Black hole chemistry, the cosmological constant and the embedding tensor

We study black-hole thermodynamics in theories that contain dimensionful constants such as the cosmological constant or coupling constants in Wald's formalism. The most natural way to deal with these constants is to promote them to scalar fields introducing a (d-1)-form Lagrange multiplier that forces them to be constant on-shell. These (d-1)-form potentials provide a dual description of them and, in the context of superstring/supergravity theories, a higher-dimensional origin/explanation. In the context of gauged supergravity theories, all these constants can be collected in the embedding tensor. We show in an explicit 4-dimensional example that the embedding tensor can also be understood as a thermodynamical variable that occurs in the Smarr formula in a duality-invariant fashion. This establishes an interesting link between black-hole thermodynamics, gaugings and compactifications in the context of superstring/supergravity theories.


Introduction
The realization in Refs. [1,2] that the cosmological constant can be considered as a thermodynamical variable in the context of black-hole physics has lead to a host of new developments encompassed in the field of black-hole chemistry. 1 As shown in [5], other constants defining a theory can also be seen as thermodynamical variables; in said reference, these constants occur as coefficients of higher-order curvature terms that are Lovelock densities.
Clearly, the same ideas can be applied to f (R) theories. Such theories can, however, be rewritten as theories of gravity coupled to a real scalar field with a non-trivial scalar potential. The form of the potential is related to the function f (R) and therefore contains the same constants as the function f (R), In Ref. [6], one of the authors proposed that the constants occurring in general scalar potentials can also be seen as thermodynamical variables in black-hole physics. In gauged supergravity, 2 though, these constants are related to the gauge coupling constants, which can be generically represented by the so-called embedding tensor. 3 This lead us to conjecture that the em-bedding tensor itself should also be regarded as a thermodynamical variable. Testing this conjecture is one of the main goals of this paper.
Wald's formalism [9][10][11] provides an efficient method to study black-hole thermodynamics, once the gauge freedoms of the fields have correctly been taken into account as explained in Refs. [12][13][14]. 4 It is only by doing this that one obtains the work terms in the first law of black-hole mechanics. However, one does not get all the work terms that are usually admitted in the literature 5 because there are no gauge symmetries associated to all of them: the gauge symmetry of the electromagnetic field gives rise to a work term of the form ΦδQ, where Q is the electric charge and Φ is the electric potential on the event horizon, but there is no additional gauge symmetry to give rise to the dual termΦδP where P is the magnetic charge andΦ is the magnetic potential on the horizon. There is no term proportional to the variation of the moduli, either, because there are no gauge symmetries associated to them. In addition, closer to our concerns in this paper, there is no term involving variations of the cosmological constant, for the same reason. 6 One can make such a term appear as in Ref. [19] but, since the action of diffeomorphisms on constants is trivial, this does not happen in a natural way and the physics behind this variation is unclear. The same happens to coupling constants.
Variations of the cosmological constant are possible in the context of supergravity/superstring theories, however. In higher-dimensional supergravity/superstring theories there are higher-rank forms which give (d − 1)-form potentials after compactification to d dimensions. These potentials are dual to constants which are determined dynamically by the equations of motion of the potentials. The main example is provided by the 3-form potential of 11-dimensional supergravity that gives rise to the cosmological constant of N = 8, d = 4 supergravity [20,21], but there are many others. Ultimately, however, it is expected that all the parameters of lower-dimensional theories (which can be collected in the embedding tensor) can be explained in a similar fashion.
The (d − 1)-form potentials dual to the constants do have an associated gauge symmetry. This suggests that the terms proportional to the variations of those constants in the first law could arise in association with the gauge symmetry of the dual (d − 1)forms. This possibility was first explored in Refs. [22,23] for the cosmological constant. We will explore it in a more systematic way here, using Wald's formalism, for the cosmological constant and for all the components of the embedding tensor in a toy model.
In Section 1, we are going to review the description of the cosmological constant in 4 A different approach to handle gauge charges based on a "solution phase space" has been proposed in Ref. [15]. 5 See, e.g. Ref. [16], in which terms proportional to the variations of the moduli are included. This inclusion has been contested in Ref. [17]. 6 One may argue that, perhaps, the procedure proposed in Refs. [12][13][14] produces a Noether-Wald charge that simply misses terms. However, as shown in Ref. [18], the Noether-Wald charge found in this way leads to a Smarr formula that also contains magnetic charges and potentials in a duality-invariant form, which suggests that nothing is missing from it. terms of a (d − 1)-form potential in the simplest setting: no matter fields. We will show that one can recover the first law of black-hole thermodynamics of Refs. [1,2] using Wald's formalism treating the gauge symmetry as in Refs. [12][13][14]. 7 Furthermore, we will derive the Smarr formula using the Komar integral as proposed in Refs. [1,5,26,27,6,18].
In Section 2 we are going to consider a more general example in 4 dimensions, with two real scalars and a 1-form field coupled to gravity. The theory is invariant under constant shifts of one of the scalars and this global symmetry can be gauged using the 1-form and its dual as gauge fields introducing at the same time two coupling constants that can be combined into a 2-component embedding tensor. This provides the opportunity to test the conjectured interpretation of the embedding tensor as a thermodynamical variable in black-hole physics. 8 As it is well known [28,29,[29][30][31][32] the consistency of this kind of electric/magnetic gaugings demands the introduction of 2-form fields which, in their turn demand the introduction of 3-form fields etc., giving rise to the so-called tensor hierarchy.
Complete tensor hierarchies have been constructed only in a few cases [33][34][35][36] and they show a one-to-one relation between the (d − 2)-forms and the global symmetries of the theory that can be gauged (just 1-dimensional in our toy model), between the (d − 1)-forms and the deformation constants of the theory (just the 2 components of the embedding tensor in our toy model) and between the d-forms and the constraints satisfied by the deformation constants of the theory (0 in our toy model). In Section 2 we will omit the construction of the tensor hierarchy of the model and we will directly introduce its fields (the 2 real scalars, the 1-form and its dual, the single 2-form dual to the Noether-Gaillard-Zumino current [38] and the two 3-forms dual to the coupling constants) and a democratic action based on the one in Ref. [30] in which all of them are present and in which the components of the embedding tensor are not constants, but functions which are forced to be constant on-shell. 9 Then, in Section 2 we will study the symmetries and define the conserved charges, including the Noether-Wald charge and the Komar charge, essentially along the lines of Ref. [41]. We will use the last two to prove the first law of black hole mechanics, to obtain the Smarr formula and to study the role the embedding tensor plays in both of them.
We will discuss our results and their implications in Section 3. Finally, the appendix contains a, not so successful, search for black hole solutions of our toy model to which our results can be applied. Unfortunately, it is very dif-ficult to find charged solutions and we only managed to find an embedding of the Schwarzschild-(a)DS solution into the model.

