3 D6 brane solutions
In order to analyse brane-flux annihilation, we first require a background with fluxes that
carry D6 charges without the presence of localized D6 branes. This is possible in massive
IIA supergravity as can be seen from the Bianchi identity for F
2
:
dF
2
= F
0
H
3
+ Nδ
6
. (3.1)
Here
F
2
is the RR 2-form field strength,
F
0
is Romans mass and
H
3
is the NSNS 3-form
field strength. Notice that the term
F
0
H
3
appears on the same footing as the D6 source
term Nδ
6
. Clearly when F
0
6= 0 this term acts as a smooth source for 6-brane charge.
A general study of backgrounds with these ingredients was carried out in [
18
]. It includes
the “massive D6” brane with flat worldvolume and non-compact transverse space [
37
] and
D6 branes with AdS
7
worldvolume and compact transverse space [
18
,
38
]. We start with
brane-flux decay in the first example but before doing so we emphasize why the
D6
-brane
is special.
3.1 Why the D6 is special
D6
branes stand out against antibranes of other dimensionality for their simplicity. This is
most obvious in the description of the backreaction of
D6
branes, which is described by
ODE’s [
18
,
33
,
38
] without having to smear the branes. A second feature-crucial for this
paper-is that the annihilation of the
H
3
-flux does not proceed via a polarisation into a
higher-dimensional object. All
Dk
branes with
k <
6 polarise into an NS5 branes [
15
].
D6
branes instead polarise into KK5 branes, which are smaller in dimension. However the KK5
assumes a circular isometry transverse to its worldvolume, one can therefore think of KK5s
as 5-branes that are smeared over a circle. This circular isometry direction will live inside
the
D6
worldvolume such that the KK5 brane looks like a 6-brane and its backreaction is
identical to that of the
D6
brane since it carries
D6
charge and tension. This is related
to the fact that
D6
solution is the only antibrane solution for which the singularity in
H
3
is consistent with the singularity of the metric. In other words: the backreaction of the
diverging
H
3
flux does not destroy the local
D6
-metric, whereas it would for
k <
6. For
instance for
k
= 3 one finds that the
H
2
3
scalar blows up near the horizon, but the horizon
of a 3-brane should be smooth AdS
5
× S
5
. Hence the 3-form fluxes necessarily destroy the
local AdS
5
throat, which is well known from the Polchinski-Strassler (PS) model [
24
]. In the
PS model the singularity at the would-be horizon can be turned into a physical singularity
corresponding to (
p, q
) 5-branes. Something similar can be expected for the
D3
solution
and indeed reference [
32
] has shown that a local 5-brane background with good singularities
it is at least not inconsistent with having ISD fluxes at the UV and a non-zero (generalised)
ADM mass.
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