Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity
Summary (2 min read)
Introduction
- In general relativity and in other gauge theories formulated on noncompact ("open") spaces, the concept of asymptotic symmetry, or "global symmetry," plays a fundamental role.
- Accordingly, the Poisson bracket of the generators of two given symmetries can differ by a constant from the generator associated with the Lie bracket of these symmetries.
- Because the theory of central charges in classical mechanics is well known [7], the authors will only discuss here the aspects which are peculiar to gauge theories and asymptotic (as opposed to exact) symmetries.
- This gives additional motivation for analyzing the canonical realization of the asymptotic symmetries on general grounds.
II. Solutions to 3-Dimensional Gravity with A < 0
- This solution will help motivate their choice of appropriate boundary conditions to be imposed on the metric in general.
- It is also interesting to note that, just as in the de Sitter case [13], a wedge cut from anti-de Sitter space provides a solution to Einstein's equations with the stress-energy tensor of a point mass.
- So the curves which serve as the T , f , f coordinate lines for the metric (2.1) can always be singled out.
- The Killing vector fields in this coordinate system are linear combinations of d/dt and d/dφ.
- In practice, the charges J[ζ] are usually determined by looking at the surface terms coming from the variation of the "volume piece" (3.1) of the Hamiltonian, namely - lim^-^ίcn^i,*- ξ\kδgίj-]^2ξiδπ il-{-(2ξίπkl-ξlπik)δgik}, (3.3) r — » oo where Gijkl =^g1/2(gikgjl + gilgjk — 2gίjgkl) and the semicolon denotes covariant differentiation within a spacelike hypersurface.
IV. The Conformal Group of Asymptotic Symmetries
- It is natural to question whether the restriction of the metric to the form (2.2) outside sources is too severe.
- Ideally, the boundary conditions could be weakened just enough so that the group of asymptotic symmetries is enlarged to the anti-de Sitter group in 2 + 1 dimensions, namely 0(2,2).
- As will be shown below, such deformation vectors have no associated charge and the generators of these deformations vanish weakly.
- To insure that spacelike surfaces initially obeying the boundary conditions (4.4) and (4.10) will preserve these boundary conditions under deformations generated by the Hamiltonian, it is necessary to impose further restrictions on the canonical variables [15].
- Actually, the dependence of ξμ on the canonical variables is not relevant in establishing (4.11) as the proper surface integral to appear in the Hamiltonian, or in evaluating the charges for a spacetime such as (4.2), because for these purposes, ξμ is only needed to leading order in 1/r.
V. The Canonical Realization of Asymptotic Symmetries
- The primary goal of this article is to point out the possible existence of central charges in the canonical realization of asymptotic symmetries.
- Then the generator obtained by computing the Poisson bracket under the assumption that ξ and η describe pure gauge deformations can only differ from the generator which would be obtained without this assumption by terms which vanish when ξ and η are pure gauge.
- Since the leading order terms of all conformal group vectors are independent of the canonical variables, it follows that δηξ μ and δξη μ make only higher order contributions to ζμ in Eq. (5.3).
- The final step in the demonstration that (5.2) is a central extension of the conformal group algebra is to show that, to leading order in 1/r, the surface deformation algebra [ζ,η~\SD coincides with the Lie algebra [£,77] for conformal group vectors ξ and η.
- In the present case where the asymptotic symmetries cannot be realized as exact symmetries of some background, it is easy to see that the central charges are not trivial.
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Citations
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Cites methods from "Central Charges in the Canonical Re..."
...Restrictive or less restrictive boundary conditions at small z (far from the brane) corresponds, as Brown and Henneaux point out in the case of AdS3, to smaller or larger asymptotic symmetry groups....
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...In this discussion we follow the work of Brown and Henneaux [22]....
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...The spirit of [22] is to determine the central charge of an AdS3 configuration by considering the commutator of deformations corresponding to Virasoro generators Lm and L−m....
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Cites background from "Central Charges in the Canonical Re..."
...It is possible to calculate the classical Poisson brackets among these generators, and one finds that this classical algebra has a central charge which is equal to [441]...
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...This problem was analyzed in detail in [441] and we will just sketch the argument here....
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...For 2+1-dimensional black holes, in the context of an old framework [67] for a relation to boundary conformal field theory which actually is a special case of the general CFT/AdS correspondence, such additional information is provided by modular invariance of the boundary conformal field theory [65,66]....
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References
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"Central Charges in the Canonical Re..." refers background or methods in this paper
...Such a theorem has been implicitly used, but not explicitly demonstrated, for example in [8,12]....
