JHEP04(2019)054
where s
1
and s
2
are ± signs and vol
S
2
denotes the volume form of S
2
with unit radius.
The self-dual five-form field strength F
5
vanishes in this background. We present the field
strengths with different signs than in [1] so that they satisfy the Bianchi identities
0 = dF
1
0 = dF
3
− H
3
∧ F
1
0 = dH
3
(2.6)
The signs s
1
, s
2
depend on the specific supergravity solution, which we discuss later in
this note.
3 Review of DGKU solutions
Here we outline the solution in [2] by D’Hoker, Gutperle, Karch, and Uhlemann (DGKU).
The spacetime takes the form of AdS
6
×S
2
×Σ, where Σ is a Riemann surface parametrized
by complex coordinates w, ¯w. The supergravity fields depend only on Σ through two
holomorphic functions A
±
(w) on Σ. For completeness we present the following quantities
which are useful to express the supergravity fields in a concise form.
κ
±
= ∂A
±
κ
2
= −|κ
+
|
2
+ |κ
−
|
2
∂B = A
+
∂A
−
− A
−
∂A
+
G = |A
+
|
2
− |A
−
|
2
+ B +
¯
B
D =
1 + R
1 − R
2
= 1 +
2|∂G|
2
3κ
2
G
(3.1)
The metric of [2] is given in the Einstein frame
ds
2
E
= f
2
6
ds
2
AdS
6
+ f
2
2
ds
2
S
2
+ 4ρ
2
|dw |
2
(3.2)
where the metric factors are
f
2
6
=
κ
2
ρ
2
√
D
f
2
2
=
κ
2
9ρ
2
1
√
D
(ρ
2
)
2
=
κ
4
6G
√
D
(3.3)
Note that to make contact with the parameterization in section 2, the metric should be
transformed into the string frame
ds
2
S
= e
φ/2
ds
2
E
(3.4)
Here the dilaton is normalized in the standard fashion to τ = χ + ie
−φ
. The solution [2]
utilizes an SU(1, 1)/U(1) parametrization of the complex scalar field in terms of B, which
is related to the axion-dilaton field via
B =
1 + iτ
1 − iτ
(3.5)
– 3 –
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