Towards a $C$-theorem in defect CFT

TL;DR

This paper investigates a potential $C$-theorem in defect conformal field theories, proposing that the regularized sphere free energy associated with defects decreases along RG flows, supported by holographic model proofs.

Contribution

It introduces a new $C$-theorem for DCFTs based on defect contributions to sphere free energy, unifying previous conjectures and providing holographic proofs.

Findings

Sphere free energy decreases along defect RG flows.

Entanglement entropy and free energy are equivalent for codimension-one defects.

Holographic models support the proposed $C$-theorem.

Abstract

We explore a $C$-theorem in defect conformal field theories (DCFTs) that unify all the known conjectures and theorems until now. We examine as a candidate $C$-function the additional contributions from conformal defects to the sphere free energy and the entanglement entropy across a sphere in a number of examples including holographic models. We find the two quantities are equivalent, when suitably regularized, for codimension-one defects (or boundaries), but differ by a universal constant term otherwise. Moreover, we find in a few field theoretic examples that the sphere free energy decreases but the entanglement entropy increases along a certain renormalization group (RG) flow triggered by a defect localized perturbation which is assumed to have a trivial IR fixed point without defects. We hence propose a $C$-theorem in DCFTs stating that the increment of the regularized sphere free energy due to the defect does not increase under any defect RG flow. We also provide a proof of our proposal in several holographic models of defect RG flows.