![](https://csdnimg.cn/release/download_crawler_static/12305751/bg4.jpg)
gravitational theory in an asymptotically anti-de Sitter space to a d-dimensional conformal
field theory on the boundary of the spacetime. Through AdS/CFT, a wide variety of
gravitational phenomena can be understood in terms of their CFT counterparts,
1
providing
a powerful tool for investigating strongly coupled systems such as quark-gluon plasmas,
condensed matter systems, and superfluids [24].
This has motivated further study into the thermodynamic phase structure of AdS
black holes, where extensive effort has been devoted to understanding the extended phase
space, in which the cosmological constant acts as a thermodynamic pressure [13, 25] via
the identification
P = −
Λ
8π
(2.1)
which can be achieved, for example, with a gauge 3-form potential [26]. The variation of
pressure further requires the presence of a conjugate volume term in the first law, which
leads to a variety of novel thermodynamic behaviour [13], including triple points [27],
reentrant phase transitions [28], and the emergence of polymer-like [12, 29] and superfluid-
like phase structures [30–32], in addition to the now commonly observed Van der Waals
transition [21]. In de Sitter space, the cosmological constant is positive, so the variable
P is better thought of as a tension rather than a pressure. This introduces qualitatively
new behaviour in the phase structure of a black hole, whose implications are the subject
of recent exploration [13, 16, 20].
For the purposes of this paper, the various thermodynamic quantities of interest are
derived from the Euclidean action for the metric g
µν
over a region M with boundary
∂M. We are interested in Einstein-Born-Infeld theory in asymptotically de Sitter space,
described by the following bulk action and boundary term:
I = −
1
16π
Z
M
d
4
x
√
g
R − 2Λ + L
BI
+
1
8π
Z
∂M
d
3
x
√
k
K − K
0
(2.2)
R is the Ricci scalar, Λ is the cosmological constant (positive for de Sitter space), K is the
trace of the extrinsic curvature of the boundary, and L
BI
is the Born-Infeld Lagrangian,
given by:
L
BI
= 4b
2
1 −
r
1 +
F
ab
F
ab
2b
2
!
. (2.3)
Here, F
ab
is the usual field strength tensor and b is the Born-Infeld parameter, representing
the maximal electromagnetic field strength in the theory, with the limit b → ∞ correspond-
ing to Maxwell electrodynamics. b can also be related to the string tension α = (2πb)
−1
in
string theory, where stretched open strings are created dynamically to suppress the electric
field to this maximal value [33].
We choose the subtraction term K
0
such that the action is normalized to I = 0 when the
mass (and therefore charge) of the black hole vanishes. This makes empty de Sitter space
the reference point for the energy, in contrast to the work of Carlip and Vaidya [19], where
1
For example, a Hawking-Page phase transition in the bulk is dual to a deconfinement transition in the
boundary field theory.
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