3D Extended BMS Algebra

• Three-dimensional extended BMS Algebra is an infinite-dimensional symmetry structure that governs asymptotic dynamics at the boundaries of flat 2+1 spacetimes.
• It combines a Virasoro algebra of superrotations with an abelian ideal of supertranslations, realized via a Poisson bracket structure and Drinfeld–Sokolov reduction.
• The bi-Hamiltonian formulation enables an integrable hierarchy with recursive Gelfand–Dikii-type methods, linking conserved charges to geometric structures in 3D gravity.

The three-dimensional extended Bondi–Metzner–Sachs (BMS₃) algebra is an infinite-dimensional symmetry structure that governs the asymptotic dynamics of locally flat (2+1)-dimensional spacetimes, with deep connections to integrability, Poisson geometry, and boundary dynamics in general relativity. This algebra can be realized as the symmetry algebra of a hierarchy of integrable systems and admits a bi-Hamiltonian structure rooted in the Drinfeld–Sokolov formalism, facilitating explicit links to the geometry and conserved charges of three-dimensional gravity.

1. Algebraic Structure and Poisson Realization

The BMS₃ algebra arises as a semi-direct sum of a Virasoro (superrotation) algebra with an abelian ideal of supertranslations. Concretely, in the canonical realization, two fields (or “boundary functions”)—denoted by $J(\phi)$ and $P(\phi)$—define the phase space, with their Poisson brackets engineered to mirror the BMS₃ commutators. The fundamental bracket is

$$\{F, G\} = \int d\phi \begin{pmatrix} \frac{\delta F}{\delta J} & \frac{\delta F}{\delta P} \end{pmatrix} \cdot D \cdot \begin{pmatrix} \frac{\delta G}{\delta J} \ \frac{\delta G}{\delta P} \end{pmatrix}$$

where $D$ is a matrix-valued differential operator encapsulating both the Virasoro structure and the abelian ideal. Fourier expansion then reproduces algebraic relations such as

$$[J_m, J_n] = (m-n) J_{m+n}, \quad [J_m, P_n] = (m-n) P_{m+n}, \quad [P_m, P_n] = 0.$$

This Poisson realization forms the foundation for the integrable hierarchies and geometric structures associated with BMS₃ systems.

2. Drinfeld–Sokolov Reduction and Zero Curvature Formulation

The boundary theory admits a gauge-theoretic interpretation via a Lax pair formalism, reflecting its deep ties to integrable systems “à la Drinfeld–Sokolov.” One introduces an $\mathfrak{isl}(2,\mathbb{R})$-valued connection, akin to the (2+1)-dimensional Poincaré algebra. By fixing the spatial component as

$$a_\phi = J(\phi) \cdot (sl(2) \text{ generator}) + \text{terms in }P,\ \partial P, \ldots,$$

the zero curvature equation, $F = da + a \wedge a = 0$, yields the dynamics of the system. This underlies the integrable character of the model, ensuring that the equations of motion are equivalent to a flatness condition for an appropriate Chern–Simons connection, which connects to the metric formulation of gravity via a explicit gauge choice.

3. Bi-Hamiltonian Structure and Integrable Hierarchies

A key structural property is the bi-Hamiltonian nature of the system: two compatible Poisson structures, $D^{(2)}$ (the “canonical” BMS₃ Poisson structure) and $D^{(1)}$ (ultralocal), organize the integrable hierarchies. The equations of motion for the $k$-th member of the hierarchy are

$$\partial_t J = D^{(2)} \left[ \frac{\delta H^{(k)}}{\delta J} \right], \quad \partial_t P = D^{(2)} \left[ \frac{\delta H^{(k)}}{\delta P} \right]$$

and, equivalently,

$$\partial_t J = D^{(1)} \left[ \frac{\delta H^{(k+1)}}{\delta J} \right], \quad \partial_t P = D^{(1)} \left[ \frac{\delta H^{(k+1)}}{\delta P} \right].$$

The hierarchy is constructed by generalizing Gelfand–Dikii polynomials: the standard KdV hierarchy arises in the $P$-sector, while the $J$-sector evolves according to a tower of generalized polynomials related by

$D^{(1)} K^{(n+1)} = D^{(2)} K^{(n)}.$

This recursive scheme provides an infinite family of mutually commuting Hamiltonians—an indicator of complete integrability.

Two core cases:

• For $k=0$, the simplest (chiral) system emerges, with conserved charges generating the full BMS₃ algebra.
• For $k=1$, after suitable field redefinitions and time scaling, one finds equations equivalent to the Hirota–Satsuma coupled KdV systems, which include the classical KdV equation for $P$ and a coupled, inhomogeneous linear equation for $J$.

4. Conserved Charges and Commutative Structures

For each $k \geq 1$, the constructed hierarchy possesses an infinite-dimensional abelian algebra of conserved charges, all in involution: $\{ Q_i, Q_j \}_{D^{(2)}} = 0, \qquad \forall\ i,j.$ For $k=0$, the conserved charges reproduce the non-abelian BMS₃ symmetry algebra. The method for producing higher charges recursively via Gelfand–Dikii-type polynomials ensures the compatibility of the involutive property across the hierarchy. These charges are explicitly realized as surface integrals in the canonical approach and correspond, in the dual gravity theory, to conserved quantities associated with asymptotic symmetries.

5. Geometric Interpretation in 3D Gravity

The hierarchy’s dynamics admit a full geometrization: the field equations, when appropriately parameterized and with suitable boundary conditions, directly describe the evolution of spacelike surfaces embedded in locally flat (2+1)-dimensional spacetimes. Here, 3D general relativity is recast as a Chern–Simons theory for $\mathfrak{isl}(2,\mathbb{R})$. Upon choosing a gauge with radial dependence captured by $b = e^{r P_2}$ and a boundary one-form $a(\phi, t)$,

$A = b^{-1}(d + a) b,$

the flatness equation $F = dA + A\wedge A = 0$ coincides with the integrable hierarchy’s equations for $J$ and $P$. The residual diffeomorphisms preserving the boundary conditions—i.e., those transformations under which $A$ maintains its asymptotic form—become the infinite-dimensional BMS₃ symmetry and guarantee Noetherianity of the conserved charges, directly linking integrable and gravitational structures.

6. Specific Integrable Models and Analytic Solutions

Within the constructed hierarchy, for $k=1$, field redefinitions and rescaling relate the equations to the Hirota–Satsuma coupled KdV system. For general $k\geq1$, the hierarchy encompasses perturbed KdV equations as specific cases. Analytic solution methods, including the dressing method and other solitonic techniques valid for generalized Gelfand–Dikii models, are explicitly realized for arbitrary $k$, leveraging the integrable and Lax pair structure of the system.

7. Physical and Theoretical Implications

The bi-Hamiltonian, integrable structure endowed with infinite sets of conserved charges and the explicit geometric realization in 3D gravity tightly links the extended BMS₃ algebra to boundary dynamics and the phase space of the gravitational field. The structure encodes not only the boundary symmetries but also the integrable content of the reduced Einstein equations with suitable boundary conditions. The recovery of BMS₃ symmetry for $k=0$ and the appearance of commuting conservation laws for higher $k$ show that the algebra plays a central role in demarcating the integrable and geometric structure of flat 3D gravity and its extensions.

This framework underlies the emergence of infinite-dimensional symmetry in the boundary dynamics of locally flat spacetimes and reveals the deep integrable and geometric content of the three-dimensional extended BMS algebra (Fuentealba et al., 2017).