### 4.3 Virasoro algebra and central charge

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

Let us define the Dirac bracket between two charges as

Here, the operator is a derivative in phase space that acts on the fields , , appearing in the charge as (104). From general theorems in the theory of asymptotic-symmetry algebras [59, 35, 36], the Dirac bracket represents the asymptotic symmetry algebra up to a central term, which commutes with each element of the algebra. Namely, one has
where the bracket between two generators has been defined in (105) and 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 . Note that, according to this rule, the central terms in the algebra aquire a factor of when operator eigenvalues are expressed in units of (or equivalently, when one performs and divide both sides of (123) by .).

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

where is a trivial central extension that can be set to 1 by shifting the background value of the charge . The non-trivial central extension is a number that is called the central charge of the Virasoro algebra. From the theorems [59, 35, 36], the central term in (123) can be expressed as a specific and known functional of the Lagrangian (or equivalently of the Hamiltonian), the background solution (the near-horizon geometry in this case) and the Virasoro generator around the background
In particular, the central charge does not depend on the choice of boundary conditions. The representation theorem leading to (124) only requires that such boundary conditions exist. The representation theorem for asymptotic Hamiltonian charges [59] was famously first applied [58] 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 , where is the radius and Newton’s constant.

For the general near-horizon solution (25) of the Lagrangian (1) and the Virasoro ansatz (107) – (108), one can prove [159, 97] 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 and , which solve the equations of motion. The central charge (125) is then given as the factor of the following expression defined in terms of the fundamental charge formula of Einstein gravity as [35]

where is the Lie derivative of the metric along and
Here, is the integration measure in dimensions and indices are raised with the metric , and is a surface at fixed time and radius . Physically, is defined as the charge of the linearized metric around the background associated with the Killing vector , obtained from Einstein’s equations [1]. Substituting the general near-horizon solution (25) and the Virasoro ansatz (107) – (108), one obtains
We will drop the factors of and from now on. In the case of the NHEK geometry in Einstein gravity, substituting (37), one finds the simple result [156]
The central charge of the Virasoro ansatz (107) – (108) around the Kerr–Newman black hole turns out to be identical to (129). We note in passing that the central charge of extremal Kerr or Kerr–Newman is a multiple of six, since the angular momentum is quantized as a half-integer multiple of . The central charge can be obtained for the Kerr–Newman–AdS solution as well [159] and the result is
where has been defined in (44).

When higher-derivative corrections are considered, the central charge can still be computed exactly, using as crucial ingredients the symmetry and the reversal symmetry of the near-horizon solution. The result is given by [20]

where the covariant variational derivative has been defined in (53) in Section 2.5. One caveat should be noted. The result [20] 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 [190] that the formalism of [35, 36] applied to the Gauss–Bonnet theory formulated using the metric only cannot reproduce the central charge (131) 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 [35, 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 [96]. 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 gauge fields and scalars, the central charge associated with the Virasoro generators along the direction , can be obtained as a straightforward extension of (128[159, 97]. One has

where the extra factor of with respect to (128) originates from integration around the extra circle (see also [151, 166] for some higher derivative corrections). Since the entropy (54) is invariant under a change of basis of the torus coordinates as (118), transforms under a modular transformation as . Now, transforms in the same fashion as the coordinate , as can be deduced from the form of the near-horizon geometry (36). Then, the central charge for the Virasoro ansatz (119) is given by

Let us now discuss the central extension of the alternative Virasoro ansatz (120) for the extremal Reissner–Nordström black hole of electric charge and mass . First, the central charge is inversely proportional to the scale 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 . Also, the central charge consists of the term of the formula (127), it then contains terms admitting three derivatives along of and, therefore, it contains a factor of . Also, the central charge is defined as an integration along and, therefore, it should contain one factor from the integration measure. Finally, the charge is inversely proportional to the five-dimensional Newton’s constant . Multiplying this complete set of scalings, one obtains that the central charge is inversely proportional to the scale .

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

One might object that (2) is not a consistent higher-dimensional supergravity uplift. Indeed, as we discussed in Section 1.2, one should supplement matter fields such as (3). However, since matter fields such as scalars and gauge fields do not contribute to the central charge (125[97], the result (134) holds for any such consistent embedding.

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

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

Let us discuss the generalization to AdS black holes. As discussed in Section 1.2, one cannot use the ansatz (2) to uplift the gauge field. Rather, one can uplift to eleven dimensions along a seven-sphere. One can then argue, as in [204], 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 (126) accounting for the gravitational contribution of the complete higher-dimensional spacetime, one obtains the central charge for the Virasoro algebra (120) as [204]

where parameters have been defined in Section 2.4.4 and is the length of the circle in the seven-sphere.

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