全息风味研究：超对称D3/D7配置与弯曲空间

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"超对称D3 / D7可在弯曲空间上提供全息风味" 这篇研究论文探讨了超对称D3/D7膜配置在全息理论中的应用，特别是在具有常曲率的空间上如何描述N = 4 $$ \mathcal{N} = 4 $$超 Yang-Mills (SYM)理论与大量N = 2 $$ \mathcal{N} = 2 $$风味自由度的耦合。D3/D7骨架配置是弦理论中的一种重要构造，它涉及到D3和D7膜的相互作用，其中D3膜位于AdS5×S5背景中，而D7膜则作为引入额外风味自由度的工具。作者系统地解决了κ对称条件，这是D7膜在AdS4切片AdS5×S5中嵌入的关键。κ对称性是确保Dp膜配置超对称性的关键条件。通过解决这些条件，他们找到了超对称嵌入的简单封闭形式，这使得在理论分析中保持对称性的可能性变得更加明确。研究中指出，直到达到临界质量，这些嵌入表现出令人惊讶的多样性。这意味着在不同的质量和参数设置下，可能有多种不同的物理行为和相。作者首次对此类嵌入的全息现象学进行了探索，这包括全息重归一化过程。全息重归一化是理解AdS/CFT对应中的边界理论的关键步骤，它涉及消除无穷大并定义合理的物理量。此外，他们还计算了单点函数，这些函数在全息对应中对应于边界理论的绿函数，对于理解和解析场论的性质至关重要。通过对不同家族的单点函数进行计算，他们尝试给出场论上的解释，这有助于将全息描述与传统的量子场论语言联系起来。为了完善超对称D3/D7配置的目录，作者还构建了类似的方法，应用于S4和dS4背景下的N = 4 $$ \mathcal{N} = 4 $$ SYM理论。S4是四维球面，dS4是四维德西特空间，它们代表了不同的拓扑和几何背景，这扩展了研究的适用范围。这项工作为理解全息原理提供了新的工具和洞察，特别是在具有非平坦背景的风味物理方面。它不仅深化了对AdS/CFT对应的理解，而且可能对强相互作用问题的研究，如QCD（量子色动力学）中的风味物理，提供新的理论框架。

JHEP11(2015)112

2.2 κ-symmetry generalities

The κ-symmetry condition projecting on those Killing spinors which are preserved by a

given brane embedding was derived in [14–16]. We follow the conventions of [14]. The

pullback of the ten-dimensional vielbein E

to the D7 worldvolume is denoted by e

(∂

)dx

, and the Cliﬀord algebra generators pulled back to the worldvolume are

denoted by γ

= e

. We follow [14] and deﬁne X

:= g

. The κ-symmetry condition

then is Γ

 = , where

det(1 + X)

∞

n=0

...j

. . . X

(n)

(p)

, (2.9a)

(n)

(p)

= (−1)

(σ

)

n+(p−3)/2

iσ

⊗ Γ

(0)

, (2.9b)

(0)

(p + 1)!

√

−det g

...i

p+1

...i

p+1

. (2.9c)

For embeddings characterized by a non-trivial slipping mode as described above, the in-

duced metric on the D7-branes reads

g =



1 + θ



dρ

+ cosh

ρ ds

AdS

+ sin

θ dΩ

. (2.10)

The pullback of the ten-bein to the D7 worldvolume is given by

= E

, a = 0 . . . 7 , e

= θ

dρ , e

= 0 . (2.11)

The κ-symmetry condition (2.9) for type IIB supergravity is formulated for a pair of

Majorana-Weyl spinors. We will ﬁnd it easier to change to complex notation, such that we

deal with a single Weyl Killing spinor without the Majorana condition.

2.2.1 Complex notation

Eq. (2.9) is formulated for a pair of Majorana-Weyl Killing spinors (

, 

), and it is the

index labeling the two spinors on which the Pauli matrices act. To switch to complex

notation we deﬁne a single Weyl spinor  = 

+ i

. With the Pauli matrices

0 −i

i 0

, σ

1 0

0 −1

, (2.12)

we then ﬁnd that iσ

(

, 

) translates to −i and σ

(

, 

) to C

. With these replacements

the action of σ

2/3

commutes with multiplication by Γ-matrices, as it should (the Γ-matrices

in (2.9) should be understood as 1

⊗ Γ). We thus ﬁnd

(n)

(7)



→ −iΓ

(0)

 for n even , J

(n)

(7)



→ −iC



(0)





for n odd . (2.13)

Note that Γ

contains an involution and does not act as a C-linear operator. To ﬁx

conventions, we choose the matrix B

deﬁned in the appendix of [23], and set C = B

. C

then is the product of four Hermitian Γ-matrices that square to 1, so we immediately get

†

= C and C

= 1. Furthermore, we have

CΓ

= (Γ

)

C , C

C = 1 . (2.14)

With (2.13) it is straightforward now to switch to complex notation in (2.9).

– 7 –

JHEP11(2015)112

2.2.2 Projection condition for our embedding ansatz

We now set up the κ-symmetry condition in complex notation for our speciﬁc ansatz for

embedding and worldvolume ﬂux. As explained above, for our analysis we do not make

an a priori restriction on the S

gauge ﬁeld to be turned on. So we set A = f (ρ)ω, with

ω a generic one-form on the S

. The ﬁeld strength is F = df ∧ ω + fdω, and we ﬁnd

the components F

ρα

= f

and F

αβ

= ∂

− ∂

. We only have 4 non-vanishing

components of F , which means that the sum in (2.9a) terminates at n = 2. We thus ﬁnd

 =

−i

det(1 + X)





1 +

ijkl



(0)

 +



(0)







. (2.15)

The pullback of the vielbein to the D7 branes has been given in (2.11) above, and we have

(0)

−1

1 + θ

Γ ,

Γ =

1 + θ

AdS

~χ

. (2.16)

The equations (2.15) and (2.16) are the starting point for our analysis in the next subsec-

tions.

2.3 Inﬁnitesimally massive embeddings

Our construction of supersymmetric embeddings will proceed in two steps. We ﬁrst want

to know what exactly the preserved supersymmetries are and what the general form of the

gauge ﬁeld is. These questions can be answered from a linearized analysis, which we

carry out in this section. With that information in hand, the full non-linear analysis will

be easier to carry out, and we come to that in the next section.

So, for now, want to solve the κ-symmetry condition in a small-mass expansion, starting

from the θ ≡ π/2, F ≡ 0 massless conﬁguration which we know as solution from the ﬂat

slicing. We expand θ =

+ δθ + . . . and analogously for f . We use f without explicit

δ, though, as it is zero for the massless embedding and there should be no confusion.

The κ-symmetry condition Γ

 =  can then be expanded up to linear order in δm. The

leading-order equation reads

(0)



(0)

= 

(0)

, Γ

(0)

= iΓ

AdS

~χ

, 

(0)

= |

θ=π/2

, (2.17)

where we use the superscript to indicate the order in the expansion in δm. For the next-

to-leading order we need to take into account that not only the projector changes, but also

the location where the Killing spinor is evaluated — the κ-symmetry condition is evaluated

on the D7s. This way we get

(0)



(1)

+ Γ

(1)



(0)

= 

(1)

, 

(1)

δθ Γ

~χ



(0)

. (2.18)

To see which supersymmetries can be preserved, if any, we need to ﬁnd out under

which circumstances the projection conditions (2.17), (2.18) can be satisﬁed. To work this

out, we note that we can only impose constant projection conditions on the constant spinor



that was used to construct the Killing spinors in (2.3): any projector with non-trivial

– 8 –

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