boxsize=0.3em,centertableaux aainstitutetext: The Blackett Laboratory,
Imperial College London, Prince Consort Road, London, SW7 2AZ, UK
bbinstitutetext: Department of Physics and Astronomy, Uppsala University, Box
516, SE-75120 Uppsala, Sweden and Centre for Geometry and Physics, Uppsala
University, Box 480, SE-75106 Uppsala, Sweden

# Gauge-invariant magnetic charges in linearised gravity

Chris Hull a Maxwell L. Hutt a,b Ulf Lindström<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Linearised gravity has magnetic charges carried by (linearised) Kaluza-Klein
monopoles. A gauge-invariant expression is found for these charges that is
similar to Penrose’s gauge-invariant expression for the ADM charges. A
systematic search is made for other gauge-invariant charges.

††preprint: Imperial-TP-2024-CH-04, UUITP-11/24

##  1 Introduction

For the linearised theory of the graviton in Minkowski space, each Killing
vector of Minkowski space gives an invariance of the theory and there is a
corresponding conserved charge $Q[k]$ that can be constructed via a
linearisation of the ADM construction Abbott1982StabilityConstant . In Ref.
HullYetAppear  analogous ‘electric’ conserved charges $Q[\kappa]$,
$Q[\lambda]$ were constructed in spacetime dimensions $d\geq 5$ for the dual
graviton theory Hull2000 ; Hull2001DualityFields  corresponding to certain
generalised Killing tensors $\kappa$, $\lambda$ that give the invariances of
the dual theory. These were then reinterpreted as ‘magnetic’ charges for the
graviton theory, for which they are topological charges.

In Ref. Penrose1982Quasi-localRelativity , covariant 2-form currents

$Y[K]_{\mu\nu}=R_{\mu\nu\alpha\beta}K^{\alpha\beta}$ (1)

were introduced and shown to be conserved in regions without sources provided
that $K_{\alpha\beta}$ is a conformal Killing-Yano (CKY) tensor. Integrating
these over a closed codimension-2 surface then gives conserved charges, which
we refer to as Penrose charges. These were studied in Ref. PAPER1  and shown
to give a linear combination of $Q[k]$, $Q[\lambda]$ and a topological charge,
with $k$ and $\lambda$ constructed from $K$. For the graviton theory, the
Penrose charges can be interpreted as giving a covariantisation of $Q[k]$.
However, the Penrose charges can also be interpreted in the dual graviton
theory as a covariantisation of the charge $Q[\lambda]$. We will discuss the
charges in the dual graviton theory in a forthcoming publication and will here
focus on the charges in the graviton theory.

Kaluza-Klein monopoles play an important role in string theory Hull:Uduality
and carry a gravitational magnetic charge Hull:braneCharges . The magnetic
charge $Q[\kappa]$ in the linearised theory is the charge carried by
linearised Kaluza-Klein monopoles HullYetAppear . The magnetic charge
$Q[\kappa]$, however, is not related to the Penrose charges. One of the aims
of this paper is to find a covariant current that gives this charge. This
leads us to generalise the Penrose construction and seek conserved 2-form
currents, linear in the Riemann tensor, of the form

$\Omega[V]_{\mu\nu}=R^{\alpha\beta\gamma\delta}V_{\mu\nu\alpha\beta|\gamma\delta}$
(2)

for some rank-6 tensor $V_{\mu\nu\alpha\beta|\gamma\delta}$ with the
symmetries111We use a vertical bar to separate groups of fully antisymmetric
indices.

$V_{\mu\nu\alpha\beta|\gamma\delta}=V_{[\mu\nu\alpha\beta]|[\gamma\delta]}$
(3)

We investigate the conditions on $V$ for this current to be conserved,
classifying the full set of such conserved 2-forms. One of our main results is
that, as well as the Penrose charges, the $\kappa$-type charges can also be
constructed as a subset of these charges, and so also have a covariant
expression.

In the quantum theory, gauge-invariant operators are particularly important.
Finding covariant forms for electric and magnetic gravitational charges, which
is our goal here, then gives gauge invariant topological operators that
generate continuous generalised symmetries.

One motivation of this work is to further understand the generalised
symmetries of the free graviton theory, following work by
Hinterbichler2023GravitySymmetries . There, it was shown that any system
containing specific electric and magnetic ‘bi-form symmetries’ which
participate in a particular mixed ’t Hooft anomaly will have a gapless phase
with a spin-2 mode in the IR spectrum. In other words, the graviton can be
viewed as the Nambu-Goldstone boson associated with the non-linear realisation
of such symmetries. Similar ideas were used in Hinterbichler:2024cxn  to show
that certain anomalies cannot be realised by any local, unitary theory.
Finding gauge-invariant expressions for the conserved quantities related to
all the (generalised) symmetries of the theory and understanding their
anomalies is, therefore, a promising avenue to gain insight into the IR
physics at play. Other work on generalised symmetries in gravity can be found
in Benedetti2022GeneralizedGraviton ; Benedetti2023GeneralizedGravitons ;
BenedettiNoether ; Gomez-Fayren2023CovariantRelativity ; Cheung:2024ypq .

Consider first the symmetries corresponding to the Killing vectors $k$. For
each $k$ there is a 1-form current

$j[k]_{\mu}\equiv T_{\mu\nu}k^{\nu}$ (4)

where the energy-momentum tensor $T_{\mu\nu}$ is the source for the linearised
Einstein equation. This can be integrated over a co-dimension one surface to
give a conserved charge that generates a 0-form symmetry. However, there is a
‘secondary current’ $J[k]_{\mu\nu}$ which satisfies

$\partial^{\nu}J[k]_{\mu\nu}=j[k]_{\mu}$ (5)

so that this 2-form current is conserved wherever the source $j$ vanishes.
However, the current $J[k]$ is not uniquely determined by (5). The ADM
construction reviewed in HullYetAppear  gives a non-covariant $J[k]$, but the
Penrose construction gives a covariant current.

Integrating $J$ over a codimension 2 surface gives a conserved charge which in
the quantum theory gives a topological operator – it is unchanged under
deformations of the surface that do not pass through regions in which $j\neq
0$. This charge generates a 1-form symmetry. Note that whereas for non-linear
gravity the surface needs to be taken at spatial or null infinity as
gravitational energy-momentum is non-local, for the linearised theory it can
be any surface as the energy-momentum is local.

There is a similar story for the $\kappa$ and $\lambda$ charges HullYetAppear
. In the dual theory, there is a primary 1-form current $j$ for each, and a
corresponding 2-form current $J$ with $\partial^{\nu}J_{\mu\nu}=j_{\mu}$, with
corresponding charges and 0-form and 1-form symmetries, similarly to the case
above. The current $J$ gives rise to a 2-form current in the graviton theory,
giving a topological charge there. In this paper we will focus on the charges
and postpone a discussion of the corresponding generalised symmetries to a
future paper.

Given the relevance of these charges to Kaluza-Klein monopoles, we also study
the gauge-invariant charges constructed from $\Omega[V]$ on toroidally
compactified spaces $\mathbb{R}^{1,D-1}\times T^{n}$ with $D+n=d$, giving them
a unified higher-dimensional origin.

Our discussion here is purely for the linearised theory. It is of course
important and interesting to consider the extension of our results to the non-
linear theory of general relativity or supergravity; we will discuss this and
related issues elsewhere.

There is a general expectation that generalised symmetries should come in dual
pairs CasiniCompleteness ; Benedetti2023GeneralizedGravitons . That is, there
should be the same number of continuous $p$-form symmetries as $(d-p-1)$-form
symmetries. This was verified for the free graviton on $d$-dimensional
Minkowski space in Benedetti2023GeneralizedGravitons  by showing that some of
the Penrose charges are trivial in $d>4$ and showing that there is the same
number of conserved $(d-2)$-form currents as of non-trivial Penrose charges.
We show here that, on topologically non-trivial backgrounds, there are more
1-form symmetries than those generated by the Penrose charges. It would,
therefore, be interesting to understand the full set of generalised symmetries
that are dual to these on such backgrounds.

### Conventions

We refer to tensors whose indices can be split into fully anti-symmetric
groups of lengths $n_{1},n_{2},\dots,n_{r}$ as ${\lfloor
n_{1},n_{2},\dots,n_{r}\rfloor}$-tensors. For example, a tensor with index
symmetries $M_{\mu\nu\rho|\sigma|\alpha}=M_{[\mu\nu\rho]|\sigma|\alpha}$ is a
${\lfloor 3,1,1\rfloor}$-tensor. Such tensors have been studied in Refs.
Medeiros2003ExoticDuality ; Bekaert2004 ; Dubois-Violette:1999iqe ; Dubois-
Violette:2000fok ; Dubois-Violette:2001wjr ; Howe2018SCKYT  where they are
sometimes referred to as ‘multi-forms’ and in the case of $r=2$ they are often
called ‘bi-forms’. Unless otherwise stated, these tensors are in _reducible_
$GL(d,\mathbb{R})$ representations. That is, they transform in a
representation which is labelled by a tensor product of single-column Young
tableaux of lengths $n_{1},n_{2},\dots,n_{r}$.

_Irreducible_ $GL(d,\mathbb{R})$ representations are instead labelled by
${[n_{1},\dots,n_{r}]}$ (note the different type of bracket). The criterion
for a reducible ${\lfloor p,q\rfloor}$-tensor, $N$, to transform in an
irreducible $GL(d,\mathbb{R})$ representation (i.e. one labelled by a single
Young tableau with two columns of lengths $p$ and $q$) is

$N_{\mu_{1}\dots\mu_{p}|\nu_{1}\dots\nu_{q}}=N_{[\mu_{1}\dots\mu_{p}]|[\nu_{1}\dots\nu_{q}]},\quad
N_{[\mu_{1}\dots\mu_{p}|\nu_{1}]\nu_{2}\dots\nu_{q}}=0$ (6)

where we have assumed $p>q$. If $p=q$ then furthermore $N$ satisfies

$N_{\mu_{1}\dots\mu_{p}|\nu_{1}\dots\nu_{p}}=N_{\nu_{1}\dots\nu_{p}|\mu_{1}\dots\mu_{p}}$
(7)

Irreducible representations of $GL(d,\mathbb{R})$ can be viewed as reducible
representations of $SO(1,d-1)\subset GL(d,\mathbb{R})$ and can, therefore, be
decomposed into a sum of irreducible representations of $SO(1,d-1)$. A
${[p,q]}$-tensor $A$ which transforms in an irreducible representation of
$SO(1,d-1)$ labelled by a Young tableau with two columns of lengths $p$ and
$q$ must satisfy eqs. (6) and (7), and must also be traceless with respect to
the $SO(1,d-1)$-invariant Minkowski metric $\eta_{\mu\nu}$:

$\eta^{\mu_{1}\nu_{1}}A_{\mu_{1}\dots\mu_{p}|\nu_{1}\dots\nu_{q}}=0$ (8)

The Young symmetriser $\mathcal{Y}$ is a linear operator which projects a
tensor onto a given irreducible representation. We will denote in the
superscript whether the projection is to an irreducible $GL(d,\mathbb{R})$ or
$SO(1,d-1)$ representation. We will denote in the subscript the Young tableau
of the irreducible representation which is being projected on to. For example,
$\mathcal{Y}_{{\ydiagram{2,1}}}^{SO}$ is the projector onto the irreducible
${[2,1]}$ representation of $SO(1,d-1)$ (so the resulting tensor is
traceless).

##  2 The charges of the graviton

We consider the Fierz-Pauli theory of a massless spin-2 field $h_{\mu\nu}$
propagating on a background spacetime $\mathcal{M}$. This is taken to be a
$d$-dimensional Minkowski space, possibly with some points or regions removed
so as to allow for non-trivial topology. In later sections we also consider
toroidal compactifications of some spatial directions. Here we present only a
summary of various conserved charges which can be associated with the
graviton, following HullYetAppear ; PAPER1 . The reader should refer to
Appendix A for detailed definitions and to Ref. PAPER1  for details.

###  2.1 Penrose charges, ADM charges and magnetic charges

We define a linearised connection $\Gamma_{\mu\nu|\rho}$ which is a
[2,1]-tensor with components

$\Gamma_{\mu\nu|\rho}=\partial_{[\mu}h_{\nu]\rho}$ (9)

The graviton has gauge transformations

$h_{\mu\nu}\to h_{\mu\nu}+\partial_{(\mu}\xi_{\nu)}$ (10)

under which the linearised Riemann tensor

$R_{\mu\nu\rho\sigma}=\partial_{\rho}\Gamma_{\mu\nu|\sigma}-\partial_{\sigma}\Gamma_{\mu\nu|\rho}$
(11)

is gauge-invariant. In the linearised theory, $\Gamma_{\alpha\beta|\gamma}$ is
related to the linearised Christoffel connection by a gauge transformation.

The ADM charges in linearised gravity have a surface integral representation
Abbott1982StabilityConstant

$Q[k]=\int\star J[k]$ (12)

where $k$ is a Minkowski space Killing vector, and $J[k]$ is the ‘secondary
current’ which satisfies

$\partial^{\nu}J[k]_{\mu\nu}=G_{\mu\nu}k^{\nu}\equiv j[k]_{\mu}$ (13)

so is conserved when $R_{\mu\nu}=0$. If the field equation
$G_{\mu\nu}=T_{\mu\nu}$ holds, then $J[k]$ is conserved away from sources. The
explicit form of $J[k]$ is given in Appendix A. In regions with sources, the
result is non-vanishing and gives the 1-form ‘primary’ conserved current
$j[k]$ which has been referred to as the Einstein current in Ref.
Lindstrom2022Geometrycurrents .

As mentioned in the previous section, the ADM charges can be related to the
covariant Penrose charges which are integrals of the Penrose 2-form (1). This
2-form is constructed from a CKY tensor, which satisfies

$\partial_{\lambda}K_{\mu\nu}=\tilde{K}_{\lambda\mu\nu}+2\eta_{\lambda[\mu}\hat{K}_{\nu]}$
(14)

with

$\tilde{K}_{\lambda\mu\nu}=\partial_{[\lambda}K_{\mu\nu]}\mbox{\quad
and\quad}\hat{K}_{\mu}=\frac{1}{d-1}\partial^{\nu}K_{\nu\mu}~{}.$ (15)

For comparison with analysis which will be used later, we give a brief
representation theory derivation of the result that the 2-form $K$ in the
Penrose 2-form $Y[K]$ must be a CKY tensor. Demanding conservation of $Y[K]$
implies that

$0=\partial^{\mu}Y[K]_{\mu\nu}=\partial^{\mu}(R_{\mu\nu\alpha\beta}K^{\alpha\beta})=R_{\mu\nu\alpha\beta}\partial^{\mu}K^{\alpha\beta}$
(16)

where we have used the contracted Bianchi identity
$\partial^{\mu}R_{\mu\nu\alpha\beta}=0$. This follows from the Bianchi
identity $\partial_{[\mu}R_{\alpha\beta]\gamma\delta}=0$ and the vacuum field
equation $R_{\mu\nu}=0$. Now we must impose a condition on
$\partial^{\mu}K^{\alpha\beta}$ such that
$R_{\mu\nu\alpha\beta}\partial^{\mu}K^{\alpha\beta}$ vanishes. This can be
done via a decomposition into irreducible representations of $SO(1,d-1)$ as
follows.

Firstly, $\partial_{\mu}K_{\alpha\beta}$ is a $GL(d,\mathbb{R})$ tensor in the
reducible ${\lfloor 1,2\rfloor}$ representation. This is a tensor product
representation, labelled by Young tableaux
${\ydiagram{1}}\otimes{\ydiagram{1,1}}$, which can instead be written as a
direct sum of irreducible $GL(d,\mathbb{R})$ representations. These can be
seen as _reducible_ representations of $SO(1,d-1)\subset GL(d,\mathbb{R})$.
Decomposing further into _irreducible_ representations of $SO(1,d-1)$ by
separating out traces, we have

$\left({\ydiagram{1}}\otimes{\ydiagram{1,1}}\,\right)_{GL}=\left(\,{\ydiagram{1,1,1}}\oplus{\ydiagram{2,1}}\,\right)_{GL}=\left(\,{\ydiagram{1,1,1}}\oplus{\ydiagram{2,1}}\oplus{\ydiagram{1}}\right)_{SO}$
(17)

where the $SO(1,d-1)$ representations are traceless by definition (as well as
satisfying the index symmetries in eqs. (6) and (7)). This can be written out
explicitly for the tensor $\partial_{\mu}K_{\alpha\beta}$ as

$\partial_{\mu}K_{\alpha\beta}=\tilde{K}_{\mu\alpha\beta}+\left(\partial_{\mu}K_{\alpha\beta}-\tilde{K}_{\mu\alpha\beta}-2\eta_{\lambda[\mu}\hat{K}_{\nu]}\right)+2\eta_{\lambda[\mu}\hat{K}_{\nu]}$
(18)

Here, the first term on the right-hand side is in the ${\ydiagram{1,1,1}}$
representation, the term in parentheses is in the traceless ${\ydiagram{2,1}}$
representation and the final term involves a vector in the ${\ydiagram{1}}$
representation. Inserting this decomposition into eq. (16), the first and last
terms vanish immediately from the Bianchi identity $R_{[\mu\nu\alpha]\beta}=0$
and the field equation $R_{\mu\nu}=0$. Therefore, the result vanishes if and
only if we impose that the term in parentheses in eq. (18) vanishes by itself.
Indeed, this immediately gives the CKY equation (14). We see from this that
the CKY equation can be equivalently written

$\mathcal{Y}_{{\ydiagram{2,1}}}^{SO}(\partial K)=0$ (19)

Arguments similar to these will be essential to our more general treatment in
section 8. This approach to defining generalised Killing tensors by the
representations which are not allowed to appear in such branchings has been
highlighted and expanded upon in Ref. Howe2018SCKYT .

An important result for analysing the Penrose charges is the following
Minkowski space decomposition of a CKY tensor penrose_rindler_1986 ;
Howe2018SCKYT ; Hinterbichler2023GravitySymmetries :

$K_{\alpha\beta}=\mathcal{A}_{\alpha\beta}+\mathcal{B}_{[\alpha}x_{\beta]}+\mathcal{C}_{\alpha\beta\gamma}x^{\gamma}+2x_{[\alpha}\mathcal{D}_{\beta]\gamma}x^{\gamma}+\frac{1}{2}\mathcal{D}_{\alpha\beta}x_{\gamma}x^{\gamma}$
(20)

where $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$ are
constant antisymmetric tensors. This is the most general form of a CKY tensor
in Minkowski space. When the Penrose charges are constructed from $K$ as in
eq. (1), the $\mathcal{B}$ and $\mathcal{D}$ terms are related to ADM charges
of the graviton, while the $\mathcal{A}$ and $\mathcal{C}$ terms are related
to magnetic charges of the graviton. It follows from this decomposition that
$\hat{K}$ is a Killing vector and $\tilde{K}$ is a closed CKY 3-form on
Minkowski space; that is, it satisfies

$\partial_{\mu}\tilde{K}_{\alpha\beta\gamma}=\frac{3}{d-2}\eta_{\mu[\alpha}\partial^{\lambda}\tilde{K}_{\beta\gamma]\lambda}$
(21)

We therefore define a Killing vector

$k_{\mu}=2(d-3)\hat{K}_{\mu}$ (22)

and a closed CKY 3-form

$\tilde{\lambda}_{\mu\nu\rho}=(-1)^{d-1}(d-1)\tilde{K}_{\mu\nu\rho}$ (23)

with coefficients chosen for later convenience. The $\mathcal{B}$\- and
$\mathcal{D}$-type CKY tensors in eq. (20) give non-zero $k$, while the
$\mathcal{C}$\- and $\mathcal{D}$-type CKY tensors give non-zero
$\tilde{\lambda}$. Notably, the $\mathcal{A}$-type CKY tensors have $k=0$ and
$\tilde{\lambda}=0$.

