Krylov complexity for Lin-Maldacena geometries and their holographic duals

Abstract: We compute the rate of growth of operator size in matrix models by probing the Lin-Maldacena class of geometries with classical probes. We consider massive point particle probes whose proper momentum equals the size of the gauge invariant operator in the matrix model. We work out the example of the BMN Plane Wave Matrix Model using the electrostatic approach and the method of background fluxes. We also work out complexities in the D2 brane as well as NS5 brane limits of the BMN matrix model along with an example of the irrelevant deformation namely the non-Abelian T-dual of $AdS_5 \times S<sup>5$. Finally, we carry out a possible calculation of the Krylov complexity on the matrix model counterpart by using a simple reduction ansatz known as the pulsating fuzzy sphere model. We outline an algorithm to define Krylov basis elements for the matrix model and compute a few Lanczos coefficients. Our analysis reveals that both the Krylov basis states as well as Lanczos coefficients are uniquely fixed in terms of the mass parameter of the matrix model.

• The paper extends holographic Krylov complexity to non-conformal Lin–Maldacena geometries by equating operator-growth rates with the proper momentum of a falling massive probe.
• It finds universal quadratic early-time growth, while late-time behavior depends on the geometry: D2 saturation, NS5 continued growth, D2-shell scaling as t^{35/24}, and divergence near a non-Abelian T-dual singularity.
• It reproduces the quadratic behavior in a pulsating fuzzy-sphere matrix-model truncation, where the first Lanczos coefficients depend on the mass deformation μ, while higher coefficients and precise gravity–field-theory matching remain open.

This paper computes holographic Krylov (spread) complexity of operator growth for the BMN Plane Wave Matrix Model (PWMM) and selected deformations thereof, using the proposal that the rate of growth of complexity in the dual quantum mechanics equals the proper momentum of a massive particle probe falling through the bulk geometry (2604.16977). Unlike earlier applications of this correspondence to conformal duals, the geometries studied here are non-conformal: the UV is the near-horizon geometry of $N$ D0 branes, and the field theory carries a mass deformation parameter $\mu$ inherited from the massive deformation of the BFSS matrix model. The author computes complexity in the electrostatic (Lin–Maldacena) description, in the background-flux (Lin) description, in the D2 and NS5 limits, for the non-Abelian T-dual of $AdS_5 \times S^5$, and finally on the matrix-model side via a pulsating fuzzy sphere truncation.

Setup: probes and proper momentum

The type IIA gravity duals of the PWMM are characterized by a harmonic function $V(\sigma,\eta)$ satisfying $\ddot{V}+\sigma^2 V''=0$ in the electrostatic $(\sigma,\eta)$ plane, with conducting disks at discrete positions $\eta_k \sim N_k$ carrying charges $Q_k \sim n_k$ encoding the fuzzy-sphere vacua $N=\sum_k n_k N_k$. The potential

$$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$

asymptotes to the D0-brane geometry, with $\mu$0 the dipole moment of the disk configuration.

A unit-mass point particle is identified with a local unitary operator $\mu$1 inserted at $\mu$2 in the matrix model; its proper radial momentum $\mu$3, defined with respect to the proper distance $\mu$4 in the Einstein-frame metric, is equated to $\mu$5. The Hamiltonian $\mu$6 is fixed by placing the particle near the asymptotic infinity $\mu$7 with vanishing initial velocity, so the probe corresponds to a heavy operator. The radial coordinate $\mu$8 is well-defined asymptotically because the electrostatic coordinates combine there into the D0 near-horizon radial direction; this identification is a structural assumption of the entire holographic computation.

Early-time growth in the electrostatic approach

Expanding the geodesic equations near the UV and solving the linearized fluctuation $\mu$9, the author finds an oscillatory approach $AdS_5 \times S^5$0 with $AdS_5 \times S^5$1, $AdS_5 \times S^5$2. The resulting complexity grows quadratically:

$AdS_5 \times S^5$3

Two features are noteworthy. First, the universal quadratic early-time growth persists despite the non-conformal, massive nature of the dual theory — the paper emphasizes this as a robust feature, later confirmed by the matrix-model calculation. Second, the leading coefficient depends on the dipole deformation $AdS_5 \times S^5$4, a dependence that reappears on the field-theory side as dependence on the mass parameter $AdS_5 \times S^5$5.

