Polymers and percolation in two dimensions and twisted N = 2 supersymmetry

Abstract

We show how a large class of geometrical critical systems including dilute polymers, polymers at the θ-point, percolation and to some extent brownian motion, are described by a twisted N = 2 supersymmetric theory with k = 1 (it is broken in the dense polymer phase that is described simply by a η, ξ system). This allows us to give for the first time a consistent conformal field theory description of these problems. The fields that were described so far by formally allowing half-integer labels in the Kac table are built and their four-point functions studied. Geometrical operators are organized in a few representations of the twisted N = 2 algebra. A noticeable feature is that in addition to Neveu-Schwarz and Ramond, a sector with quarter twists sometimes has to be introduced. The algebra of geometrical operators is determined. Fermions boundary conditions are geometrically interpreted, and the partition functions that were so far defined formally as generating functions for the critical exponents are naturally understood, sector by sector. Twisted N = 2 provides moreover a very unified description of all these geometrical models, explaining for instance why the exponents of polymers and percolation coincide. It must be stressed that the physical states in these geometrical problems are not the physical states for string theory, which are usually extracted by the BRS cohomology. In polymers for instance, Q BRS is precisely the operator that creates polymers out of the vacuum, such that the topological sector is the sector without any polymers. It seems that twisted N = 2 is the correct continuum limit (in two dimensions) for models with Parisi-Sourlas supersymmetry. Some possible explanation of this fact is advanced. As an example of application of N = 2 supersymmetry we discuss the still unsolved problem of backbone of percolation. We conjecture in particular the value D = 25 16 for the fractal dimension of the backbone, in good agreement with numerical computations. Finally some of these ideas are extended to the off-critical case. It is shown how to give a meaning to the n → 0 limit of the O( n) model S-matrix recently introduced by Zamolodchikov by introducing fermions.