There are also dual ADM charges which stem from certain invariances of the
dual graviton field HullYetAppear . The dual graviton,
$D_{\mu_{1}\dots\mu_{d-3}|\nu}$, is a mixed symmetry gauge field of rank
${[d-3,1]}$ which describes the same degrees of freedom as the graviton
$h_{\mu\nu}$. The Killing vectors of Minkowski space, which have a crucial
role in the ADM charges (12) of the graviton, are gauge parameters which leave
the graviton field invariant on all field configurations and so correspond to
global symmetries. Similarly, there are certain gauge transformations which
leave the dual graviton invariant on all field configurations. These are
analogous to the isometries generated by Killing vectors in the graviton
theory, and are similarly understood as global symmetries. The tensorial
parameters for these can be viewed as generalised Killing tensors.

The associated charges can (sometimes) be written in terms of the graviton
field and interpreted as magnetic charges for the graviton HullYetAppear .
This is most easily done with a particular choice of gauge for the graviton,
so that the resulting magnetic charges found in HullYetAppear  are given in
that particular gauge.

In $d=4$, the dual graviton is a symmetric tensor gauge field and its charges
are somewhat distinct to the case $d>4$. Here we focus on $d>4$ and will then
discuss $d=4$ in section 2.2. The magnetic charges are of two types, with
generalised Killing tensors denoted $\lambda$ and $\kappa$.

In the dual graviton theory, $\lambda$ is required to be a Killing-Yano (KY)
$(d-3)$-form. That is, it satisfies

$\partial_{\mu}\lambda_{\nu_{1}\dots\nu_{d-3}}=\partial_{[\mu}\lambda_{\nu_{1}\dots\nu_{d-3}]}$
(24)

It is a property of rank-$p$ KY forms that their Hodge dual is a closed CKY
$(d-p)$-form Frolov2008HigherdimensionalVariables ; Howe2018SCKYT . We will
denote the Hodge dual of $\lambda$ by $\tilde{\lambda}$, which is then a
closed CKY 3-form (i.e. satisfies eq. (21)).

The other type of generalised Killing tensors are the $\kappa$ tensors. In the
dual theory, these are ${[d-4,1]}$-tensors and so satisfy

$\kappa_{\mu_{1}\dots\mu_{d-4}|\nu}=\kappa_{[\mu_{1}\dots\mu_{d-4}]|\nu}$ (25)

and

$\kappa_{[\mu_{1}\dots\mu_{d-4}|\nu]}=0$ (26)

Furthermore, they satisfy a differential constraint

$\partial_{[\sigma}\kappa_{\mu_{1}\dots\mu_{d-4}]|\nu}=0$ (27)

Charges can be constructed from these generalised Killing tensors as integrals
of 2-form secondary currents $J[\lambda]$ and $J[\kappa]$, similarly to eq.
(12). We will refer to these as $\lambda$-charges and $\kappa$-charges
respectively. The secondary current $J[\lambda]$ satisfies

$\partial^{\nu}J[\lambda]_{\mu\nu}=\frac{(-1)^{d-1}}{d-1}\tilde{\lambda}^{\alpha\beta\gamma}R_{\mu\alpha\beta\gamma}\equiv
j[\lambda]_{\mu}$ (28)

and so is conserved by the Bianchi identity $R_{\nu[\alpha\beta\gamma]}=0$. An
explicit expression for $J[\lambda]$ is given in Appendix A, while $J[\kappa]$
will be discussed in section 3. In the dual description, magnetic sources can
be introduced which have the effect of sourcing the Bianchi identity, so the
‘primary’ current $j[\lambda]$ can be non-zero HullYetAppear . In the present
work we consider only the original graviton formulation of the theory and
therefore treat $R_{\nu[\alpha\beta\gamma]}=0$ as an identity.

The key result of Ref. PAPER1  is that the Penrose 2-form is related to the
secondary currents $J[k]$ and $J[\lambda]$ by

$Y[K]_{\mu\nu}=J[k]_{\mu\nu}+J[\lambda]_{\mu\nu}+\partial^{\rho}Z[K]_{\mu\nu\rho}$
(29)

where $Z[K]$ is a 3-form given by

$Z[K]^{\mu\nu\rho}=12K^{[\mu\nu}\Gamma\indices{{}^{\rho\beta]}_{|\beta}}$ (30)

Eq. (29) has been used to give a covariant generalisation of the ADM charges,
and the charges associated with the $\lambda$-invariances in Ref. PAPER1 .

In $d>4$, the Penrose charges associated with the $\mathcal{A}$\- and
$\mathcal{C}$-type CKY tensors (i.e. the KY tensors) vanish on Minkowski space
PAPER1 ; Benedetti2023GeneralizedGravitons , leaving only the $\mathcal{B}$\-
and $\mathcal{D}$-type charges which are related to the ADM charges. The
$\kappa$-charges, however, are not related to the Penrose 2-form. One of the
aims of this work is to find a manifestly gauge-invariant current constructed
in terms of the curvature and some special tensor with the property that the
current agrees with $J[\kappa]$ with the gauge choice used in HullYetAppear
and so gives a gauge-invariant generalisation of $J[\kappa]$ that can be used
in any gauge.

###  2.2 Four dimensions

Four dimensions are special due to the fact that the graviton and dual
graviton are both symmetric tensor gauge fields, as well as certain properties
of CKY tensors. Since the dual graviton is also a symmetric rank-2 gauge field
in $d=4$ its invariances are also given by the Killing vectors of Minkowski
space. Therefore, in this case, there are no independent $\lambda$ and
$\kappa$ tensors and these are only relevant for the dual graviton in $d>4$.
More precisely, the $\lambda$ tensors reduce to Killing vectors when $d=4$ and
the $\kappa$ tensors simply become constant 1-forms (which are also Killing
vectors). So the $\lambda$\- and $\kappa$-charges discussed above degenerate
in four dimensions.

Some of the charges associated with invariances of the four-dimensional dual
graviton can be written in terms of the graviton and expressed as a surface
integral of a secondary current $\tilde{J}[\tilde{k}]$ which is given in
Appendix A. Here $\tilde{k}$ is another Killing vector, independent of $k$,
which can be written in terms of a rank-2 CKY tensor as
$\tilde{k}=\star\differential{K}$ (whereas $k\propto\star\differential\star
K$). Then, in $d=4$, the Penrose 2-form is related to the secondary currents
by

$Y[K]_{\mu\nu}=J[k]_{\mu\nu}+\tilde{J}[\tilde{k}]_{\mu\nu}+\partial^{\rho}Z[K]_{\mu\nu\rho}$
(31)

Therefore, again in $d=4$, the Penrose charges give a gauge-invariant
generalisation of the ADM and dual ADM charges.

###  2.3 Penrose 2-form as a special case of $\Omega[V]$

We will later consider the more general family of conserved 2-forms
$\Omega[V]$ introduced in eq. (2). At this point, we simply demonstrate that
this more general set of 2-forms includes the Penrose 2-form $Y[K]$ for a
specific choice of $V$. Namely, choosing $V$ to be given by222This form of
$V[K]$ is closely related to the form of the Kastor-Traschen current for a KY
tensor $K$ (see Appendix of Ref. Kastor2004ConservedTensors ).

$V[K]\indices{{}_{\mu\nu\alpha\beta}^{|\gamma\delta}}=6\delta^{\gamma\delta}_{[\mu\nu}K_{\alpha\beta]}$
(32)

where
$\delta^{\gamma\delta}_{\mu\nu}\equiv\delta^{[\gamma}_{[\mu}\delta^{\delta]}_{\nu]}$
and $K$ is a CKY 2-form yields

$\Omega[V]_{\mu\nu}=6R\indices{{}^{\alpha\beta}_{\gamma\delta}}\delta^{\gamma}_{[\mu}\delta^{\delta}_{\nu}K_{\alpha\beta]}=6R\indices{{}^{\alpha\beta}_{[\mu\nu}}K_{\alpha\beta]}=R_{\mu\nu\alpha\beta}K^{\alpha\beta}=Y[K]_{\mu\nu}$
(33)

where we have used that $R_{\mu\nu}=0$ in the penultimate equality. Hence,
this particular choice of $V$ reduces $\Omega[V]$ to the Penrose 2-form.

##  3 $\kappa$-invariances and their associated charges

As mentioned in the previous section, for $d\geq 5$ the dual formulation of
linearised gravity has two types of invariances corresponding to two different
types of tensors, $\lambda$ and $\kappa$. We now discuss the $\kappa$ tensors
in more detail. These are ${[d-4,1]}$-tensors $\kappa$ satisfying eqs. (25),
(26) and (27). In particular, eq. (26) ensures the irreducibility of $\kappa$
under $GL(d,\mathbb{R})$. In the dual theory, this is essential for the dual
graviton field itself to remain irreducible. The irreducibility of the dual
graviton is necessary for the two descriptions of linearised gravity to
describe the same number of degrees of freedom. This irreducibility implies
that $\kappa$ transforms under a $GL(d,\mathbb{R})$ representation labelled by
a Young tableau with two columns of lengths $d-4$ and 1.

We define the dual of a $\kappa$ tensor, which we will denote
$\tilde{\kappa}$, by

$\tilde{\kappa}_{\alpha\beta\gamma\delta|\nu}=\frac{1}{(d-4)!}\epsilon_{\alpha\beta\gamma\delta\mu_{1}\dots\mu_{d-4}}\kappa\indices{{}^{\mu_{1}\dots\mu_{d-4}}_{|\nu}}$
(34)

This is a ${\lfloor 4,1\rfloor}$-tensor,

$\tilde{\kappa}_{\mu\nu\rho\sigma|\alpha}=\tilde{\kappa}_{[\mu\nu\rho\sigma]|\alpha}$
(35)

which satisfies

$\partial^{\mu}\tilde{\kappa}_{\mu\nu\rho\sigma|\alpha}=0$ (36)

from eq. (27). If $\kappa$ is irreducible, i.e. if it satisfies eq. (26), then
$\tilde{\kappa}$ is traceless, i.e.
$\tilde{\kappa}\indices{{}_{\mu\nu\rho\sigma|}^{\sigma}}=0$. The duality
operation in eq. (34) can be thought of as a generalisation of the Hodge dual
operation that acts on ${\lfloor p,q\rfloor}$-tensors, which we will return to
in section 5.

With a particular gauge choice for both the graviton theory and the dual
graviton theory, the connection for the dual graviton theory is dual to the
connection $\Gamma$ for the graviton theory Ref. HullYetAppear . For the
graviton, this gauge choice is

$\Gamma\indices{{}_{\mu\nu|}^{\nu}}=0$ (37)

In this gauge, the secondary current for the dual graviton theory associated
with the $\kappa$-invariances can be written locally in terms of the graviton
in this gauge for all $\kappa$ tensors as HullYetAppear

$J[\kappa]_{\mu\nu}=(-1)^{d}\frac{2}{d-1}\tilde{\kappa}_{\mu\nu\rho\sigma|\alpha}\Gamma^{\alpha\sigma|\rho}$
(38)

which satisfies

$\partial^{\nu}J[\kappa]_{\mu\nu}=\frac{(-1)^{d}}{d-1}\tilde{\kappa}_{\mu\alpha\beta\gamma|\sigma}R^{\sigma\alpha\beta\gamma}\equiv
j[\kappa]_{\mu}$ (39)

so is conserved by the Bianchi identity $R_{\sigma[\alpha\beta\gamma]}=0$. As
for $j[\lambda]$ in eq. (28), the ‘primary’ current $j[\kappa]$ can be non-
zero when sources are included in the dual formulation of the theory. Here, we
consider only the source-free case and so $j[\kappa]=0$ identically. Now,
integrating over a codimension-2 cycle $\Sigma_{d-2}$ gives a conserved charge
HullYetAppear

$Q[\kappa]=\int_{\Sigma_{d-2}}\star J[\kappa]$ (40)

which we refer to as a $\kappa$-charge.

The secondary current $J[\kappa]$ in eq. (38) can be written locally as a
total divergence

$J[\kappa]_{\mu\nu}=\partial^{\rho}Z[\kappa]_{\mu\nu\rho}$ (41)

with $Z[\kappa]$ given by

$Z[\kappa]_{\mu\nu\rho}=(-1)^{d}\frac{1}{d-1}\,\tilde{\kappa}_{\mu\nu\rho\sigma|\alpha}h^{\sigma\alpha}$
(42)

This implies that the charge $Q[\kappa]$ is topological and is non-zero only
when $h_{\mu\nu}$ is a non-globally defined field configuration. Indeed, as
$\star J[\kappa]$ is closed, the charge $\int_{\Sigma_{d-2}}\star J[\kappa]$,
for a given cycle $\Sigma_{d-2}$, depends only on the cohomology class of
$\star J[\kappa]$ and vanishes if it is an exact form. As locally $\star
J=\differential(\star Z)$, it will be exact only if $\star Z$ is a globally
defined form, which will be the case if $h_{\mu\nu}$ is globally defined. See
Refs. PAPER1 ; HullYetAppear  for further discussion.

Note that the secondary Noether current $J[\kappa]$ derived in terms of the
dual graviton in Ref. HullYetAppear  can be written locally in terms of the
graviton for any $\tilde{\kappa}$ tensors satisfying eq. (36), whereas
$J[\lambda]$ can be written locally in terms of the graviton if
$\tilde{\lambda}$ is constant.

We note from eq. (42) that $J[\kappa]$ will be conserved whether $\kappa$ is
irreducible or not (i.e. whether $\tilde{\kappa}$ is traceless or not).
Therefore, as far as the charges constructed with $\kappa$ are concerned,
there is no need for $\kappa$ to be irreducible and charges can equally well
be defined if it is reducible. The condition of irreducibility came from
requiring the charge to result from an invariance of the dual graviton theory.
Since we work directly in the graviton theory in this paper we will _not_
insist that $\kappa$ satisfy eq. (26). That is, here we require only that
$\kappa$ is a ${\lfloor d-4,1\rfloor}$-tensor (rather than an irreducible
${[d-4,1]}$-tensor). A priori, $\tilde{\kappa}$ is a ${\lfloor
4,1\rfloor}$-tensor. The reducible ${\lfloor 4,1\rfloor}$ representation of
$GL(d,\mathbb{R})$ is a tensor product
${\ydiagram{1,1,1,1}}\otimes{\ydiagram{1}}={\ydiagram{1,1,1,1,1}}\oplus{\ydiagram{2,1,1,1}}$.
The component of $\tilde{\kappa}$ in the ${\ydiagram{1,1,1,1,1}}$
representation does not contribute to the charge $Q[\kappa]$ as $Z[\kappa]$ in
eq. (42) vanishes identically when $\tilde{\kappa}$ is fully antisymmetric.
Therefore, only those $\tilde{\kappa}$-tensors in the irreducible ${[4,1]}$
representation of $GL(d,\mathbb{R})$ contribute to non-zero charges, so we
will restrict $\tilde{\kappa}$ to be a ${[4,1]}$-tensor.

##  4 $\kappa$-charge as a special case of $\Omega[V]$

###  4.1 Charges for Minkowski space

The Penrose currents $Y[K]$, which have been shown to give a gauge-invariant
construction of the ADM and $\lambda$-charges, are not related to the
$\kappa$-charges. Indeed, as we have stressed, $Y[K]$ is not the most general
covariant conserved 2-form. Instead, the most general 2-form which is linear
in the linearised Riemann tensor takes the form

$\Omega[V]_{\mu\nu}=R^{\alpha\beta\gamma\delta}V_{\mu\nu|\alpha\beta|\gamma\delta}$
(43)

where $V$ is a ${\lfloor 2,2,2\rfloor}$-tensor which must satisfy particular
constraints in order for $\Omega[V]$ to be conserved. Each $V$ satisfying
these constraints gives a co-closed 2-form which can be integrated to give a
conserved charge

$\mathbb{Q}[V]=\int_{\Sigma_{d-2}}\star\Omega[V]$ (44)

where, $\Sigma_{d-2}$ is a codimension-2 cycle. If $\Omega[V]$ is co-exact,
then $\mathbb{Q}[V]=0$ on any cycle $\Sigma_{d-2}$ and there is no
corresponding 1-form symmetry.

We have seen in section 2.3 that $\Omega[V]$ reduces to the Penrose 2-form
$Y[K]$ when $V$ is given by eq. (32), which then provides a covariantisation
of the ADM and dual ADM charges in eq. (29). We now ask if there is a choice
of $V$ for which $\Omega[V]$ agrees with the secondary current $J[\kappa]$
introduced in eq. (38), subject to the gauge choice (37). If such a
construction were possible, then the charge $\mathbb{Q}[V]$ would provide a
gauge-invariant formulation of the $\kappa$-charges.

Given a $\kappa$-tensor, suppose there exists a ${\lfloor 4,2\rfloor}$-tensor
$V[\tilde{\kappa}]$ satisfying

$\partial^{\mu}V[\tilde{\kappa}]_{\mu\nu\alpha\beta|\gamma\delta}=\frac{(-1)^{d}}{d-1}\tilde{\kappa}_{\nu\alpha\beta[\gamma|\delta]}$
(45)

Then, for such a $V$, we find

$\displaystyle\Omega[V[\tilde{\kappa}]]_{\mu\nu}$
$\displaystyle=R^{\alpha\beta\gamma\delta}V[\tilde{\kappa}]_{\mu\nu\alpha\beta|\gamma\delta}=J[\kappa]_{\mu\nu}+\partial^{\alpha}\Phi_{\mu\nu\alpha}$
(46)

where

$\Phi_{\mu\nu\alpha}=2\Gamma^{\gamma\delta|\beta}V[\tilde{\kappa}]_{\mu\nu\alpha\beta|\gamma\delta}$
(47)

This implies that $\Omega[V[\tilde{\kappa}]]$ is conserved,

$\partial^{\mu}\Omega[V[\tilde{\kappa}]]_{\mu\nu}=0$ (48)

from eq. (41), so that

$\mathbb{Q}[\kappa]\equiv\int_{\Sigma_{d-2}}\star\Omega[V[\tilde{\kappa}]]$
(49)

is a _gauge-invariant_ conserved charge constructed from the $\kappa$ tensor.
Eq. (46) gives an explicit relation between the secondary currents $J[\kappa]$
associated with the $\kappa$-invariances of the dual graviton and the
covariant 2-forms $\Omega[V[\tilde{\kappa}]]$ so that the charges are related
by

$\mathbb{Q}[\kappa]=Q[\kappa]+\int_{\Sigma_{d-2}}\differential\star\Phi$ (50)

Using eq. (42), the covariant charge $\mathbb{Q}[\kappa]$ is a total
derivative

$\mathbb{Q}[\kappa]=\int_{\Sigma_{d-2}}\differential\star(Z[\kappa]+\Phi)$
(51)

and so is a topological charge which will only be non-zero if $h_{\mu\nu}$ is
not globally defined.

