4.3 Virasoro algebra and central charge

Let us now assume in the context of the general theory (1View Equation) that a consistent set of boundary conditions exists that admit the Virasoro algebra generated by (107View Equation) – (108View Equation) as asymptotic-symmetry algebra. Current results are consistent with that assumption but, as emphasized earlier, boundary conditions have been checked only partially [156Jump To The Next Citation Point, 5, 21].

Let us define the Dirac bracket between two charges as

{ 𝒬(ζm,Λm),𝒬 (ζn,Λn )} ≡ − δ(ζm,Λm )𝒬(ζn,Λn). (122 )
Here, the operator δ (ζm,Λm ) is a derivative in phase space that acts on the fields g μν, AI μ, χA appearing in the charge 𝒬 as (104View Equation). From general theorems in the theory of asymptotic-symmetry algebras [59Jump To The Next Citation Point, 35Jump To The Next Citation Point, 36Jump To The Next Citation Point], the Dirac bracket represents the asymptotic symmetry algebra up to a central term, which commutes with each element of the algebra. Namely, one has
{𝒬 (ζm,Λm),𝒬 (ζn,Λn)} = 𝒬[(ζm,Λm ),(ζn,Λn)] + 𝒦 (ζm,Λm ),(ζn,Λn), (123 )
where the bracket between two generators has been defined in (105View Equation) and 𝒦 is the central term, which is anti-symmetric in its arguments. Furthermore, using the correspondence principle in semi-classical quantization, Dirac brackets between generators translate into commutators of quantum operators as {...} → − i[...] ℏ. Note that, according to this rule, the central terms in the algebra aquire a factor of 1βˆ•β„ when operator eigenvalues are expressed in units of ℏ (or equivalently, when one performs 𝒬 → ℏ𝒬 and divide both sides of (123View Equation) by ℏ.).

For the case of the Virasoro algebra (110View Equation), it is well known that possible central extensions are classified by two numbers c and A. The general result has the form

c 2 [β„’m, β„’n ] = (m − n )β„’m+n + --m (m − A)δm,−n, (124 ) 12
where A is a trivial central extension that can be set to 1 by shifting the background value of the charge β„’0. The non-trivial central extension c is a number that is called the central charge of the Virasoro algebra. From the theorems [59Jump To The Next Citation Point, 35Jump To The Next Citation Point, 36Jump To The Next Citation Point], the central term in (123View Equation) can be expressed as a specific and known functional of the Lagrangian β„’ (or equivalently of the Hamiltonian), the background solution ¯ ¯I A Ο• = (g¯μν,A μ,χ¯ ) (the near-horizon geometry in this case) and the Virasoro generator (ζ, Λ) around the background
c = c(β„’, ¯Ο•,(ζ,Λ)). (125 )
In particular, the central charge does not depend on the choice of boundary conditions. The representation theorem leading to (124View Equation) only requires that such boundary conditions exist. The representation theorem for asymptotic Hamiltonian charges [59] was famously first applied [58Jump To The Next Citation Point] to Einstein’s gravity in three dimensions around AdS, where the two copies of the Virasoro asymptotic-symmetry algebra were shown to be centrally extended with central charge c = -3l-- 2GN ℏ, where l is the AdS radius and G N Newton’s constant.

For the general near-horizon solution (25View Equation) of the Lagrangian (1View Equation) and the Virasoro ansatz (107View Equation) – (108View Equation), one can prove [159Jump To The Next Citation Point, 97Jump To The Next Citation Point] that the matter part of the Lagrangian (including the cosmological constant) does not contribute directly to the central charge, but only influences the value of the central charge through the functions Γ (πœƒ),α (πœƒ ),γ (πœƒ) and k, which solve the equations of motion. The central charge (125View Equation) is then given as the m3 factor of the following expression defined in terms of the fundamental charge formula of Einstein gravity as [35Jump To The Next Citation Point]

