Quasi-local charges and asymptotic symmetry generators

The quasi-local formulation of conserved charges through the off-shell approach is extended to cover the asymptotic symmetry generators. By introducing identically conserved currents which are appropriate for asymptotic Killing vectors, we show that the asymptotic symmetry generators can be understood as quasi-local charges. We also show that this construction is completely consistent with the on-shell method.


Introduction
the interesting aspects of the off-shell construction is its intimate relationship with the quasilocal construction. In the end, the ADT formalism for conserved charges was shown to give the same expression as those from the covariant phase space formalism, and thus also, those from the BBC formalism. This result naturally leads to the question on how to extend the ADT formulation to the case of asymptotic symmetry generators.
In this paper we would like to address this issue. We generalize the off-shell ADT current and potential of the exact Killing vectors to those of the asymptotic Killing vectors. In section 2, we construct the explicit form of the generalized off-shell ADT current. In section 3, we give a natural way to obtain the generalized off-shell ADT potential from the Lagrangian. And then we compare our results with those in [18,17]. In the final section, we summarize our results and give some comments on the open issues and on the future direction.

Generalized off-shell conserved currents
In this section we introduce the generalized off-shell ADT current in the spirit of the original construction of the ADT current [8,9,10,11], which is based on the equations of motion(EOM).
After reviewing the off-shell ADT current, we explain the necessity of its extension for asymptotic Killing vectors in the context of the AdS/CFT correspondence and then present its generic structure. By using the integration by parts iteratively, we give a prescription to obtain the generalized off-shell ADT current unambiguously and present its explicit form depending only on the EOM and the linearized EOM expressions.

Off-shell currents
We consider a generic theory of gravity with the action For simplicity, we focus on the theory without any matter field in the following. The EOM of the metric are given by E µν (g) = 0, whose Bianchi identity is ∇ µ E µν = 0.
In the geometry admitting a Killing vector ξ µ , the corresponding on-shell ADT current is introduced as where δ denotes the variation with respect to the metric. This on-shell current can be shown to be conserved by using EOM, Bianchi identity and the Killing property of ξ. As a result, the anti-symmetric second rank tensor Q µν = Q [µν] , which is called the on-shell ADT potential, is introduced by J µ = ∇ ν Q µν . These on-shell current and potential are highly involved for a higher curvature/derivative theories of gravity, as their EOM are very complicated. Instead, the background independent ADT current and potential have been used for TMG [26] and new massive gravity [27]. It was recognized that this background independent ADT current has not just computational convenience in some specific theories but more profound meaning in a generic theory of gravity as the off-shell extension of the on-shell ADT formulation for conserved charges [25].
The off-shell ADT current for a Killing vector ξ can be introduced as This off-shell current can be shown to be identically conserved by using the Bianchi identity and the Killing property of the vector ξ. One may note that this current reduces to the on-shell current in Eq. (2) after using the EOM, E µν (g) = 0. This identical conservation property allows us to introduce the off-shell ADT potential Q µν ADT for a Killing vector ξ as By using the above off-shell ADT potential and the one-parameter path in the solution space [25,28], one can introduce quasi-local conserved charges for the Killing vector ξ as where B may be taken in the interior region not just at the asymptotic infinity of the space-time.
One may note that quasi-local charges are computed, at the end, on the on-shell value since we have used a one-parameter path in the solution space. As was shown in [24,25] which shows us that the off-shell ADT current J µ ADT is not conserved for asymptotic Killing vectors and needs to be extended.
In this paper, we would like to generalize the above off-shell ADT current for a Killing vector ξ to the off-shell current for an asymptotic Killing vector ζ. By noting that this generalized off-shell ADT current should depend linearly on the vector ζ and reduce to the off-shell ADT current J µ ADT when ζ is taken as a Killing vector, one may take the generalized off-shell ADT current J µ ADT , without loss of generality, in the form of where £ ζ denotes the Lie derivative along ζ direction. M µν 1 ···ν k αβ 's are taken such that they satisfy the following properties where the round parenthesis denotes the total symmmetrization with a normalization factor 1/k!. One can always take these forms of M's and J µ ADT by using the fact that any commutator of ∇'s can be replaced by the Riemann tensor. The identical conservation of J µ ADT leads to severe constraints on the form of M's. Rather than solving these constraints directly, we will take a different methodology and propose a way to obtain the unambiguous form of the generalized off-shell ADT current in the following section.

