--- title: Complexity equals Volume (CV) in Holography url: https://www.emergentmind.com/topics/complexity-equals-volume-cv type: topic --- # Complexity equals Volume (CV) in Holography Complexity equals Volume (CV) is a holographic prescription that associates quantum complexity with the volume of a codimension-one bulk hypersurface anchored on a specified boundary time slice. In its standard form, the proposal is written as \[ \mathcal{C}_V=\frac{V(\Sigma)}{G_N\,\ell}, \] or equivalently \(V(\Sigma)/(G_N R)\), where \(\Sigma\) is an extremal or maximal-volume slice, \(G_N\) is Newton’s constant, and the length scale is typically taken to be the AdS radius or another reference scale [1801.01137][2008.09138]. The same idea has been extended to subregions, finite-cutoff holography, time-dependent quenches, rotating and charged black holes, de Sitter settings, higher-curvature theories, and even information-geometric analogues of Krylov complexity [1910.08082][2408.06055][2606.03049][2412.08925]. ## 1. Standard prescription and normalizations In the two-sided black-hole setting, CV assigns the complexity of a thermofield-double-like state on boundary times \((t_L,t_R)\) to the maximal spatial volume of a bulk hypersurface \(\Sigma_{t_L,t_R}\) connecting the two asymptotic boundaries [2103.13186]. For static metrics and at moments of time-reflection symmetry, the relevant hypersurface is often the \(t=0\) slice, which is extremal and, in the cases treated explicitly, maximal [1801.01137]. In subregion versions, following the entanglement-wedge construction, one replaces the full maximal slice by the codimension-zero region bounded by a boundary subregion and its RT or HRT surface [1910.08082][1803.11162]. The normalization is not unique. Several papers use the standard \(V/(G_N\ell)\) form, with \(\ell\) taken to be the AdS radius or kept as an arbitrary scale [1801.01137][1808.09917]. A refinement proposed for stationary black holes replaces this by \[ \mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f}, \] where \(A_P=4G_N\hbar\) and \(\tau_f\) is the maximal proper time from the horizon to the late-time “final slice”; this was argued to remove the apparent mismatch between large and small black holes and to recover the universal scaling \(d\mathcal{C}/dt\sim T_HS_{BH}/\hbar\) [1807.02186]. Other contexts require adapted choices: for timelike extremal slices in Schwarzschild–de Sitter, the volume is imaginary and one introduces an imaginary scale \(L_r\) so that \(\mathcal{C}_V=\mathcal{V}_{\max}/(G_NL_r)\) is real and positive [2606.03049]. A closely related observable is the complexity of formation, defined by subtracting a reference state. In AdS black holes this is often taken as the difference between the wormhole volume and the corresponding vacuum or massless-BTZ contribution, with the subtraction fixing the additive ambiguity and removing UV divergences [1801.01137][2008.09138]. ## 2. Extremal hypersurfaces, conserved quantities, and reconstruction formulas For static geometries of the form \[ ds^2=-f(r)\,dt^2+g(r)\,dr^2+r^2\,d\Sigma_{k,d-1}^2, \] the maximal slice can be described in ingoing Eddington–Finkelstein coordinates by embedding functions \(v(\lambda),r(\lambda)\). Defining \[ F(r)=f(r)\,r^{2(d-1)},\qquad G(r)=g(r)\,r^{2(d-1)}, \] the volume functional takes the form \[ V[\Sigma]=2\,\Omega_{k,d-1}\int d\lambda\,\sqrt{-F(r)\,\dot v^2+2\sqrt{F(r)G(r)}\,\dot v\,\dot r}, \] with a conserved quantity \(E\) because \(v\) appears only through \(\dot v\) [2103.13186]. The turning point \(r_{\min}\) of the slice satisfies \[ E^2=-F(r_{\min}), \] and the growth rate simplifies to \[ \frac{dV}{dt_R}=2\,\Omega_{k,d-1}\sqrt{-F(r_{\min})}, \qquad \frac{d\mathcal{C}}{dt_R} = \frac{2\,\Omega_{k,d-1}}{G_NL}\sqrt{-F(r_{\min})}, \] with the late-time limit controlled by the interior radius \(r_f\) maximizing \(\sqrt{-F(r)}\) [2103.13186]. This structure underlies a reconstruction program: given \(\dot{\mathcal C}(t)\) and the exterior geometry, one can invert Abel-type integral equations to recover the black-hole interior metric function \(f(r)\), and, together with Hartman–Maldacena entropy growth, also \(g(r)\) [2103.13186]. In BTZ, the method reconstructs the exact interior relation \(r=\sqrt{f+1}\) [2103.13186]. A