Charges and topology in linearised gravity

Covariant conserved 2-form currents for linearised gravity are constructed by contracting the linearised curvature with conformal Killing-Yano tensors. The corresponding conserved charges were originally introduced by Penrose and have recently been interpreted as the generators of generalised symmetries of the graviton. We introduce an off-shell refinement of these charges and find the relation between these improved Penrose charges and the linearised version of the ADM momentum and angular momentum. If the graviton field is globally well-defined on a background Minkowski space then some of the Penrose charges give the momentum and angular momentum while the remainder vanish. We consider the generalisation in which the graviton has Dirac string singularities or is defined locally in patches, in which case the conventional ADM expressions are not invariant under the graviton gauge symmetry in general. We modify them to render them gauge-invariant and show that the Penrose charges give these modified charges plus certain magnetic gravitational charges. We discuss properties of the Penrose charges, generalise to toroidal Kaluza-Klein compactifications and check our results in a number of examples.


Introduction
In a remarkable paper [1] Penrose introduced a conserved 2-form current Y in linearised gravity of the form where R µναβ is the linearised curvature tensor for metric fluctuations about a background Minkowski spacetime.This is conserved, i.e.
if the vacuum Einstein equations hold, i.e.R µν = 0, and K αβ is a 2-form satisfying where Tensors satisfying (the covariant version of) this equation are known as conformal Killing-Yano (CKY) tensors [2,3], so that the tensors K satisfying eq.(1.3) are the CKY tensors for Minkowski space.Penrose gave a twistorial interpretation of this equation in four dimensions [1].While Penrose focused on four dimensions, his construction extends to d dimensions.From eq. (1.2), ⋆Y is a closed (d − 2)-form and the integral of this over a (d − 2)-surface Σ defines a conserved charge This charge is unchanged under deformations of Σ that do not cross any points at which R µν = 0 and the charge provides a measure of the amount of mass/energy contained within Σ. Penrose interpreted these charges as giving covariant expressions for the total momentum and angular momentum contained within Σ.He argued that for each Minkowski space Killing vector there is a corresponding CKY tensor and that the charges Q[K] for these CKY tensors give covariant expressions for the total momentum (corresponding to the translation Killing vectors) and angular momentum (corresponding to the Lorentz Killing vectors).
Penrose went on to generalise his construction to curved spacetime.General spacetimes do not admit Killing vectors or CKY tensors, but, to construct Q[K] in eq.(1.5), ⋆Y is only needed on the surface Σ, not over the whole spacetime.Penrose constructed a 2-form ⋆Y on Σ using 'surface twistors' and proposed that this gives a quasi-local definition of momentum and angular momentum in general relativity.On taking Σ to be at null infinity he obtained the BMS momentum together with an angular momentum.
Not all the CKY tensors correspond to Killing vectors.Namely, a particular class of CKY tensors are the Killing-Yano (KY) tensors which satisfy eq.(1.4) with Kµ = 0, and do not have corresponding Killing vectors.For example, in four dimensions the space of Killing vectors is 10-dimensional while that of CKY tensors is 20-dimensional.This raises the question of the significance of the Penrose currents corresponding to the KY tensors.Penrose avoided this mismatch by imposing a hermiticity condition on the (twistor form of) his charges that left 10 real charges.This eliminated certain gravitational analogues of magnetic charge and one of the aims of this paper is to revisit this correspondence if gravitational magnetic charges of the kind analysed in Ref. [12] are included.In Ref. [6], all the Penrose charges for linearised gravity in four dimensions were associated with certain parameters (e.g. a NUT parameter) in a linearised solution, providing some insight into their significance in that case.
In this paper we consider the Penrose charges for the d-dimensional free graviton theory in Minkowski space with the Fierz-Pauli action.We construct an off-shell refinement of the Penrose currents with extra terms involving the linearised Ricci tensor that vanish on-shell.For the case in which K is a KY tensor, the improved Penrose currents are identically conserved (without using field equations) and are in fact the currents constructed by Kastor and Traschen [13].We then derive a precise relation between the charges that are given by integrating the improved Penrose currents and the momentum and angular momentum that arise from the linearisation of the ADM construction [14,15].If the graviton field is non-singular and defined on the whole of Minkowski space, then the relation is straightforward and the Penrose charges for one class of CKY tensors give the ADM charges while the remainder vanish.However, if the graviton field is not globally defined in the sense that it has Dirac string singularities or is defined in patches with transition functions involving gauge transformations, then topological or magnetic charges for gravity of the kind recently constructed in Ref. [12] can arise.In particular, if the graviton field is not globally defined, then total derivative contributions become important.We show that the standard expression for each ADM charge is only gauge-invariant up to a surface term, and this can be non-zero if the graviton is not globally defined.As a result, the standard ADM expressions are in general not gauge-invariant if the graviton field is not globally defined.We find a surface-term modification of the standard ADM expressions that is fully gauge invariant under these circumstances.We then relate the Penrose charges, which are manifestly gauge-invariant, to these covariant improved ADM charges together with certain gravitational magnetic charges.
The structure of the paper is as follows.In section 2, we review the theory of linearised gravity and outline the construction of the ADM charges and the gravitational magnetic charges introduced in Ref. [12].In section 3, we discuss Penrose's 2-form current in more detail and derive an off-shell refinement of it which is used in section 4, where the relationship between the Penrose and ADM charges is derived.We then analyse the charges constructed by integrating the Penrose 2-form current and the dual charges constructed by integrating its Hodge dual.We do this for spacetime dimensions d > 4 in sections 5 and 6.The situation in d = 4 dimensions is different and is analysed in section 7.In section 8, we discuss the charges that arise when the background Minkowski space is replaced by a product of Minkowski space with a torus, allowing the Kaluza-Klein reduction of the linearised theory.These form a more general set of charges than those that arise in Minkowski space.We first derive the charges in the dimensionally reduced theory, then give their uplift to the higher dimensional theory.Section 9 gives examples of solutions of the linearised theory, calculates their charges and checks the relationship between the Penrose charges and the ADM and magnetic charges.Finally, in section 10, we summarise our results and discuss their implications.

