Advances in High Energy Physics
where we imply (to have a systematic process) that the super
indices are chosen according to the values of and in order.
For instance, we present some of them
(1)
=
2
2
1
−
1
2
(2)
=
2
3
1
−
1
3
(3)
=
2
3
2
−
2
3
(4)
=
2
4
1
−
1
4
(5)
=
2
4
2
−
2
4
(6)
=
2
4
3
−
3
4
(7)
=
2
5
1
−
1
5
(8)
=
2
5
2
−
2
x
5
(9)
=
2
5
3
−
3
5
(10)
=
2
5
4
−
4
5
⋅⋅⋅
()
in which
2
=
∑
𝑁−1
𝑖=1
2
𝑖
.e YM eld -forms are dened by
the expression
(𝑎)
=
(𝑎)
+
1
2
(𝑎)
(𝑏)(𝑐)
(𝑏)
∧
(𝑐)
.
()
In general for we must have (−1)(−2)/2gauge
potentials. e integrability conditions
(𝑎)
+
1
(𝑎)
(𝑏)(𝑐)
(𝑏)
∧
(𝑐)
=0
()
are easily satised by using (). e YM equations
∗
(𝑎)
+
1
(𝑎)
(𝑏)(𝑐)
(𝑏)
∧∗
(𝑐)
=0
()
also are all satised. e energy-momentum tensor ()
becomes aer
(𝑁−1)(𝑁−2)/2
𝑎=1
(𝑎)
𝜆𝜎
(𝑎)𝜆𝜎
=
(
−3
)(
−2
)
2
4
()
with the nonzero components
00
=
(
−3
)(
−2
)
2
(
)
2
4
11
=−
(
−3
)(
−2
)
2
2
4
(
)
22
=−
(
−3
)(
−6
)
2
2
2
𝐴𝐴
=−
(
−3
)(
−6
)
2
2
2
Π
𝐴−2
𝑖=1
sin
2
𝑖
2<≤−1.
()
Using () and aer some simplications, one can nd that
the metric function ()has the following form given by
(
)
=1−
2
𝑛−1
−
Λ
3
2
+
2
(
−1
)
2
2𝑛−2
−
(
−1
)
2
(
−3
)
2
,
()
where =+2;onecannoteintheparticularcase
for =3that the last term of () diverges, involving an
unusual logarithmic term in Yang-Mills charge []. For this
gravity background the parameter is related to the mass of
such black hole, while and are the charges of Maxwell
eld and Yang-Mills eld, respectively. Following previous
literature[,],onecanndacloseconnectionbetweenthe
cosmological constant and pressure as =−Λ/8,leading
to the following expressions of Hawking temperature, mass,
and entropy of such black hole in terms of the horizon radius
+
=
+
4
=
−1
4
+
+
2
(
+1
)
3
+
−
(
−1
)
2
4
3
−
(
−1
)
2
2
2
2𝑛−1
+
,
()
=
𝑛
48
8
𝑛+1
+
+
3
(
−1
)
2
𝑛−3
3−
+3
𝑛−1
+
+
6
(
−1
)
2
𝑛−1
+
,
()
=
𝑛
𝑛
+
4
,
()
e Yang-Mills potential Φ
𝑄
and the electromagnetic one
Φ
𝐶
can be written as
Φ
𝑄
=
𝐶,𝑃,𝑟
+
=
𝑛
(
−1
)
8
(
3−
)
𝑛−3
+
,
()
Φ
𝐶
=
𝑄,𝑃,𝑟
+
=
𝑛
(
−1
)
4
𝑛−1
+
,
()
where
𝑛
=2
(𝑛+1)/2
/Γ((+1)/2)isthevolumeoftheunit
-sphere. In fact, according to the interpretation of the black
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