Massless Scalars and Higher-Spin BMS in Any Dimension

Xavier Bekaerta and Blagoje Oblakb
a Institut Denis Poisson, Unité Mixte de Recherche 7013,

Université de Tours, Université d’Orléans, CNRS,

Parc de Grandmont, F-37200 Tours, France.

<EMAIL_ADDRESS>
b CPHT, CNRS, École Polytechnique, IP Paris,

F-91128 Palaiseau, France.

<EMAIL_ADDRESS>
Abstract. Starting from the asymptotic kinematics of massless scalar fields
near null infinity in any spacetime dimension, we build two higher-spin
extensions of the Carrollian definition of the BMS group and its
generalisations. The first extension exhibits conformal properties reminiscent
of the singleton in Anti-de Sitter space. The second acts on the space of
radiative solutions of the d’Alembert equation, i.e. on Sachs’s representation
of BMS, which we relate to the scalar massless Poincaré representation and
extend to any Carrollian manifold. The corresponding enveloping algebra is a
higher-spin extension of BMS that can be interpreted as the asymptotic
symmetry of a putative exotic higher-spin gravity theory around Minkowski
spacetime. Along the way, we provide a pedagogical introduction to Carrollian
geometry and its relation to BMS.

###### Contents

  1. 1 Motivation and summary
  2. 2 BMS action on massless scalars
    1. 2.1 Asymptotic scalar dynamics
    2. 2.2 An interlude on densities and weights
    3. 2.3 The WRac and its avatars
    4. 2.4 The unitary Sachs module
  3. 3 Carrollian geometry and BMS groups
    1. 3.1 Principal $\bm{\mathbb{R}}$-bundles and their automorphisms
    2. 3.2 Ehresmann connections as Carrollian clocks
    3. 3.3 BMS as conformal Carroll
    4. 3.4 Carrollian volumes and generalised BMS
  4. 4 Higher-spin BMS algebras
    1. 4.1 Higher symmetries of the WRac
    2. 4.2 Higher symmetries of the Sachs module
    3. 4.3 Asymptotic higher-spin symmetries
  5. Acknowledgements

##  1 Motivation and summary

Ever since the early seminal works on conserved quantities in general
relativity [1, 2, 3, 4, 5], it has been well established that asymptotic
symmetries are crucial for the quantisation of all gauge theories. This has
become even more manifest in recent years, following a series of striking
discoveries relating Bondi-Metzner-Sachs (BMS) symmetry [2, 3, 4] to flat
space holography [6, 7, 8, 9] and scattering amplitudes [10, 11], and their
modern blend known as “celestial holography” [12, 13, 14]. As a result, the
BMS group and its extensions [6, 7, 15, 16] are now key ingredients in any
putative holographic description of quantum gravity around flat backgrounds.

The present work originates from the interplay between these concepts and
higher-spin gravity theories. Indeed, despite heavy constraints imposed by no-
go theorems on interacting higher-spins in Minkowski spacetime (see e.g. [17,
18, 19] and references therein), it is still of interest to study such systems
and their symmetries: they are relevant for string theory in the tensionless
limit [20, 21], and flat space physics more generally has key applications.
One thus naturally wonders what are the higher-spin analogues of Minkowskian
asymptotic symmetries, and in particular of the BMS group. Conversely, one may
ask if there exist Minkowskian cousins of well-known actors in the higher-spin
AdS/CFT correspondence (see e.g. the reviews [22, 23, 24]), such as the
singleton [25] that may be seen as the fundamental representation of the
higher-spin algebra, or the Flato-Fronsdal theorem [26] that provides the
decomposition of the tensor product of two singletons. The goal of this paper
is to shed light on these questions and put forward possible answers.

This is not the first time such issues are raised in the literature. For
instance, higher-spin asymptotic symmetries and their implications are well
known in spacetime dimension three, in both AdS [27, 28, 29, 30, 31] and
Minkowski backgrounds [32, 33, 34, 35, 36]. The situation is much less clear-
cut in dimensions $\geqslant 4$: in AdS, the standard higher-spin holographic
dictionary bears no obvious relation to enhanced asymptotic symmetries, while
in flat space the only known construction of higher-spin asymptotic symmetries
is that of [37, 38, 39, 40], where it was shown that Weinberg’s soft theorem
for higher-spin gauge bosons can be rephrased as the Ward identity of an
extension of standard gravitational BMS symmetry. (See also [41], whose
motivations are analogous to ours.)

Another key ingredient of this work is the mathematical language of curved
Carrollian geometry [42], based on the Carroll symmetry group [43, 44]. Its
first appearance in the physics literature dates back to [45] in the “zero
signature limit” of the Hamiltonian formulation of general relativity. More
importantly for our purposes, it was first related to BMS in [46, 47] as a
modern take on Penrose’s classical approach to asymptotic symmetries [48, 49,
50, 51]. Since then, Carrollian geometry has been ubiquitous in flat space
physics, including e.g. ultra-relativistic hydrodynamics at infinity [52, 53,
54, 55] and celestial holography [56, 57].

Despite this large body of existing knowledge, the line of thought pursued
here appears to be new. Namely, inspired by the key role of the singleton—also
known as the “Rac” [26]—in higher-spin AdS/CFT, we study the asymptotic
kinematics of a massless scalar field near null infinity [58, 59, 60] and use
it to define higher-spin BMS algebras based on Carrollian geometry. Our paper
is thus a step towards flat higher-spin holography, understood as an analogue
of the holographic dictionary between a scalar bulk field with critical mass
in AdS and the higher-spin symmetry of its boundary data. In practice, the
analysis is carried out in arbitrary bulk spacetime dimension $d+2$, involving
$d$-dimensional celestial spheres $S^{d}$, and yields two inequivalent
Minkowskian analogues of the usual singleton:

  * 1.

The Wick-rotated Rac (WRac) consists of time-independent field configurations
at null infinity that stem from “overleading” solutions of the d’Alembert
equation. It is essentially obtained upon replacing $\mathfrak{so}(d,2)$ by
$\mathfrak{so}(d+1,1)$ in the group-theoretic definition of the Rac [26] and
should therefore look familiar to higher-spin experts, as it is formally
described by the same equations up to proper changes of signature. In fact, it
shares several important qualitative features with the Rac—it has no bulk
degrees of freedom and lives on the conformal boundary—save for one cardinal
property: it is not unitarisable. Another unsatisfactory feature of the WRac
is that is not a faithful representation of BMS (nor of Poincaré); only the
Lorentz subgroup is represented faithfully.

  * 2.

The Sachs module [4] consists of time-dependent configurations on null
infinity, determined by radiative solutions of the d’Alembert equation. It
seems to be the best candidate for an analogue of the Rac in flat spacetime,
furnishing a representation of Poincaré and BMS that is both faithful and
unitary [4, 61, 62, 63, 64, 65, 66]. Its main weakness is its generality: as
we shall see, it furnishes a unitary representation not only of BMS, but of
the immensely larger group of all Carrollian bundle automorphisms.

The peculiarities of these two setups suggest that flat higher-spin holography
must be quite different from its AdS cousin. We shall proceed nonetheless and
define new Carrollian higher-spin symmetries by considering algebras of
differential operators on null infinity that preserve these structures. The
WRac will thus yield an algebra isomorphic to the standard one of bosonic
higher-spin gravity [67, 68] on a de Sitter spacetime in one less dimension,
signalling that it is “too small” to provide a suitable starting point for
higher-spin gravity in Minkowski spacetime since all supertranslations are
quotiented out. On the other hand, the symmetry algebra of the Sachs module
will define a novel candidate higher-spin extension of BMS, whose spectrum of
generators turns out to contain the BMS Killing tensors obtained in [38] from
the asymptotic symmetries of free massless higher-spin fields. Since the space
of symmetries encountered in [37, 38, 39, 40] only had a vector structure, our
present work may be seen as a way to endow it with a Lie bracket. A corollary
of this proposal is also a sharp distinction between symmetry algebras
stemming from distinct choices of fall-off conditions; choosing the “correct”
higher-spin BMS algebra for a putative higher-spin theory around Minkowski
spacetime requires the selection of certain preferred fall-offs.

The paper is organised as follows. We start in section 2 by writing the
asymptotic solution of the d’Alembert equation near null infinity. Its leading
piece then yields either the WRac, or the Sachs module of the BMS group. In
particular, we extend the Sachs Hermitian form to any dimension and relate it
to the standard Poincaré-invariant one for scalar massless unitary
representations. Section 3 is devoted to a detailed, self-contained review of
Carrollian geometry and its relation to (generalised) BMS symmetry. This also
allows us to write the Sachs inner product in a coordinate-independent way
such that invariance under bundle automorphisms becomes manifest. We stress
that sections 2–3 are devoid of higher spins and may be of general interest to
anyone working on asymptotic symmetries and flat holography. Finally, section
4 is devoted to the definition of higher-spin extensions of BMS based on
Carrollian geometry and the two aforementioned representations (WRac and
Sachs). We briefly conclude with a discussion on the fall-offs and spectra of
bulk gauge fields needed for a putative higher-spin theory to admit these
algebras as asymptotic symmetries.

##  2 BMS action on massless scalars

In this section, we solve d’Alembert’s equation $\square\Phi=0$ as an
asymptotic series in the inverse distance away from the origin in Bondi
coordinates, and find that the space of solutions supports a linear action of
the BMS algebra. We then briefly recall certain definitions pertaining to the
transformations of densities on manifolds under diffeomorphisms, before
identifying the densities that occur in the asymptotic expansion of scalars.
Thus, an “overleading” choice of fall-offs naturally leads to the WRac
solution mentioned in the introduction, while radiative fall-offs are such
that the field’s leading component defines the Sachs module. Related
considerations have been put forward in [58, 59, 60, 69] in the context of
asymptotic symmetries. The Carrollian perspective on these results is
relegated to section 3, while their higher-spin implications will be treated
in section 4.

###  2.1 Asymptotic scalar dynamics

Here we briefly recall the expression of Poincaré and BMS generators in Bondi
coordinates, then study the asymptotic expansion of solutions of the
d’Alembert equation near null infinity. This will motivate the existence of
two distinguished kinds of boundary data: the first, time-dependent but
unconstrained, will eventually lead to the Sachs module. The other, time-
independent but singular, will yield a WRac.

#### Bondi coordinates and BMS generators.

Consider retarded Eddington-Finkelstein (Bondi) coordinates $(r,u,x^{a})$ on
$(d+2)$-dimensional flat spacetime $\mathbb{R}^{d+1,1}$, where $u$ is retarded
time, $r$ is the distance away from some arbitrary spatial origin, and $x^{a}$
($a=1,\ldots\,,d$) are angles on a celestial sphere $S^{d}$ (see fig. 1). In
these terms, the Minkowski metric reads

$\text{d}s^{2}=-\text{d}u^{2}-2\,\text{d}u\,\text{d}r+r^{2}g_{ab}(x)\,\text{d}x^{a}\,\text{d}x^{b}\,,$
(1)

so $\partial_{r}$ is tangent to radial outgoing null rays. The isometries of
(1) span the Poincaré group, whose generators are vector fields
$\xi=\xi^{\mu}\partial_{\mu}$ with components (see [7, eqs. (4.7)–(4.8)] or
[37, eq. (2.13)])

$\displaystyle\xi^{r}$
$\displaystyle=-\frac{r+u}{d}\,\nabla_{a}X^{a}(x)+\frac{1}{d}\,\nabla^{2}\alpha(x)\,,$
(2) $\displaystyle\xi^{u}$
$\displaystyle=\alpha(x)+\frac{u}{d}\,\nabla_{a}X^{a}(x)\,,$ (3)
$\displaystyle\xi^{a}$
$\displaystyle=X^{a}(x)-\frac{1}{r}\,g^{ab}(x)\big{[}\partial_{b}\alpha(x)+\frac{u}{d}\,\partial_{b}\nabla_{c}X^{c}(x)\big{]}\,,$
(4)

where $X^{a}\partial_{a}$ is a conformal Killing vector field on $S^{d}$,
$\nabla$ is the Levi-Civita connection on $S^{d}$ with Laplacian $\nabla^{2}$,
and $\alpha(x)$ is any function on $S^{d}$ such that

$\tfrac{1}{d}\,\nabla^{2}\alpha+\alpha=\alpha_{0}\,,$ (5)

with $\alpha_{0}$ the zero-mode of $\alpha(x)$.111Eq. (5) can equivalently be
written as $\partial_{a}(\nabla^{2}+d)\alpha=0$, without mentioning the zero
mode. It is also equivalent to the “good-cut equation in Bondi frame”
$\nabla_{a}\nabla_{b}\alpha\propto g_{ab}$ (see e.g. [51, eq. (2.6)]). The
elements of the pair $(X,\alpha)$ respectively generate Lorentz
transformations and spacetime translations. In particular, Lorentz generators
correspond to conformal maps on the celestial sphere and induce angle-
dependent rescalings of the radial coordinate (since they leave the term
$r^{2}\text{d}\Omega^{2}$ invariant in the metric (1)). The Lie bracket of
such pairs reads

$\big{[}(X,\alpha),(Y,\beta)\big{]}=\Big{(}[X,Y],X^{a}\partial_{a}\beta-\tfrac{1}{d}\,\beta\,\nabla_{a}X^{a}-Y^{a}\partial_{a}\alpha+\tfrac{1}{d}\,\alpha\,\nabla_{a}Y^{a}\Big{)}\,,$
(6)

exhibiting the standard semi-direct sum structure of the Poincaré algebra
$\mathfrak{iso}(d+1,1)=\mathfrak{so}(d+1,1)\inplus\mathbb{R}^{d+2}$. The BMS
generalisation consists in relaxing the restriction (5) and allowing the
function $\alpha(x)$ to be arbitrary (as opposed to having only modes of
angular momentum $\ell\leqslant 1$); it is then known as a supertranslation
[2, 3, 4]. We return to this in much greater detail below, starting in section
2.4.

$u$$r$$x^{a}$$\mathscr{I}^{+}$$\mathscr{I}^{-}$•$i^{+}$•$i^{-}$

$i^{+}$$i^{-}$

Figure 1: Left: A Penrose diagram of Minkowski spacetime, displaying Bondi
coordinates $(r,u,x^{a})$ with $a=1,\cdots\\!,d$, along with the past and
future light-cones of the origin. Each circle is really a sphere $S^{d}$. The
conformal boundary of the diagram consists of future and past null infinities,
respectively denoted $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$. Right: The past
and future light-cones of the origin are related to null infinities by an
inversion $x^{\mu}\to x^{\mu}/x^{2}$, written here in inertial coordinates.
Any two segments of the same colour are mapped on one another under this map,
and the two dashed lines indicate its fixed points. See footnote 8 for a more
detailed statement.

#### Asymptotic d’Alembert equation.

Now let $\Phi$ be a complex massless scalar field in $\mathbb{R}^{d+1,1}$ with
Lagrangian density ${\cal L}=\sqrt{g}\,\nabla_{\mu}\Phi^{*}\nabla^{\mu}\Phi$,
whose equation of motion is the d’Alembert equation

$\square\Phi=\partial_{r}^{2}\Phi-2\,\partial_{r}\dot{\Phi}+\tfrac{d}{r}\,\partial_{r}\Phi-\tfrac{d}{r}\,\dot{\Phi}+\tfrac{1}{r^{2}}\nabla^{2}\Phi=0\,,$
(7)

where $\dot{\Phi}:=\partial_{u}\Phi$. Our goal is to expand this equation near
future null infinity (where $r\to\infty$ with fixed $u$ and $x$) and relate
the various degrees of the expansion to BMS representations. Indeed, in the
spirit of “asymptotic quantisation” [70, 51], the symplectic form expressed in
terms of data $\phi$ at null infinity reads

$\Omega[\delta\phi]=\lim_{r\to\infty}\Big{[}\,\,i\,r^{d}\int\text{d}u\,\text{d}^{d}x\,\sqrt{g(x)}\,\delta\Phi^{*}\wedge\delta\dot{\Phi}\,\Big{]}\,,$
(8)

where the wedge product is “vertical” and involves one-forms in field space.
The corresponding energy functional is

$E[\phi]=\frac{1}{2}\lim_{r\to\infty}\Big{[}\,r^{d}\int\text{d}u\,\text{d}^{d}x\,\sqrt{g(x)}\,|\dot{\Phi}|^{2}\,\Big{]},$
(9)

so it is clear that fall-offs at (null) infinity play a key role for the
scalar field’s phase space. Accordingly, write the solution of (7) as an
asymptotic series222The analogue of (10) for a field with mass $m>0$ involves
exponentials $e^{-mr}$. The resulting radial expansion is different from that
of the massless case; we do not consider it here.

$\Phi(r,u,x)\sim\frac{1}{r^{\Delta}}\sum_{n=0}^{\infty}\frac{\phi_{n}(u,x)}{r^{n}}\qquad\text{as
}r\to\infty\,,$ (10)

where $\phi_{0}\neq 0$ by definition and $\Delta$ is some number that we leave
free for now. Plugging this ansatz in the d’Alembert equation (7) yields [60]

$\displaystyle{\cal O}(r^{-(\Delta+1)}):$
$\displaystyle(d-2\Delta)\dot{\phi}_{0}$ $\displaystyle=0\,,$ (11)
$\displaystyle{\cal O}(r^{-(\Delta+2)}):$
$\displaystyle(d-2-2\Delta)\dot{\phi}_{1}+\Delta(d-1-\Delta)\phi_{0}-\nabla^{2}\phi_{0}$
$\displaystyle=0\,,$ (12) $\displaystyle{\cal O}(r^{-(\Delta+3)}):$
$\displaystyle(d-4-2\Delta)\dot{\phi}_{2}+(\Delta+1)(d-2-\Delta)\phi_{1}-\nabla^{2}\phi_{1}$
$\displaystyle=0\,.$ (13)

The structure of this sequence is reminiscent of the Fefferman-Graham
expansion of scalar dynamics in AdS (see e.g. [71, sec. 5]), but it is quite
distinct from it in several respects. Most crucially, it does not uniquely fix
the value of $\Delta$ since the leading-order equation (11) yields

$\Delta=d/2\qquad\text{or}\qquad\dot{\phi}_{0}=0\,.$ (14)

Here the first option imposes no constraint on $\phi_{0}$ but fixes $\Delta$
in a way that allows the energy (9) to be finite. It is indeed this first
possibility that one typically considers in the context of asymptotic
symmetries [58, 59, 60], where the corresponding solutions of the d’Alembert
equation are said to be “radiative”. By contrast, the second option in (14)
leaves $\Delta$ arbitrary, and turns out to be fairly natural from a higher-
spin perspective. Accordingly, we now investigate some immediate features of
these two families of solutions. More elaborate considerations regarding their
conformal properties and their transformation laws under BMS will be presented
in sections 2.3–2.4.