D2 and NS5 limits

The D2-brane limit ($AdS_5 \times S^5$6, dual to $AdS_5 \times S^5$7 SYM on $AdS_5 \times S^5$8) admits a consistent $AdS_5 \times S^5$9 geodesic, corresponding to motion along the $V(\sigma,\eta)$0 axis toward the conducting disk at $V(\sigma,\eta)$1. The trajectory interpolates between a non-AdS UV geometry ($V(\sigma,\eta)$2 scaling) and a two-dimensional Minkowski IR. Numerically integrating the first-order equation $V(\sigma,\eta)$3, the author finds that complexity grows quadratically initially, reaches a maximum when the particle is reflected by the disk, and then saturates as the proper momentum decreases.

The NS5 limit ($V(\sigma,\eta)$4, dual to Little String Theory on $V(\sigma,\eta)$5) is treated for both $V(\sigma,\eta)$6 and $V(\sigma,\eta)$7 geodesics. The UV is conformally non-AdS with a warp factor $V(\sigma,\eta)$8, so $V(\sigma,\eta)$9 asymptotically; the IR is again two-dimensional Minkowski. The qualitative contrast with the D2 case is explicit: the NS5-limit complexity does not saturate at late times, whereas the D2-limit complexity does. For the $\ddot{V}+\sigma^2 V''=0$0 configuration, where the probe travels between the two conducting plates at $\ddot{V}+\sigma^2 V''=0$1, the ratio of complexity growth rates $\ddot{V}+\sigma^2 V''=0$2 equals unity in the UV but drops below one in the IR — the $\ddot{V}+\sigma^2 V''=0$3 channel grows faster in the interior.

Lin solution and the D2 shell

Repeating the analysis in Lin's flux-based construction, the asymptotic (smeared) solution reproduces the same early-time structure, with $\ddot{V}+\sigma^2 V''=0$4. The more interesting result concerns the interior. Near a shell of concentric D2 branes, the geometry crosses over to the near-horizon limit of $\ddot{V}+\sigma^2 V''=0$5 flat D2 branes, and the probe trajectory obeys $\ddot{V}+\sigma^2 V''=0$6. The late-time growth rate is then

$\ddot{V}+\sigma^2 V''=0$7

a genuinely non-linear, accelerating growth in the IR, in contrast to the UV behavior and to the D2-disk saturation. This difference is attributed to the substantial modification of the interior geometry by the D2 shell; the exponent $\ddot{V}+\sigma^2 V''=0$8 is a concrete, falsifiable prediction distinguishing the interior regime.

Non-Abelian T-dual background

For the non-Abelian T-dual of $\ddot{V}+\sigma^2 V''=0$9 — dual to an irrelevant deformation of the matrix model with smeared D0 asymptotics — the consistent $(\sigma,\eta)$0 geodesic yields a first-order equation for $(\sigma,\eta)$1, with the particle falling from $(\sigma,\eta)$2 toward $(\sigma,\eta)$3. The complexity grows slowly at early times but diverges as the particle approaches the singularity near $(\sigma,\eta)$4 at late times, a behavior qualitatively distinct from all the Lin–Maldacena cases and tied directly to the singular interior of the T-dual background. The paper does not address how this divergence should be regulated or interpreted in the dual irrelevant deformation, which remains an open point.

