Complexity change under conformal transformations in AdS$_{3}$/CFT$_{2}$

TL;DR

This work computes how holographic complexity changes under small conformal transformations in AdS3/CFT2 using the Complexity=Volume proposal, performing a second-order perturbative expansion around the ground state generated by solution-generating diffeomorphisms. The authors derive a general expression for the second-order volume change, decomposing it into left/right (pure) and mixed contributions in Fourier space, and show that the change is UV-finite, nonnegative, and has distinct time-dependence structures. They illustrate the formal results with explicit examples and compare the holographic results to a field-theory fidelity-based complexity proposal, extracting nontrivial constraints on the reference state and discussing implications for gate sets and nonperturbative effects. The findings provide a concrete link between holographic complexity and field-theoretic notions of complexity, offering a framework to test and refine the choices of reference states and gates in both holographic and field-theoretic contexts.

Abstract

Using the volume proposal, we compute the change of complexity of holographic states caused by a small conformal transformation in AdS$_{3}$/CFT$_{2}$. This computation is done perturbatively to second order. We give a general result and discuss some of its properties. As operators generating such conformal transformations can be explicitly constructed in CFT terms, these results allow for a comparison between holographic methods of defining and computing computational complexity and purely field-theoretic proposals. A comparison of our results to one such proposal is given.

Figures (5)

• Figure 1: 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 $\lambda=1/\epsilon^2$ and $\tilde{\lambda}=1/\epsilon^2$ are shown as dashed (red) and dotted (blue) lines, respectively.
• Figure 2: The space of states with the reference state $\left|\mathcal{R}\right\rangle$, the groundstate $\left|0\right\rangle$ and the state $\left|\psi_{U}\right\rangle\equiv U\left|0\right\rangle$ for the generator $U$ of a small conformal transformation. The red dashed lines signify the complexities of the states $\left|\psi_{U}\right\rangle$ and $\left|0\right\rangle$ with respect to the reference state $\left|\mathcal{R}\right\rangle$.
• Figure 3: First order term in $\sigma$ (as in \ref{['Ttosecondorder']}) of the energy density $\mathcal{E}(t,x)=T_{++}(t+x) + T_{--}(t-x)$ for example 1 with $a_\pm=c_\pm=1$. This picture can be interpreted in terms of two wavepackets, one moving to the left and one moving to the right at the speed of light.
• Figure 4: Plot of the term $\mathcal{V}_{(2),\text{mixed}}(\hat{g}_+,\hat{g}_-)$ for example 1, respectively $-\mathcal{V}_{(2),\text{mixed}}(\hat{g}_+,\hat{g}_-)$ for example 2. For $t_0\rightarrow\pm\infty$, $\mathcal{V}_{(2),\text{mixed}}(\hat{g}_+,\hat{g}_-)\rightarrow0$.
• Figure 5: $\mathcal{V}_{(2),\text{mixed}}(\hat{g}_+,\hat{g}_-)$ for example 3.