Extended BMS Representations

Updated 5 January 2026

Extended BMS representations are infinite-dimensional symmetry structures at null infinity, combining supertranslations and superrotations to generalize traditional particle states.
They use a semi-direct sum of a supertranslation ideal and local conformal algebras, enabling canonical and geometric realizations that link gravitational data to celestial CFTs.
The representation theory, including induced and highest-weight modules, underpins infrared quantum gravity, holographic dualities, and suggests string-like degrees of freedom in flat space.

Extended Bondi-Metzner-Sachs (BMS) representations generalize the concept of particle states and symmetry modules in asymptotically flat spacetime, incorporating the infinite-dimensional symmetry structure of null infinity including both supertranslations and superrotations. The resulting representation theory underpins infrared aspects of quantum gravity, @@@@0@@@@, defect CFTs, and leads to novel structures such as string-like degrees of freedom in flat space quantum gravity.

1. Algebraic Structure of Extended BMS Symmetries

The extended BMS group in four dimensions consists of the semi-direct sum of the supertranslation ideal $\mathcal{T}$ and the local conformal (Virasoro) algebra on the celestial sphere $\mathcal{S} = \operatorname{Diff}(S^2)$ , i.e.,

$\mathrm{eBMS}_4 = \mathcal{S} \ltimes \mathcal{T}$

where $\mathcal{S}$ is realized as two commuting copies of the Witt (Virasoro without central charge), generated by holomorphic and anti-holomorphic vector fields $L_n$ and $\bar{L}_n$ , $n \in \mathbb{Z}$ . Supertranslations $P_{k,\ell}$ are indexed by integer or half-integer pairs and correspond to spherical harmonic modes on $S^2$ . The essential commutator relations are: $\begin{aligned} &[L_m, L_n] = (m-n) L_{m+n},\,\, [\bar{L}_m, \bar{L}_n] = (m-n)\bar{L}_{m+n}, \ &[L_m, P_{k,\ell}] = (m-k) P_{m+k,\ell},\,\, [\bar{L}_m, P_{k,\ell}] = (m-\ell) P_{k,m+\ell}, \ &[P_{k,\ell}, P_{p,q}] = 0 . \end{aligned}$ In three dimensions, the extended BMS $_3$ algebra is the semi-direct sum of the Witt algebra and the infinite-dimensional supertranslation algebra, with similar structure (Ruzziconi et al., 2 Jan 2026, Oblak, 2015, Batlle et al., 2024).

Central extensions are present in various sectors: in 3d, two independent central charges $c_1, c_2$ arise (Virasoro and supertranslation sectors), while in 4d, nonzero central terms are present in the Virasoro subalgebras at the quantum level and may arise at one-loop due to anomalies (Distler et al., 2018, Ruzziconi, 2020). For super-algebras, infinite towers of fermionic charges extend the bosonic symmetries (Fotopoulos et al., 2020, Caroca et al., 2018, Banerjee et al., 2016).

2. Canonical and Geometric Realizations

Canonical realizations are provided by symplectic charges on field theory phase space—both in bulk and at null infinity. In 4d, the gravitational phase space naturally carries these symmetry generators as surface integrals of the Bondi mass aspect and Bondi news, with symmetry transformations acting as vector fields on the solution space (Donnay et al., 2021, Ruzziconi, 2020). The charges act on gravitational data via coadjoint (Kirillov-Kostant) representations, with the coadjoint orbits classifying classical and quantum modules.

On the celestial sphere, operators are organized as primaries and descendants under the action of Virasoro generators, and supertranslations act as abelian generators that shift the conformal data (Fotopoulos et al., 2019). The quadratic Casimir of the Lorentz subgroup organizes the mode structure, both in 3d (Batlle et al., 2024) and 4d (Batlle et al., 1 Jun 2025).

In free field theory, especially for massless scalars and higher-spin extensions, the algebra can be constructed from mode expansions and OPEs of canonical fields (Banerjee et al., 2015, Campoleoni et al., 2015, Batlle et al., 2024). For instance, in 3d BMS, explicit realizations use ghost systems or important extensions such as Wakimoto representations to embed nonabelian current algebras (Banerjee et al., 2015).

3. Representation Theory: Induced and Highest-Weight Modules

The classification of unitary irreducible representations mirrors Wigner's method for the Poincaré group, now using generalized supermomenta. The procedure is:

Specify a "rest-frame" supermomentum configuration (e.g., all $P_{k,\ell}$ except the Poincaré modes vanishing).
Determine the little group stabilizing this configuration.
Choose an irrep of the little group (e.g., SO(2) in both massive and massless 4d BMS).
Build the induced representation by acting with the non-little-group generators (Ruzziconi et al., 2 Jan 2026, Oblak, 2015, Campoleoni et al., 2015).

This strategy extends to include highest-weight modules analogously to CFT:

Primary states are annihilated by positive-modes of the superrotation and supertranslation generators (Fotopoulos et al., 2019, Bagchi et al., 2017).
Descendants are obtained by acting with negative-modes.
The representation space naturally factors as a tensor product of Virasoro (or super-Virasoro in super BMS) modules with the abelian supertranslation ideal.
Null vectors arise from the structure of the extended algebra, with Kac determinant analysis applied as in standard Virasoro modules (Bagchi et al., 2017, Campoleoni et al., 2015).

