Conformal Blocks in 2d Carrollian/Galilean CFTs and Excited State Entanglement Entropy

This paper establishes the Flat/CCFT correspondence by deriving the entanglement entropy of highly excited states in 2d Carrollian/Galilean CFTs, which matches holographic results from 3D Einstein gravity and confirms the Eigenstate Thermalization Hypothesis while defining a precise dictionary between boundary and bulk parameters.

The Gist

Imagine the universe as a giant, complex movie playing on a screen. For a long time, physicists have been trying to figure out how the "screen" (the boundary of the universe) creates the "movie" (the space and time inside). A famous theory called Holography suggests that everything happening in the 3D world of gravity is actually a projection of information living on a 2D surface, much like a hologram on a credit card.

This paper tackles a very specific, tricky version of this puzzle: Flat Space Holography.

Most previous work focused on a universe that curves inward like a bowl (Anti-de Sitter space). But our actual universe is "flat" (like a sheet of paper that goes on forever). The authors wanted to see if the holographic rules still work in this flat, infinite universe.

Here is a breakdown of what they did, using simple analogies:

1. The Setting: A Flat, Noisy Room

The authors are studying a theoretical "flat" universe. In this universe, the rules of physics are described by something called Carrollian/Galilean Conformal Field Theories (C/G CFTs).

• The Analogy: Imagine a room where time and space behave differently than in our daily lives. In this room, "time" is a bit sluggish, and "space" is rigid. The authors are trying to understand how information spreads in this weird room.

2. The Problem: Heavy Weights and Entanglement

They wanted to calculate something called Entanglement Entropy.

• The Analogy: Think of "entanglement" as a deep, invisible connection between two people in a crowd. If you look at just one person, you can't understand them fully; you need to know how they are connected to the rest of the crowd. "Entropy" is a measure of how much information you are missing about that one person because of these connections.

The authors were specifically interested in what happens when you introduce a "Heavy" object into this room.

• The Analogy: Imagine the room is a calm pond. Usually, the water is flat. But if you drop a giant, heavy boulder (a "heavy state") into the pond, it creates massive waves and changes the shape of the water entirely. The authors wanted to calculate how the "connections" (entanglement) change when this heavy boulder is present.

3. The Method: The "Magic Transformation"

To solve the math, which is incredibly difficult, they used a clever trick involving Conformal Blocks.

• The Analogy: Imagine trying to measure the ripples caused by the boulder in a chaotic, stormy pond. It's too messy. The authors found a "magic transformation" (a specific mathematical coordinate change) that effectively flattens the storm.
• They showed that by changing the way you look at the coordinates (stretching and tilting the grid), the messy, heavy problem turns into a simple, clean problem that is easy to solve. It's like putting on special glasses that turn a chaotic traffic jam into a straight, empty highway.

4. The Big Discovery: The "Thermal" Surprise

When they calculated the entanglement entropy for these heavy states, they found something surprising.

• The Result: The math showed that the heavy state behaves exactly like a hot, thermal system (like a cup of coffee cooling down).
• The Meaning: This confirms a famous idea in physics called the Eigenstate Thermalization Hypothesis (ETH). It basically says: "If you look at a single, highly excited state in a quantum system, it looks just like a hot, random soup." The authors proved this happens in their flat, weird universe, just like it does in our normal universe.

5. The Grand Match: The Holographic Dictionary

The most exciting part of the paper is the "Holographic Match."

• The Analogy: The authors built a dictionary. On one side of the page, they had the math from the "boundary" (the 2D screen with the heavy boulder). On the other side, they had the math from the "bulk" (the 3D flat universe with gravity).
• The Match: They found that the numbers on the screen matched the numbers in the 3D universe perfectly.
  • The "weight" of the heavy object on the screen corresponds to the mass of a particle in the 3D universe.
  • The "charge" of the object corresponds to the spin (angular momentum) of the particle.
  • The "tilt" they calculated mathematically corresponds to the shape of a Flat Space Cosmology (a specific type of expanding universe) or a Conical Defect (a universe with a tiny hole or twist in it).

Summary

In short, this paper says:

1. We can study a flat, infinite universe using a 2D theory on its edge.
2. When we put a heavy object in this theory, it creates a specific pattern of "connections" (entanglement).
3. By using a clever mathematical trick (stretching the coordinates), we can solve this pattern easily.
4. The result proves that heavy objects in this theory act like hot, thermal systems.
5. Most importantly, the math from the 2D theory matches the gravity math of the 3D universe perfectly, giving us a new, precise dictionary to translate between the two.

This is a major step in proving that our flat universe can be understood as a hologram, just like the curved universes we studied before.

Technical Summary

Technical Summary: Conformal Blocks in 2d Carrollian/Galilean CFTs and Excited State Entanglement Entropy

Problem Statement
The emergence of spacetime from quantum information remains a central challenge in quantum gravity. While the Ryu-Takayanagi (RT) formula in AdS/CFT successfully relates boundary entanglement entropy to bulk geometry, extending this narrative to flat space holography (Flat/CCFT correspondence) presents unique difficulties due to the null nature of the asymptotic boundary and non-conserved gravitational charges. Specifically, a systematic framework for computing the entanglement entropy of highly excited states in two-dimensional Carrollian/Galilean Conformal Field Theories (C/G CFTs) has been lacking. This paper aims to fill that gap by calculating the entanglement entropy for excited states in 2d C/G CFTs and verifying its agreement with holographic calculations in three-dimensional asymptotically flat spacetimes.

