A Tale of Two Uplifts: Parisi-Sourlas with Defects

Abstract

Defects in conformal field theories (CFTs) play a key role in critical phenomena by modifying scaling behaviors and generating new universality classes. We introduce Parisi-Sourlas (PS) supersymmetry in the presence of extended operators and demonstrate that any $p$-dimensional defect in a CFT${}_d$ can be uplifted to a defect in a PS-supersymmetric CFT${}_{d+2}$. Surprisingly, there are actually two distinct uplifted defects--of dimensions $p$ and $p+2$--which reduce to the original one. We show how this reduction works for correlators with insertions both in the bulk and on the defect. As a byproduct, we find new relations between defect conformal blocks in dimensions $d$ and $d+2$. We further show that the reduction of the $p$-dimensional defect implies and extend a "global symmetry reduction" previously considered in the literature. Finally, we provide various examples of these uplifts, including perturbative computations in epsilon expansion of the uplift of the Ising magnetic line defect, as well as exact computations of observables in the four-dimensional uplift of minimal models with boundaries.

36 pages + appendices, 7 figures, 2 Mathematica notebooks, References added