Covariant Noether charges for type IIB and 11-dimensional supergravities

Historically, many different approaches for defining such conserved charges have been proposed. In order to provide context for the contributions of the present work, it will be useful to give a brief review of these various approaches and how they relate to one another. However, a full and comprehensive review of the history of conserved charges in gravitational theories is beyond the scope of this article, and the interested reader is encouraged consult the following excellent reviews: [5, 6, 7, 8, 9, 10, 11, 12].

Not surprisingly, the first considerations of conserved charges focused on asymptotically flat space-times (see [11] for a review). In the framework of the Hamiltonian formalism, the first major step forward in the task of defining conserved charges was given by the celebrated Arnowitt-Deser-Misner (ADM) definition of energy [13], which was then further complemented by the Regge-Teitelboim [14] analysis⁶⁶6The proof of the positivity of the ADM energy, under the dominant energy condition for the matter energy-momentum tensor, is a keystone result of general relativity [15, 16]. Also note that for some spacetimes with certain isometries one can define a conserved Komar quantity that is the integral of the covariant derivative of a timelike Killing vector field over a $(d-2)-$surface [17]. Sometimes, especially in the case of stationary asymptotically flat solutions, appropriate Komar integrals do match the corresponding ADM quantities [11].. The ADM mass and angular momentum are computed as integrals over spheres at spatial infinity, a calculation which in principle could be carried out, at least approximately, by a distant observer. Alternatively, one can use the Hamiltonian formalism with boundary at the null infinity; in this case we get the Bondi-Sachs mass and angular momentum [18, 19, 20, 11]. And finally, using the Hamilton-Jacobi analysis of the gravitational action functional, which can be seen as a short-cut to the full Hamiltonian analysis, quasi-local energy and angular momentum expressions (i.e. defined with respect to a bounded region of spacetime) were obtained by Brown and York [3, 21]. Reassuringly, these expressions converge to the ADM (or Bondi-Sachs) charges as spatial (or null infinity) is approached⁷⁷7Other useful quasi-local definitions of mass are the Misner-Sharp energy [22], the Hawking mass [23], the Bartnik mass [24], the Kijowski$-$Liu-Yau energy [25, 26], the Wang-Yau mass [27] among others (see [7, 10, 27, 11] for reviews on quasi-local masses).. The original Hamiltonian analysis of [13, 14] was also extended to backgrounds with more generic asymptotics by Hawking and Horowitz [28]⁸⁸8The expressions of [28, 29] reduce to the ADM expressions [13] and Abbott-Deser expression [30] when applied to the respective asymptotic backgrounds. The analysis of [3, 21, 28] requires a background subtraction procedure to get rid of divergences that arise due to the infinite spacetime volume. The same Hamiltonian and subtraction procedure can be used to compute the charges of spacetimes that are asymptotically the direct product of Minkowski spacetime and $\mathbb{R}^{p}$ (e.g. the black string or the black p-branes) [31, 32]..

The first Hamiltonian analysis of energy for asymptotically anti-de Sitter (AdS) spacetimes was provided by Abbott and Deser [30] and further developed by Henneaux and Teitelboim [33] and Chruściel and Nagy [34, 35]. This analysis yields the conserved charges for pure AdS-Einstein gravity. However, it cannot be simply extended to AdS solutions which also have matter fields present, such as scalar fields with certain fall-offs. Such cases are particularly important in the context of the AdS/CFT correspondence. In this context, Skenderis and collaborators have developed the method of holographic renormalization [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], a formalism that computes the energy and other thermodynamic observables of gravitational systems that have a dual holographic gauge theory description⁹⁹9Within holographic renormalization, the gravitational conserved charges of an asymptotically (locally) AdS solution is computed from the energy momentum tensor that one obtains when one varies the on-shell gravitational action w.r.t. the boundary metric. In spirit, this approach is thus similar to the traditional Hamiltonian analysis of, for example [3, 28]. However, this computation generically yields a divergent answer due to the infinite volume of the background. To extract the physical finite answer, it is common to use a background subtraction method (see e.g. [3, 28]); however an appropriate reference spacetime is not guaranteed to exist in general. Holographic renormalization effectively implements this background subtraction procedure in AdS in a systematic way using boundary counterterms. This is a procedure that follows naturally from the AdS/CFT duality since this is exactly the approach used in the QFT process of renormalization to remove unphysical divergencies.. A Hamilton-Jacobi approach to the method of holographic renormalization, whereby the radial coordinate plays the role of time was also developed [41, 42, 43, 44, 48, 45]¹⁰¹⁰10Positivity energy theorems for asymptotically AdS and asymptotically locally AdS spacetimes has been proven in [49, 50] using the Witten-Nester spinorial energy [16, 51].. It should be emphasized that the holographic renormalization charges agree with previous definitions of conserved charges [30, 33, 52, 28, 53, 48] when the latter are applicable, i.e. when the spacetime is conformal to the AdS solution and one considers the energy relative to AdS [48, 8]. Holographic renormalization also may be used for arbitrary asymptotically locally AdS spacetimes. Additionally, it may be shown that the holographic renormalization charges may be interpreted as Noether charges associated to asymptotic symmetries of the gravitational solution [48], and that these agree with the conserved charges computed using the covariant Noether charge approach, also called the covariant phase space approach [54, 55, 56, 20, 48].

Another important class of conserved charges are those provided by the covariant Noether charge formalism of Wald and collaborators [54, 55, 56, 20, 48], and is a systematic improvement of the covariant phase or covariant sympletic method initially discussed in [57, 58]. It also generalizes and incorporates many of the previous Hamiltonian approaches. Indeed, the covariant Noether charge formalism could equally well be called the manifestly Lorentz-covariant Hamiltonian formalism since it avoids the $(D-1)+1$ decomposition between spatial coordinates and the time coordinate that the ADM formalism requires.¹¹¹¹11Of course, with the traditional ADM and Hamiltonian approaches we can always check à posteriori that the conserved quantities obtained in the $(D-1)+1$ form are actually Lorentz-covariant. The first manifestly covariant Hamiltonian analysis has been developed by Abbott and Deser [30], which was also the first satisfactory approach to study charges in asymptotically AdS spacetimes.

