Complexity equals Action (CA)

• Complexity equals Action (CA) is a holographic proposal that defines quantum complexity as the gravitational action computed over the Wheeler–DeWitt patch.
• It combines bulk, boundary, null, and joint contributions to capture the dynamics of various spacetimes, including charged, rotating, and de Sitter cases.
• The framework provides insights into late-time growth, Lloyd-type bounds, and the influence of topology and boundary conditions on holographic complexity.

Complexity equals Action (CA) is the holographic proposal that identifies the quantum computational complexity of a boundary state with the on-shell gravitational action of the corresponding Wheeler–DeWitt (WDW) patch, normalized by $1/\pi\hbar$. In its canonical form,

$\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$

with the WDW patch defined as the union of all bulk spacelike surfaces anchored on the chosen boundary time slice, or equivalently as the domain of dependence of any anchored bulk Cauchy slice (Brown et al., 2015). Developed first for AdS black holes, the proposal has since been extended to charged, rotating, accelerating, time-dependent, higher-curvature, finite-cutoff, and de Sitter settings, where its detailed behavior depends on null-boundary terms, horizon data, and the global structure of the bulk region (Jiang, 2018, Fang et al., 2 Jun 2026).

1. Definition through the Wheeler–DeWitt patch

In the CA conjecture, the object to be evaluated is not a maximal codimension-one slice but the full WDW region. For Einstein gravity this requires the bulk Einstein–Hilbert term, Gibbons–Hawking–York terms on timelike and spacelike regulators, null-boundary contributions involving the non-affinity $\kappa$, joint terms at non-smooth intersections, and, in the Lehner et al. formulation, the null counterterm involving the expansion $\Theta\log(\ell_{\rm ct}\Theta)$ (Braccia et al., 2019, Zhang et al., 2022). A representative form used in AdS$_3$ analyses is

$$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$

with analogous generalizations in higher dimensions and in higher-curvature theories (Fu et al., 2018).

This variational structure is central rather than auxiliary. In BCFT and Vaidya calculations, delicate cancellations among bulk, regulator, joint, brane, and null-counterterm contributions determine whether subleading divergences survive and whether the final answer is reparameterization invariant (Braccia et al., 2019, Chapman et al., 2018). In higher-curvature settings, the same issue reappears in Iyer–Wald language, where the null and corner sectors are organized by Wald entropy density rather than by purely Einsteinian boundary data (Jiang et al., 2020).

2. Canonical late-time results and Lloyd-type bounds

Brown et al. derived the standard late-time benchmarks of CA in eternal AdS black holes. For uncharged Schwarzschild–AdS in any bulk dimension,

$$\frac{dS[{\cal W}]}{dt}=2M, \qquad \frac{d\mathcal C_A}{dt}=\frac{2M}{\pi\hbar},$$

which saturates the Lloyd-type bound $d\mathcal C/dt\le 2E/(\pi\hbar)$ (Brown et al., 2015). In four-dimensional Reissner–Nordström–AdS they found

$$\frac{dS[{\cal W}]}{dt}=Q^2\left(\frac1{r_-}-\frac1{r_+}\right),$$

while for rotating BTZ,

$$\frac{dS[{\cal W}]}{dt} = \frac{r_+^2-r_-^2}{4G\ell^2} = 2\sqrt{M^2-(J/\ell)^2}.$$

The same work proposed charged and rotating refinements of the complexity-growth bound involving $\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$0 and $\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$1 (Brown et al., 2015).

Subsequent tests showed that this late-time behavior is robust but not universal in the naive sense of always obeying the simplest bound. In Einstein–Born–Infeld black holes, the CA growth rate violates the generalized Lloyd bound near extremality and near charged regular spacetimes, although along fixed boundary potential curves it tends to be saturated from below when moving away from the ground state (Tao et al., 2017). By contrast, in Born–Infeld gravity on Einstein backgrounds the full on-shell WDW action rescales uniformly, and the late-time result again becomes

$\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$2

once the physical energy is defined by the Brown–York tensor (Bakhtiarizadeh et al., 2020). In Minimal Massive 3D Gravity, the nonrotating BTZ limit $\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$3 likewise yields

$\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$4

so the bound is saturated by the physical mass in that regime (Qaemmaqami, 2017).