Dualizing the cosmological constant
The action for pure gravity (described by General Relativity) coupled to a cosmological constant Λ in arbitrary dimension d is and leads to the equations of motion which can be reduced to just The dimension-dependent factor of Λ in the action has been chosen so as to arrive to this last equation. On the other hand, in the conventions that we are using, when Λ is positive (negative), the maximally symmetric solution of the above equations is the (anti-) De Sitter solution. If we interpret the cosmological term in the Einstein equations (1.2) as an energy-momentum tensor and we compare it with that of a perfect fluid −(ρ + p)u µ u ν + pg µν , we find that the perfect fluid is characterized by (1.5) In differential-form language, the action (1.1) takes the form The equations of motion that one obtains from this action, defined by the variation of the action, up to total derivatives are given by and it is not hard to see that they can be rewritten in the form which provides a check of the equivalence of the actions Eq. (1.6) and (1.1).
As it is well known [21], 10 the cosmological constant Λ can be dualized into a (d − 1)-form potential that we will denote by C. The dualization can be carried out as follows: first of all, in order to encompass both the Λ > 0 and Λ < 0 cases, we define and promote the positive constant λ to a function λ(x) that we immediately constrain to be constant by introducing a Lagrange-multiplier term in the action with the dual (d − 1)-form C playing the role of Lagrange multiplier. A general variation of this action where ϕ stands for the fields e a , λ, C, gives the equations of motion and total derivative 16πG (d) We can use the equation of motion of λ 14) to replace λ by G ≡ dC, the d-form field strength of C, arriving at the dual action The variation of the action, up to total derivatives, gives the following equations of motion (1.17b) E C = 0 is solved by a constant ⋆G. If the constant is written in the form we recover the cosmological Einstein equations Eq. (1.9). This proves the classical equivalence of the original and the dual formulations, although the second is slightly more general since the equation of motion of C can be solved by piecewise constant λ(x)s whose discontinuities can be associated to (d − 2)brane sources, which couple in a natural way to the (d − 2)-form C. 11 An important difference between C and λ is that the former has a gauge freedom, under which it transforms where χ is an arbitrary (d − 2)-form. These gauge transformations leave the field strength G and the dual action Eq. (1.15) invariant. The action Eq. (1.11) is also gauge invariant, but only up to a total derivative. In any case, this invariance is associated to a conserved charge, apparently not present in the original system. This charge can be understood in terms of the branes that source C and is directly related to λ. We study the definition of this charge in the next section.
Although the actions Eqs. (1.15) and (1.11) are equivalent, in more complex cases in which we want to dualize constants that occur in multiple places in the action, one promotes the constants to fields and adds the Lagrange-multiplier terms with the dual potentials but one does not take the next step (eliminating the constants using their equations of motion) because the resulting actions are too complicated. Thus, one stays with actions similar to Eq. (1.11) and, therefore, in what follows, we will work with it.

