⚛️ high-energy theory

Complexity measures in holographic cascading theories with multiscale dynamics

This paper systematically investigates two holographic complexity measures—complexity=volume for thermofield double states and Krylov spread complexity for single-particle states—in three-dimensional cascading gauge theories, revealing how "walking" dynamics near an intermediate conformal fixed point and the emergence of an infrared scale in the confining limit distinctly manifest in these dual descriptions.

Original authors: Carlos Nunez, Juan F. Pedraza, Javier G. Subils

Published 2026-08-12

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CC BY 4.0

Original authors: Carlos Nunez, Juan F. Pedraza, Javier G. Subils

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic video game. In this game, the rules of physics are written in a language called quantum mechanics, which is notoriously difficult to solve, especially when particles interact with extreme force. Physicists have a secret weapon to help them: a "holographic dictionary." This dictionary suggests that a difficult quantum system (like a hot soup of particles) can be translated into a much easier problem involving gravity and shapes in a higher-dimensional space. It's like trying to understand a complex 3D maze by looking at its 2D shadow; the shadow is simpler to draw, but it still holds all the secrets of the maze.

Two of the most fascinating things scientists try to measure in these quantum systems are "complexity" and "confinement." Think of complexity as a measure of how hard it is to build a specific Lego castle from a pile of loose bricks. A simple castle is easy to build (low complexity), but a massive, intricate fortress takes a long time and many steps (high complexity). In the quantum world, this tells us how "scrambled" a system has become. Confinement, on the other hand, is like a cosmic rubber band. If you try to pull two particles apart, the rubber band stretches and eventually snaps back, keeping them stuck together. This is how quarks (the building blocks of protons) are trapped inside atoms. The big question is: can we use the holographic dictionary to see how these two things—complexity and confinement—dance together as the universe changes its temperature or rules?

This paper, titled "Complexity measures in holographic cascading theories with multiscale dynamics," takes a deep dive into a specific family of theoretical universes called the B8 family. The authors, Carlos Núñez, Juan F. Pedraza, and Javier G. Subils, act like cosmic cartographers, mapping out how "complexity" behaves in these universes as they slide between two extreme states: one where the rules look the same at every size (a "conformal" state) and one where the rubber band of confinement is tightest.

They explore this using two different "lenses" or methods, each looking at a different type of quantum state.

Lens 1: The Eternal Black Hole (Thermal States)
First, they look at a "thermofield double" state. Imagine two identical universes connected by a magical tunnel (an Einstein-Rosen bridge, or wormhole). As time passes, this tunnel gets longer and longer. The authors use a rule called "Complexity = Volume" to measure how fast this tunnel grows. They found that when the universe is in a "walking" state (where it hovers near the conformal rules for a long time), the growth rate of this tunnel is very steady and predictable, like a metronome. However, as they push the universe toward the "confining" limit (where the rubber band snaps tight), the growth rate doesn't suddenly change its personality. It just smoothly transitions. This suggests that for these hot, scrambled systems, the complexity measure is a bit "blind" to the specific details of confinement; it mostly cares about the energy and the general flow of time.

Lens 2: The Bouncing Probe (Locally Excited States)
Next, they look at a single particle dropped into the vacuum of the universe. In the holographic world, this particle falls toward the center of the geometry. The authors track its "spread complexity," which is essentially how fast the particle's information spreads out. Here, the story gets much more interesting. Because the bottom of this universe is capped off like a bowl, the particle doesn't fall forever; it hits the bottom and bounces back up, like a ball in a well. This causes the complexity to oscillate (go up and down) rhythmically.

The authors discovered that the period of this bounce (how long it takes to go down and come back) is a super-sensitive detector.

When the universe is near the "walking" (conformal) state, the bounce takes a very long time.
When the universe is near the "confining" state, the bounce happens much faster.

Crucially, they tested two types of "probes": a simple point-particle and a D0-brane (a tiny, extended object). The simple particle crashed into a singularity (a mathematical tear in the fabric of space) at the bottom, causing the complexity to blow up to infinity. But the D0-brane, which interacts with the "dilaton" (a field related to the strength of forces), felt a gentle repulsive force that kept it from crashing. It bounced smoothly, and its oscillation pattern changed distinctly depending on whether the universe was confining or not.

The Big Takeaway
The paper suggests that while the "wormhole growth" method is great at spotting when a universe is "walking" (hanging out near conformal rules), it isn't very good at telling the difference between a universe that is just gapped (has a mass limit) and one that is truly confining. However, the "bouncing probe" method is a much sharper tool. The rhythm of the bounce changes dramatically depending on the nature of the ground state.

In short, the authors found that complexity is a versatile detective. It can tell us when a system is acting like a conformal fluid, but to catch a "confining" universe in the act, you need to look at how local excitations (like our bouncing probes) oscillate, not just how big the wormholes get. The B8 family of theories provided the perfect playground to separate these effects, showing us that the "rubber band" of confinement leaves a unique, rhythmic fingerprint on the quantum complexity of the system.

Technical Summary: Complexity Measures in Holographic Cascading Theories with Multiscale Dynamics

Problem Statement
This work investigates how two distinct notions of quantum complexity encode the infrared (IR) physics of a family of three-dimensional $\mathcal{N}=1$ quiver gauge theories, known as the B8 family. These theories exhibit rich multiscale Renormalization Group (RG) flows controlled by a tunable parameter $b_0 \in [0, 1]$ . The parameter interpolates between three distinct regimes:

$b_0 = 0$ : The flow terminates at the Ooguri–Park (OP) conformal fixed point.
$0 < b_0 \ll 1$ : Theories exhibit "walking" dynamics, remaining close to the conformal fixed point over a parametrically large range of scales before developing a mass gap.
$b_0 \to 1$ : Theories approach a confining limit where the Chern-Simons (CS) interactions vanish, leading to Wilson-loop confinement.

