IIB型超重力的超对称AdS6解决方案资源-CSDN文库

Open

Access

12 浏览量 2020-03-29 02:44:12 上传评论收藏 580KB PDF 举报

资源推荐

资源详情

资源评论

Eur. Phys. J. C (2015) 75:484

DOI 10.1140/epjc/s10052-015-3705-1

Regular Article - Theoretical Physics

Supersymmetric AdS

solutions of type IIB supergravity

Hyojoong Kim

1,a

, Nakwoo Kim

1,b

, Minwoo Suh

2,c

Department of Physics and Research Institute of Basic Science, Kyung Hee University, Seoul 130-701, Korea

Department of Physics, Sogang University, Seoul 121-742, Korea

Received: 14 July 2015 / Accepted: 28 September 2015 / Published online: 11 October 2015

Abstract We study the general requirement for supersym-

metric AdS

solutions in type IIB supergravity. We employ

the Killing spinor technique and study the differential and

algebraic relations among various Killing spinor bilinears to

ﬁnd the canonical form of the solutions. Our result agrees

precisely with the work of Apruzzi et al. (JHEP 1411:099,

2014), which used the pure spinor technique. Hoping to

identify the geometry of the problem, we also computed

four-dimensional theory through the dimensional reduction

of type IIB supergravity on AdS

. This effective action

is essentially a non-linear sigma model with ﬁve scalar

ﬁelds parametrizing SL(3, R)/SO(2, 1), modiﬁed by a scalar

potential and coupled to Einstein gravity in Euclidean signa-

ture. We argue that the scalar potential can be explained by

a subgroup CSO(1,1,1) ⊂ SL(3, R) in a way analogous to

gauged supergravity.

Contents

1 Introduction ..................... 1

2 Supersymmetric AdS

solutions ........... 2

2.1 Killing spinor equations ............. 2

2.2 Killing vectors .................. 2

2.3 Supersymmetric solutions ............ 3

2.4 Equations of motion ............... 5

3 Four-dimensional effective action .......... 5

3.1 Non-linear sigma model ............ 5

3.2 Scalar kinetic terms ............... 5

3.3 Scalar potential ................. 6

4 Discussions ...................... 7

Appendix A: Type IIB supergravity ........... 8

Appendix B: Gamma matrices and spinors ....... 8

B.1: Gamma matrices ................ 8

B.2: Spinors ..................... 8

e-mail: h.kim@khu.ac.kr

e-mail: nkim@khu.ac.kr

e-mail: minsuh@usc.edu

Appendix C: Spinor bilinears .............. 9

C.1: Algebraic relations ............... 9

C.2: Differential relations .............. 9

Scalar bilinears ................. 9

Vector bilinears ................. 9

Two-form bilinears ...............10

Normalization of scalar bilinears ........... 10

Appendix D: Fierz identities .............. 10

D.1: Relations of scalar bilinears ........... 10

D.2: Inner products of vector bilinears ........ 11

Appendix E: Killing vectors of ﬁve-dimensional target

space ......................... 11

References ........................ 12

1 Introduction

In recent years, there has been renewed interest in super-

symmetric AdS

solutions in D = 10 supergravity. Via

the gauge/gravity correspondence [2], such solutions should

be dual to certain D = 5 superconformal ﬁeld theories.

Five-dimensional gauge theories are perturbatively non-

renormalizable. Seiberg nonetheless argued that N = 1 super-

symmetric Sp(N ) gauge theories with hypermultiplets of

< 8 fundamental and one antisymmetric tensor represen-

tation ﬂow in the inﬁnite gauge coupling limit to supercon-

formal theories, and their SO(N

) ×U (1) global symmetry

is enhanced to E

[3–5]. Such ﬁxed point theories have a

string theory construction: in terms of the near-horizon limit

of D4–D8 brane conﬁgurations. Based on the AdS

/CFT

correspondence [6], Brandhuber and Oz identiﬁed the grav-

ity dual as a supersymmetric AdS

solution of massive

type IIA supergravity [7]. More recently this correspondence

was generalized to quiver gauge theories and AdS

orbifolds in [8].

Thanks to the development of the localization technique

[9] and its generalization to ﬁve-dimensional gauge theories

[10,11], some BPS quantities can be calculated exactly. The

123

484 Page 2 of 12 Eur. Phys. J. C (2015) 75 :484

conjectured enhancement of global symmetry to E

was

veriﬁed from the analysis of the superconformal index in

[12]. Furthermore, the S

free energy and also the

-BPS

circular Wilson loop operators are calculated and shown to

agree with the gravity side computations [13–16].

Encouraged by the successful application of localization

technique on the ﬁeld theory side, it is natural for us to look

for new supersymmetric AdS

solutions. In massive type IIA

supergravity, it was proved that the Brandhuber–Oz solution

is the unique one [17]. In type IIB supergravity, the T-dual

version of the Brandhuber–Oz solution has been known for

alongtime[18]. A new solution was obtained more recently

employing the technique of non-Abelian T-dual transforma-

tion in [19]. The dual gauge theory was investigated in [20],

but it is not completely understood yet.

