Flat Holography: Duality in Minkowski Space

• Flat holography is a framework connecting quantum gravity in asymptotically flat spacetimes with boundary field theories characterized by BMS and Carrollian symmetries.
• It employs a holographic dictionary that maps bulk data to boundary observables using methods like Mellin transforms, swing surfaces, and wedge holography, ensuring precise correlation functions.
• The approach reveals novel information-theoretic insights including entanglement volume laws, refined modular Hamiltonians, and experimental ties via flat-optical metasurfaces.

Flat holography refers to the broad class of gauge/gravity dualities that relate quantum gravity in asymptotically flat spacetimes to field theories defined on boundaries at infinity, generalizing the original AdS/CFT correspondence. In contrast to the negative curvature of AdS, flat space leads to unique features in both the dual field theory structure and the holographic dictionary, driven by the distinct asymptotic symmetries of Minkowski space and its null or spatial infinity. Contemporary research on flat holography synthesizes rigorous analysis of symmetry algebras (notably BMS and Carrollian structures), exact computations of entanglement and correlation observables, precise geometric constructions (including entanglement wedges and swing surfaces), and connections to both matrix-model dualities and experimental realizations in flat-optical systems.

1. Asymptotic Symmetries and the Structure of Flat Holography

The guiding principle of flat holography is that the quantum gravity path integral in (D+1)-dimensional asymptotically flat spacetime is dual to a D-dimensional field theory living at null infinity, governed by the Bondi-Metzner-Sachs (BMS) group or its Carrollian limit. In global four dimensions the relevant asymptotic symmetry algebra is BMS$_4$, which extends the Poincaré group $ISO(3,1)$ to include infinite-dimensional supertranslation and superrotation sectors. In three dimensions, the canonical algebra is BMS$_3$, isomorphic to the infinite-dimensional two-dimensional Galilean conformal algebra (GCA$_2$) (Bagchi et al., 2016), with generators

$$[L_m, L_n] = (m-n) L_{m+n} + \frac{c_L}{12} m(m^2-1) \delta_{m+n,0}, \quad [L_m, M_n] = (m-n) M_{m+n} + \frac{c_M}{12} m(m^2-1) \delta_{m+n,0}, \quad [M_m, M_n] = 0.$$

Here, $c_L$ and $c_M$ are central charges, with $c_M = 3/G$ in Einstein gravity.

In codimension-one implementations, the dual theory is a "Carrollian CFT" (or CCFT) living on $\mathscr{I}^+ \cong \mathbb{R}_u \times S^{d-1}$. The symmetry algebra under ultrarelativistic contraction is the conformal Carroll algebra (CCA) (Bagchi et al., 2023, Bagchi et al., 2016). Quantum field theories with such symmetry are typically ultra-relativistic or Galilean CFTs, as seen in contracted limits of Yang–Mills and Maxwell equations in four dimensions.

2. Holographic Dictionary and Correlation Functions

Flat holography admits several complementary realizations for the dictionary between bulk and boundary data:

• Covariant Surface Approach: The boundary field theory is defined at null infinity, with operator insertions parameterized by retarded time $u$ and angles $(z, \bar{z})$ on $S^2$. Carrollian and BMS$_4$ symmetries dictate the structure of correlation functions. In the 3d/2d case, BMS$_3$/GCA$_2$ Ward identities completely fix two- and three-point functions of primaries, yielding results identical in form to those of a CFT$_2$ up to details of the weights and central charges (Bagchi et al., 2014, Bagchi et al., 2016).
• Celestial (Codimension-Two) and Carrollian (Codimension-One) Approaches: In the celestial approach, the S-matrix is Mellin-transformed into a two-dimensional celestial CFT. In the Carrollian approach, as detailed in (Bagchi et al., 2023) and (Hao et al., 2023), translation invariance in $u$ and the full Poincaré group as global symmetries are manifest, and energy-momentum tensor correlation functions and higher-point structures are obtained directly from symmetry. Modified Mellin transforms yield regulated, finite three-point and two-point functions even for graviton amplitudes.
• AdS/BCFT Embedding: Flat holography can arise as a specific regime of AdS/BCFT setups with flat end-of-the-world branes (EOW branes) (Hao et al., 31 Aug 2025). In these constructions, the AdS bulk between two flat branes geometrizes the flat limit. The surviving isometry in the wedge, $ISO(1,d-1)$, is precisely the Carroll/BMS group of a CCFT$_{d-1}$. All predicted holographic correlation functions, entanglement entropy, and partition functions precisely match the expectations from the field theory side after identifying $c_M \sim (1/z_1 - 1/z_2)$.

