Holographic Complexity Equals Which Action?

TL;DR

The paper probes the holographic complexity=action (CA) proposal for charged black holes, revealing that late-time CA growth is acutely sensitive to the electric/magnetic charge composition and to the inclusion of Maxwell boundary terms.A detailed analysis of dyonic Reissner–Nordström–AdS spacetimes shows that purely magnetic charges can drive the late-time growth to zero, while suitable boundary terms can restore nonzero growth or equalize electric and magnetic contributions, depending on the choice of boundary coefficient γ.Shock-wave perturbations and switchback effects are used to test stability across charge configurations, and extensions to Einstein–Maxwell–Dilaton theories demonstrate that causal structure governs the observed behavior, with the Maxwell boundary term continuing to play a crucial role.The work then connects these higher-dimensional results to two-dimensional JT gravity and a JT-like model via dimensional reduction, demonstrating that the near-horizon/near-extremal physics governs the IR CA behavior and that the boundary terms induce a mixed CA+volume character in 2D.Overall, the study highlights the critical influence of boundary terms and causal structure on holographic complexity for charged black holes and lays groundwork for further exploration of ensembles, dualities, and lower-dimensional holographic complexity.

Abstract

We revisit the complexity$=$action proposal for charged black holes. We investigate the complexity for a dyonic black hole, and we find the surprising feature that the late-time growth is sensitive to the ratio between electric and magnetic charges. In particular, the late-time growth rate vanishes when the black hole carries only a magnetic charge. If the dyonic black hole is perturbed by a light shock wave, a similar feature appears for the switchback effect, e.g., it is absent for purely magnetic black holes. We then show how the inclusion of a surface term to the action can put the electric and magnetic charges on an equal footing, or more generally change the value of the late-time growth rate. Next, we investigate how the causal structure influences the late-time growth with and without the surface term for charged black holes in a family of Einstein-Maxwell-Dilaton theories. Finally, we connect the previous discussion to the complexity=action proposal for the two-dimensional Jackiw-Teitelboim theory. Since the two-dimensional theory is obtained by a dimensional reduction from Einstein-Maxwell theory in higher dimensions in a near-extremal and near-horizon limit, the choices of parent action and parent background solution determine the behaviour of holographic complexity in two dimensions.

Figures (10)

• Figure 1: (a) Penrose diagram for the Reissner-Nordstrom-AdS black hole (\ref{['RNBas']}). The nonextremal black holes have an outer event horizon at $r=r_{+}$ and an inner Cauchy horizon $r=r_{-}$. The shaded blue region corresponds to a typical WDW patch anchored to symmetric boundary time slices with $t_L = t_R = t/2$. (b) Penrose-like diagram of an Reissner-Nordstrom-AdS black hole with a shock wave inserted on the right boundary at the boundary time $t_R=- t_w$ -- see eq. (\ref{['shocker']}). In order to not tilt the asymptotic boundary after the shock wave, we adopt the Dray-t'Hooft prescription that the null geodesics crossing the collapsing shock wave shifts.
• Figure 2: The rate of change of complexity for the dyonic black hole given by eq. \ref{['RNBas']}, with $r_-=0.3\,r_+$, $L=0.5\,r_+$ and $\ell_\textrm{\tiny ct} = L$. We fix the parameters that determine the geometry, but vary the ratio between electric and magnetic charges. As predicted by eq. \ref{['square']}, when the charge is mostly magnetic, the growth rate of complexity approaches zero at late times. The limit $q_m \rightarrow 0$ essentially matches the top curve for $\chi=10$. Similarly the $q_e \rightarrow 0$ and the $\chi=0.1$ curves are indistinguishable on this scale.
• Figure 3: The difference of complexities of formation in the shock wave geometry as a function of the insertion time $t_w$, for a light shock wave $r_{+,2} = (1+10^{-6}) r_{+,1}$ and parameters $L= 0.5\, r_{+,2}$, $r_{-,1} = 0.5\, r_{+,2}$ and $\ell_{\text{ct}} =L$ ($r_{-,2}$ is then fixed by the condition that $q_{ T}$ is the same after the shock wave). The dashed vertical line is the scrambling time for the shock wave with these geometric parameters. We investigate the effect of varying the ratio between electric and magnetic charges: after the scrambling time, the complexity essentially remains constant for the solution with mostly magnetic charges, as predicted by eq. \ref{['VaidyaDyon']}.
• Figure 4: (a) The complexity for the shock wave geometry as a function of the insertion time $t_w$, for a light shock wave $r_{+,2} = (1+10^{-6}) r_{+,1}$ and parameters $L= 0.5\, r_{+,2}$, $r_{-,1} = 0.5\, r_{+,2}$ and $\ell_\textrm{\tiny ct} =L$. (Further, $r_{-,2}$ is determined by fixing $q_{ T}$ to be the same before and after the shock wave.) We show the result for rescaling the curves in figure\ref{['DyonShockChi']} by the factor $(1+\chi^2)/\chi^2$. We essentially see that the curves lie on top of each other, except the for smallest $\chi$, as it is more sensitive to the transient behaviour controlled by $\ell_\textrm{\tiny ct}$. (b) The influence of the transient behaviour for the complexity in the shock wave geometry. We show in solid $\chi= 0.1$ and in dot-dashed $\chi=10$ to contrast the effect of varying $\ell_\textrm{\tiny ct}$. For $\ell_\textrm{\tiny ct} =0.1\,L$, both curves are essentially on top of each other, but for $\ell_\textrm{\tiny ct} \sim L$, the curves with small $\chi$ are more sensitive to this ambiguous scale.
• Figure 5: The exponential growth for the complexity with a light shock wave, such that $r_{+,2} = (1+10^{-6}) r_{+,1}$, and also with $L= 0.5 r_{+,2}$, $r_{-,1} = 0.5 r_{+,2}$, $l_{\text{ct}} =L$. We show two examples, one for a black hole with mostly electric charge $\chi=10$ and one with mostly magnetic charge $\chi =0.015$. For the larger $\chi$, the dynamics is well approximated by an exponentially growing mode with Lyapunov exponent $\lambda_L = 2 \pi T$ until times of the order of the scrambling time (vertical black line), as in eq. \ref{['VaidyaDyon']}. For the smaller value of $\chi$, the amplitude of this initial mode is suppressed both by the energy of the shock wave and by a factor of $\chi^2$. The exponentially growing mode only dominates at very early times because it must compete with other transient effects. In the limit that the black hole has only magnetic charges ( i.e.,$\chi=0$), this exponentially growing mode is absent.