complementary formulation uses the conserved volume current \(J^a\), the future-directed unit normal to a maximal foliation. Since the trace of the extrinsic curvature vanishes on each leaf, \(\nabla_aJ^a=0\), and the flux of \(J^a\) through a hypersurface equals its volume [1807.02186]. Under the assumptions stated there, a boundary foliation determines a bulk maximal foliation without gaps, and the horizon flux of \(J^a\) yields a second-law-like statement for interior CV complexity [1807.02186]. ## 3. Exact AdS\(_3\) results, topology, and nonlocality The sharpest exact result for CV is the AdS\(_3\) wormhole formula derived at a moment of time symmetry. For orientable multi-boundary wormholes with genus \(g\) and \(n\) asymptotic regions, the time-symmetric slice is a hyperbolic surface \(\Sigma=H^2/\Gamma\) with Euler character \[ \chi=2-2g-n. \] Using the Brown–Henneaux relation \(c=3\ell/(2G)\), Gauss–Bonnet on the compact interior \(\Sigma_I\), and subtraction against \(n\) copies of the \(M=0\) BTZ black hole, one finds \[ \Delta C_V=\alpha_V\,c\,\chi,\qquad \alpha_V=-\frac{4\pi}{3}, \] or explicitly \[ \Delta C_V=-\frac{4\pi}{3}\,c\,\chi \] [1801.01137]. The derivation uses \[ 4\pi\chi=\int_{\Sigma_I}R^{(2)}\sqrt{|h|}\,d^2x = -\frac{2}{\ell^2}\,\mathrm{Vol}(\Sigma_I), \qquad \mathrm{Vol}(\Sigma_I)=-2\pi\ell^2\chi, \] so the result depends only on topology and not on any continuous moduli [1801.01137]. Because the exterior BTZ contributions cancel in the relative complexity, \(\Delta C_V\) is independent of the temperatures \(M_i\) of the outer regions and of the Fenchel–Nielsen moduli of the interior geometry [1801.01137]. This exact independence led to a strong conclusion: any circuit model reproducing CV in this regime cannot be based on strictly local gates, since locally thermofield-double-like entanglement between thermal-sized patches carries the same CV cost regardless of separation [1801.01137]. The same topology can also modify full time dependence. In the simplest Lorentzian three-boundary AdS\(_3\) wormhole, CV was computed at all times and found to grow nonlinearly and saturate at late times, in contrast to the familiar linear growth of the eternal BTZ geometry [2302.07522]. In the symmetric pair-of-pants case, the complexity of formation approaches a finite constant \(6\pi\) in the large-\(M\) limit [2302.07522]. ## 4. Time dependence, late-time growth, and thermodynamic interpretations For two-sided AdS black holes, CV typically exhibits monotone growth in time and a constant late-time slope. In the Einstein–dilaton family generated by \(A(r)=-a/r^n\), the growth rate approaches a constant from below, and numerically satisfies \[ \lim_{t\to\infty}\frac{dC_V}{dt}\le \frac{8\pi M}{d-1}, \] with saturation only at sufficiently high temperature [1808.09917]. The same paper emphasizes that, unlike CA in that model, CV behaves qualitatively as in earlier AdS black-hole studies: no overshoot of the late-time limit and no Lloyd-bound-type pathology was reported [1808.09917]. For rotating AdS black holes, the complexity of formation appears to be controlled by thermodynamic volume rather than entropy. In equal-spin Myers–Perry–AdS black holes, the proposed large-black-hole scaling is \[ \Delta C = \Sigma_g\,C_T \left(\frac{V}{V_{\mathrm{AdS}}}\right)^{\frac{D-2}{D-1}}, \] for both CV and CA, with \(V\) the thermodynamic volume and \(V_{\mathrm{AdS}}=\ell^{D-1}\) [2008.09138]. In slowly accelerating Kerr–AdS spacetimes with conical deficits, the paper computes CV only for the complexity of formation at \(t=0\), finding that it increases with the average and differential deficits near the static limit but decreases with them near extremality [2212.05902]. CV has also been adapted to de Sitter holography. In Schwarzschild–de Sitter, timelike extremal slices anchored either on a stretched horizon or on \(\mathcal I^\pm\) produce linear late-time growth in both static-patch and dS/CFT schemes, with identical asymptotic slope \[ \frac{d\mathcal{C}_V}{dt_b}\Big|_{\rm late} = \frac{i\,\Omega_{d-1}}{G_NL_r}\, r_a^{d-1}\sqrt{f(r_a)}, \] where \(r_a\) is the maximum of the effective potential \(U(r)=f(r)r^{2(d-1)}\) [2606.03049]. In global de Sitter foliations of AdS, by contrast, the maximal