The covariant charge $\mathbb{Q}[\kappa]$ is then the sum of
$Q[\kappa]=\int\differential\star Z[\kappa]$ and a further total derivative
term $\int\differential\star\Phi$. Neither $Z[\kappa]$ not $\Phi$ is gauge-
invariant, but their sum gives the gauge-invariant charge
$\mathbb{Q}[\kappa]=\int\differential\star(Z[\kappa]+\Phi)$. In section 7 we
will consider examples where the charge $\mathbb{Q}[\kappa]$ is non-zero and
will find a choice of gauge where the $\Phi$ term does not contribute, giving
$\mathbb{Q}[\kappa]=Q[\kappa]$. In other gauges, both terms contribute to
$\mathbb{Q}[\kappa]$ for these examples. However, in general the $\Phi$ term
need not vanish with the gauge condition (37), so in general the two charges
$\mathbb{Q}[\kappa]$ and $Q[\kappa]$ differ, even with the gauge choice (37).
Both are conserved topological charges given by the integral of a total
derivative, with $\mathbb{Q}[\kappa]$ a gauge-invariant charge and $Q[\kappa]$
a non-gauge-invariant Noether-type charge associated with a gauge invariance
of the dual theory. Recall that for branes, both covariant and non-covariant
charges play a role and are related but distinct Marolf:2000cb .

We can gain some further insight by analysing these charges in the dual
formulation of the theory. There, $Q[\kappa]$ is an electric charge for the
dual graviton theory, while the covariant charge $\mathbb{Q}[\kappa]$ can be
understood as a covariant charge for the dual graviton. Neither $Q[\kappa]$
nor $\mathbb{Q}[\kappa]$ is a topological charge for the dual graviton theory,
but their difference is the integral of a total derivative that will vanish
for solutions for which the dual graviton is globally defined, but can give a
non-zero topological charge more generally. Adding this topological term to
$Q[\kappa]$ then covariantises it to give the gauge-invariant charge
$\mathbb{Q}[\kappa]$. We will return to this observation in an upcoming work.

This discussion hinges on the existence of a ${\lfloor 4,2\rfloor}$-tensor
$V[\tilde{\kappa}]$ satisfying eq. (45) for a given $\kappa$-tensor. If the
background spacetime is Minkowski space, possibly with some points or regions
removed but _without_ any compactified directions, a solution is easily found
to be

$V[\tilde{\kappa}]_{\mu\nu\alpha\beta|\gamma\delta}=\frac{(-1)^{d}}{d-1}\tilde{\kappa}_{\mu\nu\alpha\beta|[\gamma}x_{\delta]}$
(52)

It is straightforward to verify that this satisfies eq. (45) for any
$\tilde{\kappa}$-tensor by using eq. (36). This choice of $V$ is clearly not
unique, and any ${\lfloor 4,2\rfloor}$-tensor $V_{0}$ satisfying
$\partial^{\mu}(V_{0})_{\mu\nu\alpha\beta|\gamma\delta}=0$ can be added to
$V[\tilde{\kappa}]$ to define another valid solution. At this point, we merely
point out that a choice of $V$ is available which relates the charges
$\mathbb{Q}[V]$ and $Q[\kappa]$ on Minkowski space. In section 8.1 we will
discuss in more generality the charges $\mathbb{Q}[V_{0}]$ which can arise
from the inclusion of such extra terms in eq. (52).

In the more general discussion of section 8, we will see that it is sufficient
to require $V$ to be an irreducible ${[4,2]}$-tensor (rather than an arbitrary
${\lfloor 2,2,2\rfloor}$-tensor) in order to construct all the distinct
charges $\mathbb{Q}[V]$. Therefore, only the irreducible ${[4,2]}$ part of
$V[\tilde{\kappa}]$ given in eq. (52) will contribute to the charges. Note
that if $V$ is irreducible, then eq. (45) implies that $\tilde{\kappa}$ is an
irreducible ${[4,1]}$-tensor. This is not a restriction as only irreducible
$\tilde{\kappa}$ tensors give non-zero charges (see section 3).

###  4.2 Choice of $V$ for solutions with compact directions

The situation is more complicated when the background spacetime has compact
directions. If $\mathcal{M}$ is the product of $T^{n}$ with
$(d-n)$-dimensional Minkowski space (or Minkowski space with some regions
removed) then $V$ must be periodic in the torus coordinates $y^{i}$ to be
well-defined. In this work, we will achieve this by requiring that
$\partial_{y^{i}}V_{\mu\nu\alpha\beta|\gamma\delta}=0$. We now discuss the
explicit constructions which give solutions of eq. (45) on such spaces.

Let us begin with the simplest case of $d=5$ and $n=1$, so
$\mathcal{M}=\mathbb{R}^{1,3}\times S^{1}$. We denote the coordinates by
$x^{\mu}=(x^{m},y)$. Crucially, the definition (52) is not well-defined on
this space since the component

$V[\tilde{\kappa}]_{\mu\nu\rho\sigma|my}=-\frac{1}{4}(\tilde{\kappa}_{\mu\nu\rho\sigma|m}y-\tilde{\kappa}_{\mu\nu\rho\sigma|y}x_{m})$
(53)

is not periodic in $y$. Indeed, for a general choice of $\tilde{\kappa}$ on a
background spacetime which has compact directions, eq. (45) may not have a
well-defined solution and the manipulations leading to eq. (46) may break
down.

However, for the five-dimensional case, there are two possible solutions to
this apparent problem which can arise for specific choices of
$\tilde{\kappa}$. The first possibility is the simple observation that
$V[\tilde{\kappa}]_{\mu\nu\rho\sigma|my}$ in eq. (53) above _is_ well-defined
if the components $\tilde{\kappa}_{\mu\nu\rho\sigma|m}=0$. That is, for
particular choices of $\tilde{\kappa}$, constructing $V[\tilde{\kappa}]$ as in
eq. (52) remains valid. The second possibility is that, while
$V[\tilde{\kappa}]$ in eq. (52) may be ill-defined for a given
$\tilde{\kappa}$, there may exist another ${[4,2]}$-tensor $V$ which solves
eq. (45). Then the manipulations leading to the crucial result in eq. (46) go
through as before. The expression given in eq. (52) is merely one possible
solution on Minkowski space. To summarise, for a given solution there may not
be a $V$ satisfying (45) for general $\kappa$, so we seek those $\kappa$ for
which there is such a $V$ and then use that $V[\kappa]$ in our construction.

Let us study a pair of cases that are relevant to the examples we consider
later. In $d=5$, the irreducible $\kappa$ tensors are in the ${\ydiagram{2}}$
representation. For the example studied in section 7.1, the $\kappa$-tensor
that gives a non-zero charge has non-vanishing components

$\kappa_{y|y}=\kappa_{0}$ (54)

where $\kappa_{0}$ is a constant. The dual (34) of this tensor has non-
vanishing components

$\tilde{\kappa}_{tabc|y}=\kappa_{0}\epsilon_{abc}$ (55)

Note that, in particular, the $\tilde{\kappa}_{\mu\nu\rho\sigma|m}$ components
all vanish and so $V[\tilde{\kappa}]$ given by eq. (52) is well-defined. This
is a realisation of the first solution outlined above.

For the charge studied in section 7.2, the relevant $\kappa$-tensor has non-
zero components

$\kappa_{t|y}=\kappa_{y|t}=\kappa_{0}$ (56)

The dual of this tensor has

$\tilde{\kappa}_{abct|t}=\tilde{\kappa}_{abcy|y}=-\epsilon_{abc}\kappa_{0}$
(57)

and all other components of $\tilde{\kappa}$ which are not related to these by
anti-symmetry vanish. In this case, $V[\tilde{\kappa}]$ defined in eq. (52) is
not well-defined on the compact direction and we must seek a different, well-
defined $V$ which satisfies eq. (45) for this $\tilde{\kappa}$. We consider
instead the ${[4,2]}$-tensor $V$ with components

$V_{abct|dt}=V_{abcy|dy}=-\frac{1}{8}\kappa_{0}\epsilon_{abc}x_{d}$ (58)

and all other components not related to these by anti-symmetry vanish. It can
be straightforwardly checked that this solves eq. (45) for $\tilde{\kappa}$
given in eq. (57). Furthermore, it has no dependence on the $S^{1}$ coordinate
$y$, so is well-defined on $\mathcal{M}$. It is a ${[4,2]}$-tensor as
$\epsilon_{[abc}x_{d]}=0$ with $a,b=1,2,3$. This is a realisation of the
second solution described above.

We now consider solutions in $d>5$ dimensions. In particular, we fix
$\mathcal{M}=\mathbb{R}^{1,3}\times T^{n}$ with $n>1$. Cases with Minkowski
space of greater than four dimensions can be analysed in an analogous manner.
With $n>1$, $V[\tilde{\kappa}]$ in eq. (52) has components

$\displaystyle V[\tilde{\kappa}]_{\mu\nu\rho\sigma|mn}$
$\displaystyle=\frac{(-1)^{d}}{2(d-1)}\left(\tilde{\kappa}_{\mu\nu\rho\sigma|m}x_{n}-\tilde{\kappa}_{\mu\nu\rho\sigma|n}x_{m}\right)$
(59) $\displaystyle V[\tilde{\kappa}]_{\mu\nu\rho\sigma|mi}$
$\displaystyle=\frac{(-1)^{d}}{2(d-1)}\left(\tilde{\kappa}_{\mu\nu\rho\sigma|m}y_{i}-\tilde{\kappa}_{\mu\nu\rho\sigma|i}x_{m}\right)$
(60) $\displaystyle V[\tilde{\kappa}]_{\mu\nu\rho\sigma|ij}$
$\displaystyle=\frac{(-1)^{d}}{2(d-1)}\left(\tilde{\kappa}_{\mu\nu\rho\sigma|i}y_{j}-\tilde{\kappa}_{\mu\nu\rho\sigma|j}y_{i}\right)$
(61)

In order for the $V[\tilde{\kappa}]_{\mu\nu\rho\sigma|mi}$ components to be
well-defined on the toroidal directions, we require
$\tilde{\kappa}_{\mu\nu\rho\sigma|m}=0$ for all $m$. For the
$V[\tilde{\kappa}]_{\mu\nu\rho\sigma|ij}$ components to be well-defined, we
require $\tilde{\kappa}_{\mu\nu\rho\sigma|i}=0$. Therefore, unless
$\tilde{\kappa}=0$ (and so $V[\tilde{\kappa}]=0$), this choice of ${\lfloor
4,2\rfloor}$-tensor $V[\tilde{\kappa}]$ is not well-defined. Therefore, we are
forced to find an alternative solution to eq. (45).

We now describe such solutions for the $\tilde{\kappa}$-tensors given above in
eqs. (55) and (57). In $d>5$ dimensions, irreducible $\kappa$-tensors are in
the ${[d-4,1]}$ representation. The $\kappa$-tensor whose dual is the
$\tilde{\kappa}$-tensor in eq. (55) has components

$\kappa_{y_{1}y_{2}\dots y_{d-4}|y_{1}}=\kappa_{0}$ (62)

in $d$-dimensions. Let us define a ${\lfloor 4,2\rfloor}$-tensor $V$ with
components

$V_{\mu\nu\rho\sigma|mi}=V[\tilde{\kappa}]_{\mu\nu\rho\sigma|mi}$ (63)

with the right-hand side given in eq. (60) and all other components vanishing.
Since $\tilde{\kappa}_{\mu\nu\rho\sigma|m}=0$ for this example, this $V$ is
well-defined. It is straightforward to verify that it satisfies eq. (45) for
$\tilde{\kappa}$ in eq. (55).

Finally, let us the consider the $d>5$ dimensional case with $\tilde{\kappa}$
given by eq. (57). The $\kappa$-tensor dual to this is

$\kappa_{ty_{2}\dots y_{d-4}|y_{1}}=\kappa_{y_{1}y_{2}\dots
y_{d-4}|t}=\kappa_{0}$ (64)

The tensor $V$ constructed in eq. (58) remains well-defined and satisfies eq.
(45) for this $\tilde{\kappa}$.

###  4.3 Condition for non-trivial charge

While a conserved current $\Omega[V]$ can be built for each ${\lfloor
4,1\rfloor}$-tensor $\tilde{\kappa}$ by using $V[\kappa]$ in eq. (52), we note
that if $\tilde{\kappa}$ is of the form

$\tilde{\kappa}_{\mu\nu\alpha\beta|\gamma}=\partial^{\sigma}\rho_{\sigma\mu\nu\alpha\beta|\gamma}$
(65)

for some ${\lfloor 5,1\rfloor}$-tensor $\rho$, then the charge
$\mathbb{Q}[\kappa]$ vanishes. This follows as

$\Omega[V[\tilde{\kappa}]]_{\mu\nu}=\frac{(-1)^{d}}{d-1}R^{\alpha\beta\gamma\delta}\partial^{\sigma}\rho_{\sigma\mu\nu\alpha\beta|\gamma}x_{\delta}=\frac{(-1)^{d}}{d-1}\partial^{\sigma}(R^{\alpha\beta\gamma\delta}\rho_{\sigma\mu\nu\alpha\beta|\gamma}x_{\delta})\equiv\partial^{\sigma}\Xi_{\mu\nu\sigma}$
(66)

where we have used $\partial_{[\sigma}R_{\alpha\beta]\gamma\delta}=0$ in the
final equality. $\Xi$ is a globally defined 3-form since it depends only on
the curvature, so
$\mathbb{Q}[\kappa]=\int\star\Omega[V[\tilde{\kappa}]]=\int\differential\star\Xi=0$
by Stokes’ theorem as the integral is assumed to be taken over a cycle.
Therefore, non-trivial $\mathbb{Q}[\kappa]$ charges only occur when the
spacetime admits $\tilde{\kappa}$-tensors satisfying eq. (36) which are not of
the form (65). This implies a notion of cohomology which we will develop in
the following sections.

##  5 Operations on ${\lfloor p,q\rfloor}$-tensors

In order to facilitate the more general discussion which follows, in
particular in section 8, we use the formalism of Ref.
Medeiros2003ExoticDuality  for multiforms, i.e. tensors of mixed Young
symmetry. In particular, we focus on ${\lfloor p,q\rfloor}$-tensors and follow
the conventions of Ref. Medeiros2003ExoticDuality . (A similar formalism was
introduced in Bekaert2004 .)

Consider a ${\lfloor p,q\rfloor}$-tensor $A$. We define a pair of exterior
derivatives $\differential_{L}$ and $\differential_{R}$ which have components

$\displaystyle(\differential_{L}A)_{\mu_{1}\dots\mu_{p+1}|\nu_{1}\dots\nu_{q}}$
$\displaystyle=\partial_{[\mu_{1}}A_{\mu_{2}\dots\mu_{p+1}]|\nu_{1}\dots\nu_{q}}$
(67)
$\displaystyle(\differential_{R}A)_{\mu_{1}\dots\mu_{p}|\nu_{1}\dots\nu_{q+1}}$
$\displaystyle=A_{\mu_{1}\dots\mu_{p}|[\nu_{1}\dots\nu_{q},\nu_{q+1}]}$ (68)

where a comma denotes a partial derivative. These generalise the standard de
Rham co-boundary operator $\differential$ which acts on differential forms. It
follows immediately that both of these differential operators are nilpotent:

$\differential_{L}^{2}=0,\quad\differential_{R}^{2}=0$ (69)

and commute, $[d_{L},d_{R}]=0$, so that $(d_{L}+d_{R})^{3}=0$. We refer to
${\lfloor p,q\rfloor}$-tensors $A$ satisfying $\differential_{L}A=0$ as
_$\differential_{L}$ -closed_ and those satisfying $\differential_{R}A=0$ as
_$\differential_{R}$ -closed_. Similarly, we refer to tensors $A$ satisfying
$A=\differential_{L}B$ for some ${\lfloor p-1,q\rfloor}$-tensor $B$ as being
_$\differential_{L}$ -exact_ and those satisfying $A=\differential_{R}C$ for
some ${\lfloor p,q-1\rfloor}$-tensor $C$ as being _$\differential_{R}$
-exact_. For example, the property (27) which the $\kappa$-tensors are
required to satisfy can be simply stated as $\kappa$ being
$\differential_{L}$-closed, i.e. $\differential_{L}\kappa=0$.

The property (69) leads to a notion of cohomology for bi-forms in flat space
Medeiros2003ExoticDuality ; Bekaert2004 ; Dubois-Violette:1999iqe ; Dubois-
Violette:2000fok ; Dubois-Violette:2001wjr  and is relevant to the theory of
higher spin fields in general Lorentz representations
Medeiros2003ExoticDuality ; deMedeiros:2003osq ; Hull2001DualityFields ;
Henneaux:2004jw . Similar cohomologies have also been employed recently in
discussions of ‘t Hooft anomalies between electric and magnetic symmetries
with fields in more exotic representations Hinterbichler2023GravitySymmetries
; Hinterbichler:2024cxn . Generalising to a curved space requires introducing
a connection and the corresponding covariant exterior derivatives
$\nabla_{L},\nabla_{R}$ are not nilpotent in general. Speculations about a
cohomology of such complexes on (A)dS have been used in discussions of duality
of partially massless representations Hinterbichler:2014xga ;
Hinterbichler:2016fgl .

The $\differential_{L}$-cohomology of ${\lfloor p,q\rfloor}$-tensors on a flat
space is the set of $\differential_{L}$-closed ${\lfloor p,q\rfloor}$-tensors
modulo those which are $\differential_{L}$-exact, with an analogous definition
for $\differential_{R}$-cohomology.

A simple generalisation of the Hodge star operation to ${\lfloor
p,q\rfloor}$-tensors is introduced as follows. Consider ${\lfloor
p,q\rfloor}$-tensors transforming under a reducible representation of
$SO(1,d-1)\subset GL(d,\mathbb{R})$. Then there are two $SO(1,d-1)$-invariant
tensors: the fully antisymmetric $\epsilon_{\mu_{1}\dots\mu_{d}}$, and the
metric $\eta_{\mu\nu}$. We define operations $\star_{L}$ and $\star_{R}$ using
the former, by

$\displaystyle(\star_{L}A)_{\mu_{1}\dots\mu_{d-p}|\nu_{1}\dots\nu_{q}}$
$\displaystyle=\frac{1}{p!}\epsilon_{\mu_{1}\dots\mu_{d-p}\alpha_{1}\dots\alpha_{p}}A\indices{{}^{\alpha_{1}\dots\alpha_{p}}_{|\nu_{1}\dots\nu_{q}}}$
(70) $\displaystyle(\star_{R}A)_{\mu_{1}\dots\mu_{p}|\nu_{1}\dots\nu_{d-q}}$
$\displaystyle=\frac{1}{q!}\epsilon_{\nu_{1}\dots\nu_{d-q}\alpha_{1}\dots\alpha_{q}}A\indices{{}_{\mu_{1}\dots\mu_{p}|}^{\alpha_{1}\dots\alpha_{q}}}$
(71)

which satisfy

$\star_{L}^{2}=(-1)^{p(d-p)+1},\quad\star_{R}^{2}=(-1)^{q(d-q)+1}$ (72)

We have already made use of this operation in defining $\tilde{\kappa}$ in eq.
(34), which can equivalently be written $\tilde{\kappa}=\star_{L}\kappa$.
Furthermore we have the adjoint derivatives $\differential_{L}^{\dagger}$ and
$\differential_{R}^{\dagger}$ which act as

$\displaystyle(\differential_{L}^{\dagger}A)_{\mu_{2}\dots\mu_{p}|\nu_{1}\dots\nu_{q}}$
$\displaystyle=\partial^{\mu_{1}}A_{\mu_{1}\dots\mu_{p}|\nu_{1}\dots\nu_{q}}$
(73)
$\displaystyle(\differential_{R}^{\dagger}A)_{\mu_{1}\dots\mu_{p}|\nu_{2}\dots\nu_{q}}$
$\displaystyle=\partial^{\nu_{1}}A_{\mu_{1}\dots\mu_{p}|\nu_{1}\dots\nu_{q}}$
(74)

which imply nilpotency $(\differential^{\dagger}_{L})^{2}=0$ and
$(\differential^{\dagger}_{R})^{2}=0$. We refer to tensors $A$ satisfying
$\differential_{L}^{\dagger}A=0$ as _$\differential_{L}$ -coclosed_, while
those satisfying $\differential_{R}^{\dagger}A=0$ are referred to as
_$\differential_{R}$ -coclosed_. Analogously, ${\lfloor p,q\rfloor}$-tensors
$A$ satisfying $A=\differential_{L}^{\dagger}B$ for some ${\lfloor
p+1,q\rfloor}$-tensor $B$ are called _$\differential_{L}$ -coexact_ and those
satisfying $A=\differential_{R}^{\dagger}C$ for some ${\lfloor
p,q+1\rfloor}$-tensor $C$ are called _$\differential_{R}$ -coexact_.

