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

Abstract

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 ${\Gamma }_k$ vertex models (based on the quantum algebra $U_{q}sl(2)$ and representation of spin $j=k/2$). We demonstrate in particular that at the $N=2$ point, the free energy of the ${\Gamma }_k$ 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. It turns out that the exponents $\nu =(k+2)/2(k+1)$ for these multicritical polymer points coincide with old phenomenological formulas due to the chemist Flory. We therefore confirm that these formulas are {\bf exact} in two dimensions, and suggest that their unexpected validity is due to non renormalization theorems for the $N=2$ underlying theories. We also discuss the status of the much discussed theta point for polymers in the light of $N=2$ renormalization group flows.

23 pages (without figures)

References in corpus (1)

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

Cited by in corpus (13)

• Polymers and percolation in two dimensions and twisted N=2 supersymmetry
• Logarithmic Minimal Models
• Singular Vectors in Logarithmic Conformal Field Theories
• Exact Scaling Functions for Self-Avoiding Loops and Branched Polymers
• Exact S-matrices for supersymmetric sigma models and the Potts model
• Spin chains with dynamical lattice supersymmetry
• The eight-vertex model and lattice supersymmetry
• Logarithmic conformal field theory with boundary
• Edge Excitations of an Incompressible Fermionic Liquid in a Disorder Magnetic Field
• Logarithmic Correlation Functions in Two Dimensional Turbulence
• Ghost Systems Revisited: Modified Virasoro Generators and Logarithmic Conformal Field Theories
• The Generic Critical Behaviour for 2D Polymer Collapse
• Off-Critical Lattice Analogues of $N=2$ Supersymmetric Quantum Integrable Models