Does boundary distinguish complexities?

Yoshiki Sato,Kento Watanabe

doi:10.1007/jhep11(2019)132

Abstract

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 AdS3/BCFT2 model as the argument by Chapman et al. . Thus, we conclude that boundary does not distinguish the complexities in general.

Highlights

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 AdS3/BCFT2 model as the argument by Chapman et al
JHEP11(2019)132 the holographic dual of the complexity is given by the gravitational action on the WheelerDeWitt (WDW) patch, CA
The CV and CA conjectures were originally proposed as holographic duals of the complexity but its definition in quantum field theory did not exist at that time

Summary

Path-integral optimization in BCFT

We will work on the path-integral optimization in BCFT to compute the optimized Liouville action proposed as the complexity of the ground state [11, 12]. 2.1 Ground state wave functional in BCFT and boundary Liouville action Consider a two-dimensional CFT on half line with a flat Euclidean metric, ds2 = δabdxadxb = dz2 + dx. For the purpose to estimate the wave functional effectively, the path integral is redundant because some high-energy degrees of freedom would be suppressed in the deep region of the bulk M. To reduce such degrees of freedom, we deform the background metric with a boundary condition keeping the wave functional.

Optimize the Liouville action

Boundary entropy

CV conjecture

CA conjecture

Findings

Discussions