The gauge conserved charge
Under the gauge transformation Eq. (1.19), the action Eq. (1.11) transforms as The total derivative is defined up to the total derivative of a total derivative, and we have made a choice that we will show is adequate to get a non-trivial result. From Eq. (1.12), instead, upon use of the Noether identity dE C = 0, we get which, together with the previous result leads to the off-shell identity and it is obvious that Given a particular solution of the equations of motion {e a , λ, C}, for each inequivalent (d − 2)-form that preserves it (i.e. for each harmonic χ h ), we can get the conserved charge contained in a closed (d − 2)-dimensional surface Σ d−2 with no boundary by integrating where we have used the fact that on-shell λ is constant.
Up to normalization constants this charge is just the volume of Σ d−2 measured in terms of the volume form χ h . Observe that the value of the charge does not change under the replacement of χ h by χ h + de for any (d − 3)-form e. Thus, it only depends on the De Rahm cohomological class of χ h , which is unique (up to normalization) on any compact, orientable Σ d−2 with no boundary. It is natural to use the induced volume form on Σ (d−2) that we will denote by Ω (d−2) Σ and, then, Thus, up to numerical constants and the volume ω Σ (not present in rationalized units) λ is the charge carried by C.

The Noether-Wald charge
The action Eq. (1.11) is also exactly invariant under diffeomorphisms and local Lorentz transformations. 12 We are interested in the Noether charge associated to the invariance under diffeomorphisms (Noether-Wald charge) and, therefore, we start by considering the variation of the action under diffeomorphisms generated by infinitesimal vector fields ξ Observe that λ must be treated as a scalar field and, therefore However, the infinitesimal transformations δ ξ of e a and C must take into account the gauge freedom of those fields as explained in Refs. [42,[12][13][14] in such a way that the invariance of the fields under those transformations for a certain parameter ξ (which we will denote by k) is a gauge-invariant statement. Since, in particular, δ k must leave invariant the metric, k is always a Killing vector. The transformations δ ξ e a and δ ξ C are combinations of standard Lie derivatives and ξ-dependent "compensating" gauge transformations where 12 Under infinitesimal diffeomorphisms it is invariant only up to a total derivative that we will take into account later.
is the (scalar) Lorentz momentum map, which satisfies for ξ = k On the other hand, P ξ is the (d − 2)-form momentum map associated to C, which is such that, for ξ = k After some massaging, we can write the transformations in the form The definitions of the momentum maps ensure that δ k e a = δ k ω ab = δ k C = 0 in a gauge-invariant fashion.
Observe that, on-shell, 13 Then, Although the existence of P k and, hence, of ω k was initially guaranteed by the invariance of G under δ k , we see here that it is also related to k being a Killing vector. Since on-shell G is, up to constants, the metric volume form, these two facts are obviously related.
Substituting the transformations Eqs. (1.33) into Eq. (1.27), using the Noether identities associated to the symmetries and performing simple manipulations we arrive at (1.38) Now we must take into account that the action Eq. (1.11) is invariant under diffeomorphisms and gauge transformations up to total derivatives: where we have used the explicit form of the compensating δ χ ξ transformation Eq. (1.29b) and the freedom that we have to add total derivatives of total derivatives to obtain a convenient expression. We arrive at the off-shell identity and it is not difficult to see that is given by (1.43)

The generalized, restricted, zeroth law
A crucial ingredient in the proof of the first law of black-hole mechanics along the lines of Refs. Refs. [42,[12][13][14] are the generalized, restricted zeroth laws. These laws are called "generalized" because they generalize the standard zeroth law of black-hole mechanics stating that the surface temperature κ is constant over the event horizon H to other thermodynamical potentials such as the electrostatic black-hole potential. On the other hand, they are are called "restricted" because their validity is restricted to the bifurcation surface BH 14 and because, rather than stating the constancy of a scalar quantity, they just state the closedness of a given differential form over BH. 15 This is enough for our purposes, though. Thus, at this point we are going to focus on solutions of the action Eq. (1.11) which describe stationary black-hole spacetimes with a cosmological constant determined by the value of λ, with bifurcate event horizons that coincide with the Killing horizon associated to a certain asymptotically timelike Killing vector k. By definition, k vanishes over the bifurcation surface which we denote by BH Thus, if all fields are regular over the horizon, it is clear that the inner products of their field strengths with k must vanish on BH : Let us consider the first of these properties. According to the definition Eq. (1.32), the (d − 2)-form P k is closed on BH. Being a (d − 2)-form on BH, it must be proportional to the induced volume form of BH, Ω BH : (1.46) Then, the closedness of P k implies that the coefficient f is a constant. This statement is one of the generalized, restricted zeroth laws of black-hole mechanics that have been used in Refs. [44,45,42,46,[12][13][14] to prove the first law. If we normalize the volume form so that its integral is equal to 1, f will be proportional to the volume of BH. This is the thermodynamical potential ("volume") associated to the thermodynamical variable λ ("pressure"). In order to make contact with the conventions of Ref. [6], it is more convenient to use the Killing co-potential (d − 2)-form ω k , which due to Eq. (1.36), the must also be closed on BH on-shell and, therefore, proportional to the volume form. Thus, we define the volume Θ λ by 16 (1.47) so that the volume Θ λ is positive for aDS black holes (signΛ < 0). Following Ref. [6], Θ λ can written as where B is a ball whose radius is that of the horizon and whose boundary is BH. This is an expression that we will generalize later on. The property Eq. (1.45b) is related to the standard zeroth law of black-hole mechanics because it implies DP k ab BH = 0 , (1.49) and because, on the bifurcation surface where n ab is the binormal to BH, with the normalization n ab n ab = −2. Since κ is constant according to the zeroth law, n ab must be covariantly constant on BH. We do not have an independent proof of this property, which is of purely geometric nature.
With this proof in hand, the zeroth law on BH (dκ BH = 0) would be a consequence of Eq. (1.49). All zeroth laws (generalized or not) would follow the same pattern since they would state that the coefficients of the expansion of certain closed (or covariantlyclosed) forms in a properly defined and normalized basis are constant as in Refs. [44,45,42,46,[12][13][14].

