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.
Highlights
Physical systems realized in nature often have some kind of random impurities
We show that the d − 2 dimensional theory is local, and that the superconformal blocks of the Parisi-Sourlas Conformal Field Theory (CFT) in d dimensions are equal to standard conformal blocks in d − 2 dimensions
We further demonstrate how dimensional reduction works by showing that the superconformal blocks of the Parisi-Sourlas CFT match the conformal blocks of a d − 2 dimensional CFT
Summary
Physical systems realized in nature often have some kind of random impurities. The presence of such impurities may change the behavior of a system. Link B in figure 2 means that the fixed point of a random field theory should possess an enhanced symmetry, called Parisi-Sourlas supersymmetry. Theories with this kind of supersymmetry have very unusual features, at least to a high energy physicist. We point out that dimensional reduction was proposed for the IR fixed point, and all along the RG flow, relating quantum field theories in d and d − 2 dimensions. Link B will be studied in the companion paper [13], where the RG flow of the random field theories is analyzed to see if it leads to supersymmetric fixed points.
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