#  Chern–Simons action and the Carrollian Cotton tensors

Olivera Mišković,1 Rodrigo Olea,2 P. Marios Petropoulos,3
David Rivera-Betancour3 and Konstantinos Siampos4 1 Instituto de Física
Pontificia Universidad Católica de Valparaíso
Universidad 330, Valparaíso, Chile 2 Departamento de Ciencias Físicas
Universidad Andres Bello
Sazié 2212, Piso 7, Santiago, Chile 3 Centre de Physique Théorique – CPHT
Ecole Polytechnique, CNRS111 _Centre National de la Recherche Scientifique_ ,
Unité Mixte de Recherche UMR 7644.
Institut Polytechnique de Paris
91128 Palaiseau Cedex, France 4 Department of Nuclear and Particle Physics
Faculty of Physics
National and Kapodistrian University of Athens
15784 Athens Greece

CPHT-RR048.072022

Abstract
In three-dimensional pseudo-Riemannian manifolds, the Cotton tensor arises as
the variation of the gravitational Chern–Simons action with respect to the
metric. It is Weyl-covariant, symmetric, traceless and covariantly conserved.
Performing a reduction of the Cotton tensor with respect to Carrollian
diffeomorphisms in a suitable frame, one discloses four sets of Cotton
Carrollian relatives, which are conformal and obey Carrollian conservation
equations. Each set of Carrollian Cotton tensors is alternatively obtained as
the variation of a distinct Carroll–Chern–Simons action with respect to the
degenerate metric and the clock form of a strong Carroll structure. The four
Carroll–Chern–Simons actions emerge in the Carrollian reduction of the
original Chern–Simons ascendant. They inherit its anomalous behaviour under
diffeomorphisms and Weyl transformations. The extremums of these Carrollian
actions are commented and illustrated.

###### Contents

  1. 1 Prologue
  2. 2 Carrollian Cotton: intrinsic features and zero-$\boldsymbol{c}$ limit
  3. 3 Carroll–Chern–Simons actions and transformation properties
  4. 4 In short

##  1 Prologue

The Cotton tensor is defined on Riemannian manifolds of arbitrary dimension,
carries three indices and is partly antisymmetric. In three dimensions, which
will be our framework, this tensor was introduced by Émile Cotton in 1899 [1]
and was formulated as a two-index symmetric tensor related to the previous by
Hodge duality:

$C_{\mu\nu}=\eta_{\mu}^{\hphantom{\mu}\rho\sigma}\nabla_{\rho}\left(R_{\nu\sigma}-\dfrac{R}{4}g_{\nu\sigma}\right).$
(1)

Here $\text{d}s^{2}=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}$ is the metric
with signature $(-++)$,
$\eta_{\mu\nu\sigma}=\mathchoice{{\hbox{$\displaystyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\epsilon_{\mu\nu\sigma}$
($\epsilon_{012}=1$), $\nabla_{\rho}$ the associated Levi–Civita connection
and $R_{\nu\sigma}$ are the components of the Ricci tensor with scalar $R$.
The combination of the latter objects inside the parentheses defines the
Schouten tensor in three dimensions.

The Cotton tensor is Weyl-covariant, and conserved as a consequence of the
first Bianchi identity and the absence of Weyl tensor

$\nabla_{\rho}C^{\rho}_{\hphantom{\rho}\nu}=0,$ (2)

irrespective of the dynamics on $g_{\mu\nu}$. In fact, the Cotton tensor
emerges as the “energy–momentum” tensor of the gravitational Chern–Simons
action:

$C^{\mu\nu}=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}}\frac{\delta
S_{\text{CS}}}{\delta g_{\mu\nu}}$ (3)

with

$S_{\text{CS}}=\frac{1}{2c}\int_{\mathscr{M}}\text{Tr}\left(\upomega\wedge\text{d}\upomega+\frac{2}{3}\upomega\wedge\upomega\wedge\upomega\right),$
(4)

where $\upomega$ is the Levi–Civita connection one-form. In this picture
$S_{\text{CS}}$ is a functional of the metric and of its derivatives.

There are numerous instances where the Cotton tensor is encountered and plays
a fine role in gravitational physics. In four-dimensional asymptotically anti-
de Sitter spacetimes, the Schouten tensor of the conformal boundary appears
explicitly at a subleading order in the Fefferman–Graham expansion of the bulk
metric, after the boundary metric and before the boundary energy–momentum
tensor. The boundary Cotton tensor itself arises as the leading term of the
Fefferman–Graham expansion of the bulk Weyl tensor. More explicitly, the
Schouten tensor appears as a gauge field associated with conformal boosts and
the Cotton tensor as the corresponding field strength in non-linearly realized
conformal group on the boundary of AdS4 (super)gravity [2]. Its presence
reveals that the boundary _is not_ conformally flat or equivalently that the
bulk is asymptotically _locally_ anti-de Sitter. Alongside, the Chern–Simons
action appears under specific circumstances as the leading-order effective
action of the boundary theory, and can serve alternatively for amending the
standard boundary conditions imposed in anti-de Sitter holography. These and
other interesting properties, such as the role of the Cotton tensor and the
occurence of the Chern–Simons action in gravitational electric–magnetic
duality, can be found in Refs. [3, 4, 5, 6, 7, 8, 9].

In an effort to design a bulk gauge that would be covariant with respect to
the conformal boundary, as the Fefferman–Graham gauge is, but at the same time
be regular for vanishing cosmological constant, as opposed to
Fefferman–Graham, a modified version of the Newman–Unti gauge was reached [10,
11, 12], inspired by fluid/gravity correspondence [13, 14]. In this gauge, the
Cotton tensor appears explicitly in the bulk metric, and its deeper role in
the spacetime reconstruction — also recognized in [15] — is more transparent.

The attempts for generalizing the gravitational holographic principle to
asymptotically flat spacetimes have abundantly fueled the interest for
Carrollian geometries [16, 17], namely for structures equipped with a
degenerate metric, as are null infinities. In this framework, one naturally
wonders how the Cotton tensor materializes within the various curvature
attributes, what sort of dynamics it conveys, and which role it plays in the
bulk reconstruction from boundary data — now defined at null infinity. Some of
these questions were accurately answered in the seminal work [18], exhibiting
some of the _Carrollian Cotton descendants_ , their dynamics inherited from
(2), as well as their occurence in the flat exegesis of the
modified/covariantized Newman–Unti gauge. Further properties have been more
recently elaborated in [19], in relation to the null-boundary manifestation of
Ehlers’ hidden Möbius group, or in defining towers of gravito-magnetic charges
exclusively from a Carrollian boundary perspective.

In the works cited earlier, the analysis of the Carrollian Cotton descendants
was circumscribed to Carrollian geometries with vanishing geometric shear, a
requirement imposed by bulk Ricci flatness. However, reaching the ultimate
radiation-flux-balance equations for asymptotically flat spacetimes in a
limiting procedure from anti-de Sitter requires to start with a non-zero
shear, as recently demonstrated in [20]. The purpose of the present note is to
present a comprehensive picture of the Carrollian Cotton tensors, while
providing at the same time the Carrollian descendants for the Chern–Simons
action, which are met in various facets of flat-asymptotic symmetries — see
e.g.[21, 22].

Our strategy can be summarized as follows: choose an adapted frame, expand in
powers of $c^{2}$ and read off the possible Carrollian dynamics (actions and
equations of motion). These usually appear in electric and magnetic, plus some
secondary occasionally non-dynamical versions. As for their Riemannian
ascendant, diffeomorphism or local Lorentz/Carroll/Weyl invariances call for a
cautious inspection involving boundary terms. This is part of our agenda.

##  2 Carrollian Cotton: intrinsic features and zero-$\boldsymbol{c}$ limit

Carroll basics are available in [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34]. The underlying geometries consist of a $d+1$-dimensional manifold
$\mathscr{M}=\mathbb{R}\times\mathscr{S}$ equipped with a degenerate metric
and its kernel vector field. We will be here restricted to $d=2$ and adopt a
metric of the form

$\text{d}\ell^{2}=a_{ij}(t,\mathbf{x})\text{d}x^{i}\text{d}x^{j},\quad
i,j\ldots\in\\{1,2\\}.$ (5)

The kernel of the metric is the _field of observers_ , here

$\upupsilon=\frac{1}{\Omega}\partial_{t}.$ (6)

The dual _clock form_ obeying $\upmu(\upupsilon)=-1$ reads

$\upmu=-\Omega\text{d}t+b_{i}\text{d}x^{i},$ (7)

and incorporates an _Ehresmann connection_ , which is the background gauge
field $\text{b}=b_{i}\text{d}x^{i}$. Notice that (5) is not the most general
degenerate metric, which could a priori have components along $\text{d}t$,
supplying $\partial_{i}$ components to the field of observers (this option is
sometimes chosen — see e.g. [20, 35]). Our choice allows for a natural
splitting of time and space coordinates, along the fibre and the base of the
Carrollian fibre bundle respectively, invariant under Carrollian
diffeomorphisms $t^{\prime}=t^{\prime}(t,\mathbf{x})$ and
$\mathbf{x}^{\prime}=\mathbf{x}^{\prime}(\mathbf{x})$. This remains
nonetheless compatible with general covariance. Restoring explicitly the
latter is possible without any conflict with the dynamics at work here, at the
expense of obfuscating the distinction of the Carrollian framework with
ordinary Riemannian situations.

The vector fields dual to the forms $\text{d}x^{i}$ are

$\hat{\partial}_{i}=\partial_{i}+\frac{b_{i}}{\Omega}\partial_{t}.$ (8)

They transform covariantly under Carrollian diffeomorphisms.222Defining the
Jacobians as $J(t,\mathbf{x})=\frac{\partial t^{\prime}}{\partial t}$,
$j_{i}(t,\mathbf{x})=\frac{\partial t^{\prime}}{\partial x^{i}}$,
$J^{i}_{j}(\mathbf{x})=\frac{\partial x^{i\prime}}{\partial x^{j}}$, the
transformations follow: $a^{ij\prime}=J^{i}_{k}J_{l}^{j}a^{kl}$,
$\Omega^{\prime}=\frac{\Omega}{J}$,
$b^{\prime}_{k}=\left(b_{i}+\frac{\Omega}{J}j_{i}\right)J^{-1i}_{\hphantom{-1}k}$
(connection-like transformation) and consequently
$\upupsilon^{\prime}=\upupsilon$, $\upmu^{\prime}=\upmu$.  More generally,
Carrollian tensors depend on time $t$ and space $\mathbf{x}$. The metric being
degenerate the spacetime indices cannot be lowered or raised. This
inconvenience can be handled by introducing a pseudo-inverse [25], but we take
instead advantage of the time-and-space splitting mentioned previously, and
consider tensors carrying only spatial indices $i,j,\ldots$ lowered and raised
with $a_{ij}$ and its inverse $a^{ij}$.333Working with an orthonormal Cartan
frame is yet another option adopted in [20]. In this perspective, the
Carrollian tensors carry again spatial indices solely, but are now organized
in representations of the $d$-dimensional orthogonal local group, subgroup of
the local Carroll group, and are raised or lowered with the identity. These
transform covariantly under Carrollian diffeomorphisms. The time index is
omitted and the corresponding object is a Carrollian scalar. Details can be
found, e.g., in the appendices of Ref. [19], of which a minimal selection will
be hosted in the present publication.

A strong Carroll structure comes with a metric-compatible and field-of-
observers-compatible connection, which is not unique due to the metric
degeneracy. Here, we use the connection inspired from an ascendant pseudo-
Riemannian spacetime equipped with Papapetrou–Randers metric

$\text{d}s^{2}=-c^{2}\left(\Omega\text{d}t-b_{i}\text{d}x^{i}\right)^{2}+a_{ij}\text{d}x^{i}\text{d}x^{j}.$
(9)

We should stress that in this metric the dependence with respect to the
velocity of light $c$ is explicit, leading thus to (5) at zero $c$. This
feature plays a pivotal role in our subsequent analysis, since the Carrollian
Cotton descendants, the corresponding Chern–Simons actions and the
conservation equations are reached respectively from Eqs. (3), (4) and (2) by
expanding in $c$. Such an expansion would receive extra contributions if every
function in (9) were itself provisioning various powers of $c$. This would not
alter the general form of the Carrollian equations, but the details would be
different.444Keeping $c$ explicit in the Papapetrou–Randers metric has been
the ruling pattern in Refs. [36, 37]. Alternative approaches for designing
descendant Carrollian dynamics can be found e.g. in [38, 39].

