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

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.

arXiv comment39 pages + appendices, multiple figures; Minor corrections

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• 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories
• Swampland Conditions for Higher Derivative Couplings from CFT
• Bootstrapping the $a$-anomaly in $4d$ QFTs
• A Generalized Nachtmann Theorem in CFT