Komar integral and Smarr formula
Before we use the Noether-Wald charge and the restricted, generalized second laws to prove the first law of black-hole mechanics [47], it is useful to test our results constructing a Komar integral [48] following Refs. [1,5,26,27,6,18] and using it to derive a Smarr formula [49] that can be tested in actual black-hole solutions.
On-shell 17 and for a Killing vector k that generates a symmetry of the whole field configuration, the Noether-Wald current defined in Eq. (1.41) satisfies On the other hand, J[k] satisfies Eq. (1.42) off-shell with ξ = k, which implies that We can write a Komar integral with volume terms, as in Ref. [26], or we can take a step further and rewrite the last two terms as total derivatives Refs. [6,18]. This is 17 Here we use the symbol . = for identities that only hold on-shell. trivial for the second term. As for the first additional term, if k generates a symmetry of the whole field configuration, and ı k L must be locally exact. Therefore, there must exist a (d − 2)-form ̟ k such that which leads to the identity Then, the Komar integral over the codimension-2 surface Σ d−2 can be defined as the integral over the In order to determine ̟ k , we first calculate the on-shell value of the Lagrangian density: tracing over the Einstein equations (1.13a) Now, using the equation of motion of λ, Eq. (1.13b), to replace ⋆λ by G and the definition of the momentum map P k in Eq. (1.32) to replace ı k G by −dP k , we get and (1.60) The Komar integral is, then . (1.61) Let us consider the anti-De Sitter case (sgn Λ < 0): if integrate the exterior derivative of the integrand over a hypersurface whose boundary is the union of a spatial section of a stationary black-hole Killing horizon (the bifurcation surface, BH, for the sake of convenience) and spatial infinity, S d−2 ∞ , Stokes' theorem tells us that For the sake of simplicity, let us consider a static, spherically symmetric black-hole solution: the Schwarzschild-aDS-Tangherlini solution [50], whose metric is given by being the volume of the unit, round, (d − 2)-sphere, and M the ADM mass. The event horizon of this solution is placed at some value r h at which W(r h ) = 0. The Hawking temperature and Bekenstein-Hawking entropy can be expressed in terms of r h even if its value cannot be determined explicitly. They are given, respectively, by [2] We can evaluate the Komar integral on a constant r surface S d−2 Observe that, since the restricted, generalized, zeroth law guarantees that, over the bifurcation surface, P k is a constant times the volume form, in general we can take that constant outside of the Komar integral K(BH). In this simple case, also λ can be taken out of the integral, but in more general cases, only the constant defining the momentum map can be taken outside the integral. The integral of λ is also the integral of ⋆G on-shell, which gives, up to normalization constants, the associated charge.
Finally, using the definition of Θ λ in Eq. (1.48) and which is the form that follows from the usual scaling and homogeneity arguments. 18

