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

TL;DR

This paper introduces two-dimensional lattice models with N=2 supersymmetry, including integrable vertex models and multicritical polymer points, providing exact results and confirming phenomenological formulas in two dimensions.

Contribution

The paper presents new geometrical lattice models with N=2 supersymmetry, including an integrable model with exact free energy and a series of multicritical polymer models matching phenomenological exponents.

Findings

Exact free energy obtained without Bethe ansatz

Lattice operators reproduce the chiral ring

Polymer exponents match Flory's formulas

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.