Crucially, for generic $0 < b_0 < 1$ , the theories possess a discrete massive spectrum and a smooth IR cap in their eleven-dimensional dual geometry, yet they do not exhibit genuine Wilson-loop confinement due to charge screening by non-zero CS interactions. The authors aim to disentangle how complexity measures respond to approximate conformality, the discrete massive sector, and genuine confinement.

Methodology
The authors employ the holographic duality to study two complementary classes of states, each associated with a different bulk observable:

Thermofield Double (TFD) States (Thermal Complexity):
- State: Finite-temperature states represented by two-sided eternal black branes.
- Observable: The Complexity=Volume (CV) prescription. The authors calculate the volume of maximal codimension-one spatial hypersurfaces anchored at the boundary times behind the black brane horizon.
- Metric: They utilize the ten-dimensional type IIA supergravity ansatz for the B8 family, solving the equations of motion numerically to determine the geometry of the black branes and their horizons.
Locally Excited States (Krylov Spread Complexity):
- State: One-particle excitations created by local operator insertions in the vacuum, represented holographically by massive probes falling radially in the ground-state geometry.
- Observable: Krylov spread complexity ( $C_K$ ), defined as the average position of the state in the Krylov basis. Following recent proposals, the growth rate of this complexity is identified with the proper radial momentum ( $P_{\bar{y}}$ ) of the bulk probe: $\dot{C}_K(t) \simeq -P_{\bar{y}}(t)$ .
- Probes: The authors analyze two types of probes to test robustness:
  - Point-like massive particles: Coupled to the Einstein-frame metric.
  - D0-branes: Coupled to the string-frame metric and the dilaton, representing physical excitations in the type IIA spectrum.
- Dynamics: They solve the geodesic equations for these probes in the zero-temperature B8 geometries to determine their radial trajectories and the resulting complexity growth.

Key Contributions and Results

Complexity of Thermal States (CV):
- Walking Dynamics: For theories with small $b_0$ (near the OP fixed point), the ratio of the late-time complexity growth rate ( $G$ ) to the thermal energy ( $E$ ) exhibits a "walking" behavior. This ratio remains approximately constant over an extended range of energies, mirroring the near-conformal RG flow.
- Confinement Insensitivity: As $b_0 \to 1$ (approaching the confining limit), the $G/E$ ratio smoothly approaches the confining result without developing qualitatively new features at the onset of confinement. The authors find that thermal complexity growth is largely insensitive to the confining nature of the ground state or the rich thermal phase transitions present in the system.
Complexity of Locally Excited States (Krylov Spread):
- Oscillatory Behavior: For generic $0 < b_0 < 1$ , the smooth IR cap of the geometry causes the radial motion of probes to be bounded, resulting in oscillations in the spread complexity growth rate.
- Probe Dependence and Singularities:
  - Massive Particles: In the ten-dimensional description, the geometry is singular at the IR end ( $r = r_s$ ). Particles reach this singularity where the warp factor vanishes, causing the complexity growth rate $\dot{C}_K$ to diverge. The authors interpret this as an artifact of the ten-dimensional singularity, which is resolved in the eleven-dimensional uplift.
  - D0-branes: Due to their coupling to the dilaton, D0-branes experience an effective potential that prevents them from reaching the singularity. They bounce smoothly at a turning point $r_f > r_s$ . Consequently, their complexity growth is smooth and continuous, vanishing twice per period (at the turning points) rather than diverging.
- Confinement Sensitivity: The period of these oscillations is controlled by the IR scale.
  - Near the conformal limit ( $b_0 \to 0$ ), the oscillation periods become parametrically long.
  - Near the confining limit ( $b_0 \to 1$ ), the behavior of the period changes qualitatively. The authors note that while the oscillations themselves diagnose the presence of a smooth IR cap and a discrete spectrum (present even in non-confining, screened phases), the specific dynamics of the D0-brane (e.g., the absence of divergence and the specific turning points) are sensitive to the transition to genuine confinement when CS interactions are switched off.
Comparison with Lattice Models:
The authors compare their holographic results with Krylov spread complexity in a confining lattice Ising model. They find a qualitative similarity: in both cases, the emergence of an IR mass scale introduces an "oscillation clock." However, they emphasize a key distinction: in the lattice model, the oscillation frequency is directly tied to the string tension (confinement strength), whereas in the B8 family, oscillations arise from the smooth IR cap (discrete spectrum) and persist even in the absence of confinement, provided the geometry is capped.

Significance and Claims
The paper claims to provide a systematic diagnostic of how complexity measures encode different aspects of IR physics in theories with multiscale dynamics. The primary significance lies in the differential sensitivity of the two complexity measures:

Thermal Complexity (CV) is shown to be a robust probe of approximate conformality (walking dynamics) but is largely insensitive to confinement.
Krylov Spread Complexity is shown to be a sensitive probe of the discrete IR spectrum and the smoothness of the IR geometry (manifested as oscillations). Furthermore, the specific behavior of extended probes (D0-branes) in the spread complexity framework appears to be more sensitive to the genuine confining nature of the ground state than thermal complexity is.

The authors conclude that the B8 family is uniquely suited to expose these distinctions because it allows for the clean separation of charge screening, discrete spectra, and Wilson-loop confinement. They assert that while oscillations in complexity are a signature of a smooth IR cap (and thus a discrete spectrum), they are not, by themselves, a sufficient diagnostic of Wilson-loop confinement. The distinction becomes apparent only when analyzing the specific response of the complexity to the limit where genuine confinement is realized.

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