If $A$ is a $\differential_{L}$-closed ${\lfloor p,q\rfloor}$-tensor, then
$\star_{L}A$ is $\differential_{L}$-coclosed. Similarly, if $B$ is a
$\differential_{L}$-exact ${\lfloor p,q\rfloor}$-tensor, then $\star_{L}B$ is
$\differential_{L}$-coexact. The analogous statements apply to $\star_{R}$.

##  6 Charges and cohomology

We have seen in section 4.3 that if $\kappa$ is $\differential_{L}$-exact,
then the charge $\mathbb{Q}[\kappa]$ vanishes. Indeed, the same is true of the
non-covariant charge $Q[\kappa]$ HullYetAppear . Hence, in order to construct
non-zero charges we require a $\kappa$-tensor in a non-trivial
$\differential_{L}$-cohomology class.

Furthermore, in order for the charge
$\mathbb{Q}[\kappa]=\int\star\Omega[V[\tilde{\kappa}]]$ to be non-zero, the
$(d-2)$-form $\star\Omega[V]$ must itself be closed and not exact, so that it
represents a non-trivial de Rham cohomology class. There are, therefore, two
different forms which must be in non-trivial cohomology classes for there to
be a non-zero charge. We consider them in turn below.

###  6.1 $\differential_{L}$-cohomology for $\kappa$-tensors

We now consider the general form for the $\kappa$-tensors, following
HullYetAppear . The $\kappa$-tensors satisfy eq. (27), which we repeat here:

$\partial_{[\mu}\kappa_{\alpha_{1}\dots\alpha_{d-4}]|\beta}=0$ (75)

i.e. they are $\differential_{L}$-closed. As pointed out previously,
$\differential_{L}$-exact $\kappa$-tensors will satisfy this automatically.
That is, if
$\kappa_{\alpha_{1}\dots\alpha_{d-4}|\beta}=\partial_{[\alpha_{1}}\rho_{\alpha_{2}\dots\alpha_{d-4}]|\beta}$
for some ${\lfloor d-5,1\rfloor}$-tensor $\rho$, then eq. (75) is trivially
satisfied.333In $d=5$ this remains true when $\kappa$ is reducible, but not
when it is irreducible. In that case, the relevant $\differential_{L}$-exact
form for a $\kappa$-tensor is
$\kappa_{\mu|\nu}=\partial_{\mu}\partial_{\nu}\psi$ for a scalar $\psi$. We
also know from section 4.3 that the charges $Q[\kappa]$ and
$\mathbb{Q}[\kappa]$ vanish in this case, so these charges depend only on the
$\differential_{L}$-cohomology class.

We begin with the case of reducible $\kappa$-tensors, which are in the
reducible ${\lfloor d-4,1\rfloor}$ representation. This is a tensor product
representation of the fully antisymmetric $(d-4)$-form representation and the
1-form representation. As such, reducible $\kappa$-tensors can be written as a
linear combination of tensors of the basic type

$\kappa^{\text{red.}}_{\alpha_{1}\dots\alpha_{d-4}|\beta}=\tau_{\alpha_{1}\dots\alpha_{d-4}}\sigma_{\beta}$
(76)

where $\tau$ is a $(d-4)$-form and $\sigma$ is a 1-form. For such a tensor,
eq. (75) is satisfied if $\tau$ is closed and $\sigma$ is constant, and will
not be $\differential_{L}$-exact only if $\tau$ is not exact. Then the non-
trivial $\kappa$-tensors are constructed from $(d-4)$-forms $\tau$ that
represent a non-trivial cohomology class, together with constant 1-forms.
Requiring non-trivial $\differential_{L}$-cohomology for $\kappa$ then relies
on having a non-trivial de Rham cohomology group $H^{d-4}(\mathcal{M})$.
General solutions then arise from taking linear combinations of terms of the
form (76). An alternative way of seeing this is to think of
$\kappa_{\alpha_{1}\dots\alpha_{d-4}|\beta}$ as being a vector-valued
$(d-4)$-form, which is closed and non-exact for each value of $\beta$.

If we instead consider irreducible $\kappa$-tensors, for which

$\kappa^{\text{irred.}}_{[\alpha_{1}\dots\alpha_{d-4}|\beta]}=0$ (77)

these can be written as a linear combination of tensors of the form

$\kappa^{\text{irred.}}_{\alpha_{1}\dots\alpha_{d-4}|\beta}=\tau_{\alpha_{1}\dots\alpha_{d-4}}\sigma_{\beta}-\tau_{[\alpha_{1}\dots\alpha_{d-4}}\sigma_{\beta]}$
(78)

This follows from the fact that all irreducible ${[p,q]}$-tensors can be
written as the Young symmetrised version of a reducible ${\lfloor
p,q\rfloor}$-tensor, and because the Young symmetriser is a linear operator.
Indeed, $\kappa^{\text{irred.}}$ is simply the Young symmetrised version of
$\kappa^{\text{red.}}$ onto the irreducible ${[d-4,1]}$ $GL(d,\mathbb{R})$
representation.

For closed $\tau$ and constant $\sigma$, we have

$\partial_{[\mu}\kappa^{\text{irred.}}_{\alpha_{1}\dots\alpha_{d-4}]|\beta}=(-1)^{d-3}\frac{d-4}{d-3}\partial_{[\mu|}\tau_{\beta|\alpha_{1}\dots\alpha_{d-5}}\sigma_{\alpha_{d-4}]}$
(79)

Demanding that this vanishes implies

$\partial_{\beta}(\tau_{[\alpha_{1}\dots\alpha_{d-4}}\sigma_{\mu]})=0$ (80)

so $\tau\wedge\sigma=\text{constant}$. This is solved by

$\tau=\tau_{0}+\beta\wedge\sigma$ (81)

where $\tau_{0}$ is a constant $(d-4)$-form and $\beta$ is a closed
$(d-5)$-form (such that $\tau$ is closed). Substituting back into eq. (78), we
see that

$\kappa^{\text{irred.}}_{\alpha_{1}\dots\alpha_{d-4}|\beta}=(\kappa_{0})_{\alpha_{1}\dots\alpha_{d-4}|\beta}+\beta_{[\alpha_{1}\dots\alpha_{d-5}}\sigma_{\alpha_{d-4}]}\sigma_{\beta}$
(82)

where
$(\kappa_{0})_{\alpha_{1}\dots\alpha_{d-4}|\beta}=(\tau_{0})_{\alpha_{1}\dots\alpha_{d-4}}\sigma_{\beta}-(\tau_{0})_{[\alpha_{1}\dots\alpha_{d-4}}\sigma_{\beta]}$
is a constant irreducible ${[d-4,1]}$ tensor. A general irreducible $\kappa$
tensor can be written as a linear combination of tensors of the form of eq.
(82). Since the constant term $\kappa_{0}$ can be written as a linear
combination of tensors of the form
$\beta_{[\alpha_{1}\dots\alpha_{d-5}}\sigma_{\alpha_{d-4}]}\sigma_{\beta}$
where $\beta$ is taken to be a constant $(d-5)$-form, we can equivalently
write the general solution as a linear combination of tensors of the form

$\kappa^{\text{irred.}}_{\alpha_{1}\dots\alpha_{d-4}|\beta}=\beta_{[\alpha_{1}\dots\alpha_{d-5}}S_{\alpha_{d-4}]|\beta}$
(83)

where $S_{\alpha|\beta}$ is a symmetric constant ${[1,1]}$-tensor. Then
$\kappa$ will be $\differential_{L}$-closed and not $\differential_{L}$-exact
if $\beta$ is closed and not exact.

Given this knowledge of the $\kappa$ tensors, we return to the question of
their $\differential_{L}$-cohomology. In order to construct non-trivial
charges $\mathbb{Q}[\kappa]$ there must exist a $\differential_{L}$-closed
$\kappa$ tensor which is not $\differential_{L}$-exact. From the
considerations above, it is clear that this is only possible if the de Rham
cohomology group $H^{d-4}(\mathcal{M})$ is non-trivial in the case of
reducible $\kappa$ tensors, and $H^{d-5}(\mathcal{M})$ is non-trivial for
irreducible ones.

Let us consider the $\kappa$-tensors given in section 4.2, which will be used
in the examples studied in section 7. The background spacetime is
$\mathbb{R}^{1,3}\times S^{1}$ with $y$ the coordinate on the $S^{1}$.
Firstly, consider the five-dimensional $\kappa$-tensor in eq. (54) with
$\kappa_{y|y}=\kappa_{0}$. Let us first treat this as an irreducible tensor.
If the $y$ coordinate were non-compact, then this would be trivial since
$\kappa_{\mu|\nu}=\partial_{\mu}\partial_{\nu}\psi$ for
$\psi=\frac{1}{2}\kappa_{0}y^{2}$. However, for $y$ periodic,
$\psi=\frac{1}{2}\kappa_{0}y^{2}$ is not a well-defined 0-form and so in this
case $\kappa$ is $d_{L}$-closed but not $d_{L}$-exact. If we treat this
$\kappa$ as a reducible tensor instead, we could write
$\kappa_{\mu|\nu}=\partial_{\mu}\xi_{\nu}$ with $\xi_{y}=\kappa_{0}y$ if $y$
were a non-compact coordinate. Again, when $y$ is periodic this is not a well-
defined 1-form and the $\kappa$-tensor is non-trivial.

The other example of interest uses the $\kappa$-tensor (56), with components
$\kappa_{y|t}=\kappa_{t|y}=\kappa_{0}$. Again, if $y$ were non-compact we
could write $\kappa_{\mu|\nu}=\partial_{\mu}\partial_{\nu}\psi$ with
$\psi=\kappa_{0}ty$. However, with $y$ periodic this is not well-defined.
Therefore, this $\kappa$-tensor is $\differential_{L}$-closed and not
$\differential_{L}$-exact. Treating $\kappa$ as a reducible ${\lfloor
1,1\rfloor}$-tensor, we could write $\kappa_{\mu|\nu}=\partial_{\mu}\xi_{\nu}$
where $\xi$ has components $\xi_{t}=\kappa_{0}y$ and $\xi_{y}=\kappa_{0}t$.
Again, this is not well-defined when $y$ is periodic, so the $\kappa$-tensor
is not $\differential_{L}$-exact. Similarly, the higher-dimensional
$\kappa$-tensors in eqs. (62) and (64) are $\differential_{L}$-closed and not
$\differential_{L}$-exact when there are compact directions.

In this work we study the graviton theory on Minkowski space with points or
regions removed, and possibly with directions toroidally compactified. Let us
first consider the case without toroidal directions. Minkowski space has
trivial cohomology, so non-trivial charges can only arise if some region is
excised from Minkowski space. A non-trivial $H^{d-4}(\mathcal{M})$ can be
generated in $\mathbb{R}^{1,d-1}$ by excising a 2-brane (with 3-dimensional
worldvolume) and so considering
$\mathcal{M}=\mathbb{R}\times(\mathbb{R}^{d-1}\setminus\mathbb{R}^{2})$ then
$H^{d-4}(\mathcal{M})=H^{d-4}(\mathbb{R}^{d-3}\setminus\\{\text{point}\\})\cong\mathbb{R}$.
A closed but non-exact $(d-4)$-form in $\mathcal{M}$ is given, for example, by
the volume form of a $(d-4)$-sphere wrapping the 2-brane. Similarly, excising
a 3-brane (with 4-dimensional worldvolume) gives non-trivial
$H^{d-5}(\mathcal{M})$.

Consider now the case of a toroidal compactification. Then
$\mathcal{M}=\mathbb{R}^{1,n}\times T^{m}$ has non-trivial
$H^{d-4}(\mathcal{M})$ provided $n\leq 3$ and has non-trivial
$H^{d-5}(\mathcal{M})$ provided $n\leq 4$. Obviously, points and/or regions
could also be removed from these compactified spaces to generate further non-
trivial cohomology groups.

###  6.2 Cohomology for $\Omega[V]$

To obtain a non-zero charge, the $(d-2)$-form $\star\Omega[V]$ should be in
non-trivial cohomology class, so $H^{d-2}(\mathcal{M})$ should be non-trivial.
In the examples we consider in the next section
$\mathcal{M}=(\mathbb{R}^{3}\setminus\\{\text{point}\\})\times\mathbb{R}\times
T^{d-4}$, i.e. the product of a spatial $\mathbb{R}^{3}$ with a point removed
with a $(d-4)$-torus and a time-like line. The $\mathbb{R}^{3}$ with a point
removed has a non-trivial 2-cycle given by a 2-sphere surrounding the point
and taking the product of this with $T^{d-4}$ gives a non-trivial cycle
$\Sigma_{d-2}$ of codimension 2 that can be integrated over. This space also
has non-trivial $H^{d-5}(\mathcal{M})$ and $H^{d-4}(\mathcal{M})$ represented
by forms on the torus and these can be used in constructing $\kappa$.

##  7 Some solutions and their charges

Ref. HullYetAppear  discussed various solutions of the linearised theory which
carry the charges $Q[k]$, $Q[\kappa]$ and $Q[\lambda]$ and the covariant
Penrose charges of these were calculated in PAPER1 . Here, we focus on the
charges $\mathbb{Q}[\kappa]$, related to the non-covariant charges $Q[\kappa]$
by eq. (49), and study solutions which carry these. We first analyse a
solution of the linearised theory derived from the Kaluza-Klein monopole
solution and then one obtained from Lorentzian Taub-NUT space.

###  7.1 Gravitational monopoles in $d\geq 5$

The linearisation of the Kaluza-Klein monopole solution considered in
HullYetAppear  is a linear superposition of a solution with mass and a
solution which carries a magnetic charge. Here we focus on the latter part of
the solution, which is in itself a valid solution of the linearised theory.
(The other part has $Q[\kappa]=0$ and $\mathbb{Q}[\kappa]=0$.)

We first consider the linearised Kaluza-Klein Ansatz on the space
$\mathbb{R}^{1,3}\times S^{1}$ with the cylinder $\mathbb{R}\times S^{1}$ at
$\vec{0}\in\mathbb{R}^{3}$ removed; we denote this space
$(\mathbb{R}^{1,3}\times S^{1})^{\bullet}$. The coordinates will be taken to
be $(t,x^{a},y)$ where $y\sim y+2\pi R_{y}$. The metric on the space is
$\eta=(-1,1,1,1,1)$. We take the graviton Ansatz

$h_{ay}=2A_{a}(x^{b})$ (84)

where the field strength $F=\differential{A}$ is given by

$F_{ab}=\epsilon_{abc}\partial_{c}\widetilde{W}$ (85)

and $\widetilde{W}$ is given by

$\widetilde{W}(x^{a})=-\frac{p}{\absolutevalue{x}}$ (86)

where $p$ is a constant parameterising the monopole charge. The non-vanishing
connection coefficients are

$\Gamma_{ab|y}=F_{ab},\quad\Gamma_{ay|b}=\partial_{a}A_{b}$ (87)

and the non-vanishing components of the linearised Riemann tensor are

$R_{abcy}=\partial_{c}F_{ab}$ (88)

It has been shown in Ref. HullYetAppear  that this solution has a non-
vanishing charge $Q[\kappa]$ corresponding to a $\kappa$-tensor with non-zero
components

$\kappa_{y|y}=\kappa_{0}$ (89)

in five dimensions, with $\kappa_{0}$ a constant. As discussed in section 6,
this is a $\differential_{L}$-closed but not $\differential_{L}$-exact
${[1,1]}$-tensor on $(\mathbb{R}^{1,3}\times S^{1})^{\bullet}$. The charge is
defined over a codimension-2 surface $\Sigma_{3}=S^{2}\times S^{1}$ where
$S^{2}$ is a 2-sphere containing the origin within a constant time
hypersurface of $\mathbb{R}^{1,3}$, and $S^{1}$ is the circle parameterised by
$y$, for which eq. (40) yields

$Q[\kappa]=8\pi^{2}R_{y}\kappa_{0}p$ (90)

We now evaluate the covariant charge $\mathbb{Q}[\kappa]$ given in eq. (49)
for this solution. First note that the Hodge dual
$\tilde{\kappa}=\star_{L}\kappa$ of the $\kappa$-tensor in eq. (89), given by
eq. (34), has non-vanishing components

$\tilde{\kappa}_{tabc|y}=\kappa_{0}\epsilon_{abc}$ (91)

This $\kappa$-tensor was discussed in section 4.2, where we showed that the
${\lfloor 4,2\rfloor}$-tensor $V[\tilde{\kappa}]$ in eq. (52) is well-defined
in this case. For the $\tilde{\kappa}$-tensor in eq. (91), $V$ has non-
vanishing components

$V_{tabc|y\alpha}=-\frac{1}{8}\kappa_{0}\epsilon_{abc}x_{\alpha}$ (92)

where $x^{\alpha}=(t,x^{a})$, which is well-defined on
$(\mathbb{R}^{1,3}\times S^{1})^{\bullet}$. Explicit calculation then yields

$\Omega[V]_{at}=p\kappa_{0}\frac{x_{a}}{\absolutevalue{x}^{3}}$ (93)

The charge $\mathbb{Q}[\kappa]$ is defined over a codimension-2 surface which
we again take as $\Sigma_{3}=S^{2}\times S^{1}$. The integration yields

$\mathbb{Q}[\kappa]=\int_{\Sigma_{3}}\star\Omega[V]=16\pi^{2}R_{y}p\kappa_{0}$
(94)

Note that this is a factor of 2 larger than $Q[\kappa]$ in eq. (90). The
difference comes from the final term in eq. (49). This solution has a non-
globally defined $h$ and so $\int\differential\star\Phi$, i.e. the final term
in eq. (49), can be non-zero. Indeed, explicit evaluation gives

$\int_{\Sigma_{3}}\differential\star\Phi=8\pi^{2}R_{y}\kappa_{0}p$ (95)

so that eqs. (90) and (95) contribute equally and sum to give
$\mathbb{Q}[\kappa]$ in eq. (94), as expected from the general result in eq.
(49).

As described in section 4, while $\mathbb{Q}[\kappa]$ is gauge-invariant,
neither of the two terms on the right-hand side of eq. (49) (that is,
$Q[\kappa]$ and $\int\differential\star\Phi$) are gauge-invariant. In the
gauge presented above (i.e. using the graviton components (84)), the two
contributions to $\mathbb{Q}[\kappa]$ given in eqs. (90) and (95) are equal.
In another gauge the two individual contributions will change, with only their
sum remaining invariant.

