Celestial Conformal Field Theory

• Celestial Conformal Field Theory is a framework that applies conformal field theory techniques to asymptotically flat spacetimes, linking twist operator correlations with cosmic brane effects.
• Entanglement Rényi entropies in CCFT are computed via the replica trick on the celestial sphere, with universal terms determined by either the sphere partition function or the Weyl anomaly.
• The holographic duality in CCFT equates twist operator insertions with cosmic brane sources, offering insights into gravitational partition functions and infrared quantum gravity features.

Entanglement Rényi entropies (EREs) in celestial holography quantify the entanglement structure of states in celestial conformal field theory (CCFT), which is conjectured to be holographically dual to scattering in asymptotically Minkowski spacetimes. In the CCFT framework, EREs are computed from the correlation functions of twist operators using the replica trick, paralleling standard CFT computations but in a spacetime and kinematical context appropriate to flat space holography. The holographic duality associates CCFT twist operator insertions with cosmic branes in the bulk, linking subregion entanglement to the gravitational partition function in the presence of these branes. For a spherical subregion, the resulting ERE takes a universal form dictated by conformal symmetry, with a key contribution determined by either the CCFT sphere partition function or its Weyl anomaly, depending on the dimension. In particular, for celestial CFTs with $d=4~\mathrm{mod}~4$, this universal term vanishes; in other cases, it is proportional to $i$ times the $d$th power of the Planck-scale regulated infrared cutoff in the bulk spacetime. This structure generalizes the well-known Ryu-Takayanagi and Rényi entropy results from AdS/CFT to the flat space/celestial CFT context (Capone et al., 12 Dec 2024).

1. Entanglement Rényi Entropy in CCFT

In a $d$-dimensional CFT, the entanglement Rényi entropy (ERE) $S_n$ for a subregion $A$ is defined via the reduced density matrix $\rho_A$ as

$$S_n = \frac{1}{1-n} \ln \mathrm{Tr} \left(\rho_A^n\right).$$

The entanglement entropy (EE) is the $n\to1$ limit of the Rényi entropies. In CCFT, the EREs are computed through $n$-sheeted covering spaces using the replica trick, which translates to a correlation function of twist operators located at the boundaries of $A$.

The authors argue that in celestial holography, the twist operators in CCFT are dual to cosmic branes inserted into the $(d+2)$-dimensional bulk spacetime. The correlator of $n$ twist operators then computes the gravitational partition function $Z_\mathrm{bulk}^{[\text{twists}]}$ in the presence of these cosmic branes, encoding the backreacted geometry.

2. Replica Trick and Twist Operators in Celestial CFT

The replica trick in CCFT is formally identical to standard CFT but takes place on the celestial sphere, the space of directions at null infinity. For a spherical subregion $A$ (a ball on $S^d$), the ERE is mapped to the two-point function of twist fields: $$\langle \mathcal{T}_n(x_1) \mathcal{T}_n(x_2) \rangle_\mathrm{CCFT}.$$ This correlator computes the partition function of the $n$-sheeted cover branched along the entangling surface. In the holographic dual, each twist insertion is associated with a cosmic brane extending into the bulk, and the corresponding partition function, to leading semiclassical order, is given by the exponential of the classical action evaluated on the backreacted geometry.

The cosmic brane tension is related to the Rényi index $n$ and the gravitational constant, mirroring the role played by the Ryu-Takayanagi or cosmic brane prescription in AdS/CFT. Universal terms in the ERE are then controlled by the bulk partition function with these branes.

3. Spherical Subregions and Universal Structure

For a spherical region $A$ in the CCFT vacuum, conformal symmetry fixes the functional form of the ERE, up to a universal constant. The key result is that the universal contribution (often called the "constant" or "topological" term) depends on whether $d$ is even or odd:

• For odd $d$: It is determined by the CCFT's sphere partition function.
• For even $d$: It is determined by its Weyl anomaly.

Explicitly, for a ball-shaped region, the universal term $S^{(\mathrm{univ})}_n$ in the ERE is proportional to (for even $d$)

$$S^{(\mathrm{univ})}_n \propto \text{Weyl anomaly of sphere partition function,}$$

while for odd $d$ it is related to the value of the regularized sphere partition function. The computation utilizes the mapping to the bulk partition function in the presence of cosmic branes spanning the causal domain of dependence of $A$.

4. Vanishing and Imaginary Universal Terms

A notable outcome is the finding that for $d\equiv4\ (\mathrm{mod}\ 4)$, the universal term in the ERE vanishes identically. For other values of $d$, it is purely imaginary and proportional to $i$ times $\Lambda^d$, where $\Lambda$ is the infrared (long-distance) cutoff in Planck units. In mathematical terms, for $d\not\equiv4\ (\mathrm{mod}\ 4)$: $S^{(\mathrm{univ})}_n \sim i (\Lambda)^{d},$ up to model-dependent coefficients. This arises from the analytic structure of the CCFT partition function and its dependence on the gravitational sector's cutoff scale. The appearance of the factor of $i$ and the power dependence on the cutoff reflect the non-unitary and topological characteristics of the universal piece in celestial holography.

5. Holographic Duality: CCFT–Cosmic Brane Correspondence

The duality between twist operators in CCFT and cosmic branes in the bulk is central to the derivation of the EREs. In this picture, the presence of a twist operator at the boundary (celestial sphere) corresponds to the insertion of a cosmic brane with prescribed tension in the bulk. The calculation of the partition function in the presence of these sources reduces (in the semiclassical regime) to finding the on-shell gravitational action for the backreacted spacetime with the cosmic brane's conical singularity.

The leading "universal" term in the ERE for a sphere then matches the leading term in the vacuum to vacuum scattering amplitude in the bulk, with the cosmic brane induced conical excess/deficit. This duality elevates the status of subregion entanglement in CCFT to a direct probe of quantum gravity in asymptotically flat space.

6. Significance and Universality of the Result

The universal ERE structure found here is dictated directly by the conformal (and, implicitly, asymptotic Lorentz) symmetry of the celestial CFT framework. The vanishing or purely imaginary character of the universal piece in different spacetime dimensions underscores both the similarities and sharp distinctions to the entanglement structures familiar from AdS/CFT. The dependence of the universal term on the CCFT sphere partition function or Weyl anomaly signals a close connection between flat space holography, quantum gravity, and conformal field theory techniques.

This analysis also confirms that, while celestial CFTs encode the same symmetries and many structural results as their AdS counterparts, their analytic, topological, and infrared properties can fundamentally differ. The matching of CCFT twist operator correlators with cosmic brane partition functions provides a robust and testable aspect of the celestial holography proposal (Capone et al., 12 Dec 2024).

7. Acknowledged Context and Collaboration

The research on EREs in celestial holography is informed by active collaboration and input from multiple experts, reflecting the developing landscape of CCFT and quantum gravity correspondence. The connection between modular entropy, cosmic branes, and flat space entanglement is underscored as a central, ongoing research issue involving both technical field theory analysis and gravitational partition function calculations (Capone et al., 12 Dec 2024).