5D梅隆场Einstein-Yang-Mills黑洞：引力自旋与等旋效应解析解
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更新于2024-07-16
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本文探讨了一种在五维支持梅隆场（Meron field）的SU(2) Einstein-Yang-Mills理论中的解析黑洞解。在这一理论中，标尺场与纯标尺场的比例是关键特征，且该解具有非平凡的拓扑电荷。值得注意的是，黑洞视界有效地屏蔽了可能存在的梅隆核心的奇异性。该解决方案的独特之处在于其度量仅依赖于一个积分常数，即阿德尔曼-达默德（ADM）质量。当在计算过程中加入适当的边界项后，这个积分常数被证明是有限的，体现了黑洞的质量属性。
此外，文章还深入研究了黑洞的热力学性质，揭示了一种类似于Reissner-Nordström情况的一阶相变，这表明理论中存在一定程度的电荷-电荷对称性。令人惊讶的是，尽管该理论基础是纯玻色子系统，但通过等旋效应，它表现出“铁电离子”行为，即使没有基本费米子成分，标量场与背景的交互也能导致空间无限远处的费米子自由度。
进一步分析，当考虑AdS 5（反德西特空间）的渐近情况时，作者从AdS/CFT（阿贝尔-杨-米尔斯/共形场论）对偶性的角度研究了这些结果的意义。这暗示着在更高维度的理论框架下，黑洞的行为及其背后的物理现象可能与量子场论中的强相互作用有深刻的联系。
这篇论文不仅提供了五维SU(2) Einstein-Yang-Mills理论中的新型黑洞解，而且还探讨了其背后的物理机制，包括引力自旋、热力学性质以及与AdS/CFT对偶性的关联，这些都是现代高能物理学和弦理论领域的重要研究课题。
JHEP06(2019)081
Therefore, in the f¨unfbein basis A = A
A
e
A
, the components of the Meron gauge field take
the simple form
A
A
=
2iλ
r
δ
a+1
A
t
a
. (2.15)
Similarly, the curvature two-form (2.11) can be written in terms of the f¨unfbein basis as
F =
1
2
F
AB
e
A
∧ e
B
, with components
F
AB
= −
4iλ (λ −1)
r
2
a
bc
δ
b+1
A
δ
c+1
B
t
a
. (2.16)
The Hodge dual of F, on the other hand, takes the form
∗ F = −
4iλ (λ −1)
r
2
t
a
e
0
∧ e
1
∧ e
a+1
. (2.17)
Using (2.13) and (2.17), equation (2.12) reduces to
2iλ (λ −1)
r
dω
a
R
− λ
a
bc
ω
b
R
∧ ω
c
R
t
a
e
0
∧ e
1
= 0 , (2.18)
which, due to the Maurer Cartan equation (2.8), is solved for
λ =
1
2
. (2.19)
2.3 Generalized hedgehog ansatz
The group element U ∈ SU(2) that corresponds to the Meron gauge potential (2.9) can be
explicitly written using the generalized hedgehog ansatz [30]. It is interesting to note that
such an approach (subsequently worked out in [31–45]) was originally developed to analyze
the Skyrme and Einstein-Skyrme models, but it also works in the case of Yang-Mills and
Einstein-Yang-Mills theories with almost no modifications (as one can see comparing [31,
39] with [6, 45]). This approach is specially designed for situations in which the group
element cannot be deformed continuously to the identity.
In order to write down the ansatz for U , we adopt the standard parametrization for
SU(2) elements:
U(x
µ
) = e
−iα
a
t
a
= cos
α
2
(x
µ
)I − 2in
a
t
a
sin
α
2
(x
µ
) , (2.20)
where I is the 2 × 2 identity and n
a
is a unit vector that can be written in the form
n
1
= sinΘcosΦ , n
2
= sinΘsinΦ , n
3
= cosΘ . (2.21)
The generalized hedgehog ansatz corresponds to the following choice for the functions α,
Θ, Φ in terms of Euler variables:
Φ =
ϕ − ψ
2
, tan Θ = tan
θ
2
csc
ϕ + ψ
2
, cot
α
2
= sec Θ tan
ϕ + ψ
2
. (2.22)
This leads to the following form for the components of the right-invariant Maurer-Cartan
form on SU(2):
ω
1
R
= −sinψdθ + cosψ sinθdϕ ,
ω
2
R
= cosψdθ + sinψ sinθdϕ , (2.23)
ω
3
R
= dψ + cosθdϕ ,
0 ≤ ψ < 4π , 0 ≤ θ < π , 0 ≤ ϕ < 2π , (2.24)
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