For extensions such as higher-spin or $N$ -extended supersymmetry, extra towers of modes for each generator (higher-spin currents, fermionic supercurrents, or internal automorphisms) are included, typically organized in induced representations with corresponding adjustments to null states, spectrum, and character formulas (Campoleoni et al., 2015, Banerjee et al., 2016, Caroca et al., 2018).

4. Extended BMS, Strings, and Holography

A central new insight is that extended BMS symmetry necessitates a shift from point-particle to string-like representations at null infinity. Nontrivial elements of the extended BMS group generate excitation of an infinite set of supermomentum modes, which, after Fourier transform, correspond to worldsheet fields $X(z)$ —the embedding of a closed string in target space (Ruzziconi et al., 2 Jan 2026). Explicitly,

$X(z) = \sum_n x_n z^{-n},\qquad \delta X(z) = -A(z) z \partial_z X(z) + \cdots$

shows the string field transformation under superrotations.

This leads directly to the conjecture that the true irreducible representations of extended BMS are described by string field theory on the celestial sphere or circle, and further, the associated symmetry underpins celestial holography where gravitational (and gauge) data at null infinity are encoded in two-dimensional CFTs, with superrotations acting as (super-)Virasoro and supertranslations as generalized Kac-Moody currents (Fotopoulos et al., 2019, Lowe et al., 2020, Batlle et al., 1 Jun 2025, Ruzziconi et al., 2 Jan 2026).

In 3d, one-loop partition functions for higher-spin and supergravity fields match vacuum characters of the relevant extended BMS algebras, providing a quantitative check of representation theory and suggesting a deep link between gravity's infra-red structure and infinite-dimensional symmetry (Campoleoni et al., 2015, Oblak, 2015, Banerjee et al., 2015, Campoleoni et al., 2016).

5. Central Extensions, Anomalies, and Quantization

For quantum representations, central charges arise in the Virasoro sectors ( $c_1$ , $c_2$ in 3d), while the abelian supertranslation ideal admits no nontrivial central extensions classically but may admit field-dependent cocycles or anomalies at the quantum level, particularly in presence of local (Virasoro) superrotations in 4d (Distler et al., 2018, Ruzziconi, 2020).

BRST quantization and cohomology for centrally extended algebras, especially of Weyl–BMS type, link the representation theory to chiral rings of topologically twisted superconformal field theories, showing that the structure is deeply intertwined with modern vertex algebra and VOA theory (Batlle et al., 2024).

Selvage approaches to quantization (coadjoint orbit and induced module) tie directly into the geometric (Covariant Phase Space) approach, and match the structure found via celestial CFTs, with detailed character formulas, Ward identities, and null state structure (Oblak, 2015, Bagchi et al., 2017, Campoleoni et al., 2016, Donnay et al., 2021, Fotopoulos et al., 2019).

6. Extensions: Supersymmetry, Higher-Spin, and Internal Symmetry

Supersymmetric and higher-spin generalizations further enlarge the BMS algebra. N-extended super-BMS algebras contain additional towers of fermionic generators (supercharges in various representations), as well as internal symmetry currents (e.g., $so(N)$ Kac-Moody currents) and their central extensions (Caroca et al., 2018, Banerjee et al., 2016). Free-field realizations via ghost systems provide explicit models for these extended structures (Banerjee et al., 2015, Banerjee et al., 2016).

Coadjoint orbit and induced representation constructions apply equally, with the representations parameterized by the orbits of the supertranslation and superrotation data, now including internal degrees of freedom (Caroca et al., 2018).

Higher-spin and hypergravity modules require adding towers of higher-spin currents (both bosonic and fermionic), for which the representation theory mimics that of "flat $\mathcal{W}_N$ " algebras: again, induced representation theory organizes the Hilbert space, with one-loop partition functions verifying the structure (Campoleoni et al., 2015, Campoleoni et al., 2016).

7. Physical Implications and Outlook

The representation theory of extended BMS has profound implications:

In quantum gravity, the string-like nature of representations captures the complete spectrum of gravitational infra-red dressings and soft sector dynamics, exceeding the Poincaré-based point-particle intuition (Ruzziconi et al., 2 Jan 2026).
In celestial holography, the extended BMS module structure underlies the mapping between 4d scattering amplitudes and 2d CFT correlators, with the full algebra dictating OPEs, Ward identities, and module spectra (Fotopoulos et al., 2019, Donnay et al., 2021, Batlle et al., 1 Jun 2025, Lowe et al., 2020).
The nontrivial interplay between induced and highest-weight modules bridges gravitational scattering states, defect CFTs, and stringy structures at null infinity.
Quantization, including BRST techniques, provides a unified language linking algebraic, geometric, and physical properties; anomalies and central extensions reflect quantum consistency and regulate infrared divergences (Batlle et al., 2024, Distler et al., 2018).

Further generalizations involve $\Lambda$ -BMS, smooth superrotations ( $\mathrm{Diff}(S^2)$ extensions), links to Carrollian field theory, and the emergence of extended $\mathcal{W}$ -algebras as the natural symmetry in flat-space holography (Ruzziconi, 2020, Batlle et al., 2024, Campoleoni et al., 2015).