![](https://csdnimg.cn/release/download_crawler_static/12266294/bgb.jpg)
entropy density and the energy density associated with the black brane background (2.2)
are given, correspondingly, by
T = N
GB
r
+
πL
2
=
r
+
√
2πL
2
p
1 + γ
GB
=
r
+
π
˜
L
2
1 + γ
GB
2
3/2
, (2.6)
s =
2π
κ
2
5
r
+
L
3
=
4
√
2π
4
L
3
κ
2
5
T
3
(1 + γ
GB
)
3/2
=
16π
4
˜
L
3
κ
2
5
T
3
(1 + γ
GB
)
3
, (2.7)
ε = 3P =
3
4
T s . (2.8)
The metric (2.2) is well defined for λ
GB
∈ (−∞, 1/4] (or γ
GB
∈ [0, ∞), with the inter-
val of positive λ
GB
corresponding to the interval γ
GB
∈ [0, 1)). We note that s/T
3
is a
monotonically decreasing function of γ
GB
in the interval γ
GB
∈ [0, ∞).
The holographic dictionary relating the coupling λ
GB
of Gauss-Bonnet gravity in D
dimensions to the parameters of the dual CFT has been thoroughly discussed in ref. [68]
(see also the comprehensive discussion of the D = 5 case in ref. [98]). For a class of four-
dimensional CFTs (usually characterized by the central charges c and a), there exists a
parameter regime (e.g. λ N
2/3
c
1 [56, 68]) in which the dual description is given by
Einstein gravity with a negative cosmological constant plus curvature squared terms treated
as small perturbations, so that e.g. the coefficient α
3
in the action (1.9) is α
3
∼ (c −a)/c ∼
1/N
c
1, as in the discussion of the superconformal N = 2 Sp(N
c
) gauge theory with
four fundamental and one antisymmetric traceless hypermultiplets by Kats and Petrov
19
[56]. For finite λ
GB
, if a dual CFT exists at all, one may relate the Gauss-Bonnet coupling
to the parameters characterizing two- and three-point functions of the energy-momentum
tensor in the CFT [68]. In particular, the holographic calculation [68] gives the central
charge c
c =
π
2
˜
L
3
κ
2
5
γ
GB
. (2.9)
Note that the central charge is a monotonically increasing non-negative function of γ
GB
in
the interval γ
GB
∈ [0, ∞), with c = 0 at γ
GB
= 0 (i.e. at λ
GB
= 1/4). Generically, we may
expect λ
GB
to be a function of both λ and N
c
at large but finite values of these parameters.
We compute the retarded two-point functions G
R
µν,ρσ
of the energy-momentum tensor
in a hypothetical finite-temperature 4d CFT dual to the Gauss-Bonnet background (2.2)
following the standard holographic recipe [99–102]. Gravitational quasinormal modes of
the background corresponding to the poles of the correlators G
R
µν,ρσ
[99, 102] have been
computed and analyzed in detail as a function of the Gauss-Bonnet parameter λ
GB
in
ref. [12]. The quasinormal spectrum at λ
GB
= 1/4 is computed analytically in section 2.4
of the present paper.
The full gravitational action needed to compute the correlators contains the Gibbons-
Hawking term and the counter-term required by the holographic renormalisation,
S = S
GB
+ S
GH
+ S
c.t.
, (2.10)
19
Other examples, as well as the string theory origins of the curvature-squared terms in the effective
action are discussed in ref. [98].
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