We now ask if it is possible to find a gauge choice where
$\mathbb{Q}[\kappa]=Q[\kappa]$ and $\int\differential\star\Phi=0$ for this
solution, such that $\mathbb{Q}[\kappa]$ gives a gauge-invariant quantity
which matches $Q[\kappa]$ in this gauge. As indicated in section 4, we will
see that such a gauge choice does exist. Consider a gauge transformation of
the graviton field $h_{\mu\nu}$ in eq. (84) to

$h^{\prime}_{\mu\nu}=h_{\mu\nu}+\partial_{(\mu}\xi_{\nu)}$ (96)

where the non-zero components of $\xi$ are taken to be

$\xi_{a}=4yA_{a}$ (97)

We note that the gauge parameter $\xi$ is not globally-defined. This is, of
course, not an issue. The gauge transformations need only be well-defined
within each chart. This is akin to a large gauge transformation winding around
the fibre direction in Maxwell theory.

The components of the gauge transformed graviton $h^{\prime}_{\mu\nu}$ are

$h^{\prime}_{ay}=4A_{a},\quad h^{\prime}_{ab}=4y\partial_{(a}A_{b)}$ (98)

One can then explicitly check that the linearised Riemann tensors of the two
gravitons are equal, $R_{\mu\nu\rho\sigma}=R^{\prime}_{\mu\nu\rho\sigma}$, as
was guaranteed. Therefore the charge $\mathbb{Q}[\kappa]$ is invariant under
the gauge transformation. However, calculating the contribution from
$\int\differential\star\Phi$ in this gauge gives a vanishing result. Indeed,
an explicit calculation of $Q[\kappa]^{\prime}$ (that is, the charge
$Q[\kappa]$ evaluated for the gauge transformed graviton
$h^{\prime}_{\mu\nu}$) verifies that, in this gauge,

$\mathbb{Q}[\kappa]=Q[\kappa]^{\prime}$ (99)

As discussed in section 4, the gravitational duality can be imposed at the
level of the connection $\Gamma$ in the gauge in which
$\Gamma\indices{{}_{\mu\nu|}^{\mu}}=0$. The transformation (97) ensures that
$h^{\prime}_{\mu\nu}$ satisfies this gauge condition. We note that eq. (99)
does not hold in this gauge when evaluated on arbitrary solutions, however it
does hold for this particular solution.

While this solution is a magnetically charged configuration from the
perspective of the graviton field, for the dual graviton description it is an
electric charge. Therefore, in the dual theory, the contribution
$\int\differential\star\Phi$ will vanish and eq. (99) follows. This would then
imply that $\int\differential\star\Phi$ should also vanish for the graviton
theory in this gauge, as we have seen directly.

##### Higher-dimensional solutions.

One can similarly construct the $\kappa$-charges associated with higher-
dimensional solutions with a background spacetime $(\mathbb{R}^{1,3}\times
T^{d-4})^{\bullet}$ for $d>5$. Let the coordinates on $T^{d-4}$ be $y^{i}\sim
y^{i}+2\pi R_{i}$, with $i=1,\dots,d-4$. We take the Ansatz (84) but with $y$
taken to be $y^{1}$. In $d$ dimensions, $\kappa$ is a ${\lfloor
d-4,1\rfloor}$-tensor. We take its non-zero components to be

$\kappa_{y_{1}y_{2}\dots y_{d-4}|y_{1}}=\kappa_{0}$ (100)

This is proportional to the volume form of $T^{d-4}$. This is of the form (82)
where $\beta$ is the volume form on the $T^{d-5}\subset T^{d-4}$ spanned by
$y_{2},\dots,y_{d-4}$, and $\sigma=dy^{1}$. This $\kappa$ is
$\differential_{L}$-closed and not $\differential_{L}$-exact. Taking its
$\star_{L}$ Hodge dual leads to the same components as in eq. (91). As
discussed in section 4.2, in $d>5$, $V[\tilde{\kappa}]$ in eq. (52) is not
well-defined for this solution. Instead, we use the ${\lfloor
4,2\rfloor}$-tensor $V$ in eq. (63). This is well-defined on $\mathcal{M}$ and
solves eq. (45) for the $\kappa$ in eq. (100). The codimension-2 integration
surface is now taken to be $\Sigma_{d-2}=S^{2}\times T^{d-4}$ with the $S^{2}$
surrounding the origin of a constant time hypersurface of $\mathbb{R}^{1,3}$.
The resulting charge is

$\mathbb{Q}[\kappa]=8\pi\kappa_{0}p\,\text{Vol}_{T^{d-4}}$ (101)

where $\text{Vol}_{T^{d-4}}=\prod_{i=1}^{d-4}2\pi R_{i}$ is the volume of the
$(d-4)$-torus. Indeed, this reduces to eq. (94) for $d=5$.

###  7.2 Lorentzian Taub-NUT in $d\geq 5$

Another example which carries the charge $Q[\kappa]$ is the product of
linearised Lorentzian Taub-NUT with a torus. The full Taub-NUT metric is
dyonic, but here we consider a linearisation only of the magnetically charged
part of the solution. In the linearised theory, this is a valid solution in
its own right. We begin with the case of $d=5$ on the background
$(\mathbb{R}^{1,3}\times S^{1})^{\bullet}$ as in the previous section. We take
a graviton field with components

$h_{at}=2A_{a}(x^{b})$ (102)

where $A_{a}(x^{b})$ satisfies

$F_{ab}=\epsilon_{abc}\partial_{c}X$ (103)

with

$X(x^{a})=-\frac{N}{\absolutevalue{x}}$ (104)

where $N$ is a constant that quantifies the NUT charge of the field
configuration. The non-zero components of the linearised Riemann tensor are

$R_{abct}=\partial_{c}F_{ab}$ (105)

This example has been studied in Ref. HullYetAppear , where it is shown that
it carries the charge

$Q[\kappa]=8\pi^{2}R_{y}N\kappa_{0}$ (106)

where the $\kappa$-tensor is chosen to be irreducible with non-zero components
$\kappa_{t|y}=\kappa_{y|t}\equiv\kappa_{0}=\text{constant}$. (More generally,
we could take a reducible $\kappa$-tensor with
$\kappa_{t|y}\equiv\kappa_{0}=\text{constant}$ and with any $\kappa_{y|t}$ and
get the same charge.) The dual is

$\tilde{\kappa}_{abct|t}=\tilde{\kappa}_{abcy|y}=-\epsilon_{abc}\kappa_{0}$
(107)

and all other components of $\tilde{\kappa}$ which are not related to these by
anti-symmetry vanish. Indeed, this $\tilde{\kappa}$ satisfies eq. (36) and is
not trivial, as discussed in section 6.

As we have noted, the form of $V$ given in eq. (52) is not necessarily well-
defined on spacetimes $\mathcal{M}$ with toroidal directions, such as the
present example. This example was studied in section 4.2, where a
${[4,2]}$-tensor $V$ was given in eq. (58) which is well-defined on
$\mathcal{M}$ and satisfies eq. (45) for this choice of $\tilde{\kappa}$. We
repeat the components of $V$ here for convenience:

$V_{abct|dt}=V_{abcy|dy}=-\frac{1}{8}\kappa_{0}\epsilon_{abc}x_{d}$ (108)

All other components not related to these by antisymmetry vanish.

We will evaluate the covariant charge $\mathbb{Q}[\kappa]$ for this choice of
$\kappa$ with the integration surface taken to be the codimension-2 cycle
$\Sigma_{3}=S^{2}\times S^{1}$ where the first factor is a 2-sphere at fixed
$r=\absolutevalue{x}$ in a constant time slice of $\mathbb{R}^{1,3}$ and the
second factor is the circle parameterised by $y$. Using eq. (103), we find

$\Omega[V]_{at}=N\kappa_{0}\frac{x_{a}}{\absolutevalue{x}^{3}}$ (109)

Then integrating over $\Sigma_{3}$ gives the charge

$\mathbb{Q}[\kappa]=\int_{\Sigma_{3}}\star\Omega[V]=16\pi^{2}R_{y}N\kappa_{0}$
(110)

We note that this is a factor of two larger than $Q[\kappa]$ in eq. (106).
Indeed, the remainder is contributed by the $\int\differential\star\Phi$ term
in eq. (49), which evaluates to

$\int_{\Sigma_{3}}\differential\star\Phi=8\pi^{2}R_{y}N\kappa_{0}$ (111)

as required.

As discussed in the previous example, the fact that the two contributions
$Q[\kappa]$ and $\int\differential\star\Phi$ in eqs. (106) and (111) are equal
is not a gauge-invariant statement. Similar considerations about gauge
dependence to those below eq. (99) are true in this case also. Implementing a
gauge transformation $h_{\mu\nu}\to
h^{\prime}_{\mu\nu}=h_{\mu\nu}+\partial_{(\mu}\xi_{\nu)}$ with gauge parameter
$\xi_{a}=4tA_{a}$ gives rise to a solution for which
$\mathbb{Q}[\kappa]=Q[\kappa]$ and $\int\differential\star\Phi=0$. In this
gauge $\Gamma\indices{{}_{\mu\nu|}^{\mu}}=0$. In the dual graviton
description, the Taub-NUT solution carries an electric charge and the
$\int\differential\star\Phi$ contribution vanishes as this measures a magnetic
charge.

##### Higher-dimensional solutions.

As in the previous sub-section, we now consider higher-dimensional solutions
with a background space $(\mathbb{R}^{1,3}\times T^{d-4})^{\bullet}$ for
$d>5$. As above, the coordinates on $T^{d-4}$ are $y^{i}$ with
$i=1,\dots,d-4$. Then we can again take the Ansatz (102) such that the
solution in question is a product of the linearised version of Lorentzian
Taub-NUT on $\mathbb{R}^{1,3}$ with a $(d-4)$-torus. Again, in order to
generate a non-trivial charge, $\kappa$ is required to be closed but not
exact. We take it to have components

$\kappa_{ty_{2}\dots y_{d-4}|y_{1}}=\kappa_{y_{1}y_{2}\dots
y_{d-4}|t}=\kappa_{0}$ (112)

This involves the volume form on $T^{d-5}\subset T^{d-4}$, so is
$\differential_{L}$-closed and not $\differential_{L}$-exact. Then evaluating
its Hodge dual gives precisely the components in eq. (107) as before. The
manipulations in eqs. (108) and (109) are unchanged, but the surface of
integration must be codimension-2 in the $d$-dimensional background space so
we take it to be $\Sigma_{d-2}=S^{2}\times T^{d-4}\equiv\Sigma_{3}\times
T^{d-5}$. Then evaluating the charge on $\Sigma_{d-2}$ yields

$\mathbb{Q}[\kappa]=\int_{\Sigma_{d-2}}\star\Omega[V]=8\pi
N\kappa_{0}\;\text{Vol}_{T^{d-4}}$ (113)

This clearly reduces to eq. (110) for $d=5$.

##  8 General construction of covariant conserved charges

In section 4, we have seen that the 2-forms $\Omega[V]$ given in eq. (43)
include the 2-forms corresponding to both the Penrose and $\kappa$-type
charges. We now give a systematic discussion of the constraints that $V$ must
satisfy in order for $\Omega[V]$ to be conserved when $R_{\mu\nu}=0$, and
examine the charges that can result. We will find that the non-trivial
solutions are in correspondence with the three types of secondary currents
mentioned previously, $J[k]$, $J[\lambda]$ and $J[\kappa]$, together with
identically conserved topological currents.

A priori, the most general gauge-invariant 2-form linear in the Riemann tensor
can be written

$\Omega[V]_{\mu\nu}=R^{\alpha\beta\gamma\delta}V_{\mu\nu|\alpha\beta|\gamma\delta}$
(114)

for some rank-6 tensor $V$ with the symmetries

$V_{\mu\nu|\alpha\beta|\gamma\delta}=V_{[\mu\nu]|[\alpha\beta]|[\gamma\delta]}$
(115)

that is, $V$ is a ${\lfloor 2,2,2\rfloor}$-tensor. We note that the
contraction with the Riemann tensor in eq. (114) projects onto the part of $V$
which is symmetric under the exchange of the $(\alpha\beta)$ and
$(\gamma\delta)$ indices. We choose not to make this symmetry explicit in our
discussion of $V$ below as it obscures other anti-symmetries of $V$ which we
wish to make manifest. We consider pure gravity in regions where

$R_{\alpha\beta}=0$ (116)

and demand the conservation of $\Omega[V]_{\mu\nu}$,

$\partial^{\mu}\Omega[V]_{\mu\nu}=(\partial^{\mu}R^{\alpha\beta|\gamma\delta})V_{\mu\nu|\alpha\beta|\gamma\delta}+R^{\alpha\beta|\gamma\delta}\partial^{\mu}V_{\mu\nu|\alpha\beta|\gamma\delta}=0$
(117)

The first term in eq. (117) will vanish via the differential Bianchi identity
$\partial_{[\mu}R_{\alpha\beta]\gamma\delta}=0$ if
$V_{\mu\nu|\alpha\beta|\gamma\delta}$ is anti-symmetric in the $\mu$,
$\alpha$, and $\beta$ indices.444Alternatively,
$V_{\mu\nu|\alpha\beta|\gamma\delta}$ could be anti-symmetric in the $\mu$,
$\gamma$, and $\delta$ indices but, as mentioned above, the contraction with
the Riemann tensor imposes that only the part of $V$ that is symmetric under
interchange of $(\alpha\beta)$ with $(\gamma\delta)$ contributes to the
current, so we choose the former case without loss of generality. This can be
achieved by restricting $V$ to be a ${\lfloor 4,2\rfloor}$-tensor, i.e.,

$V_{\mu\nu\alpha\beta|\gamma\delta}=V_{[\mu\nu\alpha\beta]|[\gamma\delta]}$
(118)

We note that in general $V$ does not have to be an ${\lfloor
4,2\rfloor}$-tensor in order for $\Omega[V]$ to be conserved. Indeed, there
are other ${\lfloor 2,2,2\rfloor}$-tensors which can be used to define a
conserved $\Omega[V]$. However, we show in appendix B that the conserved
2-forms $\Omega[V]$ which arise when $V$ is such a tensor have a specific form
which can also be accounted for by a different choice of $V$ which is a
${\lfloor 4,2\rfloor}$-tensor. Therefore, as our interest lies in classifying
the conserved 2-forms $\Omega[V]$ (rather than the tensors $V$ themselves) it
is sufficient to restrict $V$ to be a ${\lfloor 4,2\rfloor}$-tensor.

The ${\lfloor 4,2\rfloor}$ representation which $V$ transforms under is a
tensor product of the fully antisymmetric ${\ydiagram{1,1,1,1}}$ and
${\ydiagram{1,1}}$ representations, so is reducible. We also show in appendix
B that the only irreducible representation in the decomposition of the
reducible ${\lfloor 4,2\rfloor}$ representation which contributes non-
trivially to $\Omega[V]$ is the one labelled by the Young tableau
${\ydiagram{2,2,1,1}}$ (i.e. the irreducible ${[4,2]}$ representation).
Therefore, we can take $V$ to be an irreducible ${[4,2]}$-tensor; that is, it
must satisfy

$V_{[\mu\nu\alpha\beta|\gamma]\delta}=0$ (119)

With these facts in hand, the first term in eq. (117) vanishes. So the
requirement that $\Omega[V]$ is conserved reduces to

$\partial^{\mu}\Omega[V]_{\mu\nu}=R^{\alpha\beta\gamma\delta}\partial^{\mu}V_{\mu\nu\alpha\beta|\gamma\delta}=0$
(120)

We now look for constraints on $V$ such that
$\partial^{\mu}\Omega[V]_{\mu\nu}=0$ when eq. (116) is employed (that is, when
the field equations $G_{\mu\nu}=T_{\mu\nu}$ are used and sources are absent,
$T_{\mu\nu}=0$). We show in Appendix C that the most general such constraint
is

$\partial^{\mu}V\indices{{}_{\mu\nu\alpha\beta}^{|\gamma\delta}}=\omega\indices{{}_{\nu\alpha\beta}^{[\gamma|\delta]}}+\delta^{[\gamma}_{[\nu}\zeta\indices{{}_{\alpha\beta]}^{\delta]}}+\delta^{\gamma\delta}_{[\nu\alpha}\theta_{\beta]}$
(121)

where $\theta$ is a 1-form, $\zeta$ is a 3-form and
$\omega_{\nu\alpha\beta\gamma|\delta}$ is an irreducible ${[4,1]}$-tensor
which is traceless, i.e.
$\omega\indices{{}_{\nu\alpha\beta\gamma|}^{\gamma}}=0$. This constraint is
reached from representation theory considerations alone, and the reasoning is
as follows. The left-hand side of eq. (121) is $\differential_{L}^{\dagger}V$,
which is a reducible ${\lfloor 3,2\rfloor}$-tensor and so can be written as a
sum of $GL(d,\mathbb{R})$ representations. These can be branched into
$SO(1,d-1)$ representations, then eq. (120) demands that several
representations which appear in the decomposition cannot contribute. In
particular, the constraint (121) is equivalent to the statement that the only
irreducible $SO(1,d-1)$ representations which can contribute to
$\differential_{L}^{\dagger}V$ are ${\ydiagram{2,1,1,1}}$,
${\ydiagram{1,1,1}}$ and ${\ydiagram{1}}$. The tensors $\omega$, $\zeta$, and
$\theta$ are given by

$\displaystyle\omega$
$\displaystyle=\mathcal{Y}^{SO}_{{\ydiagram{2,1,1,1}}}(\differential_{L}^{\dagger}V)$
(122) $\displaystyle\zeta$
$\displaystyle=\mathcal{Y}^{SO}_{{\ydiagram{1,1,1}}}(\differential_{L}^{\dagger}V)$
(123) $\displaystyle\theta$
$\displaystyle=\mathcal{Y}^{SO}_{{\ydiagram{1}}}(\differential_{L}^{\dagger}V)$
(124)

The constraint (121) is then simply the statement that all other irreducible
$SO(1,d-1)$ representations in the decomposition of
$\differential_{L}^{\dagger}V$ must vanish. In other words, we must demand
that

$\mathcal{Y}^{SO}_{{\ydiagram{1,1,1,1,1}}}(\differential_{L}^{\dagger}V)=\mathcal{Y}^{SO}_{{\ydiagram{2,2,1}}}(\differential_{L}^{\dagger}V)=\mathcal{Y}^{SO}_{{\ydiagram{2,1}}}(\differential_{L}^{\dagger}V)=0$
(125)

(Recall that tensors in irreducible $SO(1,d-1)$ representations are traceless
so that the Young projector removes all traces.555This is the same method of
argument used to determine that $K$ must be a CKY tensor when studying the
Penrose 2-form $Y[K]$ in section 2. In that case, we explicitly wrote the
irreducible parts of $\partial K$ in terms of anti-symmetrisations and traces
(e.g. in eq. (18)). With the more complicated representations involved in the
case at hand, such formulae are unwieldy and opaque. Instead, we write
expressions in terms of $\omega$, $\zeta$ and $\theta$. These objects could be
written explicitly in terms of $V$ by evaluating the Young projectors in eqs.
(122)–(124).) The constraints (125) on $V$ are analogous to the constraints
that $k$ be a Killing vector or that $K$ be a CKY tensor, so can be
interpreted as constraining $V$ to be a particular kind of generalised Killing
tensor.