Lastly, two other major classes of methods which deserve mention are conformal methods [59, 60, 61, 62] and spinorial methods [16, 51, 63, 49, 50, 11]. A particularly relevant sub-class of conformal methods are those which which exploit the asymptotic structure of spacetimes to compute the energy. This approach is best reviewed in [5] and it has been applied to asymptotic flat backgrounds [59, 60, 61, 62] as well to asymptotically AdS spacetimes [52, 53]. Spinorial methods have been used to prove the positivity of energy both in asymptotically flat [16] and asymptotically AdS [63, 49], with our without black hole horizons (see review [11]). In [50] attempts have been made to extend positivity of energy to general asymptotically locally AdS spacetimes.

In the present study, we are particularly interested in using the covariant Noether charge formalism of Wald and collaborators [54, 55, 56, 20, 48] and its cohomological formalism extension [64, 65, 66, 67, 68, 69, 9] to compute the energy and thermodynamic potentials of gravitational theories.

An aspect of the covariant Noether charge formalism that is not completely satisfactory is the fact that it still has ambiguities associated to the fact that we can always add a total divergence to the action. Using Stokes’ theorem this amounts to have a freedom in adding a boundary term to the action and Hamiltonian that can still change the definition of charges. Lagrangian methods based on cohomological techniques attempt to eliminate this feature. That is, the cohomological formalism is the most recent attempt to define conserved charges in gravitational theories in a more mathematically rigorous way [64, 65, 66, 67, 68, 69] (see [9] for a detailed review). Essentially, when the symmetry associated to the charge (energy, angular momentum, …) under consideration is exact, this method incorporates and agrees with the covariant Noether charge (phase space) formalism of [70, 71, 72, 73]. However, in the cohomological formalism, surface charges are computed integrating surface charge one-forms. The latter are constructed directly from Euler-Lagrange derivatives of the gravitational Lagrangian, which do not depend on total divergence terms. Therefore, the cohomological approach formally avoids the above ambiguities of the Noether charge formalism (in practice, it is important to reinforce that the cohomological method yields the same charges as the covariant Noether charges if in the latter formalism we just consider the boundary terms that arise naturally when we use integration by parts to eliminate derivatives of the field that is being varied to get the equations of motion). Additionally, but not less important, the cohomological method extends the covariant phase space method in the sense that it can also be used to define conserved charges that are associated to asymptotic (as oppose to exact) symmetries (in this manuscript we shall not consider such cases). Henceforth, in a slight abuse of language, when we refer to the Noether charge formalism we will often be implicitly referring to the cohomological extension of the standard Noether approach.

It has been observed and highlighted that the covariant Noether charge and cohomological formalisms can have a limitation or undesirable feature [73, 9, 69]. Namely, there are systems where the lack of manifest integrability of the first order variation of the charges seems to impede a definition of the charges [73, 9, 69]. Indeed, these two formalisms yield an expression for the variation of a Noether charge along the moduli space of parameters that parametrize a given family of solutions. In order to get the actual conserved charge of this family of solutions, one must then integrate this charge variation along a path starting from a reference solution. For one-parameter systems this is not a problem. However, in general $-$ for systems described two or more parameters $-$ we are often not guaranteed that such a conserved charge is well-defined. In the optimal scenario, certain integrability conditions are satisfied and the existence of the desired well-defined conserved charge for the solution at hand is guaranteed [73, 9, 69]. Notably, this is the case for the Kerr solution. However, supergravity solutions have already been found where integrability conditions are not obeyed and thus it was declared that energy is simply not defined for such a system [69]. In this manuscript, we will add a few more examples $-$ the Polchinski-Strassler black brane [74] and the CGLP black brane [75].

The main aim of the present work is to observe that even when the Noether charge variations are not manifestly integrable, asymptotic scale invariance allows two thermodynamic relations to be derived which in turn imply that that the Noether variation is in fact integrable. The two relations are: a first law of thermodynamics and a Smarr relation. Asymptotic scale-invariance is a common property for gravitational solutions in the context of the gauge/gravity correspondence. In particular it holds for solutions which are asymptotically a direct product spacetime $AdS_{p}\times X^{n}$ where $X^{n}$ is a compact manifold. We illustrate our claim explicitly by considering three non-trivial examples in IIB [74] and 11-dimensional supergravity [75, 76].

The layout of this article is as follows. In Sec. 2 we start by using the covariant Noether charge formalism of [54, 55, 56, 20] to define the variation of the Noether conserved charge of any type IIB supergravity solution. Remarkably, this expression has never appeared in the literature, as far as we are aware. As an application of this expression to a manifestly integrable system, in Sec. 2.2 we compute the energy of the IIB lumpy and localized black holes of [77, 78]. Not surprisingly, we will find that this Noether energy agrees with the energy computed originally using Kaluza-Klein holographic renormalization. Next, in Sec. 2.3 we discuss the energy of the Polchinski-Strassler black brane recently constructed in [74]. This is an asymptotically AdS${}_{5}\times S^{5}$ solution of IIB supergravity where all the bosonic fields are non-trivial and that corresponds to the high-temperature phase of the gravitational-dual of the Polchinski-Strassler solution [79]. In this case, the variation of the Noether charge does not lead to an obviously integrable expression. By exploiting the asymptotic scale-invariance, we derive two additional relations, a thermodynamic first law and a Smarr relation. When these relations are satisfied, the Noether variation becomes explicitly integrable, and an energy and chemical potential may then be defined unambiguously. We note that for this example, current holographic renormalization techniques cannot be used to compute the desired charges.

