WdW-patches in AdS$_{3}$ and complexity change under conformal transformations II

TL;DR

This work extends the holographic complexity program to the action proposal (CA) in AdS$_3$/CFT$_2$ by analyzing how the groundstate complexity changes under infinitesimal local conformal transformations. It develops a detailed treatment of general WdW-patches in Poincaré AdS$_3$, including caustics, null-null joints, and the full decomposition of the action into bulk, surface, joint, and counter terms, with a σ expansion up to second order. The key finding is that, unlike the volume proposal, the CA quantity acquires terms of order $σ$ and even $σ\log(σ)$ from null joints and counter terms, which has significant implications for viable field-theory duals of CA. The results highlight qualitative differences between complexity proposals, emphasize the geometric role of caustics in dynamical WdW-patches, and constrain possible holographic realizations of CA in 1+1 dimensional CFTs.

Abstract

We study the null-boundaries of Wheeler-de Witt (WdW) patches in three dimensional Poincare-AdS, when the selected boundary timeslice is an arbitrary (non-constant) function, presenting some useful analytic statements about them. Special attention will be given to the piecewise smooth nature of the null-boundaries, due to the emergence of caustics and null-null joint curves. This is then applied, in the spirit of our previous paper arXiv:1806.08376, to the problem of how complexity of the CFT$_2$ groundstate changes under a small local conformal transformation according to the action (CA) proposal. In stark contrast to the volume (CV) proposal, where this change is only proportional to the second order in the infinitesimal expansion parameter $σ$, we show that in the CA case we obtain terms of order $σ$ and even $σ\log(σ)$. This has strong implications for the possible field-theory duals of the CA proposal, ruling out an entire class of them.

Figures (9)

• Figure 1: WdW-patch for the $t=0$ boundary slice in the Poincaré-patch. Technically, the WdW-patch would be the lightly shaded square region between the lightfronts $t=\pm z$ and the Poincaré-horizon. However, we introduce the field-theory UV cutoff $z=\epsilon$ and the IR cutoff $z=z_{max}$ near the Poincaré-horizon, shown as dashed (blue) lines. Hence, the integration-domain $\mathcal{W}$ for the action proposal, which we will still refer to as WdW-patch, is the darkly shaded region. We also mark the locations of the four spacelike joints $\mathcal{J}_1$-$\mathcal{J}_4$.
• Figure 2: A conformal diagram of the Poincaré-patch of AdS$_3$. The vertical line is the asymptotic boundary while the two diagonal lines are the two Poincaré-horizons where $t\rightarrow\pm\infty$. The two cutoff surfaces $z=\epsilon$ and $\tilde{z}=\epsilon$ are shown as dashed (red) and dotted (blue) lines, respectively. The figure is taken from Flory:2018akz.
• Figure 3: Contour plots for the functions $\partial_z\text{t}^+(z,x)$ respectively $\partial_x\text{t}^+(z,x)$ (up to numerical errors, the contours for both expressions are identical) for various boundary conditions $\text{t}^{bdy}(x)$. Top left: $\text{t}^{bdy}(x)=\frac{0.01}{1+x^2}$. Top right: $\text{t}^{bdy}(x)=\frac{-0.01}{1+x^2}$. Bottom left: $\text{t}^{bdy}(x)=\frac{0.01\cdot x}{1+x^2}$. Bottom right: $\text{t}^{bdy}(x)=\frac{0.01}{1+x^4}$. The black lines are projections of the null rays forming the lightfront down to the $x,z$-plane, and should hence be perfectly straight. Any deviation from straight line behaviour is due to numerical inaccuracies. The orange points at the boundary ($z=0$) are what we called hyperbolic points in section \ref{['sec::caustics']}, while the red points in the bulk are caustics, which are generated by the hyperbolic points. These caustics are generally the starting point of creases or null-null joints on which the function $\text{t}^+(z,x)$ is not smooth (leading to increased numerical problems). Starting from a caustic, these creases will extend from there towards the Poincaré -horizon. Those creases that we could determine analytically are marked by a dashed red line, see the discussion later in section \ref{['sec::creases']}. In the case shown on the bottom right, we see that generically, creases may collide and merge into one.
• Figure 4: Bounds relevant for the calculation of the bulk integral, not to scale. Left: Asymptotic boundary. Right: A $x=const.$ slice of the bulk, in Poincaré-patch coordinates of \ref{['Poincare']}.
• Figure 5: Possible interpretation of order $\mathcal{O}(\sigma^2)$ and $\mathcal{O}(\sigma)$ terms in $\delta\mathcal{C}$