Linearised gravity and its conserved charges
We study the spin-2 free graviton field h µν in d-dimensional Minkowski space with global Cartesian coordinates x µ and with Minkowski metric η µν = diag(−1, 1, 1 . . ., 1).We will later consider configurations in which the graviton field is singular at the locations of certain sources, and in that case we will restrict to the space M given by Minkowski space with points or regions removed, so that h µν is a non-singular field on M.
The graviton is a symmetric tensor with gauge transformation The invariant field strength is the linearised Riemann tensor where1 The Fierz-Pauli field equation with source is where G µν is the linearised Einstein tensor with R µν and R the linearised Ricci tensor and scalar respectively.The source is a symmetric tensor that is conserved, In general, solutions of this free theory need not arise as the linearisation of solutions of the full non-linear Einstein theory.
The Minkowski space Killing vectors k µ satisfy and are given by where V is a constant 1-form and Λ is a constant 2-form, corresponding to translations and Lorentz transformations respectively.Then for any Killing vector k, using eqs.(2.6) and (2.7).
Using the field equation (2.4), the current can be written in terms of the graviton field as which is a total derivative [15], where and (2.15) For a (d − 1)-dimensional hypersurface S with boundary Σ d−2 we define the charge which can be rewritten as a surface integral over the boundary If S is taken to be a hypersurface of fixed time with Σ d−2 the (d − 2)-sphere at spatial infinity, then Q[k] for a Killing vector of the form (2.8) is a conserved charge giving the linearised ADM momenta P µ and angular momenta L µν , If S is taken to be a region of a hypersurface of fixed time with boundary The Killing vectors k give invariances of the theory: any field configuration is invariant under the gauge transformation (2.1) in which ζ µ is a Killing vector.Then Q[k] is the conserved charge corresponding to the invariance under eq.(2.1) with ζ µ = k µ , and we will refer to it as an electric-type charge for the graviton.We will refer to j[k] as the primary current associated with the invariance and a 2-form current J µν with will be referred to as a secondary current.Note that if J is such a secondary current, then so is In Ref. [12], magnetic charges for linearised gravity were discussed.These are all of the form where the current is a total derivative for some totally antisymmetric Z µνρ , so that J mag is automatically a conserved 2-form current.If Z µνρ is a globally defined 3-form, the current is trivial and the charge is zero, so non-trivial magnetic charges only arise when this is not the case.
In the cases we consider here, Z µνρ has a local expression in terms of the graviton field h µν and globally defined Killing vectors or Killing tensors.If the components of h µν are non-singular functions defined over the whole of Minkowski space, so that h µν is a globally defined tensor on Minkowski space, then ⋆J mag is exact and the charge Q = Σ ⋆J mag is zero.Then, to obtain a non-trivial magnetic charge, it is necessary that h µν is not a globally defined tensor field on the entire Minkowski space.Typically for the solutions to eq. (2.4) with magnetic charges, the graviton field is not defined on the whole Minkowski space but instead on a space M which is Minkowski space with some points or regions removed, so that it can have non-trivial topology.(The regions removed from Minkowski space can be associated with the locations of magnetic sources [12].)Then to obtain non-trivial charges, h µν should be a field on M with a Dirac string singularity or it should be defined locally in patches of M with transition functions involving non-trivial gauge transformations of the form (2.1), giving a topologically non-trivial field configuration.Although h µν need not be globally defined, the field strength R µνρσ is globally defined and gauge-invariant.In cases with magnetic charges, if we try to analytically extend such a h µν defined in one patch to the whole of Minkowski space, we find Dirac string singularities.As these charges are integrals of a total derivative, they are topological charges.See Ref. [12] for further discussion.
In principle, any local 3-form Z could be used to construct such a charge.Ref. [12] focused on the charges that arise as electric charges for the dual graviton theory introduced in Ref. [16].These charges arise from invariances of the dual graviton theory that correspond to Killing vectors k in four dimensions or to generalised Killing tensors denoted κ and λ in d > 4. In regions without magnetic sources for the dual graviton, some of these electric charges for the dual graviton can be dualised to the graviton theory where they become magnetic charges given by the integral of a total derivative.We now discuss these in more detail.
In four dimensions, these electric charges for the dual graviton result in a topological charge Q[k] for the graviton theory for each Killing vector k, with currents J [k] given by eq.(2.21) with giving dual momentum and angular momentum Here P µ is the linearised version of the dual momentum or NUT 4-momentum introduced for general relativity in Refs.[17,18] and Lµν can be viewed as a dual angular momentum charge.
In d > 4, there are two types of magnetic charges for the graviton theory which correspond to electric charges for the dual graviton.The first involves a rank-(d − 3) KY tensor λ µ 1 ...µ d−3 which by definition satisfies Then a current J[λ] µν is defined by eq.(2.21) with [12] Z is a closed CKY tensor. 2 For constant KY tensors λ, the charge arises as an electric charge for the dual graviton theory [12].The magnetic charge Pµ 1 ...µn is defined as the charge constructed from J[λ] for constant λ, that is However, the integral of (2.25) gives a conserved charge for any (d − 3)-form λ, and in later sections this charge will arise for non-constant tensors λ.For non-constant λ, however, they cannot be straightforwardly interpreted as electric charges for the dual graviton due to the non-local relation of the graviton to its dual.Further discussion of this will appear in a forthcoming paper.
The other charge that arises as an electric charge for the dual graviton theory involves a generalised Killing tensor κ µ 1 ...µ d−4 |ρ which is in the GL(d) representation corresponding to a Young tableau with one column of length d − 4 and one of length 1.It satisfies the generalised Killing condition [12] In this case, the secondary current is of the form (2.21) with the 3-form Z µνρ given by giving a current J µν [κ] and a charge Q[κ].These do not play a role here as they do not correspond to Penrose charges; they will be discussed in a forthcoming paper.Finally, our analysis below will involve a total derivative current with where K is a CKY tensor and K is defined in eq.(1.4).This will be seen to arise in the relation of the ADM charges to the Penrose ones.

The Penrose currents
In this section, we discuss the Penrose currents for the free graviton in d-dimensional Minkowski space.In particular, we investigate improvement terms that make them conserved off-shell in certain cases.

The Penrose 2-form current away from sources
For any 2-form K µν there is a 2-form where R µναβ is the linearised curvature tensor (2.2).We now show that the condition for this to be conserved, i.e.
From the contracted Bianchi identity with R µν the linearised Ricci tensor, the first term on the right-hand side of eq.(3.2) vanishes if the vacuum Einstein equations R µν = 0 hold.The remaining term in eq.(3.2) will also vanish, using R µν = 0 and the Bianchi identity where a ναβ = a [ναβ] is a 3-form and b is a 1-form.Indeed, we would then have Since Y [K] is conserved in regions where R µν = 0, the value of Q[K] remains unchanged as Σ d−2 is deformed through such regions. 3This statement holds irrespective of the field equations.However, when the field equations G µν = T µν hold this statement is equivalent to saying that Q[K] is conserved in regions where T µν = 0, i.e. in regions without sources.On Minkowski space, the CKY tensors can be found explicitly.It is shown in Appendix A.2 that taking two further derivatives of eq.(3.6) leads to the integrability condition ∂ µ ∂ ν ∂ ρ K αβ = 0.It is then simple to demonstrate that the most general solution of eq.(3.6) is [19, eq. (6.4.8)] where A, B, C, and D are constant antisymmetric tensors.Therefore, in d dimensions there are d(d + 1)(d + 2)/6 rank-2 CKY tensors.Particularly, in d = 4 dimensions, there are 20 independent solutions.An important result for our analysis is that the divergence of a rank-2 CKY tensor on Minkowski space gives a Killing vector 4  Kα and the exterior derivative gives a closed CKY 3 That is, the charge is unchanged if the (d − 1)-dimensional surface swept out by Σ d−2 as it is deformed is entirely contained in a region in which Rµν = 0. 4 For CKY tensors on a curved space, it is not guaranteed that the covariant divergence Kµ = (d − 1) −1 ∇ ν Kνµ is a Killing vector but Gµν Kν remains covariantly conserved -that is, ∇ µ (Gµν Kν ) = 0 -as a result of the integrability condition (d − 2)∇ (µ Kν) = R ρ (µ K ν)ρ satisfied by CKY tensors [20].
tensor Kαβγ .We now show this.Explicitly, taking the divergence of the general rank-2 CKY tensor in eq.(3.9) gives which is the form of the Killing vectors of Minkowski space.The K's for the B-type CKY tensors are the translational Killing vectors, while the K's for the D-type CKY tensors are the Killing vectors for Lorentz transformations.We see that only CKY tensors of the B-and D-types in eq.(3.9) correspond to Killing vectors as the A-and C-type terms are divergenceless.In section 5, we will relate the Penrose currents for B-and D-type CKY tensors to the ADM currents for the corresponding Killing vectors Kµ .
A Killing-Yano (KY) 2-tensor is one that satisfies so that it is a CKY tensor whose divergence Kµ = 0.The general KY tensor is then In Minkowski space, this is co-exact: it can be written explicitly as the divergence of a 3-form where (We can explicitly verify that F in fact satisfies the rank-3 CKY equation, given in Appendix A, on Minkowski space.)In section 8, we will consider the case in which Minkowski space is replaced with R 1,d−1−n × T n where the KY tensors need not be co-exact.
The exterior derivative of the general rank-2 CKY tensor in eq.(3.9) gives which is precisely the form of a general rank-3 closed CKY tensor (see Appendix A.2 for the definition of CKY tensors of general rank).We note that only the CKY tensors of the C-and D-types in eq.(3.9) contribute to rank-3 closed CKY tensors Kµνρ .The rank-2 On Minkowski space, closed CKY tensors are in fact exact, where We can then verify that Σ α satisfies the rank-1 CKY equation; that is, Σ α is a conformal Killing vector.One important property of CKY tensors which we will use throughout is that the Hodge dual of a rank-p CKY tensor is, itself, a rank-(d − p) CKY tensor (see Appendix A).In particular, the dual of a KY tensor (i.e. a tensor of the form of eq.(3.12)) is a closed CKY tensor (i.e. of the form of eq.(3.16)) and vice-versa.This property is true of CKY tensors on any manifold, and several of the other properties discussed above are also true on more general spaces [2,3,[21][22][23].

Improved Penrose 2-form in the presence of sources
The Penrose 2-form (3.1) is conserved provided that the Ricci tensor vanishes.We now consider adding sources and suppose Einstein's equation G µν = T µν is satisfied for some conserved energy-momentum tensor T µν .As has been seen, the Penrose 2-form Y [K] is conserved in the region in which T µν = 0.
However, there exists an improvement which is conserved without use of the field equations for KY tensors.We define the 'improved Penrose 2-form' In regions without sources, this reduces to the Penrose 2-form Y [K].Now, using eq.(3.3) as well as the contracted Bianchi identity we find This is proportional to the conserved current (2.11) with Killing vector k α = 2(d − 3) Kα .
If K is a KY tensor (i.e.K = 0), Y + [K] is conserved and is precisely the current of Kastor and Traschen [13].In what follows, we will use the improved 2-form current Y + [K] for general CKY tensors.
We may now define a quantity by integrating ⋆Y + [K] over a codimension-2 cycle: If K is KY tensor (i.e. an A-or C-type CKY tensor), then Y + [K] is conserved offshell, i.e. without using the field equations or any condition on T µν , at all points and so Q + [K] is conserved.This means that the value of Q + [K] is unchanged as Σ d−2 is deformed arbitrarily (including deformations through regions where T µν = 0).For the B-and D-type CKY tensors, however, Y + [K] is only conserved at points at which R µν = 0, so that on-shell this means at points at which T µν = 0. Then for surfaces Σ d−2 that lie in a region R in which R µν = 0, the charge Q + [K] given by the integral of ⋆Y + [K] over Σ d−2 is unchanged under deformations that keep the surface in the region R.In this region In this paper, we will primarily restrict ourselves to the case where Σ d−2 ⊂ R for some region R in which R µν = 0.For general CKY tensors K, the proportionality of ∂ ν Y + [K] µν and the 1-form current j[k] µ = G µα k α , suggests a link between the improved Penrose 2-form Y + [K] and the secondary current J[k].We will discuss this in detail in the following sections.The improvement terms in eq.(3.19) are not unique and there are other improvements of the Penrose 2-form which reduce to eq. (3.1) on-shell (c.f.[24]).The particular combination in eq.(3.19) is chosen to simplify the relation with the 1-form current j[k].

