Holographic Momentum-Krylov Complexity Duality

• Holographic Momentum-Krylov Complexity Correspondence is a duality linking the time derivative of Krylov complexity in boundary QFTs to the proper momentum of a falling bulk probe.
• It employs geometric computations in diverse holographic models to quantify operator growth and diagnose phase transitions through analytic formulations and computed trajectories.
• This framework offers a universal bridge across conformal, confining, and deformed gauge theories, yielding actionable insights into quantum gravity and information dynamics.

The holographic momentum-Krylov complexity correspondence provides a geometric duality between operator growth in strongly coupled quantum systems and the proper momentum of a probe in the bulk spacetime. This correspondence systematically connects the time derivative of Krylov (or spread) complexity in boundary quantum field theories to the canonical momentum of a falling particle along a specific radial trajectory in the holographic dual. The framework offers a sharp, calculable, and universal bridge between quantum complexity diagnostics (notably Krylov complexity) and gravitational dynamics, and it interpolates between conformal, confining, and deformed holographic gauge theories.

1. Fundamental Prescription: Momentum-Complexity Correspondence

The central relation is a dictionary that matches the growth rate of Krylov complexity $C(t)$ in the boundary theory to a proper (or canonical) momentum $P_{\bar\rho}(t)$ of a massive bulk particle falling along a trajectory encoded by boundary time $t$: $\dot C(t) = -\frac{P_{\bar\rho}(t)}{\epsilon}$ where $\epsilon$ is a UV cutoff set by the field theory regulator, and $\bar\rho$ is the "proper radial coordinate" along the trajectory, defined such that $ds^2 = d\bar\rho^2$ on-shell. In Lorentzian AdS backgrounds, this reduces to: $$P_{\bar\rho}(t) = m\frac{d\bar\rho}{dt} \left[ -g_{tt}(r) + (\frac{d\bar\rho}{dt})^2 \right]^{-1/2}$$ The computation requires specifying the background geometry and the probe's motion, which may involve general radial infall, motion on submanifolds (e.g., $AdS_3$ in $AdS_5$), or extensions to include other conserved charges or non-trivial internal space navigation (Fatemiabhari et al., 24 Nov 2025, Fan, 2024, Aguilar-Gutierrez et al., 3 Jun 2025, Fatemiabhari et al., 19 Feb 2026).

This correspondence emerges convincingly across diverse holographic contexts, including:

• Pure conformal backgrounds ($AdS_n$), where it yields unbounded, typically linear or exponential complexity growth.
• Confining backgrounds with a smooth IR cap, leading to oscillatory complexity and momentum.
• Geometries with nontrivial flavor/color (e.g., quiver gauge theories), where additional proper momentum components contribute.
• Deformed backgrounds (e.g., Yang–Baxter deformations), where the bulk cutoff and initial velocity modify asymptotics but the core duality persists (Roychowdhury, 10 Jan 2026).
• The double-scaled SYK model and JT/sine-dilaton gravity, in which Krylov complexity and geodesic length (volume) are exactly matched (Heller et al., 2024, Fu et al., 26 Oct 2025).

2. Krylov Complexity: Operator Growth and Lanczos Chains

Krylov complexity $C_K(t)$ quantifies the "spread" of an operator's Heisenberg evolution in the Krylov (Lanczos) basis, constructed via recursive orthogonalization of nested commutators $[H,\cdot]$ acting on a reference operator. The basis is defined by tridiagonal Lanczos recursion, and the wavefunction amplitudes $\psi_n(t)$ satisfy a discrete Schrödinger equation: $$H|n\rangle = a_n |n\rangle + b_n |n-1\rangle + b_{n+1} |n+1\rangle$$

$$\ket{O(t)} = \sum_{n=0}^\infty i^n \psi_n(t) |O_n\rangle, \quad C_K(t) = \sum_n n |\psi_n(t)|^2$$

The structure and asymptotics of the Lanczos coefficients $b_n$ encode spectral data and dynamical properties (e.g., chaos, integrability, confinement). For integrable or free theories on non-compact space, $b_n\sim n$ yields Krylov complexity $C_K(t)\sim e^{const\, t}$, whereas in confining or compact settings $b_n$ saturates or oscillates, giving bounded or periodic complexity (Fan, 2024, Anegawa et al., 2024).

