Holographic complexity of conformal fields in global de Sitter spacetime

This paper computes and compares the holographic complexity of conformal quantum fields in global de Sitter spacetime using both volume and action prescriptions across two distinct setups: a CFT on the conformal boundary of AdS$_{d+1}$ and a CFT on a UV brane embedded within it, while contrasting these findings with results from alternative de Sitter coordinate patches.

The Gist

The Big Picture: The Universe as a Hologram

Imagine our universe is like a hologram. In physics, there's a famous idea called the Holographic Principle. It suggests that a 3D universe (like ours) can be fully described by information written on a 2D surface surrounding it, much like a 3D image is encoded on a flat 2D credit card.

This paper tries to understand how "complicated" our universe is using this holographic idea. Specifically, the authors are looking at a universe that is expanding forever (like our real universe, which is dominated by "Dark Energy"). In physics, this is called de Sitter space.

The Core Concept: What is "Complexity"?

To understand the paper, we first need to understand Quantum Complexity.

• The Analogy: Imagine you have a scrambled Rubik's Cube.
  • Entropy (a common physics measure) tells you how "messy" the cube looks.
  • Complexity asks a different question: How many specific moves (twists) does it take to turn a solved cube into this scrambled state?
• In the quantum world, "complexity" is the minimum number of steps required to build a specific quantum state from a simple starting point. The more complex the state, the more "computational work" it took to create it.

The authors want to know: As our universe expands, how does this "computational work" (complexity) change?

The Setup: The "Cosmic Mirror"

The authors use a mathematical trick called Holography (specifically AdS/CFT correspondence).

• The Problem: It's very hard to calculate complexity directly in our expanding universe (de Sitter space) because the rules of physics change with time there.
• The Solution: They imagine a "Cosmic Mirror." They take our expanding universe and say, "Let's pretend this is the reflection of a different, static universe (called Anti-de Sitter or AdS) that lives in a higher dimension."
• By calculating the complexity in the static "mirror" universe, they can figure out the complexity of our expanding universe.

They tested this in two different ways:

1. The "Volume" Method: Measuring the size of the biggest possible slice of the mirror universe.
2. The "Action" Method: Calculating the total "energy cost" (action) of the physics happening inside a specific region of the mirror universe.

The Findings: The Universe is Getting "Heavier"

Here is what they discovered, translated into everyday terms:

1. The Complexity Grows Exponentially

As time passes, the universe expands. The authors found that the complexity grows exponentially.

• The Metaphor: Imagine a balloon being inflated. As it gets bigger, the surface area grows. But in this quantum universe, it's not just the surface area growing; it's the amount of information needed to describe every new patch of space.
• Why? Because the universe is inflating, it is constantly adding new "rooms" to the house. Every time a new "room" (a Hubble volume) appears, the quantum system has to do more work to describe it. The complexity isn't just getting messy; the system is literally getting bigger and requiring more "gears" to run.

2. No "Hyper-Fast" Explosion (The Good News)

In some previous studies of smaller, static patches of the universe, scientists thought complexity might explode so fast it would become infinite in a finite amount of time (like a computer crashing instantly).

• The Result: The authors found no evidence of this "hyper-fast" explosion in the global (whole) universe.
• The Analogy: Instead of a computer crashing instantly, the universe is like a super-computer that is slowly, but steadily, adding more and more processors. It's growing fast, but it's not breaking. This suggests that the "explosion" seen in other studies might just be an illusion caused by looking at a small, limited window of the universe rather than the whole thing.

3. The "Double" Universe (The Brane Experiment)

In the second part of the paper, they added a "brane" (a membrane) to their model.

• The Analogy: Imagine you have a hologram on a piece of paper. Then, you glue a second identical piece of paper to the back of it. Now you have two universes stuck together.
• The Result: When they did this, the complexity doubled.
• The Meaning: This makes sense. If you have two universes, you need twice as many steps to describe them. Crucially, the way the complexity grows (the pattern) didn't change; it just got bigger. This tells us that the fundamental rules of how complexity evolves in an expanding universe are robust, even if you change the setup slightly.

Why Does This Matter?

1. Understanding Our Future: Since our real universe is expanding (de Sitter), understanding how complexity grows helps us understand the long-term fate of information in the cosmos.
2. Fixing the Math: It resolves a debate about whether complexity explodes instantly or grows steadily. The authors show that for the whole universe, it grows steadily but exponentially.
3. The Nature of Time: It highlights that in an expanding universe, the "difficulty" of describing the universe increases simply because the universe is getting bigger, not just because things are getting more chaotic.