In Sec. 3 we consider the low-energy classical limit of M-theory. Again, we will start by using the covariant Noether charge formalism of [54, 55, 56, 20] to compute the variation of the Noether conserved charge of a general solution to 11-dimensional supergravity, an expression that is missing in the literature. Then, in Sec. 3.2 we will consider the thermodynamics of the CGLP black brane recently constructed in [75]. This is an asymptotically $AdS_{4}\times V_{5,2}$ solution where $V_{5,2}$ is the 7-dimensional Stiefel manifold. It is the simplest finite-temperature solution that asymptotes to the Cvetič, Gibbons, Lü, and Pope (CGLP) solution [80], which has a magnetic $G_{(4)}$-flux background whose asymptotic decay describes a mass deformation of the corresponding dual $CFT_{3}$. As before, we will find that the associated Noether variation charge does not satisfy the required integrability conditions which could suggest that an energy might not be well defined for this system. However, we will again find that asymptotic scale-invariance may be used to show that the Noether variation is explicitly integrable, and we are thus able to derive the desired thermodynamic charges.

The Noether charge formalism proceeds by considering the variation of the action. As usual, the first order variation of the action (2) leads to the classical equations of motion. The variation will include boundary terms which do not affect the equations of motion, however these are fundamental for computing conserved charges. Consider first the variation with respect to the graviton, which yields:

$$\delta S_{\text{IIB}}\big{|}_{g}=\frac{1}{2\kappa_{10}^{2}}\int\mathrm{d}^{10}x\sqrt{-g}\left[R_{ab}-\frac{1}{2}\,Rg_{ab}-T^{(1)}_{ab}-T^{(3)}_{ab}-T^{(5)}_{ab}\right]\delta g^{ab}+\delta S_{g}^{(\text{GH})}\,,$$

(5)

with

	$\displaystyle T^{(1)}_{ab}=P_{a}\bar{P}_{b}+P_{b}\bar{P}_{a}-g_{ab}P_{c}\bar{P}^{c}\,,$
	$\displaystyle T^{(3)}_{ab}=\frac{1}{8}\left[(G_{(3)})_{a}^{\phantom{a}cd}(\bar{G}_{(3)})_{bcd}+(G_{(3)})_{b}^{\phantom{a}cd}(\bar{G}_{(3)})_{acd}-\frac{1}{3}g_{ab}(G_{(3)})^{abc}(\bar{G}_{(3)})_{abc}\right]\,,$
	$\displaystyle T^{(5)}_{ab}=\frac{1}{6}(F_{(5)})_{a}^{\phantom{a}cdef}(F_{(5)})_{bcdef}\,.$		(6)

$\delta S_{g}^{(\text{GH})}$ is a total divergence contribution, namely the well-known Gibbons-Hawking boundary term

$$\delta S_{g}^{(\text{GH})}=\frac{1}{2\kappa_{10}^{2}}\int\mathrm{d}^{10}x\sqrt{-g}\,g^{ab}\,\delta R_{ab}=\frac{1}{2\kappa_{10}^{2}}\int\mathrm{d}^{10}x\sqrt{-g}\,\nabla^{a}\left(\nabla^{b}\delta g_{ab}-\nabla_{a}\delta g\right).$$

(7)

It will be convenient to encode the information of this Gibbons-Hawking boundary term into a 9-form:

$$\theta(g,\delta g)=\frac{1}{2\kappa_{10}^{2}}\star\left[\left(\nabla^{b}\delta g_{ab}-\nabla_{a}\delta g\right)\mathrm{d}x^{a}\right].$$

(8)

Next, take the variation with respect to the axion/dilaton complex scalar $B$:

	$\displaystyle\delta S_{\text{IIB}}\big{|}_{B,\bar{B}}$	$\displaystyle=$	$\displaystyle\frac{1}{2\kappa_{10}^{2}}\int\delta\big{|}_{B,\bar{B}}\left[-2P\wedge\star\,\bar{P}-\frac{1}{2}\,G_{(3)}\wedge\star\,\bar{G}_{(3)}-i\,C_{(4)}\wedge G_{(3)}\wedge\bar{G}_{(3)}\right]$
		$\displaystyle=$	$\displaystyle\frac{1}{2\kappa_{10}^{2}}\int 2f^{2}\bigg{[}\left(\mathrm{d}\star\bar{P}+2i\,\bar{Q}\wedge\star\bar{P}+\frac{1}{4}\bar{G}_{(3)}\wedge\star\,\bar{G}_{(3)}\right)\delta B+{\rm c.c.}\bigg{]}+\theta\left(B,\bar{B},\delta B,\delta\bar{B}\right),$

where $Q$ is defined in (2), ${\rm c.c.}$ stands for complex conjugate, and the boundary term is

$$\theta\left(B,\bar{B},\delta B,\delta\bar{B}\right)=-\frac{1}{\kappa_{10}^{2}}f^{2}\left(\star\,\bar{P}\wedge\delta B+\star\,P\wedge\delta\bar{B}\right).$$

(10)

Consider now the variation of the action with respect to the complex scalar $A_{(2)}$. A long computation yields:¹²¹²12Recall that $F_{(5)}$ is self-dual, $\star F_{(5)}=F_{(5)}$.