Triviality of Penrose charges for KY tensors in d > 4
In this section we show that the Penrose charges for KY tensors in Minkowski space vanish for d > 4. In Ref. [8] it was shown that ⋆Y [A] and ⋆Y [C] are exact (d − 2)-forms in d > 4 when R µν = 0, and it was concluded that Q[A] and Q[C] vanish.Here we improve on their argument, showing that ⋆Y + [A] and ⋆Y + [C] are exact (d − 2)-forms off-shell, i.e. without requiring R µν = 0, and conclude that the corresponding Penrose charges Q + [A] and Q + [C] vanish identically, and this remains true even when the surface of integration is deformed through regions in which R µν = 0. Note that these results are for Minkowski space and there are modifications when considering toroidal compactifications. 5or a KY tensor K µν = f µν given by eq.(3.12), the corresponding improved Penrose 2-form Y + [f ] is the divergence of a 3-form built from the linearised curvature tensor and the rank-3 CKY tensor F defined in eq.(3.14); that is, we have for d > 4, where Y is the 3-form with components This follows from eq. (3.13) and the contracted Bianchi identities (3.3) and (3.20), as well as R [µνρ]σ = 0 but does not require assuming R µν = 0. Since the right-hand side of eq.(3.25) depends on h µν only through the linearised curvature tensor, it is globally defined irrespective of whether h µν is globally defined or not as the curvature tensor is globally defined.Therefore, in d > 4 Minkowski space, integrals of ⋆Y + [f ] over codimension-2 cycles vanish by Stokes' theorem.Then the only non-zero Penrose charges in d > 4 Minkowski space come from the CKY tensors of the B and D types (that is, those which are not KY tensors).These are precisely the ones which correspond to non-zero Killing vectors K in eq.(3.10).Note that in d = 4, eq.(3.24) is no longer valid so the improved Penrose current for KY tensors cannot be written as the divergence of a tensorial 3-form and the associated Penrose charges can be non-vanishing.
While eq. (3.24) is valid without using the field equations, in regions away where R µν = 0, the left-hand side of eq.(3.24) reduces to the Penrose 2-form Y [f ] and the final two terms of Y in eq.(3.25) vanish.For this case, Ref. [8] has given a similar covariant 3-form potential for the Penrose 2-forms associated with KY tensors.
We note that Kastor and Traschen [13] have also given a 3-form potential for the improved Penrose current when K is a KY tensor (which they refer to as a Yano current).Theirs, however, is not a covariant expression in terms of h.Therefore, if we allow nonglobal gauge-field configurations (as we are here) then we must re-examine the contributions of total derivative terms.Our 3-form potential in eq.(3.25), in contrast, is covariant.As is necessary from the Poincaré lemma, these two 3-form potentials differ by the divergence of a 4-form.Furthermore, Kastor and Traschen consider the integration of the Penrose current for a KY tensor over a (d − 2)-dimensional space Σ d−2 with a (d − 3)-dimensional boundary ∂Σ d−2 .In this case the charge Σ d−2 ⋆Y + [f ] can be written as an integral over the boundary for d > 4. For surfaces Σ d−2 that are closed this clearly vanishes, and we only consider closed surfaces in this paper.

Relation between Penrose charges and ADM charges
In this section we analyse the connection between the charges constructed from the improved Penrose 2-form Y + [K] µν and the ADM charges associated with the Killing vectors of Minkowski space k µ .We have seen in section 3 that the divergence of a CKY tensor on flat space is a Killing vector.It will be convenient to define Then eq.(3.21) gives a relation between the improved Penrose 2-form and the 1-form primary current j[k] µ = G µν k ν which can be written as The secondary current It follows from eq. ( 4.3) and the Poincaré lemma that, on contractible open sets, Y + [K] and J[k] should be related by the divergence of some locally defined 3-form Z.Moreover, from the algebraic Poincaré lemma it is to be expected that Z has a local expression in terms of the graviton field and K. Indeed, we show in appendix B.1 that where with K given in eq.(3.7).We emphasise that Z[K] is a 3-form which depends explicitly on the gauge field h µν .The divergence of eq.(4.4) then gives eq.(4.2).Integrating eq. ( 4.4) over a codimension-2 cycle Σ d−2 gives We shall generally restrict ourselves to surfaces Σ d−2 contained in a region where R µν = 0, i.e. in a region without sources if the field equations hold.Recall that Y + [K] and J[k] are only conserved in such regions since Therefore, the integral is unchanged under any deformation of the surface Σ d−2 that does not cross a region where R µν = 0.If h is a globally defined tensor, then Z is also globally defined, so that d ⋆ Z[K] = 0 and eq.(4.6) reduces to Then Q[K] depends only on K and is precisely the ADM charge for the Killing vector k = 2(d − 3) K.The general form of K is given in eq.(3.9) in terms of constant antisymmetric tensors A, B, C, and D. We will use the notation with the CKY tensor K given by the relevant term in eq.(3.9).Then if h is a globally defined tensor we have as the A-and C-type CKY tensors do not contribute to K. From subsection 3.3 we already knew this to be the case for d > 4, so here we learn that this also applies for d = 4 if h is globally defined.In four dimensions, we have where P µ , L µν are the ADM momentum and angular momentum defined in eq.(2.18).The current Y + [K] µν is invariant under the gauge transformation (2.1) while J[k] µν is not.Under the gauge transformation (2.1), Z[K] µνρ changes by so that it follows from eq. ( 4.4) that as can be explicitly verified.If h is a globally defined tensor then the gauge transformation is (2.1) with ζ a globally defined 1-form.Then δJ[k] in eq.(4.11) is the total derivative of a globally defined 3-form and it follows that the integral of this variation vanishes so that the charge Q[k] is gauge invariant.We now turn to the case in which h µν is not a globally defined field configuration, either with a Dirac string singularity or defined in patches with non-trivial transition functions involving the gauge transformation (2.1).Then Z[K] is not globally defined in general and the total derivative term d ⋆ Z[K] in (4.6) need not vanish but is instead a topological term.Moreover the variation δJ[k] µν under a gauge transformation given in eq.(4.11) is not globally defined in general6 so that the charge Q[k] defined in eq.(2.17) is no longer gauge-invariant.Then the definition of ADM charges needs modification for non-globally defined h µν .Adding the topological term to the ADM charges Q[k] as in eq.(4.6) gives a gauge-invariant result, and provides such a covariantisation.The results of subsection 3.3 place further restrictions on the topological charges for d > 4, which we discuss in the next section.In section 7 we discuss the case d = 4.