Methodology
The authors employ a field-theoretic approach centered on the computation of conformal blocks in the large central charge limit ($c_M \to \infty$). The methodology proceeds in three main stages:

1. Review of AdS$_3$/CFT$_2$ Precedents: The paper first recapitulates the standard derivation of excited state entanglement entropy in AdS$_3$/CFT$_2$. In this context, the entropy of a state created by a heavy operator is derived using the replica trick and heavy-light conformal blocks. The heavy operator's backreaction is absorbed via a conformal transformation that maps the vacuum plane to a cone (or a thermal cylinder for high energies), establishing the Eigenstate Thermalization Hypothesis (ETH).
2. Derivation of C/G Conformal Blocks: The core technical contribution involves deriving conformal blocks for 2d C/G CFTs. The authors analyze the algebra of the Carrollian/Galilean conformal group (BMS$_3$), characterized by two central charges, $c_L$ and $c_M$. They consider two regimes:
   • Light-type: All external operators have weights and charges of order $O(1)$.
   • Heavy-light type: Two external operators (creating the excited state) have weights ($H$) and charges ($\Xi$) scaling with the central charge ($H, \Xi \sim O(c_M)$), while the others remain light.
     The authors demonstrate that in specific large central charge limits, the full C/G conformal blocks reduce to global blocks. This reduction is achieved by identifying a specific C/G conformal transformation $(x, y) \to (w, z)$ that absorbs the large weight and charge dependence of the heavy operators into the transformation parameters, effectively removing the backreaction from the correlation function calculation.
3. Entanglement Entropy Calculation: Using the derived heavy-light blocks and assuming vacuum block dominance, the authors compute the entanglement entropy for a single interval in an excited state. They utilize the replica trick, where the $n$-th Rényi entropy is determined by a four-point function involving twist operators and heavy operators. The result is then mapped from the plane to the cylinder geometry.

Key Contributions and Results

• Systematic Derivation of C/G Blocks: The paper provides an analytical derivation of conformal blocks for both light and heavy-light configurations in 2d C/G CFTs. A key finding is that in the limit $c_M \to \infty$ with $H/c_M, \Xi/c_M \sim O(1)$, the full blocks reduce to global blocks via a novel transformation:
  $$\lim_{c_M \to \infty} g(x, y) = (1-w)^{\Delta(1-1/\alpha)} \alpha^{2\Delta - \Delta_r} e^{\xi \frac{z(\alpha-1)}{(w-1)^\alpha}} g_{\text{global}}(w, z)$$
  where the transformation parameters $\alpha$ and $Q$ are determined by the heavy charge $\Xi$ and weight $H$:
  $$\alpha = \sqrt{1 - \frac{24\Xi}{c_M}}, \quad Q = \frac{(\alpha^2 - 1)c_L + 24H}{2\alpha^2 c_M}$$
  Geometrically, this transformation rescales and tilts the cylinder.

• Excited State Entanglement Entropy: The authors derive the entanglement entropy for a highly excited state on a cylinder. The result takes the form of the vacuum entropy on a cylinder with a non-trivial identification $(u, \phi) \sim (u + 2\pi\alpha Q, \phi + 2\pi\alpha)$:
  $$S_A = \frac{c_L}{6} \log \left( \frac{2}{\alpha \epsilon} \sin \frac{\alpha l_\phi}{2} \right) + \frac{c_M}{6} \left( Q + \frac{\alpha(l_u - Q l_\phi)}{2} \cot \frac{\alpha l_\phi}{2} \right)$$
  When the boost charge exceeds a threshold ($\Xi > c_M/24$), $\alpha$ becomes purely imaginary, and the entropy assumes a thermal form involving hyperbolic functions, signaling the realization of the Eigenstate Thermalization Hypothesis (ETH) in these theories.

• Holographic Match: The paper establishes a precise dictionary between the boundary C/G CFT parameters and the bulk spacetime parameters in 3D Einstein gravity.

  • For conical defects (spinning particles, $M < 0$), the CFT parameters map to the mass and angular momentum of the defect.
  • For Flat Space Cosmologies (FSC) ($M > 0$), the parameters map to the mass and angular momentum of the cosmological solution.
    The field-theoretic calculation of the excited state entropy perfectly reproduces the holographic entanglement entropy computed via the swing surface proposal (the flat-space analogue of the RT formula).

Significance and Claims
The authors claim that this work provides a concrete realization of the Flat/CCFT correspondence by establishing a precise dictionary relating the weight $\Delta$ and charge $\xi$ of boundary states to the mass $m$ and angular momentum $j$ of the dual spacetime.

The primary significance lies in the consistency check it offers: the agreement between the boundary calculation (using heavy-light conformal blocks) and the bulk calculation (using the swing surface proposal) validates the swing surface proposal for excited geometries, including FSC and conical defect solutions. This confirms that the holographic entanglement entropy in flat space can be derived from the boundary C/G CFT, even for highly excited states that induce significant backreaction.

The paper notes that while their analysis relies on the highest weight representation (HWR), which characterizes heavy primary states, recent literature suggests that induced representations might be necessary for light bulk excitations. However, the authors assert that their successful holographic match using HWR provides a robust foundation for understanding the thermodynamic and geometric properties of flat space holography.