Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT

TL;DR

This paper provides non-perturbative arguments for dimensional reduction in Parisi-Sourlas supersymmetric CFTs, establishing a map between supersymmetric theories in d dimensions and ordinary CFTs in d-2 dimensions, with implications for understanding quenched disorder.

Contribution

It introduces a new operator and correlation function mapping from supersymmetric to ordinary CFTs, confirming dimensional reduction and local structure in the reduced theory.

Findings

Perfect match between superconformal and conformal blocks across dimensions

Establishment of a local, non-supersymmetric CFT in reduced dimensions

New relation between conformal blocks in different dimensions

Abstract

Quenched disorder is very important but notoriously hard. In 1979, Parisi and Sourlas proposed an interesting and powerful conjecture about the infrared fixed points with random field type of disorder: such fixed points should possess an unusual supersymmetry, by which they reduce in two less spatial dimensions to usual non-supersymmetric non-disordered fixed points. This conjecture however is known to fail in some simple cases, but there is no consensus on why this happens. In this paper we give new non-perturbative arguments for dimensional reduction. We recast the problem in the language of Conformal Field Theory (CFT). We then exhibit a map of operators and correlation functions from Parisi-Sourlas supersymmetric CFT in $d$ dimensions to a $(d-2)$-dimensional ordinary CFT. The reduced theory is local, i.e. it has a local conserved stress tensor operator. As required by reduction, we show a perfect match between superconformal blocks and the usual conformal blocks in two dimensions lower. This also leads to a new relation between conformal blocks across dimensions. This paper concerns the second half of the Parisi-Sourlas conjecture, while the first half (existence of a supersymmetric fixed point) will be examined in a companion work.