The conformal BMS group CBMS+CBMS^+, introduced by Haco, Hawking, Perry and Bourjaily, extends the Bondi-Metzner-Sachs (BMS) group of asymptotically flat spacetime at future null infinity by superrotations, a BMS dilatation EE, and an infinite family of BMS special conformal transformations CC, with closure under a modified bracket. The originating paper defined the group and computed its algebra but left its completion open: a supersymmetric extension appropriate for maximally supersymmetric conformal theories such as N=4\mathcal{N}=4 super Yang-Mills, together with the corresponding asymptotic superalgebra; a twistor representation of the group and of its supersymmetric extension (promised in a subsequent paper that never appeared); and the action of the group on conformally invariant quantum field theories — specifically on N=4\mathcal{N}=4 Yang-Mills — including the possible connection between these new symmetries, the asymptotic symmetries of Yang-Mills theory, and the infinite-dimensional symmetries known to exist for planar maximally supersymmetric Yang-Mills theory. The paper motivates, but does not construct, conserved charges for the new generators, and its conclusions do not list a charge construction as future work. None of these items is settled: substantial adjacent results exist, but the super-conformal-BMS algebra, the twistor representation, and the action of the group on conformal field theories remain open.

Background

An isolated radiating system in general relativity is modeled by an asymptotically flat spacetime, whose symmetries at future null infinity I+\mathscr{I}^+ (the boundary traced by outgoing light rays, parametrized in Bondi coordinates (u,r,xA)(u, r, x^A) by retarded time uu, radius rr and sphere angles xAx^A) form the Bondi-Metzner-Sachs (BMS) group. Besides Lorentz transformations, the BMS group contains infinitely many 'supertranslations': angle-dependent shifts u→u+f(xA)u \to u + f(x^A) of retarded time along individual light rays. Barnich and Troessaert proposed enlarging it by 'superrotations', generated by conformal Killing vectors YAY^A of the celestial two-sphere, giving the extended BMS group. In the modern infrared program, BMS Ward identities (the constraints that exact symmetries impose on correlation functions and scattering amplitudes, analogous to charge conservation) were shown to be equivalent to the soft photon and soft graviton theorems governing low-energy massless scattering, and the associated charges ('soft hair') were shown by Hawking, Perry and Strominger to act nontrivially on black hole horizons.

For theories that are not merely asymptotically flat but conformally invariant — Maxwell theory, the massless Dirac equation, classically conformal Yang-Mills theory, and in particular maximally supersymmetric N=4\mathcal{N}=4 super Yang-Mills (supersymmetry is a symmetry relating bosons and fermions; N=4\mathcal{N}=4 super Yang-Mills is the four-dimensional gauge theory with the maximal number of independent supercharges, and it carries an internal R-symmetry that rotates those supercharges) — the natural question is what replaces the BMS group at null infinity. Haco, Hawking, Perry and Bourjaily answered by solving the conformal Killing equation asymptotically at I+\mathscr{I}^+ under the usual Bondi falloff conditions. The resulting conformal BMS group CBMS+CBMS^+ has infinitesimal generators: the BMS supertranslations T(f)T(f) and superrotations R(YA)R(Y^A); an ordinary dilatation D=u∂u+r∂rD = u\partial_u + r\partial_r; a 'BMS dilatation' E=u22∂u+r(u+r)∂rE = \frac{u^2}{2}\partial_u + r(u+r)\partial_r; and a new infinite family of 'BMS special conformal transformations' C(ZA)C(Z^A) labeled by strictly conformal Killing vectors ZAZ^A of the sphere (the flat-space special conformal transformations sit inside this family, and the flat conformal generator KuK_u equals 2E2E). The group is strictly larger than both the flat-space conformal group and the extended BMS group. Its algebra — schematically [T,R]∼T[T, R] \sim T, [T,D]∼T[T, D] \sim T, [R,R]∼R[R, R] \sim R, [C,D]∼C[C, D] \sim C, [D,E]∼E[D, E] \sim E, [E,R]∼C[E, R] \sim C, and [R,C]∼E[R, C] \sim E (or ∼C\sim C when the superrotation is generated by a Killing vector), with [T,T]=[C,T]=[C,C]=[T,E]=[C,E]=[E,E]=0[T, T] = [C, T] = [C, C] = [T, E] = [C, E] = [E, E] = 0 — closes only under a modified bracket: because the vector fields are themselves metric-dependent, the ordinary Lie commutator must be corrected by the variation of the vector fields under the metric perturbation they generate, a correction that matters precisely in commutators involving TT. The construction is related to the conformal Carroll group of Duval, Gibbons and Horvathy (Carrollian geometry is the c→0c \to 0 structure intrinsic to null surfaces; the BMS group is a conformal extension of the Carroll group). A supersymmetric relative, the super-BMS group of Awada, Gibbons and Shaw for conformal supergravity, already existed in 1986, but — as the source itself notes — without any investigation of asymptotically conformal transformations.

