具有$$AdS_{p+1}$$AdSp+1真空的p分支作为$$R^2$$R2重力模型资源-CSDN文库

Open

Access

18 浏览量 2020-04-10 18:14:32 上传评论收藏 407KB PDF 举报

资源推荐

资源详情

资源评论

Eur. Phys. J. C (2019) 79:633

https://doi.org/10.1140/epjc/s10052-019-7139-z

Regular Article - Theoretical Physics

p-Branes with AdS

p+1

vacuum as models of R

gravity

A. A. Zheltukhin

1,2,a

Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine

Nordita, KTH Royal Institute of Technology and Stockholm University, 106 91 Stockholm, Sweden

Received: 14 January 2019 / Accepted: 13 July 2019 / Published online: 29 July 2019

Abstract Branes with constant mean curvature of their

hyper-worldsheets of codimension 1 are treated as the

Nambu-Goldstone ﬁelds of the broken Poincare symme-

try. Mapping of their action into quadratic curvature grav-

ity action with spontaneously generated gravity, is shown.

Equation for the brane potential extremals and its solution

describing hyper-ws of constant curvature are found. For

membranes in R

1,3

this extremum is shown to be a saddle

3-dim. hypersurface which deﬁnes classically instable vac-

uum.

1 Introduction

Invention of the Green–Schwarz superstring and kappa

symmetry led to the construction of a renormalizable ten-

dimensional theory of all fundamental interactions including

quantum gravity [1]. The local kappa symmetry acting on

two-dimensional worldsheet of superstring is a key element

of the theory which provides the balance between the bosonic

and fermionic degrees of freedom. The very existence of this

symmetry imposes essential restrictions on the dimension

of the ambient supersymmetric space-time in which strings

move. The admissible dimensions of the space-time are

D = 3, 4, 6, 10, but only D = 10 turns out to be consistent in

the quantum picture resulting in ﬁve possible renormalizible

theories [2]. The latter are known as type I, type IIA, type

IIB supplemented by two theories originating from heterotic

strings invariant under SO(32) and E8 × E8 gauge symme-

tries, respectively. These ﬁve theories are based on different

types of strings: type I theory includes both open and closed

strings, while types IIA and IIB include only closed strings.

All these theories are entangled by S and T duality transfor-

mations. In particular, S-duality transforms type I theory to

the SO(32) heterotic one, and type IIB theory into itself.

The latter is transformed into type IIA theory under T -duality

e-mail: aaz@nordita.org

transformations [3]. These theories are believed to be limiting

cases of one unknown theory which consistently describes

quantum gravity. Herewith, the D = 10, N = 2a super-

gravity, in particular, arises as low energy limit of Type IIA

superstring. In this limit which corresponds to the inﬁnitely

large string tension the discussed ﬁve theories yield different

10-dimensional versions of supergravity. The later belongs

to ﬁeld theories associated with point-like particles treated as

idealized geometric objects with the dimension p = 0. So,

the transition from point-like particles to one-dimensional

extended objects (p = 1) cures D = 10 supergravity from

ultraviolet divergencies. However, this scenario works only

for the dimension D = 10 that implies the Kaluza-Klein

compactiﬁcation of six excess coordinates in order to restore

the physical dimension D = 4. But such a compactiﬁca-

tion creates the Landscape problem [4] that means loss of

uniqueness in the choice of the physically relevant universes

describing SU(3) × SU(2) × U(1) invariant interactions

and three families of quarks and leptons [5]. Unfortunately,

the list of such universes contains billions of candidates.

Another problem is the existence of unique N = 1 super-

gravity in eleven dimensions which cannot be derived from

the superstring theories in 10-dimensional space-time. At

the same time the 10-dimensional IIA superstring is derived

from 11-dimensional supermembrane [6] through simulta-

neous dimensional reduction in both D = 11 space-time

and 3-dimensional ( p = 2) supermembrane worldsheet [7].