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...In that case, the Hamiltonian generator //[ξ] of a given asymptotic symmetry ξ differs from a linear combination of the constraints φA(x) of the canonical formalism by a surface term J[ξ\ which is such that H\_ζ] has well defined functional derivatives [8],...
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...The procedure for obtaining the global charges of a gauge theory within the Hamiltonian formalism has been well established [8]....
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...This gives an additional motivation for imposing extra boundary conditions to eliminate the supertranslation ambiguities [8,4]....
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...The asymptotic symmetry group is the infinite dimensional "Spi group" [4] as long as the behavior of the gravitational variables at spatial infinity is not restricted by means of parity conditions as in [8]....
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1,094 citations
"Central Charges in the Canonical Re..." refers background in this paper
...The asymptotic symmetries are by definition those gauge transformations which leave the field configurations under consideration asymptotically invariant, and recently, it has been explicitly shown that they are essential for a definition of the total ("global") charges of the theory [1,2]....
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Frequently Asked Questions (12)
Q2. What is the motivation for considering the spacetime just described?
The motivation for considering the spacetime just described is that it is the analogue of the conic geometry for 2 + 1 gravity with Λ = 0 [12], for which the wedge α Φ 1 is related to total energy and the jump A Φ 0 is related to total angular momentum.
Q3. What is the symmetry group associated with the boundary conditions?
Then the asymptotic symmetries coincide with the Killing vector fields d/dt and d/dφ, and the asymptotic symmetry group associated with these boundary conditions is R x SO(2).
Q4. What is the simplest way to determine the surface deformations?
Then the Hamiltonian,HK]=Jd-x^(x)^(x)-hJK], (3.2)will have well defined variational derivatives, and may be used as the generator of the allowed surface deformations.
Q5. What are the nonzero components of the canonical variables needed for computing expression (3.3?
The only nonzero components of the canonical variables needed for computing expression (3.3) arer κ*4 ,2r*(3.5)which gives- δJ[_ζ] = 4π[(3)<f <Sα - (3)ξφδ(aA)
Q6. What is the definition of the asymptotic symmetries?
The asymptotic symmetries are canonically realized by the "factor group" of surface deformation generators, which is defined by identifying two Hamiltonian generators if they describe the same asymptotic (conformal group) deformation and differ only by a pure gauge deformation.
Q7. What is the simplest way to write the Hamiltonian?
for the boundary conditions and asymptotic symmetries of a gravitation theory to be acceptable, it must be possible to write the Hamiltonian as the usual linear combination of constraints [14]fd"x^(x)Jfμ(x) (3.1)plus an appropriate surface term J[_ζ].
Q8. What is the effect of identifying the points with the points?
By identifying the points (Γ= f ', r = r',φ = φ') with the points (Γ= t' — 2πA, f = r', $ — φ' + 2πα) for all f', r' and φ', this will have the effect of removing a "wedge" of coordinate angle 2π(l — α) and introducing a "jump" of 2πA in coordinate time.
Q9. What is the simplest way to evaluate a deformed surface?
In this case, the central charge K[ξ, η] reduces to the value of the charge J\\_ζ] on the surface deformed by η.To evaluate ./[£] on the deformed surface, the expression (4.11) can be greatly simplified by specializing to r, φ coordinates and using the known asymptotic form of the canonical variables.
Q10. What is the logical connection between the Poisson bracket and the canonical variables?
The critical step in this analysis is to recognize that the "volume" term of the Poisson bracket (5.2) may be calculated by assuming that ξ and η are pure gauge, in which case the charges vanish.
Q11. What is the lapse and shift in the canonical formalism?
From (4.3,4.4), the lapse and shift are determined to be9 \\ (4.9)Kso that the asymptotic behavior of the canonical variables is given by Eqs. (4.4) along withπrr = 0(l/r), π^ = 0(l/r2), π++ = 0(l/r5). (4.10)However, in the canonical formalism, the spacelike surfaces are evolved according to Hamiltonian evolution, which generally differs from Lie transport unless the spatial Einstein equations (3)Gij = Λgij hold.
Q12. What is the lapse and shifts for the spacetime coordinate system?
The lapse and shifts and computed straightforwardly from (2.2); in particular,Γ Γ2 + R2 HI/2 Γ A2R2 Π-1/2αA(r2 + R2) ''r2(a2R2-A2)-A2R2'and, since (3)<f = 0, the component ξr = 0 always.