For a thorough study, the authors of [1] investigated the

general form of supersymmetric AdS

solutions of type IIB

supergravity, using the pure spinor approach. They found

that the four-dimensional internal space is a ﬁbration of S

over a two-dimensional space, and they also showed that the

supersymmetry conditions boil down to two coupled partial

differential equations. Of course, any solution of the PDEs

provides a supersymmetric AdS

solution at least locally. In

particular, the two explicit solutions mentioned above can be

reproduced as speciﬁc solutions to the PDEs. But otherwise

these non-linear coupled PDEs are so complicated that cur-

rently it looks very hard, if not impossible, to obtain more

AdS

solutions by directly solving the PDEs.

The objective of this article is to procure additional insight

into this problem, using alternative methods. In the ﬁrst part

we use the Killing spinor approach which is probably more

better known and has been successfully applied to many sim-

ilar problems; see e.g. [21–23]. Following the standard pro-

cedure we work out the algebraic and differential constraints

which should be satisﬁed by various spinor bilinears and

derive the supersymmetric conditions. In the end, we con-

ﬁrm that our results are in precise agreement with that of [1].

Secondly, via dimensional reduction of the bosonic sector of

the D = 10 action on AdS

, we present a four-dimensional

effective theory action, which turns out to be a non-linear

sigma model of ﬁve scalar ﬁelds coupled to gravity. The

scalar ﬁelds parametrize the coset space SL(3, R)/SO(2, 1).

Also there is a non-trivial scalar potential, which breaks the

global sl (3, R) symmetry to a certain subalgebra. Although

in this paper we do not present new solutions, we believe the

identiﬁcation of the D = 4 effective action will prove useful

in the construction of explicit solutions and their classiﬁca-

tions.

This paper is organized as follows. Section 2 contains an

analysis on the supersymmetry conditions for AdS

solu-

tions. In Sect.3, we study the four-dimensional effective the-

ory from dimensional reduction on AdS

. Technical details

are relegated to the appendices.

2 Supersymmetric AdS

solutions

2.1 Killing spinor equations

We consider the most general supersymmetric AdS

solu-

tions of type IIB supergravity. We take the D = 10 metric as

a warped product of AdS

with a four-dimensional Rieman-

nian space M

= e

AdS

+ ds

, (2.1)

where U is a warp factor. To respect the symmetry of AdS

we should set the ﬁve-form ﬂux to zero. The complex three-

form ﬂux G is non-vanishing only on M

. The warp factor

U , the dilation φ and the axion C, are functions on M

and

of course independent of coordinates in AdS

To preserve some supersymmetry, we require the van-

ishing of supersymmetry transformations of the gravitino

and the dilatino i.e. δψ

= 0,δλ= 0. With the gamma

matrix decomposition (B.1) and the spinor ansatz (B.8),

we reduce the ten-dimensional Killing spinor equations to

four-dimensional ones. There are two differential and four

algebraic-type equations:

1±

npq

(γ

npq

+ 2γ

npq

)ξ

2±

= 0, (2.2)

2±

∗

npq

(γ

npq

+ 2γ

npq

)ξ

1±

= 0, (2.3)

ime

−U

1∓

+ ∂

U γ

1±

−

npq

2±

= 0, (2.4)

ime

−U

2∓

+ ∂

U γ

2±

−

∗

npq

1±

= 0, (2.5)

2±

npq

1±

= 0, (2.6)

∗

1±

∗

npq

2±

= 0, (2.7)

where

1±

=(∇

−

)ξ

1±

2±

= (∇

)ξ

2±

(2.8)

With the assumption that there exists at least one nowhere-

vanishing solution to the equations in the above, we can

construct various spinor bilinears. Then the supersymmetric

condition is translated into various algebraic and differen-

tial relations between the spinor bilinears. We have recorded

them in Appendix C.1 and C.2.

2.2 Killing vectors

We ﬁrst need to study the isometry of the four-dimensional

Riemannian space M

. We note that the following two com-

plex vectors satisfy the Killing equation ∇

= 0:

123

Eur. Phys. J. C (2015) 75 :484 Page 3 of 12 484

1−

+ ξ

2−

, ξ

2−

+ ξ

1−

(2.9)

If these vectors are to provide a true symmetry of the full

ten-dimensional solution as well, we need to check if

U = (d i

+ i

d) U = K

∂

U = 0, (2.10)

where L

is a Lie derivative along the Killing vector K .

From (C.4) and (C.6), we ﬁnd that in fact only three of them

satisfy the above condition. Hence, the true Killing vectors

are

≡ Re (ξ

2−

+ ξ

1−

), (2.11)

≡ Im (ξ

2−

+ ξ

1−

), (2.12)

≡ Re (ξ

1−

+ ξ

2−

). (2.13)

Using (2.6) and (2.7), we have P

= 0, which implies

that

φ = L

C = 0, (2.14)

where i = 1, 2, 3. Also we obtain i

∗ G = 0from(2.40)

and (2.41), and i

d ∗G = 0 using the equation of the motion

for G,

thus

∗ G = 0. (2.15)

Hence, we conclude that K

describe symmetries of the full

ten-dimensional solutions.