Any solution of $\differential_{L}^{\dagger}V=0$ can be added to a solution of
eq. (121) to generate another solution. We will first analyse solutions of the
homogeneous equation $\differential_{L}^{\dagger}V=0$ and then analyse
particular solutions of eq. (121). The general solution of (121) is then given
by a particular solution plus a solution of the homogeneous equation.

###  8.1 Solutions of $\differential_{L}^{\dagger}V=0$

We first consider ${[4,2]}$-tensors $V_{0}$ for which
$\differential_{L}^{\dagger}V_{0}=0$, that is

$\partial^{\mu}(V_{0})_{\mu\nu\alpha\beta|\gamma\delta}=0$ (126)

If $V_{0}$ is $\differential_{L}$-coexact, then

$(V_{0})_{\mu\nu\alpha\beta|\gamma\delta}=\partial^{\sigma}\hat{V}_{\sigma\mu\nu\alpha\beta|\gamma\delta}$
(127)

for some ${\lfloor 5,2\rfloor}$-tensor $\hat{V}$, and eq. (126) is satisfied.
In this case we have

$\Omega[V_{0}]_{\mu\nu}=-\partial^{\alpha}\left(R^{\sigma\beta\gamma\delta}\hat{V}_{\sigma\mu\nu\alpha\beta|\gamma\delta}\right)$
(128)

where the differential Bianchi identity
$\partial^{[\alpha}R^{\sigma\beta]\gamma\delta}=0$ has been used. That is,
$\Omega[V_{0}]$ is the divergence of a globally defined 3-form (since the
3-form depends only on the curvature) so the charge
$\mathbb{Q}[V_{0}]=\int\star\Omega[V_{0}]=0$ by Stokes’ theorem as the
integral is taken over a cycle.

Therefore, non-trivial charges can only arise from
$\differential_{L}$-coclosed ${[4,2]}$-tensors (satisfying eq. (126)) that are
not $\differential_{L}$-coexact (satisfying eq. (127)), and the charge depends
only on $V_{0}$ modulo a $\differential_{L}$-coexact piece. As a result, the
charge corresponds to the $\differential_{L}$-cohomology class of the
${\lfloor d-4,2\rfloor}$-tensor $\star_{L}V_{0}$ and this can only be non-zero
if the manifold has non-trivial $H^{d-4}$ (see section 5).

If $V_{0}$ is $\differential_{L}$-coclosed then there is a locally-defined
(i.e. defined in a contractible patch) ${\lfloor 5,2\rfloor}$-tensor $\hat{V}$
satisfying (127) so that $\Omega[V_{0}]$ is a total derivative

$\Omega[V_{0}]_{\mu\nu}=\partial^{\rho}W_{\mu\nu\rho}$ (129)

where the 3-form $W$ can be read off from (128). If $V_{0}$ is not
$\differential_{L}$-coexact, then $\hat{V}$ will not be a globally-defined
gauge-invariant tensor. Recalling the discussion of section 3, if $W$ has non-
covariant dependence on the graviton field $h$ then this extra term is not co-
exact when $h$ is non-globally defined. Therefore these terms can produce non-
trivial charges for such field configurations.

Alternatively, via similar manipulations to those in eq. (46) we find that,
for any $V_{0}$ satisfying eq. (126)

$\Omega[V_{0}]_{\mu\nu}=2\partial^{\alpha}\left(\Gamma^{\gamma\delta|\beta}(V_{0})_{\mu\nu\alpha\beta|\gamma\delta}\right)~{},$
(130)

giving an alternative expression for $W$.

Then given any solution $V$ to (121), a new solution is obtained by taking
$V\to V+V_{0}$ where $V_{0}$ is any solution of (126). This changes
$\Omega[V]_{\mu\nu}\to\Omega[V+V_{0}]_{\mu\nu}=\Omega[V]_{\mu\nu}+\partial^{\rho}W_{\mu\nu\rho}$
so that the current is changed by an identically conserved term
$d_{L}^{\dagger}W$. This will change the charge by the integral of a total
derivative, and this change will be non-trivial if $W$ is not globally-
defined.

###  8.2 Four dimensions

We briefly note that in four dimensions the form of $V$ simplifies as it can
then be written

$V_{\mu\nu\alpha\beta|\gamma\delta}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}v_{\gamma\delta}$
(131)

for some 2-form $v$. Now

$\Omega[V]_{\mu\nu}=\frac{1}{2}R^{\alpha\beta\gamma\delta}\epsilon_{\mu\nu\alpha\beta}v_{\gamma\delta}=(\star
R)_{\mu\nu\gamma\delta}v^{\gamma\delta}$ (132)

In regions where $R_{\mu\nu}=0$, the Riemann tensor is equal to the Weyl
tensor, $R_{\mu\nu\rho\sigma}=C_{\mu\nu\rho\sigma}$, so

$\Omega[V]_{\mu\nu}=(\star
C)_{\mu\nu\rho\sigma}v^{\rho\sigma}=C_{\mu\nu\rho\sigma}(\star
v)^{\rho\sigma}=R_{\mu\nu\rho\sigma}u^{\rho\sigma}$ (133)

where we have used $\star C\star=-C$ and defined $u=\star v$. This is of the
same form as the Penrose 2-form and is conserved if and only if $u$ is a CKY
tensor. Therefore, in $d=4$, the Penrose 2-form is the most general conserved
2-form current.

Indeed, in four dimensions we have seen in section 2 that all the ADM and dual
ADM charges of the graviton are already included in the Penrose charges, and
the $\kappa$-invariances are not relevant. In all that follows, we restrict to
$d>4$.

###  8.3 Relation of covariant and primary currents

For $d>4$, inserting the general constraint (121) for the divergence of $V$
into eq. (120) implies

$\displaystyle\begin{split}\partial^{\mu}\Omega[V]_{\mu\nu}&=R\indices{{}^{\alpha\beta}_{\gamma\delta}}\left(\omega\indices{{}_{\nu\alpha\beta}^{[\gamma|\delta]}}+\delta^{[\gamma}_{[\nu}\zeta\indices{{}_{\alpha\beta]}^{\delta]}}+\delta^{\gamma\delta}_{[\nu\alpha}\theta_{\beta]}\right)\\\
&=R^{\alpha\beta\gamma\delta}\omega_{\nu\alpha\beta\gamma|\delta}-\frac{1}{3}R\indices{{}^{\alpha\beta\gamma}_{\nu}}\zeta_{\alpha\beta\gamma}-\frac{2}{3}G_{\nu\alpha}\theta^{\alpha}\end{split}$
(134)

As guaranteed by the derivation above, this vanishes by the algebraic Bianchi
identity $R_{[\alpha\beta\gamma]\delta}=0$ and the field equation
$R_{\mu\nu}=0$.

Note that, if we do not impose the Bianchi identity and field equations, the
three contributions to eq. (134) are of precisely the same form as the 1-form
primary currents $j[k]$, $j[\lambda]$ and $j[\kappa]$ defined in eqs. (13),
(28) and (39) respectively. Indeed, if we allow sources for both the graviton
and its dual then $R_{\mu\nu}\neq 0$ and $R_{[\mu\nu\rho]\sigma}\neq 0$
Hull2001DualityFields . The Poincaré lemma then implies that the covariant
2-form $\Omega[V]$ is related to the secondary currents $J[k]$, $J[\lambda]$
and $J[\kappa]$ up to the divergence of a 3-form locally. We will see a
concrete realisation of this in the following.

###  8.4 Sub-cases

A full solution of the constraint (121) for $V$ will be given in section 8.5.
In order to illustrate the methods used, we first solve several simpler sub-
cases. We begin with the case where only $\zeta$ and $\theta$ are non-zero in
section 8.4.1. Next, we consider the case where only $\omega$ is non-zero in
section 8.4.2, and finally we include all three terms in the constraint in
section 8.5. As discussed above, $V$ is only fixed up to the addition of a
solution $V_{0}$ of the homogeneous equation (126), which adds a derivative
term $\partial^{\rho}W_{\mu\nu\rho}$ to $\Omega[V]_{\mu\nu}$.

####  8.4.1 Case 1: $\omega=0$ — Penrose charges

Firstly, we study solutions to the constraint (121) with $\omega=0$. We will
show that, up to the addition of a $\differential_{L}$-coclosed tensor, the
unique solution for $V$ takes the form of the solution presented in eq. (32),
for which $\Omega[V]=Y[K]$, so that we recover the Penrose charges.

Given the role that the Penrose solution (32) will play, we first verify that
it solves the constraint (121). This must be the case, since we know $Y[K]$ is
conserved. With $V$ given by eq. (32), using the CKY condition (14), we find

$\displaystyle\partial^{\mu}V\indices{{}_{\mu\nu\alpha\beta}^{|\gamma\delta}}$
$\displaystyle=6\delta^{[\gamma}_{[\mu}\delta^{\delta]}_{\nu}\partial^{\mu}K_{\alpha\beta]}$
$\displaystyle=6\delta^{[\gamma}_{[\mu}\delta^{\delta]}_{\nu}\tilde{K}\indices{{}_{\alpha\beta]}^{\mu}}+12\delta^{[\gamma}_{[\mu}\delta^{\delta]}_{\nu}\delta^{\mu}_{\alpha}\hat{K}_{\beta]}$
$\displaystyle=-3\delta^{[\gamma}_{[\nu}\tilde{K}\indices{{}_{\alpha\beta]}^{\delta]}}+3(d-3)\delta^{[\gamma}_{[\nu}\delta^{\delta]}_{\alpha}\hat{K}_{\beta]}$
(135)

So $V$ given in eq. (32) is indeed a solution of the constraint (121) with

$\omega_{\nu\alpha\beta\gamma|\delta}=0,\quad\zeta_{\mu\nu\rho}=-3\tilde{K}_{\mu\nu\rho},\quad\theta_{\mu}=3(d-3)\hat{K}_{\mu}$
(136)

From the properties of CKY tensors discussed in section 2, we see that $\zeta$
is a rank-3 closed CKY tensor and $\theta$ is a Killing vector in this case.

We now turn to the main goal of this sub-section, to study solutions of the
constraint (121) with $\omega=0$. That is, we wish to find the
${[4,2]}$-tensors $V$ which satisfy

$\partial^{\mu}V\indices{{}_{\mu\nu\alpha\beta}^{|\gamma\delta}}=\delta^{[\gamma}_{[\nu}\zeta\indices{{}_{\alpha\beta]}^{\delta]}}+\delta^{[\gamma}_{[\nu}\delta^{\delta]}_{\alpha}\theta_{\beta]}$
(137)

In terms of irreducible $SO(1,d-1)$ representations, this is the sub-case of
eq. (121) where, as well as the constraints of eq. (125), we also have

$\mathcal{Y}^{SO}_{{\ydiagram{2,1,1,1}}}(\differential_{L}^{\dagger}V)=0$
(138)

We begin by finding integrability conditions on $\zeta$ and $\theta$ which
follow from eq. (137). Taking the divergence of eq. (137) gives

$\displaystyle
0=\partial^{\nu}\left(\delta^{[\gamma}_{[\nu}\zeta\indices{{}_{\alpha\beta]}^{\delta]}}+\delta^{[\gamma}_{[\nu}\delta^{\delta]}_{\alpha}\theta_{\beta]}\right)$
(139)

Expanding the anti-symmetrisations, we have

$0=\frac{1}{6}\left(\partial_{\gamma}\zeta_{\alpha\beta\delta}-3\eta_{\gamma[\alpha}\partial^{\nu}\zeta_{\beta\delta]\nu}+2\eta_{\delta[\alpha|}\partial_{\gamma}\theta_{|\beta]}+\eta_{\gamma[\alpha}\eta_{\beta]\delta}\partial^{\nu}\theta_{\nu}\right)-(\gamma\leftrightarrow\delta)$
(140)

Now, anti-symmetrising this equation over all the indices immediately gives

$\partial_{[\gamma}\zeta_{\alpha\beta\delta]}=0$ (141)

so $\zeta$ is a closed 3-form. Furthermore, contracting eq. (140) with
$\eta^{\alpha\gamma}$ yields

$0=(3-d)\partial^{\alpha}\zeta_{\alpha\beta\delta}+(2-d)\left(\partial_{\delta}\theta_{\beta}-\eta_{\delta\beta}\partial^{\alpha}\theta_{\alpha}\right)$
(142)

and further contracting with $\eta^{\beta\delta}$ gives

$\partial^{\alpha}\theta_{\alpha}=0$ (143)

Substituting this into eq. (142) and symmetrising over $\beta$ and $\delta$
gives

$\partial_{(\delta}\theta_{\beta)}=0$ (144)

so we find that $\theta$ is a Killing vector. Eq. (142) then implies

$(3-d)\partial^{\alpha}\zeta_{\alpha\beta\delta}=(2-d)\partial_{\beta}\theta_{\delta}$
(145)

Substituting this back into eq. (140) and rearranging yields

$\left(\partial_{\gamma}\zeta_{\alpha\beta\delta}-\frac{3}{d-2}\eta_{\gamma[\alpha}\partial^{\nu}\zeta_{\beta\delta]\nu}\right)-(\gamma\leftrightarrow\delta)=0$
(146)

Antisymmetrising this equation over the $\beta$, $\gamma$ and $\delta$ indices
and using eq. (141) then finally gives

$\partial_{\alpha}\zeta_{\beta\gamma\delta}-\frac{3}{d-2}\eta_{\alpha[\beta}\partial^{\nu}\zeta_{\gamma\delta]\nu}=0$
(147)

which is the condition for $\zeta$ to be a rank-3 closed CKY tensor.
Therefore, it follows from (137) that $\theta$ must be a Killing vector and
and $\zeta$ must be a rank-3 closed CKY tensor. Furthermore, they are related
non-locally by eq. (145).

From the discussion of section 2, we know that $\theta$ and $\zeta$ can be
written as the divergence and curl of a CKY 2-form respectively (see Appendix
A of Ref. PAPER1  for further details). The construction works as follows. A
general Minkowski space Killing vector and rank-3 closed CKY tensor can be
parameterised by

$\displaystyle\begin{split}\theta_{\mu}&=\alpha_{\mu}+\beta_{\mu\nu}x^{\nu}\\\
\zeta_{\mu\nu\rho}&=\gamma_{\mu\nu\rho}+\tilde{\beta}_{[\mu\nu}x_{\rho]}\end{split}$
(148)

where $\alpha$, $\beta$, $\tilde{\beta}$ and $\gamma$ are constant
antisymmetric tensors. The relation between $\zeta$ and $\theta$ in eq. (145)
then implies that

$\tilde{\beta}_{\mu\nu}=-\frac{3}{d-3}\beta_{\mu\nu}$ (149)

Now, introducing a rank-2 CKY tensor $K$ as in eq. (20) and setting its
components to

$\mathcal{B}_{\mu}=-\frac{2}{d-3}\alpha_{\mu},\quad\mathcal{C}_{\mu\nu\rho}=-\frac{1}{3}\gamma_{\mu\nu\rho},\quad\mathcal{D}_{\mu\nu}=\frac{1}{3(d-3)}\beta_{\mu\nu},$
(150)

we find that $\theta$ and $\zeta$ defined in eq. (148) are related to the CKY
tensor $K$ precisely as in eq. (136). The constant $\mathcal{A}_{\mu\nu}$ term
in the CKY decomposition is left undetermined. Substituting the relations in
eq. (136) into (137) gives

$\partial^{\mu}V\indices{{}_{\mu\nu\alpha\beta}^{|\gamma\delta}}=-3\delta^{[\gamma}_{[\nu}\tilde{K}\indices{{}_{\alpha\beta]}^{\delta]}}+3(d-3)\delta^{[\gamma}_{[\nu}\delta^{\delta]}_{\alpha}\hat{K}_{\beta]}$
(151)

Now, reversing the manipulations in eq. (135) implies

$\partial^{\mu}V\indices{{}_{\mu\nu\alpha\beta|}^{\gamma\delta}}=\partial^{\mu}\left(6\delta^{[\gamma}_{[\mu}\delta^{\delta]}_{\nu}K_{\alpha\beta]}\right)$
(152)

such that, up to the addition of a $\differential_{L}$-coclosed tensor, $V$ is
given by the solution presented in eq. (32) which yields the Penrose charges.

Let us now discuss the charges $\mathbb{Q}[V]$ which result from these
solutions of eq. (137). With $V$ satisfying eq. (152), the divergence of
$\Omega[V]$ is given by

$\partial^{\mu}\Omega[V]_{\mu\nu}=R\indices{{}^{\alpha\beta}_{\gamma\delta}}\partial^{\mu}\left(6\delta^{\gamma}_{[\mu}\delta^{\delta}_{\nu}K_{\alpha\beta]}\right)=\partial^{\mu}\left(R\indices{{}^{\alpha\beta}_{\gamma\delta}}\delta^{\gamma}_{[\mu}\delta^{\delta}_{\nu}K_{\alpha\beta]}\right)=\partial^{\mu}Y[K]_{\mu\nu}$
(153)

where we have used the differential Bianchi identity
$\partial_{[\mu}R_{\alpha\beta]\gamma\delta}=0$ in the second equality and eq.
(33) in the final equality. The Poincaré lemma then implies that $\Omega[V]$
is given locally by the Penrose 2-form up to the divergence of a 3-form, i.e.

$\Omega[V]_{\mu\nu}=Y[K]_{\mu\nu}+\partial^{\rho}W_{\mu\nu\rho}$ (154)

Up to the addition of such a term, we see that when $\omega=0$ in the
constraint (121) _the unique solution for $\Omega[V]$ is given by the Penrose
2-form $Y[K]$_.

####  8.4.2 Case 2: $\zeta=\theta=0$ — $\kappa$-type charges

We now consider the simplification of the constraint (121) in which $\theta=0$
and $\zeta=0$, but $\omega$ is non-zero. In this case, the constraint becomes

$\partial^{\mu}V_{\mu\nu\alpha\beta|\gamma\delta}=\omega_{\nu\alpha\beta[\gamma|\delta]}$
(155)

In terms of irreducible $SO(1,d-1)$ representations, this is the sub-case of
eq. (121) where, as well as the constraints of eq. (125),
$\differential_{L}^{\dagger}V$ also satisfies

$\mathcal{Y}^{SO}_{{\ydiagram{1,1,1}}}(\differential_{L}^{\dagger}V)=\mathcal{Y}^{SO}_{{\ydiagram{1}}}(\differential_{L}^{\dagger}V)=0$
(156)

A straightforward integrability condition of eq. (155) is found by taking a
further divergence of eq. (155),

$\partial^{\nu}\omega_{\nu\alpha\beta[\gamma|\delta]}=0$ (157)

Antisymmetrising this equation on the $\beta$ and $\delta$ indices then
implies

$\partial^{\nu}\omega_{\nu\alpha\beta\gamma|\delta}=0$ (158)

We note that this is the property obeyed by $\tilde{\kappa}$ in eq. (36),
which points to a relation between the covariant charges related to
$\Omega[V]$ and the $\kappa$-charges. Recalling that $\omega$ is also required
to be traceless, it satisfies all the same properties required of a
$\tilde{\kappa}$ tensor (see section 3). We conclude that the tensor $\omega$
appearing in the constraint (155) must be a $\tilde{\kappa}$-tensor.