	$\displaystyle\delta S_{\text{IIB}}\big{|}_{A_{(2)},\bar{A}_{(2)}}\hskip-8.5359pt$	$\displaystyle=$	$\displaystyle\hskip-5.69046pt\frac{1}{2\kappa_{10}^{2}}\int\delta\big{|}_{A_{(2)},\bar{A}_{(2)}}\left[-\frac{1}{2}\,G_{(3)}\wedge\star\,\bar{G}_{(3)}-4F_{(5)}\wedge\star\,F_{(5)}-i\,C_{(4)}\wedge G_{(3)}\wedge\bar{G}_{(3)}\right]$		(11)
		$\displaystyle=$	$\displaystyle\hskip-5.69046pt\frac{1}{2\kappa_{10}^{2}}\int\frac{1}{2f}\bigg{[}\left(\mathrm{d}\star\bar{G}_{(3)}+i\,\bar{Q}\wedge\star\,\bar{G}_{(3)}-\bar{P}\wedge\star\,G_{(3)}-4\,i\,\bar{G}_{(3)}\wedge\star\,F_{(5)}\right)\wedge\delta A_{(2)}+{\rm c.c.}\bigg{]}$
			$\displaystyle\hskip-5.69046pt+\theta\left(A_{(2)},\bar{A}_{(2)},\delta A_{(2)},\delta\bar{A}_{(2)}\right),$

where the boundary term is

$$\theta\left(A_{(2)},\bar{A}_{(2)},\delta A_{(2)},\delta\bar{A}_{(2)}\right)=\frac{1}{4\kappa_{10}^{2}}\Big{[}i\star F_{(5)}\wedge\bar{A}_{(2)}-f\left(\star\,\bar{G}_{(3)}-\bar{B}\star G_{(3)}\right)-2i\,C_{(4)}\wedge\bar{F}_{(3)}\Big{]}\wedge\delta A_{(2)}+{\rm c.c.}$$

(12)

Finally, consider the variation of the action with respect to the real R-R field $C_{(4)}$ which gives:

	$\displaystyle\delta S_{\text{IIB}}\big{|}_{C_{(4)}}$	$\displaystyle=$	$\displaystyle\frac{1}{2\kappa_{10}^{2}}\int\delta\big{|}_{C_{(4)}}\left[-4F_{(5)}\wedge\star\,F_{(5)}-i\,C_{(4)}\wedge G_{(3)}\wedge\bar{G}_{(3)}\right]$		(13)
		$\displaystyle=$	$\displaystyle\frac{4}{\kappa_{10}^{2}}\int\left(\mathrm{d}\star F_{(5)}-\frac{i}{8}G_{(3)}\wedge\bar{G}_{(3)}\right)\wedge\delta C_{(4)}+\theta\left(C_{(4)},\delta C_{(4)}\right)$

with the boundary term given by

$$\theta\left(C_{(4)},\delta C_{(4)}\right)=-\frac{4}{\kappa_{10}^{2}}\star F_{(5)}\wedge\delta C_{(4)}\,.$$

(14)

Requiring that the variations (5), (2.1), (11), (13) vanish leads to the equations of motion of type IIB supergravity:

	$\displaystyle R_{ab}=P_{a}\bar{P}_{b}+P_{b}\bar{P}_{a}+\frac{1}{6}(F_{(5)})_{a}^{\phantom{a}cdef}(F_{(5)})_{bcdef}$
	$\displaystyle\qquad\qquad+\frac{1}{8}\left[(G_{(3)})_{a}^{\phantom{a}cd}(\bar{G}_{(3)})_{bcd}+(G_{(3)})_{b}^{\phantom{a}cd}(\bar{G}_{(3)})_{acd}-\frac{1}{6}g_{ab}(G_{(3)})^{abc}(\bar{G}_{(3)})_{abc}\right]\,,$		(15a)
	$\displaystyle\mathrm{d}\star P-2i\,Q\wedge\star\,P+\frac{1}{4}G_{(3)}\wedge\star\,G_{(3)}=0\,,$		(15b)
	$\displaystyle\mathrm{d}\star G_{(3)}-i\,Q\wedge\star\,G_{(3)}-P\wedge\star\,\bar{G}_{(3)}+4i\,G_{(3)}\wedge\star F_{(5)}=0\,,$		(15c)
	$\displaystyle\mathrm{d}\star F_{(5)}-\frac{i}{8}G_{(3)}\wedge\bar{G}_{(3)}=0\,,$		(15d)

where we contracted (5)-(2.1) with the inverse metric to get the Ricci scalar $R$ and then inserted this quantity back into (5) to get the trace reversed equation of motion for the graviton (15a).

It follows from (15a) that the on-shell Ricci volume form is $\star\mathcal{R}=2P\wedge\star\bar{P}+\frac{1}{4}G_{(3)}\wedge\star\bar{G}_{(3)}$ . Moreover, the self-duality condition (3) implies that on-shell $F_{(5)}\wedge\star F_{(5)}=F_{(5)}\wedge F_{(5)}=0$. Using these relations on (2) one finds that the on-shell 10-form Lagrangian reads

$$\mathcal{L}\big{|}_{on-shell}=-\frac{1}{2\kappa_{10}^{2}}\left(\frac{1}{4}G_{(3)}\wedge\star\,\bar{G}_{(3)}+i\,C_{(4)}\wedge G_{(3)}\wedge\bar{G}_{(3)}\right).$$

(16)

Given a diffeomorphism vector generator $\xi$, we can construct the associated sympletic Noether current 9-form [54, 55, 20]:

	$\displaystyle J=\Theta\left(g,\mathcal{L}_{\xi}g\right)$	$\displaystyle+\Theta\left(P,\bar{P},\mathcal{L}_{\xi}P,\mathcal{L}_{\xi}\bar{P}\right)+\Theta\left(A_{(2)},\bar{A}_{(2)},\mathcal{L}_{\xi}A_{(2)},\mathcal{L}_{\xi}\bar{A}_{(2)}\right)$
		$\displaystyle+\Theta(C_{(4)},\mathcal{L}_{\xi}C_{(4)})-\operatorname{\iota}_{\xi}\mathcal{L}\big{|}_{on-shell}\,,$		(17)

where $\Theta(\phi_{i},\mathcal{L}_{\xi}\phi_{i})\equiv\theta(\phi_{i},\mathcal{L}_{\xi}\phi_{i})$ for $\phi_{i}=\{g,P,\bar{P},A_{(2)},\bar{A}_{(2)},C_{(4)}\}$, i.e. we make the replacements $\delta\phi_{i}\to\mathcal{L}_{\xi}\phi_{i}$ on the boundary terms (8), (10), (12) and (14). Here, $\mathcal{L}_{\xi}\phi_{i}$ is the Lie derivative of the field $\phi_{i}$ along the diffeomorphism generator $\xi$. Also, $\operatorname{\iota}_{\xi}\mathcal{L}\big{|}_{on-shell}$ is the interior product of $\xi$ with the 10-form (16).

