II类超引力下Minkowski解的N=2超对称与AdS d解决方案研究

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标题"最小通量Minkowski分类"探讨的是在II类超重力背景下，N=2超对称理论与特定SU(2) R对称的Minkowski四维解的系统性研究。这种分类主要关注的是寻找和理解那些具有这两种额外对称性的宇宙学解，它们在弦理论和引力理论中具有重要意义。在这个研究中，作者Niall T. Macpherson和Alessandro Tomasiello深入分析了一系列相对简单的偏微分方程（PDE），这些方程能够简化复杂的物理现象。他们的工作涵盖了多个子案例，包括但不限于相交的brane系统，这是高维空间中的基本构造，它们的相互作用在弦理论的宇宙模型中起着关键作用。特别是，他们着重于AdS (Anti-de Sitter) 空间的解决方案，AdS空间因其与量子场论的阿贝尔-杨斯-卡哈拉德-塞曼（AdS/CFT）对应关系而备受关注，这是一对理论框架，描述了在强耦合领域和弱耦合领域的物理行为之间的联系。其中，特别值得注意的是对AdS d解决方案的研究，d值分别为5、6和7。近年来，一个重要的发现是通用的大规模AdS 7解决方案，它可能揭示了新的物理现象或者提供了对某些高维理论的新理解。这些解决方案在引力理论和宇宙学模型中可能有深远的影响。此外，通过假设内部六维空间的紧凑性，研究者们发现了具有局限性的D形和O型平面解决方案。D-branes和O-planes是弦理论中的基本物体，它们的存在对理解空间时间的拓扑和量子引力有着至关重要的作用。这些局部化的解决方案展示了理论如何处理在不同维度中物质和能量的分布，对探索弦理论的微观结构和宇宙的几何性质至关重要。这篇研究不仅深化了对II类超重力下Minkowski解决方案的理解，而且还提供了新的洞察力，特别是在AdS/CFT对应、扩展超对称和通量紧凑化等前沿理论领域。通过这些分类和简化，研究者们有望推动理论物理学的进一步发展，并为未来的实验验证提供理论指导。

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JHEP09(2017)126

after deﬁning functions C, B

and forms B

, F , G on M

. One obtains in this way several

equations for Ψ, Ψ

ˆγ

Ψ and

ˆγ

, which we give in appendix B. One of the consequences is

that the zero-form parts of both Ψ and

ˆγ

must vanish:

= (

ˆγ

)

= 0 . (3.6)

A spinor η in four dimensions deﬁnes an associated basis

{η, η

, ˆγη, ˆγη

} (3.7)

for the space of spinors. Thus we can expand η

on the basis associated to η

= η , η

= a

η + aˆγη + bη

+ b

ˆγη

. (3.8)

(2.9) makes (3.7) orthogonal and η have constant norm, and since |η

| = |η

|, it follows

that |a

+ |b

+ |a|

+ |b|

= 1. However since η

2†

= a

and η

ˆγη

= b

, we must set

= b

= 0 to satisfy (3.6); so we have

|a|

+ |b|

= 1 . (3.9)

One expects from (3.6) an orthogonal SU(2) structure on M

. We can parameterize it as

in [30, section 3.2] using the spinors to deﬁne the vielbein

= η

†

−

, w

= η

−

. (3.10)

The pure spinors Φ

on M

then read

∧ E

∧ e

∧E

, Φ

−

∧ e

∧E

)

(3.11)

where

= b(e

− y

) + i(aw + bv

= −(e

d(y

+ iy

) − (y

+ iy

), (3.12)

= ia(e

− y

) + (bw −av

) .

Here v = v

+ iv

, w = w

+ iw

, and the y

are the three “embedding coordinates” of S

deﬁned in (A.22).

In terms of the parameterization in (3.8), with a

= b

= 0, the pure spinors on M

can be written as

odd

= −

iav

∧ e

w∧w−

∧w

, Ψ

even

= −

∧ (bw −av

) ∧ e

w∧w

, (3.13a)

odd

ˆγ



bw −av



∧ e

w∧w

, Ψ

even

ˆγ

w∧w−

∧w

, (3.13b)

odd

= −



aw + bv



∧ e

w∧w

even

= −

w∧w+

∧w

, (3.13c)

odd

ˆγ

= −

ibv

∧ e

w∧w+

∧w

even

ˆγ

= −

∧ (aw + bv

) ∧ e

w∧w

. (3.13d)

– 6 –

JHEP09(2017)126

We can actually take b to be real. To see this, notice that (3.13) and (B.5b) yield the

1-form constraint



3A+2C−Φ



+ 2be

3A+2C−Φ

= 0. (3.14)

This implies that bdb = bdb and so b = |b|e

iβ

for dβ

= 0. Now, sending w → e

iβ

in (3.13) is equivalent to sending b, b → |b| everywhere they appear and multiplying

and

ˆγ

by e

iβ

in (B.5a)–(B.5b). However, since β

is constant, these phases leave these

conditions unchanged; so without loss of generality we can set β

= 0. We now expand

a ≡ a

+ ia

, so that (3.9) now reads

+ a

+ b

= 1 . (3.15)

Notice that the analysis in this part of the paper is very similar to the AdS

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

× M

product, by

topologically ﬁbering part of the M

over the S

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

, 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

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

– 7 –

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II类超引力下Minkowski解的N=2超对称与AdS d解决方案研究

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