#### Recursive solution and GJMS operators.

Let us first analyse the branch of solutions obtained by choosing $\Delta=d/2$
in (14). Then eq. (11) is automatically satisfied, with $\phi_{0}$ to be
thought of as an arbitrary initial condition on null infinity; we shall
typically assume that $\dot{\phi}_{0}$ goes to zero in the far past and future
in order for the energy (9) to be finite, but this is not essential here. The
subsequent hierarchy of equations (12), (13), etc. is then solved in terms of
time-independent “integration functions” $F_{1}(x)$, $F_{2}(x)$, etc. and time
integrals of $\phi_{0}$. For instance, eqs. (12)–(13) yield

$\displaystyle\phi_{1}(u,x)$
$\displaystyle=F_{1}(x)+\tfrac{1}{2}\int_{0}^{u}\text{d}v\,\big{(}\nabla^{2}-\tfrac{d}{2}(\tfrac{d}{2}-1)\big{)}\phi_{0}(v,x)\,,$
(15) $\displaystyle\phi_{2}(u,x)$
$\displaystyle=F_{2}(x)-\tfrac{u}{4}\big{(}\nabla^{2}-(\tfrac{d}{2}+1)(\tfrac{d}{2}-2)\big{)}F_{1}(x)$
$\displaystyle\quad+\tfrac{1}{8}\int_{0}^{u}\text{d}v\int_{0}^{v}\text{d}w\,\prod_{j=0}^{1}\big{(}\nabla^{2}-(\tfrac{d}{2}+j)(\tfrac{d}{2}-j-1)\big{)}\phi_{0}(w,x)\,,$
(16)

where $F_{1}(x)$ and $F_{2}(x)$ are smooth but otherwise arbitrary. Similar
expressions hold for all $\phi_{n}$’s, $n\geqslant 1$: each starts with an
integration function $F_{n}(x)$, then involves a polynomial in $u$ where the
coefficient of $u^{k}$ is linear in $F_{n-k}(x)$ ($k=1,\cdots\\!,n-1$), and
ends with an $n$-fold time integral of $\phi_{0}(u,x)$, acted upon by the
differential operator

$P_{2n}:=\prod_{j=0}^{n-1}\Big{(}\nabla^{2}-(\tfrac{d}{2}+j)(\tfrac{d}{2}-j-1)\Big{)}$
(17)

of order $2n$. For $n=1$, this is the Yamabe operator on $S^{d}$, i.e. the
conformally covariant completion of the Laplacian. (See e.g. [67, sec. 3] for
a concise review.) More generally, (17) is the so-called GJMS operator of
order $2n$ on $S^{d}$ [72, 73]. It is conformally covariant when acting on a
scalar primary field of scaling dimension $\tfrac{d}{2}-n$ on $S^{d}$ (see
section 2.2 for the terminology). More generally, it is Weyl-covariant when it
acts on conformal densities of weight $n-\tfrac{d}{2}$ on an arbitrary curved
manifold.

#### Truncated hierarchy.

We now turn to the branch of solutions of eq. (7) obtained by choosing the
second option in (14): $\phi_{0}(u,x)=\phi_{0}(x)$. Then the power $\Delta$ is
arbitrary and the hierarchy of equations starting with (12)–(13) can again be
integrated iteratively in terms of successive time integrals of $\phi_{0}$ and
time-independent integration functions $F_{n}(x)$, similarly to eqs.
(15)–(16). The key difference with respect to the choice $\Delta=d/2$ is that
(i) the energy (9) is now generally either infinite or zero,333Since
$\dot{\phi}_{0}=0$, one might be tempted to conclude that (9) vanishes; the
reason this is not the case is because subleading terms in the expansion (10)
are time-dependent, and contribute to (9). and (ii) the differential operators
involved in the hierarchy of solutions are no longer conformally covariant,
since they do not take the GJMS form (17). This second issue can in fact be
cured in one specific class of choices of $\Delta$: if

$\Delta=\frac{d}{2}-N$ (18)

for some positive integer $N$, then the expansion (10) can be split as

$\Phi(r,u,x)\sim\frac{1}{r^{d/2-N}}\sum_{n=0}^{N-1}\frac{\phi_{n}(u,x)}{r^{n}}+\frac{1}{r^{d/2}}\sum_{n=0}^{\infty}\frac{\phi_{N+n}(u,x)}{r^{n}}$
(19)

where $\phi_{N}(u,x)$ is again completely unconstrained while
$\phi_{0},\phi_{1},\ldots\,,\phi_{N-1}$ satisfy a closed system of equations
that can be solved explicitly. To see this, note that (18) truncates the first
$N$ equations of the hierarchy (11)–(12)–(13)–$\,\cdots$ so that

$\displaystyle\dot{\phi}_{0}$ $\displaystyle=0\,,$ (20) $\displaystyle
2(N-1)\dot{\phi}_{1}+(\tfrac{d}{2}-N)(\tfrac{d}{2}+N-1)\phi_{0}-\nabla^{2}\phi_{0}$
$\displaystyle=0\,,$ (21) $\displaystyle
2(N-2)\dot{\phi}_{2}+(\tfrac{d}{2}-N+1)(\tfrac{d}{2}+N-2)\phi_{1}-\nabla^{2}\phi_{1}$
$\displaystyle=0\,,$ (22) $\displaystyle\;\,\,\vdots$
$\displaystyle(\tfrac{d}{2}-N+N)(\tfrac{d}{2}+N-N-1)\phi_{N-1}-\nabla^{2}\phi_{N-1}$
$\displaystyle=0\,.$ (23)

The solution can again be found by writing $\phi_{1},\ldots\,,\phi_{N-1}$ as
described around eq. (16), with the difference that time integrals can now be
carried out explicitly. Furthermore, the last equation (23) involves no time
derivatives, so it boils down to a pure constraint that needs to hold at all
times, eventually yielding the set of conditions

$\prod_{j=0}^{N-1-k}\Big{(}\nabla^{2}-(\tfrac{d}{2}+j)(\tfrac{d}{2}-j-1)\Big{)}F_{k}(x)=0\qquad\forall\,k=0,1,\cdots\\!,N-1,$
(24)

where $F_{0}:=\phi_{0}(x)$. Each integration function $F_{k}$ must therefore
lie in the kernel of a GJMS operator (17) of order $2(N-k)$. In general, such
kernel conditions have no smooth solution, so most $F_{k}$’s are either
singular, or vanish. As a result, the “overleading” branch of solutions of (7)
built in this way is similarly either singular or trivial; requiring that
solutions be smooth then sets $\phi_{0}=\phi_{1}=\cdots=\phi_{N-1}=0$ in the
expansion (19), reducing it to what one would have obtained by choosing
$\Delta=d/2$ in the first place. This is yet another sense in which the choice
$\Delta=d/2$ is “canonical” in (14), supplementing the energy argument above.

An exception occurs for $d$ even and $N\geqslant d/2\geqslant 1$: in that
case, the GJMS kernel condition (24) involves factors of the form
$\nabla^{2}+\ell(\ell+d-1)$, where $\ell=j-\tfrac{d}{2}+1\geqslant 0$ in terms
of the index $j$ used in (24). Each such factor does have a non-trivial smooth
kernel that consists of spherical harmonics with angular momentum $\ell$,
where “angular momentum” is meant in the sense of the index $\ell$ in the
value $\ell(\ell+d-1)$ of the quadratic Casimir of SO($d+1$). In terms of the
field $\Phi$ solving the d’Alembert equation (7), each such spherical harmonic
is multiplied by some power of $r$ and some positive power of $u$. It is thus
perfectly possible—at least in even spacetime dimensions—to build smooth
solutions of d’Alembert that do not satisfy the canonical boundary condition
$\Phi={\cal O}(r^{-d/2})$. One should keep in mind, however, that their energy
(9) is typically infinite and that their spacetime dependence is heavily
constrained, since the set of such smooth solutions is finite-dimensional.

#### Combining the two branches.

To summarise, assuming that the only powers of $r$ in the expansion (10) are
integers modulo $d/2$, the space of solutions of the d’Alembert equation (7)
splits in two branches, roughly in the spirit of AdS/CFT:

$\Phi(r,u,x):=R(r,u,x)+S(r,u,x)\qquad\text{(on-shell)}.$ (25)

Here the second term has a radial expansion dictated by the canonical choice
$\Delta=d/2$ in (14), namely

$S(r,u,x)\sim\frac{1}{r^{d/2}}\sum_{n=0}^{\infty}\frac{\phi_{n}(u,x)}{r^{n}}\qquad\text{with}\qquad\phi_{0}(u,x):=\lim_{r\to\infty}[\,r^{\frac{d}{2}}\,S(r,u,x)\,]\,,$
(26)

where $\phi_{0}(u,x)$ is unconstrained while the subleading terms
$\phi_{n\geqslant 1}$ are fixed by $\phi_{0}$ and integration functions
$F_{n}(x)$, in a hierarchy that begins with eqs. (15)–(16). The energy (9) of
$S$ is solely determined by the leading term $\phi_{0}$, and it is finite
provided $\dot{\phi}_{0}$ decays to zero sufficiently fast at early and late
times. We shall refer to this canonical branch of solutions as the Sachs
module of BMS, in reference to the work [4] where it was first studied from
the perspective of asymptotic symmetries; this is also why we write this
branch as “$S$” in eq. (25). By contrast, the first term on the right-hand
side of (25) typically has an “overleading” and finite radial expansion

$R(r,u,x)=\sum_{n=1}^{N}\frac{\psi_{n}(u,x)}{r^{d/2-n}}\,,$ (27)

where the $\psi_{n}$’s are again determined by integration functions $F_{k}$,
but these are now constrained by GJMS kernel conditions of the form (24). The
corresponding solutions are generally singular and have infinite energy. Non-
singular, non-trivial solutions only occur in special circumstances
($N\geqslant d/2\in\mathbb{N}$), then spanning a finite-dimensional set of
field configurations given by spherical harmonics. While this overleading
branch of solutions is somewhat awkward from the viewpoint of asymptotic
symmetries, it is quite natural from a higher-spin perspective that likens it
to the singleton, or Rac, in AdS [26, 74], whose boundary behaviour is indeed
constrained by a d’Alembertian condition. In the flat case, this condition
becomes a set of Euclidean Laplacian constraints (24) on celestial spheres, so
we shall refer to this branch of solutions as the Wick-rotated Rac (WRac) of
Minkowski space; hence the notation “$R$” in eq. (25).

The remainder of this section is devoted to a detailed group-theoretic
analysis of these leading and subleading components: following a brief aside
(section 2.2) on densities and weights, the WRac is discussed in section 2.3
while the BMS transformation law of the various terms of the expansion (10),
and in particular the unitary Sachs module, are studied in section 2.4.

###  2.2 An interlude on densities and weights

This work is concerned with transformation laws of various (mostly scalar)
fields on celestial spheres under the BMS group, and in particular under
celestial diffeomorphisms or conformal maps, so it is crucial to fix our
conventions in that respect. What follows therefore spells out our definitions
for Radon densities, volumic densities, conformal densities, conformal
primaries, and the accompanying terminology.

A word of caution may be in order here. The abstract content of this
subsection may seem off-putting to physicists, especially as we start with
some measure-theoretic concepts that are not normally covered in the physics
literature. However, to the extent that our first objective is to understand
BMS transformations of the $\phi_{n}$’s in the expansion (10), it is essential
to define an action of diffeomorphisms on densities when the underlying metric
(e.g. that of a celestial sphere) is not flat. This problem turns out to
require a bit of mathematical machinery that, to our knowledge, has been
overlooked in the literature on BMS; we attempt to fill that gap here, as it
will even play a Carrollian role in section 3.4. (For an introduction to
measure theory, see e.g. [75].)

#### Radon densities.

Consider an orientable $d$-dimensional manifold $\mathscr{M}$ and endow it
with a measure $\mu$.444For any measurable set
$\mathscr{N}\subseteq\mathscr{M}$, one writes
$\mu(\mathscr{N}):=\int_{\mathscr{N}}\text{d}\mu$. The latter is equivalent to
a volume form $\varepsilon$, i.e. a nowhere-vanishing top form; given local
coordinates $x^{a}=(x^{1},\cdots,x^{d})$, any such form reads555We abuse
notation slightly by using the same letter $\varepsilon$ to denote both the
volume form $\varepsilon$ and its “density function” $\varepsilon(x)$. We also
write $\text{d}\mu$ for the volume form corresponding to the measure $\mu$.
This notation is common but rather abusive since this differential form is not
exact in general.

$\text{d}\mu=\varepsilon=\varepsilon(x)\,\text{d}^{d}x$ (28)

where $\varepsilon(x)$ is some strictly positive function on $\mathscr{M}$ and
$\text{d}^{d}x:=\text{d}x^{1}\wedge\cdots\wedge\text{d}x^{d}$. This applies in
particular when $\mathscr{M}$ carries a metric $g_{ab}$, in which case
$\varepsilon(x)=\sqrt{g(x)}:=\sqrt{\det g_{ab}(x)}$.

Now suppose some group acts on $\mathscr{M}$ by orientation-preserving
diffeomorphisms $x\to x^{\prime}=F(x)$. In general, such transformations
deform the measure in the sense that $F^{*}\varepsilon\neq\varepsilon$ : the
pullback by $F$ need not preserve the volume form. However, it is certainly
true that $F^{*}\varepsilon=\rho_{F}\,\varepsilon$ (or, equivalently, in
coordinates $\text{d}\mu\big{(}F(x)\big{)}=\rho_{F}(x)\,\text{d}\mu(x)$ ) for
some positive function $\rho_{F}(x)$ known as the Radon-Nikodym derivative of
$\mu$ [76, sec. 18.4], proportional to the Jacobian of $F$. It is indeed
straightforward to compute $\rho_{F}$ for any diffeomorphism acting on a
volume form: writing the diffeomorphism as $x\to F(x)$, eq. (28) yields

$\rho_{F}(x)=\frac{\text{d}\mu\big{(}F(x)\big{)}}{\text{d}\mu(x)}=\frac{\varepsilon\big{(}F(x)\big{)}}{\varepsilon(x)}\,\bigg{|}\frac{\partial
F}{\partial x}\bigg{|}\,,$ (29)

where $|\partial F/\partial x|:=|\det(\partial x^{\prime a}/\partial x^{b})|$
is the Jacobian of the map $x\to x^{\prime}=F(x)$ in local coordinates. It is
then straightforward to verify the cocycle property $\rho_{F_{1}\circ
F_{2}}=F_{2}^{*}(\rho_{F_{1}})\,\rho_{F_{2}}$ [77, sec. 3.2.3], either from
the pullback definition $F^{*}\varepsilon=\rho_{F}\,\varepsilon$, or from the
coordinate expression (29). Again, all this applies to measures induced by
metrics, in which case

$\rho_{F}(x)=\sqrt{\frac{g\big{(}F(x)\big{)}}{g(x)}}\,\,\bigg{|}\frac{\partial
F}{\partial x}\bigg{|}\,.$ (30)

We stress that the presence of the same metric $g$ in both the numerator and
the denominator of (30) is not a typo: the metric was not transformed with the
usual transformation law under the diffeomorphism $x\to x^{\prime}$ because
the measure density $\varepsilon(x)=\sqrt{g(x)}$ is understood as being fixed
in the present setting. In fact, had the metric been transformed following the
usual formula

$g^{\prime}_{ab}(x^{\prime})=\frac{\partial x^{c}}{\partial x^{\prime
a}}\frac{\partial x^{d}}{\partial x^{\prime b}}\,g_{cd}(x)\,,$ (31)

the naïve candidate $\sqrt{g^{\prime}(x^{\prime})/g(x)}\,|\partial
x^{\prime}/\partial x|$ would have equalled one. By contrast, the actual
Radon-Nikodym derivative (30) is typically a highly non-trivial function on
$\mathscr{M}$.