Analysis of Penrose charges in d > 4
We have seen in the last section that for globally defined h µν the Penrose charges give the ADM charges, while if h µν is not globally defined, then the Penrose charges give the naive ADM charges (2.17) plus a topological charge.In this section, we further investigate the general case in which h µν need not be not globally defined for dimensions d > 4.
In dimensions d > 4, the results of subsection 3.3 restrict the charges.The right-handside of eq.(3.24) is a globally defined total derivative even if h µν is not globally defined, so the CKY tensors with coefficients A or C in eq.(3.9) give a current Y + [K] µν that is co-exact and so give zero charge and Neither of the two terms on the right hand side are gauge-invariant in general but their sum is, so that adding an identically conserved term d † Z to J gives a gauge-invariant current.Thus Y + [B] µν can be viewed as an improved ADM current and the integral of this gives an improved ADM charge . This is unchanged under deformations of Σ d−2 which do not cross regions where T µν = 0.This agrees with the usual ADM charge when h is globally defined and is gauge-invariant even when h is not globally defined, so that it provides a natural improved definition of the ADM charge, which we denote P µ , for a constant Killing vector k µ .It can be written as The Penrose 2-form Y [K] for the B-type CKY tensors can be written Under a gauge transformation (2.1) this transforms as a 1-form gauge field Since we assume that Σ d−2 is contained in a region where R µν = 0, we have , so the Penrose charge for the B-type CKY tensors is which is the electric charge for the 1-form gauge field b µ .Now, consider the D-type CKY tensors, These have both K and K non-zero: (5.11) from eqs. (3.10) and (3.15), so that k is a Killing vector generating a Lorentz transformation.Then As before, adding the identically conserved improvement term d † Z to J[k] gives a gaugeinvariant current.Integrating this gives a covariant improved ADM angular momentum which we denote L µν , This charge agrees with the ADM angular momentum when h is globally defined and is gauge invariant even when h is not globally defined.Note that in this case eq.(4.6) can also be written as where Q[λ] is given by eq.(2.27) with λµνρ = (−1) d−1 (d − 1) Kµνρ and W [D] is given by with K a D-type CKY tensor.The variation of the total derivative term involving W [D] cancels the gauge variations of both Q[k] and Q[λ].However, there seems to be no gaugeinvariant way of separating the total charge d We will return to this point in a future publication, where we will show that this term can alternatively be understood as the covariantisation of one of the magnetic charges discussed in Ref. [12] that arises as an electric charge for the dual graviton.
6 Dual charges in d > 4 For each K, the Penrose 2-form current Y [K] is conserved on-shell and can be integrated over a (d−2)-cycle to give a conserved charge Q It is a topological charge that generates a 1-form symmetry [11].The dual If this is the case, it can be integrated over a 2-cycle to give a conserved dual charge q[K].This dual charge then generates a (d − 3)-form symmetry [11].
We now consider the conditions on K for the closure of the Penrose 2-form.Taking the curl of eq.(3.1) and using the CKY equation (3.6), we find where we have used the differential Bianchi identity ∂ We can form conserved charges by integrating Y [K] over a 2-cycle Σ 2 which is contained in a region where R µν = 0, where K µν = σ µν is a closed CKY tensor, The charge q[A] for the A-type closed CKY tensors can be written as where This is the magnetic charge for the 1-form gauge field a and is non-zero only when h is not a globally defined gauge field configuration.
For the B-type closed CKY tensors, the charge can be written which is the magnetic charge for the 1-form b defined in eq.(5.7).These charges also vanish unless the gauge field is not globally defined.There are d(d+1)/2 such dual charges q[A], q[B] in d dimensions.There are d(d+1)(d+ 2)/6 CKY tensors but, as discussed in section 3.3, on Minkowski spacetime the Penrose charges are trivial when K is a KY tensor, so that only Q[B] and Q[D] are non-trivial.Then the number of non-trivial Penrose charges is d(d+1)/2 also (for d > 4).This is in accordance with the discussion of Ref. [8], which argued that the equality between the number of 1form and (d − 3)-form symmetries was to be expected for higher-form symmetries that are charged under continuous spacetime symmetries and that this required Q[A] and Q[C] to be trivial.For example, in the present case, the 1-form symmetries generated by the Q[K] transform non-trivially under the Lorentz group as the CKY tensors K carry Lorentz indices [9].This stems from the principle that higher-form symmetries always come in dual pairs [25].It will be seen in section 8 that for spacetimes that are the product of Minkowski space with a torus the classification of charges and dual charges is slightly different, but the equality between the numbers of 1-form symmetries and (d − 3)-form symmetries remains.

Penrose charges in d = 4
The case of d = 4 is special due to several properties of the Riemann tensor and the CKY tensors.

No independent dual charges in d = 4
We have seen that for any CKY tensor K the Penrose current Y [K] is conserved in regions where R µν = 0, i.e. on-shell in regions away from sources.However, in d = 4, Y [K] is also closed on-shell in these regions.This follows from eq. (6.1) which, in d = 4, implies This result is derived in Appendix B.2.The right-hand side of eq.(7.1) vanishes by the field equations in regions without sources.Note that this result holds only in four dimensions.
However, in regions where sources are present, Y [K] is only closed when K is a closed CKY tensor.
As Y [K] is both closed and co-closed in the absence of sources, we can build charges by integrating Y [K] or ⋆Y [K] over 2-cycles.The integral ⋆Y [K] gives the Penrose charges while Y [K] gives further conserved charges.However, in four dimensions, these charges are not independent of the Penrose charges.This follows from the duality of the CKY tensors in four dimensions, as we now show.
First, we recall from section 3 that the Hodge dual of a CKY tensor is also a CKY tensor.So in four dimensions the Hodge dual of a rank-2 CKY tensor K is another rank-2 CKY tensor ⋆K, giving a conserved current Y [⋆K].In four dimensions there is a 20dimensional vector space of CKY 2-tensors K and the dual tensors ⋆K form the same 20-dimensional space; Hodge duality is an automorphism of this space.The set of currents Y [⋆K] is precisely the same as the set of currents Y Next, the closure of Y [K] is equivalent to the co-closure of its Hodge dual, which can be written in terms of another CKY 2-tensor ⋆K.Now, when R µν = 0, the Riemann tensor and Weyl tensor, W µναβ , are equal.Therefore, where we have used the property that ⋆W ⋆ = −W .Hence As a result, in the absence of sources, Hodge duality doesn't give any new currents: Y [⋆K], ⋆Y [K] and hence ⋆Y [⋆K] all give the same set of currents as the Y [K].The equivalence of these charges was checked in Ref. [6] for a specific solution of the linearised vacuum field equations.The dual charges q[K] defined in eq.(6.2) are then related to the Penrose charges in d = 4 by Similarly, in four dimensions the double dual Riemann tensor ⋆R⋆ has many of the same properties as the Riemann tensor in the absence of sources so we could construct the conserved 2-form currents (⋆R⋆) µναβ K αβ but, again, this reproduces the same set of currents; this follows from (⋆R⋆) µναβ K αβ = (⋆R) µναβ (⋆K) αβ .

Relation of Penrose and secondary currents
In four dimensions, eq.(4.4) becomes where Note that in d = 4, K is a closed CKY 3-form and hence k = ⋆ K is a Killing vector that is, in general, different from K, as is explicitly shown in appendix A.2.
As discussed in section 4, if h µν is globally defined then so is Z[K] and the integral of eq.(7.7) gives so that the Penrose charges give the ADM charges.On the other hand, if h µν is not globally defined then the Penrose charges give the naive ADM charges (2.18) plus a topological term (4.6).
Note that comparing with eqs.(2.22) and (5.16) we also have with W [K] given by eq. ( 5.16), so that integrating over a 2-cycle Σ 2 gives This suggests that the Penrose charge for a CKY tensor K gives the ADM charge for the Killing vector k given by eq. ( 4.1) plus the dual ADM charge for the Killing vector k given by eq.(7.9) plus a topological charge associated with the 3-form W [K].However, we saw in the last section that for d > 4 the topological term serves to covariantise the naive ADM charges.Our aim in this section is to analyse the situation for d = 4.The improved Penrose 2-form Y + [K] depends on a CKY tensor K, which is given in terms of the constant antisymmetric tensors A, B, C and D in eq.(3.9).The results of subsection 3.3 restrict the charges in dimensions d > 4, so that the only non-trivial Penrose charges arise for the B and D tensors.However, in d = 4 that result does not apply and there are potential Penrose charges for all four tensors A, B, C and D.
From the results of section 3, only the B and D terms contribute to the Killing vector k µ = 2(d − 3) Kµ , whereas only the C and D terms contribute to the Killing vector kµ = (⋆ K) µ .Therefore, the Penrose charges for the CKY tensors with only A and C non-zero do not contribute to the charges Q[k] and only appear in the topological terms An explicit four-dimensional solution to the linearised Einstein equations is given in Ref. [6] and has charges corresponding to all four types of CKY tensor.The parts of the solution which couple to the B-and D-type CKY tensors are globally defined field configurations (they are linearised Schwarzschild and Kerr solutions respectively).However, the parts of the solution which couple to the A-and C-type CKY tensors are, indeed, not globally defined (they are a linearised C-metric and Taub-NUT space respectively).We review part of this solution in section 9.

Analysis of Penrose charges in d = 4
We now analyse the Penrose charges for the four types of CKY tensor in eq.(3.9).
The A-type CKY tensors are constant 2-forms and so give k µ = 0 and kµ = 0. Therefore, eq.(7.11) simplifies to give from eq. (7.8).As discussed above, these charges are non-zero only for non-globally defined h µν configurations.We will study an example of such a solution in section 9.The Penrose 2-form for A-type CKY tensors can be written as where a is the 1-form defined in eq. ( 6.4).The surface Σ 2 is required to be in a region where Then in terms of a, we can write the Penrose charge as which is an electric charge for the 1-form gauge field a.We define 2-form charges M µν by writing this charge as The B-type CKY tensors in eq.(3.9) give the constant translation Killing vectors If h µν is globally defined, then the integral of the total derivative term d ⋆ Z[B] vanishes and the B-type Penrose charges give the ADM momenta in eq.(2.18).As discussed in section 5 for d > 4, if h µν is not globally defined the total derivative term serves to covariantise the result to give a gauge-invariant definition of the ADM momentum P µ : As for the A-type charges, the Penrose 2-form for the B-type CKY tensors can be written as where b is the 1-form defined in eq.(5.7).Then the Penrose charge for the B-type CKY tensors is which is the electric charge for the 1-form b µ .The C-type CKY tensors in eq.(3.9) give k µ = 0 while kµ = (⋆C) µ are constant Killing vectors.Then eq.(7.11) yields The dual momenta Q[ k], given in eq.(2.23), are not gauge-invariant in general and the term involving W [C] serves to covariantise them.Then the Penrose charge gives a gaugeinvariant definition of the dual momenta which we denote Pµ , In particular, the dual mass (which is related to the four-dimensional NUT charge) is the Penrose charge for (⋆C) µ = −δ t µ .The D-type CKY tensors are the only ones to yield both non-zero Kµ = 1  2 k µ = D µν x ν and kµ = (⋆D) µν x ν , so both Q[k] and Q[ k] contribute to eq. (7.11).For a given D, the two Killing vectors k µ and kµ are Lorentz Killing vectors giving Lorentz transformations with parameters Λ µν = D µν and Λµν = (⋆D) µν .As in section 5, the charge Q[D] gives the ADM angular momentum when h µν is globally defined and the total derivative term serves to covariantise the definition of the angular momentum when h µν is not globally defined.This yields the improved definition of angular momentum that is gauge-invariant even when h µν is not globally defined.

