Random geometry and non-unitary quantum field theories

The main subjects of study are conformally invariant spatially extended objects, such as percolation clusters, domain walls in spin systems and self-avoiding walks. We wish to describe these objects as precisely as possible, in terms of critical exponents and correlation functions. A substantial part of the work takes place in two dimensions, but certain techniques can also be extended to higher dimensions.

After an introduction to the geometrical objects and their phenomenology, we clarify their relations to lattice models of loops and spins. In the continuum limit we find links to a quantum field theory of Liouville type, and to other results coming from high-energy physics.

The study of correlation functions of two, three and four points lead to the concepts of fusion and conformal bootstrap. Probabilistic approaches such as stochastic Loewner evolution (SLE) and its variants provide another angle of attack. A crucial algebraic understanding is obtained by identifying the symmetries of the models, both on the lattice (affine Temperley-Lieb algebra) and in the continuum limit (interchiral conformal symmetry).

In many cases of physical interest the corresponding representations are indecomposable, implying that the correlation functions are described by logarithmic conformal field theories.