The Parisi-Sourlas Uplift and Infinitely Many Solvable 4d CFTs

TL;DR

This paper explores how 2d CFTs can be uplifted to 4d theories with Parisi-Sourlas supersymmetry, revealing new relations between conformal blocks and enabling bootstrap of unknown observables.

Contribution

It demonstrates that any 2d CFT can be uplifted to a 4d PS SUSY CFT, establishing new relations between conformal blocks and expanding bootstrap methods.

Findings

Mapped 2d four-point functions to 43 4d functions related by SUSY

Found 43 non-trivial relations between conformal blocks across dimensions

Identified infinite conserved currents in uplifted minimal models

Abstract

Parisi-Sourlas (PS) supersymmetry is known to emerge in some models with random field type of disorder. When PS SUSY is present the $d$-dimensional theory allows for a $d-2$-dimensional description. In this paper we investigate the reversed question and we provide new indications that any given CFT$_{d-2}$ can be uplifted to a PS SUSY CFT$_{d}$. We show that any scalar four-point function of a CFT$_{d-2}$ is mapped to a set of 43 four-point functions of the uplifted CFT$_{d}$ which are related to each other by SUSY and satisfy all necessary bootstrap axioms. As a byproduct we find 43 non trivial relations between conformal blocks across dimensions. We test the uplift in generalized free field theory (GFF) and find that PS SUSY is a powerful tool to bootstrap an infinite class of previously unknown GFF observables. Some of this power is shown to persist in perturbation theory around GFF. We explain why all diagonal minimal models admit an uplift and we show exact results for correlators and CFT data of the $4d$ uplift of the Ising model. Despite being strongly coupled $4d$ CFTs, the uplifted minimal models contain infinitely many conserved currents and are expected to be integrable.