The relativistic Papapetrou–Randers metric (9) infers a convenient although
non-orthonormal Cartan mobile frame
$\left\\{\text{e}_{\hat{0}}=\frac{1}{c\Omega}\partial_{t},\text{e}_{\hat{\imath}}=\hat{\partial}_{i}\right\\}$
and coframe
$\left\\{\uptheta^{\hat{0}}=-c\upmu,\uptheta^{\hat{\imath}}=\text{d}x^{i}\right\\}$.
The hatted indices $\left\\{\hat{0},\hat{\imath}\right\\}$ are meant to
distinguish this frame from the coordinate coframe
$\left\\{\uptheta^{0}=\text{d}x^{0}=c\text{d}t,\uptheta^{i}=\text{d}x^{i}\right\\}$
and frame
$\left\\{\text{e}_{0}=\frac{1}{c}\partial_{t},\text{e}_{i}=\partial_{i}\right\\}$.
In order to avoid cluttering of symbols and comply with the conventions used,
e.g., in Ref. [19, 36, 37], we will keep the hat exclusively on the time
direction. For the Carrollian side, we will rather use
$\left\\{\text{e}_{\hat{t}}=\frac{1}{\Omega}\partial_{t},\text{e}_{\hat{\imath}}=\hat{\partial}_{i}\right\\}$
and
$\left\\{\uptheta^{\hat{t}}=-\upmu,\uptheta^{\hat{\imath}}=\text{d}x^{i}\right\\}$,
and ignore the hat on the spatial indices.

The relativistic (affine) connection one-form elements
$\upomega^{\mu}_{\hphantom{\mu}\nu}=\Gamma^{\mu}_{\rho\nu}\uptheta^{\rho}$
with $\Gamma^{\mu}_{\rho\nu}$ the Levi–Civita connection coefficients read:

$\displaystyle\upomega_{\hat{0}i}$ $\displaystyle=$ $\displaystyle
c\left(\varphi_{i}\upmu+\varpi_{ij}\text{d}x^{j}\right)-\frac{1}{c}\hat{\gamma}_{ij}\text{d}x^{j},$
(10) $\displaystyle\upomega_{ij}$ $\displaystyle=$
$\displaystyle\left(c^{2}\varpi_{ij}-\hat{\gamma}_{ij}\right)\upmu+a_{il}\hat{\gamma}_{jk}^{l}\text{d}x^{k}$
(11)

with $\upomega_{\hat{0}i}=-\upomega_{i\hat{0}}$. These expressions disclose
two Carrollian tensors, the vorticity and the acceleration

$\varpi_{ij}=\partial_{[i}b_{j]}+b_{[i}\varphi_{j]},\quad\varphi_{i}=\dfrac{1}{\Omega}\left(\partial_{t}b_{i}+\partial_{i}\Omega\right),$
(12)

revealed inside

$\text{d}\upmu=\upvarpi+\upvarphi\wedge\upmu=\varpi_{ij}\text{d}x^{i}\wedge\text{d}x^{j}+\varphi_{i}\text{d}x^{i}\wedge\upmu.$
(13)

We have also introduced the symmetric symbols

$\hat{\gamma}^{i}_{jk}=\dfrac{a^{il}}{2}\left(\hat{\partial}_{j}a_{lk}+\hat{\partial}_{k}a_{lj}-\hat{\partial}_{l}a_{jk}\right)$
(14)

dubbed Carroll–Levi–Civita connection coefficients, and the Carrollian tensors

$\hat{\gamma}_{ij}=\frac{1}{2\Omega}\partial_{t}a_{ij}=\xi_{ij}+\frac{1}{d}a_{ij}\theta,\quad\theta=\dfrac{1}{\Omega}\partial_{t}\ln\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}},$ (15)

which define the geometric Carrollian shear (traceless) $\xi_{ij}$ and the
Carrollian expansion $\theta$ ($\hat{\gamma}_{ij}$ is the extrinsic curvature
of the spatial section $\mathscr{S}$). All these enter the _Carrollian
connection_ adopted here:

$\hat{\upomega}^{\hat{t}}_{\hphantom{\hat{t}}\hat{t}}=\hat{\upomega}^{\hat{t}}_{\hphantom{\hat{t}}i}=\hat{\upomega}^{i}_{\hphantom{\hat{t}}\hat{t}}=0,\quad\hat{\upomega}^{i}_{\hphantom{i}j}=-\hat{\gamma}^{i}_{\hphantom{i}j}\upmu+\hat{\gamma}_{jk}^{i}\text{d}x^{k}$
(16)

(the hat signals this connection is Carrollian as opposed to the Riemannian
displayed in (10) and (11)).

It is important to stress that the Carrollian connection (16) has been
designed so as to _define a parallel transport that respects the time-and-
space splitting_ mentioned above, embracing distinct time and space Carrollian
covariant derivatives $\frac{1}{\Omega}\hat{D}_{t}$ and $\hat{\nabla}_{i}$.
Both are metric-compatible:
$\frac{1}{\Omega}\hat{D}_{t}a_{ij}=\hat{\nabla}_{k}a_{ij}=0$.

For $d=2$, the $\mathscr{S}$-Hodge duality is induced by
$\eta_{ij}=\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\epsilon_{ij}$. This
duality is involutive on Carrollian vectors as well as on two-index symmetric
and traceless Carrollian tensors:

$\ast\\!V_{i}=\eta^{l}_{\hphantom{l}i}V_{l},\quad\ast
W_{ij}=\eta^{l}_{\hphantom{l}i}W_{lj}.$ (17)

In particular555We use here the conventions of Ref. [18], namely
$\epsilon_{12}=-1$, convenient when using complex coordinates
$\\{\zeta,\bar{\zeta}\\}$, where
$\text{d}\ell^{2}=\frac{2}{P^{2}}\text{d}\zeta\text{d}\bar{\zeta}$ with
$P=P(t,\zeta,\bar{\zeta})$ a real function. In this case
$\eta_{\zeta\bar{\zeta}}=\nicefrac{{-\text{i}}}{{P^{2}}}$,
$\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower 0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}=\nicefrac{{\text{i}}}{{P^{2}}}$ and the
volume form reads
$\frac{1}{2}\eta_{ij}\text{d}x^{i}\wedge\text{d}x^{j}=\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}=\frac{\text{d}\zeta\wedge\text{d}\bar{\zeta}}{\text{i}P^{2}}$.
Notice that $\eta^{il}\eta_{jl}=\delta^{i}_{j}$ and $\eta^{ij}\eta_{ij}=2$ so
that
$\upmu\wedge\text{d}x^{i}\wedge\text{d}x^{j}=-\eta^{ij}\text{d}t\,\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}\Omega=\text{i}\,\eta^{ij}\text{d}t\frac{\text{d}\zeta\wedge\text{d}\bar{\zeta}}{P^{2}}\Omega$.

$\ast\\!\varpi=\frac{1}{2}\eta^{ij}\varpi_{ij}\Leftrightarrow\varpi_{ij}=\ast\varpi\eta_{ij}.$
(18)

Weyl transformations act as

$a_{ij}\to\frac{1}{\mathcal{B}^{2}}a_{ij},\quad
b_{i}\to\frac{1}{\mathcal{B}}b_{i},\quad\Omega\to\frac{1}{\mathcal{B}}\Omega$
(19)

with $\mathcal{B}=\mathcal{B}(t,\mathbf{x})$. The expansion $\theta$ and the
acceleration $\varphi_{i}$ transform as connections under these rescalings:

$\varphi_{i}\to\varphi_{i}-\hat{\partial}_{i}\ln\mathcal{B},\quad\theta\to\mathcal{B}\theta-\frac{d}{\Omega}\partial_{t}\mathcal{B}.$
(20)

They play the role of Weyl connections in time and space Carroll–Weyl
covariant metric-compatible derivatives.666On the one hand, the Carrollian
covariant derivatives act as ordinary derivatives on scalars, whereas on
Carrollian vectors $V^{i}$ we obtain:
$\frac{1}{\Omega}\hat{D}_{t}V^{i}=\frac{1}{\Omega}\partial_{t}V^{i}+\hat{\gamma}^{i}_{\hphantom{i}j}V^{j}$
and
$\hat{\nabla}_{j}V^{i}=\hat{\partial}_{j}V^{i}+\hat{\gamma}^{i}_{jk}V^{k}$. On
the other hand Weyl–Carroll covariant derivatives on weight-$w$ fields are as
follows:
$\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\Phi=\frac{1}{\Omega}\hat{D}_{t}\Phi+\frac{w}{d}\theta\Phi$,
$\frac{1}{\Omega}\hat{\mathscr{D}}_{t}V^{l}=\frac{1}{\Omega}\hat{D}_{t}V^{l}+\frac{w-1}{d}\theta
V^{l}$ (both of weight $w+1$), and
$\hat{\mathscr{D}}_{j}\Phi=\hat{\partial}_{j}\Phi+w\varphi_{j}\Phi$,
$\hat{\mathscr{D}}_{j}V^{l}=\hat{\nabla}_{j}V^{l}+(w-1)\varphi_{j}V^{l}+\varphi^{l}V_{j}-\delta^{l}_{j}V^{i}\varphi_{i}$
(with unaltered weights). More details on the latter are given in [19, 36,
37]. The weights of $\varpi_{ij}$ and $\xi_{ij}$ are $-1$; hence the
connection transforms as follows:

$\begin{cases}\upomega^{\hat{0}}_{\hphantom{\hat{O}}i}\to\frac{1}{\mathcal{B}}\left(\upomega^{\hat{0}}_{\hphantom{\hat{O}}i}+c\hat{\partial}_{i}\ln\mathcal{B}\,\upmu-\frac{1}{c}\frac{1}{\Omega}\partial_{t}\ln\mathcal{B}\,a_{ij}\text{d}x^{j}\right)\\\
\upomega^{i}_{\hphantom{i}j}\to\upomega^{i}_{\hphantom{i}j}-\delta^{i}_{j}\text{d}\ln\mathcal{B}+\eta^{i}_{\hphantom{i}j}\ast\\!\hat{\partial}_{k}\ln\mathcal{B}\,\text{d}x^{k},\end{cases}$
(21)

and similarly for the Carrollian connection:

$\hat{\upomega}^{i}_{\hphantom{i}j}\to\hat{\upomega}^{i}_{\hphantom{i}j}-\delta^{i}_{j}\text{d}\ln\mathcal{B}+\eta^{i}_{\hphantom{i}j}\ast\\!\hat{\partial}_{k}\ln\mathcal{B}\,\text{d}x^{k}.$
(22)

Curvature tensors are defined through the action of covariant-derivative
commutators on various fields. Considering for instance a Carrollian vector
field $V^{i}$, we obtain for $d=2$:777We set
$\hat{R}_{ijk}=\left(\hat{\nabla}_{i}+\varphi_{i}\right)\hat{\gamma}_{jk}-\left(\hat{\nabla}_{k}+\varphi_{k}\right)\hat{\gamma}_{ij}=-\hat{R}_{kji}$
and $\hat{R}_{ijkl}=\hat{K}\left(a_{ik}a_{jl}-a_{il}a_{jk}\right)$, handier
than $\hat{r}_{ijk}$ and $\hat{r}_{ijkl}$ previously introduced in Refs. [19,
36, 37].

