十维超引力与球形Dpbrane解的全息对偶

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"这篇研究论文探讨了球形麸在十维超重力理论中的新解，这些解描述了Dp球体与球形世界体积之间的相互作用。通过超对称性，这些解决方案与（p+1）维的最大超对称Yang-Mills理论在S p + 1空间上的全息对偶性得以建立。球体的有限大小在规范理论中充当了红外截止，这在超重力解决方案中体现为几何形状的平滑上限。在高能（紫外线）区域，球体大小的影响消失，背景渐近趋近于描述平坦Dp球体近似水平极限的已知超重力解决方案。研究人员还计算了球形麸解决方案的壳作用，并且发现它与最近关于S p + 1上的最大超对称Yang-Mills理论的自由能的超对称定位结果相一致。" 这篇由Nikolay Bobev、Pieter Bomans和Fridrik Freyr Gautason共同撰写的论文发表在JHEP08(2018)029期刊上，由Springer出版，接收日期为2018年7月7日，接受日期为2018年7月25日，最终发表于2018年8月8日。该研究属于开放访问，允许读者免费获取全文。研究的核心在于构建了一组具有16个超电荷的十维超重力新解，这些解揭示了Dp球体（一种弦理论中的对象）在球形世界体积上的物理效应。在全息原理的框架下，这些解决方案与在S p + 1空间上的（p+1）维最大超对称Yang-Mills理论建立了对偶关系。这种对偶性是弦理论和量子场论之间的一种联系，使得高维问题可以被简化为低维问题进行研究。在红外区域，球体的半径限制了理论的行为，提供了自然的红外截止，这在超重力解的几何结构中清晰可见。而在高能区域，球体的大小不再影响理论行为，背景逐渐过渡到描述平坦Dp球体近似水平极限的标准超重力解决方案。此外，作者还计算了这些球形麸解的壳作用，这是量子场论中计算物理量的重要工具，其结果与基于超对称定位方法得出的最大超对称Yang-Mills理论的自由能结果吻合，进一步证实了这些解的正确性和实用性。这篇论文的工作深化了我们对Dp球体、全息对偶以及超重力理论的理解，对于理解高维物理和量子场论的红外及紫外线行为具有重要意义。同时，它也为理论物理学家提供了一个新的工具来研究复杂物理系统的性质。

JHEP08(2018)029

The full ten-dimensional supergravity solution describing Dp-branes with ﬂat world-

volume, with p ≤ 6, in asymptotically ﬂat space and in string frame is (see for example [38])

= H

−1/2

p+1

+ H

1/2

9−p

, (3.1)

= g

(3−p)/4

, (3.2)

p+1

= (g

−1

vol

p+1

. (3.3)

Here ds

p+1

and ds

9−p

denote the ﬂat metrics on R

1,p

and R

9−p

respectively, vol

p+1

is the

volume form on R

1,p

and H is a harmonic function on R

9−p

. The harmonic function has

isolated singularities at the position of the branes. For a single stack of Dp-branes at the

origin we have ds

9−p

= dr

+ r

dΩ

8−p

, with dΩ

8−p

the unit radius metric on a round 8 −p

sphere. The harmonic function in this case is

H = 1 +

6−p

7−p

, (3.4)

where V

n−1

= 2π

n/2

/Γ(n/2) is the volume of the unit radius n-sphere. The fundamental

charge of a Dp-brane is given by

2π

(2π`

)

p+1

, (3.5)

and the Yang-Mills coupling constant of the worldvolume gauge theory is

(2π)

(2π`

)

. (3.6)

The constants in (3.4) must satisfy a Dirac quantization condition. Indeed, integrating the

magnetic ﬁeld strength over dΩ

8−p

leads to

2κ

?dC

p+1

N(7 − p)V

8−p

2κ

6−p

= N , (3.7)

where the integer N is interpreted as the number of Dp-branes.

The ﬁeld theory limit of Dp-branes in a holographic context was ﬁrst studied in [9]

(see also [25]). Introducing the dimensionless radial coordinate U = r/(2π`

), this limit is

equivalent to zooming in on the near-horizon region of the branes:

N U

p−7

2πV

6−p

 1 . (3.8)

Using (3.8), the metric and dilaton simplify to

= (gU )

7−p

p+1

+ (gU)

p−7



+ U

dΩ

8−p



, (3.9)

= g

(gU)

(p−3)(7−p)

, (3.10)

p+1

= g

−1

(gU)

7−p

vol

p+1

, (3.11)

The ten-dimensional Newton constant is related to the string length `

through 4πκ

= (2π`

)

therefore 2κ

6−p

= 2π. Note that in our conventions the Newton constant does not depend on g

– 8 –

JHEP08(2018)029

where we have introduced g as

(2π`

p−7

2πV

6−p

. (3.12)

For 0 < p < 3 at high energies, U  1, the string coupling becomes small indicating

that the theory is free in the UV. As discussed in section 2 this is the expected UV behavior

of maximal SYM theory in 1 < d < 4 dimensions. Conversely for 3 < p < 7 the dilaton

increases at high energies indicating that the ﬁeld theory is strongly coupled. This again ﬁts

nicely with the fact that for d > 4 the SYM theory is not renormalizable. Clearly the case

p = 3 is special since the string coupling is constant throughout the solution and the metric

is that of AdS

×S

. This is the well-known holographic dual description of the conformal

N = 4 SYM theory in d = 4. The background in (3.9)–(3.11) possesses ISO(1, p)×SO(9−p)

isometry for p 6= 3 and SO(2, 4)×SO(6) for p = 3. This is the same as the global symmetry

group of the SYM theories discussed in section 2. It is therefore clear that this near-horizon

solution nicely exhibits the physics we expect from a holographic dual to SYM on ﬂat space.

