Loop models and boundary CFT

Abstract

We present a range of exact techniques within two-dimensional conformal field theory (CFT), using the Q-state Potts and the O(n) models as exploratory tools. Both are equivalent to models of oriented loops, which act as level lines of a height model. The height model can be treated via a geometrical Coulomb gas construction, giving access to exact bulk and boundary properties. We detail the derivation of critical exponents and relate their physical interpretation to properties of clusters and loops. The underlying Temperley-Lieb algebra is discussed, and we show how the various topological sectors should be combined to yield exact continuum limit partition functions. These give access to probabilistic results, in particular to crossing formulae in percolation. Finally, we discuss how these results can be extended to classes of new conformally invariant boundary conditions, in which the weights of boundary-touching loops are modified. © 2012 Springer-Verlag Berlin Heidelberg.