It can be shown that $\mathrm{d}J=-E_{i}\,\mathcal{L}_{\xi}\phi_{i}$, where $E_{i}$ stands for the equations of motion (summation convention holds here) [54, 55, 20]. Therefore, on-shell ($E_{i}=0$) the current is closed, i.e. $\mathrm{d}J=0$ for all $\xi$. It follows that there is a Noether charge 8-form $\widetilde{Q}_{\xi}$ locally constructed from $\{\xi,\phi_{i}\}$, such that on-shell one has $J=\mathrm{d}\widetilde{Q}$, since in these conditions $\mathrm{d}J=\mathrm{d}^{2}\widetilde{Q}_{\xi}=0$ [54, 55, 20].¹³¹³13$\widetilde{Q}_{\xi}$ is defined uniquely up to the addition of a closed form $\mathrm{d}\chi$.

To evaluate (2.1) it is useful to recall the definition of Lie derivative of a scalar¹⁴¹⁴14This is Cartan’s formula for $p=0$. and of a torsion-free metric tensor (with a Levi-Civita connection), as well as Cartan’s formula for the Lie derivative of a $p$-form $A_{(p)}$:

	$\displaystyle\mathcal{L}_{\xi}B=\xi^{a}\nabla_{a}B=\operatorname{\iota}_{\xi}dB\,,$
	$\displaystyle\mathcal{L}_{\xi}g_{ab}=2\nabla_{(a}\xi_{b)}\,,$
	$\displaystyle\mathcal{L}_{\xi}A_{(p)}=\mathrm{d}\operatorname{\iota}_{\xi}A_{(p)}+\operatorname{\iota}_{\xi}\mathrm{d}A_{(p)}\,.$		(18)

Using these relations, the identity $G_{(3)}\wedge\bar{G}_{(3)}=F_{(3)}\wedge\bar{F}_{(3)}$, the equations of motion (15), and the $p$-form identities listed in the appendix in (83), one finds:¹⁵¹⁵15To get $\Theta\left(g,\mathcal{L}_{\xi}g\right)$ we further use the commutator relation $\left[\nabla_{a},\nabla_{b}\right]\xi_{c}=R_{abcd}\xi^{d}$

	$\displaystyle\Theta\left(g,\mathcal{L}_{\xi}g\right)$	$\displaystyle=$	$\displaystyle\star\Big{(}\frac{1}{\kappa_{10}^{2}}\left[\nabla^{b}\nabla_{(a}\xi_{b)}-g^{bc}\nabla_{a}\nabla_{(b}\nabla_{c)}\right]\mathrm{d}x^{a}\Big{)}$		(19)
		$\displaystyle=$	$\displaystyle\frac{1}{9!}\left[\frac{1}{\kappa_{10}^{2}}\varepsilon_{a_{1}\cdots a_{9}a}\left(\nabla_{b}\nabla^{[b}\xi^{a]}+R^{a}_{\>\>b}\xi^{b}\right)\right]\mathrm{d}x^{a_{1}}\wedge\cdots\wedge\mathrm{d}x^{a_{9}}\,,$

	$\displaystyle\Theta\left(B,\bar{B},\mathcal{L}_{\xi}B,\mathcal{L}_{\xi}\bar{B}\right)$	$\displaystyle=$	$\displaystyle-\frac{1}{\kappa_{10}^{2}}\,f^{2}\left(\star\bar{P}\wedge\operatorname{\iota}_{\xi}dB+\star P\wedge\operatorname{\iota}_{\xi}d\bar{B}\right)$		(20)
		$\displaystyle=$	$\displaystyle\frac{1}{9!}\left[\frac{1}{\kappa_{10}^{2}}\varepsilon_{a_{1}\cdots a_{9}a}\left(-\bar{P}^{a}P_{b}-P^{a}\bar{P}_{b}\right)\xi^{b}\right]\mathrm{d}x^{a_{1}}\wedge\cdots\wedge\mathrm{d}x^{a_{9}}\,,$

	$\displaystyle\Theta\left(C_{(4)},\mathcal{L}_{\xi}C_{(4)}\right)$	$\displaystyle=$	$\displaystyle-\frac{4}{\kappa_{10}^{2}}\star F_{(5)}\wedge\left[\operatorname{\iota}_{\xi}\mathrm{d}C_{(4)}+\mathrm{d}\left(\operatorname{\iota}_{\xi}C_{(4)}\right)\right]$		(21)
		$\displaystyle=$	$\displaystyle\frac{1}{9!}\left[\frac{1}{\kappa_{10}^{2}}\varepsilon_{a_{1}\cdots a_{9}a}\left(-\frac{1}{6}(F_{(5)})^{ac_{1}\cdots c_{(4)}}(F_{(5)})_{bc_{1}\cdots c_{4}}\right)\xi^{b}\right]\mathrm{d}x^{a_{1}}\wedge\cdots\wedge\mathrm{d}x^{a_{9}}$
			$\displaystyle+\frac{1}{4\kappa_{10}^{2}}\big{(}-i\star F_{(5)}\wedge\bar{F}_{(3)}\wedge\operatorname{\iota}_{\xi}A_{(2)}+i\star F_{(5)}\wedge A_{(2)}\wedge\operatorname{\iota}_{\xi}\bar{F}_{(3)}+{\rm c.c.}\big{)}$
			$\displaystyle+\frac{4}{\kappa_{10}^{2}}\mathrm{d}\left(\star F_{(5)}\wedge\operatorname{\iota}_{\xi}C_{(4)}\right)-\frac{4}{\kappa_{10}^{2}}\mathrm{d}\star F_{(5)}\wedge\operatorname{\iota}_{\xi}C_{(4)}\,,$