Construction
We introduce the generalized off-shell ADT current for an asymptotic Killing vector ζ as The second term J µ ∆ is introduced to preserve the off-shell conservation of the ADT current for an asymptotic Killing vector ζ, such that its divergence cancels the right-hand side of Eq. (6).
To obtain J µ ∆ explicitly, let us consider an n-th order derivative theory of gravity. Generically the linearized EOM expression E µν can be written as where the coefficient functions f 's satisfy The above form of the linearized EOM may be regarded as a generic derivative expansion since the commutators of the covariant derivatives can always be replaced by Riemann tensors.
Because of the symmetrization, the order of covariant derivatives does not matter. From now on, we will always order the covariant derivatives in the increasing ρ i -numbering just for the convenience.
As noted above, the additional current J µ ∆ is designed to satisfy the following relation By using the expression in (10), the first term in the left-hand side can be written generically in the form as where we performed the integration by parts iteratively to arrive at the second equality. The explicit forms of F αβ and H ρ are given by 1 As shown in the appendix A, it turns out that By combining the equations, (12), (13) and the relation (15), we can naturally identify the In general, the identical conservation condition alone does not give us the unique expression of the additional current J ρ ∆ . In the above, we have fixed this ambiguity by declaring that J ρ ∆ is constructed only by f functions as in Eq.s (14) and (16). In other words, by performing the integration by parts successively, J ρ ∆ can be rewritten in the following form of where S ρρ ′ should be a symmetric tensor over indices ρ and ρ ′ . This is the criterion for fixing the ambiguity in our approach.
As shown in appendix B, one can obtain more useful expression of F αβ in Eq. (14) as and J ρ ∆ in Eq. (16) can be written in the form of This expression of J ρ ∆ (£ ζ g, δg) is one of our main results. By construction, the additional current J ρ ∆ depends only on the EOM. It is manifestly covariant and conserved when g satisfies the EOM and δg does the linearized EOM. Furthermore, it turns out to be a symplectic current as will be shown in the next section. Now, it is straightforward to see that this current is antisymmetric about its arguments: One might say that the demanded condition of J µ ∆ in Eq. (12) implies the above antisymmetric property. We would like to emphasize that the demanded condition does not warrant the above antisymmetric property since it does not fix the ambiguity which allows the addition of total derivative terms. In contrast, we constructed the explicit form of J ρ ∆ (£ ζ g, δg) in Eq. (B.6) with manifestly antisymmetric property without any ambiguity.
A few leading order terms of J ρ ∆ are given explicitly as For example, in the case of Einstein gravity, the expression of J ρ ∆ (£ ζ g, δg) from our formula is given by where It is interesting to note that the expression of J µ ∆ is exactly half of the expression of the so-called invariant symplectic current W µ at (E.15) in [19] up to the sign convention.
In the following section, we explore the connection of our construction, which depends only on the EOM, to the covariant phase space approach, and then show how to obtain the above generalized off-shell ADT potential from the Lagrangian.

Off-shell potential and asymptotic symmetry generators
In this section, we compare our construction to the covariant phase space approach [12,13,14] and indicate the difference explicitly. Then we present the method to obtain the generalized off-shell ADT potential from the given Lagrangian. As a specific example, we consider a generic higher curvature theory of gravity. We also give some comments on the relation of our construction to a mathematical construction based on the so-called variational bi-complex [18,20].