3. Horizon-charge formulas beyond Einstein gravity

A major development of the CA program is the Iyer–Wald reformulation of late-time action growth for arbitrary diffeomorphism-invariant theories. For stationary black holes with multiple Killing horizons, the general result can be written as

$\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$5

where $\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$6 are the angular momenta and $\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$7 is the matter-sector contribution defined by the Noether-charge decomposition (Jiang, 2018). For $\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$8 matter this reduces to

$\mathcal C_A=\frac{I_{\rm WDW}}{\pi\hbar},$9

which recovers the familiar Einstein–Maxwell formula as a special case (Jiang, 2018).

This framework clarifies that the late-time CA rate is controlled by horizon charges and chemical potentials rather than by asymptotic AdS structure alone. The derivation does not require AdS asymptotics and extends to higher-curvature theories and arbitrary stationary backgrounds (Jiang, 2018). It also explains why the Brown et al. and Lehner et al. prescriptions agree at late times: the difference between the two action-counting methods comes only from the boundary term on the horizon segments, but both give the identical late-time result once horizon and singularity contributions are assembled (Jiang et al., 2019).

Additional matter couplings modify the horizon formula in a controlled way. In five-dimensional charged supersymmetric black holes of minimal gauged supergravity, the electromagnetic Chern–Simons term adds an anomaly-sensitive correction,

$\kappa$0

so the late-time complexity growth carries information about the chiral anomaly of the dual theory (Jiang et al., 2020). For charged accelerating AdS black holes, conical deficits generate extra terms proportional to the pole tensions $\kappa$1, and in the $\kappa$2 limit these deficit terms disappear, recovering the ordinary charged-AdS result (Jiang et al., 2021).

4. Time dependence, collapse, and the switchback effect

In Vaidya geometries, CA becomes explicitly dynamical. For thin null-shell collapse in AdS, the null-fluid action can be chosen so that the shell is on-shell inert, $\kappa$3, and the shell’s direct contribution to $\kappa$4 vanishes (Chapman et al., 2018). Nevertheless, the null-boundary counterterm is essential: without it, the one-sided black-hole growth rate does not approach the eternal-black-hole value; with it,

$\kappa$5

and the full time dependence interpolates smoothly between early and late regimes (Chapman et al., 2018).

Charged AdS–Vaidya collapse exhibits the same structural feature. The action growth rate and the slope of the complexity of formation agree with the switchback effect for light shocks, but only after including the particular counterterm on the null boundaries (Jiang, 2018). In the light-shock regime, the scrambling time is

$\kappa$6

and the formation slope is approximately zero for $\kappa$7, then becomes constant for $\kappa$8 (Jiang, 2018).

The higher-curvature, multiple-horizon generalization is especially sharp. In a Vaidya geometry with a light shockwave, the slope of the complexity of formation satisfies

$\kappa$9

where $\Theta\log(\ell_{\rm ct}\Theta)$0 is the unperturbed late-time growth rate (Jiang et al., 2020). The null-boundary counterterm is not optional in this analysis: unlike the eternal-black-hole late-time rate, the switchback kink depends crucially on that term (Jiang et al., 2020).

A different dynamical probe arises in the local-quench setup dual to a point particle falling in AdS$\Theta\log(\ell_{\rm ct}\Theta)$1. There, after subtracting the vacuum piece, the CA complexity difference is

$\Theta\log(\ell_{\rm ct}\Theta)$2

with early-time slope $\Theta\log(\ell_{\rm ct}\Theta)$3, exactly saturating the Lloyd bound, while at late times $\Theta\log(\ell_{\rm ct}\Theta)$4 (Ageev et al., 2018).