Summary in One Sentence

The authors used a holographic "mirror" to show that as our expanding universe grows, the amount of information (complexity) needed to describe it grows exponentially because the universe is constantly adding new space, but it does so in a stable, predictable way rather than exploding into chaos.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the challenge of defining and computing holographic quantum computational complexity for Conformal Field Theories (CFTs) living on a global de Sitter (dS) spacetime.

• Context: While holographic complexity has been extensively studied for CFTs on static or asymptotically Anti-de Sitter (AdS) boundaries, the case of time-dependent, expanding de Sitter backgrounds remains less understood.
• Key Challenges:
  • Time Dependence: Global de Sitter space lacks a global timelike Killing vector, meaning the Hamiltonian is time-dependent, and energy is not conserved. This complicates the definition of states and the evolution of complexity.
  • IR Divergences: QFTs in de Sitter space suffer from ubiquitous infrared (IR) divergences, particularly for massless fields. The authors aim to probe how quantum information measures (complexity) respond to these long-distance effects.
  • Holographic Framework: Unlike the standard AdS/CFT, dS holography is not fully settled. The authors utilize a specific setup where global dS is realized as a slicing of higher-dimensional AdS, allowing the use of standard holographic conjectures (Volume and Action).
• Comparison: The study contrasts results with previous work on static patch dS holography (where complexity exhibits "hyperfast" growth) and conformal (Poincaré) dS patches, aiming to determine if hyperfast growth is a universal feature of dS or specific to the static patch/observer horizon.

2. Methodology

The authors employ two primary holographic prescriptions to compute complexity ($\mathcal{C}$) for a CFT living on a $d$-dimensional global de Sitter boundary:

A. Geometric Setup

• Bulk Geometry: $(d+1)$-dimensional AdS spacetime ($AdS_{d+1}$) foliated by global $d$-dimensional de Sitter slices ($dS_d$).
• Metric:
  $$ds^2 = dr^2 + \sinh^2 r \left( -dt^2 + \cosh^2 t \, d\Omega_{d-1}^2 \right)$$
  Here, $r$ is the radial coordinate in AdS, and $t$ is the global de Sitter time. The boundary CFT lives at the asymptotic limit $r \to \infty$ (regulated by a cutoff $\Lambda$).
• Regulator: The UV cutoff of the boundary CFT ($\epsilon$) is related to the IR cutoff of the bulk ($\Lambda$) via $\epsilon = e^{-\Lambda}$.

B. Complexity = Volume (CV) Conjecture

• Definition: $\mathcal{C}_V = \frac{V_{\Sigma}}{G_N L}$, where $V_{\Sigma}$ is the maximal volume of a spacelike hypersurface anchored at the boundary time $t = t_*$.
• Approach:
  • The authors derive the Euler-Lagrange equation for the extremal surface $t(r)$.
  • Analytic Case: For the special case of $t_* = 0$ (time-reflection symmetry), they derive a closed-form analytical solution involving hypergeometric functions.
  • General Case: For $t_* \neq 0$, the equations are non-linear and lack integrals of motion. The authors resort to numerical analysis to determine the maximal volume as a function of $\Lambda$ and $t_*$.

C. Complexity = Action (CA) Conjecture

• Definition: $\mathcal{C}_A = \frac{I_{WDW}}{\pi \hbar}$, where $I_{WDW}$ is the on-shell gravitational action evaluated over the Wheeler-DeWitt (WDW) patch.
• Approach:
  • The WDW patch is the domain of dependence of the boundary time slice, bounded by null hypersurfaces.
  • The authors compute the action contributions: Bulk Einstein-Hilbert term, Gibbons-Hawking-York (GHY) boundary terms, Joint terms (intersections of null boundaries), and the LMPS counter-terms (to remove reparameterization ambiguity).
  • They perform the integration analytically for general $d$ and specific boundary times, separating cases for even and odd dimensions due to different divergence structures.

D. Braneworld Extension

• The authors extend the analysis to a dS braneworld scenario (similar to the Karch-Randall setup).
• Setup: A brane with non-zero tension $T$ is inserted at a finite radial position $r = r_B$ in $AdS_{d+1}$. Two copies of AdS are glued across the brane ($Z_2$ symmetry).
• Induced Gravity: The induced geometry on the brane is global de Sitter. The boundary CFT now lives on this brane, coupled to an induced higher-derivative gravity theory.
• Goal: To determine if the backreaction of the brane tension and the doubling of the bulk geometry alters the qualitative features of complexity.