The first law and black-hole chemistry
We are ready to proof the first law of black-hole mechanics in this theory using Wald's formalism [9,10,10].
We consider field configurations that describe asymptotically flat, stationary, blackhole spacetimes admitting a timelike Killing vector k whose bifurcate Killing horizon coincides with the black hole's event horizon H. k, then, will be given by a linear combination with constant coefficients Ω n of the timelike Killing vector associated to stationarity, t µ ∂ µ and the [ 1 (1.70) The constant coefficients Ω n are the angular velocities of the horizon. The starting point of the proof is the fundamental relation [9][10][11] valid for on-shell field configurations ϕ satisfying the equations of motion and perturbations of the fields δϕ satisfying the linearized equations of motion.
We are going to integrate this relation over the hypersurface Σ defined as the space bounded by infinity and the bifurcation sphere BH on which k = 0. Therefore, its boundary, δΣ, has two disconnected pieces: a (d − 2)-sphere at infinity, S d−2 ∞ , and the bifurcation sphere BH. Using Stokes theorem and taking into account that k = 0 on BH, we obtain the relation where we have added conventional minus signs that take into account the minus sign in our definitions of the variations of the fields under diffeomorphisms.
As explained in Ref. [11,45], the right-hand side can be identified with δM − Ω n δJ n , where M is the total mass of the black-hole spacetime and J n are the independent components of the angular momentum.
Using the explicit form of the Noether-Wald charge Eq. (1.43) The right-hand side of this identity is expected to be of the form TδS + ΦδQ for some charges Q and potentials Φ and/or "pressures" ϑ and "volumes" Θ ϑ . In this expression λ plays the role of charge or pressure, while Θ λ , defined in Eq. (1.47), plays the role of conjugate potential or volume. Using Eqs. (1.18) and (1.47) (sign Λ = −1) (1.74) Using also Eq. (1.50) we arrive at δM = TδS + Ω n δJ n + Θ λ δλ . (1.75) The (unconventional, in black-hole chemistry literature) factor of λ present in the definition of Θ λ can be absorbed in δ. However, when the cosmological constant arises as the square of another, more fundamental constant, as in gauged supergravity, this form of the first law is more natural. Also, in these theories, the coupling constant is often associated to (d − 1)-form potentials coming from higher dimensions [20].
A a matter of fact, it is always possible to introduce a (d − 1)-form potential dual to the coupling constants, masses or any other parameters occurring in the action. These (d − 1)-forms are part of what is known as the tensor hierarchy of the theory. At the level of the action they can always be introduced in the same way we introduced the potential dual to the cosmological constant: promoting first the parameters to fields and introducing the dual (d − 1)-form potentials as Lagrange multipliers that constrain the fields/parameters to be constant. Intuitively, we expect terms in the first law and Smarr formula associated to all those (d − 1) potentials and, henceforth, to all those coupling constants, masses and other parameters.
In the next section we are going to consider a very simple model inspired by gauged supergravity, in which we can test these ideas.

A more general example
In this section we want to consider a more general model which essentially describes two scalars φ 1 , φ 2 and a 1-form field A coupled to gravity, represented by the Vierbein e a , in d = 4. In this model, the invariance under constant shifts of φ 2 has been gauged using a combination of the 1-form A and its dualÃ as gauge fields with two coupling constants ϑ and ϑ. By consistency, it is necessary to introduce a 2-form B which can be taken to be the dual of the Noether current j associated to the invariance under constant shifts of φ 2 , so no additional degrees of freedom are added to the theory. Actually, one can write an action for all these fields which gives the expected equations of motion plus the duality relations between j and B and between A andÃ (see Ref. [30]).
One can go further, dualizing the two coupling constants into 2 3-forms C andC as we have done with the cosmological constant in the previous section, completing the tensor hierarchy as in Refs. [33,34]. Again, this introduces no new local degrees of freedom.
Before introducing the action that describes this system, we introduce some notation: the coupling constants, their dual 3-forms, the 1-form and its dual and the the 0and 2-form gauge parameters are collected in symplectic vectors ϑ M , C M , A M , σ M , χ M as follows: (2.1) The field strengths are defined as and δ M. is 1 forC and zero for C. Under the gauge transformations where the potential is assumed to be a function of φ 1 only and of the two coupling constants ϑ,θ which are needed for dimensional reasons; for the same reason they must appear in it quadratically, so that the potential is a homogeneous function of degree two, i.e. the potential satisfies The second and third terms in the second line of the action are gong to be referred to as "additional"; they are topological and do not contain kinetic terms.
The equations of motion are defined by the general variation of the action (2.7) The equations of the 3-forms are just so that the ϑ M are (piecewise) constant on-shell, as intended.
The Einstein equations only involve the field strengths of the fundamental fields φ 1 , φ 2 and A because the additional terms are all topological: (2.9) Observe that dual fieldsÃ, B occur in the energy-momentum tensor through the field strengths Dφ 2 and F. The additional terms do not involve the scalars, either, and, therefore Furthermore, they do not involve A and, therefore, Now, let us consider the equations of motion of the dual fields which give duality relations. The equation of motion ofÃ, gives the duality relation between φ 2 (the current j) and B (its field strength H) on-shell, when dθ = 0. The equation of motion of B, is the duality relation betweenÃ and A.
The equations of motion of the components of the embedding tensor are (2.14) On-shell these equations are the duality relations between the components of the embedding tensor ϑ M and the 3-forms C M as given in [33] In the framework of this theory, these duality relations are only non-trivial when the corresponding component of the embedding tensor occurs in the scalar potential. However, it is clear that if those parameters also occur as coefficients of terms of higher order in the Riemann curvature, the duality relations will be non-trivial as well.
Once the duality relations implied by the equations of motion of the dual fields A, B, C M and the embedding tensor ϑ M are taken into account, the action we are studying describes a very simple model of a vector field and two scalars, one of which is charged with respect to the vector field, its dual or a combination of both, coupled to gravity. The use of the dual vector field as a gauge field is, perhaps, unusual, and demands the presence of the 2-form B, but we can always eliminate this aspect of the model by settingθ = 0.
Finally, Θ receives contributions from the variations of e a , φ 1 , φ 2 , A M , ϑ M but not from those of B or C M , which occur in the action with no derivatives: As we have mentioned, the action is invariant under gauge transformations up to a total derivative that takes the form This total derivative is only defined up to total derivatives and we can make use of this freedom to obtain gauge-invariant results, if need be.