Krylov complexity in the pulsating fuzzy sphere model

On the matrix-model side, the author reduces the bosonic PWMM to the $(\sigma,\eta)$5 pulsating fuzzy sphere — two coupled anharmonic oscillators with Hamiltonian $(\sigma,\eta)$6, $(\sigma,\eta)$7 containing the $(\sigma,\eta)$8-dependent harmonic, quartic, and cubic $(\sigma,\eta)$9 terms. Choosing a normalized Gaussian seed operator $\eta_k \sim N_k$0 with $\eta_k \sim N_k$1, the Liouvillian orbit $\eta_k \sim N_k$2 is constructed explicitly for $\eta_k \sim N_k$3. Since this basis is not orthogonal (e.g., $\eta_k \sim N_k$4), a Gram–Schmidt (Lanczos) procedure is applied to build the Krylov basis $\eta_k \sim N_k$5 that tri-diagonalizes $\eta_k \sim N_k$6.

The first two Lanczos coefficients are computed analytically:

• $\eta_k \sim N_k$7, linear in the mass parameter for the entire range of deformations;
• $\eta_k \sim N_k$8, which scales as $\eta_k \sim N_k$9 for $Q_k \sim n_k$0 and linearly as $Q_k \sim n_k$1 for $Q_k \sim n_k$2, with a minimum at a critical $Q_k \sim n_k$3 solving $Q_k \sim n_k$4.

Solving the Schrödinger equation on the Krylov chain via a Green's function with Neumann boundary conditions, the early-time complexity is

$Q_k \sim n_k$5

reproducing the quadratic growth found holographically, with the coefficient controlled by $Q_k \sim n_k$6 in exact analogy to the $Q_k \sim n_k$7-dependence on the gravity side. The paper notes that only $Q_k \sim n_k$8 and $Q_k \sim n_k$9 are computed analytically; the extension to $N=\sum_k n_k N_k$0 for $N=\sum_k n_k N_k$1 (needed for the coefficient $N=\sum_k n_k N_k$2 in closed form and for late-time behavior) is outlined but not executed, and the algorithm for arbitrary $N=\sum_k n_k N_k$3 is left open. It is also an assumption, not a derivation, that the Gaussian seed with $N=\sum_k n_k N_k$4 is the appropriate choice of initial operator.

Limitations and open questions

Several caveats are stated or implicit in the analysis. The identification of proper radial momentum with $N=\sum_k n_k N_k$5 is a proposal whose validity in non-conformal, massive-deformation backgrounds is tested here only qualitatively; no precise matching of coefficients between the gravity result (governed by $N=\sum_k n_k N_k$6) and the matrix-model result (governed by $N=\sum_k n_k N_k$7) is established. The matrix-model calculation is restricted to the bosonic $N=\sum_k n_k N_k$8 truncation of the full supersymmetric $N=\sum_k n_k N_k$9 theory, to early times, and to the first two Lanczos coefficients. Whether the Lanczos coefficients grow linearly with $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$0 — the hallmark of chaos in the operator-growth framework — is not determined, even though the BMN matrix model is known to exhibit chaos; nor is it established whether the linear scaling of $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$1 and $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$2 with $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$3 at large deformation, where the system becomes integrable, is a universal diagnostic of the integrable/non-integrable transition. The late-time matrix-model complexity, which should be compared against the $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$4 interior scaling and the saturation behavior found holographically, remains uncomputed.

Conclusion

The paper extends the holographic Krylov complexity program to the non-conformal Lin–Maldacena class of geometries and their matrix-model duals. Its main quantitative results are the universal quadratic early-time growth $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$5 on both sides of the duality, the distinct late-time behaviors across limits (saturation for D2, continued growth for NS5, $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$6 scaling near the D2 shell, and divergence near the non-Abelian T-dual singularity), and analytic expressions for the first Lanczos coefficients of the pulsating fuzzy sphere model, which are fixed entirely by the mass parameter $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$7. The work leaves open the construction of a general Lanczos algorithm at large $$V_{PWMM} = V_0\Big(\eta\sigma^2 - \frac{2}{3}\eta^3\Big) + \frac{P\,\eta}{(\eta^2+\sigma^2)^{3/2}}$$8, the full supersymmetric matrix-model treatment, and a quantitative gravity–matrix-model matching of the complexity coefficients.