JT引力与EMD理论的嵌入：弦论与M理论的全息对偶揭示
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更新于2024-09-04
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本文探讨了在弦理论和M理论的背景下，二维贾德金-塔利(JT)引力与 SYK模型红外固定点之间的全息对应关系。JT引力通常被认为是复杂量子系统中涌现出来的简单引力理论，而 SYK模型因其随机矩阵理论的特性，被广泛用于研究量子信息和黑洞热力学。
作者通过对一维电磁动力学(EMD)理论的一般维度上的准一致Kaluza-Klein(KK)缩减，揭示了一种精确的嵌入方式，使得JT引力能够在高维空间中得以体现。这种嵌入对于理解弦理论中的低维引力行为至关重要，因为它提供了一个从微扰理论到全非线性理论的有效途径。
对于维度 D=4和D=5，这些EMD理论本身可以直接嵌入到超引力中，这进一步加深了理论之间的联系。在这个过程中，超引力理论不仅作为JT引力的更高维度升级，还为我们探索更深层次的物理现象，如黑洞物理和量子引力提供了新视角。
作者特别关注到，一类JT引力的解可以通过提升转化为时间依赖的带电极值黑洞，这是一个重要的进展，因为这使得理论能够描述更复杂的物理情境，比如D1/D5-脑模型。在这个模型中，世界面不再是传统的闵可夫斯基时空，而是更为特殊的米尔恩宇宙。米尔恩宇宙的特点在于其膨胀性质，这对于理解量子引力中的非平凡拓扑结构和时间方向的量子效应具有重要意义。
总结来说，这篇论文通过精确的嵌入方法，不仅连接了二维JT引力和弦/M理论，而且还展示了如何通过提升解决来描述更丰富的物理现象，如时间依赖的带电黑洞和米尔恩背景下的D1/D5-脑模型。这些发现有助于深化我们对量子引力、黑洞信息以及时空本质的理解。
Eur. Phys. J. C (2018) 78 :791 Page 3 of 11 791
L =
√
−g
R−
1
2
(∂φ)
2
−V (φ)−
1
4
e
aφ
F
2
−
1
4
e
bφ
F
2
, (2.1)
where F = dA, F = dA and the dilatonic coupling con-
stants satisfy
ab =−
2(D − 3)
D − 2
. (2.2)
The scalar potential V is inspired by those in gauge super-
gravities, given in terms of a super-potential W [58]
V =
dW
dφ
2
−
D − 1
2(D − 2)
W
2
,
W =
√
2(D − 2)
2
a
(D − 2)a
2
+ 2(D − 3)
g
be
−
1
2
a φ
− ae
−
1
2
bφ
.
(2.3)
For reasons that will become apparent, in this paper, we are
particularly interested in the case with
a =
2(D−3)
2
(D−1)(D−2)
, b =−
2(D−1)
D−2
. (2.4)
The potential is thus
V =−(D − 1)g
2
e
−
2(D−3)φ
√
2(D−1)( D−2)
+ (D − 3)e
2φ
√
2(D−1)( D−2)
=−(D − 1)g
2
e
−aφ
+ (D − 3)e
−
1
2
(a+b)φ
. (2.5)
Note that if we set A = 0 and also turn off the scalar potential,
the remainder of the theory is simply the Kaluza–Klein the-
ory with A being the Kaluza–Klein vector. The EMD theories
were inspired by gauged supergravities. In D = 4 and 5, the
Lagrangians are the consistent truncations of the bosonic sec-
tor of the respective gauged STU models. Their embeddings
in M-theory and type IIB strings via Kaluza–Klein sphere
reductions were given in [59].
We now make a constant shift of φ, and redefine
φ =
˜
φ + c , A
μ
=
˜
A
μ
e
−
1
2
ac
, A
μ
=
˜
A
μ
e
−
1
2
bc
,
g
2
e
−ac
= g
2
2
, e
−
1
2
(a+b)c
g
2
= g
2
1
. (2.6)
Dropping the tilde, the Lagrangian takes the same form as
(2.1), but with V now given by
V =−(D − 1)
g
2
2
e
−aφ
+ (D − 3)g
2
1
e
−
1
2
(a+b)φ
. (2.7)
This allows us to set the parameter g
1
and g
2
to zero inde-
pendently.
2.2 Conformal transformation
We now make a conformal transformation [60]
g
μν
= e
−
2
(D−1)(D−2)
φ
˜g
μν
, (2.8)
the Lagrangian, after dropping the tildes, becomes
L =
√
−g
R + (D − 1)(D − 3)g
2
1
−
1
4
−1
F
2
+
3
(D − 1)g
2
2
−
1
4
F
2
. (2.9)
where
= e
−
D−2
2(D−1)
φ
. (2.10)
The conformal transformation (2.8) is such that the dila-
ton’s kinetic term is absent and the equation of motion for
becomes algebraic. This can be generally done in supegrav-
ities or gauged supergravities and such a conformally trans-
formed theory was referred as the f (R)-version of supergrav-
ity in [60]. In this paper we shall also refer (2.9) as gravity
in the f (R)-frame.
In order to make contact with JT gravity through dimen-
sional reduction, we take A = 0 and g
2
= 0, the resulting
EMD theory in the f (R)-frame is
L =
√
−g
R +(D −1)(D −3)g
2
1
−
1
4
−1
F
2
. (2.11)
In four and five dimensions, the theories can be obtained
from taking appropriate limit of the bosonic sector of the
STU gauged supergravity models. When g
1
= 0, they can
be truncated consistently from supergravities and hence can
be embedded in string and M-theory.
2.3 JT gravity from Kaluza–Klein reduction
In this subsection, we show that JT gravity can be obtained
from this EMD theory (2.11) by the consistent Kaluza-
Klein reduction. The internal space is taken to be (D − 2)-
dimensional Einstein-space d
2
D−2,k
with R
ij
= (D −
3)kg
ij
, where k =−1, 0, 1. The corresponding Ricci scalar
is given by
R
k
= (D − 2)(D − 3)k. (2.12)
In order for the reduction to be consistent, we take all the sin-
glet of the isometry group of the internal space. The reduction
ansatz is thus given by
ds
2
D
=ds
2
2
+ϕ(x)
2
d
2
D−2,k
, A
μ
=A
μ
(x), =(x),
(2.13)
where μ = 0, 1 and x
μ
are respectively the indices and
coordinates of the metric ds
2
2
. The Kaluza–Klein reduction
in the Einstein frame down to dimensions higher than or
equal to three was obtained in [61]. We find that the reduced
Lagrangian from (2.11) in two dimensions is
123
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