In view of this observation, the N = 2a, D = 10 super-

gravity is also derived from the N = 1, D = 11 super-

membrane. As a result, supermembranes and superstrings

get equal status in the frame of mysterious 11-dimensional

M-theory [8

]. But the story does not end there, in view of

the tendency to further growth of the dimension p of rele-

vant extended objects as shown in [9]. Therein, the set of the

numbers p and D, ensuring the presence of the kappa sym-

metry on a ( p +1)-dimensional hyper-worldsheet embedded

into D-dimensional Minkowski space, was revealed. This set

includes, in particular, the values p = 2, D = 4, 5, 7, 11;

123

633 Page 2 of 11 Eur. Phys. J. C (2019) 79 :633

p = 3, D = 6, 8; p = 4, D = 9 and p = 5, D = 10

corresponding to the embeddings of membranes and 3-, 4-,

5-branes into the mentioned higher-dimensional Minkowski

spaces. The pair p = 5, D = 10 describes the heterotic

5-brane associated with the soliton solution of the heterotic

string equations found in [10]. In M-theory the supermem-

brane and super 5-brane are called M2 and M5 branes. M-

branes together with superstrings are treated as fundamen-

tal constituents of supersymmetric 11-dimensional M-theory

unifying gravity with other fundamental forces. Thus, both

branes and strings play key roles in theoretical understanding

of fundamental interactions, phenomenology of elementary

particle physics, black holes, dark matter and AdS/CFT cor-

respondence. A comprehensive review of these and other

problems of contemporary physics in which branes play a

vital role may be found in [11]. The breakthrough role of

branes in attempts to reveal new physics sharpens the prob-

lem of their quantum consistency. Solution of this prob-

lem has to shed a new light on quantization of gravity.

Until now quantization of extended objects remains an open

problem even for the case of (super)membranes (p = 2)

[12]. This uncertainty in the quantum status of the brane

paradigma requires new tools for analyzing its viability. A

weak progress in this matter is explained by a complicated

non-linear character of brane equations and constraints. The

non-linearity problem is typical of the theories unifying the

Yang-Mills, gravitational and other ﬁelds covariant under dif-

feomorphisms, gauge and internal symmetries. To solve the

problem, we need to deepen understanding of the structure

of brane non-linearities.

Here we assume that the non-linearities can be converted

into the geometric structures known from gravity and gauge

theories. For branes, the proof of this assumption allows to

implement the BRST-BFV quantization and others methods

used for general non-linear systems. We expect that the non-

linearities of ordinary p-branes can be described by a com-

bination of scalars built from the curvature tensor of ( p + 1)-

dimensional hyper-worldsheets (h-ws).

It implies a map of

non-linear terms into those known from (p + 1)-dim. f (R)

gravity which generalizes the Starobinsky 4-dim. model [13].

Construction of such a map will also give an additional moti-

vation for the hypothesis that our world is a 3-brane embed-

ded into higher-dimensional spaces [14–17].

The inherent non-linearities prevent the use of string har-

monic oscillators for brane quantization. On the other hand,

it is known that for some non-linear systems the transition

to new dynamical variables, similar to the action-angle vari-

ables, permits to linearize equations of motion and to solve

them. Then, the general solution makes it possible to perform

quantization in terms of the initial data. Such a possibility

To be short we will further use the abbreviation h-ws for hyper-

worldsheet(s).

necessitates search for variables alternative to the world vec-

tors of brane h-ws. In string theory Regge and Lund [18]

used the ﬁrst and the second fundamental forms of string

worldsheets embedded into 4-dim. Minkowski space as new

variables (see also [19,20]). Generalization of this idea for

strings in D-dim. Minkowski space, accompanied with treat-

ment of these variables as gauge multiplets , allowed to trans-

form string equations into an exactly integrable system of

PDEs [21,22]. So, it is natural to use such geometric variables

for p-branes to verify whether brane non-linearities are inte-

grable. For this purpose we apply the renowned differential-

geometric approach describing hypersurfaces embedded into

Riemannian spaces based on the Maurer-Cartan structure

equations [23–26]. This approach was represented in terms

of gauge theory and spontaneous symmetry breaking in [27]

for the Dirac (fundamental) p-branes sweeping minimal h-

ws. The spontaneous symmetry breaking (see [28] and refs.

there) appears due to embedding of hypersurfaces in space-

time. In the case a h-ws 

mi n

p+1

swept by a fundamental p-

brane [29]inD-dim. Minkowski space causes spontaneous

breaking of its Poincare symmetry ISO(1, D − 1) to the

subgroup ISO(1, p) × SO(D − p − 1) [30–33]. The break-

ing yields the Nambu-Goldstone (N-G) ﬁelds identiﬁed with

the coefﬁcients l

μν

of the second fundamental form of 

mi n

p+1

with arbitrary codimention. These coefﬁcients form a N-G

tensor multiplet of the local group SO(D − p − 1).TheY-M

multiplet B

of this group, together with the N-G multiplet

and the metric g

μν

, compose a set of ( p+1)-dim. h-ws ﬁelds.