Comparison with the covariant phase space
It has been known that there are several ways to construct conserved charges for an asymptotic Killing vectors [2,29,18,28,21]. Though we have constructed quasi-local conserved charges through the additional current J µ ∆ explicitly, it is useful to find the connection of our construction with the covariant phase space method which is based on the Lagrangian. For simplicity, we will focus on a covariant theory of gravity. It is straightforward to include gravitational Chern-Simons terms, which will be omitted.
By applying another variation to Eq. (24) and using δ 1 δ 2 I[g] = δ 2 δ 1 I[g] with δ 1 δ 2 g µν = δ 2 δ 1 g µν , one can show that this symplectic current satisfies the following relation This relation shows us that ω µ is a conserved current when g and δg satisfy the EOM and the linearized EOM, respectively. Furthermore, one may notice that the symplectic current ω µ (g ; £g, δg) satisfies the same divergence relation with our additional current J µ ∆ (£g, δg) in Eq. (12). This means that the difference between them should be, at most, a total derivative as 2 √ −gJ µ ∆ (g ; £g, δg) = ω µ (g ; £g, δg) + ∂ ν √ −gA µν (g ; £g, δg) , where A µν denotes a certain antisymmetric second rank tensor determined by the given Lagrangian. Note that this relation tells us that 2 √ −gJ µ ∆ is symplectic up to a total derivative. And then, the generalized off-shell ADT current J µ ADT can be written in terms of ω µ as In order to see the relation between our construction and the covariant phase space approach, one may note that J µ ADT vanishes whenever g and δg satisfy the EOM and the linearized EOM, respectively. Under this condition, the generalized off-shell ADT current J µ ADT reduces to the symplectic current ω µ up to a total derivative. Hence, the on-shell difference between our construction and the covariant phase space formalism resides in the total derivative term A µν . Now, we give a recipe to determine A µν from the surface term Θ µ . By recalling the form of J µ ∆ obtained in Eq. (17), the definition of ω µ given in Eq. (25)

and the relation between them in
Eq. (27), one can set where one may notice that the total derivative part is composed of two pieces, symmetric S µν and antisymmetric A µν . This relation gives us a definite way to obtain A µν from the variation of the surface term Θ µ . Schematically, the Θ term is given by Θ(δg) ∼ ∇ · · · ∇δg. Therefore, one can see that δΘ(£g) ∼ ∇ · · · δ∇£g + ∇ · · · ∇δ£g ∼ ∇ · · · δ∇£g + ∇ · · · ∇£δg , where we have used £δg = δ£g. To relate this form to the expression of £Θ(δg), it is useful to note the following identity for arbitrary tensor T µ 1 ···µ k : Though we have constructed J µ ADT and Q µν ADT from the EOM expression, it is much better to obtain the generalized off-shell ADT Q µν ADT from the given Lagrangian. By doing this, one can compare more clearly our construction to the covariant phase space approach or the BBC formalism. To this purpose, it is useful to recall the Noether current and potential. The off-shell Noether current J µ under the general diffeomorphism ζ can be introduced as This current can be shown to be identically conserved and thus the associated potential K µν can be defined by Now we derive the relation among the generalized off-shell ADT current, the K µν term and the Θ µ term for the asymptotic Killing vectors, following the steps in the case of exact Killing vectors [24]. By taking a generic metric variation of the off-shell Noether current, one can obtain where we have used the generic variation of the Lagrangian in Eq. (24) and the definition of Lie derivative on the Θ term as This leads to the identity for the general diffeomorphism ζ: As a result, our final expression of the generalized off-shell ADT potential Q µν ADT is given by which is identical with the off-shell ADT potential Q µν ADT except for the additional term A µν . A couple of comments are in order.
• The generalized off-shell ADT current J µ ADT depends only on the EOM, while the Noether current, J µ depends on the Lagrangian and the symplectic current, ω µ does on the surface term Θ µ .
• When ζ is an exact Killing vector, the corresponding generalized current J µ ADT reduces to the original off-shell ADT current J µ ADT .
• When the background metric and the linearized metric δg satisfy EOM and the linearized EOM, respectively, the generalized current J µ ADT reduces to a symplectic current J µ ∆ .
• Though the above Q µν ADT is ambiguous up to the total derivative term ∂ ρ U µνρ by construction, it is irrelevant in our discussion on the asymptotic symmetry generators.
Quasi-local charge for an asymptotic Killing vector ζ may be defined just like Eq. (3). Its infinitesimal variation under an asymptotic Killing vector η is given by where · · · denotes irrelevant total derivative terms. In the last equality, we have used the off-shell Noether current J µ in Eq. (31) along with the relation The asymptotic symmetry algebra can be constructed by using the above variational form of quasi-local charges. Since the detailed steps for this construction is completely parallel to those given, for instance, in [18,20,22], we will omit those.