	$\displaystyle\hskip-17.07182pt\Theta\left(A_{(2)},\bar{A}_{(2)},\mathcal{L}_{\xi}A_{(2)},\mathcal{L}_{\xi}\bar{A}_{(2)}\right)$
	$\displaystyle\hskip 14.22636pt=\frac{1}{4\kappa_{10}^{2}}\Big{[}i\star F_{(5)}\wedge\bar{A}_{(2)}-f\left(\star\,\bar{G}_{(3)}-\bar{B}\star G_{(3)}\right)-2i\,C_{(4)}\wedge\bar{F}_{(3)}\Big{]}\wedge\Big{[}\operatorname{\iota}_{\xi}F_{(3)}+\mathrm{d}\left(\operatorname{\iota}_{\xi}A_{(2)}\right)\Big{]}+{\rm c.c.}$
	$\displaystyle\hskip 5.69046pt=\frac{1}{9!}\left[\frac{1}{\kappa_{10}^{2}}\varepsilon_{a_{1}\cdots a_{9}a}\left(-\frac{1}{8}(G_{(3)})^{acd}(\bar{G}_{(3)})_{bcd}-\frac{1}{8}(\bar{G}_{(3)})^{acd}(G_{(3)})_{bcd}\right)\xi^{b}\right]\mathrm{d}x^{a_{1}}\wedge\cdots\wedge\mathrm{d}x^{a_{9}}$
	$\displaystyle\hskip 19.91684pt+\frac{1}{4\kappa_{10}^{2}}\Big{[}i\star F_{(5)}\wedge\bar{F}_{(3)}\wedge\operatorname{\iota}_{\xi}A_{(2)}+\left(i\star F_{(5)}\wedge\bar{A}_{(2)}-2i\,C_{(4)}\wedge\bar{F}_{(3)}\right)\wedge\operatorname{\iota}_{\xi}F_{(3)}+{\rm c.c.}\Big{]}$
	$\displaystyle\hskip 19.91684pt+\mathrm{d}\left[\frac{1}{4\kappa_{10}^{2}}\left[f\left(\star\,\bar{G}_{(3)}-\bar{B}\star G_{(3)}\right)-i\star F_{(5)}\wedge\bar{A}_{(2)}+2i\,C_{(4)}\wedge\bar{F}_{(3)}\right]\wedge\operatorname{\iota}_{\xi}A_{(2)}+{\rm c.c.}\right],$		(22)

and

	$\displaystyle-\operatorname{\iota}_{\xi}\mathcal{L}\big{|}_{on-shell}$	$\displaystyle=$	$\displaystyle\frac{1}{9!}\left[\frac{1}{\kappa_{10}^{2}}\varepsilon_{a_{1}\cdots a_{9}a}\left(\frac{1}{48}g^{a}_{\phantom{a}b}(G_{(3)})_{cde}(\bar{G}_{(3)})^{cde}\right)\xi^{b}\right]\mathrm{d}x^{a_{1}}\wedge\cdots\wedge\mathrm{d}x^{a_{9}}$		(23)
			$\displaystyle+\frac{i}{2\kappa_{10}^{2}}G_{(3)}\wedge\bar{G}_{(3)}\wedge\operatorname{\iota}_{\xi}C_{(4)}+\frac{i}{2\kappa_{10}^{2}}\Big{(}C_{(4)}\wedge\bar{F}_{(3)}\wedge\operatorname{\iota}_{\xi}F_{(3)}+{\rm c.c.}\Big{)}\,.$

Notice that in (19)-(23) we have opted to display some of the contributions explicitly in terms of the 9-form components. This is because most of these contributions add-on to build the equation of motion for the graviton (15a) and thus will not contribute to the final current (as displayed, it is easy to track these contributions). The only such contribution that survives after using the graviton equation of motion is the first term in (19) that can be rewritten as:

$\displaystyle\frac{1}{9!}\left[\frac{1}{\kappa_{10}^{2}}\varepsilon_{a_{1}\cdots a_{9}a}\nabla_{b}\nabla^{[b}\xi^{a]}\right]\mathrm{d}x^{a_{1}}\wedge\cdots\wedge\mathrm{d}x^{a_{9}}=\frac{1}{2\kappa_{10}^{2}}\mathrm{d}\star\mathrm{d}\xi\,.$

(24)

Adding all contributions (19)-(23), using the equations of motion (15a) and (15d), and (24) one finally finds that the sympletic Noether current 9-form (2.1) is given by

$\displaystyle J=\mathrm{d}\widetilde{Q}_{\xi}\,,$

(25)

where we have defined the Noether 8-form charge

	$\displaystyle\widetilde{Q}_{\xi}$	$\displaystyle\equiv$	$\displaystyle\frac{1}{2\kappa_{10}^{2}}\star\mathrm{d}\xi+\frac{4}{\kappa_{10}^{2}}\star F_{(5)}\wedge\operatorname{\iota}_{\xi}C_{(4)}$		(26)
			$\displaystyle+\frac{1}{4\kappa_{10}^{2}}\Big{[}\left(f\left(\star\,\bar{G}_{(3)}-\bar{B}\star G_{(3)}\right)-i\star F_{(5)}\wedge\bar{A}_{(2)}+2i\,C_{(4)}\wedge\bar{F}_{(3)}\right)\wedge\operatorname{\iota}_{\xi}A_{(2)}+{\rm c.c.}\Big{]}\,.$

Consistent with the discussion below (2.1), we find that on-shell the current $J$ is indeed conserved, $\mathrm{d}J=\mathrm{d}^{2}\widetilde{Q}_{\xi}=0$.