Their invariant action S

Di r

and the potential V

Di r

(l, g) built

in [27] are given by Eqs. (4) and (5) from Sect. 2. The poten-

tial encodes the non-linearities of fundamental p-branes in

terms of l

μν

and g

μν

. So, V

Di r

(l, g) expressed through the

curvature tensor of 

mi n

p+1

has to uncover the required geo-

metric structure of brane non-linearities.

This scheme was realized in [34] for fundamental p-

branes sweeping minimal h-ws with codimension 1 (i.e.

D = p + 2). In this case the gauge ﬁeld B

is absent.

The Dirac membrane p = 2, D = 4 is an example of such

branes. It describes the bosonic sector of the supermembrane

from the above discussed list of branes invariant under the

kappa-symmetry [9]. The case p = 3, D = 5 selects fun-

damental 3-branes living in 5-dim. Minkowski space. The

3-brane potential follows from Eq. (5)

Di r

p=3



−

Sp(l

)Sp(l

) + c



p=3

. (1)

As shown in [34], V

Di r

is expressed through the squared

scalar curvature R of the h-ws 

mi n

p+1

by Eq. (10)from

Sect. 3.

It means that action (4) with codim 1 encodes a

particular case of ( p + 1)-dim. gravity theories quadratic

The formula (1) is the same for any ( p + 1)-dim. minimal h-ws 

mi n

p+1

with codim 1.

123

剩余10页未读，继续阅读

评论收藏

内容反馈

weixin_38569219

粉丝: 4
资源: 984

具有$$ AdS_ {p + 1} $$ AdSp + 1真空的p分支作为$$ R ^ 2 $$ R2重力模型

最新资源

具有$$ AdS_ {p + 1} $$ AdSp + 1真空的p分支作为$$ R ^ 2 $$ R2重力模型

重力求解程序

Visual_DSP++中文开发手册

ADSP-21364.rar_ADSP_VisualDSP_VisualDsp++例程_adsp-

Spartan3ADSP_MB_Example_10_1

ADSP-21062_FIR.zip_ADSP_ADSP-21060_ADSP滤波_adsp2489 FIR_fir滤波器快速

arith_encoder_JPEG2000.rar_ADSP_ADSP BF533_JPEG2000 C++_bf5_jpeg

FFT.rar_ADSP_adsp21 fft_fft ADSP

ADSP-BF533_hwr_rev3.6_处理器_ADSP_应用手册_BF53XX_

ADSP-21375.rar_ADSP_ADSP-21469 开发板 例程_adsp-_开发板例程

wm_adsp.rar_ADSP

adsp++的vdk介绍

ADSP.rar_ADSP_adsp101

ADSP-BF531.rar_ADSP_ADSP BF531_ADSP-BF_ADSP-BF512_BF533-based vi

ADSP_BF561中文资料

ADSP-21489_Test20180321.rar_21489_21489底层程序_ADSP 21489_ADSP 音频底层

ADSP-21369.rar_ADSP-2116I_ADSP-21469 开发板 例程_ADSP-TS101_VisualDSP

21489_dsp_overuxg_ADSP_锁相环dsp_ADSP-21489_

ADSP_BF53系列中文手册

ADSP.zip_ADSP_ADSP快速收敛算法实现_constantly7zk_saleyme

ADSP.rar_ADSP

ADSP-BF533.rar_ADSP_ADSP-BF53123_ADSP-bf561_BF533 video_adsp-bf5

adsp_bf533_ads1243

Audio_Class.rar_ADSP-BF_audio ADSP_audio class_class_usb audio

ADSP SLIDES_hardly8dd_Help!_ADSP_

ADSP-BF533_EZ_KIT_Lite_Assembly.zip_ADSP-BF51_adsp-bf533_bf533

Power_On_Self_Test.rar_ADSP_self_self test

adsp.rar_ADSP

adsp-sol.zip_ADSP_adsp manual_solution manual_zip

ADSP-218xDSP

最新资源

ADSP-21375.rar_ADSP_ADSP-21469 开发板例程_adsp-_开发板例程

ADSP-21369.rar_ADSP-2116I_ADSP-21469 开发板例程_ADSP-TS101_VisualDSP