时空翻倍的几何低能有效动作资源-CSDN文库

Open

Access

200 浏览量 2020-04-07 09:52:41 上传评论收藏 315KB PDF 举报

资源详情

资源评论

Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 930 (2018) 135–154

www.elsevier.com/locate/nuclphysb

Geometric low-energy effective action in a doubled

spacetime

Chen-Te Ma

a,∗

, Franco Pezzella

Department of Physics and Center for Theoretical Sciences, National Ta iwa n University, Ta ipe i 10617, Taiwan, ROC

Istituto Nazionale di Fisica Nucleare – Sezione di Napoli, Complesso Universitario di Monte S. Angelo ed. 6,

via Cintia, 80126 Napoli, Italy

Received 1

September 2017; accepted 7 March 2018

Available

online 12 March 2018

Editor: Leonardo

Rastelli

Abstract

The

ten-dimensional supergravity theory is a geometric low-energy effective theory and the equations of

motion for its ﬁelds can be obtained from string theory by computing β functions. With d compact dimen-

sions,

an O(d, d; Z) geometric structure can be added to it giving the supergravity theory with T-duality

manifest. In this paper, this is constructed through the use of a suitable star product whose role is the one to

implement the weak constraint on the ﬁelds and the gauge parameters in order to have a closed gauge sym-

metry

algebra. The consistency of the action here proposed is based on the orthogonality of the momenta

associated with ﬁelds in their triple star products in the cubic terms deﬁned for d ≥ 1. This orthogonality

holds also for an arbitrary number of star products of ﬁelds for d = 1. Finally, we extend our analysis to the

double sigma model, non-commutative geometry and open string theory.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Quantum gravity is expected to unify all theories of the fundamental interactions by com-

bining quantum mechanics and general relativity. String theory is a candidate to provide a

Corresponding author.

E-mail

addresses: yefgst @gmail .com (C.-T. Ma), franco .pezzella @na .infn .it (F. Pezzella).

https://doi.org/10.1016/j.nuclphysb.2018.03.004

Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

136 C.-T. M a, F. Pezzella / Nuclear Physics B 930 (2018) 135–154

self-consistent framework for quantum gravity. A crucial role at this aim is played by T-duality

and S-duality.

String theory

, whose action is deﬁned in the two-dimensional world-sheet space, explores

the target-space theory and its low-energy limit via one-loop β-functions [1,2] and α



correc-

tions. One-loop β-functions provide the equations of motion satisﬁed, in the target space, by the

background ﬁelds with which the string interacts and, in particular, the one associated with the

graviton ﬁeld, present in the spectrum of closed strings, generates the Einstein gravity equation.

The main goal of double ﬁeld theory (DFT) [3–6]i

s to manifestly incorporate T-duality, i.e.

the O(d, d; Z) invariance in the target space with d compact dimensions, as a global symmetry

of the low-energy ﬁeld theory deriving from closed strings living in a D-dimensional spacetime

which is the product of a Mink

owski n-dimensional ﬂat space M =R

n−1,1

with a d-dimensional

torus T

(n +d = D). Then, the ﬁelds of DFT live in the product of M with a 2d-dimensional

doubled torus containing both the torus T

, parametrized by the original compact coordinates

, and its dual

, parametrized by the dual coordinates ˜x

. The ﬁeld content of DFT involves

the metric ﬁeld g

, the Kalb–Ramond ﬁeld B

(i, j = 1, ...D) and a dilaton, i.e. the massless

bosonic sector of the closed string. Since these ﬁelds depend on x

and ˜x

simultaneously,

DFT is expected to have gauge invariance both under diffeomorphisms on the former and dual

diffeomorphisms on the latter, i.e. a gauge invariance under doubled diffeomorphisms.

DFT is still to be fully constructed. In [3], an action for such theory is gi

ven only to cubic

order in the ﬂuctuations of the above mentioned ﬁelds of the massless sector of closed strings

around a ﬁxed background. In this framework, the invariance under linearized doubled diffeo-

morphisms is based on the so-called weak constr

aint, i.e. f ≡ 2∂

∂

f = 0, that has to be

satisﬁed by ﬁelds and gauge parameters that are, therefore, required to live in the kernel of the

second-order differential operator . The weak constraint has a stringy interpretation since it

arises from the level matching condition of closed string theory. The gauge parameters are the

