A Tale of Two Uplifts: Parisi-Sourlas with Defects

TL;DR

This work establishes two consistent Parisi–Sourlas uplifts (PSU and TSU) for any $p$-dimensional defect in a CFT$_d$, embedding the DCFT into a PS CFT$_{d+2}$ while ensuring dimensional reduction back to the original defect. It derives new defect conformal block relations across dimensions, proves non-perturbative decoupling and diagrammatic reduction, and demonstrates a global symmetry reduction $OSp(m|n)\to O(m-n)$ within the TSU framework. The authors validate the construction with explicit correlator uplift rules for one-point, bulk–defect, and bulk–bulk two-point functions, as well as three-point functions, and illustrate through examples including the Wilson–Fisher line defect and conformal boundaries in minimal models. The results provide a robust, nontrivial foundation for the PS uplift of extended probes and open avenues for applications in bootstrap, exact computations, and potential realizations in random-field systems 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.

Figures (7)

• Figure 1: Diagram of relations between the fixed point of a random field theories in $d+2$ dimensions (RF$_{d+2}$ f.p.) in the presence of a defect, a PS CFT$_{d+2}$ with defects and a pure CFT$_{d}$ with a defect.
• Figure 2: Two different PS uplifts of the same $(p,q)$ defect theory.
• Figure 3: Dimensional reduction.
• Figure 4: Extended $p$-dimensional defects in CFT$_d$.
• Figure 5: Parallel space uplift (PSU).