JHEP08(2015)090
The relevant type IIA gravity action, in string frame, reads
S =
1
2k
2
0
Z
d
10
x
√
−g
e
−2φ
R + 4(∂φ)
2
−
1
2
|F
4
|
2
−
1
2
|F
2
|
2
. (2.2)
Here 2k
2
0
= (2π)
7
l
8
s
where l
s
≡
√
α
0
is the string length, F
4
= dC
3
is the RR four-form
which is magnetically sourced by the N
c
D4-branes, F
2
= dC
1
, and φ is the dilaton.
The θ-backreacted background [4, 5] (see appendix A for a detailed re-derivation based
on the 1d reduced action) is a solution of the equations of motion derived from the action
above. The string frame metric reads
ds
2
10
=
u
R
3/2
p
H
0
dx
µ
dx
µ
+
f
√
H
0
dx
2
4
+
R
u
3/2
p
H
0
du
2
f
+ u
2
dΩ
2
4
, (2.3)
where
f = 1 −
u
3
0
u
3
, H
0
= 1 −
u
3
0
u
3
Θ
2
1 + Θ
2
. (2.4)
The background also includes a running dilaton, a RR one-form and a four-form field
strength given by
e
φ
= g
s
u
R
3/4
H
3/4
0
, C
1
=
Θ
g
s
f
H
0
dx
4
, F
4
= 3R
3
ω
4
, (2.5)
with the flux quantization condition fixing the value of R as
Z
S
4
F
4
= 8π
3
l
3
s
g
s
N
c
, R = (πg
s
N
c
)
1/3
l
s
. (2.6)
Here Θ is a parameter which, as it will be clear in a moment, is proportional to θ. The
original background at zero θ angle, found in [1], can be readily obtained from the one
above setting Θ = 0. Let us recall that in the expression above µ = 0, 1, 2, 3 are the
1 + 3 Minkowski directions where the Yang-Mills theory is defined, dΩ
2
4
is the metric of
a S
4
of radius one, u is the transverse radial coordinate u ∈ [u
0
, ∞) - holographically
mapped to the RG scale of the dual field theory - x
4
is the compact coordinate of length
2πM
−1
KK
and R is a curvature radius. The isometry group of S
4
is mapped into a global
SO(5) symmetry group in the dual field theory, which acts non-trivially on the Kaluza-
Klein massive modes (signaling that these are, in fact, not decoupled in the limit we are
considering). Finally g
s
is the string coupling and ω
4
is the volume form of the transverse
S
4
, of volume V
S
4
= 8π
2
/3.
Notice also that, crucially, the S
x
4
circle shrinks to zero size when u = u
0
. Absence of
conical singularities at u = u
0
is guaranteed if
u
0
=
4R
3
9
M
2
KK
1
1 + Θ
2
, (2.7)
which thus implies that the (x
4
, u) subspace has the topology of a disk.
Reliability of the background requires e
φ
to be small: when this condition is violated
we should better make use of the 11d (“M-theory”) completion of the model. As it was
– 5 –
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