Example: higher curvature gravity
In this section, we consider the higher curvature gravity as specific examples to apply our formulation of the generalized quasi-local charges. In the higher curvature gravity L = L(R, R 2 , R µν R µν , · · ·), it is very useful to regard the Riemann tensor as an independent variable and to introduce Then, one can show that the EOM expression E µν , the off-shell Noether potential K µν and the surface term Θ µ take the canonical forms of Now, it is sufficient to obtain A µν in order to find the quasi-local charge for an asymptotic Killing vector. To use our procedure in obtaining A µν given in Eq. (29), one may note that, after the repeated integration by parts, the antisymmetric part A µν comes only from the variation of covariant derivatives in Θ µ (£ ξ g). Specifically, the relevant part for the first term of Θ µ (£ ξ g) in (37) comes from the metric variation of the covariant derivative and is given by 2P µ(αβ)ν δ∇ ν £ ζ g αβ = −4P µ(αβ)ν δΓ ρ ν(α £ ζ g β)ρ + · · · = −2∇ ν P µ(νβ)α g ρσ δg σα £ ζ g ρβ − P µ(αβ)σ g ρν δg σα £ ζ g ρβ + · · · = ∇ ν − 3 4 P µναρ g βσ (£ ζ g αβ δg ρσ − δg αβ £ ζ g ρσ ) + g ρ[µ P ν](αβ)σ δg αβ £ ζ g ρσ where · · · denotes the irrelevant terms, which are either symmetric part over µν-indices, S µν , or non total derivative part, V αβ | ρ . Just like the first term, the second term of Θ µ (£ ξ g) in (37) leads to the relevant terms as Combining the above results, one can show that It is very interesting to notice the complete consistency with the result given in [23]. This shows us that our construction in the case of the higher curvature theory of gravity is identical with the mathematical construction through the, so-called, horizontal homotopy operator in the BBC formalism. This equivalence also tells us that the central charge in the asymptotic symmetry algebra should be the same as the one in the BBC formalism. It may be straightforward to check that our construction leads to the same results with the BBC formalism even in any higher derivative theory of gravity.
Just for concreteness, we present some detailed expressions in Einstein gravity. In this case, P -tensor is given by and it turns out that For any diffeomorphism paramter ζ (h µν = δg µν ) in Einstein gravity and so, the generalized off-shell ADT potential for an asymptotic Killing vector ζ in Einstein gravity is given by

Conclusion
We have constructed the generalized off-shell ADT current and potential by using only the EOM and the linearized EOM expression. By connecting this construction to the covariant phase space method, we have presented a definite way to obtain the off-shell ADT current and potential from the given Lagrangian. Our construction can be applied to a generic higher derivative theory of gravity even with gravitational Chern-Simons terms. We have also indicated the relation of our construction to the BBC formalism. As a specific example, we have presented the generalized off-shell ADT potential explicitly for a higher curvature theory of gravity. Our construction shows us the usefulness of the off-shell or quasi-local formulation of conserved charges.
In view of the generic structure of the generalized off-shell ADT current, our construction corresponds to the choice of a further symmetric form as follows This fact can be inferred from Eq. (17) by noting that we can perform the iterative integration by parts on £ ζ g instead of δg. By performing the integration by parts further on the term S ρρ ′ , one can show that M µν 1 ···νnαβ takes the above form. In this sense, our construction may be regarded as the most symmetric one.
As was shown in the higher curvature theory of gravity, our final results are equivalent to those from the BBC formalism. To compare our construction to the BBC formalism, it is useful to recall that the BBC construction of the asymptotic symmetry generators starts from the so-called on-shell vanishing current S µ (ζ) = √ −gE µν ζ ν . One may note that our construction of the generalized off-shell ADT potential Q µν ADT given in Eqs. (9) and (23) can be rewritten as This shows that our generalized off-shell ADT potential Q µν ADT corresponds to the potential k µν in the BBC formalism [18,20]. In conjunction with the equivalence in the case of a higher curvature theory of gravity, this form strongly indicates the formal equivalence between our construction of the generalized off-shell ADT potential and the BBC formalism, though it does not prove the equivalence. On the other hand, there are some differences between two constructions. In our construction we have not used an on-shell condition on the metric g and do not need any canonical choices of the surface term Θ µ or the off-shell Noether potential K µν . Any ambiguity in such terms should be canceled in Eq. (34) by construction. In the BBC formalism, a priori ambiguous quantities like the Θ-term are fixed by the horizontal homotopy operator while we do not need such canonical choice. However, it seems very plausible to expect the formal equivalence between them in consideration of the final expression for the potentials. It would be very interesting to prove their equivalence.
As a further direction, it would be also very interesting to study physics in [30,31,32] by using our construction.