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.

Recently, Chapman et al. argued that holographic complexities for defects distinguish action from volume. Motivated by their work, we study complexity of quantum states in conformal field theory with boundary. In generic two-dimensional BCFT, we work on the path-integral optimization which gives one of field-theoretic definitions for the complexity. We also perform holographic computations of the complexity in Takayanagi's AdS/BCFT model following by the "complexity = volume" conjecture and "complexity = action" conjecture. We find that increments of the complexity due to the boundary show the same divergent structures in these models except for the CA complexity in the AdS 3 /BCFT 2 model as the argument by Chapman et al. Thus, we conclude that boundary does not distinguish the complexities in general.

We consider the Schwinger effect in confining phase by using a holographic setup.The potential analysis is performed for the confining D3-brane and D4-brane backgrounds. We find the critical electric field above which there is no potential barrier and the system becomes unstable catastrophically. An intriguing point is that no Schwinger effect occurs when the electric field is smaller than the confining string tension. *

We consider universal aspects of a holographic Schwinger effect in general backgrounds with an external homogeneous electric field. The argument is based on the potential analysis developed in our previous work. Under some conditions, there always exists a critical electric field, above which the potential barrier vanishes and the system becomes unstable catastrophically. The critical value agrees with the one obtained from the Dirac-Born-Infeld action. For general confining backgrounds, we show that the Schwinger effect does not occur when the electric field is weaker than the confining string tension.Comment: 17 pages, v2:accepted versio