So far, we have taken $\xi$ to be just a diffeomorphism generator. Until otherwise stated, we now take $\xi$ to be a Killing vector field [54, 55, 56, 20] or an asymptotically Killing vector field [83, 84, 85, 86, 68, 9, 69]. Moreover, consider a solution of IIB supergravity. It depends on one or more parameters $m_{k}$ (say, with $k=1,\cdots$) and we are interested in considering variations $\delta$ along this moduli space of solutions. The variation $\omega_{\xi}$ (in the moduli space) of the charge associated to $\xi$ is then [54, 55, 56, 20, 84, 85, 86, 68, 9]

$$\omega_{\xi}\equiv\sum_{i}\left[\delta\widetilde{Q}_{\xi}(\phi_{i})-\widetilde{Q}_{\delta\xi}(\phi_{i})-\operatorname{\iota}_{\xi}\theta(\phi_{i},\delta\phi_{i})-E_{\mathcal{L}}\left(\mathcal{L}_{\xi}\phi_{i},\delta\phi_{i}\right)\right].$$

(27)

The last contribution vanishes when $\xi$ is a Killing vector, i.e. when $\mathcal{L}_{\xi}\phi_{i}=0$. We are interested inly in such cases so we set $E_{\mathcal{L}}\left(\mathcal{L}_{\xi}\phi_{i},\delta\phi_{i}\right)=0$ in the rest of our discussion. The third contribution is the interior product of the sum of the boundary terms $\theta(\phi_{i},\delta\phi_{i})$ given in (8), (10), (12) and (14). Finally, the first and second contributions in (27) are given by

	$\displaystyle\delta\widetilde{Q}_{\xi}=\partial_{m_{k}}\!\widetilde{Q}_{\xi}\,\mathrm{d}m_{k}\,,$
	$\displaystyle\widetilde{Q}_{\delta\xi}=\widetilde{Q}_{\partial_{m_{k}}\!\xi\,\mathrm{d}m_{k}}\,.$		(28)

Thus, $\widetilde{Q}_{\delta\xi}$ vanishes when the Killing vector $\xi^{a}$ is independent of the solution parameters $m_{k}$. This is certainly the case for the Killing vector fields $\xi=\partial_{t}$ and $\xi=\partial_{\psi_{j}}$ or $\xi=\partial_{w_{j}}$ responsible for time, rotational or translational symmetries, respectively. Therefore, $\widetilde{Q}_{\delta\xi}=0$ for the cases we are interested.

To sum, for each Killing vector field $\xi$ describing time, translational or rotational isometries, one can associate a 8-form Noether charge whose variation, in the moduli space of solutions, is given by

	$\displaystyle\omega_{\xi}$	$\displaystyle=$	$\displaystyle\sum_{k}\left(\partial_{m_{k}}\!\widetilde{Q}_{\xi}\,\mathrm{d}m_{k}\right)$
			$\displaystyle-\operatorname{\iota}_{\xi}\theta(g,\delta g)-\operatorname{\iota}_{\xi}\theta\left(B,\bar{B},\delta B,\delta\bar{B}\right)-\operatorname{\iota}_{\xi}\theta\left(A_{(2)},\bar{A}_{(2)},\delta A_{(2)},\delta\bar{A}_{(2)}\right)-\operatorname{\iota}_{\xi}\theta\left(C_{(4)},\delta C_{(4)}\right),$

with $\widetilde{Q}_{\xi}$ given in (26) and the several boundary terms $\theta(\phi_{i},\delta\phi_{i})$ given by (8), (10), (12) and (14), respectively. As a check of our computation, one can explicitly confirm, using the equations of motion (15) and the fact that $\xi$ is a Killing vector, that $\omega_{\xi}$ is indeed a closed $8-$form, that is to say $\mathrm{d}\omega_{\xi}=0$.

Equation (2.1) is an universal expression within IIB supergravity and one of our main contributions. It gives the variation (in the moduli or parameter space) of the conserved Noether charge associated to a Killing vector field $\xi$ (when $\xi$ is independent of the moduli). Note, however, that we have not yet discussed the physical interpretation for the associated Noether charge. For example, we might be tempted but we cannot say that $\omega_{\xi}$ with $\xi=\partial_{t}$ describes the variation of the energy; in general it does not. To find this physical interpretation and the energy of a solution we need some physical input from the system at hand. We illustrate the usefulness of (2.1) for two distinct systems in the two next subsections. In the first case the system is manifestly integrable, meaning that the integration of (2.1) (with $\xi=\partial_{t}$) yields immediately the energy. In the second case, an inspection of (2.1) will suggest that we cannot define an energy for the system. However, using the asymptotic scale-invariance of the system we will be able to compute the energy and thermodynamic potential of the system unambiguously.

In this subsection we use (2.1) to find the energy of solutions of type IIB supergravity that are asymptotically AdS${}_{5}\times S^{5}$ and that were previously constructed in [77, 78]. They describe lumpy and localized black holes, i.e. black holes that are deformed along the polar direction of the $S^{5}$. The equations of motion allow such a solution if the AdS₅ and $S^{5}$ radii, which we will denote by $L$, are the same. In these solutions, the only fields that are present are the graviton $g$ and the RR self-dual 5-form $F_{(5)}=\mathrm{d}C_{(4)}$, i.e. $B=0$ and $A_{(2)}=0$.