3. Entanglement Structure and Swing-Surface Holography

Entanglement and mixed-state correlation measures provide powerful probes of holographic duality in flat space:

• Entanglement Entropy and Rényi Entropy: In 3d flat space, the holographic entanglement entropy for a boundary interval is given by the length of a null-anchored "swing surface": a spacelike geodesic in the bulk, connected to the boundary via null geodesics following the modular flow (Jiang et al., 2017, Bagchi et al., 2014, Hao et al., 31 Aug 2025). The entanglement entropy in BMSFT (c_L=0, c_M=3/G) for a boundary interval $A$ of lengths $(l_u, l_\phi)$ is

$$S_{EE} = \frac{c_M}{6} \left( \frac{l_u}{l_\phi} - \frac{\epsilon_u}{\epsilon_\phi} \right),$$

with $\epsilon_{u,\phi}$ UV cutoffs (Jiang et al., 2017). In TMG and GMMG, an additional $c_L$-dependent term enters, and the same swing-surface prescription applies (Setare et al., 2022).

• Balanced Partial Entanglement (BPE) and Entanglement Wedge Cross-section (EWCS): The BPE is a measure defined via a fine-grained entanglement contour on the field theory side, analytically matched with the minimal EWCS in the bulk: the minimal geodesic splitting the entanglement wedge of two boundary intervals (Basu, 2022). In pure gravity, one finds BPE(A:B) = EWCS(A:B), and in TMG, a topological crossing term appears.
• Reflected Entropy: The reflected entropy $S_R(A:B)$ in a Galilean CFT (GCFT$_2$), calculated via a double-replica twist operator technique, is shown to equal twice the entanglement wedge cross-section $E_W$, both in field theory and holographically in the bulk (Basak et al., 2022, Setare et al., 2022):

$S_R(A:B) = 2\, E_W(A:B),$

valid in multiple backgrounds (vacuum, finite-temperature, and finite-size orbifold) under large-$c$ and single-block dominance.

• Modular Hamiltonians, First Law, and Modular Chaos Bound: The modular Hamiltonian for a single interval in BMSFT can be computed from symmetry, and its bulk dual is realized via Iyer–Wald covariant charges associated with the swing surface (Apolo et al., 2020). The first law $\delta S = \delta \langle K_A \rangle$ holds, and the modular chaos Lyapunov bound is saturated, analogous to the AdS/CFT context.

4. Explicit Realizations: Wedge Holography, Carrollian Fluids, Celestial Correspondence, and Solvable Models

Flat holography supports a wide range of explicit constructions:

• Wedge Holography: By embedding two EOW branes as hyperbolic or de Sitter submanifolds in flat $(d+1)$-dimensional spacetime, one obtains a wedge holography regime (Ogawa et al., 2022). The induced boundary theory is a non-unitary Euclidean CFT on $S^{d-1}$ with imaginary central charge, directly calculable from the on-shell action. Gluing hyperbolic and de Sitter wedges and taking suitable limits incorporates celestial holography features.
• Carrollian Fluid/Gravity Correspondence: Starting from AdS$_4$ and taking a zero-speed-of-light limit, a conformal Carrollian structure emerges at the boundary (null infinity) (Ciambelli et al., 2018). The dual field theory is a Carrollian (ultra-relativistic) fluid specified by spatial metric, time, and scale factor. Explicit solutions reconstruct Kerr–Taub–NUT and Robinson–Trautman spacetimes from Carrollian fluid data.
• Celestial and Carrollian CFTs: The celestial holography approach treats the S-matrix as correlators in a 2d Euclidean CFT on $S^2$ at null infinity, leveraging Mellin transformations of plane wave amplitudes (Hao et al., 2023). In the Carrollian framework, the modified Mellin transform with $e^{i \omega u}$ renders translation invariance manifest and avoids anomalies in conformal weights, producing time-dependent correlation functions compatible with the full BMS group (Bagchi et al., 2023).
• Solvable 2d Flat Holography Models: Exact dual pairs are constructed using BF-like $\mathcal{N}=1$ supergravity in flat space and double-scaled Hermitian matrix models (Rosso, 2022). The total partition function receives contributions only at disk and cylinder topologies, matching precisely between bulk and boundary, and the spectral form factor and density of states can be computed exactly.
• Metasurface Flat Holography: Flat-optics implementations realize flat holography in experiment, employing non-interleaved TiO$_2$ metasurfaces encoding holographic phase profiles in orthogonal polarization and color channels (Yueqiang et al., 2019). These allow massive information multiplexing and vectorial holographic imaging, demonstrating applications of flat-holography concepts in optics.