Having fixed the terminology, we are now ready to state the following
definition: a Radon density with weight w on $\mathscr{M}$ is a function
$\phi(x)$ that transforms under any diffeomorphism $F:x\to x^{\prime}$ as
$\phi\to F\cdot\phi:=(\rho_{F^{-1}})^{\texttt{w}}\,\phi\circ F^{-1}$. More
explicitly, writing $F\cdot\phi:=\phi^{\prime}$ and using $\rho_{F^{-1}}\circ
F=\rho_{F}^{-1}$, the transformation law reads

$\phi^{\prime}(x^{\prime})=\rho_{F}(x)^{-\texttt{w}}\,\phi(x)\stackrel{{\scriptstyle\text{\eqref{s12t}}}}{{=}}\left(\frac{g(x)}{g(x^{\prime})}\right)^{\texttt{w}/2}\bigg{|}\,\det\left(\frac{\partial
x^{\prime a}}{\partial x^{b}}\right)\,\bigg{|}^{-{\texttt{w}}}\,\phi(x)\,.$
(32)

Here we chose to write the Radon-Nikodym derivative (30) only for the special
case where the measure on $\mathscr{M}$ is inherited from a metric; this is by
far the most relevant situation we will encounter below. In particular, it is
essential for later reference to work out the action of vector fields on
densities from the infinitesimal version of eq. (32): letting $x^{\prime
a}=x^{a}+\epsilon X^{a}(x)$ and defining $\delta\phi:=-\lim_{\epsilon\to
0}\tfrac{1}{\epsilon}(\phi^{\prime}-\phi)$, the expansion of (32) yields

$\delta\phi={\cal L}_{X}\phi+\texttt{w}\,\nabla_{a}X^{a}\,\phi$ (33)

where ${\cal L}_{X}\phi:=X^{a}\partial_{a}\phi$ and $\nabla_{a}X^{a}$ is the
covariant divergence of the vector field $X$. More generally, for an arbitrary
volume form $\varepsilon$ on $\mathscr{M}$, the factor $\nabla_{a}X^{a}$ would
be replaced by the divergence $\text{div}(X)$ defined thanks to the Lie
derivative

${\cal L}_{X}\varepsilon=\text{div}(X)\,\varepsilon\,.$ (34)

In coordinates, it follows from (28) that
$\text{div}(X)=\varepsilon(x)^{-1}\partial_{a}\big{(}\varepsilon(x)X^{a}(x)\big{)}=\partial_{a}X^{a}(x)+X^{a}(x)\,\partial_{a}\big{(}\ln\varepsilon(x)\big{)}$.
For a scalar field ($\texttt{w}=0$), eq. (33) reduces to
$\delta\phi=X^{a}\partial_{a}\phi$. An example of scalar density with non-zero
weight is provided by (super)translations, whose weight $\texttt{w}=-1/d$ is
visible in the BMS bracket (6). More generally, we shall see in section 2.4
that eqs. (32)–(33) provide the basic form of the BMS transformation law of
the $\phi_{n}$’s in the expansion (10).

One readily verifies that the definition (32) furnishes a representation of
the group of orientation-preserving diffeomorphisms of $\mathscr{M}$, in the
sense that $F_{1}\cdot(F_{2}\cdot\phi)=(F_{1}\circ F_{2})\cdot\phi$.
Equivalently, the commutator of two transformations of the form (33) generated
by $X,Y$ is itself of the form (33) generated by the Lie bracket $[X,Y]$. The
vector space carrying this representation is the set $C^{\infty}(\mathscr{M})$
of all smooth functions on $\mathscr{M}$. In what follows we denote this space
by $C^{\infty}_{\texttt{w}}(\mathscr{M})$ to stress the weight w; in
particular, $C^{\infty}_{0}(\mathscr{M})=C^{\infty}(\mathscr{M})$ is the usual
space of scalar fields on $\mathscr{M}$. Note that the generalisation to
tensor-valued densities is straightforward: simply add the appropriate
Jacobian matrices on the right-hand side of (32). In terms of infinitesimal
transformations, eq. (33) then remains valid with ${\cal L}_{X}\phi$ the usual
Lie derivative of a tensor field.

We stress that the transformation laws (32)–(33) are the most important ones
for our purposes, and for BMS generally: all densities in sections 3–4 will be
Radon densities in the sense defined here, unless explicitly stated otherwise.
Accordingly, the hasty reader may skip the next few paragraphs and go straight
to section 2.3. For now, we present a few related notions of densities, both
for completeness and for comparison with other patches of the literature.

#### Volumic densities.

By contrast with Radon densities, a (volumic) density with weight w is a
scalar field $\varphi(x)$ whose transformation law reads666The adjective
“volumic” is very much non-standard. We use it only in this subsection, for
the sole purpose of clearly distinguishing inequivalent concepts of densities:
Radon, volumic, and conformal.

$\varphi^{\prime}(x^{\prime})=\bigg{|}\,\det\left(\frac{\partial x^{\prime
a}}{\partial x^{b}}\right)\,\bigg{|}^{-{\texttt{w}}}\,\varphi(x)\,.$ (35)

More generally, a tensor-valued (volumic) density of weight w is a tensor
field whose usual transformation law under reparametrisations involves an
extra Jacobian factor to the power w, exactly as in the scalar definition
(35). The corresponding infinitesimal transformation law is given by (33)
except that the divergence of $X$ on the right-hand side now involves standard
derivatives instead of covariant ones:

$\delta\varphi={\cal L}_{X}\varphi+\texttt{w}\,\partial_{a}X^{a}\,\varphi\,,$
(36)

where ${\cal L}_{X}$ is the Lie derivative along $X$ acting on the tensor
field $\varphi$; for a scalar field ($\texttt{w}=0$) it reduces to
$\delta\varphi=X^{a}\partial_{a}\varphi$. Scalar volumic densities of weight
$\texttt{w}=1$ on a manifold $\mathscr{M}$ (not necessarily orientable) are
the objects that can be integrated over $\mathscr{M}$ in a coordinate-
independent way.

It is clear from these definitions that volumic densities and Radon densities
are closely related notions: their transformations (32) and (35) are identical
save for a slightly different convention in handling the metric-induced
measure. This is why we denote their weights by the same symbol w. Note that
the transformation law (35) can be defined on any manifold, with or without
metric or measure. This is perhaps the reason why volumic densities—normally
just called densities${}^{\text{\ref{remvol}}}$—are more familiar in the
literature.

#### Conformal densities.

Independently of a density’s behaviour under diffeomorphisms, one can also
define a notion of weight under Weyl transformations. The transformation law
of a (scalar or tensor) conformal density $\psi$ of conformal weight $w$ under
Weyl transformations is given by

$g_{ab}(x)\to
g^{\prime}_{ab}(x)=\Omega^{2}(x)\,g_{ab}(x)\,,\qquad\psi^{\prime}(x)\,=\,\Omega(x)^{w}\,\psi(x)\,.$
(37)

This is the notion of weight mentioned below the GJMS operator (17).

Note that a field may well be a volumic density and a conformal density
simultaneously. For instance, the metric $g_{ab}$ is a tensor density of
volumic weight zero (since it is a covariant tensor field whose transformation
law under diffeomorphisms is (31)) and conformal weight two (since its
transformation law under Weyl transformations is (37)). Similarly, the volume
density $\sqrt{g}$ on a manifold of dimension $d$ is a scalar volumic density
with weight $\texttt{w}=1$ in the sense of eqs. (31) and (35), and conformal
weight $w=d$ in the sense of (37).

#### Conformal primaries.

Finally, one may compose a Weyl transformation $g_{ab}\to\Omega^{2}g_{ab}$ and
a conformal map $x\to x^{\prime}$ such that the metric remains unchanged:
$g^{\prime}_{ab}(x^{\prime})=g_{ab}(x)$, thereby fixing the Weyl parameter to
$\Omega(x)=|\det(\partial x^{\prime}/\partial x)|^{1/d}$. As a result, the
corresponding conformal transformation law of a scalar density $\chi$ with
volumic weight w (in the sense of (35)) and conformal weight $w$ (in the sense
of (37)) reads

$x\to
x^{\prime},\qquad\chi(x)\to\chi^{\prime}(x^{\prime})=\bigg{|}\,\det\left(\frac{\partial
x^{\prime a}}{\partial x^{b}}\right)\,\bigg{|}^{-\texttt{w}+w/d}\chi(x).$ (38)

The same would hold for a tensor density up to additional Jacobian matrices.
This is by definition the transformation law of a conformal primary under a
conformal transformation. In particular, on a conformally flat manifold, we
say that $\chi$ has scaling dimension $\Delta$ if it transforms under
dilations as

$x\to x^{\prime}=\lambda
x\,,\qquad\chi^{\prime}(x^{\prime})\,=\,\lambda^{-\Delta}\,\chi(x)\,,$ (39)

with $\lambda>0$. Note that the scaling dimension of a tensor field is not
independent of its ranks and weights: if $\chi$ is $r$ times contravariant and
$s$ times covariant, a volumic density of weight w, and a conformal density of
weight $w$, then its scaling dimension is $\Delta=d\,\texttt{w}-w+s-r$.

For instance, the GJMS operator (17) is conformally covariant when it acts on
a conformal density of weight $w=n-\tfrac{d}{2}$; in particular, on the
conformally flat sphere $S^{d}$, such a density is a conformal primary of
scaling dimension $\Delta=\tfrac{d}{2}-n$. Another example is provided by the
fluctuation $h_{ab}$ of the metric around a conformally flat background, which
has rank $s=2$ and is a primary field of scaling dimension zero (since
$w=s=2$). On the celestial sphere, both the metric and its determinant are
conformal primary fields of scaling dimension $\Delta=0$. We will soon
encounter similar transformation laws for the terms of the scalar expansion
(10); indeed, our notation in (39) is consistent with that in (10), as
$\phi_{0}$ will turn out to have scaling dimension $\Delta$.

###  2.3 The WRac and its avatars

Having reviewed the terminology of densities and weights, let us now return to
the massless scalar field of section 2.1. Consider the simplest singular case
$\Delta=\frac{d}{2}-1$ among the choices (18), whereupon the boundary field

$\phi_{0}(x):=\lim_{r\to\infty}\big{[}\,r^{\frac{d}{2}-1}\Phi(r,u,x)\,\big{]}$
(40)

is time-independent by virtue of (14) and the perturbative form of
d’Alembert’s equation truncates immediately. There is then a single GJMS
kernel condition (24) that applies to $\phi_{0}$ itself, and it states that
$\phi_{0}$ lies in the kernel of the aforementioned Yamabe operator (i.e. (17)
with $n=1$):

$\left(\nabla^{2}-\frac{d-2}{4(d-1)}\,{{\cal R}}\right)\phi_{0}=0\,,$ (41)

where ${\cal R}=d(d-1)$ is the scalar curvature of the unit sphere $S^{d}$.
This condition generally has no non-zero smooth solutions, so $\phi_{0}$ is
singular at best. A virtue of eq. (41), however, is that it is invariant under
Weyl transformations of the metric provided $\phi_{0}$ is a conformal density
with weight $w=-\Delta=1-d/2$. This will indeed turn out to be the case: see
eq. (43) below.

We shall return to the Yamabe equation (41) in great detail in section 4.1,
but for now it is worth stressing some properties of the corresponding bulk
field $\Phi$. Since the expansion (27) truncates to a single term
$\Phi(r,u,x)=r^{1-d/2}\phi_{0}(u,x)$, one may equally well investigate it at
large $r$, finite $r$, or even small $r$. Actually, in Cartesian coordinates
$x^{\mu}$, such singular configurations form a class of solutions of (7) that
are homogeneous of degree $1-\frac{d}{2}$: they solve both $\Box\Phi=0$ and
$(x^{\mu}\partial_{\mu}+\tfrac{d}{2}-1)\Phi=0$.777Note that homogeneity means
$(u\partial_{u}+r\partial_{r}+\tfrac{d}{2}-1)\Phi=0$ in Bondi coordinates, so
the ansatz $\Phi(r,u,x)=r^{1-d/2}\phi_{0}(u,x)$ implies
$\partial_{u}\phi_{0}=0$ at $u\neq 0$, which is indeed satisfied by the WRac.
This agrees with the fact that the solutions $\phi_{0}$ of the Yamabe equation
on $S^{d}$ admit an “ambient” description as massless scalars in Minkowski
spacetime with a suitable homogeneity degree. (This goes back to Dirac [78];
see also [67, sec. 3] or [79, sec. 3.5] for reviews.) In such cases, the
celestial sphere $S^{d}$ is usually realised as a projectivised light-cone
through the origin, i.e. as the set of past- or future-oriented null
directions, seen as the base space of a fibre bundle

$\mathbb{R}^{d+1}\backslash\\{0\\}\twoheadrightarrow
S^{d}:\bm{x}\mapsto[\bm{x}]=\\{\lambda\,\bm{x}\in\mathbb{R}^{d+1}\,|\,\lambda>0\\}\,.$
(42)

It is geometrically appealing that one can equivalently see the celestial
sphere as projectivised null infinity; this is really no surprise, as the past
(resp.) future light-cone through the origin and future (resp. past) null
infinity are related by an inversion $x^{\mu}\to x^{\mu}/x^{2}$ provided one
restricts attention to the interior $x^{2}\leqslant 0$ (see fig. 1).888To be
precise, the inversion is actually discontinuous on the light-cone: time-like
points near the past/future light-cone are mapped on the time-like part of
future/past null infinities, whereas space-like points near the past/future
light-cone are mapped on the space-like part of past/future null infinities.

Note that this ambient interpretation is standard in higher-spin AdS/CFT,
where the scalar singleton [25], usually called “Rac” [26], is instrumental:
it is the minimal unitary irreducible representation (uir) of
$\mathfrak{so}(d,2)$, described from the CFTd perspective as a primary scalar
field solving d’Alembert’s equation on the boundary of AdSd+1, with a scaling
dimension that saturates the unitarity bound. (See [80] for a review of the
relevant representations, and [79] for a review on singletons in AdS/CFT.) The
ambient version of this construction consists in seeing the singleton as the
space of homogeneous solutions of the Laplace equation on the embedding space
$\mathbb{R}^{d,2}$ with a “two-time” signature. The analogy with the singular
field $\phi_{0}$ appearing in (41) is immediate: the field $\Phi$ defines a
space of homogeneous solutions of d’Alembert’s equation in
$\mathbb{R}^{d+1,1}$, so one is tempted to think of $\phi_{0}$ as the
“singleton” of a higher-spin dSd+1/ECFTd correspondence [81] involving de
Sitter space and a Euclidean CFT defined on the celestial sphere $S^{d}$. This
reiterates our motivation expressed above to refer to $\phi_{0}$ as a “Wick-
rotated Rac”.

As explained in section 2.1, these remarks extend to singular solutions with
$\Delta=\frac{d}{2}-N$ for integers $N>1$. The Yamabe condition (41) is then
replaced by GJMS conditions (24), and the boundary fields
$\phi_{0},\phi_{1},\cdots\\!,\phi_{N-1}$ may be seen as Wick-rotated, now
time-dependent, higher-order analogues of the Rac,999See [82] on the
holographic link between higher-order Rac and partially-massless higher-spin
gravity. with respective weights ranging from
$\texttt{w}=\Delta/d=\frac{1}{2}-N/d$ to $\texttt{w}=\frac{1}{2}-1/d$.

###  2.4 The unitary Sachs module

In its strictest sense, the BMS group is the semi-direct product between the
Lorentz group and the vector group of supertranslations. Lorentz
transformations are thus seen as (globally well-defined) conformal
transformations of celestial spheres [83], while supertranslations are scalar
densities with volumic weight $\texttt{w}=-1/d$, in the sense of eqs.
(32)–(33), acting on retarded time as angle-dependent shifts [4]. We now show
how this group of transformations affects the components of the asymptotic
expansion (10) of a massless scalar field satisfying the d’Alembert equation
near null infinity. Superrotations [6] are initially discarded for simplicity,
but we will eventually see that diffeomorphisms of celestial spheres may be
included at no cost, showing that Sachs’s module [4] is really a unitary
representation of the generalised BMS group of [15, 16]. Further extensions to
“super-projectable” transformations and beyond will be addressed in sections
3–4.

#### BMS generators again.

BMS transformations are diffeomorphisms of spacetime: they are generated by
vector fields that depend on an infinity of parameters, generalising the
finite-dimensional set of Poincaré-generating vector fields [7]. In Minkowski
spacetime and in terms of Bondi coordinates, any such BMS vector field
$\xi=\xi^{\mu}\partial_{\mu}$ has components (2)–(4) with a completely
arbitrary supertranslation function $\alpha(x)$. The Poincaré subalgebra
consists of the same vector fields with the added constraint (5), whereupon
supertranslations reduce to standard spacetime translations. The weight
$\texttt{w}=-1/d$ of supertranslations under Lorentz transformations—and
diffeomorphisms more generally—can then be read off from the Lie bracket (6),
whose right-hand side involves transformations (33) for $\alpha,\beta$.

Now act with a vector field of the form (2)–(4) on the radial expansion (10),
with an arbitrary parameter $\Delta$, to read off the BMS transformation laws

$\displaystyle\delta\phi_{0}$
$\displaystyle=\big{(}\alpha+\tfrac{u}{d}\,\nabla_{a}X^{a}\big{)}\dot{\phi}_{0}+X^{a}\partial_{a}\phi_{0}+\tfrac{\Delta}{d}\,\nabla_{a}X^{a}\,\phi_{0}\,,$
(43) $\displaystyle\delta\phi_{n}$
$\displaystyle=\big{(}\alpha+\tfrac{u}{d}\,\nabla_{a}X^{a}\big{)}\dot{\phi}_{n}+X^{a}\partial_{a}\phi_{n}+\tfrac{\Delta+n}{d}\,\nabla_{a}X^{a}\,\phi_{n}$
$\displaystyle\quad+\big{(}\tfrac{u}{d}\,\nabla_{a}X^{a}+\tfrac{1}{d}\,\nabla^{2}\alpha\big{)}(\Delta+n-1)\phi_{n-1}-g^{ab}\big{(}\partial_{b}\alpha+\tfrac{u}{d}\,\partial_{b}\nabla_{a}X^{a}\big{)}\partial_{a}\phi_{n-1}\,,$
(44)

where $n\geqslant 1$. Recall that the fields $\phi_{n}$ ($n\geqslant 0$) are
in general functions of both $u$ and $x$. Eq. (43) means for instance that
$\phi_{0}$ has weight $\texttt{w}=\Delta/d$ under conformal maps, reducing to
$\texttt{w}=1/2$ for $\Delta=d/2$: this will play a role below.