volume scales with the spatial volume of the boundary slice, so at fixed cutoff \(C_V(t_*)\propto (\cosh t_*)^{d-1}\); it is even in \(t_*\), minimized at \(t_*=0\), and grows exponentially at large \(|t_*|\) rather than exhibiting any finite-time divergence [2604.21408]. ## 5. Subregion CV, deformations, and phase-sensitive phenomenology In the subregion version of CV, the relevant object is the codimension-zero region of the entanglement wedge on a fixed time slice. In AdS\(_3\)/CFT\(_2\), this was used to motivate a field-theoretic interpretation via \(T\bar T\) flow: the deformation acts as a reversible circuit, the layer density \[ d[\mu]=\frac{12\pi}{c\mu^2}\,d\mu \] maps through \(\mu=16\pi G_N/r_c^2\) to the radial measure \(d[\mu]=-r_c\,dr_c\), and \[ C(\rho_A)\propto \int d[r]\,\mathrm{Area}(r) \] reproduces the wedge volume underlying subregion CV [1910.08082]. Boundary conditions can change the UV structure. In AdS\(_3\)/BCFT\(_2\), the global CV complexity on the \(t=0\) slice is \[ C_V=\frac{R^2}{G_N l} \left[ \frac{L}{\epsilon} +\cot\alpha\log\frac{z_{\rm IR}}{\epsilon} -\frac{L}{z_{\rm IR}} \right], \] so the boundary introduces a subleading logarithmic divergence with coefficient \(- (R^2/G_Nl)\cot\alpha\) [1910.03489]. The same work shows that subregion CV can jump discontinuously across RT phase transitions; for an interval at critical distance \(d_c\), the jump is \[ \Delta C_V^{(b/c)} = \frac{R^2}{G_N l} \left[ 2\cot\alpha\log\!\big(\cot(\alpha/4)\big)+\pi \right] \] [1910.03489]. In time-dependent states, CV exhibits distinct behavior in full and subregion settings. After a local quench in AdS\(_3\)/CFT\(_2\), the global CV deviation obeys the early-time expansion \[ \Delta\mathcal C_V(t)\approx 16\pi h\left(1+\frac{t^2}{2\alpha^2}\right), \] then continues to grow unboundedly, while the subregion CV of an interval shows early quadratic growth, an intermediate nearly linear rise, and a late return to equilibrium [1803.11162]. The paper interprets this non-monotonic subregion behavior as “effective-complexity-like,” since it can decrease while entanglement entropy and integrated entanglement density continue to increase [1803.11162]. Holographic superconductors provide another phase-sensitive test. For Einstein–Maxwell–scalar models, the superconducting phase was found numerically to have smaller complexity of formation than the unstable normal phase below \(T_c\), and at low temperature the thermal contribution scales as \(T^\alpha\), with \(\alpha=d-1\) for \(m^2=0\), independently of \(q\) [1902.07586]. In subregion CV for a \(2+1\)-dimensional holographic superconductor, the strip complexity has a single UV divergence, grows linearly with large strip width, and tracks phase transitions; in the first-order case it develops an “S” curve, while in the second-order case it shows a kink at \(T_c\) [1903.00613]. At finite cutoff, the \(T\bar T\)-deformed theory yields a particularly geometric correction. The difference between the deformed and undeformed CV complexities is \[ \Delta\mathcal C_V = \mathcal C_V^{(T\bar T)}-\mathcal C_V^{(0)} = \frac{\rho_c}{G_N}\,W_\sigma, \] where \[ W_\sigma=\frac{1}{2(d-1)^2}\int_\sigma d^{d-1}y\,\sqrt h\,K^2 \] is the Willmore, or bending, energy of the time-constant slice \(\sigma\) [2408.06055]. ## 6. Generalizations, ambiguities, and ongoing problems Several works replace the bare volume by higher-curvature or more general geometric functionals. In hyperscaling-violating black branes, generalized volume-complexity takes \[ \mathcal C_{\rm gen} = \max_{\partial\Sigma=\Sigma_\tau} \left[ \frac{1}{G_NL}\int_\Sigma d^{d+1}\sigma\,\sqrt h\,F_1 \right], \] with \(F_1=1+\lambda L^4 C^2\) in the main example, and retains linear late-time growth whenever the associated effective potential has an interior maximum [2207.05287]. In doubly holographic island setups, the near-brane expansion motivates a generalized CV functional involving Wald-like curvature terms and extrinsic-curvature corrections, reducing in Einstein gravity to a generalized island volume on the brane [2010.16398]. Plain CV can also be too restrictive as an interior probe. For multi-horizon Bardeen–AdS black holes, standard CV with \(a(r)=1\) probes only the outermost interior band \(r_{h_2}