Counting and duality
In four dimensions, there are 20 CKY tensors, and hence 20 Penrose charges, while there are only 10 Killing vectors, and hence only 10 ADM charges.This mismatch was one of the puzzles considered in Ref. [1].We have seen that the 4 charges Q[B] correspond to the 4-momentum and the 6 charges Q[D] correspond to the angular momentum, and in each of these cases the Killing vectors are proportional to K.There is another map from CKY tensors K to Killing vectors, with the Killing vectors given by k = ⋆ K, as in eq.(7.9), suggesting that the remaining 10 Penrose charges Q[A], Q[C] could be related to the Killing vectors k.This turns out to be the case for the C-type CKY tensors but not for the A-type ones.For the C-type CKY tensors, the vectors k are constant and so are the translation Killing vectors.However, the A-type CKY tensors have vanishing k and k and so the Penrose charges Q[A] are not related to any of the charges based on Killing vectors.It is the D-type CKY tensors for which k are the Lorentz transformation Killing vectors.
In addition to the 10 charges Q[B], Q[D] corresponding to the ADM charges there are the 10 charges Q[A], Q[C] which we have seen correspond to the 10 KY tensors (that is, the A-and C-type CKY tensors).As discussed in section 6, higher-form symmetries are expected to come in dual pairs with equality between the number of 1-form and (d − 3)form symmetries [25].If d = 4, the duality is between two 1-form symmetries.In this case, the four charges Q[B] are dual to the four charges Q[C] the six charges Q[A] are dual to the six charges Q[D].This gives a pairing between the 4-momentum P µ and the dual 4-momentum Pµ together with a pairing between the angular momentum L µν and the charges M µν defined in eq.(7.16).While we treat the theory classically here, this pairing can also be understood in a canonical quantisation framework [7,9].
It is interesting to compare this with the pairing for d > 4 in section 6 in which is the magnetic charge (6.3) for the potential a defined in eq.(6.4).Comparing with eq. ( 7.15), it is also the electric charge for the dual potential ã defined by eq. ( 6.4) with A replaced by ⋆A: which satisfies dã = ⋆da (7.27) As we can use (7.16) to write so that the pairing of L µν with M µν indeed corresponds to a pairing of Q[D] with q[A].
In d = 4, from eqs. (7.17) and (7.22), so that the pairing between the 4-momentum P µ and the dual 4-momentum Pµ can be viewed as a pairing between Q[B] and q[B].
In the absence of sources, the free graviton theory has a dual formulation in terms of a dual graviton hµν and the dual 4-momentum Pµ can be interpreted as the ADM 4-momentum for the dual graviton theory [12].8 Penrose charges in Kaluza-Klein theory 8.1 Linearised gravity on the product of Minkowski space with a torus Kaluza-Klein monopole solutions are BPS states carrying magnetic charges that are an important part of the spectrum of supergravity and M-theory.In particular, the graviphotons which arise from compactification on tori have magnetic monopole solutions whose uplift to the gravitational theory carry the gravitational magnetic charges discussed in Refs.[12,26].
In this section, we analyse the Penrose charges for a background which is the product of Minkowski space with a torus and in the next section we will evaluate these charges for linearised Kaluza-Klein monopoles and other solutions.We will show that the higher-form symmetries of the graviton and the graviphoton fields in the dimensionally reduced theory are unified from the higher-dimensional perspective, thus giving an interpretation of the uplift of these symmetries.
We focus on solutions of linearised gravity on M = R 1,D−1 × T n with D + n = d.We denote the coordinates x µ = (x m , y i ) where y i are periodic coordinates on T n , y i ∼ y i + 2πR i , and the x m are Cartesian coordinates on R 1,D−1 .The metric on M is the Minkowski metric η µν .
There are local solutions of the Killing equation (2.7) of the form (2.8) but those with explicit dependence on y i are not globally defined on M. The globally defined Killing vectors are the constants k µ = V µ and the vectors k µ with k m = Λ mn x n and k i = 0 corresponding to Lorentz transformations on R 1,D−1 .These give rise to conserved charges P µ and L mn as before, but now there are no charges L im and L ij .The ADM energy for Kaluza-Klein theories was discussed in Ref. [27].
Similarly, there are local CKY tensors of the form (3.9) but only those with no explicit dependence on y i are globally defined on M. The CKY tensors on M are then of the form (3.9) with the only non-zero parameters being A µν , giving a constant 2-form on M, C mnp , giving a constant 3-form on R 1,D−1 , and B y , giving a constant scalar on R 1,D−1 .From these CKY tensors we can construct corresponding Penrose currents and charges, as before.Note that, as there are no CKY tensors with B m = 0 or D mn = 0, it appears that P m and L mn cannot be expressed as Penrose charges.We will discuss below how these charges do in fact have a covariant expression in the dimensionally reduced theory.
Of particular interest are configurations in which the graviton field is independent of the toroidal coordinates, For these, as we shall see, more conserved charges can be defined.Such configurations can be dimensionally reduced to a field theory in D dimensions and the charges are most easily analysed in the dimensionally reduced theory.We analyse the currents and charges in the dimensionally reduced theory in the following subsections, and then examine their lift back up to d dimensions.

Kaluza-Klein Ansatz
We take a Kaluza-Klein Ansatz of the following form for the d-dimensional graviton h µν where φ ≡ n i=1 φ (ii) .We see that the D-dimensional theory is governed by a D-dimensional graviton hmn , n one-form graviphoton fields A (i) m , and n(n + 1)/2 scalars φ (ij) , which are arranged in a symmetric n × n matrix.We take all fields to be independent of the compact dimensions; that is, ∂ i h µν = 0.It immediately follows that the d-dimensional linearised Riemann tensor satisfies The components of the linearised Riemann tensor are where Rmnpq = −2∂ [m hn][p,q] is the curvature of the D-dimensional graviton and m is the field strength of the i th graviphoton.The Ansatz in eq. ( 8.2) is chosen so that G mn = Ḡmn , where Ḡmn is the Einstein tensor for the D-dimensional graviton hmn .Then the d-dimensional field equations G µν = T µν imply the D-dimensional equations Denoting the various components of the energy-momentum tensor as the equations of motion for the D-dimensional fields can be written Therefore, in the D-dimensional theory we see that T mn is the source for the graviton hmn , the T mi = j (i) m are electric sources for the graviphotons, and the scalars are sourced by both T ij components and the trace T µ µ .

2-form currents in the absence of sources
As in section 3, we will first consider the conserved 2-form currents in a region where Rmn = 0 and ∂ p F (i) pm = 0.If the field equations (8.7) hold, this is a region where T µν = 0. We can build two such 2-forms in the D-dimensional theory, for a 2-form Kmn and n one-forms ξ (ij) are symmetric in the m, n indices and do not lead to charges of the type discussed here.)By the same arguments as in section 3, the 2-form current Y [ K] is conserved in regions where Rmn = 0 provided that Kmn is a CKY tensor on R 1,D−1 .The conservation of X[ξ] follows similar lines.We have The first term vanishes when ∂ p F (i) pm = 0, viz.
The other term in eq. ( 8.9) similarly vanishes provided that ξ (i) satisfies where This is the condition for the ξ (i) to be closed conformal Killing vectors (closed CKVs) on D-dimensional Minkowski space.The general solution for where the n (i) are constant one-forms and m (i) are constants.Then is a constant.

