JHEP07(2016)004
one can neve rt he l e ss use t he reduced Lagrangian (1.3). In fact in t hi s case, one can even
ignore th e I
2
term. This was indeed done in many previous works, for example [
12–15].
Since one of our purposes is to construct dyonic black holes in higher dimensions, the La-
grangian (
1.3) is not suitable for this purp ose and we shall use the original Lagrangian (1.1)
instead for all our constructions.
The paper is organized as follows. In section
2, we review the EBI theory and then
derive the equations of motion for all dimensions. In section
3, we obtain the exact dyonic
(A)dS black hole solutions in four dimensions with a generic topological horiz on. Then
we study the global structure, black hole thermodynamics and the singularity structures.
In section
4, we generalize the results to all even dimensions. We conclude the paper
in section
5.
2 EBI and its equations of motion
In this secti on, we consider the EBI theory. The Lagrangian of BI theory can be naturally
generalized to cur ved spacetimes and the Lagrangian i s given by
L = −b
2
s
−det
g
µν
+
F
µν
b
+ b
2
q
−det (g
µν
) , (2.1)
where g
µν
is the metric. The Lagrangian of the EBI theory with a bare cosmological
constant Λ
0
can be wri t t en by
L =
√
−g (R −2Λ
0
) − b
2
s
−det
g
µν
+
F
µν
b
, (2.2)
where Λ
0
= Λ − b
2
/2. Here, Λ is the effective cosmological constant. The variation of
Lagrangian (
2.2) gives rise to
δL =
√
−g (−E
µν
δg
µν
+ E
ν
A
δA
ν
+ ∇
µ
J
µ
) , (2.3)
where g = det(g
µν
), J
µ
is the surface term and
E
µν
= G
µν
+ g
µν
Λ
0
+
b
2
2
√
−h
√
−g
h
−1
(µν)
, (2.4)
E
ν
A
= ∇
µ
√
−h
√
−g
b
h
−1
[µν]
, (2.5)
in which G
µν
= R
µν
− g
µν
R/2, h
µν
= g
µν
+ F
µν
/b, h ≡ det(h
µν
), and (h
−1
)
µν
denotes the
inverse of h
µν
, satisfying
(h
−1
)
µρ
h
ρν
= δ
µ
ν
, h
νρ
(h
−1
)
ρµ
= δ
µ
ν
. (2.6)
We further defined
h
−1
(µν)
=
1
2
h
−1
µν
+
h
−1
νµ
,
h
−1
[µν]
=
1
2
h
−1
µν
−
h
−1
νµ
. (2.7)
The equations of motion are then gi ven by E
µν
= 0 and E
µ
A
= 0. These equations are
derived from the original Lagrangian (
2.2) of the EBI theory an d hence are applicable in
all dimensions and for all char ge configurations.
– 3 –
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