Now let us study the Lie bracket of the Killing vectors.

Using (C.13) and (C.19), the Fierz identities (D.2) and the

normalization (C.28), we show that the three Killing vectors

satisfy an SU(2) algebra,

, K

]=

ijk

. (2.16)

This SU(2) isometry of the four-dimensional Riemannian

space corresponds to the SU(2)

R-symmetry of dual ﬁve-

dimensional ﬁeld theory. Then we construct a 3 × 3 matrix,

whose elements are the inner products of the Killing vectors

(D.9), and ﬁnd that this matrix is singular

det (K

· K

) = 0. (2.17)

This guarantees that K

are the Killing vectors of S

.The

radius l of the two-sphere is given by

= (K

)

+ (K

)

+ (K

)

= 2



− 4(ξ

)(ξ

)



. (2.18)

The equation of the motion for G is d ∗ G = (−6dU +iQ) ∧∗G +

P ∧∗G

∗

2.3 Supersymmetric solutions

We have showed that once we require the supersymme-

try conditions, then the four-dimensional Riemannian space

should contain S

. Now we focus on the remaining two-

dimensional space. We start with two one-forms L

and L

from (C.11):

≡ e

U +

(ξ

+ ξ

)∂

C − me

−

=−i ∂



U −

(ξ

− ξ

)



, (2.19)

≡ Im



2−

+ ξ

1−



−

∂



U +

(ξ

+ ξ

)



, (2.20)

where

= Re



2−

− ξ

1−



. (2.21)

Using the Fierz identities, one can show that the one-forms

and L

are orthogonal to the Killing vectors,

· L

= K

· L

= 0. (2.22)

Together with L

C = 0, the one-form L

is also orthogonal

to the Killing vectors. Then we introduce coordinates z and

z =−3mi e

U −

(ξ

− ξ

y = 3me

U +

(ξ

+ ξ

(2.23)

Since L

z = i

dz ∼ K

· L

= 0 and similarly L

y =

0, the coordinates z and y are independent of the sphere

coordinates. In terms of the coordinates z and y, the one-

forms are

ydC − me

−

dz, (2.24)

−

dy. (2.25)

Then we calculate inner products of the one-forms L

and

, hoping to be able to ﬁx the remaining two-dimensional

metric. However, we cannot immediately calculate the inner

products involving L

, because it includes dC. The resolution

is that we consider the one-form L

deﬁned in (2.21) instead.

From (C.15) and (C.16), we have

d(e

4U−

) = e

4U+

dC ∧ L

, (2.26)

d(e

4U+

) = 0. (2.27)

We introduce another coordinate w and write L

−4U−

dw. (2.28)

123

剩余11页未读，继续阅读

评论收藏

内容反馈

weixin_38607282

粉丝: 3
资源: 973

IIB型超重力的超对称AdS 6解决方案

最新资源

IIB型超重力的超对称AdS 6解决方案

在IIB型超重力中关联AdS6解决方案

六维N =（1，1）测量的超重力下的超对称AdS 6真空

IIB型超重力在S 4上对N = 2 * $$ \ mathcal {N} = {2} ^ {\ ast} $$的精确全息

关于10和11维的超对称AdS 6解决方案

η形变的AdS 5×S 5超弦的T对偶的IIB型超重力解

不带D3黄铜的IIB型超重力超对称AdS 5解决方案

IIB型超重力I中弯曲的AdS 6×S 2：局部解决方案

T 1，1上来自IIB型超重力的AdS 5真空

N = 2 $$ \ mathcal {N} = 2 $$类型IIB超重力的AdS4超对称解决方案

来自物质耦合F（4）的超重力的超对称AdS 6黑洞

半对称超重力和（1，0）SCFT的边际算符的超对称AdS 7背景

超对称AdS 3超重力背景和全息

关于非超对称超重力解的稳定性

扭曲的IIB缩减量出的六维超重力

弯曲的AdS 6中的IIB型7臂：分区功能，臂腹板和探针限制

S折和（非）超对称Janus解

压扁超对称AdS5黑洞的边界

纤维GK几何形状和超对称AdS解决方案

AdS6×S2扭曲时空中的大规模自旋2激发

无用的D3骨架以及AdS 5黑洞，AdS 5孤子与“软壁”重力解决方案之间的插值解决方案

IIB型绕线束上的磁通压缩

增压超级地层

关于4D规范超重力中的量子熵函数

超对称RG流动和II型球面压实产生的Janus

Sasaki-Einstein流形的Janus和J折解

宇宙计量gauge魔术模型的超对称解

N $$ \ mathcal {N} $$ = 2个超对称S形折叠

最新资源