Holographic Complexity for Defects Distinguishes Action from Volume

Abstract: We explore the two holographic complexity proposals for the case of a 2d boundary CFT with a conformal defect. We focus on a Randall-Sundrum type model of a thin AdS$_2$ brane embedded in AdS$_3$. We find that, using the "complexity=volume" proposal, the presence of the defect generates a logarithmic divergence in the complexity of the full boundary state with a coefficient which is related to the central charge and to the boundary entropy. For the "complexity=action" proposal we find that the complexity is not influenced by the presence of the defect. This is the first case in which the results of the two holographic proposals differ so dramatically. We consider also the complexity of the reduced density matrix for subregions enclosing the defect. We explore two bosonic field theory models which include two defects on opposite sides of a periodic domain. We point out that for a compact boson, current free field theory definitions of the complexity would have to be generalized to account for the effect of zero-modes.

Holographic Complexity for Defects Distinguishes Action from Volume

The paper "Holographic Complexity for Defects Distinguishes Action from Volume" explores the intriguing differences in holographic complexity proposals, namely the "complexity=volume" (CV) and "complexity=action" (CA), particularly within the context of conformal defects in a two-dimensional boundary conformal field theory (CFT). The authors utilize a Randall-Sundrum type model featuring a thin AdS$_2$ brane embedded in AdS$_3$ as their primary framework for analysis. This study uncovers marked discrepancies between the two proposals, highlighting the effects of defects on holographic complexity measures.

In examining the CV proposal, the study reveals that the presence of a conformal defect leads to a logarithmic divergence in the complexity of the boundary state. This divergence is characterized by a coefficient related to the central charge and the boundary entropy, specifically tied to the concept of Affleck-Ludwig boundary entropy. Notably, this demonstrates an increase in complexity attributable to the defect, offering empirical evidence of how defects can influence holographic complexity. On the other hand, the CA proposal intriguingly shows that complexity remains unaffected by the defect's presence—suggesting that these two approaches compute fundamentally different aspects of complexity.

Further exploration in the paper focuses on subregions that encompass a defect. For the CV proposal, the results indicate that a subregion's complexity is significantly influenced by defects, yielding a logarithmic divergence akin to that observed for the entire boundary state. However, the CA proposal maintains its stance; the subregion's complexity remains unaffected by defects, reinforcing the dramatic difference between the two holographic complexity measures.

From a field theory perspective, the paper examines models of free bosonic field theories incorporating conformal defects, notably evaluating the complexity using Gaussian methods tailored to such systems. The results are mixed: while some model computations align with the CA proposal—exhibiting defect-independent complexity—others reflect the CV proposal's divergent characteristics attributable to defects.

The paper's findings underscore the necessity for refined approaches in understanding holographic complexity, especially in the presence of conformal defects. The identified divergences provoke questions regarding the nature of computational complexity within quantum systems, suggesting potential paths for future research to confirm these results in models embedded in tangible string backgrounds or alternative theoretical constructs.

Summarily, this paper stimulates further contemplation and research into the nontrivial implications of defects in holographic frameworks, emphasizing a need for clear distinctions between propositions when calculating system complexities—both theoretically and practically within the field of quantum physics.