A general ansatz for such a solution which is static, preserves the $SO(4)$ symmetry of AdS₅ and the $SO(5)$ subgroup of the S⁵ (while eventually breaking its $SO(6)$ symmetry) and that has horizon topology $S^{8}$ is [77]

	$\displaystyle\mathrm{d}s^{2}=\frac{L^{2}}{\left(1-y^{2}\right)^{2}}\Bigg{[}-y^{2}\left(2-y^{2}\right)\,G(y)\,Q_{1}\,\mathrm{d}t^{2}$
	$\displaystyle\hskip 85.35826pt+\frac{4y_{+}^{2}}{\left(2-y^{2}\right)G(y)}\,Q_{2}\,\left[\mathrm{d}y+(1-y^{2})^{2}Q_{3}\,\mathrm{d}x\right]^{2}+y_{+}^{2}Q_{5}\,\mathrm{d}\Omega_{3}^{2}\Bigg{]}$
	$\displaystyle\hskip 28.45274pt+L^{2}\left[Q_{4}\,\frac{4\mathrm{d}x^{2}}{2-x^{2}}+Q_{6}\,\left(1-x^{2}\right)^{2}\mathrm{d}\Omega_{4}^{2}\right]\,,$		(30a)
	$\displaystyle C_{(4)}=\frac{L^{4}y_{+}^{4}}{\sqrt{2}}\frac{y^{2}\left(2-y^{2}\right)}{\left(1-y^{2}\right)^{4}}\,H(y)\,Q_{7}\,\mathrm{d}t\wedge\mathrm{d}S_{(3)}-\frac{L^{4}}{\sqrt{2}}\,W\,\mathrm{d}S_{(4)}\,.$		(30b)

where $\left\{Q_{I},W\right\}$ ($I=1,\ldots,7$) are functions of the $S^{5}$ polar coordinate $x$ and of the radial coordinate $y$ and, in addition, of the (single) black hole parameter $y_{+}$ that essentially gives the temperature of the black hole. One has $x\in[-1,1]$, with $x=\pm 1$ corresponding to the north and south poles of the S⁵, and $y\in[0,1]$ with $y=0$ being the horizon location and $y=1$ being the boundary of the AdS₅.

If we set $Q_{1}=Q_{2}=Q_{4}=Q_{5}=Q_{6}=Q_{7}=W=1$ and $Q_{3}=0$, we recover the familiar global AdS₅-Schwarzschild$\times$S⁵ solution which preserves the $SO(6)$ symmetry group of the $S^{5}$. This is best seen if we apply the coordinate transformations $r=\frac{r_{+}}{1-y^{2}}\,,\tilde{x}=x\sqrt{2-x^{2}}$ (and $y+=r_{+}/L$) which allow to rewrite the solution in its standard form, namely

	$$\mathrm{d}s^{2}=-f\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f}+r^{2}\mathrm{d}\Omega_{3}^{2}+L^{2}\left(\frac{\mathrm{d}\tilde{x}^{2}}{1-\tilde{x}^{2}}+(1-\tilde{x}^{2})\mathrm{d}\Omega_{4}^{2}\right)\,,$$		(31a)
	$$F_{\mu\nu\rho\sigma\tau}=\epsilon_{\mu\nu\rho\sigma\tau},\qquad F_{abcde}=\epsilon_{abcde}\,,$$		(31b)
	$$\hbox{where}\quad f=1+\frac{r^{2}}{L^{2}}-\frac{r_{+}^{2}}{r^{2}}\left(\frac{r_{+}^{2}}{L^{2}}+1\right)\,,$$

and $\epsilon_{\mu\nu\rho\sigma\tau}\,\mathrm{d}y^{\mu}\wedge\cdots\wedge\mathrm{d}y^{\tau}$ and $\epsilon_{abcde}\,\mathrm{d}x^{a}\wedge\cdots\wedge\mathrm{d}x^{e}$ are the volume forms of the AdS₅ and $S^{5}$ base spaces, respectively.

Besides including the AdS₅-Schwarzschild$\times$S⁵ as a special solution, the ansatz (30) also describes black hole solutions that preserve the full $SO(4)$ symmetry of the S³ but only an $SO(5)$ symmetry of the S⁵, that is to say, that allow deformations along the polar direction $x$ that break $SO(6)$ down to $SO(5)$. For such ‘lumpy’ black hole solutions one has $\left\{Q_{I},W\right\}=\left\{Q_{I}(x,y;y_{+}),W(x,y;y_{+})\right\}$ but it is still true that the AdS₅ or S⁵ radius $L$ drops out of the equations of motion, so that the black hole is described by the single parameter $y_{+}$. The temperature of the black hole is [77]

$$T=\frac{1}{L}\frac{2y_{+}^{2}+1}{2\pi y_{+}}\,.$$

(32)

This background is a particularly good example to illustrate the practical use of (2.1) to obtain straightforwardly the energy of a ‘simple’ system that depends on a single parameter ($y_{+}$) and for which the variation of the Noether charge is manifestly integrable. For $\xi=\partial_{t}$, (2.1) reduces in the present case to

	$\displaystyle\omega_{\xi}$	$\displaystyle=$	$\displaystyle\partial_{y_{+}}\!\widetilde{Q}_{\xi}\,\mathrm{d}y_{+}-\operatorname{\iota}_{\xi}\theta(g,\delta g)-\operatorname{\iota}_{\xi}\theta\left(C_{(4)},\delta C_{(4)}\right)$		(33)
		$\displaystyle=$	$\displaystyle\frac{1}{2\kappa_{10}^{2}}\Big{\{}\partial_{y_{+}}\Big{[}\star\mathrm{d}\xi+8\star F_{(5)}\wedge\operatorname{\iota}_{\xi}C_{(4)}\Big{]}$
			$\displaystyle-\operatorname{\iota}_{\xi}\Big{[}\left(\nabla^{b}\partial_{y_{+}}g_{ab}-g^{bc}\nabla_{a}\partial_{y_{+}}g_{bc}\right)dx^{a}-8\star F_{(5)}\wedge\partial_{y_{+}}C_{(4)}\Big{]}\Big{\}}\,\mathrm{d}y_{+}$

where we collected $\theta(g,\delta g)$ and $\theta\left(C_{(4)},\delta C_{(4)}\right)$ from (8) and (14), respectively.