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

TL;DR

The paper shows that twisted N=2 supersymmetry with k=1 provides a complete conformal-field-theory description of key two-dimensional geometrical transitions, including dense/dilute polymers, percolation, and related processes. It constructs sectorized partition functions (NS, Ramond, Z4) and computes four-point functions for L-leg operators using an eta,xi free-field realization with c = -2 for dense polymers and a twisted N=2 structure for dilute polymers, linking lattice results to continuum CFT via Coulomb-gas formulations. A central result is the unifying explanation for the coincidence of polymer and percolation exponents and the prediction D = 25/16 for the percolation backbone, with broader implications for the role of Parisi–Sourlas supersymmetry in two dimensions. The framework also connects to off-critical extensions, screening structures, and modular invariants, offering a robust tool for analyzing a wide class of two-dimensional critical geometries within twisted N=2 and related algebras.

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.