5. Novel Information-Theoretic and Quantum Matter Aspects

Flat holography introduces qualitatively new features relative to AdS:

• Entanglement "Volume Law": In contrast to the area law for ground states of local CFTs, flat holography (for boundary on $S^d$) generically produces a maximal entanglement "volume law" scaling, reflecting the intrinsic non-locality of the dual field theories (Li et al., 2010). Suitable non-local quadratic actions, e.g. $$S[\phi] = \frac{1}{2} \int_{S^d} \phi e^{-\Delta} \phi$$, reproduce this scaling.
• Trivial Bulk-to-Boundary Correlators: After suitable non-local counterterms, all finite-connected correlation functions vanish, signaling that any infinitesimal subsystem is maximally entangled with its complement (Li et al., 2010).
• Trace Anomalies and RG Flows: Flat-holographic duals exhibit anisotropic stress tensors and distinctive trace anomalies proportional to the Ricci scalar, $\langle T^i{}_i \rangle = \frac{C_M}{4\pi}R$ (Fareghbal et al., 2015, Fareghbal et al., 2018), with monotonic C-functions defined along radial evolution, decreasing from UV to IR for unitary interpretations.
• Black Hole and Cosmological Entropy: The Cardy-like state-counting in 2d GCFTs with central charge $c_M$ precisely reproduces Bekenstein–Hawking entropy for 3d flat cosmological horizons (shifted-boost orbifolds of $\mathbb{R}^{1,2}$) (Bagchi et al., 2012), and the first law of thermodynamics is mirrored in the dual field theory variation of quantum numbers.

6. Generalizations, Limit Constructions, and Further Developments

Flat holography encompasses several generalized constructions:

• AdS/BCFT with Flat Brane Embeddings: By varying the brane tension in AdS, one interpolates between AdS/BCFT and fully flat wedge holography. The type I/II setups correspond respectively to null BCFTs (causal diamonds), CFT + flat gravity, and their linear combination yields Carrollian CFTs as pure flat-space duals (Hao et al., 31 Aug 2025).
• Extensions to GMMG and TMG: In topologically massive or generalized minimal massive gravity, higher-derivative contributions introduce additional central charges but do not modify the minimal-surface (EWCS) prescription for holographic entanglement and reflected entropy, up to universal constants (Setare et al., 2022).
• Higher Dimensional and Carrollian Gauge Theories: Carrollian symmetry extends to higher dimensions, with explicit representations in ultra-relativistic limits of gauge theories (Maxwell, Yang–Mills) and infinite symmetry enhancement (conformal Carrollian algebra), underpinning the structural properties required for holographic duality to persist (Bagchi et al., 2016).

7. Outlook and Open Problems

Flat holography has become a mature subject with a deep interplay between symmetry, quantum information, and geometry. Outstanding topics include:

• Complete classification of Carrollian field theories with desired BMS invariance;
• Non-perturbative definition of CCFTs and extension of flat holography to dynamical (non-conformal, non-unitary) matter;
• Direct derivations of quantum-corrected holographic information measures, such as reflected entropy, beyond semi-classical block dominance;
• Explicit realization of non-local dual field theories in higher dimensions and their relation to Minkowski scattering amplitudes;
• Generalization to supersymmetric, matter-coupled, and higher-spin theories.

The framework of flat holography as realized via Carrollian, BMS-invariant, and non-local CCFTs provides both a rigorous extension of holographic duality beyond AdS and a platform for new insights into quantum gravity, information, and experiment-inspired flat-optical analogs (Hao et al., 31 Aug 2025, Bagchi et al., 2023, Rosso, 2022, Yueqiang et al., 2019).