JHEP03(2018)054
can be located at different points along their transverse directions, e.g. some at y
6
= 0 and
others at y
6
= π, etc. In [60], we considered N = 1, corresponding to only parallel sources;
we are interested here in N > 1. This excludes the case p = 3, for which the whole M
are the internal transverse directions and there are no internal parallel directions. In other
words, in the following, considering p = 3 forces to take N = 1. As each set I is defined
by its parallel and transverse one-forms, we denote them as {e
a
||
I
} and {e
a
⊥
I
}.
Each source action is given by the sum S
DBI
+S
W Z
, detailed in appendix A. We do not
consider higher order corrections to those actions, as e.g. in [72]. Only S
DBI
contributes
to the Einstein equation and the dilaton equation of motion (e.o.m.). It does through the
energy momentum tensor T
MN
(here in 10d curved indices) and its trace T
10
= g
MN
T
MN
.
It is defined as
1
p
|g
10
|
X
sources
δS
DBI
δg
MN
= −
e
−φ
4κ
2
10
T
MN
, (2.5)
with the constant κ
10
. We will rather use flat indices, T
AB
= e
M
A
e
N
B
T
MN
, and for each
source in the sum, we will further decompose onto the different directions with projectors:
the 4d flat directions α, and the internal a
||
and a
⊥
. We show in appendix A that for each
single source, T
a
⊥
b
⊥
= e
M
a
⊥
e
N
b
⊥
T
MN
= 0. We then obtain
T
AB
= δ
α
A
δ
β
B
T
αβ
+
X
I
δ
a
||
I
A
δ
b
||
I
B
T
I
a
||
I
b
||
I
, (2.6)
and explicit expressions for T
αβ
and T
I
a
||
I
b
||
I
are derived in appendix A. The
P
sources
in (2.5)
gets decomposed into
P
I
P
sources∈I
. Those are present within T
αβ
, while T
I
a
||
I
b
||
I
only
contains
P
sources∈I
. The trace T
10
also gets a natural decomposition into traces for each
set I: T
10
=
P
I
T
I
10
, and expressions for those quantities can be found in appendix A. One
eventually shows that
T
αβ
= η
αβ
T
10
p + 1
, T
I
a
||
I
b
||
I
= δ
a
||
I
b
||
I
T
I
10
p + 1
. (2.7)
Finally, with these definitions, the contribution to the dilaton e.o.m. is given by
1
p
|g
10
|
X
sources
δS
DBI
δφ
= −
e
−φ
2κ
2
10
T
10
p + 1
. (2.8)
We now focus on the fluxes Bianchi identities (BI). There is only one BI which includes
a source term, because of the single size p: the corresponding sourced flux is denoted by
the internal form F
k
with 0 ≤ k = 8 − p ≤ 5. As explained in appendix A, the BI can be
written in terms of the previous quantities as
dF
k
− H ∧ F
k−2
=
ε
p
p + 1
X
I
T
I
10
vol
⊥
I
, (2.9)
with F
−1
= F
−2
= 0 and ε
p
= (−1)
p+1
(−1)
[
9−p
2
]
. We now project the BI on each vol
⊥
I
.
To that end, we introduce the same notation as in [60]: given a form G, the projected form
obtained by keeping only its components entirely along directions of vol
⊥
I
is denoted G|
⊥
I
,
– 8 –
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