ector ﬁelds ξ

, x

, ˜x

) and

, x

, ˜x

) generating, respectively, the linearized gauge

transformations on the metric tensor and the Kalb–Ramond ﬁelds. Subsequently, in [7]a mani-

festly background independent action has been constructed for the ﬁeld E

= g

+ B

and for

the dilaton d. Further aspects of this action have been studied in [8]. Such formulation results to

be O(D, D) invariant and it has been shown equivalent to the generalized metric formulation [9],

still O(D, D) invariant, where the invariance under doubled diffeomorphisms is based, this time,

on the so-called str

ong constraints, i.e. a generalization of the weak constraint to any product

of ﬁelds and gauge parameters. In the case of d compact dimensions, the O(D, D) symmetry

structure breaks to the O(d, d; Z) symmetry preserving the periodic boundary conditions.

So fa

r, a non-trivial theory that is invariant under doubled diffeomorphisms without using

the above mentioned constraints has not been found yet. One could try to formulate a theory in

terms of ﬁelds automatically projected in the kernel of  through a suitably deﬁned projector

operator, the star product, which takes an arbitrary ﬁeld or a g

auge parameter to that kernel.

This operator has to be used also in the gauge transformations in order to ensure that the gauge

variations are allowed variations of the ﬁelds and shows, in general, a non-associative property

that does not give any problems to the invariance of the action that, instead, could be spoiled by

the non-closure of the g

auge algebra. In other words, it may happen that, due to the non-closure

of the gauge algebra, one should not neglect a total derivative term generated by applying a gauge

transformation. The total derivative term or boundary term is also related to the global geometry

of DFT [10–12]. Furthermore, a closed g

auge algebra in DFT is also important to ensure the

closure of the supersymmetry algebra [13–15].

C.-T. M a, F. Pezzella / Nuclear Physics B 930 (2018) 135–154 137

It has been already observed that DFT is expected to be a candidate for a supergravity theory

with manifest T-duality and that the target space could be understood from a world-sheet theory

by computing one-loop β-functions. An interesting work, in this direction, has been to compute

the one-loop β-functions in the double sigma model without using an

y constraint [16–20].

The bosonic sigma model gi

ves the dynamics of bosonic closed strings and of bosonic open

strings satisfying Neumann and Dirichlet boundary conditions. In particular, a non-commutative

geometry appears in the quantization of the bosonic sigma model with the endpoints of open

strings attached to a brane, in the presence of a B-ﬁeld. A

Seiberg–Witten map in the target space

also comes from a non-commutative geometry and it ensures the equivalence of a commutative

description and a non-commutative description. A useful study of the non-commutative side con-

sists in computing α



corrections to all orders through the Moyal product. Non-commutative

geometry could also be understood through an O(D, D) generalized metric [21], being the

Dirac–Born–Infeld (DBI) theory [22,23]the low-energy effective ﬁeld theory of the double

sigma model for the open string.

As already claimed, the issue of DFT is the loss of the closure of the g

auge algebra without

using the strong constraint. This latter means that it is always necessary to eliminate half of the

degrees of freedom so one could not have any new physics beyond the ordinary supergravity the-

ory. An interesting wo

rk [24] related to relaxing the strong constraint consists in using a suitable

product, the ◦-product, exhibiting properties of commutativity, associativity and distributivity

and that allows to get the closure of the gauge algebra, but the theory deﬁned in this way could

not reduce to the cubic action [3]

obtained from closed string ﬁeld theory.

The main goal of this paper is to ﬁnd a DFT

-inspired string low-energy effective theory in the

doubled spacetime, embodying in it the aforementioned star product. The latter, differently from

the ◦-product of [24], does not exhibit associativity that can be shown, however, to be recovered

under particular conditions. At this aim, we ﬁrst consider the case d =1o

f a spacetime with only

one doubled space-dimension. In this case, the star product has associativity without imposing

any particular condition. We ﬁnd here that a closed gauge algebra can be obtained and that

the action, uniquely determined by the gauge symmetry, results to be the sum of the ordinary

super

gravity theory and the dual supergravity [25].

The e

xtension to arbitrary d can be again understood from the triple product [25] that again

gives the orthogonality of the momenta, just like in d = 1. The result implied by the triple prod-

ucts suggests that the cubic action can be obtained, up to a boundary term, from a DFT deﬁned

through the star product without loosing an

y ﬂuctuations of the ﬁelds. Imposing the orthogonal-

ity of momenta also gives a closed gauge algebra. This can be extended to an arbitrary number

of ﬁelds and the results do not contradict the ones already known from closed string ﬁeld theory.

