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.