$\displaystyle\left[\hat{\nabla}_{i},\frac{1}{\Omega}\hat{D}_{t}\right]V^{j}$
$\displaystyle=$
$\displaystyle\hat{R}^{j}_{\hphantom{j}ik}V^{k}+\hat{\gamma}_{i}^{\hphantom{i}k}\hat{\nabla}_{k}V^{j}-\varphi_{i}\frac{1}{\Omega}\hat{D}_{t}V^{j},$
(23) $\displaystyle\left[\hat{\nabla}_{k},\hat{\nabla}_{l}\right]V^{i}$
$\displaystyle=$
$\displaystyle\hat{R}^{i}_{\hphantom{i}jkl}V^{j}+\ast\varpi\eta_{kl}\frac{2}{\Omega}\hat{D}_{t}V^{i},$
(24)

from which we further define

$\hat{R}^{k}_{\hphantom{k}ikj}=\hat{R}_{ij}=\hat{K}a_{ij}.$ (25)

One reaches the same tensors by computing the Carrollian curvature two-form
starting from the connection (16):888The torsion and curvature two-forms are
$\mathcal{T}^{\mu}=\text{d}\uptheta^{\mu}+\upomega^{\mu}_{\hphantom{\mu}\nu}\wedge\uptheta^{\nu}=\frac{1}{2}T^{\mu}_{\hphantom{\mu}\nu\rho}\uptheta^{\nu}\wedge\uptheta^{\rho}$
and
$\mathcal{R}^{\mu}_{\hphantom{\mu}\nu}=\text{d}\upomega^{\mu}_{\hphantom{\mu}\nu}+\upomega^{\mu}_{\hphantom{\mu}\rho}\wedge\upomega^{\rho}_{\hphantom{\rho}\nu}=\frac{1}{2}R^{\mu}_{\hphantom{\mu}\nu\rho\lambda}\uptheta^{\rho}\wedge\uptheta^{\lambda}$.

$\hat{\mathcal{R}}^{\hat{t}}_{\hphantom{\hat{t}}j}=0,\quad\hat{\mathcal{R}}^{i}_{\hphantom{i}j}=\hat{R}^{i}_{\hphantom{i}kj}\upmu\wedge\text{d}x^{k}+\frac{1}{2}\hat{R}^{i}_{\hphantom{i}jkl}\text{d}x^{k}\wedge\text{d}x^{l}.$
(26)

It should be emphasized that the last terms in the commutators (23), (24) (and
(28), (29) below) betray the presence of torsion in the Carroll (or the
Carroll–Weyl) connection adopted here. This torsion is encoded in the tensors
$\varphi_{i}$, $\varpi_{ij}$, $\theta$ and $\xi_{ij}$:

$\hat{\mathcal{T}}^{\hat{t}}=\varphi_{i}\upmu\wedge\text{d}x^{i}-\ast\varpi\eta_{ij}\text{d}x^{i}\wedge\text{d}x^{j},\quad\hat{\mathcal{T}}^{i}=\hat{\gamma}^{i}_{\hphantom{i}j}\text{d}x^{j}\wedge\upmu.$
(27)

Introducing torsion is the price to pay for maintaining metric compatibility,
while ensuring the time-and-space splitting and a non-trivial interplay
between the base and the fibre. As a bonus, the field of observers is
parallelly transported, $\hat{\nabla}_{\upupsilon}\upupsilon=0$, and the time
fibres are geodesics for the Carrollian manifold
$\mathscr{M}=\mathbb{R}\times\mathscr{S}$.

Likewise, we obtain the curvature tensors for the Carroll-Weyl connection:

$\displaystyle\left[\hat{\mathscr{D}}_{i},\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\right]V^{j}$
$\displaystyle=$
$\displaystyle\hat{\mathscr{S}}^{j}_{\hphantom{j}ik}V^{k}-(w-1)\hat{\mathscr{R}}_{i}V^{j}+\xi_{i}^{\hphantom{i}k}\hat{\mathscr{D}}_{k}V^{j},$
(28)
$\displaystyle\left[\hat{\mathscr{D}}_{k},\hat{\mathscr{D}}_{l}\right]V^{i}$
$\displaystyle=$
$\displaystyle\hat{\mathscr{S}}^{i}_{\hphantom{i}jkl}V^{j}-(w-1)\hat{\mathscr{A}}\eta_{kl}V^{i}+\ast\varpi\eta_{kl}\frac{2}{\Omega}\hat{\mathscr{D}}_{t}V^{i},$
(29)

where $w$ stands for the weight of the vector field $V^{j}$. Furthermore,
setting

$\hat{\mathscr{R}}_{i}=\frac{1}{\Omega}\partial_{t}\varphi_{i}-\frac{1}{2}\left(\hat{\partial}_{i}+\varphi_{i}\right)\theta,\quad\hat{\mathscr{A}}=\ast\varpi\theta-\eta^{ij}\hat{\nabla}_{i}\varphi_{j}$
(30)

defines unambiguously the Carroll–Weyl curvature tensors
$\hat{\mathscr{S}}^{j}_{\hphantom{j}ik}$ and
$\hat{\mathscr{S}}^{i}_{\hphantom{i}jkl}$.999Again those are slightly
different from $\hat{\mathscr{R}}^{j}_{\hphantom{j}ik}$ and
$\hat{\mathscr{R}}^{i}_{\hphantom{i}jkl}$ used earlier in [19, 36, 37]:
$\hat{\mathscr{S}}^{j}_{\hphantom{j}ik}=\hat{\mathscr{D}}^{j}\xi_{ik}-\hat{\mathscr{D}}_{k}\xi^{j}_{\hphantom{j}i}+\delta^{j}_{i}\hat{\mathscr{R}}_{k}-a_{ik}\hat{\mathscr{R}}^{j}$
and
$\hat{\mathscr{S}}^{i}_{\hphantom{i}jkl}=\hat{\mathscr{K}}\left(a_{ik}a_{jl}-a_{il}a_{jk}\right)$.
Tracing finally we obtain

$\hat{\mathscr{S}}^{k}_{\hphantom{k}ikj}=\hat{\mathscr{S}}_{ij}=\hat{\mathscr{K}}a_{ij},\quad\hat{\mathscr{K}}=\hat{K}+\hat{\nabla}_{k}\varphi^{k}.$
(31)

In summary, the independent Carroll–Weyl curvature tensors in $d=2$ are
$\hat{\mathscr{K}}$, $\hat{\mathscr{A}}$ and $\hat{\mathscr{R}}_{i}$ of
weights $2$, $2$ and $1$. They obey

$\displaystyle\frac{2}{\Omega}\hat{\mathscr{D}}_{t}\ast\\!\varpi+\hat{\mathscr{A}}$
$\displaystyle=$ $\displaystyle 0,$ (32)
$\displaystyle\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\hat{\mathscr{K}}-a^{ij}\hat{\mathscr{D}}_{i}\hat{\mathscr{R}}_{j}-\hat{\mathscr{D}}_{i}\hat{\mathscr{D}}_{j}\xi^{ij}$
$\displaystyle=$ $\displaystyle 0,$ (33)
$\displaystyle\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\hat{\mathscr{A}}+\eta^{ij}\hat{\mathscr{D}}_{i}\hat{\mathscr{R}}_{j}$
$\displaystyle=$ $\displaystyle 0,$ (34)

the last two being the Carroll–Weyl–Bianchi identities.

Riemann, Ricci and Carrollian descendants of the Riemannian metric (9) are
yield using the prescription already exploited in Refs. [18, 19] — or [36, 37]
for general energy–momentum tensors. In a first step, this consists in
reducing the relativistic tensors with respect to Carrollian diffeomorphisms.
Next, one expands the latter in powers of $c$, and at each power bona-fide
Carrollian tensors emerge. Following this prescription for the Riemann
curvature two-form of (9) we find

$\displaystyle\mathcal{R}^{\hat{0}}_{\hphantom{\hat{0}}i}$ $\displaystyle=$
$\displaystyle
c\left[c^{2}\ast\\!\varpi^{2}a_{ik}+\hat{\nabla}_{(i}\varphi_{k)}+\varphi_{i}\varphi_{k}-2\ast\\!\varpi\eta_{j(i}\hat{\gamma}_{k)}^{\hphantom{k)}j}-\frac{1}{c^{2}}\left(\frac{1}{\Omega}\hat{D}_{t}\hat{\gamma}_{ik}+\hat{\gamma}_{ij}\hat{\gamma}^{j}_{\hphantom{j}k}\right)\right]\upmu\wedge\text{d}x^{k}$
(35)
$\displaystyle-c\left[\hat{\partial}_{i}\ast\\!\varpi+2\varphi_{i}\ast\\!\varpi-\frac{1}{c^{2}}\eta^{mn}\hat{\nabla}_{m}\hat{\gamma}_{ni}\right]\frac{1}{2}\eta_{kl}\text{d}x^{k}\wedge\text{d}x^{l},$
$\displaystyle\mathcal{R}^{i}_{\hphantom{i}j}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{R}}^{i}_{\hphantom{i}j}-\eta^{i}_{\hphantom{i}j}\left[c^{2}\left(\hat{\partial}_{k}\ast\\!\varpi+2\varphi_{k}\ast\\!\varpi\right)+\eta^{mn}\varphi_{m}\hat{\gamma}_{nk}\right]\upmu\wedge\text{d}x^{k}$
(36)
$\displaystyle+\eta^{i}_{\hphantom{i}j}\left[3c^{2}\ast\\!\varpi^{2}+\frac{1}{2c^{2}}\eta^{mn}\eta^{rs}\hat{\gamma}_{mr}\hat{\gamma}_{ns}\right]\frac{1}{2}\eta_{kl}\text{d}x^{k}\wedge\text{d}x^{l},$

where various Carrollian tensors emerge besides the Carrollian curvature two-
form read off in (26). No torsion is available for the relativistic
Levi–Civita connection at hand.

The Riemannian scalar curvature $R$ reads:101010For arbitrary $d$ we find:
$R=c^{2}\varpi^{ij}\varpi_{ij}+\hat{R}-2\left(\hat{\nabla}_{k}+\varphi_{k}\right)\varphi^{k}+\frac{1}{c^{2}}\left(\xi^{ij}\xi_{ij}+\frac{d+1}{d}\theta^{2}+\frac{2}{\Omega}\partial_{t}\theta\right)$.

$\frac{R}{2}=c^{2}\ast\\!\varpi^{2}+\hat{K}-\left(\hat{\nabla}_{k}+\varphi_{k}\right)\varphi^{k}+\frac{1}{c^{2}}\left(\xi^{2}+\frac{3}{4}\theta^{2}+\frac{1}{\Omega}\partial_{t}\theta\right),$
(37)

where $\xi^{2}=\frac{1}{2}\xi^{ij}\xi_{ij}$. It captures various Carrollian
curvature scalars, which infer Carrollian avatars of the Einstein–Hilbert
action
$\int_{\mathscr{M}}\frac{\text{d}x^{3}}{c}\mathchoice{{\hbox{$\displaystyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}R$. We will not
elaborate on this aspect of Carrollian gravitational dynamics that would
deserve a thorough comparison with Refs. [38, 39].

