy to point particles, which propagate on the shortest distance between two points, one requires that the area swept out by the string is minimal. Such kind of problems have been investigated by mathematicians, in particular geometers, for a long time: A minimal surface is a surface in a (curved) space, which has minimal area. A classical example for minimal surfaces are soap films, which are clamped on a frame. In order to deal with this problem mathematically, one applies the calculus of variation: One considers the so-called area functional, which associates a number (area) to every surface in space. Employing the methods of geometric analysis one can find those surfaces, which minimize the surface area. These are called critical points of the area functional. One big advantage of the area functional is the fact, that it is bounded from below since the area of a surface cannot be negative. For such kind of variational problems there exist powerful mathematical tools. The variational problems that arise in string theory are also formulated in terms of differential geometry. However, these are more complicated since they are unbounded from below. The first part of the project focuses on the investigation of geometric and analytic properties of various variational problems from string theory. To handle the unbounded functionals it will be necessary to develop new mathematical regularization methods. These transform unbounded functionals to bounded ones, for which a large number of mathematical tools exist. The difficult question will be whether the "regularized problem" also provides information on the original problem. In the second part of the project we want to study the equations that govern the dynamics of a superstring in a curved space. Formally, these equations comprise of a linear and a non-linear wave equation. Linear wave equation model the unperturbed expansion of waves, i.e. sound waves. On the other hand, the solutions of non-linear wave equations may develop singularities: In the case of sound waves these would correspond to the occurrence of supersonic waves. For both equations that govern the dynamics of a superstring there already exists an extensive number of results in the mathematical literature both in analysis and geometry. In the course of this project we want to cleverly combine and extend the existing methods. The nonlinearities appearing here are manageable from a mathematical point of view. For this reason one can expect that the equations can have both global solutions, but also solutions that develop a singularity. It will be exciting to investigate in which cases singularities will occur. In addition, we want to explore how the geometry of the surrounding space influences the dynamics of the superstring.

The project Geometric Variational Problems from String Theory investigated various action functionals originating in string theory with rigorous mathematical methods. In theoretical physics string theory is a promising candidate for a so-called theory of everything. Such a theory is able to describe all phenomena in our universe in a unified fashion. The central assumption in string theory is that elementary particles are modelled by small one -dimensional objects, which are called strings. If such a string evolves through our universe then it sweeps out an area which is referred to as the worldsheet of the string. Similar to the requirement that point particles are described by geodesics, which are curves of minimal length, one requires that the worldsheet of a string should have minimal surface area. The universe in which the string evolves with respect to time can be modelled by the Einstein equations of general relativity and, in general, it will be a curved space. Already due to this f act one can imagine that it will not always be possible for the string to sweep out a surface of minimal area such that the resulting mathematical problem will be a demanding one. On the mathematical side these difficulties manifest itself in the fact that the equation governing the dynamics of a string is given by a nonlinear partial differential equation. The presence of nonlinearities in such kind of equations often leads to the fact that their solutions only exist for a finite time and then form a singularity. Such kind of solutions would not be of interest for theoretical physics as they describe an unstable string which cannot represent a stable elementary particle. An additional difficulty comes from the fact that most of the realistic models employed in string theory contain additional terms which lead to the so-called superstring theories. One central result of this project states that the equations which govern the dynamics of a large class of superstring theories admit global solutions (these are solutions which exist for all times) under the assumption that the worldsheet of the string expands sufficiently fast. The requirement that the worldsheet expands rapidly suppresses the nonlinear contributions in the equations governing the dynamics of a superstring and thus the solutions can exist for all times. Morever, in the course of the project a number of conditions, for example on the curvature of the universe, was found under which the equations describing a superstring do not admit a solution. Such conditions are important for theoretical physics as they give hints in which situations the equations of string theory cannot be solved.