Gauge conserved charges
We are going to study the effect of all the independent gauge transformations simultaneously. We will denote all of them by δ g . From the general variation of the action Eq. (2.7) we get Substituting the above δ g variations and the expressions for the equations of motion and operating, we arrive at (2.21) On the other hand, the action is only gauge invariant up to the total derivative in Eq. (2.17) that we can write, for the sake of convenience, in the form (2.22) and combining this result with the previous one we arrive at the off-shell identity which implies, locally Now we must identify the Killing parameters σ M , Λ, χ M that generate transformations that leave invariant all the fields.
When the fields are on-shell, these Killing parameters give rise to several independent conserved charges in the 3-volume V with compact boundary ∂V: Electric charge: associated to β, which can set to 1: 19 If we deform ∂V without crossing any sources (i.e. points at which the equations of motion are not satisfied.), the difference between the charges will be, via Stokes theorem, the volume integral whose integrand vanishes on-shell.

2-form charge: associated to the function α
3-form charge: associated to the space which we are going to integrate over 20 , it is just its volume (surface) We can also define a magnetic charge which is conserved in the same sense as the electric one thanks to the Bianchi identity instead of the equations of motion. This charge can be combined with the electric one in a symplectic vector

Transformations of the fields
As usual, we want to define transformations δ ξ that annihilate all the fields of a given solution in a gauge-invariant way for certain parameters ξ = k which are, in particular, Killing vectors. We have to combine standard Lie derivatives and k-dependent ("compensating") gauge transformations into gauge-covariant Lie derivatives. It is convenient to start by analyzing the 2-form B through its gauge-invariant 3form field strength H = dB. Due to the Bianchi identity dH = 0 (2.34) When ξ = k, there must exist a momentum map 1-form P k such that Now, the transformation of B is the Lie derivative plus a gauge transformation with a ξ-dependent 1-form parameter Λ ξ (2.36) When ξ = k we can use the definition of the momentum map 1-form P k to get which is solved by the choice Then, we define where the 1-form P ξ is the momentum map 1-form P k when ξ = k. With these definitions, δ k B = 0 automatically and in a gauge-invariant fashion.
Let us now consider the gauge-invariant 2-form field strengths F M : On-shell and for ξ = k upon use of the definition of P k . Then, locally, there must exist momentum maps P M k such that The transformation of the 1-forms A M is (minus) their Lie derivative plus a gauge transformation with a ξ-dependent 1-form parameter Λ ξ which has to be the same we determined before and a gauge transformation with ξ-dependent 0-form parameters (2.43) which is solved by the choice and where, when ξ = k, P M ξ and P ξ are, respectively, the momentum map 0-and 1-forms. Again, δ k A M = 0 automatically and in a gauge-invariant form . φ 1 is a scalar, and transforms in the standard way This transformation is assumed to vanish for ξ = k. The scalar φ 2 is, however, a gauge field. It is convenient to analyze, first, its covariant derivative, which is, actually, gauge-invariant. (2.49) On-shell and for ξ = k the following identity must hold where σ M ξ defined in Eq. (2.46). The transformation of φ 2 is a combination of (minus) the Lie derivative and a gauge transformation with parameter σ M and δ k φ 2 vanishes identically by virtue of Eq. (2.50). Let us consider, finally, the 3-forms. As usual, it is convenient to study their 4form field strengths first. They are gauge-invariant on-shell only. By assumption, and because these are 4-forms in 4 dimensions defining the momentum map 2-forms P M G k . The transformations of the 3-forms C M must be a combination of their Lie derivatives and gauge transformations with the parameters σ M ξ , Λ ξ that we have already determined and, possibly χ M ξ : (2.53) On-shell and for ξ = k We can show that the last two terms are, locally, total derivative: (2.55) Thus, we define the 2-form X M 2 k by Absorbing it in the definition of P M G k , which now satisfies we conclude that which is solved by Then, we arrive at the definition which vanishes automatically for ξ = k.
Summarizing, the transformations that we are going to consider are and the momentum maps 0-, 1-, and 2-forms satisfy DP k ab = −ı k R ab , (2.62a) (2.63)

Transformation of the action
Substituting the above transformations of the fields in 64) integrating by parts and using the Noether identities we are left with (2.66) Under these transformations, the action transforms into the integral of a total derivative, that we have chosen so as to obtain a final gauge-invariant result: (2.67) Equating this result for δ ξ S with the one in Eq. (2.65) we arrive to the identity Simplifying this expression we get The second term in this formula, in parenthesis, should be compared with the 2form charge associated to gauge transformations Eq. (2.25).