Cotton Carrollian relatives are reached following the above pattern. The
reduction of $C^{\mu\nu}$ is straightforward: Carrollian scalars and vectors
emerge from $C^{\hat{0}\hat{0}}$ and $C^{\hat{0}i}$, while
$C^{ij}-\frac{C^{\hat{0}\hat{0}}}{2}a^{ij}$ leads to symmetric and traceless
Carrollian tensors. They are readily decomposed in powers of $c$ as follows:

$\displaystyle\frac{1}{c}C^{\hat{0}\hat{0}}$ $\displaystyle=$ $\displaystyle
c^{2}\gamma+\varepsilon+\frac{\zeta}{c^{2}}+\frac{\tau}{c^{4}},$ (38)
$\displaystyle C^{\hat{0}i}$ $\displaystyle=$ $\displaystyle
c^{2}\psi^{i}+\chi^{i}+\frac{z^{i}}{c^{2}},$ (39)
$\displaystyle\frac{C^{\hat{0}\hat{0}}a^{ij}}{2c}-\frac{C^{ij}}{c}$
$\displaystyle=$
$\displaystyle\Psi^{ij}+\frac{X^{ij}}{c^{2}}+\frac{Z^{ij}}{c^{4}}.$ (40)

With this, any Carrollian structure supplied with the connection at hand, is
naturally endowed with ten Weyl-covariant Carrollian Cotton descendants. These
are

  * •

four weight-$3$ scalars:

$\gamma=8\ast\\!\varpi^{3},\quad\varepsilon=\left(\hat{\mathscr{D}}_{l}\hat{\mathscr{D}}^{l}+2\hat{\mathscr{K}}\right)\ast\\!\varpi,\quad\zeta=\hat{\mathscr{D}}_{i}\hat{\mathscr{D}}_{j}\ast\\!\xi^{ij},\quad\tau=\ast\xi_{ij}\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\xi^{ij};$
(41)

  * •

three weight-$2$ forms:

$\displaystyle\psi_{i}$ $\displaystyle=$ $\displaystyle
3\eta_{ji}\hat{\mathscr{D}}^{j}\ast\\!\varpi^{2},$ (42)
$\displaystyle\chi_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta_{ji}\hat{\mathscr{D}}^{j}\hat{\mathscr{K}}+\frac{1}{2}\hat{\mathscr{D}}_{i}\hat{\mathscr{A}}-2\ast\\!\varpi\left(\hat{\mathscr{R}}_{i}+2\hat{\mathscr{D}}^{j}\xi_{ij}\right)+3\hat{\mathscr{D}}^{j}\left(\ast\varpi\xi_{ij}\right),$
(43) $\displaystyle z_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta_{ij}\hat{\mathscr{D}}^{j}\xi^{2}-\hat{\mathscr{D}}^{j}\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\ast\\!\xi_{ij}-\ast\xi_{ij}\hat{\mathscr{D}}_{k}\xi^{jk};$
(44)

  * •

three weight-$1$ traceless and symmetric two-index covariant tensors:

$\displaystyle\Psi_{ij}$ $\displaystyle=$
$\displaystyle-2\ast\\!\varpi^{2}\ast\\!\xi_{ij}+\hat{\mathscr{D}}_{i}\hat{\mathscr{D}}_{j}\ast\\!\varpi-\frac{1}{2}a_{ij}\hat{\mathscr{D}}^{k}\hat{\mathscr{D}}_{k}\ast\\!\varpi-\eta_{ij}\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\ast\\!\varpi^{2},$
(45) $\displaystyle X_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta_{ki}\hat{\mathscr{D}}^{k}\left(\hat{\mathscr{R}}_{j}+\hat{\mathscr{D}}^{l}\xi_{jl}\right)+\frac{1}{2}\eta_{kj}\hat{\mathscr{D}}_{i}\left(\hat{\mathscr{R}}^{k}+\hat{\mathscr{D}}_{l}\xi^{kl}\right)$
(46)
$\displaystyle-\frac{3}{2}\hat{\mathscr{A}}\xi_{ij}-\hat{\mathscr{K}}\ast\\!\xi_{ij}+3\frac{\ast\varpi}{\Omega}\hat{\mathscr{D}}_{t}\xi_{ij},$
$\displaystyle Z_{ij}$ $\displaystyle=$ $\displaystyle
2\ast\\!\xi_{ij}\xi^{2}-\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\ast\\!\xi_{ij}.$
(47)

As for the conservation equation (2), it supplies the following Carrollian
decompositions:

$\nabla_{\rho}C^{\rho}_{\hphantom{\rho}\hat{0}}=c^{2}\mathcal{D}_{\text{Cot}}+\mathcal{E}_{\text{Cot}}+\frac{\mathcal{F}_{\text{Cot}}}{c^{2}}+\frac{\mathcal{W}_{\text{Cot}}}{c^{4}}=0,$
(48)

and

$\frac{1}{c}\nabla_{\rho}C^{\rho
i}=c^{2}\mathcal{I}_{\text{Cot}}^{i}+\mathcal{G}_{\text{Cot}}^{i}+\frac{\mathcal{H}_{\text{Cot}}^{i}}{c^{2}}+\frac{\mathcal{X}_{\text{Cot}}^{i}}{c^{4}}=0.$
(49)

All identities are Weyl-covariant with

$\displaystyle\mathcal{D}_{\text{Cot}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\gamma-\hat{\mathscr{D}}_{i}\psi^{i},$
(50) $\displaystyle\mathcal{E}_{\text{Cot}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\varepsilon-\hat{\mathscr{D}}_{i}\chi^{i}+\Psi_{ij}\xi^{ij},$
(51) $\displaystyle\mathcal{F}_{\text{Cot}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\zeta-\hat{\mathscr{D}}_{i}z^{i}+X_{ij}\xi^{ij},$
(52) $\displaystyle\mathcal{W}_{\text{Cot}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\tau+Z_{ij}\xi^{ij},$ (53)

and

$\displaystyle\mathcal{I}_{\text{Cot}}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hat{\mathscr{D}}^{i}\gamma+2\ast\\!\varpi\ast\\!\psi^{i},$
(54) $\displaystyle\mathcal{G}_{\text{Cot}}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hat{\mathscr{D}}^{i}\varepsilon-\hat{\mathscr{D}}_{j}\Psi^{ij}+2\ast\\!\varpi\ast\\!\chi^{i}+\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\psi^{i}+\psi_{j}\xi^{ij},$
(55) $\displaystyle\mathcal{H}_{\text{Cot}}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hat{\mathscr{D}}^{i}\zeta-\hat{\mathscr{D}}_{j}X^{ij}+2\ast\\!\varpi\ast\\!z^{i}+\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\chi^{i}+\chi_{j}\xi^{ij},$
(56) $\displaystyle\mathcal{X}_{\text{Cot}}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hat{\mathscr{D}}^{i}\tau-\hat{\mathscr{D}}_{j}Z^{ij}+\frac{1}{\Omega}\hat{\mathscr{D}}_{t}z^{i}+z_{j}\xi^{ij}.$
(57)

Interpreting the Cotton Carrollian descendants is possible along the same
lines as for the ordinary Riemannian Cotton tensor. The main differences are
that a Carrollian geometry has a fibre-bundle structure and a wider freedom
for its affine connection. This blurs to some extent the concept of conformal
flatness, which is the feature emerging when the Cotton vanishes in three-
dimensional Riemannian manifolds, and more options emerge.

Vanishing geometric shear

From Eq. (15), when $\xi_{ij}=0$ the time dependence in the metric $a_{ij}$ is
factorized:
$a_{ij}(t,\mathbf{x})=\text{e}^{2\sigma(t,\mathbf{x})}\bar{a}_{ij}(\mathbf{x})$.
Moreover, in two dimensions $\bar{a}_{ij}(\mathbf{x})$ is necessarily
proportional to $\delta_{ij}$, hence choosing complex coordinates, the metric
on the two-dimensional surface $\mathscr{S}$ is recast as

$\text{d}\ell^{2}=\frac{2}{P^{2}}\text{d}\zeta\text{d}\bar{\zeta}$ (58)

with $P=P(t,\zeta,\bar{\zeta})$ a real function. Consequently, a subset of the
Carroll–Cotton tensors vanish, as it is inferred from Eqs. (41), (44), (47):
$\zeta$, $\tau$, $z_{i}$ and $Z_{ij}$.

Vanishing Carrollian vorticity

In the Eq.(13), for $\ast\varpi=0$ we find :

$\text{d}\upmu=\upvarphi\wedge\upmu.$ (59)

Using Fröbenius criterion we are instructed that $\upmu$ is proportional to an
exact form. We can choose appropriately the time coordinate so that the
Ehresmann connection b vanishes, leading to

$\upmu=-\Omega(t,\mathbf{x})\text{d}t.$ (60)

The vanishing Carrollian Cotton tensors are now $\gamma$, $\varepsilon$,
$\psi_{i}$ and $\Psi_{ij}$ (see (41), (42), (45)).

Vanishing Carrollian shear and vorticity

This merges the two previous situations and the Carrollian structure is of the
form

$\text{d}\ell^{2}=\frac{2}{P(t,\zeta,\bar{\zeta})^{2}}\text{d}\zeta\text{d}\bar{\zeta},\quad\upmu=-\Omega(t,\zeta,\bar{\zeta})\text{d}t.$
(61)

Despite the factorization of the metric and of the clock form, not all
Carroll–Cotton tensors are zero. We find indeed from (30), (31),

$\hat{\mathscr{R}}_{\zeta}=\frac{1}{\Omega}\partial_{t}\partial_{\zeta}\ln(\Omega
P),\quad\hat{\mathscr{A}}=0,\quad\hat{\mathscr{K}}=2P^{2}\partial_{\zeta}\partial_{\bar{\zeta}}\ln(\Omega
P)$ (62)

($\hat{\mathscr{R}}_{\bar{\zeta}}$ is the complex conjugate of
$\hat{\mathscr{R}}_{\zeta}$), and using (43), (46) we obtain for the
Carroll–Cotton:

$\begin{cases}\chi_{\zeta}=\frac{\text{i}}{2}\hat{\mathscr{D}}_{\zeta}\hat{\mathscr{K}}=\frac{\text{i}}{\Omega^{2}}\partial_{\zeta}\left((\Omega
P)^{2}\partial_{\zeta}\partial_{\bar{\zeta}}\ln(\Omega P)\right)\\\
X_{\zeta\zeta}=\text{i}\hat{\mathscr{D}}_{\zeta}\hat{\mathscr{R}}_{\zeta}=\frac{\text{i}}{\Omega}\frac{1}{(\Omega
P)^{2}}\partial_{\zeta}\left((\Omega
P)^{2}\partial_{t}\partial_{\zeta}\ln(\Omega P)\right),\quad
X_{\zeta\bar{\zeta}}=0\end{cases}$ (63)

with $\chi_{\bar{\zeta}}=\bar{\chi}_{\zeta}$ and
$X_{\bar{\zeta}\bar{\zeta}}=\bar{X}_{\zeta\zeta}$. The tensors in (63) vanish
if $\Omega P$ is constant. This result could have been anticipated by noticing
that the Papapetrou–Randers (see Eq. (9)) pseudo-Riemannian ascendant of (61)
is conformally flat provided $\Omega P$ be constant.

Concluding, in Carrollian geometry, “conformal flatness” concerns separately
the base and the fibre of the bundle $\mathscr{M}=\mathbb{R}\times\mathscr{S}$
with distinct vanishing Carroll–Cotton tensors.

##  3 Carroll–Chern–Simons actions and transformation properties

General-covariant actions
$S=\frac{1}{c}\int_{\mathscr{M}}\text{d}^{d+1}x\mathchoice{{\hbox{$\displaystyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\mathcal{L}$ on
Riemannian spacetimes $\mathscr{M}$ lead to covariantly conserved
energy–momentum tensors
$T^{\mu\nu}=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule
height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{-g\,}$}\lower
0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}}\frac{\delta S}{\delta
g_{\mu\nu}}$. When the action is furthermore Weyl-invariant, $T^{\mu\nu}$ is
Weyl-covariant of weight $d+3$ with $T^{\mu}_{\hphantom{\mu}\mu}=0$ and
$\mathscr{D}_{\mu}T^{\mu\nu}=\nabla_{\mu}T^{\mu\nu}=0$.

The Chern–Simons action (4) _is not_ invariant under frame transformations.
The latter may be induced by diffeomorphisms in coordinate frames or be more
general (e.g., local Lorentz transformations on orthonormal frames, when
applicable):

$\uptheta\to\uptheta^{\prime}=\Uplambda\uptheta,$ (64)

where $\uptheta$ is a column matrix encoding all $\uptheta^{\mu}$s and
$\Uplambda$ a square matrix with entries $\Lambda^{\mu}_{\hphantom{\mu}\nu}$.
With this, genuine tensors are invariant — their components transform with
$\Lambda^{\mu}_{\hphantom{\mu}\nu}$ and
$\Lambda^{-1\mu}_{\hphantom{-1\mu}\nu}$ though. The torsion (if any) and
curvature two-forms transform homogeneously

$\mathcal{T}^{\prime}=\Uplambda\mathcal{T},\quad\mathcal{R}^{\prime}=\Uplambda\mathcal{R}\Uplambda^{-1},$
(65)

whereas the connection one-form acquires an extra non-homogeneous
piece:111111In terms of connection coefficients the transformation reads:
$\Gamma^{\rho\prime}_{\mu\nu}=\Lambda^{-1\alpha}_{\hphantom{-1\alpha}\mu}\Lambda^{-1\beta}_{\hphantom{-1\beta}\nu}\Lambda^{\rho}_{\hphantom{\rho}\gamma}\Gamma^{\gamma}_{\alpha\beta}-\Lambda^{-1\alpha}_{\hphantom{-1\alpha}\mu}\text{e}_{\alpha}\left(\Lambda^{\rho}_{\hphantom{\rho}\gamma}\right)\Lambda^{-1\gamma}_{\hphantom{-1\gamma}\nu}$.
For coordinate-frame transformations induced by diffeomorphisms, $\Uplambda$
is the Jacobian matrix J with entries $\frac{\partial x^{\mu\prime}}{\partial
x^{\nu}}$.