Improvement in the presence of sources
We now relinquish the constraint that Rmn = 0 and ∂ p F (i) pm = 0.As seen in section 3.2, in the presence of sources we can add improvement terms to the 2-form currents which involve Ricci tensors.These improvements were useful in writing the relations between the Penrose charges and the ADM charges in section 5. We now give the relevant improvement terms for Kaluza-Klein solutions, the conserved 2-form currents for which are given in eq.(8.8) with Kmn a CKY tensor and ξ However, X[ξ] mn is of a different form.We find that the relevant improved 2-form is Their divergences are given by where j[ k] m = Ḡmn kn , with Ḡmn the Einstein tensor of the D-dimensional graviton, and The right-hand side of both equations (8.17) and (8.18) vanishes on-shell in regions where T µν = 0 from the field equations (8.7), as was required by the arguments of the previous subsection.
As seen in previous sections, this gives a relation between the 2-form currents Y + [ K] and X + [ξ] and the 2-forms F (i) mn and J[ k] mn (whose divergence gives the 1-form current j[ k] m ).The relation is found in the same manner as before.We have a 2-form secondary current J[ k] mn , given by eq.(2.13) for the D-dimensional graviton instead of the d-dimensional one, such that Then by the Poincaré lemma, eqs.(8.17) and (8.18) imply relations between the 2-forms Y + [ K] and X + [ξ], and the 2-forms J[ k] and F (i) .Indeed, we finds where Z and ∆ are 3-forms given by The form of Z is the same as that of Z in eq.(4.5), here evaluated for the D-dimensional graviton.
Notice that when ⋆X + [ξ] is integrated over a cycle, the total derivative term ∂ p ∆ mnp will vanish as ∆ depends on A (i) m only through the curvature F (i) mn , which is globally defined.

Analysis of the charges
For the D-dimensional dimensionally reduced theory, conserved charges are constructed by integrating ⋆Y + [ K] and ⋆X + [ξ] over a (D − 2)-dimensional surface Σ D−2 .We now require that Σ D−2 is contained in a region of R 1,D−1 where Rmn = 0 and ∂ p F (i) pm = 0, so that the following charges are conserved in the sense that Σ D−2 can be arbitrarily deformed within that region.We define where ⋆ is the Hodge dual on R 1,D−1 .
These charges can be written in terms of J[ k] and F (i) using eq.(8.22).Recalling that the ∆ contribution in eq.(8.22) vanishes when integrated, we have where are the ADM charges of the D-dimensional graviton.Eq. (8.27) is the D-dimensional version of the result we had previously for d dimensions, and the analysis of the previous sections then immediately applies here, with different results for the cases D = 4 and D > 4.
The new feature of these results is the set of Q[ξ] charges.Eq. (8.28) equates these charges to the electric charges for the graviphotons.These generate electric U (1) 1-form symmetries [11].Recalling eqs.(8.14) and (8.19), we note that this charge is associated with the m (i) -type terms in ξ (i) .The n (i) -type terms in eq. ( 8.13) contribute to X + [ξ] only via ∆ in eq. ( 8.24), which vanishes when integrated.
As a result, the charges associated with the D-dimensional fields in the Kaluza-Klein reduction of d-dimensional linearised gravity on an n-torus include the ADM charges of the D-dimensional graviton, the electric charges of the graviphotons, together with the magnetic charges for the D-dimensional graviton arising when hmn is a non-globally defined gauge field configuration.

Dual charges
As noted in section 6, in d dimensions there are dual charges in the linearised graviton theory which are supported on 2-dimensional surfaces and which are parameterised by closed CKY tensors σ.We now show that the same is true of the Kaluza-Klein theory.
For the case D = 4, we find that Y [ K] is closed and co-closed for all CKY tensors Kmn and X[ξ] is closed and co-closed for all closed CKVs ξ (i) m .This results from the duality properties of CKY tensors in four dimensions discussed in section 7.As found previously, the dual charges q[K] are already included in the set of Penrose charges Q For D > 4, we find that Y [ K] mn is closed when Kmn = σmn is a closed CKY tensor.We also finds that X[ξ] is closed for all closed CKVs ξ (i) .Therefore, we can integrate over a 2-cycle Σ 2 in a region of R 1,D−1 away from sources to give conserved charges.Integrating Y [σ] over a 2-cycle gives the charge q[σ] for the D-dimensional graviton which is familiar from section 6. Integrating X[ξ] over a 2-cycle yields new charges.We write where we have used eq.(8.4) in the first equality and mn] = 0 in the second.Integrating over a 2-cycle, the final term vanishes by Stokes' theorem as it is the exterior derivative of a globally defined 1-form.Therefore, integrating Y [σ] and X[ξ] over a 2-cycle yields charges The charges F (i) are the magnetic charges for the graviphotons, which generate (D − 3)form symmetries [11].Recall from eq. (8.14) that ξ(i) is only non-zero for m (i) -type closed CKVs ξ m , so the constant n (i) -type terms in eq. ( 8.13) yield vanishing charges as they contribute only to the total derivative term in eq.(8.30), which integrates to zero.
Finally, we return to the expectation that higher-form symmetries should come in dual pairs that was discussed in section 6.We have seen that the Kaluza-Klein type solutions considered in this section have charges supported on codimension-2 cycles and dual charges supported on 2-dimensional cycles (generating 1-form and (D − 3)-form symmetries respectively).When D = 4, both types of cycles are 2-dimensional and the two types of charges are not independent.They are generated by charges corresponding to four-dimensional CKY tensors K mn and four-dimensional closed CKVs ξ m produce vanishing charges.Therefore, there are 20 + n non-trivial charges in D = 4.
When D > 4, the charges q[σ] defined on 2-cycles are built only from the closed Ddimensional CKY tensors and the charges q[ξ] are built only from the m (i) -type closed CKVs.Again, the charges associated with the constant n (i) -type closed CKVs are trivial.The Penrose charges Q[K] are built from the B-and D-type D-dimensional CKY tensors and the charges Q[ξ] are built from the m (i) -type closed CKVs.The remaining CKY tensors and the constant closed CKVs produce trivial charges.So, again, there is the expected duality between the dimension-2 and codimension-2 charges (generating (D − 3)-form and 1-form symmetries respectively), with 1  2 D(D + 1) + n of each.

Uplift to d dimensions
We now ask whether these 2-form currents in D dimensions and their associated charges have a unified origin in d dimensions.First we introduce a 2-form K on M with components with K ij unrestricted.(K ij does not enter in the following analysis and can be chosen arbitrarily, e.g.K ij = 0.) We will consider the object which are the mn components of the d-dimensional 2-form where the only non-zero components of V µνρσ are This is seen as follows.We note that the first of eqs.(8.4) can be written from which it follows that Upon comparison with eq.(8.8), this gives eq. ( 8.33).
The charge corresponding to a given K is found by integrating ⋆Y [K ] over a codimension-2 cycle in a region of M away from sources.We take this cycle to fully wrap the n-torus For example, we may take Σ D−2 to be the (D − 2)-sphere at spatial infinity in R 1,D−1 .We then have where Q[ K] and Q[ξ] were defined in eqs.(8.25) and (8.26) respectively, and is the volume of the torus.Therefore we see that the charges generating the higher-form symmetries in the D-dimensional theory have an uplift to the d-dimensional theory.Then, from eqs. (8.27) and (8.28), the charges on the right-hand side of eq. ( 8.39) are related to the ADM and dual ADM charges of the D-dimensional graviton so eq.(8.39) gives a covariant d-dimensional origin for these charges.
Note that the terms involving φ in eq. ( 8.34) can be absorbed into a field redefinition of the d-dimensional graviton.We define h ′ µν to have components where hmn is defined in eq.(8.2).The curvature tensor R ′ for h ′ is so that now Y [K ] in eq. ( 8.34) can be written which is of the form of a Penrose current.
However, the current is not quite a Penrose current of the type discussed in previous sections as K is not a CKY tensor in d dimensions.Instead, it satisfies and That is, its K mn components are those of a D-dimensional CKY tensor, while its K mi components are those of n closed CKVs in D-dimensions.This leads to the conservation of the current (8.43) as R ′ µνρσ satisfies in the absence of sources, from eq. (8.7).The fact that its trace with the D-dimensional metric η pq vanishes allows the possibility of terms involving η pq on the right hand side of ∂ n K µν in the conditions for conservation of the current.The analysis of section 3 can be extended to allow for such possibilities in which restricting to a special set of configurations allows a more general set of conserved charges.For configurations in which the curvature is traceless with respect to some symmetric tensor Π µν , i.e.

Penrose charges for various solutions
In this section we evaluate the charges discussed in the previous sections for various solutions of the linearised Einstein equations.The linearisation arises from writing the full metric as g µν = ḡµν + h µν with ḡµν a solution of the Einstein equations and then the Fierz-Pauli equations for h µν are the terms in the Einstein equation for g µν that are linear in h µν , and involve the metric connection ∇µ for the background metric ḡµν .In this paper, we take ḡµν to be the Minkowski metric.We have so far used Cartesian coordinates for which ∇µ = ∂ µ , but in this section we also use spherical polar coordinates.
In the linearised theory, linear superpositions of solutions are again solutions, while of course this is not the case in the non-linear Einstein theory.
We discuss a solution with electric gravitational charges (i.e.ADM mass and angular momentum) which correspond to the B-and D-type CKY tensors, a solution with magnetic gravitational charge corresponding to the C-type CKY tensors, a solution which carries the (d− 3)-form symmetry charge q[K], and finally a solution which carries the A-type Penrose charge.

