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

TL;DR

This paper demonstrates how twisted N=2 supersymmetry provides a comprehensive conformal field theory framework for two-dimensional geometrical phase transitions like polymers and percolation, including explicit calculations and new predictions.

Contribution

It introduces a complete conformal field theory description of 2D phase transitions using twisted N=2 supersymmetry, including four-point functions and operator products.

Findings

Computed four-point functions of operators with half-integer Kac labels.

Built modular invariant partition functions incorporating various fermionic sectors.

Conjectured the fractal dimension of the percolation backbone as D=25/16.

Abstract

It is shown how twisted N=2 (k=1) provides for the first time a complete conformal field theory description of the usual geometrical phase transitions in two dimensions, like polymers, percolation or brownian motion. In particular, four point functions of operators with half integer Kac labels are computed, together with geometrical operator products. In addition to Ramond and Neveu Schwartz, a sector with quarter twists has to be introduced. The role of fermions and their various sectors is geometrically interpreted, modular invariant partition functions are built. The presence of twisted N=2 is traced back to the Parisi Sourlas supersymmetry. It is shown that N=2 leads also to new non trivial predictions; for instance the fractal dimension of the percolation backbone in two dimensions is conjectured to be D=25/16, in good agreement with numerical studies.