$\upomega^{\prime}=\Uplambda\left(\upomega-\text{X}\right)\Uplambda^{-1},\quad\text{X}=\Uplambda^{-1}\text{d}\Uplambda.$
(66)

Using

$\text{d}\upomega^{\prime}=\Uplambda\left(\text{d}\upomega+\text{X}\wedge\upomega+\upomega\wedge\text{X}-\text{X}\wedge\text{X}\right)\Uplambda^{-1}$
(67)

we find generally

$\text{Tr}\left(\upomega^{\prime}\wedge\text{d}\upomega^{\prime}+\frac{2}{3}\upomega^{\prime}\wedge\upomega^{\prime}\wedge\upomega^{\prime}\right)=\text{Tr}\left(\upomega\wedge\text{d}\upomega+\frac{2}{3}\upomega\wedge\upomega\wedge\upomega\right)+\text{d}\text{Tr}\left(\text{X}\wedge\upomega\right)-\frac{1}{3}\text{Tr}\left(\text{X}\wedge\text{d}\text{X}\right).$
(68)

Let us focus on infinitesimal transformations of the form

$\Uplambda=\mathbb{1}+\Upxi\quad\Rightarrow\quad\text{X}=\text{d}\Upxi.$ (69)

Under the latter, the Chern–Simons action (4) transforms as

$\delta_{\Upxi}S_{\text{CS}}=\frac{1}{2c}\oint_{\partial\mathscr{M}}\text{Tr}\left(\text{d}\Upxi\wedge\upomega\right)=-\frac{1}{2c}\oint_{\partial\mathscr{M}}\text{Tr}\left(\Upxi\text{d}\upomega\right),$
(70)

where we have assumed that $\partial\partial\mathscr{M}=\emptyset$. The last
expression can be alternatively expressed in terms of the curvature
$\mathcal{R}=\text{d}\upomega+\upomega\wedge\upomega$, and provides the
anomaly, be it gravitational, Lorentz or mixed, depending on the frame and the
transformation performed.

In view of the Carrollian applications, we would like to elaborate on the
behaviour (70) in Papapetrou–Randers coframe
$\left\\{\uptheta^{\hat{0}}=-c\upmu,\text{d}x^{i}\right\\}$, for Carrollian
diffeomorphisms mapping $\left\\{t,\mathbf{x}\right\\}$ onto
$\left\\{t^{\prime}(t,\mathbf{x}),\mathbf{x}^{\prime}(\mathbf{x})\right\\}$.
This guarantees the form stability of the metric (9) and of the time-and-space
splitting advertised earlier. Under these transformations we find — see
footnote 2:

$\Lambda^{\hat{0}}_{\hphantom{\hat{0}}\hat{0}}=1,\quad\Lambda^{\hat{0}}_{\hphantom{\hat{0}}j}=0,\quad\Lambda^{i}_{\hphantom{i}\hat{0}}=0,\quad\Lambda^{i}_{\hphantom{i}j}=J^{i}_{\hphantom{i}j}=\frac{\partial
x^{i\prime}}{\partial x^{j}}.$ (71)

Carrollian diffeomorphisms are generated by vector fields
$\upxi=\xi^{\hat{0}}(t,\mathbf{x})\frac{1}{c\Omega}\partial_{t}+\xi^{i}(\mathbf{x})\hat{\partial}_{i}$
so that

$\Xi^{\hat{0}}_{\hphantom{\hat{0}}\hat{0}}=0,\quad\Xi^{\hat{0}}_{\hphantom{\hat{0}}j}=0,\quad\Xi^{i}_{\hphantom{i}\hat{0}}=0,\quad\Xi^{i}_{\hphantom{i}j}=\partial_{j}\xi^{i}.$
(72)

With this, in the present framework, the Chern–Simons variation (70) reads:

$\delta_{\upxi}S_{\text{CS}}=-\frac{1}{2c}\oint_{\partial\mathscr{M}}\partial_{j}\xi^{i}\text{d}\upomega^{j}_{\hphantom{j}i}.$
(73)

Assuming a Levi–Civita connection (10), (11), and using Eq. (16), Eq. (73) is
recast as:

$\delta_{\upxi}S_{\text{CS}}=-\frac{c}{2}\oint_{\partial\mathscr{M}}\partial_{j}\xi^{i}\text{d}\left(\varpi^{j}_{\hphantom{j}i}\upmu\right)-\frac{1}{2c}\oint_{\partial\mathscr{M}}\partial_{j}\xi^{i}\text{d}\hat{\upomega}^{j}_{\hphantom{j}i}.$
(74)

Being topological, the anomalous contributions (74) neither compromise the
covariant conservation of the Cotton tensor nor spoil its conformal
properties. Regarding this last statement, one should notice that Weyl
rescalings are also anomalous. In the Papapetrou–Randers frame, using the
Levi–Civita connection (21) for infinitesimal transformations
$\mathcal{B}=1+\lambda$, we find:

$\delta_{\lambda}S_{\text{CS}}=\frac{1}{c}\oint_{\partial\mathscr{M}}\text{d}x^{i}\wedge\left(\frac{1}{2}\text{d}a_{ij}a^{jk}\hat{\partial}_{k}\lambda+\left(\varpi_{ij}\text{d}x^{j}+\varphi_{i}\upmu\right)\frac{1}{\Omega}\partial_{t}\lambda\right)\,.$
(75)

A more detailed analysis of the associated boundary dynamics should shed some
light on the Weyl anomaly, on a similar fashion as in[40].

Consider now a dynamical system on a Carrollian manifold
$\mathscr{M}=\mathbb{R}\times\mathscr{S}$ described with an action
$S=\int_{\mathscr{M}}\text{d}t\,\text{d}^{d}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\Omega\mathcal{L}$,
functional of $a_{ij}$, $\Omega$ and $b_{i}$. The associated _Carrollian
momenta_ , which replace the corresponding relativistic energy–momentum tensor
$T^{\mu\nu}$ are now (see [41, 42])

$\begin{cases}\Pi^{ij}=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}\Omega}\frac{\delta
S}{\delta a_{ij}}\\\
\Pi^{i}=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}\Omega}\frac{\delta
S}{\delta b_{i}}\\\
\Pi=-\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}}\left(\frac{\delta
S}{\delta\Omega}+\frac{b_{i}}{\Omega}\frac{\delta S}{\delta
b_{i}}\right).\end{cases}$ (76)

These are the _energy–stress tensor_ , the _energy current_ and the _energy
density_. Invariance under Carrollian diffeomorphisms impose two conservation
equations [37], respectively in time and space directions

$\displaystyle\left(\frac{1}{\Omega}\partial_{t}+\theta\right)\Pi+\left(\hat{\nabla}_{i}+2\varphi_{i}\right)\Pi^{i}+\Pi^{ij}\hat{\gamma}_{ij}$
$\displaystyle=$ $\displaystyle 0,$ (77)
$\displaystyle\left(\hat{\nabla}_{j}+\varphi_{j}\right)\Pi^{j}_{\hphantom{j}i}+2\Pi^{j}\varpi_{ji}+\Pi\varphi_{i}+\left(\frac{1}{\Omega}\partial_{t}+\theta\right)P_{i}$
$\displaystyle=$ $\displaystyle 0,$ (78)

involving a fourth Carrollian momentum dubbed _momentum current_ , $P_{i}$. In
the present formalism based on the Carrollian data (5) and (7) the latter is
not defined directly through a variation of the action with respect to some
conjugate variable. It is however inevitable, and this can be verified
whenever a microscopic action is available in terms of fundamental fields as,
e.g., in [43].121212In fact, one could chose a different description for the
degenerate metric $\text{d}\ell^{2}$, where $\text{d}x^{i}$ are traded for
$\text{d}x^{i}-w^{i}\text{d}t$. The Carrollian momenta conjugate to the
variables $w^{i}$ would then be $P_{i}$ — see e.g. [44].

When the Carrollian action is Weyl-invariant $\Pi^{ij}$, $\Pi^{i}$, $P^{i}$
and $\Pi$ are Weyl-covariant with weights $d+3$, $d+2$, $d+2$ and $d+1$, and
obey the trace condition:

$\Pi^{ij}=\Upsilon^{ij}+\frac{\Pi}{d}a^{ij},\quad\Upsilon^{i}_{\hphantom{i}i}=0.$
(79)

Equations (77) and (78) are thus traded for

$\displaystyle\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\Pi+\hat{\mathscr{D}}_{i}\Pi^{i}+\Upsilon^{ij}\xi_{ij}$
$\displaystyle=$ $\displaystyle 0,$ (80)
$\displaystyle\frac{1}{d}\hat{\mathscr{D}}_{j}\Pi+\hat{\mathscr{D}}_{i}\Upsilon^{i}_{\hphantom{i}j}+2\Pi^{i}\varpi_{ij}+\left(\frac{1}{\Omega}\hat{\mathscr{D}}_{t}\delta^{i}_{j}+\xi^{i}_{\hphantom{i}j}\right)P_{i}$
$\displaystyle=$ $\displaystyle 0.$ (81)

Expanding the Chern–Simons action (4) in powers of $c$ in the
Papapetrou–Randers background (9) equipped with Levi–Civita connection (10)
and (11), delivers four distinct Carrollian avatars of the Chern–Simons
dynamics,

$S_{\text{CS}}=c^{3}S_{\text{CCS}}^{\text{pm}}+cS_{\text{CCS}}^{\text{m}}+\frac{1}{c}S_{\text{CCS}}^{\text{e}}+\frac{1}{c^{3}}S_{\text{CCS}}^{\text{pe}},$
(82)

possessing four sets of Weyl-covariant Carrollian momenta of the type (76),
(79), obeying four sets of conformal Carrollian conservation equations (80),
(81). As we have already anticipated, these Carroll–Chern–Simons actions are
in general anomalous under Carrollian diffeomorphisms and Weyl transformations
with topological anomalies. The Carrollian momenta and Carrollian conservation
equations are precisely those recovered in Sec. 2 when decomposing the
Riemannian Cotton tensor and its divergence. This is summarized as follows.

Paramagnetic Carroll–Chern–Simons

This stems out of the $c^{3}$-order term in the Chern–Simons action:

$S_{\text{CCS}}^{\text{pm}}=4\int_{\mathscr{M}}\text{d}t\,\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}\Omega\ast\\!\varpi^{3}.$ (83)

The associated momenta are $\Pi=2\gamma$ in (41), $\Pi^{i}=2\psi^{i}$ in (42),
$\Upsilon^{ij}=0$ and Eqs. (80), (81) are now

$\mathcal{D}_{\text{Cot}}=0,\quad\mathcal{I}_{\text{Cot}}^{i}=0,$ (84)

see (50) and (54). From these equations and comparison with (81) we infer that
the momentum $P^{i}$ vanishes. For the paramagnetic Carroll–Chern–Simons
action131313The variation vanishes for covariant actions based on genuine
scalars $\Phi$ as
$S[\Phi]=\int_{\mathscr{M}}\text{d}t\,\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\Omega\Phi$.
$\delta_{\upxi}S_{\text{CCS}}^{\text{pm}}=0$ in agreement with (74), where no
term of order $c^{3}$ is present. This action is also manifestly Weyl-
invariant, in line with (75).

