Renormalization Group Flows, the $a$-Theorem and Conformal Bootstrap

TL;DR

This paper links renormalization group flows to conformal field theories via spectral deformations, providing a new proof of the 4d a-theorem and framing the 6d a-theorem as a conformal bootstrap problem.

Contribution

It introduces a novel approach connecting RG flows to spectral deformations of generalized free CFTs and offers a new proof of the 4d a-theorem, reinterpreting the 6d case as a bootstrap problem.

Findings

Provided a simple proof of the 4d a-theorem.

Reinterpreted the 6d a-theorem as a conformal bootstrap problem.

Linked RG flows to spectral deformations of CFTs in a new framework.

Abstract

Every renormalization group flow in $d$ spacetime dimensions can be equivalently described as spectral deformations of a generalized free CFT in $(d-1)$ spacetime dimensions. This can be achieved by studying the effective action of the Nambu-Goldstone boson of broken conformal symmetry in anti-de Sitter space and then taking the flat space limit. This approach is particularly useful in even spacetime dimension where the change in the Euler anomaly $ a_{UV}-a_{IR}$ can be related to anomalous dimensions of lowest twist multi-trace operators in the dual CFT. As an application, we provide a simple proof of the 4d $a$-theorem using the dual description. Furthermore, we reinterpret the statement of the $a$-theorem in 6d as a conformal bootstrap problem in 5d.