One of the most important problems in string theory is to connect it with what has been measured in high-energy experiments. A lot of progress has been made, but we still have many problems to solve and up to now there is no direct evidence that the elementary particles that we observe are strings. Below we briefly review the situation and state some of the outstanding problems.
Around 1985 it was established that there are five different ten-dimensional string theories: Type I strings, type IIA strings, type IIB strings, E8×E8 heterotic strings and SO(32) heterotic strings. They are all supersymmetric and each of them unifies gauge theories with gravity in a consistent quantum theory valid at a very short length scale called the string scale. This was a major breakthrough in the long-standing search for a consistent quantum theory of gravity but left many questions unanswered.
The five string theories are in-equivalent in string perturbation theory where it is assumed that string interactions are weak, but in the 1990's it was recognized that, at the non-perturbative level, they are all part of a single eleven-dimensional theory called M-theory. The synthesis of the five different string theories into a single underlying theory is a fascinating story that is far from fully understood and remains a major area of research in string theory today.
If we want to connect string theory with experiments we first of all need to explain why observed spacetime has only four dimensions rather than ten. This is usually achieved by compactifying six of the ten dimensions on a compact six-dimensional manifold that is sufficiently small to have avoided detection so far. Since supersymmetry has not been observed so far in particle experiments, we also need to understand how it gets broken on the way from the string scale to the length scale probed by current experiments.
It is often assumed that supersymmetry is preserved at the characteristic length scale of the compact manifold and is instead broken by some effect in four-dimensional field theory at lower energies. In this case the compact manifold must satisfy rather stringent mathematical conditions which string theorists have studied in detail. In string compactifications the four-dimensional physics depends not only on the string length but also on the shape and the size of the compact manifold. The parameters characterizing a particular compactification are called moduli and their values together with that of another string theory field, called the dilaton, must somehow be determined in order to make contact with observed particle physics.
Until a few years ago it was not known how to determine the moduli because their potential was flat at each order of string perturbation theory with no particular values favored. It turns out, however, that a potential can be generated for the moduli by introducing fluxes of closed string gauge fields along different directions inside the compact manifold. The minima of this potential correspond to favored values of the moduli which in turn determine couplings and particle masses in the four-dimensional field theory.
While this allows us in principle to predict various features of particle physics from a given string model, the moduli can be fixed in a huge number of ways and this leads to an enormous number of different predictions that are all equally valid a priori. This spells trouble for any theory, such as string theory, which is supposed to predict from first principles the behavior of the elementary particles that we observe in high-energy experiments. The multitude of potential minima for the moduli goes under the name of the string landscape problem and has occupied many string theorists in recent years.
The program of connecting string theory to particle phenomenology faces many challenges in addition to the landscape problem. There are essentially two approaches to string compactification. The first, that we refer to as top-down, is mostly based on the heterotic string theory and assumes that both the string length and the size of the compact manifold are of the order of the Planck length. The second one, that we call bottom-up, is based on so called Dirichlet-branes (D-branes) of type I and type II theories and allows much larger values for the string length and the size of the compact manifold, even as large as the length scale that will be probed in the upcoming experiments at the Large Hadron Collider at CERN in Switzerland.
In the second approach, which is also referred to as brane-world compactification, the gauge theories of the SM and of the MSSM are defined inside the world-volume of stacks of D-branes. Gauge fields then correspond to open strings with both ends attached to branes in a particular stack, while quarks and leptons correspond to open strings having their two end-points attached to two different stacks of D branes. In order to have chiral matter, the two stacks of D branes must be at angles or carry different magnetizations in the compact extra-dimensions. Simple models of this type, where the compact geometry is a flat six-dimensional torus, can be studied in considerable detail and can serve as prototypes for more general string compactifications. Several technical issues need to be addressed in order to make these models fully consistent. So called orientifold planes are introduced to enable the cancellation of certain field theory tadpoles and supersymmetry can be partially or completely broken by introducing orbifold singularities into the compact geometry. Various bells and whistles can be added to make the models more interesting and closer to the physics of the real world but at the same time they become more complicated and less amenable to exact treatment.
Semi-realistic models that are string extensions of the Standard Model of particle physics and of the Minimal Supersymmetric Standard Model have been constructed using both the top-down and bottom up approaches to string compactification, making many string theorists guardedly optimistic that successful string phenomenology is waiting around the corner, or the next corner, or the next.