Magnetic Carroll–Chern–Simons

The $c$-order provides

$S_{\text{CCS}}^{\text{m}}=\frac{1}{2}\int_{\mathscr{M}}\upmu\wedge\left(\hat{\upomega}^{i}_{\hphantom{i}j}\wedge\text{d}x^{k}\hat{\mathscr{D}}_{k}\ast\\!\varpi+\ast\varpi\text{d}\hat{\upomega}^{i}_{\hphantom{i}j}\right)\eta^{j}_{\hphantom{j}i}+\int_{\mathscr{M}}\text{d}t\,\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}\Omega\left[\ast\varpi\hat{\nabla}_{i}\varphi^{i}-\varphi^{i}\hat{\mathscr{D}}_{i}\ast\\!\varpi\right].$
(85)

Now $\Pi=2\varepsilon$ in (41), $\Pi^{i}=2\chi^{i}$ in (43),
$\Upsilon^{ij}=-2\Psi^{ij}$ in (45) and Eqs. (80), (81) are

$\mathcal{E}_{\text{Cot}}=0,\quad\mathcal{G}_{\text{Cot}}^{i}=0,$ (86)

see (51), (55) and (81), from which $P^{i}=2\psi^{i}$, Eq. (42). The
Carrollian-diffeomorphism transformation of (85) is performed with the help of
the rules for the connection (66) and its exterior differential (67) in the
infinitesimal version (69) — $\upmu$, $\upvarphi$ and $\ast\varpi$ are
invariant, whereas $\eta^{j}_{\hphantom{j}i}$ transforms ordinarily. Only the
first integral contributes with

$\delta_{\upxi}S_{\text{CCS}}^{\text{m}}=-\frac{1}{2}\oint_{\partial\mathscr{M}}\partial_{j}\xi^{i}\text{d}\left(\varpi^{j}_{\hphantom{j}i}\upmu\right).$
(87)

Unsurprisingly this expression coincides with the magnetic ($c$) order in
(74).

Although the action (85) contains explicitly the Carrollian Weyl connection
$\upvarphi$, it turns out to be Weyl-invariant, as expected from (75), which
features only the order $\nicefrac{{1}}{{c}}$.

Electric Carroll–Chern–Simons

The order $\nicefrac{{1}}{{c}}$ is as follows:

$\displaystyle S_{\text{CCS}}^{\text{e}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathscr{M}}\text{Tr}\left(\hat{\upomega}\wedge\text{d}\hat{\upomega}+\frac{2}{3}\hat{\upomega}\wedge\hat{\upomega}\wedge\hat{\upomega}\right)+\int_{\mathscr{M}}\text{d}t\,\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}\Omega\bigg{[}\varphi^{i}\eta^{kl}\hat{\nabla}_{k}\hat{\gamma}_{li}$
(88)
$\displaystyle\left.+\ast\\!\varpi\eta^{kl}\eta^{ij}\hat{\gamma}_{ki}\hat{\gamma}_{lj}-\ast\varpi\left(\frac{1}{\Omega}\hat{D}_{t}\hat{\gamma}^{i}_{\hphantom{i}i}+2\hat{\gamma}_{ij}\hat{\gamma}^{ij}\right)+\hat{\gamma}^{i}_{\hphantom{i}k}\eta^{kl}\left(\hat{\nabla}_{(l}\varphi_{i)}+\varphi_{l}\varphi_{i}\right)\right]$

with $\Pi=2\zeta$ in (41), $\Pi^{i}=2z^{i}$ in (44), $\Upsilon^{ij}=-2X^{ij}$
in (46) and Eqs. (80), (81) are now

$\mathcal{F}_{\text{Cot}}=0,\quad\mathcal{H}_{\text{Cot}}^{i}=0,$ (89)

see (52) and (56). Comparing with Eq. (81) we find the momentum
$P^{i}=2\chi^{i}$, given in (43). As for the magnetic case, the electric
Carroll–Chern–Simons action transforms under Carrollian diffeomorphisms. Only
the first integral in (88) is anomalous and its transformation is readily
determined using the generic expression (73) with the connection
$\hat{\upomega}$:

$\delta_{\upxi}S_{\text{CCS}}^{\text{e}}=-\frac{1}{2}\oint_{\partial\mathscr{M}}\partial_{j}\xi^{i}\text{d}\hat{\upomega}^{j}_{\hphantom{j}i}=-\frac{1}{2}\oint_{\partial\mathscr{M}}\partial_{j}\xi^{i}\hat{\mathcal{R}}^{j}_{\hphantom{j}i}+\frac{1}{2}\oint_{\partial\mathscr{M}}\partial_{j}\xi^{i}\hat{\upomega}^{j}_{\hphantom{j}k}\wedge\hat{\upomega}^{k}_{\hphantom{k}i}.$
(90)

This agrees with the order-$\nicefrac{{1}}{{c}}$ term in the variation (74),
which is indeed the electric order. Under a Weyl rescaling, the behaviour of
(88) is

$\delta_{\lambda}S_{\text{CCS}}^{\text{e}}=\oint_{\partial\mathscr{M}}\text{d}x^{i}\wedge\left(\frac{1}{2}\text{d}a_{ij}a^{jk}\hat{\partial}_{k}\lambda+\left(\varpi_{ij}\text{d}x^{j}+\varphi_{i}\upmu\right)\frac{1}{\Omega}\partial_{t}\lambda\right),$
(91)

also read off in (75), which contains a single order in $c$. This boundary
term does not affect the field equations, which are Weyl-covariant.

Paraelectric Carroll–Chern–Simons

Finally, the order $\nicefrac{{1}}{{c^{3}}}$ gives

$S_{\text{CCS}}^{\text{pe}}=-\int_{\mathscr{M}}\text{d}t\,\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}\eta^{kl}\hat{\gamma}^{i}_{\hphantom{i}k}\hat{D}_{t}\hat{\gamma}_{il}=-\int_{\mathscr{M}}\text{d}t\,\text{d}^{2}x\mathchoice{{\hbox{$\displaystyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{a\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}\ast\\!\xi^{ij}\hat{\mathscr{D}}_{t}\xi_{ij}$
(92)

leading to $\Pi=2\tau$ in (41), $\Pi^{i}=0$ and $\Upsilon^{ij}=-2Z^{ij}$ in
(47). Equations (80) and (81) reduce to

$\mathcal{W}_{\text{Cot}}=0,\quad\mathcal{X}_{\text{Cot}}^{i}=0,$ (93)

see (53) and (57), which exhibit using (81) $P^{i}=2z^{i}$, Eq. (44). In the
case at hand the action is Carroll-diffeomorphism invariant, hence
$\delta_{\upxi}S_{\text{CCS}}^{\text{pe}}=0$ in agreement with (74), which
does not feature any order-$\nicefrac{{1}}{{c^{3}}}$ term. It is also Weyl-
invariant.

The present analysis calls for several comments. We have reached four
Carroll–Chern–Simons actions following a precise guideline, which consists in
choosing a Papapetrou–Randers background for a relativistic theory that
enables a subsequent Carrollian reduction organized in powers of $c$. The
names given to the four Carroll–Chern–Simons actions follow the pattern
already used for the scalar field or the Einstein–Hilbert action in Refs. [38,
39, 43, 44, 45].141414It was suggested in [46] to use type I and II for
electric and magnetic, respectively; an alternative option could be time-like
and space-like. As pointed out in the introduction, besides electric and
magnetic actions, non-dynamical replicas sometimes appear. This phrasing seems
less appropriate in the present context because we are dealing anyhow with
topological actions, and the terms paramagnetic and paraelectric are better
qualified. It should be added in passing that the various patterns used for
reaching Carrollian actions — algebra gauging, strict zero-$c$ limit or our
reduction method — sometimes deliver different though equally consistent
results (see also e.g. [47, 48, 49, 50]). Specific approaches for the search
of Carrollian topological theories might also be designed, like Carrollian
analogues of topological Riemannian terms [51]. In our scheme, the topological
nature of the Carroll–Chern–Simons descendants is pledged by the original
pseudo-Riemannian Chern–Simons action.

A noticeable feature of ours is that all but the paraelectric dynamics break
off-shell local Carroll-boost invariance because of non-vanishing energy
fluxes $\Pi^{i}$. The latter emerge in the expansion of the “heat current”
$C^{\hat{0}i}$ and originate from the original relativistic theory.151515Not
every Lorentz-boost-invariant relativistic theory is expected to lead to a
Carroll-boost-invariant Carrollian relative, as explained in the footnote 40
of [19]. Examples of this sort are numerous. The magnetic Carrollian scalar
dynamics naturally accommodates such a current, as opposed to the electric
instance, where it is absent [43]. One may choose to discard it at the expense
of facing a constrained theory [44, 45]. When the Carrollian manifold is the
null-infinity conformal hypersurface in asymptotically flat spacetimes, such
currents are unavoidably generated by outgoing (or incoming) gravitational
radiation [18, 19, 20]. No alternative exists in those cases, other than
accepting the physical consequences, which are the non-conservation of charges
[19, 37].

Extremums of the pseudo-Riemannian Chern–Simons action have vanishing Cotton
tensor. These are conformally flat three-dimensional pseudo-Riemannian
spacetimes. In the Carrollian framework, the nature of the extremums is
different for each of the four available dynamics. We will not perform a
systematic resolution of the equations in each instance, but rather provide
some generic features.

The paramagnetic Carroll–Chern–Simons dynamics (83) requires geometries with
$\gamma=\psi^{i}=0$, hence (using (41), (42)) with vanishing vorticity
$\ast\varpi$. These geometries are equipped with a clock form (60). Since the
dynamics at hand is Weyl-invariant, $\Omega$ can be set to $1$ by a Weyl
transformation, which further simplifies the clock form to $\upmu=-\text{d}t$,
thus leading to zero $\varphi_{i}$ and $\hat{\mathscr{A}}$. The fibre of the
Carrollian manifold is trivial but the basis metric $a_{ij}(t,\mathbf{x})$ is
utterly arbitrary.

Extremums of the magnetic Carroll–Chern–Simons action (85) are reached with
$\varepsilon=\psi^{i}=\Psi^{ij}=\chi^{i}=0$. The three first equations are
satisfied with $\ast\varpi=0$ (see (41), (42) and (45)). This might not be the
most general solution, but it has the virtue of setting $b_{i}=0$ and
$\hat{\partial}_{i}=\partial_{i}$, and as explained for the paramagnetic
action, $\Omega=1$ and $\varphi_{i}=\hat{\mathscr{A}}=0$. The last magnetic
equation $\chi_{i}=0$ imposes thus (see (43)) $\hat{\mathscr{K}}=\hat{K}=K$ be
a function of time only. This is a severe constraint on the metric
$a_{ij}(t,\mathbf{x})$, which admits non trivial solutions besides those
reached by introducing a factorized time dependence on a two-dimensional
metric with constant curvature having thus zero shear.161616An example is
$\text{d}\ell^{2}=k(t)\,\text{e}^{b(y)}\text{d}x^{2}-\frac{1}{4K(t)}\left(b^{\prime}(y)\right)^{2}\text{d}y^{2}$
with arbitrary $b(t)$, positive $k(t)$ and $K(t)<0$. For this metric the shear
does not vanish.

The paraelectric instance (92) selects $\tau=z^{i}=Z^{ij}=0$. Using (41), (44)
and (47) a typical solution emerges with zero shear $\xi_{ij}$, corresponding
to three-dimensional Carrollian bundles with a base metric proportional to the
Euclidean metric as in (58), and arbitrary clock form and Ehresmann
connection.

Finally, the electric Carroll–Chern–Simons dynamics (88) leads to
$\zeta=z^{i}=X^{ij}=\chi^{i}=0$. The two first equations are satisfied with
$\xi_{ij}=0$ (see Eqs. (41), (44)) — we do not exclude less restrictive
solutions. This simplifies the base metric as in the paraelectric case, but
further conditions remain stemming out of (43) and (46). In holomorphic
coordinates defined in (58) these conditions read:

$\begin{cases}\chi_{\zeta}=\frac{\text{i}}{2}\hat{\mathscr{D}}_{\zeta}\hat{\mathscr{K}}+\frac{1}{2}\hat{\mathscr{D}}_{\zeta}\hat{\mathscr{A}}-2\ast\\!\varpi\hat{\mathscr{R}}_{\zeta}=0\\\
X_{\zeta\zeta}=\text{i}\hat{\mathscr{D}}_{\zeta}\hat{\mathscr{R}}_{\zeta}=0,\end{cases}$
(94)

and set an interplay between the base and the fibre. Non-trivial solutions
exist as the time-independent instance with
$\Omega=\Omega(\zeta,\bar{\zeta})$, $b_{i}=b_{i}(\zeta,\bar{\zeta})$ and
$a_{ij}=a_{ij}(\zeta,\bar{\zeta})$. There $\theta=0$,
$\varphi_{\zeta}=\partial_{\zeta}\ln\Omega$, $\hat{\mathscr{R}}_{\zeta}=0$,
$\hat{\mathscr{A}}=0$, $X_{\zeta\zeta}=0$, while
$\ast\varpi=\frac{\text{i}\Omega
P^{2}}{2}\left(\partial_{\zeta}\frac{b_{\bar{\zeta}}}{\Omega}-\partial_{\bar{\zeta}}\frac{b_{\zeta}}{\Omega}\right)$
and
$\chi_{\zeta}=\frac{\text{i}}{2\Omega^{2}}\partial_{\zeta}\left(\Omega^{2}\hat{\mathscr{K}}\right)$.
The vanishing of the latter delivers the family of shear-free Carrollian
manifolds that define the null boundaries of stationary algebraically special
asymptotically flat spacetimes, as displayed in detail in Refs. [18, 19].

