Instability of complex CFTs with operators in the principal series

TL;DR

This paper demonstrates that certain complex conformal field theories with operators in the principal series are inherently unstable, linking this instability to tachyonic modes in the dual AdS space, using harmonic analysis and effective action methods.

Contribution

It provides a direct proof of the instability of d-dimensional CFTs with principal series operators, extending the understanding of AdS/CFT correspondence and conformal stability.

Findings

Instability of CFTs with principal series operators proven.

Connection to tachyonic modes in AdS space established.

Explicit examples include melonic tensor models and fishnet models.

Abstract

We prove the instability of $d$-dimensional conformal field theories (CFTs) having in the operator-product expansion of two fundamental fields a primary operator of scaling dimension $h=d/2+i r$, with non-vanishing $ r\in\mathbb{R}$. From an AdS/CFT point of view, this corresponds to a well-known tachyonic instability, associated to a violation of the Breitenlohner-Freedman bound in AdS${}_{d+1}$; we derive it here directly for generic $d$-dimensional CFTs that can be obtained as limits of multiscalar quantum field theories, by applying the harmonic analysis for the Euclidean conformal group to perturbations of the conformal solution in the two-particle irreducible (2PI) effective action. Some explicit examples are discussed, such as melonic tensor models and the biscalar fishnet model.