In holographic models, explicit analytic results exist in several cases:

• For $SL(2)$ subsectors and $AdS_3/CFT_2$ primaries: $C_K(t)$ grows sinusoidally or linearly (Fatemiabhari et al., 24 Nov 2025, Aguilar-Gutierrez et al., 3 Jun 2025).
• In the deconfined (black hole) phase: $b_n \sim \alpha n$ leads to $C_K(t) \sim e^{2\alpha t}$, with $\alpha=\pi/\beta$ in $AdS_3$/BTZ at temperature $T=1/\beta$ (Fan, 2024).
• In confining geometries: $b_n$ oscillates, $C_K(t)$ is bounded and periodic (Fatemiabhari et al., 19 Feb 2026).

3. Holographic Realizations: Examples and Explicit Computations

a. Conformal and Integrable Sectors:

• In $AdS_3$ slicing of $AdS_5$, the $SL(2)$ sector of ${\cal N}=4$ SYM is exactly tractable. There, radial geodesic motion simplifies to:

$$\tanh\rho(t) = \tanh\rho_0 \cos t, \qquad P_{\bar\rho}(t) = m\tanh\rho_0 \sin t$$

This duals exactly to sinusoidal Krylov complexity growth in the $SL(2)$ chain (Fatemiabhari et al., 24 Nov 2025).

b. Full ${\cal N}=4$ SYM Dynamics:

• Allowing nontrivial motion in both radial directions in the $AdS_5$-sliced-by-$AdS_3$ geometry, the proper momentum is:

$$P_{\bar\rho}(t) = \frac{H}{\cosh r \cosh\rho} \sqrt{1 - \frac{\cosh^2 r \cosh^2\rho}{\cosh^2 r_0 \cosh^2\rho_0}}$$

This description encodes the Krylov complexity dynamics of the full theory, with the $SL(2)$ case as a special limit (Fatemiabhari et al., 24 Nov 2025).

c. Confining Gauge Theories:

• In the Anabalón–Ross soliton (IR capped): $\dot K(t) = P_{y}(t)$ is oscillatory; the period and amplitude are set by the IR mass gap and UV cutoff, respectively. Exact analytic solutions in terms of elliptic functions are available (Fatemiabhari et al., 27 Nov 2025, Fatemiabhari et al., 19 Feb 2026).

d. Quiver Gauge Theories:

• In warped/fibered AdS backgrounds (e.g., AdS$_3$/AdS$_2$ quivers), the trajectory includes both radial and "quiver" components $(r(t),\eta(t))$. Early-time growth is quiver-structure dependent; at late times, $\dot C_K(t)$ reverts to universal pure-AdS growth as $\eta(t)$ motion damps (Fatemiabhari et al., 16 Dec 2025).

e. Deformed Geometries:

• Yang–Baxter deformations yield a shifted initial radial position, finite radial span, and suppressed maximum momentum. The momentum-complexity equality persists, with the proper dictionary (Roychowdhury, 10 Jan 2026).

4. Universal Features and Diagnostic Power

A summary of robust qualitative phenomena:

• Conformal Cases: Unbounded linear or exponential growth of $C_K(t)$, matching pure AdS or thermal black hole backgrounds.
• Confining Holography: Oscillatory, bounded complexity and momentum, with frequency $\Omega$ set by the mass gap or cap, and amplitude controlled by the UV cutoff. This oscillatory behavior is universal among confining geometries and matches Krylov complexity in integrable-confined quantum spin chains (e.g., Ising with longitudinal field) (Fatemiabhari et al., 19 Feb 2026, Fatemiabhari et al., 27 Nov 2025).
• Phase Transitions: Krylov complexity provides an order parameter for the confinement/deconfinement transition, with exponential (deconfined) or oscillatory (confined) late-time behavior (Anegawa et al., 2024).
• Effect of Conserved Charges: Additional charges (e.g., angular momentum, R-charge) alter the frequency/amplitude but preserve qualitative periodicity (Fatemiabhari et al., 19 Feb 2026, Fatemiabhari et al., 27 Nov 2025).