Five-dimensional linearised Myers-Perry black hole solution
An example of a solution which carries electric gravitational charges is the linearised fivedimensional Myers-Perry black hole metric.This solution is not well-defined at the origin of the spatial R 4 , so we are considering its linearisation around a background space which is R 1,4 with the line r = 0 removed.We denote this space (R 1,4 ) • and parameterise it by spherical coordinates {t, r, θ, φ, ψ}.
The linearised Myers-Perry metric has non-vanishing components where M is the mass parameter of the black hole and J 1 , J 2 are parameters corresponding to the rotation of the black hole in the φ and ψ directions respectively (that is, in the planes spanned by the Cartesian coordinates x 1 , x 2 and x 3 , x 4 respectively).
Calculating the Penrose charges for the different types of CKY tensors yields where B 0 , D 12 and D 34 are the components in Cartesian coordinates.Firstly, we note that the charges Q[A] and Q[C] vanish.This is in agreement with the discussion of section 3.3 as the A-and C-type CKY tensors are KY tensors and so do not contribute to the Penrose charges when d > 4.
The Penrose charge Q[B 0 ] is the ADM mass, in agreement with eq.(4.9) as the B-type CKY tensors correspond to translational Killing vectors.The Penrose charge Q[D] gives the two independent ADM angular momenta, which also agrees with eq.(4.9) as the D-type CKY tensors correspond to rotational Killing vectors.Note that the rotation parameters J 1 and J 2 are picked out by the components of the CKY parameter D µν in the planes orthogonal to the rotation axis.
For the solution in eq. ( 9.1), the total derivative contributions to the Penrose charges vanish so the Penrose charges give precisely the ADM charges in this case.

Linearised Lorentzian Taub-NUT
We now consider an example of a graviton configuration in four dimensions with non-trivial topology that carries magnetic gravitational charge.We denote the coordinates on R 1,3 by x m = (t, x α ) with α = 1, 2, 3. We will consider a background spacetime (R 1,3 ) • which, similarly to the previous section, is defined as R 1,3 with the line given by x α = 0 removed.Consider the Ansatz where A α is a 1-form connection on R 3 \ {0} which is independent of t with field strength We choose A so that the field strength is given by a potential V with and we choose V to be corresponding to a source at x α = 0 with strength N , referred to as the NUT charge.The non-zero components of the curvature tensor are then This is the linearisation of the four-dimensional Lorentzian Taub-NUT solution [28][29][30] and was referred to as a 'gravitypole' in Ref. [31].This solution was also discussed in Ref. [6].
The dual ADM mass is the charge Q[ k] in eq.(2.23) with k the constant timelike Killing vector km = δ m t and is proportional to the NUT charge We have seen in section 7 that this charge is related to the C-type CKY tensors, e.g. in eq.(7.22).To that end, consider a C-type CKY tensor K mn = λmnp x p (9.9) where λmnp has non-zero components λαβγ = −ǫ αβγ (9.10) Then km is related to K mn by eq.(7.9), viz., km = 1 3! ǫ mnpq λnpq =⇒ kt = 1, kα = 0 (9.11) We evaluate the Penrose charge by integrating over a 2-sphere at constant r and t.From eqs. (9.5) and (9.7), we find Then integrating over the 2-sphere gives The result is independent of r and t, reflecting the topological nature of the charge.Note that the result in eq. ( 9.13) is a factor of 2 larger than Q[ k] in eq.(9.8).The remaining contribution comes from the topological d ⋆ W [C] term in eq.(7.22).In particular, for K given by eq. ( 9.9), we find Hence the results in eqs.(9.8), (9.13) and (9.15) are consistent with eq.(7.22).
In the linearised gravity theory, this can be simply extended to multi-centred solutions with for some sources labelled by s of strength N s at positions x s ∈ R 3 .Provided the 2-sphere on which the charge is defined is large enough so as to contain all the sources, we recover the result (9.13) above with N replaced by s N s .This can be shown directly via a slightly more involved integration.We can also consider this as a part of a higher dimensional Kaluza-Klein type solution.Namely, we consider a solution in d = 4 + n dimensions on the space (R 1,3 × T n ) • which is defined to be R 1,3 × T n with the cylinders where x = x s ∈ R 3 removed.The coordinates on the full space are denoted x µ = (x m , y i ), as in section 8, with x m = (t, x α ) as above and the coordinates on T n are periodic with y i ∼ y i + 2πR i .The solution is given by simply taking the only non-zero components of h µν to be h αt = 2A α (x β ) where A α satisfies eqs.(9.4), (9.5), and (9.6) so the higher-dimensional solution is the product of the linearised Lorentzian Taub-NUT space with a torus.
In section 8.7, we have seen that for Kaluza-Klein solutions of this type, the ADM and dual ADM charges on (R 1,3 ) • are related to charges Q[K ] via eq.(8.39).Here, K µν is a 2-form on (R 1,3 × T n ) • related to K mn by eq.(8.32).This can be verified explicitly by calculating Q[K ] for the Lorentzian Taub-NUT solution above.The integration surface is a codimension-2 cycle in d dimensions, which we take to be Σ = S 2 × T n where S 2 is a 2-sphere of constant t and r within (R 1,3 ) • .From the higher-dimensional perspective, the charge could then be interpreted as that of an n-brane fully wrapping the torus.Similar manipulations to those above yield which is indeed related to Q[K] in eq. ( 9.13) by a factor of the volume of the n-torus, as expected from eq. (8.39).

Linearised Kaluza-Klein monopole
In Ref. [12] a solution was considered with a source carrying both mass m and a topological charge p.When m = |p| this is a linearisation of the Kaluza-Klein monopole solution.In the linearised theory, this can be regarded as a superposition of a solution with mass m and a solution with topological charge p, and further more there are superpositions of such solutions with multiple sources at different locations.In this subsection, we will consider multi-source solutions with magnetic charges and show that these carry the charges defined on 2-cycles introduced in section 8.6.Consider a background spacetime given by R 1,3 × S 1 with the cylinder {0 ∈ R 3 } excluded.We denote this space by (R 1,3 × S 1 ) • and its coordinates by x µ = (x m , y).The coordinates on R 1,3 are x m = (t, x α ) as in the previous subsection, and the coordinate on the S 1 is y ∼ y + 2πR y .We take a Kaluza-Klein Ansatz where A α is a 1-form connection on R 3 \ {0} whose field strength (9.4) satisfies This is a similar solution to that of section 9.2, with the roles of t and y reversed.Upon reduction over the S 1 fibre, this gives a four-dimensional solution with a Dirac monopole of strength p at x = 0.The five-dimensional Riemann tensor has non-zero components In particular, for a single monopole at the origin, we can take the Dirac monopole solution of F θφ = p sin θ and all other components equal to zero.Note that R a delta-function at the point x = 0, so we exclude this point from R 3 and exclude the corresponding cylinder x = 0 from R 1,3 × S 1 to give (R 1,3 × S 1 ) • .
From the discussion in section 8.6, for Kaluza-Klein type solutions, we can construct the charges q[σ] and q[ξ] where σmn is a closed CKY tensor and ξ (i) are closed CKVs in D = 4 dimensions.For the linearised solution above h mn = 0, so q[σ] = 0.However, we find where we have written the closed CKV as ξ m = n m +mx m as in eq.(8.13), whose divergence is ξ = m from eq. (8.14).Here we have taken Σ 2 to be a 2-sphere at fixed r = (x α x α ) 1/2 , t, and y.The fact that the charge q[ξ] is topological (i.e.unchanged by small deformations of the surface on which it is defined) is manifest as the result is independent of r, t, and y.
Again this can be generalised simply in the linearised theory to a solution with multiple sources of strengths p s at locations x s ∈ R 3 .In this case all the locations x s of the sources should be removed from the background manifold and the potential is The result in eq. ( 9.22) is modified simply by replacing p by s p s , provided that Σ 2 is large enough such that r > |x s | for all s.The manipulations leading to this result are similar to those that lead to eq. (9.17).
From eq. (9.22), for this example the charge reduces to the first Chern number of the graviphoton field A m = h my /2 evaluated on Σ 2 .Therefore, only when h (and therefore A) is a non-globally defined gauge field configuration will these charges be non-zero.The fact that the charge is non-zero and proportional to p in the example above reflects the non-trivial winding of the gauge field configuration around the compact direction.