5. Topology, boundaries, and nonlocality

One of the most distinctive CA results concerns multiboundary AdS$\Theta\log(\ell_{\rm ct}\Theta)$5 wormholes. For wormholes with $\Theta\log(\ell_{\rm ct}\Theta)$6 asymptotic regions and genus $\Theta\log(\ell_{\rm ct}\Theta)$7 in the causal-shadow region, the complexity relative to $\Theta\log(\ell_{\rm ct}\Theta)$8 copies of the $\Theta\log(\ell_{\rm ct}\Theta)$9 BTZ black hole is

$_3$0

so that

$_3$1

The coefficient is independent of temperature and of the Fenchel–Nielsen moduli of the hyperbolic interior, because the exterior contributions cancel in the difference and the interior volume is fixed by Gauss–Bonnet (Fu et al., 2018). Since $_3$2 for nontrivial wormholes, $_3$3 implies that adding handles lowers the CA complexity. The same analysis argues that any dual CFT gate set realizing this complexity cannot be local, because the complexity is blind to how far apart the thermally sized entangled patches are (Fu et al., 2018).

Physical boundaries modify CA differently from CV. In AdS$_3$4/BCFT$_3$5, the full on-shell action reduces to

$_3$6

so the leading $_3$7 divergence is the same as in boundary-less AdS$_3$8, there is no subleading $_3$9 divergence depending on the brane angle $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$0, and the boundary enters through a finite term $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$1 (Braccia et al., 2019). This contrasts with CV and CV2.0, where a subleading logarithmic divergence survives (Braccia et al., 2019).

Conical deficits furnish another geometric sensitivity test. For charged accelerating black holes, the late-time rate gains two new terms proportional to the deficit-induced tensions $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$2, which can be rewritten as $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$3; these vanish smoothly when the deficits are removed (Jiang et al., 2021). For slowly accelerating Kerr–AdS black holes, the CA growth rate acquires deficit corrections that can dominate in the regime $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$4, so the large-black-hole result no longer scales purely as $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$5 (Zhang et al., 2022). Taken together, these examples show that CA is sensitive to global topology, boundary conditions, and localized geometric defects, but not always in the same way as CV.

6. Divergences, renormalization, and alternative formulations

The original CA prescription is not free of structural difficulties. In perturbative Einsteinian cubic gravity and non-perturbative Einstein–Weyl gravity, the naive late-time CA rate becomes divergent because bulk and spacelike-cap contributions near the singularity develop power-law blowups as the inner cutoff $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$6 (Jiang et al., 2019). This led to a modified proposal in which one drops the bulk and spacelike-singularity pieces and retains only the null-segment and joint contributions,

$$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$7

Its late-time growth rate is then

$$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$8

and the same construction reproduces the switchback effect in Vaidya geometry (Jiang et al., 2019).

Finite-cutoff holography provides a different route to an unambiguous CA prescription. In the holographic dual of a $$I_{\rm WDW} = \int_W \frac{R-2\Lambda}{16\pi G}\sqrt{|g|}\,d^3x +\frac{1}{8\pi G}\sum \int K\,d\Sigma +\frac{1}{8\pi G}\sum \int \kappa\,dS +\frac{1}{8\pi G}\sum \int \epsilon\,a\,dS,$$9-deformed CFT, one evaluates the WDW action with a timelike cutoff, null counterterms, and an additional volume counterterm chosen so that the leading divergence is a positive volume law (Astaneh, 2024). The resulting difference between the deformed CA complexity and the renormalized undeformed one is

$$\frac{dS[{\cal W}]}{dt}=2M, \qquad \frac{d\mathcal C_A}{dt}=\frac{2M}{\pi\hbar},$$0

namely the bending (Willmore) energy of the time-constant slice $$\frac{dS[{\cal W}]}{dt}=2M, \qquad \frac{d\mathcal C_A}{dt}=\frac{2M}{\pi\hbar},$$1 (Astaneh, 2024).

Outside AdS, CA can behave qualitatively differently. In Schwarzschild–de Sitter, both the static-patch prescription restricted to the stretched horizon and the dS/CFT prescription at future and past infinity give vanishing CA growth rates, because the regularized action of the restricted WDW region remains finite and its time derivative cancels at late time (Fang et al., 2 Jun 2026). This suggests that the familiar linear late-time growth of CA is not a generic feature of all holographic settings, but a property tied to specific causal and asymptotic structures.