To conclude this section, we observe that for Carrollian manifolds with closed
clock form $\upmu$ (i.e. vanishing vorticity and acceleration, following
(13)), the paramagnetic and magnetic Carroll–Chern–Simons actions vanish,
whereas the electric action simplifies as
$S_{\text{CCS}}^{\text{e}}=\frac{1}{2}\int_{\mathscr{M}}\text{Tr}\left(\hat{\upomega}\wedge\text{d}\hat{\upomega}+\frac{2}{3}\hat{\upomega}\wedge\hat{\upomega}\wedge\hat{\upomega}\right)$.
Demanding the absence of torsion, Eq. (27), makes furthermore the paraelectric
vanish, and trivializes to some extent the electric action by removing time
derivatives, and effectively downgrading the geometry to its two-dimensional
traits with no time dependence.

##  4 In short

In the present note, we have considered strong Carroll structures equipped
with a Carrollian connection designed to respect the time-and-space splitting
inferred by the chosen Papapetrou–Randers frame. In this setup, we have
reached four distinct Carroll–Chern–Simons actions, expanding the original
ascendant pseudo-Riemannian Chern–Simons in powers of $c$, which amounts to
performing a reduction under Carrollian diffeomorphisms. The variation of
these actions with respect to the Carrollian geometric data, i.e., the metric
and the clock form, yield four sets of three Carrollian Cotton tensors, which
obey Carrollian conservation equations.

Two out of the four Carroll–Chern–Simons actions are truly invariant under
Carrollian diffeomorphisms and Weyl transformations: the paramagnetic and
paraelectric avatars. Extremizing the former leads to Carrollian manifolds
with trivial fibres, while the extremums of the latter capture Carrollian
geometries with conformally Euclidean spatial sections and arbitrary clock
forms.

The magnetic and electric actions are more intruiguing. At the first place
Carroll diffeomorphisms are broken by boundary terms, and so are Weyl
transformations for the electric case. In the pseudo-Riemannian framework, and
from a holographic perspective, these phenomena are likened to anomalies. Our
understanding of Carrollian dynamics is still too poor, let alone holography
on boundaries of Carrollian spacetimes, to venture into such interpretations.
The question is however relevant.

The magnetic extremums embrace paramagnetic Carrollian manifolds with purely
time-dependent base curvature. Examples of such spaces do exist, but
conceivable applications remain unexplored. Extremizing the electric action
reveals Carrollian spacetimes with conformally Euclidean spatial sections and
a non-trivial interplay between the base and the fibre. These geometries are
tailor-made for describing null infinity in Ricci-flat four-dimensional
spacetimes.

Along with an effort to further understand the general solutions, the
properties and the applications of the Carroll–Chern–Simons actions at hand,
one should not dismiss the importance of generalizing our approach, possibly
by considering more general Carrollian connections, or alternative techniques
for disclosing Carrollian dynamics. The physics at null infinity of
asymptotically flat spacetimes can provide a possible playground for
challenging potential findings, another being that of black-hole horizons, as
already mentioned. Among the standard tools for analyses of this sort, one
finds the search for charges and conservation, encoded in the (conformal)
Carroll group.

## Acknowledgements

We would like to thank our colleagues Andrea Campoleoni, Sangmin Choi, Simon
Pekar, Anastasios Petkou and Matthieu Vilatte for useful discussions. The work
of David Rivera-Betancour was funded by Becas Chile (ANID) Scholarship No.
72200301. Marios Petropoulos thanks Olivera Mišković and Rodrigo Olea for
financial support and hospitality in Instituto de Física, Pontificia
Universidad Católica de Valparaíso and Departamento de Ciencias Físicas,
Universidad Andrés Bello, Santiago. Olivera Mišković and Rodrigo Olea thank
the Centre de Physique Théorique of the Ecole Polytechnique for hospitality
during the completion of this work. This work has been funded in part by
Anillo Grant ANID/ACT210100 _Holography and its Applications to High Energy
Physics, Quantum Gravity and Condensed Matter Systems_ and FONDECYT Regular
Grants 1190533, 1230492 and 1231779. David Rivera-Betancour thanks the
programme Erasmus+ of the Institut Polytechnique de Paris as well as the
Aristotle University of Thessaloniki and the Kapodistrian University of Athens
for hosting him with this fellowship. The _Third Carroll Workshop_ held in the
Aristotelian University of Thessaloniki in October 2023 is also acknowledged
for providing an utmost creative framework, where ideas related to the present
work have been exchanged.

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# Factoring Multidimensional Data to Create a Sophisticated Bayes Classifier

Anthony LaTorre Electronic address: tlatorre9 at gmail dot com Department of
Physics, University of Chicago, Chicago IL 60637  Division of Math, Physics,
and Astronomy, California Institute of Technology, Pasadena, CA 91125

###### Abstract

In this paper we derive an explicit formula for calculating the marginal
likelihood of a given factorization of a categorical dataset. Since the
marginal likelihood is proportional to the posterior probability of the
factorization, these likelihoods can be used to order all possible
factorizations and select the “best” way to factor the distribution from which
the dataset is drawn. The best factorization can then be used to construct a
Bayes classifier which benefits from factoring out mutually independent sets
of variables.

##  1 Introduction

We wish to investigate the most efficient way to compute the probability of
some variable conditioned on other variables, i.e. to calculate
$P(y\mid\bm{x})$. This can be done using the definition of conditional
probability which says:

$P(y\mid\bm{x})=\frac{P(y,\bm{x})}{P(\bm{x})}.$ (1)

The denominator in Equation 1 is a scaling factor which doesn’t depend on $y$,
so our focus will be the estimation of $P(y,\bm{x})$, which can also be
written as

$P(y,\bm{x})=P(y)P(\bm{x}\mid y).$ (2)

Since we assume $y$ is a single variable, we can efficiently estimate $P(y)$,
so the problem can also be reduced to estimating $P(\bm{x}\mid y)$.

The simplest approach to computing $P(y,\bm{x})$ would be to construct an $N$
dimensional histogram, assuming we have $N-1$ random variables in $\bm{x}$,
and computing the relative frequency of events ending up in the bin
$(y,\bm{x})$. This method has the advantage that we don’t have to make any
assumptions about the distribution from which the data is drawn. The drawback
is that the error on the prediction of $y$ will scale exponentially like
$K^{N/2}$, where $K$ is the number of categories or bins for each variable.
This makes intuitive sense since for a dataset with a fixed number of samples,
the number of bins grows exponentially with the number of variables, and
therefore we should expect a large error because the statistics in each bin
will be low.

One common solution to this problem is to use a Naive Bayes classifier [1]
where all the $X_{i}$ variables are assumed to be mutually independent
conditional on $y$. With this assumption we can rewrite Equation 1 as

$P(y\mid\bm{x})=\frac{1}{P(\bm{x})}P(y)\prod_{i}P(x_{i}\mid y).$ (3)

The error on the prediction in this case only grows like $\sqrt{2N}$. This
also makes intuitive sense since we have reduced the calculation from looking
up the probability in a $N$ dimensional histogram to a product of $N$
1-dimensional histograms. However, the accuracy of this estimate will depend
on how good the independent approximation is.

In this paper, we propose a solution to this problem by asking how the
distribution $P(\bm{x}\mid y)$ can be factored. For example, supposing we had
just two variables $X_{1}$ and $X_{2}$, we can ask whether it can be factored
as $P(x_{1}\mid y)P(x_{2}\mid y)$ or whether we need to consider the full
probability $P(x_{1},x_{2}\mid y)$. To distinguish between these two models,
we consider ordering them by the posterior probability of the factorization
[2]. For the example just described this involves calculating two
probabilities: $P(M_{I}\mid D)$ and $P(M_{D}\mid D)$, where $M_{I}$ stands for
the independent model which can be factored and $M_{D}$ stands for the
dependent model. In particular, we show that for a categorical distribution,
there is a simple analytical solution for computing the marginal
likelihood111By marginal likelihood we mean the likelihood of the model
independent of the specific parameters governing the model. of a given model,
i.e. $P(D\mid M)$ which is proportional to the posterior probability $P(M\mid
D)$ up to a constant scaling factor and some prior over the models.

For the general case of $N$ variables, we have to consider all possible ways
to factor $P(\bm{x}\mid y)$. This is equivalent to considering all possible
groupings of the variables, where within each group the variables are
considered to be dependent, but the different groups are treated as
independent. For example, all possible groupings of 4 variables are shown in
Table 1.

$(X_{1},X_{2},X_{3},X_{4})$
---
$(X_{1}),(X_{2},X_{3},X_{4})$
$(X_{2}),(X_{1},X_{3},X_{4})$
$(X_{3}),(X_{1},X_{2},X_{4})$
$(X_{4}),(X_{1},X_{2},X_{3})$
$(X_{1},X_{2}),(X_{3},X_{4})$
$(X_{1},X_{3}),(X_{2},X_{4})$
$(X_{1},X_{4}),(X_{2},X_{3})$
$(X_{1}),(X_{2}),(X_{3},X_{4})$
$(X_{1}),(X_{3}),(X_{2},X_{4})$
$(X_{1}),(X_{4}),(X_{2},X_{3})$
$(X_{2}),(X_{3}),(X_{1},X_{4})$
$(X_{2}),(X_{4}),(X_{1},X_{3})$
$(X_{3}),(X_{4}),(X_{1},X_{2})$
$(X_{1}),(X_{2}),(X_{3}),(X_{4})$
Table 1: All possible groupings of 4 variables. Each group represents a
different way to factor the probability distribution
$P(x_{1},x_{2},x_{3},x_{4})$.

The rest of the paper is organized as follows: in Section 2 we consider the
general problem of how to factor a distribution $P(\bm{x})$ by computing the
marginal likelihood $P(D\mid M)$ for a given factorization. In Section 3 we
explicitly calculate these terms for two binary random variables and then give
a formula for the Bayes factor to distinguish between them. Finally, in
Section 4 we give concluding remarks.

##  2 Calculation

In this section, we consider the general problem of factoring a distribution
of $N$ categorical random variables labelled $(X_{0},X_{1},\ldots,X_{N})$. To
do so, we choose the “best” factorization where the ordering of the different
factorizations is determined by the posterior probability of a given
factorization. If we label a given factorization as a model $M$, we wish to
compute $P(M\mid D)$ where $D$ stands for the data. This posterior is related
to the marginal likelihood $P(D\mid M)$ as follows:

$P(M\mid D)=\frac{P(D\mid M)P(M)}{P(D)}.$ (4)

The first term in the numerator of Equation 4 is the marginal likelihood, the
second term in the numerator represents any prior we may have, and the
denominator is simply a scaling factor which is not relevant if we only
consider ordering the models. So, to order the models we need to compute the
marginal likelihood $P(D\mid M)$ and then we can multiply it by any prior we
might have (or simply use the likelihood directly if we have no preference for
any particular factorization).

We will represent each possible factorization as a particular grouping of the
$N$ variables. For example, the possible groupings for 4 random variables are
shown in Table 1. For a given factorization $M$, we will label each group
within the factorization as $g_{i}$, where $i$ runs over all the different
groups. For example, one possible factorization of 4 random variables is
$((X_{1},X_{2}),(X_{3}),(X_{4}))$. For this case, the group $(X_{1},X_{2})$
will be referred to as $g_{0}$, the next group $(X_{3})$ as $g_{1}$, and
$(X_{4})$ as $g_{2}$.