Linearised C-metric solution
As noted in section 7.3, since the constant A-type CKY tensors neither contribute to Killing vectors K nor to closed CKY tensors K, the Penrose charges Q[A] are not related to the ADM charges.
An example of a solution with non-zero Penrose charge Q[A] has been given in Ref. [6, eq.(4.89)],where it was said to arise as a linearisation of the C-metric solution to general relativity.It has a discontinuity along the z-axis and this discontinuity can be remedied by the addition of a pure gauge solution for z > 0 and a different one for z < 0, with the two solutions related by a gauge transformation.This example and the example studied in section 9.2 are in accordance with our conclusion that the Penrose charges correspond to other charges as well as the ADM charges, and that the total derivative term in eq.(4.6) must be included for topologically non-trivial gauge field configurations.

Conclusion and outlook
We have seen that for any CKY tensor K, the corresponding improved Penrose 2-form current Y + [K] is conserved in regions without sources so that ⋆Y + [K] is a closed (d − 2)form that can be integrated over a (d − 2)-cycle Σ contained in a region where R µν = 0 to give a charge Q[K].This gives a topological charge that depends only on the cohomology class of ⋆Y + [K] and the homology of Σ.To obtain non-trivial charges, the space on which the graviton field is defined cannot be the whole of Minkowski space but must be Minkowski space with some regions removed, so that Σ can have non-trivial homology.The excluded regions are associated with the locations of sources.
In Minkowski space, the CKY tensors are given in terms of constant forms A, B, C, D by eq.(3.9).If d > 4 then ⋆Y + [A] and ⋆Y + [C] define trivial cohomology classes and so the charges Q[A] and Q[C] vanish.They also vanish in d = 4 if the graviton field is globally defined.The remaining Penrose charges Q[B] and Q[D] then give the standard ADM momentum and angular momentum when the graviton field is globally defined.In the case where it is not globally defined, these charges give improved gauge-invariant versions of the ADM charges.
The most interesting case is that in which d = 4 and the graviton is not globally defined.Then Q[B] and Q[D] give the covariantised ADM momentum and angular momentum as before, but now Q[A] and Q[C] can be non-zero and give magnetic-type charges for the graviton.Each magnetic charge can be expressed as the integral of a closed form which is locally exact; however it is the exterior derivative of a form which is not gauge-invariant and so is not globally defined in general.The charge Q[C] gives the NUT momentum of the linearised theory while the charge Q[A] is the electric charge for the gauge potential a µ defined in eq.(6.4).In the case in which the 2-form A is basic, i.e. it defines a 2-plane, then a µ can be thought of as the projection of the connection Γ onto that 2-plane.
As a particularly interesting application of our findings, we have considered the Kaluza-Klein setting, in which d-dimensional Minkowski space is replaced with the product of Ddimensional Minkowski space with a torus of dimension d−D.With an appropriate Kaluza-Klein Ansatz to reduce from d to D dimensions, we have found Penrose charges together with graviphoton electric charges in D dimensions.In this case, interesting gravitational magnetic charges arise for D = 4.
The Penrose charges may be regarded as generators of 1-form symmetries while the dual charges given by integrating the Hodge duals of the Penrose currents may be regarded as generators of (d − 3)-form symmetries.We have checked that the number of 1-form symmetries equals the number of (d−3)-form symmetries in Minkowski space, in accordance with the discussion of Refs.[8,25], and show that this remains true on the product of Minkowski space with a torus.
We have presented a unified framework for discussing charges in linearised gravity and the corresponding currents.In particular, global properties and dualities are discussed.Although some of the results have appeared previously, they are understood here in a wider context that facilitates generalisations.In particular, the triviality of Penrose charges associated with Killing-Yano tensors in dimensions d > 4 was shown in Ref. [8].We have extended the relation of Ref. [8] to an off-shell identity and gave a systematic construction using the properties of conformal Killing-Yano tensors.The equality between the numbers of 1-form and (d − 3)-form symmetries is expected on general grounds in Ref. [25] and was seen explicitly for linearised gravity on Minkowski space in Ref. [8].Here we confirm their results and extend them to Kaluza-Klein compactifications of the linearised graviton theory.A relation between the Penrose charges and the ADM charges was anticipated in Ref. [1] and seen in the examples studied in Ref. [6], but the general relation given here is novel and makes a number of properties explicit.
In addition to the charges discussed here, the linear graviton theory also has the magnetic charges of Ref. [12] given by integrating the currents (2.21) with Z given by eq.(2.30), constructed from κ-tensors satisfying eq.(2.29).These do not arise from the Penrose charges and it would be interesting to find a covariant origin for them similar to the Penrose construction.
A natural issue is the generalisation of our discussion here to the non-linear theory to give covariant charges for general relativity and supergravity.This will be addressed in a forthcoming paper.
where R µν is the Ricci tensor of the full spacetime.Similar integrability conditions follow for higher-rank CKY tensors.
Two further definitions of note are: • A Killing-Yano (KY) tensor is a CKY which is co-closed; that is, Kν 2 ...νp = 0, hence the final term in eq.(A.1) vanishes.
• A closed conformal Killing-Yano (closed CKY) tensor is a CKY which is closed; hence the first term in eq.(A.1) vanishes.
Throughout the discussion, the following facts will be of frequent importance [21,22]: • The Hodge dual of a CKY tensor is a CKY tensor.
• The Hodge dual of a KY tensor is a closed CKY tensor, and vice-versa.
A CKY tensor of rank 1 is simply a conformal Killing vector and, similarly, a KY tensor of rank 1 is a Killing vector.Not all manifolds admit Killing vectors and even fewer admit CKY tensors, although there are examples of spaces with no isometries which admit a rank-2 KY tensor [32].

A.2 CKY tensors of Minkowski space
We now describe the CKY tensors of Minkowski space, M = R 1,d−1 , though all these results apply to flat space of any metric signature.Consider a CKY 2-tensor K, which satisfies eq.(3.6).An integrability condition of the CKY equation is that K must satisfy ∂ µ ∂ ν ∂ ρ K αβ = 0.This is seen as follows.The Riemann curvature tensor vanishes for Minkowski space and hence the integrability condition in eq.(A.5) becomes ∂ (µ Kν) = 0 (A.6) so Kµ is a Killing vector on Minkowski space.That is, the general solution for Kµ is simply where u is a constant 1-form and v is a constant 2-form.Noting that Kµ is proportional to the divergence of K µν , this implies that K µν is at most quadratic in x µ .Inserting the most general quadratic Ansatz quickly leads to the conclusion that the most general solution is that given in eq.(3.9).Analogous arguments hold for higher-rank CKY tensors on flat space.Namely, integrability conditions of eq.(A.1) on flat space imply that Kµ 1 ...µ p−1 is a rank-(p − 1) KY tensor.Just as the Killing vectors, these are at most linear in x µ and so one again finds that the CKY tensors K µ 1 ...µp are at most quadratic in x µ .Substituting the most general quadratic Ansatz into the CKY equation then implies that a CKY tensor of rank p can be parameterised by four constant forms, denoted A, B, C and D, of rank p, p + 1, p − 1 and p respectively, as We recall from the discussion in section 3 that Kµνρ = ∂ [µ K νρ] is a closed CKY 3-form, and that Kµ = 1 d−1 ∂ ν K νµ is a Killing vector.In terms of K and K, the CKY equation for K is given in eq.(3.6), which we repeat here for convenience, Now, from eq. (2.2), we have where we have integrated by parts in the second equality and then used the CKY equation (B.1) in order to write the result in terms of K and K.We have collected total divergence terms into a 3-form Λ 1 given by Λ µνα

1 )
Consider next the B-type CKY tensors, K µν = B [µ x ν] .From eq. (3.10), these correspond to constant translational Killing vectors Kµ = − 1 2 B µ and give Kµνρ = 0. Then Z[B] µνρ = 12B [µ x ν Γ ρβ] |β (5.2) [ρ R µν]αβ = 0 in the first equality and the algebraic Bianchi identity R α[µνρ] = 0 in the second.It is clear that this vanishes for closed CKY tensors K, for which K = 0.For d > 4, Y [K] is closed only when K is a closed CKY tensor.However, in d = 4 we will see in section 7.1 that Y [K] is closed for all CKY tensors in regions where R µν = 0. We note in passing that (in d > 4) the improved Penrose 2-form Y + [K] is not closed unless K is closed and R µν = 0, in which case it is simply equal to the Penrose 2-form Y [K].The closed CKY tensors are given in eq.(3.16): they are the A-and B-type CKY tensors.
It is natural to regard the d charges Q[B] as pairing with the d charges q[B] and the d(d − 1)/2 charges Q[D] as pairing with the d(d − 1)/2 charges q[A].

m
closed CKVs.The 2-form current Y [ K] mn is the D-dimensional Penrose 2-form current for the CKY tensor Kpq on R 1,D−1 .Therefore, the improvement terms are the D-dimensional version of those considered previously in eq.(3.19),
Note that this is different from Einstein gravity where the charge is only conserved if Σ d−2 is taken to be the sphere at infinity and the fields satisfy suitable boundary conditions.
is contained within a region without sources (where j[k] vanishes).Then Q[k] gives the momentum and angular momentum in the (d − 1)-dimensional region S bounded by Σ d−2 .