Geometrical lattice models for N = 2 supersymmetric theories in two dimensions

We introduce in this paper two-dimensional lattice models whose continuum limit belongs to the N = 2 series. The first kind of model is integrable and obtained through a geometrical reformulation, generalizing results known in the k = 1 case, of the Γk models (based on the quantum algebra Uqsl(2) and representation of spin j = k/2). We demonstrate in particular that at the N = 2 point, the free energy of the Γk vertex model can be obtained exactly by counting arguments, without any Bethe ansatz computation, and we exhibit lattice operators that reproduce the chiral ring. The second class of models is more adequately described in the language of twisted N = 2 supersymmetry, and consists of an infinite series of multicritical polymer points, which should lead to experimental realizations. The presence of N = 2 in that case is traced back to the Parisi-Sourlas supersymmetry of the lagrangians usually used to replace n → 0 limits. Boundary conditions as well as fermionic and bosonic variables are geometrically interpreted. Moreover it turns out that the exponents ν = (k + 2)2(k + 1) for these multicritical polymer points coincide with the old phenomenological formulas of Flory. We therefore confirm that these formulas are exact in two dimensions, and suggest that their unexpected validity due to non-renormalization theorems for the N = 2 underlying theories. We also discuss the status of the much discussed θ-point for polymers in the light of N = 2 renormalization group flows.