Towards a C-theorem in defect CFT

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.

From the Weyl Anomaly to Entropy of Two-Dimensional Boundaries and Defects

“…Eq (14). agrees with a key result of ref [15]. that the coefficient of ln ε in S def EE includes a contribution from the stress-tensor one-point function.…”

“…that the coefficient of ln ε in S def EE includes a contribution from the stress-tensor one-point function. We have extended that result, showing that this contribution is proportional to the defect central charge d 2 .For defects of various dimensions, refs [15,. 33, 34] found that the defect's universal contribution to EE need not obey a defect c-theorem.…”

How a -Type Anomalies Can Depend on Marginal Couplings

“…We useD for the bulk connection and D for the induced connection. 5 This term (13) cannot stand by itself since it does not satisfy the WZ consistency condition [9]. This condition follows from the Abelian nature of the Weyl symmetry and requires that two Weyl transformations commute, namely [δ, δ ]W = 0.…”

“…(We discuss the uniqueness of the solution below.) 5 One can easily include a connection Γ K IJ in the space of couplings. The difference is Weyl invariant by itself and hence doesn't affect our result.…”

“…The a-type surface anomaly that is the focus of this letter must decrease under boundary RG flow [4]. There are arguments that a and F quantities associated with hypersurfaces should also have a monotonicity property [5][6][7]. Another way of characterizing the a-anomaly is through the trace of the stress tensor.…”