The probability distribution for each group $g_{i}$, can be specified by the
variables $p_{i,j}$ labeling the probability of landing in the
$j^{\mathrm{th}}$ bin for group $g_{i}$. When referring to these probabilities
for a general group $g$ we will also sometimes denote these probabilities as
$p_{g,j}$. The number of bins in each group is equal to

$\eta_{i}=\prod_{X\in g_{i}}|X|.$

where $|X|$ is the number of possible values for the random variable $X$. For
the previous example, assuming each variable can take on only two discrete
values, there would be 4 probabilities for the first group,
$p_{0,0},p_{0,1},p_{0,2},p_{0,3}$, two probabilities for the second group
$p_{1,0},p_{1,1}$, and two probabilities for the last group $p_{2,0},p_{2,1}$.
We will sometimes denote these values as vectors, i.e. $\bm{p_{0}}$,
$\bm{p_{1}}$, and $\bm{p_{2}}$. When referring to a general group $g$, we will
simply write $\bm{p_{g}}$.

We begin by considering the marginal likelihood and expanding it in terms of
the probabilities $p_{i,j}$

$\displaystyle P(D\mid M)$
$\displaystyle=\int_{p_{0,1}}\int_{p_{0,2}}\cdots\int_{p_{m,\eta_{m}}}P(D\mid
p_{i,j},M)P(p_{i,j}\mid M),$ (5)

where $m$ is equal to the number of groups. The two terms on the right are the
full likelihood and the prior for the model parameters $p_{i,j}$ for the
factorization $M$.

For the second term in Equation 5, we assume a flat222The Dirichlet
distribution with $\bm{\alpha}=\bm{1}$ is completely flat and so is the only
non-informative prior where our only constraint is that the sum of the
probabilities must add to 1. It also happens to be a good choice since it is
the conjugate prior of the multinomial distribution. Dirichlet prior for the
probabilities within each group,

$P(p_{i,j}\mid M)=\prod_{g\in M}\mathrm{Dir}(\bm{p_{g}},\bm{1}).$ (6)

The Dirichlet distribution is given by the equation:

$\mathrm{Dir}(\bm{\alpha},\bm{x})=\frac{1}{B(\bm{\alpha})}\prod_{i}x_{i}^{\alpha_{i}-1}$
(7)

where $B$ is the multivariate Beta function given by

$\displaystyle B(\bm{\alpha})$
$\displaystyle=\frac{\prod_{i}\Gamma(\alpha_{i})}{\Gamma(\sum_{i}\alpha_{i})}.$
(8)

The first term in Equation 5 is the full likelihood of observing a dataset
$D$, which is given by the multinomial distribution

$P(D\mid p_{i,j},M)=\frac{N!}{\prod_{i}n_{i}!}\prod_{i}p_{i}^{n_{i}}.$ (9)

Note that in this last expression, the $i$ variable runs over all possible
combinations of the values for the $\bm{X}$ variables, $n_{i}$ represents the
number of samples with outcome $i$, and $p_{i}$ represents the probability of
getting the outcome $i$. Table 2 shows all the terms in this equation for a
factorization of 4 random variables as $(X_{1},X_{2}),(X_{3}),(X_{4})$. Next,
we rewrite Equation 9 grouping the $p$ variables for groups together

$P(D\mid p_{i,j},M)=\frac{N!}{\prod_{i}n_{i}!}\prod_{g\in
G}\prod_{i}p_{g,i}^{n_{g,i}}$ (10)

where the $n_{g,i}$ term is equal to the number of data values which have the
$i^{\mathrm{th}}$ combination of values for the group $g$ marginalized over
all the other variables. This is equivalent to the $i^{\mathrm{th}}$ bin in
the flattened histogram of the dataset when only looking at the variables in
the group $g$.

i | $\bm{x}$ | $p_{i}\rightarrow p_{i,j}$
---|---|---
0 | 00 0 0 | $p_{0}=p_{0,0}\cdot p_{1,0}\cdot p_{2,0}$
1 | 00 0 1 | $p_{1}=p_{0,0}\cdot p_{1,0}\cdot p_{2,1}$
2 | 00 1 0 | $p_{2}=p_{0,0}\cdot p_{1,1}\cdot p_{2,0}$
3 | 00 1 1 | $p_{3}=p_{0,0}\cdot p_{1,1}\cdot p_{2,1}$
4 | 01 0 0 | $p_{4}=p_{0,1}\cdot p_{1,0}\cdot p_{2,0}$
5 | 01 0 1 | $p_{5}=p_{0,1}\cdot p_{1,0}\cdot p_{2,1}$
6 | 01 1 0 | $p_{6}=p_{0,1}\cdot p_{1,1}\cdot p_{2,0}$
7 | 01 1 1 | $p_{7}=p_{0,1}\cdot p_{1,1}\cdot p_{2,1}$
8 | 10 0 0 | $p_{8}=p_{0,2}\cdot p_{1,0}\cdot p_{2,0}$
9 | 10 0 1 | $p_{9}=p_{0,2}\cdot p_{1,0}\cdot p_{2,1}$
10 | 10 1 0 | $p_{10}=p_{0,2}\cdot p_{1,1}\cdot p_{2,0}$
11 | 10 1 1 | $p_{11}=p_{0,2}\cdot p_{1,1}\cdot p_{2,1}$
12 | 11 0 0 | $p_{12}=p_{0,3}\cdot p_{1,0}\cdot p_{2,0}$
13 | 11 0 1 | $p_{13}=p_{0,3}\cdot p_{1,0}\cdot p_{2,1}$
14 | 11 1 0 | $p_{14}=p_{0,3}\cdot p_{1,1}\cdot p_{2,0}$
15 | 11 1 1 | $p_{15}=p_{0,3}\cdot p_{1,1}\cdot p_{2,1}$
Table 2: Table showing the 16 terms in the likelihood expression for four
binary random variables. The second column shows the different possible values
for the random variables $X_{1}$, $X_{2}$, $X_{3}$, and $X_{4}$ as a binary
string. The third column shows the values for the probability written as they
appear in Equation 9 and then expanded in terms of the probabilities for the
factorization $(X_{1},X_{2}),(X_{3}),(X_{4})$ in Equation 10.

Plugging in the prior (Equation 6) and likelihood (Equation 10) into Equation
5 we get

$\displaystyle P(D\mid M)$
$\displaystyle=\int_{p_{0,1}}\int_{p_{0,2}}\cdots\int_{p_{m,\eta_{m}}}\frac{N!}{\prod_{i}n_{i}!}\prod_{g\in
M}\mathrm{Dir}(\bm{p_{g}},\bm{1})\prod_{i}p_{g,i}^{n_{g,i}}.$ (11)

The last product in this expression can be written as

$\mathrm{Dir}(\bm{p_{g}},\bm{1})\prod_{i}p_{g,i}^{n_{i}}=\frac{1}{B(\bm{1})}\prod_{i}p_{g,i}^{n_{i}}=\mathrm{Dir}(\bm{p_{g}},\bm{1}+\bm{n})\frac{\mathrm{B}(\bm{1}+\bm{n})}{\mathrm{B}(\bm{1})}.$
(12)

Plugging this back into Equation 11 we get

$\displaystyle P(D\mid M)$
$\displaystyle=\int_{p_{0,1}}\int_{p_{0,2}}\cdots\int_{p_{m,\eta_{m}}}\frac{N!}{\prod_{i}n_{i}!}\prod_{g\in
M}\mathrm{Dir}(\bm{p_{g}},\bm{1}+\bm{n})\frac{\mathrm{B}(\bm{1}+\bm{n_{g}})}{\mathrm{B}(\bm{1})}$
(13)

which simplifies to

$\displaystyle P(D\mid M)$
$\displaystyle=\frac{N!}{\prod_{i}n_{i}!}\prod_{g\in
M}\frac{\mathrm{B}(\bm{1}+\bm{n_{g}})}{\mathrm{B}(\bm{1})}.$ (14)

Note that the length of the vector $\bm{1}$ in the denominator of Equation 14
is equal to $\eta_{i}$ for group $i$, i.e. it has the same length as
$\bm{p_{g}}$ and $\bm{n_{g}}$.

Equation 14 is the main result of this paper which gives an explicit formula
for the calculation of the marginal likelihood of a given factorization. There
exist efficient routines for calculating the multivariate Beta function, so
calculating the marginal likelihood for a given factorization is
straightforward. In practice, it is useful to compute the log of Equation 14
when comparing factorizations since otherwise the factorials can produce
numbers which do not fit into standard 32 or 64 bit floats and integers.

##  3 Two Binary Variables

$x_{1}$ | $x_{2}$ | $n_{i}$
---|---|---
0 | 0 | $n_{1}$
0 | 1 | $n_{2}$
1 | 0 | $n_{3}$
1 | 1 | $n_{4}$

Table 3: Table showing the mapping between the values for the random variables
$\bm{x}$ and the numbers in Equation 15.

.

In the simple case of 2 binary random variables, we get the result

$\displaystyle P(D\mid M_{I})$
$\displaystyle=\frac{(n_{1}+n_{2})!(n_{1}+n_{3})!(n_{2}+n_{4})!(n_{3}+n_{4})!}{n_{1}!n_{2}!n_{3}!n_{4}!(N+1)(N+1)!}$
(15) $\displaystyle P(D\mid M_{D})$ $\displaystyle=\frac{6}{(N+3)(N+2)(N+1)}$
(16)

where $M_{I}$ stands for the independent hypothesis, and $M_{D}$ stands for
the dependent hypothesis, the $n_{i}$ are the numbers of samples with the
values shown in Table 3, and $N$ is the total number of samples.

We can use these values to compute the Bayes Factor

$\displaystyle K=\frac{P(D\mid M_{I})}{P(D\mid
M_{D})}=\frac{(n_{1}+n_{2})!(n_{1}+n_{3})!(n_{2}+n_{4})!(n_{3}+n_{4})!(N+3)(N+2)}{n_{1}!n_{2}!n_{3}!n_{4}!(N+1)!6}.$

For values of $K$ less than 1 the data favor the correlated hypothesis, and
for values greater than 1 it favors the independent hypothesis.

##  4 Conclusion

We have given an explicit formula for calculating the marginal likelihood of a
given factorization of a dataset of categorical random variables. This formula
can be used to choose the best way to factor the conditional probability
$P(\bm{x}\mid y)$ and thus gives us a way to construct a Bayes classifier
which takes full advantage of mutually independent sets of variables.

Although it is possible in principle sort through all possible factorization
using Equation 14, the number of calculations of the multivariate Beta
function grows like the size of the powerset of the number of variables. This
means that in practice it is prohibitive to sort through all factorizations
for more than approximately 30 variables (where one would need to calculate
over a billion terms). However, we suspect that for real world data there are
ways to considerably reduce the number of factorizations to check by using
heuristics.

One particular area where we hope this technique will be of use is in high
energy physics analyses. For these analyses it is very common to define a
broad range of cuts designed to remove background events and keep events of
interest. Although multiple cuts are used, in the end the ultimate goal is
often to define a total cut efficiency and sacrifice, i.e. to determine what
percentage of background events are successfully cut and what fraction of
signal events are “accidentally” cut. If you have a full Monte Carlo
simulation of both background and signal it can be relatively easy to do this,
but often some of the background events cannot be simulated due to complexity
or time constraints. Therefore, it is often necessary to use a combined fit to
a sideband and the region of interest to fit for the cut efficiencies. The
number of terms in this fit will grow exponentially as the number of cuts
increases, and therefore this technique can be used to construct a model of
for the cut efficiencies with considerably fewer parameters.

## Acknowledgments

Thanks to Kevin Labe and Marie Vidal for multiple reviews of this paper.

## References

  * [1] David J. Hand and Keming Yu.  Idiot’s bayes – not so stupid after all?  International Statistical Review, 69:385–398, 2001.
  * [2] Edwin T. Jaynes.  Probability Theory: The Logic of Science.  Cambridge University Press, Cambridge, England, first edition, 2003.