Einstein || cJ = 12irli→m∞ 𝒬 Lm [β„’L −m¯g;¯g ]|m3, (126 )
where β„’L−m ¯g is the Lie derivative of the metric along L−m and
1 ∫ ( 1 𝒬EiLnmstein[h;g] ≡ ------ dSμν ξ νDμh + ξμD σhσν + ξσD νhσμ + -hD νξ μ 8πGN S 2 1- μσ ν 1-νσ μ ) + 2h D σξ + 2h D ξσ . (127 )
Here, --1---√--- α1 αd−2 dS μν = 2(d−2)! − gπœ–μνα1...αd−2dx ∧ ⋅⋅⋅ ∧ dx is the integration measure in d dimensions and indices are raised with the metric gμν, h ≡ gμνhμν and S is a surface at fixed time and radius r. Physically, QEiξnstein[h;g] is defined as the charge of the linearized metric h μν around the background gμν associated with the Killing vector ξ, obtained from Einstein’s equations [1]. Substituting the general near-horizon solution (25View Equation) and the Virasoro ansatz (107View Equation) – (108View Equation), one obtains
∫ --3k- π cJ = G ℏ d πœƒα(πœƒ)Γ (πœƒ)γ(πœƒ). (128 ) N 0
We will drop the factors of GN and ℏ from now on. In the case of the NHEK geometry in Einstein gravity, substituting (37View Equation), one finds the simple result [156Jump To The Next Citation Point]
cJ = 12J . (129 )
The central charge of the Virasoro ansatz (107View Equation) – (108View Equation) around the Kerr–Newman black hole turns out to be identical to (129View Equation). We note in passing that the central charge cJ of extremal Kerr or Kerr–Newman is a multiple of six, since the angular momentum is quantized as a half-integer multiple of ℏ. The central charge can be obtained for the Kerr–Newman–AdS solution as well [159Jump To The Next Citation Point] and the result is
12ar+- cJ = Δ , (130 ) 0
where Δ0 has been defined in (44View Equation).

When higher-derivative corrections are considered, the central charge can still be computed exactly, using as crucial ingredients the SL (2,ℝ ) × U(1) symmetry and the (t,Ο• ) reversal symmetry of the near-horizon solution. The result is given by [20Jump To The Next Citation Point]

∫ δcovL cJ = − 12k ------πœ–abπœ–cdvol(Σ), (131 ) Σ δRabcd
where the covariant variational derivative δcovβˆ•δRabcd has been defined in (53View Equation) in Section 2.5. One caveat should be noted. The result [20Jump To The Next Citation Point] is obtained after auxiliary fields are introduced in order to rewrite the arbitrary diffeomorphism-invariant action in a form involving at most two derivatives of the fields. It was independently observed in [190Jump To The Next Citation Point] that the formalism of [35Jump To The Next Citation Point, 36Jump To The Next Citation Point] applied to the Gauss–Bonnet theory formulated using the metric only cannot reproduce the central charge (131View Equation) and, therefore, the black-hole entropy as will be developed in Section 4.4. One consequence of these two computations is that the formalism of [35Jump To The Next Citation Point, 36, 90] is not invariant under field redefinitions. In view of the cohomological results of [35], this ambiguity can appear only in the asymptotic context and when certain asymptotic linearity constraints are not obeyed. Nevertheless, it has been acknowledged that boundary terms in the action should be taken into account [237, 164]. Adding supplementary terms to a well-defined variational principle amount to deforming the boundary conditions [56, 266, 213] and modifying the symplectic structure of the theory through its coupling to the boundary dynamics [96Jump To The Next Citation Point]. Therefore, it remains to be checked if the prescription of [96] to include boundary effects would allow one to reconcile the work of [190] with that of [20].