Generalized, restricted, zeroth laws
We just need to adapt the discussion in Section 1.3 to the model at hand, which has more fields. On the bifurcation surface BH we have These equations are equivalent to the equations that the Killing parameters discussed on page 2.1 must satisfy: first of all, the second equation implies that P k = h + dα, where h is a harmonic 1-form on the bifurcation surface and α and arbitrary function. However, the first equation tells us that h has to be removed from that identity and P M k = ϑ M α + β M for β M which is constant over the bifurcation surface. The last equation implies that β M = ϑ M β if ϑ M = 0, but it is arbitrary when ϑ M = 0. The third equation takes the form and, summarizing, we have 21 where γ M is a constant symplectic vector and Ω BH is the volume 2-form of the bifurcation surface; from Eq. (2.71d) we have The components of the constant vector β M can be interpreted as the electric and magnetic potentials over the bifurcation surface and the fact that they are constant is the generalized zeroth law restricted to the bifurcation surface. It is unclear whether this property can be extended to the whole event horizon in this particularly complex model or, at least, it is unclear how to prove it. However, we will not need this proof. As the β M are the thermodynamical potentials associated to the electric and magnetic charges, we will denote them by Observe that Eq. (2.74) becomes a constraint on these potentials: The components of the constant vector γ M are the thermodynamical potentials ("volumes") associated to the thermodynamical variables ϑ M ("pressures"). Again, in order to make contact with the conventions of Ref. [6], we can define the potentials where V BH is the volume of the bifurcation surface The fact that the vector Θ M is constant over the bifurcation surface is another generalized, restricted, zeroth law There is no role for the function α: there are no conserved charges associated to the gauge transformations δ Λ and α will also drop out of the Smarr formula.

Komar integral and Smarr formula
We are now ready to construct the Komar integral for this theory along the lines explained in Section 1.4. It provides a highly non-trivial check of the Noether-Wald charge.
Let us consider a field configuration that satisfies the all the equations of motion and an infinitesimal diffeomorphism ξ = k that generates a symmetry of the whole field configuration. Then, since Θ(ϕ, δ ξ ϕ) is linear in δ ξ ϕ, it vanishes when ξ = k and, since the equations of motion are satisfied, so does Θ(ϕ, δ ξ ϕ). Then, from from the definition Eq. (2.69) we find that 22 22 As before, we use . = for identities that only hold on-shell.

J[k]
On the other hand, by construction, and, thus, the total derivative in Eq. (2.67) evaluated for ξ = k, which coincides with the on-shell value of J[k] in Eq. (2.79), must vanish identically and, locally, there is a 2-form ̟ k such that and, on-shell (⋆F =F and dϑ M = 0) Combining this result with the other terms and operating, we get (2.85) Let us now consider the term involving the scalar potential: using the fact that the potential is a homogeneous function of the embedding tensor, Eq. (2.6), and the on-shell ϑ M equation of motion, Eq. (2.14), we obtain After use of the definition of the 2-form momentum map P M G k in Eq. (2.62d), we arrive at (2.87) Combining this result with Eq. (2.70b) we obtain the Komar charge This is a manifestly (formally) symplectic-invariant result [18] that reduces to the result obtained in Section 1.4 if we eliminate the scalar and 1-form fields; it reduces to the Einstein-Maxwell result upon setting the embedding tensor to zero.
We can now proceed as in Section 1.4 to derive the Smarr formula through the identity between the Komar integrals over the bifurcation surface and at spatial infinity Eq. (1.62). At infinity and we obtain the Smarr formula In order to check this formula we need explicit analytic solutions of the equations of motion of this model, but this is quite difficult, as the attempt made in the appendix shows. However, it is clear that, had we considered an additional cosmologicalconstant parameter λ in the theory, we would simply have obtained an additional term −λΘ λ in the above formula. That formula should remain valid when the embedding tensor is set to zero, in which case the theory reduces to the cosmological Einstein-Maxwell theory. Still, it is quite difficult to check the Smarr formula explicitly because the radius of the horizon is the solution of a quartic equation, except in particular cases such as for cold black holes, which have extremal, zero temperature horizons [51].
Observe that, due to the constraint Eq. (2.76), only one combination of the electric and magnetic potentials occurs in the above formula and, henceforth, only one combination of the electric and magnetic charges does.

The first law and black-hole chemistry
It is not necessary to repeat here all the steps that lead to the first law (2.92) Observe that, as usual, only the variation of the electric charge and its associated electric potential occur in the first law. This could be due to a limitation of the techniques that we are using. Nevertheless, if the magnetic counterpart of the ΦδQ term was present, due to the constraint Eq. (2.76), there would be a combination of electric and magnetic charges the mass of the black whole would be independent of. We will comment upon this point in the discussion section.