It is important, for future reference, to write down the finite coordinate
transformations at null infinity generated by BMS vector fields (2)–(4).
Accordingly, let $F$ be a diffeomorphism of $S^{d}$ obtained from the flow of
$X^{a}\partial_{a}$ in (4), and let $\alpha$ be a finite supertranslation
function. Then the corresponding BMS transformation of $\mathscr{I}$ is
$(u,x)\to(u^{\prime},x^{\prime})$ with

$u^{\prime}=\rho_{F}(x)^{1/d}u+\alpha(x^{\prime})\qquad\text{and}\qquad
x^{\prime a}=F^{a}(x)\,,$ (45)

where $\rho_{F}(x)$ is the Radon-Nikodym derivative (30) of the measure on
$S^{d}$ under $F$. As we shall explain in sections 3.3–3.4, this is actually a
conformal transformation of null infinity as a whole when $F$ is a conformal
map, and it is a generalised conformal transformation when $F$ is an arbitrary
diffeomorphism. The field transformation $\phi_{0}\to\,{\cal
U}_{(F,\alpha)}\phi_{0}$ that accompanies this map can then be found by
integrating eq. (43) to obtain

$\Big{(}\,{\cal
U}_{(F,\alpha)}\phi_{0}\Big{)}\big{(}u^{\prime},x^{\prime}\big{)}=\rho_{F}(x)^{-\Delta/d}\,\phi_{0}(u,x)\,,$
(46)

where the notation anticipates that we wish to think of $\,{\cal
U}_{(F,\alpha)}$ as a unitary operator. This equation is nothing but the
transformation law (32) of a density with weight $\texttt{w}=\Delta/d$, up to
the presence of an extra transformation of retarded time. We shall see shortly
that unitarity occurs for $\texttt{w}=1/2$, i.e. for the radiative fall-off
condition (26). More generally, it is clear at this point that the radial
expansion (10) furnishes a representation of BMS, organised as a hierarchy of
the form (43)–(44).101010It would be amusing to find a $w_{1+\infty}$-like
structure in the sequence (43)–(44) when $d=2$, as has been achieved in
gravity [84, 85, 86]. We will not attempt to do that here.

#### Sachs module.

Let us now show that the group action (46) with $\Delta=d/2$ furnishes a
massless irreducible unitary representation of the BMS group. To achieve this
we take a somewhat unexpected route: we start by recalling the construction of
(massless) Poincaré representations in momentum space, then perform a Fourier
transform along retarded time to recover eq. (46) and argue that any massless
Poincaré representation lifts to a representation of BMS, and even to a
representation of the generalised BMS group where superrotations are allowed
to be generic diffeomorphisms of celestial spheres [15, 16]. The argument is
somewhat similar to that of [87, 88, 89, 90].

Consider a massless scalar uir of the Poincaré group in spacetime dimension
$d+2$. The orbit (mass shell) is the future light-cone (42) of null momenta
with positive energy: ${\cal O}=\\{(|{\bf q}|,{\bf
q})\in\mathbb{R}^{+}\times\mathbb{R}^{d+1}\\}\cong\mathbb{R}^{+}\times S^{d}$.
The Hilbert space $L^{2}({\cal O},\mu)$ consists of square-integrable
wavefunctions with a Hermitian form111111We write wavefunctions on ${\cal O}$
as $\Phi$, $\Psi$, etc. Despite the similar notation, this has nothing to do
with the bulk field $\Phi$ that solved the d’Alembert equation (7) in section
2.1.

$\langle\Phi|\Psi\rangle:=\int_{\mathbb{R}^{d+1}}\frac{\text{d}^{d+1}{\bf
q}}{|{\bf q}|}\,\Phi^{*}({\bf q})\Psi({\bf q})$ (47)

where we chose the Lorentz-invariant measure for later convenience. The
Poincaré group is represented unitarily on this Hilbert space, according to

$\big{(}\,\,{\cal U}_{(F,\alpha)}\Phi\big{)}\big{(}F({\bf
q})\big{)}:=e^{\,i\,\langle\,F(q)\,,\,\alpha\,\rangle}\,\Phi({\bf q})\,,$ (48)

where $F$ is a Lorentz transformation, $\alpha$ is a translation and $\langle
q,\alpha\rangle:=q_{\mu}\,\alpha^{\mu}$ is the bilinear pairing (scalar
product) of spacetime momenta and translations. Our goal is to rephrase eqs.
(47)–(48) in terms of Bondi-like coordinates on the orbit ${\cal O}$.

Accordingly, let $E=|{\bf q}|>0$ be the energy of the momentum vector ${\bf
q}$ and let ${\bf x}\in S^{d}\subset\mathbb{R}^{d+1}$ be a unit vector such
that ${\bf q}=E\,{\bf x}$, so that $q=(E,E{\bf x})$. Then the Hermitian form
(47) can be recast as

$\langle\Phi|\Psi\rangle=\int_{0}^{\infty}\text{d}E\,E^{d-1}\int_{S^{d}}\text{d}^{d}{\bf
x}\,\sqrt{g({\bf x})}\,\Phi^{*}(E,{\bf x})\Psi(E,{\bf x})$ (49)

where $\text{d}^{d}{\bf x}\,\sqrt{g({\bf x})}$ is the standard volume form on
the unit sphere $S^{d}$. Note that the wavefunctions $\Phi$ and $\Psi$ vanish
for $E\leqslant 0$ since their support is the positive-energy zero-mass shell
${\cal O}$. Now introduce a notion of “retarded time” $u$ thanks to the
Fourier transform

$\phi(u,{\bf
x}):=\int\limits_{-\infty}^{+\infty}\tfrac{\text{d}E}{\sqrt{2\pi}}\,E^{\frac{d}{2}-1}\,e^{-iEu}\,\Phi(E,{\bf
x})\,,\qquad\Phi(E,{\bf
x})=\frac{1}{E^{\frac{d}{2}-1}}\int\limits_{-\infty}^{+\infty}\tfrac{\text{d}u}{\sqrt{2\pi}}\,e^{iEu}\,\phi(u,{\bf
x})\,,$ (50)

where the power $E^{\frac{d}{2}-1}$ will be justified shortly from an
asymptotic argument. Setting this aside for now, eq. (50) allows us to rewrite
the inner product (47)–(49) in a way that exactly coincides with Sachs’s
Hermitian form introduced in the early days of BMS symmetry [4] (see also [91,
thm 12.3]), namely

$\langle\phi|\psi\rangle:=\langle\Phi|\Psi\rangle=i\int\limits_{-\infty}^{+\infty}\text{d}u\int\limits_{S^{d}}\text{d}^{d}{\bf
x}\,\sqrt{g({\bf x})}\,\,\phi^{*}(u,{\bf x})\,\partial_{u}\psi(u,{\bf x})\,.$
(51)

We stress that we have thus matched an inner product of quantum wavefunctions,
in a Poincaré uir, with what was understood in [4] as an inner product of
radiative solutions (26) of the d’Alembert equation. The matching goes
further: the quantum expectation value of the Poincaré Hamiltonian in a state
$|\phi\rangle$ is nothing but the energy (9) of the corresponding classical
field configuration (26).

The justification of the weighted Fourier transform (50) stems from the
asymptotic behaviour of massless fields in Bondi coordinates—a property that
was instrumental in the seminal works relating asymptotic symmetries to soft
theorems [10] (see also [11, Sec. 9, Ex. 4]). Indeed, consider the on-shell
bulk field

$\displaystyle\Phi(r,u,{\bf x})$
$\displaystyle=\frac{1}{(2\pi)^{(d+1)/2}}\int_{\mathbb{R}^{d+1}}\frac{\text{d}^{d+1}{\bf
q}}{|{\bf q}|}\,\,e^{-i(u+r)\,|{\bf q}|\,+\,ir\,{\bf x}\cdot{\bf
q}}\,\Phi({\bf q})$ (52)
$\displaystyle=\int\limits_{0}^{+\infty}\frac{\text{d}E}{\sqrt{2\pi}}\,E^{d-1}\,e^{-iEu}\int_{S^{d}}\frac{\text{d}^{d}\hat{\bf
q}\,\sqrt{g(\hat{\bf
q})}}{(2\pi)^{d/2}}\,e^{iEr\,(\cos\theta-1)}\,\Phi(E,\hat{\bf q})\,,$ (53)

where $E=|{\bf q}|>0$ is the norm of an integrated momentum and $\hat{\bf
q}:={\bf q}/|{\bf q}|$ so that $\text{d}^{d+1}{\bf
q}=E^{d}\,\text{d}E\,\text{d}^{d}\hat{\bf q}\,\sqrt{g(\hat{\bf q})}$, while
$\theta\in[0,\pi]$ denotes the angle between ${\bf q}$ and the spatial
direction ${\bf x}$. We stress that ${\bf q}$ and ${\bf x}$ are unrelated at
this stage. However, near null infinity $r\to\infty$, the stationary phase
approximation of (52) results in the collinearity $\theta\approx 0$, i.e.
${\bf q}\approx E\,{\bf x}$ for non-zero $E$. The integral over the celestial
sphere then satisfies the asymptotic behaviour

$\displaystyle\int_{S^{d}}\text{d}^{d}\hat{\bf q}\,\sqrt{g(\hat{\bf
q})}\,e^{iEr\,(\cos\theta-1)}\,\Phi(E,\hat{\bf q})$
$\displaystyle=\int_{0}^{\pi}\text{d}\theta\,(\,\sin\theta\,)^{d-1}\int_{S^{d-1}}\text{d}\Omega\,e^{iEr\,(\cos\theta-1)}\,\Phi(E,\hat{\bf
q})$
$\displaystyle\sim\int_{0}^{\pi}\text{d}\theta\,\theta^{d-1}\int_{S^{d-1}}\text{d}\Omega\,e^{-iEr\theta^{2}/2}\,\Phi(E,{\bf
x})$ $\displaystyle\propto\frac{1}{(Er)^{d/2}}\,\Phi(E,{\bf x})$ (54)

where we factorised the measure on $S^{d}$ as $\text{d}^{d}\hat{\bf
q}\,\sqrt{g(\hat{\bf q})}=(\sin\theta)^{d-1}\text{d}\theta\,\text{d}\Omega$,
with $\text{d}\Omega$ the standard measure on $S^{d-1}$. In the last line we
also neglect irrelevant constant factors (such as the volume of $S^{d-1}$)
that can be eliminated by a redefinition of $\Phi$. Thus eq. (53) yields

$\Phi(r,u,{\bf x})\,\stackrel{{\scriptstyle
r\to\infty}}{{\sim}}\,r^{-\frac{d}{2}}\int\limits_{-\infty}^{+\infty}\tfrac{\text{d}E}{\sqrt{2\pi}}\,E^{\tfrac{d}{2}-1}\,e^{-iEu}\,\Phi(E,{\bf
x})=r^{-d/2}\phi(u,x)\,,$ (55)

which is to say that the asymptotic behaviour of the field $\Phi(r,u,{\bf x})$
in position space identifies with the weighted Fourier transform of the
wavefunction $\Phi(E,{\bf x})$ in momentum space. Up to the factor $r^{-d/2}$,
this is exactly the Fourier transform introduced in (50). (As before, we let
$\Phi(E,{\bf x})$ be supported in the region $E>0$.)

Let us now return to wavefunctions with the Hermitian form (47). Since all
Poincaré transformations act unitarily on their Hilbert space, let us rewrite
their action in terms of coordinates $(u,x)$, and identify the transformation
of a wavefunction with that given by (46) for a field at infinity. We start
from the unitary operator (48), now applied to the wavefunction (50), which
yields

$\big{(}\,{\cal U}_{(F,\alpha)}\,\phi\big{)}(u,{\bf
x})=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\text{d}E\,E^{\frac{d}{2}-1}\,e^{-iEu}\,e^{i\,\langle\,(E,E{\bf
x})\,,\,(\alpha^{0},\bm{\alpha})\,\rangle}\,\,\Phi\big{(}F^{-1}(E,{\bf
x})\big{)}\,.$ (56)

Here the argument of $\Phi$ on the right-hand side involves the action of a
Lorentz transformation $F$ on a point $(E,{\bf x})$ on the light-cone.
Specifically, $F^{-1}(E,{\bf x})=\big{(}E\,f({\bf x}),F^{-1}({\bf x})\big{)}$,
where ${\bf x}\to F^{-1}({\bf x})$ is a conformal transformation of $S^{d}$
and $f({\bf x})$ is a positive function on $S^{d}$ that can be found thanks to
the fact that the measure $E^{d-1}\text{d}E\,\sqrt{g({\bf
x})}\,\text{d}^{d}{\bf x}$ is Lorentz-invariant. Indeed, one has
$f(x)=\big{(}\rho_{F^{-1}}(x)\big{)}^{-1/d}$ in terms of the Radon-Nikodym
derivative (30). Furthermore, the exponent in the integrand of (56) is
$\langle(E,E{\bf x}),(\alpha^{0},\bm{\alpha})\rangle=E\alpha^{0}+E\,{\bf
x}\cdot\bm{\alpha}:=E\,\alpha({\bf x})$, where $\alpha({\bf x})$ is the
supertranslation function on the celestial sphere implementing the spacetime
translation $\alpha$. Using all this in (56) gives

$\big{(}\,{\cal U}_{(F,\alpha)}\,\phi\big{)}(u,{\bf
x})=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\text{d}E\,E^{\frac{d}{2}-1}\,e^{-iEu}\,e^{iE\alpha({\bf
x})}\,\Phi\big{(}E\,f({\bf x}),F^{-1}({\bf x})\big{)}\,.$ (57)

The coordinates ${\bf x}$ are mere spectators here, so one can change the
integration variable into $\tilde{E}:=E\,f({\bf x})$ to find

$\displaystyle\big{(}\,{\cal U}_{(F,\alpha)}\,\phi\big{)}(u,{\bf x})$
$\displaystyle=\big{(}\rho_{F^{-1}}({\bf
x})\big{)}^{1/2}\phi\Big{(}\big{(}\rho_{F^{-1}}({\bf
x})\big{)}^{1/d}\big{(}u-\alpha({\bf x})\big{)},F^{-1}({\bf x})\Big{)}\,.$
(58)

This is it: upon evaluating the left-hand side at $(u^{\prime},x^{\prime})$
given by (45) instead of $(u,x)$, eq. (58) coincides with the scalar field’s
transformation law (46) with $\Delta=d/2$. In other words, the radiative
branch (26) of solutions the d’Alembert equation, endowed with the Sachs form
(51) and acted upon by BMS transformations according to eq. (46) with
$\Delta=d/2$, is equivalent to a scalar massless uir of the Poincaré group.
Conversely, any such uir lifts to a corresponding uir of BMS, since eqs.
(46)–(58) make sense for any supertranslation $\alpha$ and define a
representation that leaves the Sachs form invariant.121212More generally, all
faithful uirs of the Poincaré group ISO(3,1) lift to uirs of BMS4.
Equivalently, all uirs of BMS4 induced from BMS little groups corresponding to
Poincaré little groups of faithful uirs of ISO(3,1) remain irreducible upon
restriction. The proof is a corollary of [66, Th. 7], which states that the
branching rules for the restriction from BMS to Poincaré are obtained from the
branching rules of the corresponding little groups. Note that this result may
seem to contradict earlier statements in [61, 62, 63, 64], but the latter rely
on overly restrictive choices of topology for the functional space of
supertranslations, whereas the “nuclear” topology of [66] is better suited for
physical applications. (See also [87, 88, 89] for massive scalars.) Perhaps
more strikingly, the lift holds regardless of the restriction to conformal
transformations: it even applies when Lorentz transformations are enhanced to
arbitrary diffeomorphisms of $S^{d}$. Indeed, eq. (46) always defines a
representation of the extended group $\text{Diff}(S^{d})\ltimes
C_{\texttt{w}}^{\infty}(S^{d})$, where the first factor consists of
“superrotations” in the sense of [15, 16, 92] and $\texttt{w}=-1/d$ for
supertranslations. Unitarity with respect to the Sachs form (51) then holds
provided $\Delta=d/2$. We shall return to this example in great detail in
sections 3.4 and 4.2.

#### The non-unitary WRac.

We conclude this section by briefly returning to the WRac of section 2.3,
given by (40) in terms of the bulk field. It is then straightforward to verify
that the space of such (singular) fields also carries a representation of the
BMS group, simply by exponentiating the transformation law (43) to obtain (46)
with $\Delta=\frac{d}{2}-1$. This representation, however, is not faithful,
since $\phi_{0}(x)$ is now time-independent so that all (super)translations
act trivially. Furthermore, the lack of smoothness and the weight
$\texttt{w}=\Delta/d=\frac{1}{2}-1/d$ prevent the existence of a natural
Hermitian form on such fields that would be BMS-invariant (or even Poincaré-
invariant, for that matter). We will nevertheless return to this singular
construction in section 4.1, as it is a natural cousin of the singleton in
AdS/CFT.

As a final comment, note that the saddle-point analysis of eqs. (52)–(55)
naïvely suggests that all massless scalars satisfy the fall-offs $\Phi={\cal
O}(r^{-d/2})$, apparently ruling out the overleading fall-offs of the WRac.
The way out of this paradox can be found by going back to eq. (53), which
shows that saddle point of the Fourier transform in momentum space occurs
either for collinear momenta, or at zero energy. It so happens that the WRac
realizes this second possibility, in accordance with the time-independence
ensured by the equation of motion $\dot{\phi}_{0}=0$ in (14). (By the way,
this is true not only of the WRac, but of any massless field that satisfies
$\Phi={\cal O}(r^{-\Delta})$ with $\Delta\neq d/2$.)

##  3 Carrollian geometry and BMS groups

Our approach has been pragmatic so far: starting from the massless field
equation (7), we found that its space of radiative solutions carries a unitary
representation of the BMS group. In order to relate this structure to higher-
spin transformations, we now adopt a more intrinsic perspective based on
Carrollian geometry. The latter may be seen as an ultra-relativistic limit of
Lorentzian geometry when the speed of light vanishes and the metric becomes
degenerate due to the appearance of one vanishing eigenvalue. This motivates
the definition of automorphisms analogous to “isometries” or “conformal maps”,
except that they must preserve this null direction. Accordingly, we now review
the description of BMS transformations as Carrollian conformal maps, following
the intrinsic (i.e. purely from the boundary) and geometric (i.e. global and
coordinate-free) approach of Penrose [48], translated in Carrollian language
in [46]. We start with generalities on principal $\mathbb{R}$-bundles, their
automorphisms and connections; we then introduce Carrollian spacetimes,
isometries and conformal maps; finally, we define Carrollian volume forms and
their symmetries to revisit the Sachs module (51) from a Carrollian viewpoint.

