![](https://csdnimg.cn/release/download_crawler_static/12285892/bg7.jpg)
shown in [1], in fact, the 10d solution can be obtained starting from an AdS
7
× S
4
planar
black hole solution of 11d supergravity, reducing it on the M-theory circle and performing
a double Wick rotation. We will make use of this picture in the following.
As it will be reviewed below, the integral of C
1
over S
x
4
at u → ∞, is fixed to be
proportional to θ by the holographic dictionary. Then, since the S
x
4
circle shrinks, a
trivial solution like C
1
∼ θdx
4
is not allowed. Regularity requires in fact that C
1
goes to
zero at u = u
0
, a condition which is precisely satisfied by the solution above.
The UV ’t Hooft coupling and the θ angle of the gauge theory, can be related to the
gravity parameters by considering the low energy limit of the D4-brane action
S
D4
= −τ
4
Tr
Z
d
4
x dx
4
e
−φ
p
−det(G + F) + τ
4
Z
C
5
+
1
2
τ
4
Z
C
1
∧ TrF ∧ F , (2.8)
where F
αβ
≡ 2πα
0
F
αβ
is proportional to the gauge field strength, τ
4
= (2π)
−4
l
−5
s
, C
5
is
the electric five-form sourced by the branes (its field strength F
6
is the Hodge dual to
F
4
) and G
αβ
is the induced metric on the world-volume. Expanding the action to second
order in derivatives, considering the UV asymptotics u → ∞, integrating over the compact
x
4
direction and comparing the resulting 4d action with the Minkowski version of (2.1),
one gets
6
λ = g
2
N
c
≡ 2λ
4
≡ 2g
2
Y M
N
c
= 4πg
s
N
c
l
s
M
KK
, θ + 2πk =
1
l
s
Z
S
x
4
C
1
, (2.9)
where k is an integer. In the above expression we have introduced the parameters g
Y M
and λ
4
which are often referred to as the (UV) 4d gauge and ’t Hooft coupling of the
holographic model in the literature, despite the fact that they differ for a factor of 2 from
the standard bare ones appearing in eq. (2.1). As we have observed above, λ
4
∼ T
s
/M
2
KK
parameterizes how far the holographic model is from pure Yang-Mills.
The second relation in (2.9) defines θ mod 2π integer shifts (since the integral of C
1
is gauge invariant only modulo 2πZ) realizing the expected multi-branched feature of the
dual gauge theory vacuum. Moreover, together with (2.5), it implies that the bare θ angle
is related to the Θ parameter of the background by
Θ ≡
λ
4
4π
2
θ + 2kπ
N
c
=
λ
8π
2
θ + 2kπ
N
c
. (2.10)
As it is also suggested by eq. (2.1), in the ’t Hooft limit, the corrections to the physics
due to the θ parameter, w.r.t. the θ = 0 case, are actually weighed by the combination
appearing in Θ. Moreover, since this parameter depends on k, what we actually get on the
gravity side is an infinite family of solutions corresponding to possible field theory vacua.
As it was shown in [5] the curvature invariants of the background remain small if
|Θ| λ
1/4
4
. A similar constraint is obtained by studying the tension of the domain walls
(identified with wrapped D6-branes) separating the various k-vacua in the dual field theory:
if |Θ| is too large, the (metastable) vacua become unstable. Our results will thus be reliable
provided Θ satisfies these bounds. Actually, as we will see in the following section, on the
6
Remember that
R
TrF ∧ F ≡ (1/2)
R
d
4
xTrF
˜
F .
– 6 –