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

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 nondisordered 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.

More on holographic correlators: twisted and dimensionally reduced structures

“…This turned out to be related to the linear combination one needs in order to repackage bosonic conformal blocks into superblocks of the Parisi-Sourlas superconformal algebra. As shown in [15,19], correlation functions in d dimensions with this supersymmetric property can be interpreted as those of a theory in d − 2 dimensions. For the theories studied here, we were able to use this relation twice so that the…”

“…Identifying these hidden structures not only leads to more compact expressions, but also suggests new symmetry properties of the bulk theory. In particular, we will show that the correlators exhibit an emergent dimensional reduction structure, which is closely related to the Parisi-Sourlas supersymmetry [14,15]. The second aspect concerns certain protected subsectors of the superconformal field theories.…”

“…For this reason, theories with Parisi-Sourlas supersymmetry have very unusual features, from a high energy physicist's point of view. Recently, this supersymmetry was scrutinized in [15] using conformal bootstrap techniques, where a number of kinematical results were derived. In particular, [15] showed that the Parisi-Sourlas supersymmetry leads to a beautiful identity for conformal blocks across different spacetime dimensions…”

“…There are two ways one can have a contribution of order N −3 from potentially singular terms in the s-JHEP04(2021)008 channel OPE: either the exchange of a generator W (p) , or that of : W (k 3 ) W (k 4 ) :. 15 The latter appears with an OPE coefficient of order N −3 in W (k 1 ) ×W (k 2 ) , and order 1 in W (k 3 ) × W (k 4 ) , hence we are allowed to ignore it only if it is a regular term, which happens if k 1 + k 2 ≤ k 3 + k 4 . A similar story applies to the t-channel OPE, which leads to the requirement that k 2 + k 3 ≤ k 1 + k 4 .…”

“…Using crossing symmetry one can extend these results to any 15 The prefactor chosen for the four-point function is such that…”