Noether-Wald charge in supergravity: the fermionic contribution

We study the invariance of N = 1, d = 4 supergravity solutions under diffeor-mophisms and show that, in order to obtain consistent conditions (“Killing equa-tions”) invariant under local supersymmetry transformations, one has to perform supersymmetry transformations generated by the superpartner of the vector that generates standard diffeomorphisms, just as a superspace analysis indicates. Using these transformations, we construct a Noether-Wald charge of N = 1, d = 4 super-gravity with fermionic contributions which is diff- Lorentz-and supersymmetry-invariant (up to a total derivative).


Introduction
The determination of the gauge parameters whose associated transformations leave all the fields of a given configuration invariant (reducibility or Killing parameters) is a basic ingredient in the characterization of the solutions of a given theory and in the definition of their conserved charges [1].Since Wald's approach to the first law of black-hole thermodynamics is based on the properties of the conserved charge associated to the invariance under diffeomorphisms (the so-called Noether-Wald charge) [2], the definition and properties of the Killing parameters is also a fundamental in that context.
In diff-invariant theories with a spacetime metric, the parameters must be Killing vectors but, in presence of other gauge symmetries, one has to take into account the gauge transformations induced by the diffeomorphism [3]. 1 Crucially, the combined diff-gauge transformations (expressed through covariant Lie derivatives) leave invariant the fields in a gauge-covariant fashion.In Wald's approach, the induced gauge transformations give rise to the work terms in the first law of black-hole mechanics [6,7]. 2  In most theories the spacetime metric is a pure tensor under diffeomorphisms and its transformation does not induce any gauge transformations: the vector fields that generate the isometry are standard Killing vectors.In theories with local supersymmetry, though, the metric is no longer a pure tensor and it is necessary to take into account the local supersymmetry transformations induced by the diffeomorphism: the generator will not satisfy the standard Killing vector equation, but a supercovariant generalization [9].The superpartner of this equation (in pure N = 1, d = 4 supergravity) will express the invariance of the gravitino and must take into account the induced local Lorentz and supersymmetry transformations [9].The induced supersymmetry transformation parameters will give rise to additional, fermionic, terms in the Noether-Wald charge with potential implications for the thermodynamics of the black-hole solutions of these theories. 3 The superspace analysis of the problem 4 indicates that these two equations expressing the invariance of the metric and gravitino must involve a superpartner of the Killing vector that we can call Killing spinor, even though the equation it satisfies is not the standard one and our spinor is fermionic.Thus, one is lead to consider superdiffeomorphisms generated by a Killing supervector superfield whose leading components are the Killing vector and spinor. 5Their equations transform covariantly under all the symmetries of the theory and Killing parameters remain Killing after any of those transformations.
As a result, the Noether-Wald charge will also be invariant, up to toal derivatives, under all the local symmetries of the theory.This is an essential, sine qua non, property of the Noether-Wald charge.All the modifications from the standard result (the Komar charge) vanish for vanishing fermionic fields and, therefore, they do not affect the thermodynamics of purely bosonic black-hole solutions but they may have interesting consequences for black holes with non-trivial fermionic fields, as we will discuss in Section 4. We start by reviewing the action and equations of motion of N = 1, d = 4 supergravity in Section 2, and stuyding its symmetries and associated conserved charges (including the Noether-Wald charge) in Section 3.

N = 1, d = 4 supergravity
The 1 st -order action of N = 1, d = 4 supergravity [18, 19] in the conventions of [20]  is [19] 6 where are the Vierbein, gravitino, spin connection and gamma matrix 1-forms, respectively, and D is the (exterior) Lorentz-covariant derivative De a ≡ de a − ω a b ∧ e b , (2.3a) Ψ is the gravitino field strength 2-form.The curvature and torsion 2-forms R ab , T a can be defined through the identities The total (Lorentz and general) covariant derivative is ∇ and it satisfies the first Vierbein postulate ∇e a = De a − Γ µν a dx µ ∧ dx ν = 0 . (2.5) The solution to the equation of motion of the spin connection is ) ) where ω(e) is Levi-Civita spin connection and Using this solution in Eq. ( 2.1) we obtain the 1.5-order action S[e a , ω ab (e, ψ), ψ] [23].Its variation is given by where are, respectively, the Vierbein and gravitino equations of motion and the pre-symplectic 3-form.We have simplified E using the Fierz identity T a ∧ γ 5 γ a ψ = 0

Symmetries and conserved charges Local Lorentz symmetry
The 1.5-order action is exactly invariant under local Lorentz transformations where σ ab = −σ ba .Using the associated Noether identity we find the off-shell conserved current where Lorentz charge 2-form Q[σ] is given by Following [1] we can construct a conserved charge for each κ ab L generating a Lorentz transformation that leaves invariant all the fields of a given solution of the equations of motion, δ κ L e a = δ κ L ψ = 0, as the integral over a closed 2-surface Σ: Observe that the conservation of the Lorentz charge only needs the invariance of the gravitino and spin connection.Thus, the above formula can give non-trivial conserved Lorentz charges even if δ κ L e a = 0.