Discussion
In this paper we have shown how the variations of the cosmological constant and other dimensionful constants occurring in a theory of gravity can be consistently dealt with and understood in the framework of Wald's formalism and how they enter the first law of black-hole thermodynamics and the Smarr formula. In the example that we have completely worked out in Section 2, the constants that we have considered can be seen as components of the embedding tensor (a very simple one since there is only a 1-dimensional symmetry to be gauged) and our result proves the conjectured role of the embedding tensor as a thermodynamical variable.
A very interesting aspect of the Smarr formula is that, if it is general enough and it includes all the charges a black hole can carry and all the moduli of the theory under consideration, then it has to be invariant under all the duality transformations. Observe that duality transformations act on the moduli and charges but leave the mass, temperature and entropy invariant because the Einstein metric is left invariant by them. In Ref. [18] we showed that, in the context of pure N = 4, d = 4 supergravity, indeed, the term involving the electric and magnetic potentials and charges is formally symplectic invariant. This automatically implies its invariance under the SO(6)×SL(2, R) duality group of N = 4, d = 4 supergravity since all the 4-dimensional duality groups act on the 1-form fields as a subgroup of the symplectic group [38]. The same happens in the very simple example that we have considered here but we have also seen that the term involving the embedding tensor and its conjugate thermodynamical potential is also electric-magnetic duality invariant as it should, according to the general arguments given above. In more general models the embedding tensor is denoted by ϑ A M , where the index A runs over the Lie algebra of the symmetry group of the theory. The terms that must occur in the first law and in the Smarr formula must be, respectively, of the form Thus, in general 4-dimensional theories with an arbitrary number of 1-form fields labeled by I, we expect the first law and the Smarr formula to take the general form 23 We expect to verify the validity of this general formula in more general models of gauge supergravity in forthcoming works.
Concerning the particular model that we have constructed and studied in Section 2 to test these ideas, as we pointed out before, only one combination of the electric and magnetic potentials may occur in the first law. Therefore, there would be a combination of electric and magnetic charges the mass of the black holes of this theory would not depend on. In order to check this quite unusual property it is necessary to find the most general black-hole solutions of the theory. This is a very complicated problem. In the appendix we have managed to find solutions with one charge (the embedding of the Reissner-Nordström-(A)DS black hole in this theory) for a particularly simple choice of embedding tensor, but these solutions are not general enough to check whether this property, predicted by the first law, that is true. 24 Further work in this direction is necessary and under way.

A Searching for solutions
We would like to have a black-hole solution of the theory introduced in Section 2 in order to test the general results that we have derived. For the sake of simplicity, we set ϑ = 0 (electric gauging) and we set ϑ = g, constant. We can set B = 0 (so F = dA)and ignore C. The equations of motion that remain to be solved are E a = ı a ⋆ (e c ∧ e d ) ∧ R cd + 1 2 ı a dφ 1 ⋆ dφ 1 + dφ 1 ∧ ı a ⋆ dφ 1 + 1 2 ı a Dφ 2 ⋆ j + Dφ 2 ∧ ı a ⋆ j equated to zero. In the search for solutions, it is convenient to express these equations in component language: We are interested in static, spherically-symmetric solutions with a metric of the form where λ and R are functions of r to be determined and dΩ 2 (2) = dθ 2 + sin 2 θdϕ 2 . (A.4) The timelike Killing vector is k = ∂ t and we assume that it generates a diffeomorphism that leaves invariant all the fields. This means that If we assume that the electromagnetic field is electric and we work in the gauge in which the only non-trivial component is A t and it is only a function of r, then the scalars φ 1,2 only depend on r as well and Now it is the turn of the Einstein equations. We can, first, take the trace R + 1 2 (∂φ 1 ) 2 + 1 2 (φ 1 ) 2 (Dφ 2 ) 2 − 2V = 0 , (A. 10) and use it in the original equations to simplify them where we have eliminated an integration constant through a shift of r and where the integration constant a will be set to 1 to give metric at spatial infinity the standard normalization. We are left with the following equations: The first equation is solved by for some other integration constant that we call, again, a. Then the other two equations take the form 2 (λr) ′ − 2 + 1 2 a 2 r −2 + 2Λr 2 = 0 , (A.20a) r 2 λ ′ ′ − 1 2 a 2 r −2 + 2Λr 2 = 0 . (A.20b) Combining these two equations we can eliminate the terms that depend on a: We can integrate it immediately: which corresponds to the Reissner-Nordström-(anti-)De Sitter (RN(A)DS) metric [50]. As a matter of fact, substituting the above value of λ in either of the previous equations, we find that c = a 2 /4 , (A. 23) and, since a is, up to constants, the electric charge, the identification of the solution with the RN(A)DS solution is confirmed.