3. Key Results

A. Volume Complexity ($\mathcal{C}_V$)

• Time Dependence: Numerical analysis reveals that the maximal volume (and thus complexity) grows exponentially with boundary time $t_*$:
  $$V_{\Sigma} \sim e^{(d-1)t_*}$$
• Cutoff Dependence: The volume also scales exponentially with the bulk IR cutoff $\Lambda$:
  $$V_{\Sigma} \sim e^{(d-1)\Lambda}$$
• Physical Interpretation: This exponential growth is attributed to the expansion of the spatial volume of the de Sitter slice ($\sim \sinh^{d-1}\Lambda \cosh^{d-1}t_*$). Unlike systems with fixed degrees of freedom, the number of degrees of freedom in the CFT increases as the universe inflates (Hubble volume grows).
• Lloyd Bound: The exponential growth suggests a potential violation of the Lloyd bound (linear growth in time) for systems where degrees of freedom are added during time evolution.
• No Hyperfast Growth: Crucially, the complexity does not diverge at finite time (no "hyperfast" growth), contrasting with results from static patch holography.

B. Action Complexity ($\mathcal{C}_A$)

• Analytic Solution: The authors successfully derive an analytic expression for the on-shell action for general $d$ and $t_*$.
• Leading Behavior: The leading divergence matches the volume conjecture:
  $$\mathcal{C}_A \sim \epsilon^{-(d-1)} \cosh^{d-1}(t_*) \sim e^{(d-1)\Lambda} e^{(d-1)t_*}$$
• Logarithmic Divergence: A distinct feature is found for odd-dimensional de Sitter spacetimes: the complexity exhibits a logarithmic divergence ($\ln \epsilon$), similar to the behavior observed in AdS/CFT for odd boundary dimensions. This is absent in even dimensions.
• Agreement: The leading time-dependence and cutoff dependence of $\mathcal{C}_A$ are consistent with $\mathcal{C}_V$.

C. Braneworld Results

• Doubling Effect: When a tensional brane is inserted, the bulk geometry consists of two glued copies of AdS.
• Quantitative Change: The complexity in the braneworld setup is exactly twice the complexity of the single-copy setup ($\mathcal{C}_{brane} = 2 \mathcal{C}_{no-brane}$).
• Qualitative Invariance: The qualitative features (exponential time growth, divergence structure, presence of log terms in odd dimensions) remain unchanged.
• Interpretation: The insertion of a purely tensional brane does not alter the intrinsic quantum entanglement structure or the scaling laws of complexity; it merely doubles the available bulk degrees of freedom.

4. Significance and Conclusions

• Resolution of Hyperfast Growth Debate: The paper provides strong evidence that "hyperfast" complexity growth (exponential divergence at finite time) is not a universal property of de Sitter holography. It appears to be specific to the static patch description (where the boundary is a cosmological horizon). In global de Sitter, complexity grows exponentially with time but remains finite for any finite $t_*$.
• Role of Degrees of Freedom: The exponential growth of complexity is identified as a consequence of the expanding spatial volume and the increasing number of degrees of freedom in the inflating background, rather than just the scrambling of a fixed set of degrees of freedom.
• Universality of UV Structure: The UV divergence structure of complexity in global dS (power-law and logarithmic terms) mirrors that of AdS/CFT, suggesting that these features are robust across different holographic dualities.
• Braneworld Implications: The study demonstrates that the "double holography" setup (braneworld) preserves the fundamental scaling of complexity, offering a consistent framework to study CFTs coupled to dynamical gravity in de Sitter space.

5. Future Outlook

The authors suggest several directions for future work:

• Investigating the Lloyd bound in the context of time-varying degrees of freedom.
• Extending the analysis to FLRW spacetimes and black holes in global de Sitter (finite temperature CFTs).
• Applying the dS/dS correspondence and $T\bar{T}$ deformations to study complexity under holographic RG flows.
• Exploring connections between complexity and the index of inclusion in von Neumann algebras of observables.

In summary, this work establishes a rigorous framework for computing holographic complexity in global de Sitter space, revealing that while the complexity grows rapidly due to cosmic expansion, it retains the structural features of AdS complexity and avoids the pathologies (hyperfast growth) associated with static patch descriptions.