We refer to [9] and references thereof for further support of this holographic duality.

Our goal is to generalize the solutions in (3.9)–(3.11) and construct supergravity back-

grounds which correspond to spherical Dp-branes and provide a holographic description

of maximal SYM on S

p+1

. This necessitates an understanding of how to construct super-

gravity solutions for D-branes with Euclidean worldvolume. This was addressed in several

papers by Hull [39–41] where he argued that there are Euclidean branes, or E-branes, not

of regular type II string theory but of the so-called type II

∗

string theory. The existence

of a low-energy supergravity limit of these type II

∗

string theories can be deduced inde-

pendently from a supergravity point of view [42]. The type II

∗

supergravity theories admit

E-brane solutions

for which the brane worldvolume is Euclidean and the time direction

is transverse to the brane worldvolume, i.e. E-branes resemble instantons. The E-brane

solutions can be obtained from the Dp-brane solutions above by analytically continuing

the time direction of the brane worldvolume into a spatial coordinate and at the same

time analytically continuing the polar angle of the sphere transverse to the brane into a

time-like coordinate. This analytic continuation results in changing the worldvolume of

the brane from R

1,p

to R

p+1

and the transverse S

8−p

sphere in (3.9) to de Sitter space,

8−p

. The analytic continuation does not only aﬀect the metric, but also changes the R-R

ﬁelds. In [39–41] all R-R ﬁelds are taken to be real with “wrong sign” kinetic terms. In this

paper we use an equivalent formulation in which all R-R ﬁelds are imaginary with “usual

sign” kinetic terms. To ensure supersymmetry the Killing spinors for these E-branes have

to satisfy rather unusual reality conditions, this is explained in some detail in appendix B.

Finally we note that solutions of the Lorentzian type IIA

∗

string theory should uplift to

solutions of the so-called M

∗

theory, see [39–41], which has the somewhat exotic (2, 9)

signature of the metric, i.e. two time-like and nine spatial dimensions.

The real constant g will be identiﬁed with the coupling constant of the (p + 2)-dimensional gauged

supergravity theory in which the brane solutions can be eﬀectively described.

Note that in the notation of [39–41] an E(p + 1)-brane is the Euclidean version of a Dp-brane.

– 9 –

JHEP08(2018)029

4 Spherical branes

To construct the near-horizon solution for Euclidean Dp-branes wrapped on spheres we can

utilize intuition from the ﬁeld theory discussion in section 2 and make a suitable ansatz

for the ten-dimensional metric. The total isometry group of the solution should be a

direct product of the isometry group of the S

p+1

which the Dp-brane is wrapping and the

R-symmetry group of the Yang-Mills ﬁeld theory in p + 1 dimensions:

SO(p + 2) ×SU(1, 1) ×SO(6 − p) . (4.1)

A ten-dimensional metric ansatz that implements these symmetries is given by

= ∆

+ R

dΩ

p+1

+ e



dθ

+ P cos

θ d

Ω

+ Q sin

θ dΩ

5−p

i

. (4.2)

The functions A, B, ∆, P , and Q depend on r and θ and satisfy suitable positivity con-

ditions such that the metric has the correct signature, while R is a constant that sets the

radius of S

p+1

. The metric on a round n-sphere is denoted by dΩ

with volume form vol

Clearly the dΩ

p+1

and dΩ

5−p

factors in the metric realize the SO(p + 2) × SO(6 − p) part

of the isometry group in (4.1). The non-compact SU(1, 1) factor in the R-symmetry of the

SYM theory is realized as the isometry group of two-dimensional de Sitter space with metric

Ω

= −dt

+ cosh

t dψ

, (4.3)

where ψ is 2π-periodic.

Note that for P = Q = 1 the metric in (4.2) simpliﬁes signiﬁcantly, namely the

metric transverse to the worldvolume of the branes is the round metric on dS

8−p

which

has SO(1, 8 −p) isometry group, i.e. the same as for the Euclidean Dp-branes in ﬂat space

discussed at the end of section 3. Even without having an explicit solution for spherical

branes, intuition from ﬁeld theory suggests that for values of the radial coordinate much

larger than the scale set by R the supergravity solution should reduce precisely to the

Euclidean Dp-brane background with P = Q = 1. This is suggested by the UV limit in the

ﬁeld theory where the curvature of S

p+1

should play no role in the dynamics of the SYM

theory and it should reduce to that in ﬂat space.

In addition to the metric (4.2), we also have to make an ansatz for the type II NS-NS

and R-R form ﬁelds and the dilaton. Dp-branes are electrically charged under C

p+1

and

it is natural to expect there to be a component of C

p+1

along the dΩ

p+1

worldvolume of

the branes. We should also allow for all other form ﬁelds in the supergravity to have non-

zero values as long as they preserve the isometry group in (4.1). In addition the dilaton

can be an arbitrary function of r and θ. With this ansatz at hand one should analyze

carefully the equations of motion and the supersymmetry variations of the ten-dimensional

supergravity theory imposing that the background preserves 16 out of the 32 supercharges.

This analysis should result in a system of coupled non-linear partial diﬀerential equations

for the unknown functions in the ansatz. It is fair to assume that without any further

insight it will be diﬃcult to solve explicitly this system of equations. Fortunately progress

can be made by employing a well-tested strategy in top-down holography, namely reduce

– 10 –

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十维超引力与球形Dpbrane解的全息对偶

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