We stress that the language reviewed here will be crucial in section 4, as it
simplifies computations whose expression in coordinates is otherwise unwieldy.
Still, the presentation will unavoidably be quite mathematical, so we have
attempted to frame it in a pedagogical manner accessible to theoretical
physicists. It is self-contained, save for basic notions of differential
geometry and fibre bundles that are not reviewed in any depth; we refer e.g.
to [93, 94, 95] for the necessary background. A pedagogical introduction to
Carrollian conformal geometry is also provided e.g. in [56, 96]. Our
terminology and notation will mostly follow [97, appendix A].

###  3.1 Principal $\bm{\mathbb{R}}$-bundles and their automorphisms

Here we review elementary concepts, needed later, on principal
$\mathbb{R}$-bundles. We start from the notion of fundamental vector field to
define projectable, super-projectable, and invariant vector fields as
generators of suitable automorphisms. (These families of vector fields will
provide the basic building blocks of higher-spin differential operators in
section 4.) We conclude by defining basic and invariant tensor fields, crucial
for the later definition of Carrollian clocks (section 3.2), metrics (section
3.3) and volumes (section 3.4).

#### Ray bundle and fundamental vector field.

The structure underlying any Carrollian geometry is a principal
$\mathbb{R}$-bundle—a ray bundle—whose $(d+1)$-dimensional total space
$\mathscr{M}_{d+1}$ is a Carrollian “spacetime” while the $d$-dimensional base
manifold $\bar{\mathscr{M}}_{d}:=\mathscr{M}_{d+1}/\mathbb{R}$ is Carrollian
“space”. We let

$\pi\,:\,\mathscr{M}_{d+1}\twoheadrightarrow\bar{\mathscr{M}}_{d}$ (59)

be the projection from the bundle to its base, such that the preimage
$\pi^{-1}(x)\cong\mathbb{R}$ of any $x\in\bar{\mathscr{M}}$ is a fibre in
$\mathscr{M}$. The additive group $\mathbb{R}$ acts freely (and
properly131313Requiring the $\mathbb{R}$-action to be “proper” ensures that
the quotient $\mathscr{M}/\mathbb{R}$ is a manifold. This rules out
pathological cases such as a torus acted upon by $\mathbb{R}$ via translations
along an irrational slope.) on $\mathscr{M}$ by Carrollian time translations
whose orbits are fibres; each fibre may thus be seen as the worldline of a
test mass (see fig. 2). Note that our convention for dimensions matches that
of section 2: the $d$-dimensional celestial sphere
$\bar{\mathscr{M}}_{d}=S^{d}$ is the base manifold of $(d+1)$-dimensional null
infinity $\mathscr{M}_{d+1}=\mathscr{I}_{d+1}^{\pm}\cong
S^{d}\times\mathbb{R}$, which is indeed Carrollian. In what follows we will
lighten notation by omitting the dimension subscript in $\mathscr{M}_{d+1}$
and $\bar{\mathscr{M}}_{d}$.

Given the principal $\mathbb{R}$-bundle
$\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}$, its fundamental vector field
is the vector field $\xi\in\mathfrak{X}(\mathscr{M})$ whose flow consists of
fibrewise translations; its integral curves are orbits of the
$\mathbb{R}$-action on $\mathscr{M}$, i.e. Carrollian worldlines. In
particular, $\xi$ vanishes nowhere, and it is vertical in the sense that
$\pi_{*}(\xi)=0$ in terms of the pushforward of the projection (59).
Conversely, choosing a nowhere-vanishing vertical vector field $\xi$ on a
fibre bundle $\pi:\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}$ with fibres
$\pi^{-1}(x)\cong\mathbb{R}$ is equivalent to choosing a free action of
$\mathbb{R}$ on $\mathscr{M}$, and automatically endows $\mathscr{M}$ with the
structure of a principal $\mathbb{R}$-bundle. The data defining a principal
$\mathbb{R}$-bundle will therefore be denoted as a pair $(\mathscr{M},\xi)$
from now on.

Figure 2: A principal $\mathbb{R}$-bundle
$(\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}},\xi)$. Each curve is a fibre,
and all points of that fibre are mapped by the projection (59) on a single
point on the base $\bar{\mathscr{M}}$. The bundle $\mathscr{M}$ is locally
isomorphic to a product $\bar{\mathscr{M}}\times\mathbb{R}$, but in the case
at hand this is even true globally, thanks to the global existence of a smooth
$(\mathbb{R},+)$ group action.

Locally, one can always find adapted coordinates $(u,x^{a})$ on $\mathscr{M}$
with $u\in\mathbb{R}$ and $x^{a}$ ($a=1,\ldots\\!,d$) coordinates on (a patch
in) $\bar{\mathscr{M}}$, such that Carrollian time translations read $u\to
u+\text{cst}$. The corresponding fundamental vector field is
$\xi=\partial_{u}$, with integral curves labelled by their initial point
$x^{a}=x^{a}_{0}$ and parametrised by null time $u$. The projection (59) then
simply reads

$\pi\,:\,\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}\,:\,(u,x^{a})\mapsto
x^{a}\,.$ (60)

Note that any principal $\mathbb{R}$-bundle is trivial, i.e. it is globally
true that $\mathscr{M}\cong\bar{\mathscr{M}}\times\mathbb{R}$ as fibre
bundles, and the fibre coordinate $u$ is always globally well-
defined.141414This crucially relies on the principal bundle structure,
requiring the existence of a globally well-defined action of $\mathbb{R}$ on
$\mathscr{M}$. It would not be true for arbitrary bundles with fibres
homeomorphic to $\mathbb{R}$ (think e.g. of a Möbius band, which is crucially
not a principal $\mathbb{R}$-bundle). In this sense, it is often simpler to
just work in a chart that makes the splitting between $\bar{\mathscr{M}}$ and
$\mathbb{R}$ manifest, as in the Bondi coordinates of section 2. Still,
sticking to intrinsic geometry (as opposed to local charts) reveals the bundle
structure underlying Carrollian manifolds. This will simplify some tedious
computations involving differential operators in section 4. In what follows we
therefore go back and forth between local coordinates and global statements.

#### Examples: null infinity and light-cone.

Most importantly for our purposes, future and past null infinities
$\mathscr{I}^{\pm}\cong S^{d}\times\mathbb{R}$, on the conformal boundary of
compactified Minkowski spacetime $\mathbb{R}^{d+1,1}$, are typically described
in terms of a retarded/advanced time coordinate $u\in\mathbb{R}$ along with
angular coordinates $x^{a}$ on a celestial sphere $S^{d}$ (see fig. 1 above).
Both are thus trivial $\mathbb{R}$-bundles over $S^{d}$ with a projection (60)
in terms of Bondi coordinates. The fundamental vector field $\xi=\partial_{u}$
generates time translations $u\to u+\text{cst}$, so its congruence of integral
curves spans a cone with its tip at $i^{\pm}$ removed.

Another aforementioned example of Carrollian structure is provided by light-
cones (say at the origin) in Minkowski spacetime, which may be seen as images
of null infinity by the inversion $x^{\mu}\to x^{\mu}/x^{2}$. Each light-cone
$\mathscr{N}\cong S^{d}\times\mathbb{R}$ is thus a principal bundle (42) whose
trivial structure is obvious. Furthermore, it can be used as the celebrated
Möbius model of conformally flat geometry, i.e. the projective null cone
identified with the celestial sphere, seen as a conformally flat manifold.
(See e.g. [67, sec. 3] or [79, sec. 2.2] for reviews.) This makes conformal
symmetry manifest since Lorentz transformations of $\mathbb{R}^{d+1,1}$ induce
conformal transformations of $S^{d}$, as we used below eq. (56). We shall
return to conformal (BMS) maps on Carrollian manifolds in section 3.3, but
this first requires that we define families of vector fields generating
suitable symmetries of $\mathbb{R}$-bundles.

#### (Super-)projectable and invariant vector fields.

The bundle structure (59) and the fundamental vector field $\xi$ readily
motivate the definition of several subspaces of vector fields on
$\mathscr{M}$; these will be crucial in section 4 to build differential
operators spanning higher-spin symmetries. Accordingly, a vector field
$X\in\mathfrak{X}(\mathscr{M})$ is projectable if ${\cal L}_{\xi}X$ is
vertical, i.e. if

${\cal L}_{\xi}X=f\,\xi$ (61)

for some $f\in C^{\infty}(\mathscr{M})$. It is super-projectable if ${\cal
L}_{\xi}f=0$, and it is invariant if $f=0$. All three such subsets of vector
fields span Lie algebras, respectively denoted
$\mathfrak{X}_{\text{pro}}(\mathscr{M})$,
$\mathfrak{X}_{\text{spro}}(\mathscr{M})$ and
$\mathfrak{X}_{\text{inv}}(\mathscr{M})$, with the obvious inclusions

$\mathfrak{X}_{\text{inv}}(\mathscr{M})\subset\mathfrak{X}_{\text{spro}}(\mathscr{M})\subset\mathfrak{X}_{\text{pro}}(\mathscr{M})\subset\mathfrak{X}(\mathscr{M})\,.$
(62)

Now for the geometric interpretation. First, the projectability condition (61)
ensures that the flow of $X$ maps fibres on fibres, i.e. that $X$ generates
automorphisms of $\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}$, seen as a
fibre bundle without any extra structure. In particular, any projectable
vector field $X$ has a well-defined projection $\bar{X}:=\pi_{*}(X)$ on
$\bar{\mathscr{M}}$. The super-projectability condition ${\cal L}_{\xi}f=0$ is
somewhat tricky to justify for now, but we will return to it shortly in local
coordinates; we will also see in sections 3.3–3.4 that it specifies the class
of vector fields that includes all conformal/BMS transformations and their
generalisations. Finally, the invariance condition ${\cal L}_{\xi}X=0$ states
that $X$ generates automorphisms of
$\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}$, seen as a principal
$\mathbb{R}$-bundle.

In coordinates $(u,x^{a})$ such that $\xi=\partial_{u}$, any projectable
vector field reads $X=X^{a}(x)\partial_{a}+g(u,x)\partial_{u}$ where
$g(u,x)=\alpha(x)+\int_{0}^{u}\text{d}v\,f(v,x)$ and the functions $X^{a}(x)$,
$\alpha(x)$, $f(u,x)$ are arbitrary. It is super-projectable if $f(u,x)$ is
independent of $u$, i.e. if

$X=X^{a}(x)\,\partial_{a}+\big{(}\alpha(x)+uf(x)\big{)}\partial_{u}\,,$ (63)

and it is invariant if all its components are time-independent (i.e. $f=0$).
Note the similarity between eq. (63) and the $(u,a)$ components (3)–(4) of the
BMS vector field, whose $r\to\infty$ limit is clearly super-projectable on
null infinity: this was actually the reason for introducing super-projectable
vector fields in the first place.

Note that we could have started from the definition of bundle automorphisms as
maps that preserve the fibre bundle structure
$\pi:\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}$; each such map is a
diffeomorphism $F:\mathscr{M}\to\mathscr{M}$ such that there exists a base
diffeomorphism $\bar{F}:\bar{\mathscr{M}}\to\bar{\mathscr{M}}$ for which

$\bar{F}\circ\pi=\pi\circ F\,,$ (64)

ensuring that $F$ maps fibres on fibres (see e.g. [93]). In adapted
coordinates $(u,x^{a})$, it takes the form of a smooth transformation

$(u,x)\to F(u,x)=\big{(}u^{\prime}(u,x),x^{\prime}(x))$ (65)

where $x\to x^{\prime}(x)$ is any diffeomorphism of the base space
$\bar{\mathscr{M}}$ and $u^{\prime}$ is any function of $(u,x)$; see the
examples in fig. 3. Vector fields whose flow consists of such maps are
precisely the projectable ones defined above, while super-projectable vector
fields generate diffeomorphisms (65) for which $u^{\prime}=f(x)u+\alpha(x)$
with $f(x)>0$. Similarly, we could have defined principal bundle automorphisms
as bundle automorphisms $F:\mathscr{M}\to\mathscr{M}$ that are equivariant
with respect to the action of $\mathbb{R}$ on $\mathscr{M}$ and thus leave the
fundamental vector field invariant in the sense that $F_{*}(\xi)=\xi$. In the
notation (65), one then has $u^{\prime}=u+\alpha(x)$. The corresponding vector
fields are invariant, as defined above.

The reason we avoided this route (from finite transformations to their
infinitesimal counterparts) is due to the standard recipe for higher-spin
extensions of symmetries, which consists in first considering infinitesimal
spacetime symmetries realised as first-order differential operators, then
allowing for higher-order operators by turning to the enveloping algebra. It
is indeed this approach that will be pursued in section 4, and it exclusively
involves infinitesimal generators in Lie algebras as opposed to finite
transformations in Lie groups.

Figure 3: Given the bundle $\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}$,
any bundle automorphism maps $\bar{\mathscr{M}}$ to itself smoothly, but
generally shuffles its points so that different fibres are mapped on one
another (left). Furthermore, automorphisms locally change the origin of the
Carrollian time coordinate (right). They also affect the local normalisation
of the fundamental vector field (not shown here). In the BMS context, local
translations along fibres are supertranslations, while superrotations are
automorphisms projecting onto conformal maps of the base space: see sections
3.3–3.4.

$~{}\xleftarrow{~{}~{}~{}~{}~{}~{}}~{}$
$~{}\xrightarrow{~{}~{}~{}~{}~{}~{}}~{}$

#### Basic and invariant tensor fields.

The notion of invariance, and other similar constraints, is readily extended
to covariant tensor fields (e.g. differential forms) instead of vectors.
Accordingly, a covariant tensor field $T_{\mu\nu\cdots}$ on $\mathscr{M}$ is
called invariant if ${\cal L}_{\xi}T=0$, horizontal if any of its contractions
with $\xi$ vanishes, and basic if it is both invariant and horizontal. In this
last case, $T$ is the pullback by the projection (59) of a tensor $\bar{T}$ on
the base manifold $\bar{\mathscr{M}}$; in other words, $T$ is basic iff
$T=\pi^{*}\bar{T}$.

For example, a function $f$ on $\mathscr{M}$ is invariant if ${\cal
L}_{\xi}f=0$; in local coordinates $(u,x^{a})$, it is any time-independent
function $f(x)$. Similarly, a one-form $A$ on $\mathscr{M}$ is invariant if
its components in adapted coordinates are $u$-independent (i.e. if
$A=f(x)\,\text{d}u+A_{a}(x)\,\text{d}x^{a}$); it is horizontal if its $u$
component vanishes, so basic one-forms read $A=A_{a}(x)\text{d}x^{a}$ in local
coordinates. Other examples of basic tensor fields will be provided by
Carrollian metrics in section 3.3.

###  3.2 Ehresmann connections as Carrollian clocks

The concepts presented so far in section 3 apply to any principal
$\mathbb{R}$-bundle, without any Carrollian spacetime interpretation save for
time translations. We now push this intuition much further by defining
Carrollian clocks and horizontal surfaces, all based on a choice of Ehresmann
connection on $\mathscr{M}$. This eventually leads to the notion of Euler
vector fields, and it will allow us to write the algebras (62) as semi-direct
sums that start to resemble BMS. The spacetime picture will later be further
completed by the addition of a Carrollian metric in section 3.3. We refer e.g.
to [97, 56, 55] for more on the natural appearance of Ehresmann connections in
Carrollian geometry.

#### Carrollian clocks.

Let $\mathscr{M}\twoheadrightarrow\bar{\mathscr{M}}$ be a principal
$\mathbb{R}$-bundle with fundamental vector field $\xi$, and let
$p\in\mathscr{M}$ be a point. Then a non-zero vector $v$ tangent to
$\mathscr{M}$ at $p$ is time-like if it is vertical, i.e. if
$v\propto\xi_{p}$; otherwise $v$ is space-like. Thus the space of time-like
vectors has measure zero within the set of all tangent vectors on a Carrollian
manifold, which should be contrasted with Lorentzian manifolds where time-like
vectors are as “generic” as space-like ones.

Now, one typically wishes to have a global prescription splitting any tangent
vector, anywhere in $\mathscr{M}$, as the sum of a time-like piece and a
space-like “horizontal” one. This can be achieved with an Ehresmann connection
on $(\mathscr{M},\xi)$, i.e. an invariant one-form
$A\in\Omega^{1}(\mathscr{M})$ such that $i_{\xi}A=1$ everywhere. In the
Carrollian context, such a connection may be called a Carrollian clock, since
the normalization $i_{\xi}A=1$ suggests the interpretation of $A$ as a field
of clocks “dual” to the fundamental vector field $\xi$ that generates time
translations. In adapted coordinates $(u,x^{a})$ such that $\xi=\partial_{u}$,
any clock reads $A=\text{d}u+A_{a}(x)\text{d}x^{a}$ with arbitrary time-
independent components $A_{a}(x)$.