In five-dimensional Einstein gravity coupled to U (1) gauge fields and scalars, the central charge associated with the Virasoro generators along the direction ∂ Ο•i, i = 1,2 can be obtained as a straightforward extension of (128View Equation[159Jump To The Next Citation Point, 97Jump To The Next Citation Point]. One has

∫ π cΟ•i = 6πki dπœƒα (πœƒ)Γ (πœƒ)γ(πœƒ), (132 ) 0
where the extra factor of 2π with respect to (128View Equation) originates from integration around the extra circle (see also [151Jump To The Next Citation Point, 166] for some higher derivative corrections). Since the entropy (54View Equation) is invariant under a SL (2,β„€ ) change of basis of the torus coordinates (Ο•1,Ο•2) as (118View Equation), cΟ• i transforms under a modular transformation as k i. Now, k i transforms in the same fashion as the coordinate Ο• i, as can be deduced from the form of the near-horizon geometry (36View Equation). Then, the central charge for the Virasoro ansatz (119View Equation) is given by
c(p1,p2) = p1cΟ•1 + p2cΟ•2. (133 )

Let us now discuss the central extension of the alternative Virasoro ansatz (120View Equation) for the extremal Reissner–Nordström black hole of electric charge Q and mass Q. First, the central charge is inversely proportional to the scale R χ set by the Kaluza–Klein direction that geometrizes the gauge field. One can see this as follows. The central charge is bilinear in the Virasoro generator and, therefore, it gets a factor of 2 (Rχ ). Also, the central charge consists of the 3 n term of the formula (127View Equation), it then contains terms admitting three derivatives along χ of e−inχβˆ•R and, therefore, it contains a factor of R −χ3. Also, the central charge is defined as an integration along χ and, therefore, it should contain one factor R χ from the integration measure. Finally, the charge is inversely proportional to the five-dimensional Newton’s constant G5 = (2πR χ)G4. Multiplying this complete set of scalings, one obtains that the central charge is inversely proportional to the scale R χ.

Using the simple embedding of the metric and the gauge fields in a higher-dimensional spacetime (2View Equation), as discussed in Section 1.2, and using the Virasoro ansatz (120View Equation), it was shown [159Jump To The Next Citation Point, 146, 77Jump To The Next Citation Point] that the central charge formula (126View Equation) gives

6Q3 cQ = ---- . (134 ) R χ
One might object that (2View Equation) is not a consistent higher-dimensional supergravity uplift. Indeed, as we discussed in Section 1.2, one should supplement matter fields such as (3View Equation). However, since matter fields such as scalars and gauge fields do not contribute to the central charge (125View Equation[97Jump To The Next Citation Point], the result (134View Equation) holds for any such consistent embedding.

Similarly, we can uplift the Kerr–Newman black hole to five-dimensions, using the uplift (2View Equation) – (3View Equation) and the four-dimensional fields (25View Equation) – (39View Equation). Computing the central charge (132View Equation) for the Virasoro ansatz (120View Equation), we find again13

6Q3 cQ = ---- . (135 ) R χ

Under the assumption that the U(1) gauge field can be uplifted to a Kaluza–Klein direction, we can also formulate the Virasoro algebra (119View Equation) and associated boundary conditions for any circle related by an SL (2,β„€ ) transformation of the torus 2 U (1). Applying the relation (133View Equation) we obtain the central charge

( ) Q3 c(p1,p2) = p1cJ + p2cQ = 6 p1(2J) + p2--- . (136 ) R χ

Let us discuss the generalization to AdS black holes. As discussed in Section 1.2, one cannot use the ansatz (2View Equation) to uplift the U (1) gauge field. Rather, one can uplift to eleven dimensions along a seven-sphere. One can then argue, as in [204Jump To The Next Citation Point], that the only contribution to the central charge comes from the gravitational action. Even though no formal proof is available, it is expected that it will be the case given the results for scalar and gauge fields in four and five dimensions [97]. Applying the charge formula (126View Equation) accounting for the gravitational contribution of the complete higher-dimensional spacetime, one obtains the central charge for the Virasoro algebra (120View Equation) as [204Jump To The Next Citation Point]

r2+-−--a2 cQ = 6Qe ΞΔ R , (137 ) 0 χ
where parameters have been defined in Section 2.4.4 and 2πR χ is the length of the U (1) circle in the seven-sphere.

The values of the central charges (129View Equation), (130View Equation), (131View Equation), (132View Equation), (133View Equation), (135View Equation), (136View Equation), (137View Equation) are the main results of this section.

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