Furthermore, we now show that _all_ $\tilde{\kappa}$-tensors can be written as
in eq. (155) for some choice of $V$ on Minkowski space. In fact, the relevant
choice for $V$ which gives rise to any given $\tilde{\kappa}$ tensor has
already been given in eq. (52), which we repeat here for convenience,

$V[\tilde{\kappa}]_{\mu\nu\alpha\beta|\gamma\delta}=\frac{(-1)^{d}}{d-1}\tilde{\kappa}_{\mu\nu\alpha\beta|[\gamma}x_{\delta]}$
(159)

For this choice of $V$, it then follows from eq. (45) that eq. (155) is
satisfied with

$\omega_{\nu\alpha\beta\gamma|\delta}=\frac{(-1)^{d}}{d-1}\tilde{\kappa}_{\nu\alpha\beta\gamma|\delta}$
(160)

Hence, the tensors $\omega$ appearing in constraint (155) are in one-to-one
correspondence with $\tilde{\kappa}$-tensors, and for each such tensor we can
construct a conserved 2-form $\Omega[V]$.

We recall from the discussion in section 4.2 that the form of $V$ given in eq.
(159) is not necessarily well-defined when $\mathcal{M}$ has toroidal
directions due to the explicit coordinate dependence of $V$. If $\mathcal{M}$
does have compact directions, it may be that there is no well-defined $V$
which satisfies eq. (155) for a given $\tilde{\kappa}$. We have, however, seen
examples is section 4.2 of $\kappa$-tensors on such spacetimes and have
constructed ${[4,2]}$-tensors $V$ which satisfy eq. (45) for these
$\kappa$-tensors.

Let us now study the charges $\mathbb{Q}[V]$ associated with the solutions of
eq. (155). With $V$ given by eq. (52), these are the same covariant charges
constructed in section 4, so
$\mathbb{Q}[V[\tilde{\kappa}]]=\mathbb{Q}[\kappa]$. In section 4.3, we saw
that $\kappa$ must represent a non-trivial $\differential_{L}$-cohomology
class if the charge $\mathbb{Q}[\kappa]$ is to be non-zero.

Therefore, when we impose the simplification that $\zeta=0$ and $\theta=0$ in
the constraint (121), the non-trivial charges $\mathbb{Q}[\kappa]$ are
labelled by $\tilde{\kappa}$ tensors which are $\differential_{L}$-coclosed
but not $\differential_{L}$-coexact, and so correspond to the
$\differential_{L}$-cohomology of ${\lfloor d-4,1\rfloor}$-tensors. These
charges are related to the charges $Q[\kappa]=\int\star J[\kappa]$ by eq.
(49).

###  8.5 The general case

We now consider the full constraint (121) with all components $\omega$,
$\zeta$ and $\theta$ non-zero. We determine integrability conditions on these
tensors by similar manipulations to those in the sub-cases considered in
previous sections. In section 8.4.1, we addressed the case where $\omega=0$
and found that integrability conditions impose that $\zeta$ and $\theta$ must
be a closed CKY 3-form and a Killing vector respectively. They ultimately
appeared in a combination which yielded the Penrose 2-form. Including $\omega$
will alter the manipulations somewhat, but the result is similar. We emphasise
again that we are aiming to find particular solutions to eq. (121), to which
we can always add a $\differential_{L}$-coclosed ${[4,2]}$-tensor to give
another solution.

We begin as in eq. (139), by taking the divergence $\partial^{\nu}$ of the
constraint (121), giving

$-\partial^{\nu}\omega_{\nu\alpha\beta[\gamma|\delta]}=\frac{1}{6}\left(\partial_{\gamma}\zeta_{\alpha\beta\delta}-3\eta_{\gamma[\alpha}\partial^{\nu}\zeta_{\beta\delta]\nu}+2\eta_{\delta[\alpha|}\partial_{\gamma}\theta_{|\beta]}+\eta_{\gamma[\alpha}\eta_{\beta]\delta}\partial^{\nu}\theta_{\nu}\right)-(\gamma\leftrightarrow\delta)$
(161)

Antisymmetrising over all the free indices then implies

$\partial_{[\gamma}\zeta_{\alpha\beta\delta]}=-3\partial^{\nu}\omega_{\nu[\alpha\beta\gamma|\delta]}$
(162)

which is in contrast to eq. (141) when $\omega=0$. Contracting eq. (161) with
$\eta^{\alpha\gamma}$, the left-hand side vanishes as $\omega$ is traceless.
So eq. (142) is unchanged by the presence of $\omega$. Then the manipulations
of eqs. (143) to (145) go through unchanged. In particular, this implies that
_$\theta$ is a Killing vector even when $\omega\neq 0$_.

Following the same manipulations which led to eq. (147) now gives

$\partial_{\alpha}\zeta_{\beta\gamma\delta}-\frac{3}{d-2}\eta_{\alpha[\beta}\partial^{\nu}\zeta_{\gamma\delta]\nu}=3\partial^{\nu}\omega_{\nu\beta\gamma\delta|\alpha}$
(163)

which recovers eq. (147) when $\omega=0$. Therefore, the presence of $\omega$
implies that $\zeta$ is no longer a rank-3 closed CKY tensor.

Contracting eq. (163) with $\partial^{\beta}$ annihilates the right-hand side,
giving

$\partial_{\alpha}\partial^{\beta}\zeta_{\beta\gamma\delta}=0$ (164)

Therefore,

$\partial^{\beta}\zeta_{\beta\gamma\delta}=c_{\gamma\delta}$ (165)

is a constant 2-form. It then follows that $\zeta$ is at most linear in the
coordinates, up to the addition of a co-closed 3-form. The most general such
object is then

$\zeta_{\beta\gamma\delta}=a_{\beta\gamma\delta}+b_{\beta\gamma\delta|\sigma}x^{\sigma}+\alpha_{\beta\gamma\delta}$
(166)

where $\alpha$ is a co-closed 3-form, $a$ is a constant 3-form and $b$ is a
constant ${\lfloor 3,1\rfloor}$-tensor with
$b\indices{{}_{\gamma\delta\beta|}^{\beta}}=c_{\gamma\delta}$ to satisfy eq.
(165). The standard way to proceed with such manipulations is to insert eq.
(166) back into eq. (163) to constrain the form of $b$. However, this is less
useful in the present case due to the non-zero right-hand side. Instead, a
neater way forward is to decompose $b$ into $SO(1,d-1)$ representations,

$b_{\beta\gamma\delta|\sigma}=\bar{b}_{\beta\gamma\delta|\sigma}+\frac{3}{d-2}b^{\prime}_{[\beta\gamma}\eta_{\delta]\sigma}$
(167)

where $b^{\prime}_{\alpha\beta}=b\indices{{}_{\alpha\beta\gamma|}^{\gamma}}$
is the trace of $b$, and $\bar{b}$ is traceless. Now, eq. (166) gives

$\zeta_{\beta\gamma\delta}=a_{\beta\gamma\delta}+\frac{3}{d-2}b^{\prime}_{[\beta\gamma}x_{\delta]}+\bar{b}_{\beta\gamma\delta|\sigma}x^{\sigma}+\alpha_{\beta\gamma\delta}$
(168)

However, using the fact that $\bar{b}$ is traceless, we note that the
$\bar{b}$ term can be written as the total divergence of a 4-form,

$\bar{b}_{\beta\gamma\delta|\sigma}x^{\sigma}=\frac{4}{d-2}\partial^{\alpha}\left(x_{[\alpha}\bar{b}_{\beta\gamma\delta]|\sigma}x^{\sigma}\right)$
(169)

and so this term can be absorbed into a redefinition of $\alpha$. Therefore,
the most general form of $\zeta$ is

$\zeta_{\beta\gamma\delta}=\zeta^{0}_{\beta\gamma\delta}+\alpha_{\beta\gamma\delta}$
(170)

where
$\zeta^{0}_{\beta\gamma\delta}=a_{\beta\gamma\delta}+\frac{3}{d-2}b^{\prime}_{[\beta\gamma}x_{\delta]}$,
which is the general form of a rank-3 closed CKY tensor on flat space. So
$\zeta$ has contributions from a closed CKY 3-form and a co-closed 3-form.

Substituting eq. (170) into eq. (163) then implies

$\partial^{\nu}\omega_{\nu\beta\gamma\delta|\sigma}=\frac{1}{3}\partial_{\sigma}\alpha_{\beta\gamma\delta}$
(171)

This can be rewritten in the form

$\partial^{\nu}\pi_{\nu\beta\gamma\delta|\sigma}=0$ (172)

where

$\pi_{\nu\beta\gamma\delta|\sigma}=\omega_{\nu\beta\gamma\delta|\sigma}-\frac{4}{3}\eta_{\sigma[\nu}\alpha_{\beta\gamma\delta]}$
(173)

We see that, in contrast to the results of section 8.4.2, $\omega$ is no
longer $\differential_{L}$-coclosed. Instead, eq. (172) suggests that $\pi$ is
the correct object to be associated with $\tilde{\kappa}$ in this case.
However, we see that $\pi$ is not traceless:

$\pi\indices{{}_{\nu\beta\gamma\sigma|}^{\sigma}}=\frac{d-3}{3}\alpha_{\nu\beta\gamma}$
(174)

As mentioned in section 3, the current $J[\kappa]$ in eq. (38) is conserved
irrespective of whether $\tilde{\kappa}$ is traceless or not. When
$\tilde{\kappa}$ is traceless, this charge corresponds directly to an
invariance of the dual graviton. Now, we are forced allow a non-zero trace for
$\tilde{\kappa}$ in order to associate it with $\pi$, which is
$\differential_{L}$-coclosed. With this subtlety addressed, we make the
association

$\pi_{\nu\beta\gamma\delta|\sigma}=\frac{(-1)^{d}}{d-1}\tilde{\kappa}_{\nu\beta\gamma\delta|\sigma}$
(175)

Then, as in section 8.4.2, a solution for $V$ which is related to this $\pi$
by eq. (45) is given by $V[\tilde{\kappa}]$ in eq. (159). Again, if
$\tilde{\kappa}$ is $\differential_{L}$-coexact then the associated charge
$\mathbb{Q}[\kappa]$ vanishes and non-trivial charges associated with
$\tilde{\kappa}$ correspond to the $\differential_{L}$-cohomology of ${\lfloor
d-4,1\rfloor}$-tensors. As noted in section 4, we point out that the $V$ given
in eq. (52) is not well-defined when $\mathcal{M}$ has compact dimensions and
in that case it is not guaranteed that all $\tilde{\kappa}$ tensors have a
corresponding $V[\tilde{\kappa}]$.

Therefore, we find that including all three terms in the constraint (121) does
not greatly change the conclusions reached in the previous two sections.
Namely, $\theta$ must be a Killing vector, $\zeta$ comprises a rank-3 closed
CKY tensor and a co-closed 3-form $\alpha$ which combines with $\omega$ in
such a way that the ${\lfloor 4,1\rfloor}$-tensor $\pi$ is
$\differential_{L}$-coclosed and so must be a $\tilde{\kappa}$ tensor
(although now it is not traceless).

The covariant charges of $\Omega[V]$ associated with these tensors are then
related to the secondary currents $J[\kappa]$, $J[\lambda]$ and $J[k]$ as
follows. By analogous manipulations to eq. (46), up to the addition of a
coclosed 2-form, we have

$\displaystyle\Omega[V]_{\mu\nu}$
$\displaystyle=-2\Gamma^{\gamma\delta|\beta}\partial^{\alpha}V_{\mu\nu\alpha\beta|\gamma\delta}+\partial^{\alpha}\Phi_{\mu\nu\alpha}$
$\displaystyle=-2\Gamma^{\gamma\delta|\beta}\left(\omega_{\mu\nu\beta\gamma|\delta}+\eta_{\gamma[\mu}\zeta_{\nu\beta]\delta}+\eta_{\gamma[\mu}\theta_{\beta}\eta_{\nu]\delta}\right)+\partial^{\alpha}\Phi_{\mu\nu\alpha}$
$\displaystyle=-2\Gamma^{\gamma\delta|\beta}\left(\omega_{\mu\nu\beta\gamma|\delta}+\eta_{\gamma[\mu}\alpha_{\nu\beta]\delta}+\eta_{\gamma[\mu}\zeta^{0}_{\nu\beta]\delta}+\eta_{\gamma[\mu}\theta_{\beta}\eta_{\nu]\delta}\right)+\partial^{\alpha}\Phi_{\mu\nu\alpha}$
(176)

where $\Phi$ is the 3-form given in eq. (47). We have used the constraint
(121) in the second equality and eq. (170) in the third. Now, writing the
Killing vector $\theta$ as the divergence of a CKY 2-form $K$ and writing the
closed CKY 3-form $\zeta^{0}$ as its curl, the final two terms of the previous
equation rearrange into the Penrose current as explained in section 8.4.1. The
first two terms in the previous equation can also be straightforwardly
rearranged, yielding

$\Omega[V]_{\mu\nu}=J[\kappa]_{\mu\nu}+Y[K]_{\mu\nu}+\partial^{\alpha}\Phi_{\mu\nu\alpha}$
(177)

where $\zeta^{0}$ and $\theta$ are given in terms of $K$ by eq. (136) and
$\pi$ is related to $\tilde{\kappa}$ by eq. (175). Then from eq. (29), we have

$\Omega[V]_{\mu\nu}=J[k]_{\mu\nu}+J[\lambda]_{\mu\nu}+J[\kappa]_{\mu\nu}+\partial^{\alpha}(\Phi_{\mu\nu\alpha}+Z[K]_{\mu\nu\alpha})$
(178)

with $k$ and $\tilde{\lambda}$ given by eqs. (22) and (23) respectively (they
are proportional to $\theta$ and $\zeta$). We recall from Refs.
Benedetti2023GeneralizedGravitons ; PAPER1  that the Penrose 2-form $Y[K]$ is
co-exact in $d>4$ when $K$ is a KY tensor. So non-trivial charges result only
from the $\mathcal{B}$\- and $\mathcal{D}$-type CKY tensors in the $Y[K]$
contribution to eq. (177).

In summary, in this section we have studied solutions to the general
constraint on $V$ in eq. (121). We have looked for particular solutions to
this equation, to which one can always add a ${[4,2]}$-tensor which satisfies
eq. (126). Such extra terms can contribute to the charges $\mathbb{Q}[V]$ on
spaces with non-trivial cohomology (see section 8.1). We then analysed the
conserved 2-forms $\Omega[V]$ associated with the solutions of eq. (121), and
their relation to the secondary currents $J[k]$, $J[\lambda]$ and $J[\kappa]$.
One can always add a coclosed 2-form to $\Omega[V]$ without spoiling
conservation. If this 2-form is locally the divergence of a non-covariant
3-form, it can produce non-trivial charges. Since the addition of such a term
is arbitrary, we only classify the charges $\mathbb{Q}[V]$ up to the addition
of such terms.

##  9 Dimensional reduction of charges in Kaluza-Klein theory

In the previous section we have given a general discussion of the gauge-
invariant 2-form currents which are conserved in the linearised graviton
theory in $d$ dimensions. We now consider a toroidal compactification of this
theory on $T^{n}$ and ask whether the conserved quantities in $D\equiv d-n$
dimensions can be related to those in $d$ dimensions.

Concretely, we consider $\mathcal{M}=\mathbb{R}^{1,D-1}\times T^{n}$ and
denote the coordinates $x^{\mu}=(x^{m},y^{i})$ where $y^{i}\sim y^{i}+2\pi
R_{i}$. The metric on $\mathcal{M}$ is the Minkowski metric $\eta_{\mu\nu}$.

###  9.1 Kaluza-Klein Ansatz

We will focus in particular on compactifications where the graviton components
are independent of the compact directions:

$\partial_{i}h_{\mu\nu}=0$ (179)

and take an Ansatz of the form

$h_{\mu\nu}\quad\longrightarrow\quad
h_{mn}=\bar{h}_{mn}-\frac{2}{D-2}\eta_{mn}\phi,\quad h_{mi}=2A_{m}^{(i)},\quad
h_{ij}=2\phi^{(ij)}$ (180)

where $\phi\equiv\sum_{i=1}^{n}\phi^{(ii)}$. We will refer to $\bar{h}_{mn}$
as the $D$-dimensional graviton. There are also $n$ one-form fields
$A^{(i)}_{m}$ which are the graviphotons and $n(n+1)/2$ scalars $\phi^{(ij)}$
sitting in a symmetric matrix. Eq. (179) implies

$\partial_{i}R_{\mu\nu\rho\sigma}=0$ (181)

The components of the $d$-dimensional linearised Riemann tensor can be written

$\begin{split}R_{mnpq}&=\bar{R}_{mnpq}-\frac{2}{D-2}\left(\eta_{m[p}\partial_{q]}\partial_{n}\phi-\eta_{n[p}\partial_{q]}\partial_{m}\phi\right)\\\
R_{mnpi}&=\partial_{p}F^{(i)}_{mn}\\\
R_{minj}&=\partial_{m}\partial_{n}\phi^{(ij)}\end{split}$ (182)

where $\bar{R}_{mnpq}=-2\partial_{[m}\bar{h}_{n][p,q]}$ is the $D$-dimensional
linearised Riemann tensor and
$F^{(i)}_{mn}=\partial_{m}A^{(i)}_{n}-\partial_{n}A^{(i)}_{m}$ is the field
strength of the $i^{\text{th}}$ graviphoton.

Here we will consider the theory in the absence of sources, in which case the
$d$-dimensional equations of motion imply that the $D$-dimensional fields obey

$\begin{split}\bar{G}_{mn}&=0\\\ \partial^{p}F^{(i)}_{pm}&=0\\\
\partial_{m}\partial^{m}\phi^{(ij)}&=0\end{split}$ (183)

See Ref. PAPER1  for the a discussion of these equations in the presence of
sources.

###  9.2 2-form currents in the $D$-dimensional theory

In the dimensionally reduced theory, there are several 2-form currents which
can be constructed from the fields $\bar{h}_{mn}$, $A^{(i)}_{m}$ and
$\phi^{(ij)}$. Firstly, we construct the $D$-dimensional 2-forms

$\bar{\Omega}[\bar{V}]_{mn}=\bar{R}^{abcd}\bar{V}_{mnab|cd}$ (184)

where $\bar{V}$ is a ${[4,2]}$-tensor in $D$-dimensions. The considerations of
section 8 then apply and $\bar{V}$ must satisfy

$\partial^{m}\bar{V}\indices{{}_{mnab}^{|cd}}=\bar{\omega}\indices{{}_{nab}^{[c|d]}}+\delta^{[c}_{[n}\zeta\indices{{}_{ab]}^{d]}}+\delta^{cd}_{[na}\theta_{b]}$
(185)

where, $\bar{\omega}$ is a traceless ${[4,1]}$-tensor, $\zeta$ is a 3-form and
$\theta$ is a 1-form. Consistency then implies that $\theta$ is a Killing
vector, and $\bar{\omega}$ and $\zeta$ assemble into a rank-3 closed CKY
tensor and a $\tilde{\kappa}$-tensor as in section 8.5.

In a similar manner, we can construct conserved 2-forms involving the
graviphotons. To do so, we contract the $R^{abci}$ component of the
$d$-dimensional Riemann tensor (182) with some object. This ensures the result
will be gauge-invariant when we uplift to $d$ dimensions.

There are, in fact, multiple ways to do this but it is sufficient to consider
a 2-form of the form

$\Delta[\chi]_{mn}=R^{abci}\chi^{(i)}_{mnc|ab}=\partial^{c}F^{(i)ab}\chi^{(i)}_{mnc|ab}$
(186)

for some ${[3,2]}$-tensors $\chi^{(i)}$. We could also consider a 2-form
constructed like $R^{abci}\xi^{(i)}_{mnab|c}$ for some ${[4,1]}$-tensors
$\xi^{(i)}$, however we find that these merely produce a subset of the charges
of those produced by the $\Delta[\chi]_{mn}$ above.