The standard terminology of connections and their curvature carries over to
the Carrollian realm: a clock is locally synchronised if the connection $A$ is
flat ($\text{d}A=0$), and it is globally synchronised if $A=\text{d}f$ is
exact. Note that the curvature $\text{d}A$ is a basic two-
form.151515$i_{\xi}(\text{d}A)={\cal L}_{\xi}A=0$ and ${\cal
L}_{\xi}(\text{d}A)=\text{d}i_{\xi}\text{d}A=0$ thanks to Cartan’s magic
formula ${\cal L}_{\xi}=\text{d}\circ i_{\xi}+i_{\xi}\circ\text{d}$. This is
actually manifest in adapted coordinates, where
$\text{d}A=\partial_{[a}A_{b]}(x)\,\text{d}x^{a}\wedge\text{d}x^{b}$ only
involves the base form $A_{a}(x)\text{d}x^{a}$. In this sense, the only
obstruction that may prevent a locally synchronised clock from being globally
synchronisable occurs if the first de Rham class of $\bar{\mathscr{M}}$ (i.e.
its first Betti number) happens to be non-trivial. The total space
$\mathscr{M}$ plays no role for this issue, in accordance with our comment
above that all principal $\mathbb{R}$-bundles are trivial.

#### Horizontal vectors and simultaneity.

Given a principal $\mathbb{R}$-bundle $(\mathscr{M},\xi)$ with an Ehresmann
connection $A$, a vector field $X\in\mathfrak{X}(\mathscr{M})$ is horizontal
if $i_{X}A=0$. The condition $i_{\xi}A=1$ then ensures that any horizontal
vector is space-like; in adapted coordinates, any horizontal vector reads
$X=X^{a}(u,x)\big{(}\partial_{a}-A_{a}(x)\partial_{u}\big{)}$ with arbitrary
components $X^{a}(u,x)$. The notion of horizontality also allows us to define
a linear lift $\ell$ from vector fields on the base $\bar{\mathscr{M}}$ to
invariant horizontal vector fields on $\mathscr{M}$:

$\ell:\mathfrak{X}(\bar{\mathscr{M}})\hookrightarrow\mathfrak{X}_{\text{inv}}(\mathscr{M}):\bar{X}\mapsto\ell(\bar{X})\,,$
(66)

where $X=\ell(\bar{X})$ is the unique invariant and horizontal vector field
such that $\pi_{*}(X)=\bar{X}$. In adapted coordinates, this amounts to the
“minimal coupling” prescription $\ell:X^{a}(x)\,\partial_{a}\mapsto
X^{a}(x)\big{(}\partial_{a}-A_{a}(x)\partial_{u}\big{)}$. Note that the lift
(66) is a morphism of Lie algebras iff the connection is flat; this is most
easily verified in adapted coordinates.

Suppose now that one picks a point in $\mathscr{M}$. The set of horizontal
vectors at that point is a $d$-dimensional subspace of the $(d+1)$-dimensional
tangent space at that point, so the Carrollian clock $A$ actually defines a
distribution of $d$-planes in $\mathscr{M}$. Each such plane is a local choice
of “absolute space”. When the clock is locally synchronised ($\text{d}A=0$),
the distribution turns into a foliation of $\mathscr{M}$ thanks to the
Frobenius theorem (see e.g. [94, Chap. 11] for details). If, in addition, the
clock is globally synchronised ($A=\text{d}f$), then horizontal vector fields
are those that solve ${\cal L}_{X}f=0$. In adapted coordinates, one has
$f(u,x)=u+g(x)$ and horizontal vectors read
$X=X^{a}(u,x)(\partial_{a}-\partial_{a}g(x)\partial_{u})$, so the leaves of
the foliation are submanifolds specified by an equation of the form
$u=\text{cst}-g(x)$. This defines a global Carrollian time
$u^{\prime}:=f(u,x)=u+g(x)$.

To conclude, suppose one has chosen two clocks $A$ and $A^{\prime}$. Then
their difference $A^{\prime}-A$ is necessarily basic. Accordingly, any change
of clock mapping $A\to A^{\prime}=A+\pi^{*}\bar{B}$ with
$\bar{B}\in\Omega^{1}(\bar{\mathscr{M}})$ is called a local Carrollian boost.
In terms of the distribution of space-like $d$-planes specified by a clock,
the boost’s effect is to locally tilt the planes, i.e. to locally change the
notion of simultaneity. We will use such boosts in section 3.4 to prove that
invariant volume forms on $\mathscr{M}$ induced by a volume form on
$\bar{\mathscr{M}}$ are independent of one’s choice of Carrollian clock.

#### Euler vector fields.

The notion of horizontality defined by a clock will soon allow us to unveil
the semi-direct structure of projectable and invariant vector fields
introduced in section 3.1. In order to also include super-projectable vector
fields in this analysis, an aside is now needed on Euler vector fields.
Namely, let $A=\text{d}f$ be a globally synchronised clock and let
$\Sigma\subset\mathscr{M}$ be the corresponding leaf at Carrollian time $f=0$.
Then an Euler vector field is a super-projectable vector field $\eta$ on
$\mathscr{M}$ such that

${\cal L}_{\eta}\xi=-\xi\,,\qquad{\cal
L}_{\eta}A=+A\,,\qquad(i_{\eta}A)|_{\Sigma}=0\,.$ (67)

Any such vector field generates dilations of Carrollian time. Indeed, in
adapted coordinates, it reads $\eta=u\partial_{u}+X^{a}(x)\partial_{a}$ with
arbitrary components $X^{a}(x)$. Euler vector fields are thus not unique; in
fact the difference between two Euler vector fields $\eta,\eta^{\prime}$ is an
invariant horizontal vector field $Y:=\eta^{\prime}-\eta$ since ${\cal
L}_{\xi}Y={\cal L}_{\xi}\eta^{\prime}-{\cal L}_{\xi}\eta=0$ and
$i_{Y}A\big{|}_{\Sigma}=i_{\eta^{\prime}}A\big{|}_{\Sigma}-i_{\eta}A\big{|}_{\Sigma}=0$
(the latter equality extends to $\mathscr{M}$ since $A$ and $Y$ are both
invariant). It is natural to fix this ambiguity by calling canonical Euler
vector field the unique Euler vector field $\eta$ such that ${\cal
L}_{\eta}X=0$ for any invariant horizontal vector field
$X\in\ell\big{(}\mathfrak{X}(\bar{\mathscr{M}})\big{)}$, where $\ell$ is the
horizontal lift (66). The canonical Euler vector field simply reads
$\eta=u\,\partial_{u}$ in adapted coordinates.

Together, the fundamental vector field $\xi$ and any Euler vector field $\eta$
span the affine algebra $\mathfrak{igl}(1)$ of the real line, where $\xi$ and
$\eta$ respectively generate fibrewise translations and dilations. In fact, a
principal $\mathbb{R}$-bundle endowed with a global section $\Sigma$ can be
identified with a line bundle (i.e. a vector bundle with one-dimensional
fibre) with zero section $\Sigma$, in which case there is a canonical action
of the affine algebra $\mathfrak{igl}(1)$ on each fibre. This affine structure
will now allow us to write the algebra of super-projectable vector fields as a
semi-direct sum.

#### Algebras of (super-)projectable and invariant vector fields.

The arsenal of clocks and horizontal vectors allows us to gain some insight in
the structure of the algebras (62) defined above; as it happens, this will
also eventually provide the semi-direct structure of BMS transformations.
Accordingly, note first that all three algebras
$\mathfrak{X}_{\text{pro}}(\mathscr{M})$,
$\mathfrak{X}_{\text{spro}}(\mathscr{M})$ and
$\mathfrak{X}_{\text{inv}}(\mathscr{M})$ of section 3.1 admit an ideal
consisting of vertical vector fields. This ideal is isomorphic to
$C^{\infty}(\mathscr{M})$ in the projectable case, since any vertical vector
field reads as $f\,\xi$ in terms of some function $f$ on $\mathscr{M}$; it is
isomorphic to $\\{f\in C^{\infty}(\mathscr{M})\,|\,{\cal
L}^{2}_{\xi}f=0\\}\cong C^{\infty}(\bar{\mathscr{M}})\otimes\mathfrak{igl}(1)$
in the super-projectable case, where $\mathfrak{igl}(1)$ is spanned by the
fundamental vector field $\xi$ and the canonical Euler vector field $\eta$;
and it is isomorphic to $C_{\text{inv}}^{\infty}(\mathscr{M})=\\{f\in
C^{\infty}(\mathscr{M})\,|\,{\cal L}_{\xi}f=0\\}\cong
C^{\infty}(\bar{\mathscr{M}})$ in the invariant case. It follows that the
quotients

$\frac{\mathfrak{X}_{\text{pro}}(\mathscr{M})}{C^{\infty}(\mathscr{M})}\cong\frac{\mathfrak{X}_{\text{spro}}(\mathscr{M})}{C^{\infty}(\bar{\mathscr{M}})\otimes\mathfrak{igl}(1)}\cong\frac{\mathfrak{X}_{\text{inv}}(\mathscr{M})}{C^{\infty}(\bar{\mathscr{M}})}\cong\mathfrak{X}(\bar{\mathscr{M}})$
(68)

are Lie algebras, all three isomorphic to the algebra
$\mathfrak{X}(\bar{\mathscr{M}})$ of vector fields on the base. It is tempting
to deduce from this the semi-direct sum expressions

$\displaystyle\mathfrak{X}_{\text{pro}}(\mathscr{M})$
$\displaystyle\cong\mathfrak{X}(\bar{\mathscr{M}})\inplus
C^{\infty}_{0}(\mathscr{M})\,,$ (69)
$\displaystyle\mathfrak{X}_{\text{spro}}(\mathscr{M})$
$\displaystyle\cong\mathfrak{X}(\bar{\mathscr{M}})\inplus\big{(}C^{\infty}_{0}(\bar{\mathscr{M}})\otimes\mathfrak{igl}(1)\big{)}\,,$
(70) $\displaystyle\mathfrak{X}_{\text{inv}}(\mathscr{M})$
$\displaystyle\cong\mathfrak{X}(\bar{\mathscr{M}})\inplus
C^{\infty}_{0}(\bar{\mathscr{M}})\,,$ (71)

where the subscript in $C^{\infty}_{0}$ indicates that
$\mathfrak{X}(\bar{\mathscr{M}})$ acts on functions via the Lie derivative,
i.e. according to eqs. (33)–(36) with $\texttt{w}=0$. Indeed, the
decompositions (69)–(71) are correct but not canonical, as they require a
choice of flat connection $A$. To see this, consider projectable vector fields
for definiteness: in order to write any such vector field $X$ as a pair
$(\bar{X},f)\in\mathfrak{X}(\bar{\mathscr{M}})\inplus
C^{\infty}(\mathscr{M})$, one needs both the projection $\bar{X}=\pi_{*}(X)$
and a horizontal lift (66) that defines the function $f$ by
$f\,\xi:=X-\ell(\pi_{*}(X))$. Conversely, the same data is needed in order to
associate with any pair $(\bar{X},f)\in\mathfrak{X}(\bar{\mathscr{M}})\inplus
C^{\infty}(\mathscr{M})$ a vector field on $\mathscr{M}$ via
$X=\ell(\bar{X})+f\xi$. This lift only exists once a clock has been chosen,
and it is a Lie algebra morphism only if the clock is synchronisable. This is
why we could not yet write the isomorphisms (69)–(71) in section 3.1.

The expressions (69)–(71) also provide information on the groups of
automorphisms generated by projectable, super-projectable and invariant vector
fields. These groups are semi-direct products, and they respectively read

$\text{Diff}(\bar{\mathscr{M}})\ltimes
C_{0}^{\infty}(\mathscr{M}),\qquad\text{Diff}(\bar{\mathscr{M}})\ltimes\big{\\{}\text{Maps
}\bar{\mathscr{M}}\to\text{IGL(1)}\big{\\}},\qquad\text{Diff}(\bar{\mathscr{M}})\ltimes
C_{0}^{\infty}(\bar{\mathscr{M}}).$ (72)

The corresponding transformations in coordinates $(u,x^{a})$ were written
around (65). These semi-direct products manifestly resemble the BMS group, but
we are not quite there yet. First, BMS vector fields in (2)–(4) are super-
projectable but not invariant, so they generate automorphisms of null infinity
seen as a fibre bundle, but not as a principal $\mathbb{R}$-bundle. In this
sense, the first two groups in (72) are too large, while the third is overly
restrictive. Second, the Abelian normal subgroups in (72) consist of functions
on $\mathscr{M}$ or $\bar{\mathscr{M}}$ (as in the “warped Virasoro group”
where $\bar{\mathscr{M}}=S^{1}$ [98, 99]), while the Abelian subgroup needed
for BMS consists of densities with a suitable weight: recall the bracket (6)
or the weight $-1/d$ appearing in the BMS action in (45). Defining BMS thus
requires an extra, conformal, structure, to which we now turn.

###  3.3 BMS as conformal Carroll

Having reviewed the basics of principal $\mathbb{R}$-bundles, we now consider
genuine Carrollian physics by endowing these bundles with degenerate metrics.
This will lead to Carrollian isometries whose natural generalisation [46, 47]
will allow us to introduce the Carrollian cousins of conformal groups, and in
particular BMS algebras. The groups obtained in this way generally have no
infinite-dimensional non-Abelian factor, save for exceptional circumstances
(one- or two-dimensional celestial spheres) where conformal generators admit
an infinite-dimensional enhancement. The generalisation to non-conformal
diffeomorphisms of the base and “generalised” BMS [15, 16] will be treated in
section 3.4.

#### Carrollian spacetimes.

An (invariant) Carrollian metric is a symmetric, covariant, rank-two tensor
field $\gamma_{\mu\nu}$ on $\mathscr{M}$ which is basic (${\cal
L}_{\xi}\gamma=0$ and $\gamma(\xi,\cdot)=0$), positive semi-definite
($\gamma(X,X)\geqslant 0$) and whose radical is spanned by $\xi$ (i.e.
$\gamma(X,\cdot)=0$ iff $X$ is vertical). Since it is basic, it can be written
as the pullback $\gamma=\pi^{*}g$ of a Riemannian (non-degenerate, positive-
definite) metric $g$ on the base $\bar{\mathscr{M}}$. Thus, in adapted
coordinates $(u,x^{a})$ with $\xi=\partial_{u}$, a Carrollian metric reads

$\gamma=g_{ab}(x)\,\text{d}x^{a}\,\text{d}x^{b}$ (73)

where $g_{ab}$ can be any Riemannian metric on the base. A triplet
$(\mathscr{M},\xi,\gamma)$ consisting of a principal $\mathbb{R}$-bundle
$\mathscr{M}$ with fundamental vector field $\xi$ and a Carrollian metric
$\gamma$ will be called a Carrollian spacetime. A typical example of
Carrollian metric is again provided by null infinity, whose metric can be read
off from the limit $r\to\infty$ of the Minkowski metric (1) restricted to a
surface with constant $r$. The result is nothing but eq. (73), i.e. the
pullback by the projection (60) of the metric on a celestial sphere, up to a
divergent normalisation.

Note that the invariance condition on the Carrollian metric may be relaxed in
full generality [47], which is crucial in certain cases. For instance, the
induced metric on a light-cone in Minkowski spacetime is not invariant,
although it obeys all the other axioms above. Moreover, in Penrose’s approach
the metric structure at null infinity is only defined up to a conformal
factor, so a more general definition may seem in order (see e.g. [49, sec.
2]). In practice the restriction to invariant metrics at null infinity adopted
here entails no loss of generality (see e.g. [51, sec. II.B.2] or [100, sec.
II]), so we stick to it.

A final comment: in contrast to Riemannian geometry, there is no unique
torsionless affine connection compatible with a given Carrollian structure
$(\xi,\gamma)$—the reason essentially lies in the fact that the metric is not
invertible. This is yet another instance where the usefulness of Ehresmann
connections becomes manifest in Carrollian geometry: a unique notion of
parallel transport on a Carrollian spacetime $(\mathscr{M},\xi,\gamma)$
becomes privileged only once it is further endowed with a clock $A$, cf. the
“special” connection in [97, app. A]. This is not crucial for our purposes, so
we will not dwell on that point.

#### Carrollian isometries.

Given a Carrollian spacetime $(\mathscr{M},\xi,\gamma)$, its Carrollian
isometries are principal bundle automorphisms that preserve the Carrollian
metric. Any such automorphism projects on an isometry of the base
$(\bar{\mathscr{M}},g)$. Thus, a vector field $X\in\mathfrak{X}(\mathscr{M})$
generates Carrollian isometries iff it is invariant and satisfies in addition
the Killing equation on $\bar{\mathscr{M}}$, that is,

${\cal L}_{X}\xi=0\qquad\text{and}\qquad{\cal L}_{\bar{X}}g=0\,.$ (74)

In adapted coordinates $(u,x^{a})$, any such vector field reads
$X=X^{a}(x)\,\partial_{a}+\alpha(x)\,\partial_{u}$, where
$\bar{X}=X^{a}(x)\,\partial_{a}$ and $\alpha(x)$ are, respectively, a Killing
vector field and an arbitrary function on $\bar{\mathscr{M}}$ (recall eq.
(63)). The algebra of Carrollian isometry generators thus has a structure of
semi-direct sum

$\mathfrak{isom}(\mathscr{M})\cong\mathfrak{isom}(\bar{\mathscr{M}})\inplus
C^{\infty}_{\texttt{w}}(\bar{\mathscr{M}})$ (75)

provided a synchronised clock has been chosen to define a horizontal lift
(66). Note that the weight w of fibrewise translations under diffeomorphisms
of $\bar{\mathscr{M}}$ is arbitrary in (75), since the Killing equation in
(74) implies $\nabla_{a}X^{a}=0$ so that the coefficient of w in the
transformation laws (33)–(36) vanishes. The simplest choice is $\texttt{w}=0$,
but we will soon see that conformal invariance yields instead
$\texttt{w}=-1/d$, as in the BMS bracket (6).

The Carrollian isometry group with Lie algebra (75) has a structure of semi-
direct product $\text{Isom}(\bar{\mathscr{M}})\ltimes
C^{\infty}_{\texttt{w}}(\bar{\mathscr{M}})$. The BMS group to be introduced
now will extend it, in the same way that conformal maps extend standard
isometries.

