String Theory and Conformal Field Theory

A major issue of contemporary theoretical physics is to unify gravitation with the other fundamental forces of Nature, by developing a consistent quantum theory of gravitation. The most promising candidate for achieving this goal is String Theory, which has attracted much attention in the past two decades. It is based on the assumption that the fundamental objects in Nature are not pointlike, as in traditional Quantum Field Theories, but rather one-dimensional extended objects, i.e. strings, whose length is so small that they can't be observed in present day experiments. The usual elementary particles, like electrons and quarks, correspond in this picture to low-energy excitations of the fundamental strings.

A most striking feature of String Theory is that it gives precise predictions concerning the structure of space-time. Indeed, one can show that strings can propagate consistently in flat space-time only if the dimension is 26 (for bosonic strings) or 10 (for superstrings). As this is in sharp contradiction with the everyday experience that space-time is 4 dimensional, one has to suppose that some of the dimensions of physical space-time are curved up, so that they can't be observed with the experimental apparatus available: this is the subject of string compactifications.

Conformal Field Theory (CFT for short) is closely related to String Theory, since the propagation of fundamental strings can be described by a suitable two-dimensional Conformal Field Theory, allowing the computation of string scattering amplitudes and other important data using techniques of CFT. But CFT is an important and interesting subject in itself, with many physical applications (universality classes of critical phenomena in 2 dimensions, quantum Hall effect, conformal turbulence, percolation, etc), and with many links to mathematics (Vertex Operator Algebras, vector-valued modular functions and forms, Moonshine, Modular Tensor Categories, etc).

The current research activities at Eotvos University are focused towards a better understanding of Conformal Field Theory, with the long term goal to achieve a (partial) classification of all consistent theories. This involves in particular the study of the modular representation (or even better, the Modular Tensor Category) associated to a CFT, the investigation of its symmetries (Galois-action, simple currents, etc) and of the conformal characters. The highlights of this research include the proof of the congruence subgroup property (according to which the kernel of the modular representation is a congruence subgroup for rational CFTs), the discovery of the trace formulas for mapping classes of finite order, and the development of a general procedure for determining conformal characters from the knowledge of the modular representation using the theory of vector-valued modular forms.

Another line of research, which is closely related to the above, is the study of orbifold models in general, and of permutation orbifolds in particular. This general theory, besides providing a general recipe for constructing new consistent CFTs from old one, has turned out to be an invaluable tool in the study of the general properties of CFTs through the so-called Orbifold Covariance Principle. In particular, it is at the heart of the proof of the congruence subgroup property alluded to before. The theory of permutation orbifolds has many exciting connections with several branches of modern mathematics, e.g. the theory of 2-dimensional groups.