We must require that $\Delta[\chi]$ is conserved. This will be the case
provided that the $\chi^{(i)}$ obey a differential condition. This can be
derived analogously to the manipulations in Appendix C as follows.
$\partial^{m}\chi^{(i)}_{mnc|ab}$ is, a priori, in the
${\ydiagram{1,1}}\otimes{\ydiagram{1,1}}$ representation of
$GL(d,\mathbb{R})$. This can be decomposed into irreducible representations of
$SO(1,d-1)$. Demanding conservation of $\Delta[\chi]$ then implies that only
some of these irreducible representations can appear. In this case, the
allowed $SO(1,d-1)$ representations are ${\ydiagram{1,1,1,1}}$ and $\bullet$.
Therefore, we demand that $\chi^{(i)}$ satisfies

$\partial^{m}\chi^{(i)}_{mnc|ab}=\gamma^{(i)}_{ncab}+(\eta_{a[n}\eta_{c]b}\delta^{(i)}-a\leftrightarrow
b)$ (187)

where
$\gamma^{(i)}=\mathcal{Y}^{SO}_{{\ydiagram{1,1,1,1}}}(\differential_{L}^{\dagger}\chi^{(i)})$
are 4-forms and
$\delta^{(i)}=\mathcal{Y}^{SO}_{\bullet}(\differential_{L}^{\dagger}\chi^{(i)})$
are 0-forms. Contracting the $c$ and $b$ indices and then taking the
divergence of this equation implies $\partial_{a}\delta^{(i)}=0$, so
$\delta^{(i)}$ are constants. Given this, taking a divergence of eq. (187)
implies that the $\gamma^{(i)}$ are coclosed.

We can also construct conserved 2-forms from the scalar fields $\phi^{(ij)}$.
The components of the Riemann tensor involving the scalars is given in the
final line of eq. (182). The most general conserved 2-form that can be built
involving this gauge-invariant combination of derivatives of $\phi^{(ij)}$ is
of the form

$\Psi[\mu]_{mn}=\partial^{a}\partial^{b}\phi^{(ij)}\mu^{(ij)}_{mna|b}$ (188)

where $\mu^{(ij)}$ is a ${[3,1]}$-tensor for each $i,j$. Similar arguments to
above show that conservation of this 2-form implies that $\mu^{(ij)}$ must
satisfy

$\partial^{m}\mu^{(ij)}_{mna|b}=\nu^{(ij)}_{nab}$ (189)

where $\nu^{(ij)}$ is a 3-form for each $i,j$, from which it follows that
$\nu^{(ij)}$ is coclosed.

Therefore, we have conserved 2-forms $\bar{\Omega}[\bar{V}]_{mn}$ in (184)
where $\bar{V}$ satisfies (185), $\Delta[\chi]_{mn}$ in (186) where
$\chi^{(i)}$ satisfies (187), and $\Psi[\mu]_{mn}$ in (188) where $\mu^{(ij)}$
satisfies (189).

### Charges in the $D$-dimensional theory

Let us now consider the charges associated with the conserved 2-forms
$\bar{\Omega}[\bar{V}]_{mn}$, $\Sigma[\xi]_{mn}$ and $\Psi[\mu]_{mn}$
constructed above. The construction of $\bar{\Omega}[\bar{V}]$ is exactly
analogous to that of $\Omega[V]$ in section 8, just in $D$ dimensions.
Therefore, there are associated charges

$\bar{\mathbb{Q}}[\bar{V}]=\int_{\Sigma_{D-2}}\star\bar{\Omega}[\bar{V}]$
(190)

where $\Sigma_{D-2}$ is a codimension-2 cycle in $D$-dimensions. The
discussion of the $d$-dimensional charges $\mathbb{Q}[V]$ in section 8.5
carries through analogously here.

Next we consider the charges constructed from the conserved 2-forms
$\Delta[\chi]$. From eq. (187), we have

$\displaystyle\begin{split}\Delta[\chi]_{mn}&=\partial^{c}(F^{(i)ab}\chi^{(i)}_{mnc|ab})-F^{(i)ab}(\gamma^{(i)}_{mnab}+2\eta_{a[m}\eta_{n]b}\delta^{(i)})\\\
&=\partial^{c}(F^{(i)ab}\chi^{(i)}_{mnc|ab})-F^{(i)ab}\gamma^{(i)}_{mnab}-\delta^{(i)}F^{(i)}_{mn}\end{split}$
(191)

The first term is the divergence of a globally defined 3-form and so will
vanish by Stokes’ theorem when integrated over a cycle. Therefore, the related
charge is

$\bar{\mathbb{Q}}[\chi]\equiv\int_{\Sigma_{D-2}}\star\Delta[\chi]=-2\int_{\Sigma_{D-2}}\star\gamma^{(i)}\wedge
F^{(i)}-\delta^{(i)}\int_{\Sigma_{D-2}}\star F^{(i)}$ (192)

The first term on the right-hand side is a magnetic charge which depends only
on the cohomology class of the closed $(D-4)$-form $\star\gamma^{(i)}$. It is
carried by $(D-4)$-branes wrapping cycles which are Poincaré dual to the
cohomology class $[\star\gamma^{(i)}]$ HullYetAppear . The second term on the
right hand side is the electric charge of the graviphotons, which generates
the standard electric $U(1)$ 1-form symmetry of Maxwell theory
Gaiotto2015GeneralizedSymmetries .

Finally we consider the charges which are constructed from the 2-forms
$\Psi[\mu]$, related to the scalars $\phi^{(ij)}$. The charge is

$\bar{\mathbb{Q}}[\mu]=\int_{\Sigma_{D-2}}\star\Psi[\mu]$ (193)

From the definition (188), stripping off derivatives and using eq. (189)
yields

$\Psi[\mu]_{mn}=\partial^{a}\left(\partial^{b}\phi^{(ij)}\mu^{(ij)}_{mna|b}-\phi^{(ij)}\nu^{(ij)}_{mna}\right)$
(194)

The first term inside the parentheses is globally defined since the gauge
transformations of the scalars $\phi^{(ij)}$ are simply shifts by a constant.
Therefore, this term does not contribute to the charge. The charge can then be
written

$\bar{\mathbb{Q}}[\mu]=-\int_{\Sigma_{D-2}}\differential\star(\phi^{(ij)}\nu^{(ij)})$
(195)

which is a topological charge and can be non-zero only when $\phi^{(ij)}$ are
not globally defined. Physically, this measures the winding of the scalars
around the compact directions.

###  9.3 Embedding in $d$-dimensional theory

Having constructed the $D$-dimensional charges which can be constructed from
the fields $\bar{h}$, $A^{(i)}$, and $\phi^{(ij)}$, we now wish to embed them
into the $d$-dimensional charges $\Omega[V]$ which can be constructed from the
$d$-dimensional graviton $h$. Firstly, we define a ${[4,2]}$-tensor $V$ in
$d$-dimensions with components

$V_{mnab|cd}=\bar{V}_{mnab|cd},\quad
V_{mnai|cd}=V_{mncd|ai}=\frac{1}{4}\chi^{(i)}_{mna|cd},\quad
V_{mnai|bj}=\frac{1}{4}\mu^{(ij)}_{mna|b}$ (196)

and a $d$-dimensional ${[2,2]}$-tensor $\mathscr{P}_{\mu\nu\rho\sigma}$ with
components

$\mathscr{P}_{mnpq}=\frac{2}{D-2}\left(\eta_{m[p}\partial_{q]}\partial_{n}\phi-m\leftrightarrow
n\right)$ (197)

and all other components vanishing. Now consider the object

$\Omega[V]_{mn}=\bar{\Omega}[\bar{V}]_{mn}+\Delta[\chi]_{mn}+\Psi[\mu]_{mn}$
(198)

This is the $mn$ component of the tensor

$\Omega[V]_{\mu\nu}=(R^{\alpha\beta\gamma\delta}+\mathscr{P}^{\alpha\beta\gamma\delta})V_{\mu\nu\alpha\beta|\gamma\delta}$
(199)

This can be seen by expanding

$\displaystyle\Omega[V]_{mn}$
$\displaystyle=(R^{abcd}+\mathscr{P}^{abcd})V_{mnab|cd}+4R^{abci}V_{mnab|ci}+4R^{aibj}V_{mnai|bj}$
(200)

and substituting in eqs. (182) and comparing with the definitions (184), (186)
and (188).

Therefore, if we choose to integrate $\star\Omega[V]$ over a $(d-2)$-cycle
given by

$\Sigma_{d-2}=\Sigma_{D-2}\times T^{n}$ (201)

then

$\mathbb{Q}[V]=\int_{\Sigma_{d-2}}\star\Omega[V]=\text{Vol}_{T^{n}}\left(\bar{\mathbb{Q}}[\bar{V}]+\bar{\mathbb{Q}}[\chi]+\bar{\mathbb{Q}}[\mu]\right)$
(202)

Therefore, we see that the gauge-invariant charges
$\bar{\mathbb{Q}}[\bar{V}]$, $\bar{\mathbb{Q}}[\chi]$, and
$\bar{\mathbb{Q}}[\mu]$ which are constructed in the dimensionally reduced
$D$-dimensional theory have a unified gauge-invariant origin in the original
$d$-dimensional theory.

##### Acknowledgements.

CH is supported by the STFC Consolidated Grants ST/T000791/1 and ST/X000575/1.
MLH is supported by a President’s Scholarship from Imperial College London. UL
gratefully acknowledges a Leverhulme visiting professorship to Imperial
College as well as the hospitality of the theory group at Imperial.

##  Appendix A Some definitions and conventions

The secondary current $J[k]$ introduced in eq. (12) can be written

$J[k]^{\mu\nu}=\partial_{\sigma}\mathcal{K}^{\mu\nu|\rho\sigma}k_{\rho}-\mathcal{K}^{\mu\sigma|\rho\nu}\partial_{\sigma}k_{\rho}$
(203)

where

$\mathcal{K}^{\mu\nu|\rho\sigma}=-3\eta^{\mu\nu\alpha|\rho\sigma\beta}h_{\alpha\beta}$
(204)

and

$\eta^{\mu\nu\rho|\alpha\beta\gamma}=\eta^{\alpha\sigma}\eta^{\beta\kappa}\eta^{\gamma\lambda}\delta^{\mu}_{[\sigma}\delta^{\nu}_{\kappa}\delta^{\rho}_{\lambda]}$
(205)

The secondary current $J[\lambda]$ in eq. (28), corresponding to the
$\lambda$-invariances, can be explicitly constructed in terms of the dual
graviton $D$ and, when $\tilde{\lambda}$ is constant, can be rewritten in
terms of the graviton $h_{\mu\nu}$. In this case, the secondary current is

$J[\lambda]_{\mu\nu}=\partial^{\rho}Z[\lambda]_{\mu\nu\rho}$ (206)

where

$Z[\lambda]_{\mu\nu\rho}=(-1)^{d+1}\frac{4}{d-1}\tilde{\lambda}_{[\sigma\mu\nu}h_{\rho]}{}^{\sigma}$
(207)

When $\tilde{\lambda}$ is not constant, there is another contribution to
$Z[\lambda]$ in the dual theory which cannot be written locally in terms of
the graviton $h_{\mu\nu}$. In any case, the current $J[\lambda]$ in eq. (206)
is clearly conserved for any 3-form $\tilde{\lambda}$ so, in the graviton
theory, we will take this as the definition of $J[\lambda]$ for all
$\tilde{\lambda}$.

In $d=4$, the $\lambda$ tensors reduce to Killing vectors (since a rank-1 KY
tensor is simply a Killing vector). The secondary current
$\tilde{J}[\tilde{k}]$ which appears in eq. (31), associated to the dual
graviton’s invariances in four dimensions, can be written

$\tilde{J}[\tilde{k}]_{\mu\nu}=\partial^{\rho}Z[\tilde{k}]_{\mu\nu\rho}$ (208)

where

$Z[\tilde{k}]_{\mu\nu\rho}=-\epsilon_{\mu\nu\rho\sigma}\tilde{k}^{\tau}h\indices{{}_{\tau}^{\sigma}}$
(209)

##  Appendix B Decomposition of $V$ into $\text{GL}(d,\mathbb{R})$
representations

As discussed in section 8, the most general form that the tensor $V$ in eq.
(43) can take is that of a reducible ${\lfloor 2,2,2\rfloor}$-tensor,
satisfying eq. (115). That is, it transforms in a $GL(d,\mathbb{R})$
representation characterised by a tensor product Young tableau

$\ydiagram{1,1}\otimes\ydiagram{1,1}\otimes\ydiagram{1,1}$ (210)

We wish to impose constraints on $V$ such that
$\partial^{\mu}\Omega[V]_{\mu\nu}=0$. From eq. (117), we have

$\partial^{\mu}\Omega[V]_{\mu\nu}=(\partial^{\mu}R^{\alpha\beta|\gamma\delta})V_{\mu\nu|\alpha\beta|\gamma\delta}+R^{\alpha\beta|\gamma\delta}\partial^{\mu}V_{\mu\nu|\alpha\beta|\gamma\delta}=0$
(211)

This must vanish for all field configurations satisfying the field equation
$R_{\mu\nu}=0$ and the Bianchi identities

$R_{[\mu\nu\rho]\sigma}=0,\quad\partial_{[\alpha}R_{\beta\gamma]\rho\sigma}=0,\quad\partial^{\mu}R_{\mu\nu\rho\sigma}=0$
(212)

The last of these equations follows from the second by using $R_{\mu\nu}=0$.
The fact that this must vanish for all such configurations implies that the
two terms in eq. (211) must vanish independently. The second term in eq. (211)
will vanish provided $V$ satisfies some differential condition which will be
determined later (the details of this calculation can be found in Appendix C).
The vanishing of the first term, however, places a constraint on the
symmetries of $V$ (i.e. the representation which $V$ transforms in) such that
it projects $\partial^{\mu}R^{\alpha\beta|\gamma\delta}$ onto the field
equations and/or Bianchi identities in such a way that the result vanishes. We
determine this constraint in this Appendix, proceeding in three stages as
follows.

  1. 1.

Firstly, since we are only interested in tensors $V$ for which $\Omega[V]$ is
non-zero, we will decompose the naïve tensor product representation (210)
which $V$ transforms in into irreducible $GL(d,\mathbb{R}$) representations.
Then, by contracting with the Riemann tensor to form
$\Omega[V]_{\mu\nu}=R^{\alpha\beta\gamma\delta}V_{\mu\nu\alpha\beta|\gamma\delta}$
we will determine which of these can contribute to $V$ in a way which gives a
non-trivial $\Omega[V]$ from which charges can be constructed.

  2. 2.

Having found the allowed irreducible $GL(d,\mathbb{R})$ representations which
$V$ can transform in, we will insert these into eq. (211) and demand that the
first term vanishes (the second term will be considered in Appendix C). Some
of the representations which $V$ can transform in immediately project
$\partial^{\mu}R^{\alpha\beta|\gamma\delta}$ onto a Bianchi identity, and so
vanish. For example, this is the case if $V$ transforms in the irreducible
[4,2] representation, then
$\partial^{\mu}R^{\alpha\beta|\gamma\delta}V_{\mu\nu\alpha\beta|\gamma\delta}=\partial^{[\mu}R^{\alpha\beta]|\gamma\delta}V_{\mu\nu\alpha\beta|\gamma\delta}=0$
from the second Bianchi identity in eq. (212). However, the first term of eq.
(211) could also vanish by the third identity in eq. (212) if $V$ has a factor
of the Minkowski metric, i.e. is a sum of terms each of which is the tensor
product of the Minkowski metric with another tensor. For example, if
$V_{\mu\nu|\alpha\beta|\gamma\delta}=\eta_{\alpha[\mu}A_{\nu]|\beta|\gamma\delta}-(\alpha\leftrightarrow\beta)$
for some ${\lfloor 1,1,2\rfloor}$-tensor $A$, then
$\partial^{\mu}R^{\alpha\beta|\gamma\delta}V_{\mu\nu|\alpha\beta|\gamma\delta}=2\partial_{\alpha}R^{\alpha\beta|\gamma\delta}A_{\nu|\beta|\gamma\delta}=0$.

A systematic method to analyse all the ways in which the Minkowski metric can
arise in $V$ is to decompose the irreducible $GL(d,\mathbb{R})$
representations into irreducible $SO(1,d-1)$ representations by separating out
all possible traces within $V$. We can then find all the $SO(1,d-1)$
representations that can contribute to $V$ in a manner such that the first
term in eq. (211) vanishes. Indeed, this is precisely the same logic employed
in section 2 to show that $K$ must be a CKY in order for the Penrose 2-form
$Y[K]$ to be conserved.

  3. 3.

Since we are ultimately interested in the set of conserved 2-forms
$\Omega[V]$, rather than the tensors $V$ themselves, we will ask whether all
the allowed representations which $V$ can transform in produce distinct
2-forms $\Omega[V]$. We will see that, while there are other representations
which are allowed in the decomposition of $V$, the form of $\Omega[V]$ which
arises can always be constructed with a $V$ in the [4,2] $GL(d,\mathbb{R})$
representation. Therefore, it will be sufficient to simply demand that $V$ is
a [4,2]-tensor.

#####  $GL(d,\mathbb{R})$ decomposition of $V$.

We begin with the first stage outlined above. Decomposing the tensor product
representation in eq. (210) into irreducible $GL(d,\mathbb{R})$
representations gives

${\ydiagram{1,1}}\otimes{\ydiagram{1,1}}\otimes{\ydiagram{1,1}}=\left(\;{\ydiagram{1,1,1,1}}\oplus{\ydiagram{2,1,1}}\oplus{\ydiagram{2,2}}\;\right)\otimes{\ydiagram{1,1}}=\left(\;{\ydiagram{1,1,1,1}}\otimes{\ydiagram{1,1}}\;\right)\oplus\left(\;{\ydiagram{2,1,1}}\otimes{\ydiagram{1,1}}\;\right)\oplus\left(\;{\ydiagram{2,2}}\otimes{\ydiagram{1,1}}\;\right)$
(213)

The three tensor product representations on the right-hand side of eq. (213)
can themselves be decomposed similarly. The first of them gives

$\displaystyle\ydiagram{1,1,1,1}\otimes\ydiagram{1,1}$
$\displaystyle=\ydiagram{1,1,1,1,1,1}\oplus\ydiagram{2,1,1,1,1}\oplus\ydiagram{2,2,1,1}$
(214)

However, we note that only the final diagram gives a non-zero contribution to
$\Omega[V]$, because of the Bianchi identity
$R_{[\alpha\beta\gamma]\delta}=0$. This can be seen explicitly by decomposing
the component of $V$ in the ${\ydiagram{1,1,1,1}}\otimes{\ydiagram{1,1}}$
representation as

$V^{{\ydiagram{1,1,1,1}}\otimes{\ydiagram{1,1}}}_{\mu\nu\alpha\beta|\gamma\delta}=V^{{\ydiagram{1,1,1,1,1,1}}}_{\mu\nu\alpha\beta\gamma\delta}+V^{{\ydiagram{2,1,1,1,1}}}_{\mu\nu\alpha\beta[\gamma|\delta]}+V^{{\ydiagram{2,2,1,1}}}_{\mu\nu\alpha\beta|\gamma\delta}$
(215)

where the three terms on the right-hand side are the Young projections of