A complexity/fidelity susceptibility g-theorem for AdS$_3$/BCFT$_2$

TL;DR

The paper uses a holographic Kondo model within the AdS/BCFT framework to investigate how bulk geometric data evolve along RG flows. It proves that bulk volume decreases monotonically along the RG flow for AdS$_3$/BCFT$_2$ models of Takayanagi's type and supports this with numerical results in the holographic Kondo setup, where the temperature is lowered and the embedding moves, causing volume loss. It then discusses a possible interpretation of bulk volume as a measure of complexity or fidelity susceptibility in the dual BCFT, proposing a BCFT g-theorem analogue, and contrasting this with the impurity entropy g-theorem. The work also analyzes the finiteness and model-dependence of the relative complexity, highlights assumptions about energy conditions, and outlines directions for extending the results to higher dimensions and to other holographic complexity proposals.

Abstract

We use a recently proposed holographic Kondo model as a well-understood example of AdS/boundary CFT (BCFT) duality, and show explicitly that in this model the bulk volume decreases along the RG flow. We then obtain a proof that this volume loss is indeed a generic feature of AdS/BCFT models of the type proposed by Takayanagi in 2011. According to recent proposals holographically relating bulk volume to such quantities as complexity or fidelity susceptibility in the dual field theory, this suggests the existence of a complexity or fidelity susceptibility analogue of the Affleck-Ludwig g-theorem, which famously states the decrease of boundary entropy along the RG flow of a BCFT. We comment on this possibility.

Figures (4)

• Figure 1: Setup for the holographic description of a BCFT according to Takayanagi:2011zk. The asymptotically AdS bulk spacetime $N$ has the conformal boundary $M$ and additional boundary $Q$. The defect $P$ is the intersection of $M$ and $Q$. We have used standard coordinates $t,x,z$ as in \ref{['BTZ']}, where $t,x$ are boundary directions and $z$ increases into the bulk. The figure is taken from Erdmenger:2014xya.
• Figure 2: Embedding profiles $x_+(z)$ for the embedding of the brane $Q$ into the bulk spacetime \ref{['BTZ']}. Note that the bulk spacetime $N$ is located to the right of the curves, at larger $x$-values. See also figure \ref{['fig::NMQP']} again. At $T=T_c$, the scalar field vanishes everywhere and the embedding is known to be given by a constant tension solution \ref{['backgroundSolutionGaugeAndEmbedding']}. As the temperature is lowered (or $\mu$ is increased), the scalar field condenses and the brane bends to the right. The figure is presented as in Erdmenger:2015spo.
• Figure 3: Relative complexity as defined in \ref{['ComplexityIntegral']} shown as a function of $T/T_c$. Although, as discussed in the text, in the Kondo model this quantity is by definition finite as $\epsilon\rightarrow0$, we have calculated these points using an explicit cutoff $\epsilon=10^{-10}$ in order to avoid numerical problems.
• Figure 4: Geometric setup for the proof of volume loss along a boundary RG flow parametrised by $\mu$.