The source paper computed the group and its algebra, but constructed no charges and established no action on quantum field theory, and its conclusions explicitly defer the rest: the generalization to conformal theories with maximal supersymmetry is called 'a natural road ahead', with the 'exciting possibility' of connecting the asymptotic symmetries of Yang-Mills theory (Strominger) to the infinite-dimensional symmetries of planar maximally supersymmetric Yang-Mills theory (the planar or large-color limit, in which the number of gauge-theory colors tends to infinity and scattering amplitudes simplify); a twistor representation (twistor space is Penrose's complexified null-geometric reformulation of massless kinematics, in which conformal symmetries act linearly) of the group and its supersymmetric extension was promised for a subsequent paper; and the introduction flags studying the action of the conformal BMS group on N=4\mathcal{N}=4 Yang-Mills as 'an obvious next step'. The question below is that deferred completion, in the source's own generality.

Problem Statement

Complete the conformal BMS symmetry framework at future null infinity in the generality in which it was posed by Haco, Hawking, Perry and Bourjaily:

  1. Supersymmetric extension. Construct the supersymmetric extension of the conformal BMS group appropriate for conformal theories with maximal supersymmetry (for example, N=4\mathcal{N}=4 super Yang-Mills), and compute the corresponding asymptotic superalgebra: the full set of fermionic generators and their brackets with the bosonic generators T(f)T(f), R(YA)R(Y^A), DD, EE and C(ZA)C(Z^A).

  2. Twistor representation. Construct a twistor representation of the conformal BMS group and of its supersymmetric extension — an explicit action of these symmetries on twistor space that reproduces their action on null infinity and their algebra — as promised in the source's conclusions.

  3. Action on conformal quantum field theories and amplitudes. Determine how the conformal BMS symmetries act on conformally invariant quantum field theories, and establish the action on maximally supersymmetric N=4\mathcal{N}=4 Yang-Mills theory, which the source flags as an obvious next step. Determine also the connection between these new symmetries and the infinite-dimensional symmetries of soft-particle scattering: establish whether and how the conformal BMS symmetries relate to the asymptotic symmetries of Yang-Mills theory and to the infinite-dimensional symmetries known to exist for planar maximally supersymmetric Yang-Mills theory.

These three deliverables form one program with a common objective: turning the conformal BMS group from a formally defined asymptotic symmetry group into a physically realized symmetry of conformal massless theories, where items 1 and 2 give the mathematical completion of the symmetry structure and item 3 gives its physical content, each being defined relative to the same group CBMS+CBMS^+. The three parts are jointly posed. The target throughout is the conformal BMS group itself, not a neighboring structure (the ordinary or extended BMS group, the Weyl BMS group, BMS plus superdilations, or a purely Carrollian conformal algebra). A construction of conserved charges for the new generators EE and CC is motivated by the source's introduction but is not part of its deferred program, and is out of scope.

References

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