#### Carrollian conformal structure.

An equivalence class $(\mathscr{M},[\xi,\gamma])$ of Carrollian spacetimes,
with respect to the equivalence relation

$(\xi,\gamma)\sim(\Omega^{-1}\,\xi\,,\Omega^{2}\,\gamma)$ (76)

where $\Omega=\pi^{*}\bar{\Omega}$ is any positive invariant function, is
called a Carrollian conformal structure.161616It is also called a universal
structure in Penrose’s approach to asymptotically flat spacetimes, since it is
the natural kinematical structure at null infinity; see e.g. [51, sec. II.B.2]
or [100, sec. II]. Its projection on the base defines a conformal structure
$(\bar{\mathscr{M}},[g])$ for the class of Riemannian metrics
$g^{\prime}\sim\overline{\Omega}^{2}\,g$. The key point of (76) is to link the
conformal weight of $\gamma$ to that of $\xi$ in such a way that
$\gamma\otimes\xi\otimes\xi$ be “Weyl-invariant”. This will eventually fix the
weight $\texttt{w}=-1/d$ of BMS supertranslations under celestial
diffeomorphisms.

Note that one could also define a “Lifschitz” scaling where fundamental vector
fields are identified as $\xi\sim\Omega^{-z}\,\xi$ for some exponent
$z\in\mathbb{R}$ instead of (76) [47]. This leads to a different weight of
“supertranslations” under diffeomorphisms of the base space and no longer
corresponds to the structure normally encountered in asymptotically flat
gravity, but it is conceivable that it would find applications elsewhere,
similarly to Lifschitz symmetries in generalisations of AdS/CFT [101, 102]. We
will not study this situation in depth, but we shall from time to time mention
the generalisation of conformal (BMS) results to the Lifschitz case since our
geometrical approach makes the extension straightforward.

A bundle automorphism of $(\mathscr{M},\xi,\gamma)$ that preserves a given
Carrollian conformal structure, so that $\xi^{\prime}=\Omega^{-1}\,\xi$
(rescaling of Carroll time) and $\gamma^{\prime}=\Omega^{2}\,\gamma$ (local
rescaling of the metric) is a Carrollian conformal map.171717To be precise, if
$F:\mathscr{M}\to\mathscr{M}$ is the automorphism, we write the transformed
fundamental vector field and metric as $\xi^{\prime}:=F_{*}\xi$ and
$\gamma^{\prime}:=(F^{-1})^{*}\gamma$. Its projection on the base is a
conformal transformation of $(\bar{\mathscr{M}},g)$, with
$g^{\prime}=\overline{\Omega}^{2}\,g$. (One could similarly define a
Carrollian Lifschitz map by replacing the first condition with
$\xi^{\prime}=\Omega^{-z}\,\xi$.)

#### Carrollian conformal vector fields.

Having chosen a Carrollian conformal structure $(\mathscr{M},[\xi,\gamma])$,
pick a representative in the equivalence class (76), i.e. a Carrollian
spacetime $(\mathscr{M},\xi,\gamma)$. Then any projectable vector field
$X\in\mathfrak{X}(\mathscr{M})$ such that

${\cal L}_{X}\xi=-f\,\xi\qquad\text{and}\qquad{\cal
L}_{X}\gamma=2f\,\gamma\,,$ (77)

for some function $f$ on $\mathscr{M}$, is a Carrollian conformal vector field
that generates Carrollian conformal maps of $(\mathscr{M},\xi,\gamma)$. One
can check that $X$ is also a Carrollian conformal vector field for any
representative $(\xi^{\prime},\gamma^{\prime})$ in the equivalence class (76),
i.e. ${\cal L}_{X}\xi^{\prime}=-f^{\prime}\,\xi^{\prime}$ and ${\cal
L}_{X}\gamma^{\prime}=2f^{\prime}\,\gamma^{\prime}$ with $f^{\prime}=f+{\cal
L}_{X}(\ln\Omega)$. Note that the conditions (77) imply ${\cal L}_{\xi}f=0$
since $\gamma$ is basic by definition, so $f=\pi^{*}\bar{f}$ is invariant and
$X$ is automatically super-projectable. The projection $\bar{X}=\pi_{*}(X)$ is
a conformal Killing vector field of $(\bar{\mathscr{M}},g)$, since the
condition ${\cal L}_{X}\gamma=2f\,\gamma$ projects to the conformal Killing
equation ${\cal L}_{\bar{X}}g=2\bar{f}g$ where
$\bar{f}=\tfrac{1}{d}\nabla\cdot\bar{X}$. Accordingly, the conditions (77) are
equivalent to

${\cal L}_{X}\xi=-f\,\xi\,,\qquad{\cal L}_{\bar{X}}g=2\bar{f}\,g\,,$ (78)

which will be called the Carrollian conformal Killing equations of
$(\mathscr{M},[\xi,\gamma])$.

The structure of Carrollian conformal vector fields is most easily revealed in
adapted coordinates $(u,x^{a})$, where super-projectable vector fields take
the form (63). The second conformal condition in (78) then leaves the function
$\alpha(x)$ in (63) arbitrary, while fixing
$f(x)=\tfrac{1}{d}\nabla_{a}X^{a}(x)$, where $d$ is the dimension of the base
manifold and $\nabla$ is its Levi-Civita connection. Any Carrollian conformal
vector field can thus be written as

$X=X^{a}(x)\partial_{a}+\big{(}\alpha(x)+\tfrac{u}{d}\nabla_{a}X^{a}(x)\big{)}\partial_{u}$
(79)

for some conformal Killing vector $\bar{X}=X^{a}(x)\partial_{a}$ of the base
manifold. The coincidence with the components of BMS vector fields (3)–(4) is
manifest. It readily follows that the space of Carrollian conformal vector
fields is a Lie algebra with bracket (6) whose abstract structure is that of a
semi-direct sum

$\mathfrak{conf}(\mathscr{M})\cong\mathfrak{conf}(\bar{\mathscr{M}})\inplus
C^{\infty}_{\texttt{w}}(\bar{\mathscr{M}})\qquad\text{with}\qquad\texttt{w}=-1/d,$
(80)

where the subscript w stresses that local fibrewise translations $u\to
u+\alpha(x)$ are densities with weight $\texttt{w}=-1/d$ under diffeomorphisms
of $\bar{\mathscr{M}}$, in the sense of eq. (33).

Note for completeness that the “Lifschitz” generalisation of the above
definitions is immediate: since the identification is now declared to be
$\xi\sim\Omega^{-z}\xi$, the Lifschitz version of eqs. (78) is obtained by
replacing the first condition by ${\cal L}_{X}\xi=-zf\,\xi$ while leaving the
rest unchanged. The coordinate expression of a “Lifschitz Carrollian conformal
vector field” then takes the form (79) with the factor $u/d$ replaced by
$z\,u/d$, which in turn implies that fibrewise translations $\alpha(x)$ are
densities with weight $-z/d$ instead of $-1/d$. This is so similar to the
standard structure of Carrollian conformal maps that we will no longer return
to the Lifschitz example in what follows.

#### Example: BMS algebras.

The definition of Carrollian conformal structures is motivated by the example
of null infinity $\mathscr{I}$ at the boundary of asymptotically flat
spacetimes, whose Carrollian conformal maps are really BMS transformations
[46, 47, 48]. Thus, by definition, Carrollian conformal vector fields on null
infinity $\mathscr{I}\cong\mathbb{R}\times S^{d}$ span the (extended) BMS
algebra, i.e. the semi-direct sum (80) with a base manifold given by the
celestial sphere:

$\mathfrak{bms}_{d+2}\cong\mathfrak{conf}(S^{d})\inplus
C^{\infty}_{\texttt{w}}(S^{d})\qquad\text{with}\qquad\texttt{w}=-1/d.$ (81)

Fibrewise translations $u\to u+\alpha(x)$ spanning the Abelian ideal
$C^{\infty}_{\texttt{w}}(S^{d})$ are called supertranslations, where the
function $\alpha$ transforms as a density of weight $\texttt{w}=-1/d$ under
diffeomorphisms of $S^{d}$; consequently, it is a scalar conformal primary of
scaling dimension $\Delta=+1$ under the conformal algebra of $S^{d}$. The
latter always contains the Lorentz algebra $\mathfrak{so}(d+1,1)$, whose
elements are interpreted as bulk Lorentz transformations; the two actually
coincide for $d\geqslant 3$, but an infinite-dimensional enhancement to
superrotations occurs for $d=1,2$ [6, 7, 103], so that

$\mathfrak{conf}(S^{d})\cong\left\\{\begin{array}[]{cc}\mathfrak{so}(d+1,1)&\text{for}\quad
d\geqslant 3,\\\ \mathfrak{X}(S^{1})\oplus\mathfrak{X}(S^{1})&\text{for}\quad
d=2,\\\ \mathfrak{X}(S^{1})&\text{for}\quad d=1.\end{array}\right.$ (82)

The terminology is justified by the fact that supertranslations (respectively
superrotations) extend finite-dimensional bulk spacetime translations
(respectively rotations and boosts). A minor subtlety is that the embedding
$\mathfrak{so}(d+1,1)\subset\mathfrak{bms}_{d+2}$ is not canonical—it is
equivalent to picking a “good cut” [104, 50, 105, 106] at null infinity, or
equivalently a bulk point in Minkowski spacetime—but we will not dwell on
that.

BMS algebras as defined in (81) will be the starting point of our higher-spin
symmetry construction in section 4.1. Before moving to that topic, however, a
generalisation of Carrollian conformal maps is required.

###  3.4 Carrollian volumes and generalised BMS

In the context of scattering amplitudes and asymptotic symmetries, a natural
generalisation of the BMS groups defined above consists in extending conformal
superrotations to arbitrary (generally non-conformal) diffeomorphisms of
celestial spheres [15, 16, 92]. We now review the Carrollian formulation of
this enhancement, generalising to arbitrary dimensions the $d=2$ case treated
in [15, sec. 4]. The key point is to require “generalised BMS transformations”
(gBMS for short181818We stress that the shorthand “gBMS” does not refer to the
global BMS group (where superrotations are restricted to Lorentz
transformations).) to preserve a volumic structure rather than a metric
structure, but this requires a few geometric preliminaries. Accordingly, we
now review notions of Carrollian volume forms and their interplay with
Carrollian clocks, then introduce generalised conformal maps. As an
application, the Hermitian form (51) is recast in a geometric (i.e.
coordinate-independent) way, generalising the Sachs inner product [4] to any
Carrollian measured space. This will make generalised BMS symmetry manifest
and provides a key prerequisite for the higher-spin considerations of section
4.2, where the Sachs Hermitian form will be used to define higher-spin
algebras admitting a unitary representation.

#### Carrollian measured spaces.

Let $(\mathscr{M},\xi)$ be a principal $\mathbb{R}$-bundle and assume
$\mathscr{M}$ is orientable. Then an invariant volume form on
$(\mathscr{M},\xi)$ is a nowhere-vanishing top form
$\varepsilon\in\Omega^{d+1}(\mathscr{M})$ such that ${\cal
L}_{\xi}\varepsilon=0$. In adapted coordinates $(u,x^{a})$, any invariant
volume form reads
$\varepsilon=\pm\varepsilon(x)\,\text{d}u\wedge\text{d}^{d}x$, where the
function $\varepsilon(x)$ is strictly positive on $\bar{\mathscr{M}}$. A
triplet $(\mathscr{M},\xi,\varepsilon)$ made of a principal
$\mathbb{R}$-bundle with orientable total space $\mathscr{M}$ and fundamental
vector field $\xi$ , together with an invariant volume form $\varepsilon$,
will be called a Carrollian measured space since the volume form is really a
measure (recall section 2.2).

It is intuitively clear that a Carrollian measured space
$(\mathscr{M},\xi,\varepsilon)$ induces a volume form on the base manifold
$\bar{\mathscr{M}}$. This is not as trivial as for Carrollian metrics: the
contraction ${\cal V}:=i_{\xi}\varepsilon$ does not vanish, so $\varepsilon$
is not basic and cannot be written as the pullback of a form on
$\bar{\mathscr{M}}$. Note, however, that ${\cal V}\in\Omega^{d}(\mathscr{M})$
is closed ($\text{d}{\cal V}=0$) thanks to the Cartan formula along with the
invariance condition ${\cal L}_{\xi}\varepsilon=0$; also note that it is
horizontal ($i_{\xi}{\cal V}=0$) since $i_{\xi}$ is nilpotent. It follows that
${\cal V}$ is basic and uniquely determined by a volume form
$\bar{\varepsilon}\in\Omega^{d}(\bar{\mathscr{M}})$ on the base manifold:

${\cal V}\,:=\,\pi^{*}\bar{\varepsilon}\,\in\,\Omega^{d}(\mathscr{M})\,.$ (83)

In adapted coordinates where
$\varepsilon=\pm\varepsilon(x)\,\text{d}u\wedge\text{d}^{d}x$, one simply has
${\cal V}=\pm\varepsilon(x)\,\text{d}^{d}x$. Combined with the notion of
clocks introduced in section 3.2, this elementary observation will now allow
us to define volume forms on Carrollian manifolds starting from a volume form
on the base manifold.

#### Inducing volume forms from the base.

Let $\bar{\varepsilon}$ be a volume form on $\bar{\mathscr{M}}$ and define
${\cal V}$ by (83). If in addition $\mathscr{M}$ is endowed with an Ehresmann
connection $A$, one can define an invariant volume form
$\varepsilon:=A\wedge{\cal V}$ on $\mathscr{M}$ (indeed, $\varepsilon$ is
clearly a nowhere vanishing top form on $\mathscr{M}$ and ${\cal
L}_{\xi}\varepsilon={\cal L}_{\xi}A\wedge{\cal V}+A\wedge{\cal L}_{\xi}{\cal
V}=0$ since $A$ and ${\cal V}$ are both invariant). The volume form thus
obtained is independent of the choice of connection, since any two Ehresmann
connections differ by a horizontal form, and any horizontal $(d+1)$-form
automatically vanishes. In this sense, any Ehresmann connection does the job
equally well; in what follows we will systematically use this to induce a
Carrollian measured space $(\mathscr{M},\xi,\varepsilon)$ from the
(equivalent) data $(\mathscr{M},\xi,\bar{\varepsilon})$ made of a principal
$\mathbb{R}$-bundle $(\mathscr{M},\xi)$ and a volume form $\bar{\varepsilon}$
on the orientable base space $\bar{\mathscr{M}}$.

For example, consider a Carrollian spacetime $(\mathscr{M},\xi,\gamma)$ that
projects down to a Riemannian space $(\bar{\mathscr{M}},g)$. The metric $g$ on
$\bar{\mathscr{M}}$ determines its canonical volume form
$\bar{\varepsilon}:=*1\in\Omega^{d}(\bar{\mathscr{M}})$, so an invariant
volume form $\varepsilon=A\wedge{\cal V}$ with ${\cal
V}=\pi^{*}\bar{\varepsilon}$ can be defined once any Carrollian clock $A$ has
been chosen. This provides another definition (valid in any dimension) of the
volume form used e.g. in [51, sec. II.B.4] or [100, sec. III.C]. We shall
refer to it as the Carrollian volume form of the canonical Carrollian measured
space $(\mathscr{M},\xi,\varepsilon)$ induced by $(\mathscr{M},\xi,\gamma)$,
in the same way that any Riemannian metric determines a canonical volume form.
In adapted coordinates, the construction is nearly trivial since the
Carrollian volume form is simply $\text{d}u\wedge\text{d}^{d}x\,\sqrt{g(x)}$ .

It is useful, for future reference, to know how bundle automorphisms affect
invariant volume forms. Namely, let $F:\mathscr{M}\to\mathscr{M}$ be an
automorphism, so that it admits a projection
$\bar{F}:\bar{\mathscr{M}}\to\bar{\mathscr{M}}$ satisfying (64). Then note
that

$F^{*}{\cal V}=\pi^{*}(\rho_{\bar{F}})\,{\cal V}$ (84)

in terms of the Radon-Nikodym derivative (29) of the base volume form under
$\bar{F}$. This elementary fact will soon allow us to define the Carrollian
analogue of the Radon densities defined in section 2.2.

#### Campiglia-Laddha structures.

We are finally ready to define the equivalence class of Carrollian measured
spaces that generalises the conformal class (76) and will eventually lead to
generalised BMS transformations. Thus, we call Campiglia-Laddha structure an
equivalence class $(\mathscr{M},[\xi,\varepsilon])$ of Carrollian measured
spaces, with respect to the relation

$(\xi,\varepsilon)\sim(\Omega^{-1}\,\xi\,,\Omega^{d+1}\,\varepsilon)$ (85)

where $\Omega=\pi^{*}\bar{\Omega}$ is any positive invariant function. These
conditions can equivalently be expressed in terms of the volume form
$\bar{\varepsilon}$ on the base, namely

$(\xi,\bar{\varepsilon})\sim(\Omega^{-1}\,\xi\,,\bar{\Omega}^{d}\bar{\varepsilon})\,.$
(86)

To the best of our knowledge, the first discussion of such equivalence classes
in their own right, characterising generalised BMS transformations, appeared
in [15, sec. 4.1] for $d=2$; hence the terminology. In particular, any
Carrollian conformal structure $(\mathscr{M},[\xi,\gamma])$ yields a canonical
Campiglia-Laddha structure $(\mathscr{M},[\xi,\varepsilon])$ by virtue of the
above construction of the Carrollian volume form, since $A\sim\Omega A$ (due
to the condition $i_{\xi}A=1$) and ${\cal V}\sim\Omega^{d}{\cal V}$ (due to
$\bar{\varepsilon}\sim\bar{\Omega}^{d}\bar{\varepsilon}$).

