Bethe Forum 2014

Scientific Programme

Discrete symmetries and their stringy origin

Discrete symmetries play an important role for model-building and phenomenology. Some examples are 1) discrete flavor symmetries used in order to explain the observed patterns of masses and mixings of quarks and leptons and 2) discrete symmetries that forbid unwanted operators (like those that induce rapid proton decay). As quantum-gravitational effects are believed to break discrete symmetries in a way that their phenomenological properties are lost, these symmetries need to be protected against such effects. In the traditional approach this is achieved when the discrete symmetry originates as a discrete remnant of a (spontaneously) broken (anomaly-free) gauge symmetry. For example, a Z_N symmetry can originate from a U(1) gauge symmetry, where the symmetry-breaking is induced by an object of charge N. Also a non-Abelian discrete symmetry can originate from a gauge symmetry, but typically the breaking is more involved: large representations of the non-Abelian gauge symmetry are needed to induce the breaking.

There is a second possibility to protect discrete symmetries against gravitational effects by relating their origin to the relevant energy scale: the scale where quantum effects of gravity become important, i.e. the planck scale. Hence, this approach needs a consistent theory of quantum gravity and string theory is a prime candidate. Within string theory discrete symmetries (Abelian and non-Abelian) can naturally originate from both, the traditional approach described above and, furthermore, from the compactification from ten to four dimensions. In the later case one can partially understand the origin of the discrete symmetry intuitionally from properties of the six-dimensional compactification space. But in addition to this intuition there can be vanishing couplings and relations between coupling strengths that only show up in the explicit computation of string amplitudes. Some of them can be interpreted as an enlargement of the intuitive symmetry, but some might not. Intersecting D-branes and heterotic orbifold compactifications provide consistent frameworks that allow to study the origin of discrete symmetries from string theory directly. Recently, discrete R-symmetries have been re-examined in this context. But a full study of couplings and their interpretation as discrete symmetries remains as an open problem.

The aim of this workshop is to bring together experts on discrete symmetry phenomenology with experts on their possible string theory origin.

Non-geometry, asymmetric orbifolds and model building

String compactifications have mostly focussed on geometrical constructions. Prime examples that preserve a certain amount of target space supersymmetry are Calabi-Yau and orbifold compactifications. However, it has been realized very early on that string theory also admits constructions that do not admit any (easy) geometrical interpretation, for example asymmetric orbifolds, free-fermions and Gepner models.

Recently the field of non-geometrical string compactifications has revived. One motivation for this has been the search for constructions that provide a build-in mechanism for moduli stabilization. Another reason is that it has been realized that maximal (N=8) supergravities in four dimensions admit many gaugings, but only a small subset of those can be associated with compactifications of 10D supergravity. Some of the other gauge supergravities can be obtained by applying T-dualities to the geometrical compactifications. Hence, one expects that there must be some sort of lift of these 4D gauged supergravities to 10D string theory. They go under the name of non-geometrical flux backgrounds.

Since an underlying idea is that various configurations of fluxes are related by T-dualities, it would be useful to have a formulation of the low-energy theory of string theory that is T-duality covariant. Here, double field theory enters the scene: it is a construction in which the number of coordinates are doubled to make T-duality manifest.

Double field theory is one attempt to have a definite stringy description of non-geometry. Another approach is to use asymmetric orbifolds. Even though these orbifolds do not have a simple geometrical interpretation, they provide exactly solvable string solutions. The connection between them and the non-geometric fluxes has recently been investigated. In addition some first attempts have been made to do model building on such backgrounds. Furthermore, a natural description of asymmetric Z2xZ2 orbifolds are free-fermionic constructions. Also, quite recently, there has been a full classification of all symmetric orbifold geometries compatible with heterotic N=1 or more supersymmetry in four dimensions using the language of cristallography. It would therefore be very interesting to obtain a similar classification of asymmetric orbifolds. In addition the techniques to determine the nature of the non-geometrical fluxes might also be applicable to more involved non-geometrical string constructions like Gepner models.

In this workshop we would like to bring together experts on the various aspects of non-geometry and exact string constructions to share their recent results and discuss how some of the open questions mentioned above can be addressed.