Towards a $C$-theorem in defect CFT

TL;DR

This work proposes a unified C-theorem for defect CFTs by promoting the universal part of the defect free energy, $\tilde{D} = \sin(\\pi p/2) \, log|\\langle {\cal D}^{(p)} \rangle|$, to a monotone quantity along defect RG flows, thereby bridging the BCFT and DCFT theorems into a single framework. It systematically compares defect free energy and defect entropy, showing that their equality holds for codimension-one defects but not in general, and demonstrates that defect entropy need not decrease along flows, motivating the defect-free-energy criterion. The authors support the proposal with conformal perturbation theory and Wilson-loop defect tests in multiple field theories, and provide a robust holographic proof via NEC-based arguments in several DCFT models, including domain-wall, probe-brane, and AdS/BCFT setups. The results unify existing monotonicity theorems (g-, b-, F-theorems) within a DCFT context and offer a path toward stronger, perhaps relative-entropy-based, proofs, with potential implications for defect dynamics and IR fixed-point structure.

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.

Figures (5)

• Figure 1: (Left) A dimension-one conformal defect $\mathcal{D}^{(1)}$ in Lorentzian flat spacetime. The spherical subsystem $A$ of radius $R$ surrounds the defect. (Center, Right) A codimension-one defects $\mathcal{D}^{(d-1)}$ as an interface (Center) and a boundary (Right). The subsystem $A$ intersects with the defect in these cases.
• Figure 2: The locations of the entanglement surface $\Sigma$ and the conformal defect $\mathcal{D}^{(p)}$ in the hyperbolic coordinates \ref{['EHyper']}. The hyperbolic space $\mathbb{H}^{p-1}$ is fibered on each point of the base.
• Figure 3: The conformal defect $\mathcal{D}^{(1)}$ on $\mathbb{S}^{d}$. For $d=2$, it winds along the equator of $\mathbb{S}^{2}$ ($\tau$-direction).
• Figure 4: The defect free energy (Left) and the defect entropy (Right) of the 1/2-BPS Wilson loop in the 4$d$${\cal N}=4$ SYM. The $N=2$ cases are shown. The defect free energy is positive for any $\lambda$ while the defect entropy can be negative.
• Figure 5: The bulk AdS space $\mathcal{B}$ is surrounded by the union $\mathcal{M} \cup \mathcal{Q}$ of the boundary and bulk hemispheres. It is bipartited by the Ryu-Takayanagi surface anchored on the entanglement surface $\Sigma$.

Theorems & Definitions (1)

• Conjecture