![](https://csdnimg.cn/release/download_crawler_static/12336949/bg9.jpg)
We can actually take b to be real. To see this, notice that (3.13) and (B.5b) yield the
1-form constraint
d
e
3A+2C−Φ
b
+ 2be
3A+2C−Φ
v
2
= 0. (3.14)
This implies that bdb = bdb and so b = |b|e
iβ
0
for dβ
0
= 0. Now, sending w → e
iβ
0
w
in (3.13) is equivalent to sending b, b → |b| everywhere they appear and multiplying
˜
Ψ
and
˜
Ψ
ˆγ
by e
iβ
0
in (B.5a)–(B.5b). However, since β
0
is constant, these phases leave these
conditions unchanged; so without loss of generality we can set β
0
= 0. We now expand
a ≡ a
1
+ ia
2
, so that (3.9) now reads
a
2
1
+ a
2
2
+ b
2
= 1 . (3.15)
Notice that the analysis in this part of the paper is very similar to the AdS
5
classification
in [31]. In particular, we are getting an orthogonal SU(2) structure, just like in that case.
It would be interesting to explore to what extent this gets generalized if one abandons
our assumption, made in section 2 that the metric is locally an S
2
× M
4
product, by
topologically fibering part of the M
4
over the S
2
.
Since we have a
−1
and b
−1
appearing in (3.13), we should treat the cases where these
vanish individually, before performing a general analysis for arbitrary non zero a and b.
This is what we will now proceed to do.
4 Classification
We have obtained a pure spinor parameterization (3.13) in terms of three real functions a
1
,
a
2
, b. Some of the cases when one or more of these parameters vanish have to be studied
separately. This gives rise to a ramified structure; we will study it in detail in this section.
Sections 4.1 and 4.2 are not strictly needed in the ramification, in the sense that their
results can be obtained as limits from sections 4.5 and 4.4 respectively. Sections 4.3, 4.4
and 4.5 are instead all substantially different, and have to be dealt with separately. We
give a brief summary in section 4.6.
Before we begin, let us make some general comments about our methods. Using the
supersymmetry equations, we get local expressions for the metric and the fluxes, as well
as some PDEs. We then impose the Bianchi identities to be satisfied almost everywhere:
this yields some additional PDEs. When we interpret the resulting solutions physically, we
often find that they in fact involve one or more localized object, such as a D-brane or an
orientifold; not surprisingly, this is always the case when the internal space M
6
is compact.
One then needs to check, in each example, that the Bianchi identities are not only valid
away from these localized sources, but also on them, with the appropriate delta-like source
term included. Practically speaking, this is best done (just like in electromagnetism) by
checking the integral version of the Bianchi identities. In fact, the behaviour of the metric
and of the other fields near a localized object is a good guide to which delta-like terms
are present; this is because the local behaviour is in fact usually locally identical (via a
change of coordinates) to that of a brane or a brane system in flat space. In this section,
we will find the local form of the fields and the PDEs to be solved. A detailed treatment
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