In contrast to conformal structures, the extension of the equivalence (85) to
a Lifschitz-like relation $\xi\sim\Omega^{-z}\,\xi$ also requires a
modification of the relation imposed on Carrollian volumes. This is because
the normalisation condition $i_{\xi}A=1$ for Carrollian clocks requires
$A\sim\Omega^{z}A$ in the Lifschitz case, which in turn implies that the
volume form must now be identified as
$\varepsilon\sim\Omega^{d+z}\,\varepsilon$ instead of the second condition in
(85). (By contrast, the Lifschitz generalisation of the conformal equivalence
(76) only affected the fundamental vector field, not the metric.) Aside from
this modification, the discussion of “Lifschitz-generalised” conformal maps
and vector fields is entirely analogous to that presented in section 3.3 for
the conformal case, so we will not dwell on this extension of the formalism.

Similarly to section 3.3, any diffeomorphism of $\mathscr{M}$ that preserves a
given Campiglia-Laddha structure $(\mathscr{M},[\xi,\varepsilon])$, so that
$\xi^{\prime}=\Omega^{-1}\,\xi$ and
$\varepsilon^{\prime}=\Omega^{d+1}\,\varepsilon$ for any representative
$(\xi,\varepsilon)$, will be called a generalised conformal map.191919With the
notation of footnote 17, one has $\xi^{\prime}:=F_{*}\xi$ and
$\varepsilon^{\prime}:=(F^{-1})^{*}\varepsilon$. It is necessarily a bundle
automorphism (since $\xi^{\prime}=\Omega^{-1}\,\xi$) and its projection on the
base can be any orientation-preserving diffeomorphism of $\bar{\mathscr{M}}$
as per eq. (86). In adapted coordinates $(u,x^{a})$, any generalised conformal
map $F:\mathscr{M}\to\mathscr{M}$ reads

$(u,x)\to(u^{\prime},x^{\prime})=\big{(}\rho_{\bar{F}}(x)^{1/d}u+\alpha(x),\bar{F}(x)\big{)}$
(87)

for some diffeomorphism $\bar{F}:\bar{\mathscr{M}}\to\bar{\mathscr{M}}$, with
$\rho_{\bar{F}}=\Omega^{d}$ the corresponding Radon-Nikodym derivative (29) on
the base and $\alpha(x)$ any “supertranslation” function. The similarity with
the BMS transformation (45) is striking indeed. We stress that such maps are
much less constrained than conformal maps, thanks to the fact that the
identifications (85) are much broader than those defining conformal structures
in (76). Indeed, any volume form on the base (with fixed orientation) is
solely determined by a single positive function, so the second identification
in (85) simply says that any two volume forms belong to the same equivalence
class (provided they define the same orientation). Put differently, the orbit
of a volume form is the set of all possible (oriented) volume forms,202020One
should still keep in mind that the equivalence class applies to pairs
$(\xi,\varepsilon)$ so, strictly speaking, the various volume forms
$\varepsilon^{\prime}$ in the orbit of $\varepsilon$ are distinguished by
their partner $\xi^{\prime}$ inside the pair
$(\xi^{\prime},\varepsilon^{\prime})\sim(\xi,\varepsilon)$. and a Campiglia-
Laddha structure as defined by (85) generally contains many distinct conformal
structures as defined by (76). The only exception occurs for one-dimensional
base manifolds ($d=1$), where conformal maps and their generalisations
coincide.

#### Generalised conformal vector fields.

Again mimicking section 3.3, it is natural to call generalised conformal
vector field any projectable vector field $X\in\mathfrak{X}(\mathscr{M})$ such
that

${\cal L}_{X}\xi=-f\,\xi\qquad\text{and}\qquad{\cal
L}_{X}\varepsilon=(d+1)f\,\varepsilon$ (88)

for some function $f$ on $\mathscr{M}$. Equivalently,

${\cal L}_{X}\xi=-f\,\xi\qquad\text{and}\qquad{\cal
L}_{\bar{X}}\bar{\varepsilon}=d\,\bar{f}\,\bar{\varepsilon}$ (89)

in terms of the base volume form. This is the volumic generalisation of the
conformal conditions (77)–(78). In particular, any
$X\in\mathfrak{X}(\mathscr{M})$ that satisfies eqs. (88) generates generalised
conformal maps on $\mathscr{M}$. As in section 3.3, eqs. (88) imply ${\cal
L}_{\xi}f=0$ since $\varepsilon$ is invariant by definition, so $X$ is
necessarily super-projectable. However, in contrast to section 3.3, the
projection $\bar{X}=\pi_{*}(X)$ is now an arbitrary vector field on
$\bar{\mathscr{M}}$. Note that the last condition in (89) implies
$\bar{f}=\tfrac{1}{d}\,\text{div}\bar{X}$ with the divergence defined by (34),
in agreement with the special case $\bar{f}=\tfrac{1}{d}\nabla\cdot\bar{X}$
obtained above (79) for Carrollian conformal vector fields.

In adapted coordinates $(u,x^{a})$, any generalised conformal vector field
takes the form (79) with an arbitrary vector field $X^{a}\partial_{a}$ on the
base manifold. It readily follows that the space of generalised conformal
vector fields is a Lie algebra, with a bracket (6) that generalises that of
Poincaré and BMS vector fields. Abstractly, the algebra is a semi-direct sum

$\mathfrak{gconf}(\mathscr{M})\cong\mathfrak{X}(\bar{\mathscr{M}})\inplus
C^{\infty}_{\texttt{w}}(\bar{\mathscr{M}})\qquad\text{with}\qquad\texttt{w}=-1/d,$
(90)

where the subscript stresses once more that local fibrewise translations $u\to
u+\alpha(x)$ are densities with weight $\texttt{w}=-1/d$ under diffeomorphisms
of $\bar{\mathscr{M}}$, in the sense of eq. (33).

#### Example: Generalised BMS.

It is immediate to apply the definition of generalised conformal maps and
vector fields to null infinity $\mathscr{I}\cong S^{d}\times\mathbb{R}$. The
algebra (90) with $\bar{\mathscr{M}}=S^{d}$ is then referred to as a
generalised BMS algebra [15, 16] (see also [107])

$\mathfrak{gbms}_{d+2}\cong\mathfrak{X}(S^{d})\inplus
C^{\infty}_{\texttt{w}}(S^{d})\qquad\text{with}\qquad\texttt{w}=-1/d,$ (91)

where the Abelian ideal $C^{\infty}_{\texttt{w}}(S^{d})$ is still spanned by
supertranslations, while the subalgebra $\mathfrak{X}(S^{d})$ generates
arbitrary diffeomorphisms of celestial spheres that generalise the
superrotations introduced around eq. (81). Conversely, the conformal BMS
algebras (82) are Lie subalgebras of generalised BMS algebras, leading to the
hierarchy

$\mathfrak{iso}(d+1,1)\subset\mathfrak{bms}_{d+2}\subseteq\mathfrak{gbms}_{d+2}\subset\mathfrak{X}_{\text{spro}}(\mathscr{I}_{d+1})\subset\mathfrak{X}_{\text{pro}}(\mathscr{I}_{d+1})\,,$
(92)

where the second inclusion is an equality only for
$\mathfrak{bms}_{3}=\mathfrak{gbms}_{3}$. The higher-spin extensions of the
symmetry groups involved in this sequence will be studied in detail in section
4.2. However, this first requires that we return to the Sachs form (51) and
rephrase it in a manifestly Carrollian and coordinate-independent way after
defining suitable Carrollian densities.

#### Carrollian Hermitian form.

Consider a Carrollian measured space $(\mathscr{M},\xi,\varepsilon)$ and
define the basic form ${\cal V}=i_{\xi}\varepsilon=\pi^{*}\bar{\varepsilon}$
induced from the volume form $\bar{\varepsilon}$ on $\bar{\mathscr{M}}$, as in
(83). Then, for any two functions $\phi$ and $\psi$ on $\mathscr{M}$, the
wedge product of $\phi^{*}\text{d}\psi$ and ${\cal V}$ is a top form given by

$\phi^{*}\text{d}\psi\wedge{\cal V}=(\phi^{*}\,{\cal
L}_{\xi}\psi)\,\varepsilon\,,$ (93)

as follows from the relation ${\cal
L}_{\xi}\psi=i_{\xi}(\text{d}\psi)$.212121Indeed, ${\cal
L}_{\xi}\psi\,\varepsilon=i_{\xi}(\text{d}\psi)\wedge\varepsilon=-\text{d}\psi\wedge
i_{\xi}\varepsilon=-\text{d}\psi\wedge{\cal V}$ where the second equality
holds because the $(d+2)-$form $\text{d}\psi\wedge\varepsilon=0$ vanishes
identically (its degree exceeds $\text{dim}(\mathscr{M})=d+1$). This top form
allows us to define the integral

$\langle\phi|\psi\rangle\,:=\,i\int_{\mathscr{M}}\phi^{*}\,\text{d}\psi\wedge{\cal
V}\,=\,i\int_{\mathscr{M}}(\phi^{*}\,{\cal L}_{\xi}\psi)\,\varepsilon\,,$ (94)

which is a Hermitian form when $\phi,\psi$ satisfy suitable boundary
conditions since the property
$\langle\phi|\psi\rangle=\langle\psi|\phi\rangle^{*}$ follows from integration
by parts of the middle integral, along with $\text{d}{\cal V}=0$. We shall
refer to it in full generality as the Carrollian Hermitian form of
$(\mathscr{M},\xi,\varepsilon)$. In adapted coordinates $(u,x^{a})$, it
reproduces the earlier Sachs inner product (51), generalising the result in
[4] to any dimension and any Carrollian measured space. It may roughly be seen
as a “matrix element” of the Carroll Hamiltonian $i\xi$ between the
“asymptotic data” $\phi$ and $\psi$.

Note in passing that the Hermitian form (94) is intimately related to the
symplectic form (8) applied to radiative solutions ($\Phi={\cal O}(r^{-d/2})$)
of the d’Alembert equation (7). Indeed, the symplectic form (8) may be recast
in Carrollian terms as

$\Omega[\delta\phi]=i\int_{\mathscr{M}}\big{(}\delta\phi^{*}\wedge\text{d}(\delta\phi)\big{)}\wedge{\cal
V}=i\int_{\mathscr{M}}\big{(}\delta\phi^{*}\wedge{\cal
L}_{\xi}(\delta\phi)\big{)}\wedge\varepsilon\,,$ (95)

where $\phi$ is understood as the leading term of the expansion (10) with
$\Delta=d/2$. The notation in (95) may be confusing: the wedge product that
follows $\delta\phi^{*}$ involves differential forms in field space, while
that in $\varepsilon=A\wedge{\cal V}$ involves forms on $\mathscr{M}$.
(Fortunately, this is the only time this issue will affect us.)

#### Carrollian densities.

Having defined Carrollian measured spaces, it is straightforward to define
notions of densities analogous to those of section 2.2. Let therefore
$(\mathscr{M},\xi,\varepsilon)$ be a Carrollian measured space with base
volume form $\bar{\varepsilon}$ and let $F:\mathscr{M}\to\mathscr{M}$ be a
bundle automorphism so that the identity (84) holds. This allows us to define
a (scalar) Carrollian density with weight w, as a function $\phi$ on
$\mathscr{M}$ that transforms under any bundle automorphism $F$ according to
$\phi\to\phi^{\prime}:=F\cdot\phi$ with

$F\cdot\phi:=(\pi^{*}\rho_{\bar{F}^{-1}})^{\texttt{w}}\,\,(F^{-1})^{*}\phi\,,$
(96)

where $\rho_{\bar{F}^{-1}}$ is the Radon-Nikodym derivative (29) of the base
volume form $\bar{\varepsilon}$ under $\bar{F}^{-1}$. This applies in
particular to any generalised conformal map, in which case
$\varepsilon^{\prime}=(F^{-1})^{*}\varepsilon=\Omega^{d+1}\,\varepsilon$ and
$\xi^{\prime}=F_{*}\xi=\Omega^{-1}\xi$ with
$\Omega=(\pi^{*}\rho_{\bar{F}^{-1}})^{1/d}$. We stress that eq. (96) is a
straightforward analogue of the transformation of densities and primary fields
encountered e.g. in CFT. This is manifest in adapted coordinates $(u,x^{a})$,
where any bundle automorphism takes the form (65) so that the transformation
law (96) becomes

$\phi^{\prime}\big{(}u^{\prime}(u,x),x^{\prime}(x)\big{)}=\rho_{\bar{F}}(x)^{-\texttt{w}}\,\phi(u,x)\,,$
(97)

which is nothing but the generalisation of the BMS transformation law (46) to
any bundle automorphism, with $\texttt{w}=\Delta/d$. Indeed, eqs. (46) and
(97) coincide in the special case where $F$ is a generalised conformal map
(87).

The infinitesimal transformation corresponding to eq. (96) can be found in the
same way as for standard densities in section 2.2. Thus, for any projectable
vector field $X$, eq. (96) yields

$\delta\phi={\cal L}_{X}\phi+\texttt{w}\,\text{div}(\bar{X})\,\phi$ (98)

where the divergence is defined by (34). When $\bar{\mathscr{M}}$ carries a
metric, $\text{div}(\bar{X})=\nabla_{a}\bar{X}^{a}$ as before. It is again
manifest that eq. (98) reduces to the infinitesimal BMS transformation law
(43) when expressed in adapted coordinates and applied to a generalised
conformal vector field.

#### Manifest generalised BMS symmetry of the Sachs form.

We showed in section 2.4 that BMS transformations acting on densities with
weight $\texttt{w}=1/2$ leave the Sachs form (51) invariant. This remains true
even when the conformal assumption is relaxed so that BMS is enhanced to
generalised BMS, allowing for any diffeomorphism of the base manifold. Indeed,
if the wavefunctions $\phi,\psi$ appearing in the Sachs form (94) transform as
Carrollian densities (96) with weight $\texttt{w}=1/2$, then any generalised
conformal map of $\mathscr{M}$ sends
$\varepsilon\to\varepsilon^{\prime}=\Omega^{d+1}\varepsilon$,
$\xi\to\xi^{\prime}=\Omega^{-1}\xi$, $\phi\to\phi^{\prime}=\Omega^{-d/2}\phi$
and $\psi\to\psi^{\prime}=\Omega^{-d/2}\psi$ with
$\Omega=(\pi^{*}\rho_{\bar{F}^{-1}})^{1/d}$, which manifestly implies that the
top form $(\phi^{*}\,{\cal L}_{\xi}\psi)\,\varepsilon$ and the Carrollian
Hermitian form (94) are invariant under generalised conformal transformations.
One can actually be even more general: any bundle automorphism leaves the
Sachs form invariant provided $\phi,\psi$ are Carrollian densities with weight
$\texttt{w}=1/2$. The proof is immediate in terms of the middle expression in
(94) upon using the transformation law (96) with $\texttt{w}=1/2$. We will
also confirm this independently in section 4.2 in terms of vector fields.

Following section 2.4, a corollary of these observations is a neat sequence of
symmetry enhancements summarised by the inclusions (92). Indeed, we saw below
eq. (26) that any scalar massless uir of the Poincaré group lifts to a
massless uir of the BMS group, but we now know that this even lifts to a
massless uir of the generalised BMS group, and even further to the group of
all bundle automorphisms. As counter-intuitive as it may sound, the Hilbert
space of radiative modes of a massless particle on Minkowski spacetime carries
a uir of all Carrollian symmetries of null infinity. In section 4.2, this
property will be extended to the corresponding higher-spin algebras.

##  4 Higher-spin BMS algebras

Having established our playground, we now turn to the construction of higher-
spin extensions of (generalised) BMS algebras. Our strategy will be the same
in all cases and follows a standard higher-spin pattern now applied to
Carrollian structures:

  * 1.

Start by considering suitable Lie algebras of vector fields on a Carrollian
manifold—typically the Carrollian conformal Killing vector fields (standard
BMS) of section 3.3, or the generalised conformal vector fields (generalised
BMS) of section 3.4.

  * 2.

Build the corresponding, much larger, universal enveloping algebras
(“$\mathscr{U}$”) whose elements are suitable families of differential
operators of arbitrarily high order.

  * 3.

In each case, $\mathscr{U}$ admits an ideal (an annihilator
“$\mathscr{U}_{0}$”) related to the fact that some of its elements are trivial
in the corresponding realisation (e.g. all powers of the conformal Laplacian
in the case of the WRac). Mod out this annihilator to obtain higher-spin
algebras of the general form $\mathscr{U}/\mathscr{U}_{0}$. This quotient is
actually automatic for the Sachs module, but in that case one further
restricts attention to operators that are Hermitian with respect to the form
(51).

As announced in the introduction, this investigation is motivated by the
desire to define a Minkowskian analogue of the Rac in AdS. We shall focus on
the two structures encountered in section 2: the WRac and the Sachs module,
respectively studied in sections 4.1 and 4.2. Both will provide candidate
higher-spin extensions of BMS, and both have overlaps with the asymptotic
higher-spin symmetries of [37, 38, 39, 40], investigated in our language in
section 4.3. However, none of them will reproduce the “flat Rac” obtained in
[41] from a genuine Minkowskian limit of the Rac in AdS space. This suggests
that more work is required in order to understand the relation between scalar
kinematics at infinity, Carrollian geometry, and flat limits—an issue that we
will not address here.

Note that we will rely heavily on the conventions and Carrollian language of
section 3, as this will allow us to avoid local coordinates when studying
differential operators on Carrollian spacetimes. We will also use various
generic properties of differential operators that will not be reviewed at
length. (We refer e.g. to [108, 109] for detailed introductions.) In
particular, all differential operators are henceforth understood to act on
scalar fields $f\in C^{\infty}(\mathscr{M})$.222222Note that these fields are
allowed to be densities with non-zero weight, which will manifest itself in
the presence of zeroth-order factors $\propto\text{div}\,X$ in first-order
differential operators specified by a vector field $X$, as in the
transformation law (33). Homogeneous differential operators of orders zero and
one in the partial derivatives are, respectively, functions (acting by local
multiplication) and vector fields (acting by local derivation), while