Local supersymmetry
Under the local supersymmetry transformations the 1.5-order action is only invariant up to a total derivative [19] Using the associated Noether identity [19,20] iγ a E a ∧ ψ + DE = 0 , (3.9) we find the supercharge 2-form Again [1], we can construct a conserved supercharge for each local supersymmetry parameter κ S leaving invariant a solution {e a , ψ} of the equations of motion, δ κ S e a = δ κ S ψ = 0, via the integral 8 which, as in the Lorentz charge case, only demands κ S to be a Killing spinor satisfying the Killing spinor equation (KSE) The on-shell local supersymmetry algebra acting on all the fields of the theory is where the parameters of the Lie derivative and local Lorentz and supersymmetry transformations are

Diffeomorphisms
When the fields under consideration have some kind of gauge freedom, infinitesimal diffeomorphisms δ ξ x µ = ξ µ induce ("compensating") gauge transformations that need to be taken into account.In this theory σ ξ ab and ǫ ξ are ξ-related local Lorentz and supersymmetry parameters which can be determined when ξ = k such that δ k = 0 on all the fields.
It is natural to start with the metric which, in non-supersymmetric theories, is just a general tensor so that δ k g µν = −£ k g µν = 0, the Killing vector equation (KVE) for k.However, in supergravity, the metric transforms under supersymmetry, £ k g µν = 0 would not be invariant, and, according to the general prescription, we must write [9] where ∇ µ (e) is the Levi-Civita covariant derivative.The superspace analysis suggests that ǫ ξ is related to the supersymmetric partner of the vector ξ, the spinor λ, by When the spinor λ is the superpartner of k, it will be denoted by κ.Thus, (k, κ) satisfy the following generalization of the KVE [9] ∇ with integrability condition where we have defined the supersymmetric generalization of the Lorentz momentum map or Killing bivector [5] associated to (k, κ) P k,κ ab by or by a 1-form equation Observe that this equation assumes Eq. (3.17

Covariance of the Killing equations
The main goal of this paper is to obtain results compatible with all the symmetries of the theory.Thus, we must prove that the equations δ k,κ g µν = 0 (3.17 It can be shown that the local on-shell supersymmetry algebra Eqs.(3.12), (3.13) also holds on k and κ with these supersymmetry transformations.An important intermediate result needed to prove this is or in the equivalent form that follows naturally from the on-shell superfield formalism Using this result one gets a simple and transparent supersymmetry transformation of the Lorentz momentum map equation in the form (3.20): For the supersymmetry momentum map equation Eq. (3.25) one finds The Noether-Wald charge The Noether-Wald charge is the 2-form associated to the invariance under diffeomorphisms.In order to get consistent results, we must use the δ ξ,λ transformations constructed in the previous section with compensating Lorentz and supersymmetry transformations.The later and the standard diffeomorphisms only leave the action invariant under diffeomorphisms up to a total derivative.Thus, we must take into account a total derivative term of the form On the other hand, if we use the general variation of the action Eq.(2.7), we arrive to the identity Using the Noether identity associated to local Lorentz invariance Eq. (3.2), the Noether identity associated to diff invariance: and the supersymmetry Noether identity Eq. (3.9), we are lead to the off-shell converved current This current is a total derivative as can be seen by using Eq.(3.3) with σ ab replaced by −P ξ ab and using the Fierz identities.The result is is the Noether-Wald charge 2-form we were after.Indeed, it is manifestly invariant under diffeomorphisms and local Lorentz transformations.Furthermore, for Killing parameters (k, κ) it can be shown that it is on-shell closed and supersymmetry-invariant up to a total derivative ) is, thus, the supersymmetric generalization of the Komar charge.Observe that this charge contains terms corresponding to the standard Komar charge (a Lorentz charge 2-form Eq. (3.4) for the momentum map) plus terms corresponding to the supercharge Eq. (3.10) neither of which is separately invariant under supersymmetry.

Discussion
Our definition of invariance under (super-) diffeomorphisms and the supersymmetric Noether-Wald charge we have derived from it provide a solid basis to study the supersymmetric thermodynamics of the black holes of N = 1, d = 4 supergravity.Unavoidably, and contrary to the results of [10], when they are present, fermions (gravitini) must play a role in it if local supersymmetry remains unbroken.As a general rule, gauge fields with gauge freedoms are expected to contribute to the Noether-Wald charge and, probably, other kinds of fermions will not contribute, although this needs to be proven.
In order to develop this supersymmetric thermodynamics (themodynamics in superspace, actually, because, when fermions are present, bosonic fields such as the metric have body and soul) many notions of Lorentzian geometry need to be extended to this realm: how are event horizons defined and characterized in this setting?(How do particles move in this space?)Do they coincide with (super-) Killing horizons?If so, can one generalize the definition of surface gravity to them? etc.
Another important ingredient to make this possible generalization relevant is the existence of black-hole solutions with gravitino hair.It was conjectured in [24] and proven in [25] that the only ones in N = 1, d = 4 (Poincaré) supergravity are those whose gravitini can be removed by a supersymmetry transformation as in [26].In N = 2, d = 4 (Poincaré) supergravity things are different [27,28] and a solution with non-pure-gauge gravitini built over the body of the extremal Reissner-Nordström black hole exists [29].This solution provides a testing ground for the ideas presented here [30].
The work presented here can obviously be extended to N = 1, 2, d = 4 AdS supergravity and higher-dimensional generalizations and it can also be applied to the study of the superalgebras of symmetry of solutions (the prescriptions used in [31][32][33]20] should follow from our definitions).Work in this direction is currently under way [30].
The definitions of differential form and exterior derivatives in superspace differ form those used in the spacetime component approach of the main text.In particular, differential forms in superspace are defined as and the exterior differential acts from the right, that is (see [34] as well as e.g.[39,38] and references therein).The same applies to the contraction of differential forms These conventions decrease the number of sign factors in supertensor expressions.The superspace torsion and curvature 2-forms are defined by [34] T where, as indicated above, exterior derivative acts from the right. 10 The bosonic and fermionic covariant derivatives D A = (D a , D α ) appear in the decomposition of the covariant differential as .8)space there and in [36,37]. 10Thus, the relation with the notation in the main text and in [20] is ω ab → −ω ab and R cd ab → −R cd ab , but T a → T a and R ab → R ab .