5. Extended Correspondences and Quantum Gravity Regimes

Double-Scaled SYK and 2D Gravity:

• In DSSYK/JT/sine-dilaton gravity, Krylov complexity exactly equals (up to normalization) the bulk geodesic length (complexity=volume) at both semiclassical and quantum disk levels, including leading $1/G_N$ quantum corrections:

$$C_K(t) = \frac{1}{2|\ln q|} L_{\text{quantum}}(t;q,\beta)$$

The correspondence extends to higher moments (capturing replica wormholes), logarithmic Krylov complexity (probing the saddle structure), and spread entropy (matching von Neumann entropy after tracing out baby-Universes) (Heller et al., 2024, Fu et al., 26 Oct 2025).

Switchback Effects:

• The presence of shockwave perturbations in the bulk (e.g., boundary operator quench) is manifest in Krylov complexity as a delay in linear growth (the "switchback" effect), in precise analogy to classical holographic complexity (Heller et al., 2024).

6. Practical Implications and Open Directions

The momentum-Krylov complexity correspondence:

• Provides a calculable entry in the AdS/CFT dictionary that concretely links quantum operator growth to gravitational dynamics.
• Serves as a diagnostic for confining dynamics, information scrambling, and phase transitions in strongly coupled field theories.
• Suggests that Krylov complexity is sensitive to the global and local properties of the holographic dual (geometry, mass gap, spectral density, deformations).
• Extends to top-down string constructions (flavored quivers, Yang–Baxter deformations) and lower/higher-dimensional models.

A plausible implication is that Krylov complexity and its geometric dual can act as universal probes of quantum information dynamics, complementary to entanglement and out-of-time-order correlators, with scope to test quantum gravity phenomena (e.g., topology change, baby universes) in dual field theories.

7. Table: Archetypal Holographic Realizations

Holographic Dual Geometry	Krylov Complexity Regime	Proper Momentum Behavior
$AdS_3$ Slicing / $SL(2)$ Subsector	Sinusoidal or linear growth	$P_{\bar\rho} \sim \sin t$
Generic Confined (e.g., Anabalón–Ross)	Oscillatory, bounded	$P_y(t)$ periodic, finite
AdS-Schwarzschild / Black Hole	Exponential growth	$P_r(t) \sim e^{\lambda t}$
Quiver/Deformed (YB, flavor, etc.)	Early-time uplift, late universal	Damped to universal

The table summarizes the primary qualitative outcomes found in top-down holographic settings, as documented in (Fatemiabhari et al., 24 Nov 2025, Fatemiabhari et al., 27 Nov 2025, Fatemiabhari et al., 16 Dec 2025, Fatemiabhari et al., 19 Feb 2026, Roychowdhury, 10 Jan 2026).

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References:

• "Holographic Krylov complexity in ${\cal N}=4$ SYM" (Fatemiabhari et al., 24 Nov 2025)
• "Holographic Krylov complexity in confining gauge theories" (Fatemiabhari et al., 27 Nov 2025)
• "Holographic Krylov Complexity for Conformal Quiver Gauge Theories" (Fatemiabhari et al., 16 Dec 2025)
• "Krylov Complexity, Confinement and Universality" (Fatemiabhari et al., 19 Feb 2026)
• "Momentum-Krylov complexity correspondence" (Fan, 2024)
• "Krylov operator complexity in holographic CFTs: Smeared boundary reconstruction and the dual proper radial momentum" (Aguilar-Gutierrez et al., 3 Jun 2025)
• "Holographic Krylov complexity for Yang-Baxter deformed supergravity backgrounds" (Roychowdhury, 10 Jan 2026)
• "Krylov spread complexity as holographic complexity beyond JT gravity" (Heller et al., 2024)
• "Toward Krylov-based holography in double-scaled SYK" (Fu et al., 26 Oct 2025)
• "Krylov Complexity in Quantum Field Theory" (Adhikari et al., 2022)
• "Krylov complexity as an order parameter for deconfinement phase transitions at large $N$" (Anegawa et al., 2024)