We mainly use the weak constraint to obtain the orthogonality of momenta when d = 1

and the

triple star products for arbitrary d. The analysis of constraints does not depend on the construc-

tion of an action and a gauge symmetry. In other words, this implies that the orthogonality of

momenta should be an unavoidable condition by using the weak constraint for d = 1

and triple

star products for arbitrary d. We also construct a double sigma model in terms of the star product.

T-duality for non-geometric ﬂux

es is deﬁned by non-commutativity parameters. The transfor-

mations between commutative and non-commutativity parameters are physical transformations

of DFT that are shown to be actually related by an O(D, D) transformation. The star product

is made commutative by imposing the orthogonality of momenta, which could maintain all the

theoretical properties of DFT

We ﬁrst re

view the properties of the projector in Sec. 2, discuss the case for d =1 and do the

extension to arbitrary d respectively in Sec. 3 and Sec. 4. A study of DFT and the double sigma

138 C.-T. M a, F. Pezzella / Nuclear Physics B 930 (2018) 135–154

model is exhibited in Sec. 5 and Sec. 6. We discuss T-duality in a non-commutative space [26]in

Sec. 7. The open string theory is shown in Sec. 8. Finally, we discuss the results and conclude in

Sec. 9.

2. Properties of the projector

A role of the projector we are going to discuss in this section [3]is to keep the ﬁelds and

gauge parameters in ker , i.e. the kernel of the second-order differential operator  ≡2∂

∂

Through the projector, some constraints are imposed on momenta of ﬁelds and gauge parameters.

The weak constraint could be rewritten, in general, in an O(d, d) covariant form:

∂

f = 0, (1)

where

∂

≡



∂



,m= 1, 2, ···,d,

M =1, 2, ···, 2d. (2)

Here, we denote the doubled indices by the uppercase letters and the non-doubled indices by the

lower-case letters. The doubled indices are raised or lowered by an O(d, d) invariant metric:

η =



0 I

I 0



. (3)

Let us introduce the following notation in the doubled spacetime. We use X

and K

for de-

noting, respectively:

≡



˜x



and K

≡





(4)

where p

are the momenta conjugate to x

while the windings ω

can be considered as the

conjugate to the dual coordinate ˜x

Before introducing our projector

, let us ﬁrst observe that, given a general double ﬁeld A, one

could introduce a Fourier series for it, along the compact dimensions:

A ≡



iKX

(5)

where KX ≡ K

. A canonical projection of the ﬁeld A into a ﬁeld satisfying the weak

constraint can be deﬁned through the following star product denoted by ∗:

A ∗ 1 ≡



iKX

KK,0

, (6)

where KK ≡ K

. This actually gives

∂

(A ∗ 1) = 0. (7)

In other words, the star product imposes the weak constraint on A, projecting it into A ∗1 satis-

fying the constraint [A ∗ 1] = 0.

It is also possible to deﬁne the star product of constrained ﬁelds A ∗1

and B ∗1as follows:

A ∗ B =B ∗ A =



K,K



i(K+K



KK,0



. (8)

剩余19页未读，继续阅读

评论收藏

内容反馈

时空翻倍的几何低能有效动作

评论0

最新资源

时空翻倍的几何低能有效动作

评论0

最新资源

相关推荐

领先的6D低能有效动作，N = 11 $$ \ mathcal {N} = \ left（1,1 \ right）$$ SYM理论

向量/张量背景下6D超多重理论中的领先低能有效动作

时空的有限几何玩具模型作为纠错码

构建弯曲时空“几何结构”的逻辑批判

高维超越完备时空的交叠对冲分布与非庸正几何

分形几何学(教程).pdf

论文研究 - 时间的几何解释和时空的新模型

基于几何算法的车辆GPS轨迹时空数据挖掘.pdf

计算几何计算几何计算几何

立体几何图形制作软件《几何图霸》V2.0

几何讲义++第一学期++解析几何.pdf

几何光学PPT几何光学PPT几何光学PPT几何光学PPT

专题二 巧用图形的翻折解决几何问题 2020年中考数冲刺几何难点突破 专题汇编(原卷版).docx

高等几何PPT高等几何PPT高等几何PPT高等几何PPT高等几何PPT

特殊的几何形状和时空签名

论文研究 - 时空欧几里得模型

黎曼几何引论 上册

几何学教程立体几何卷.pdf

高等几何高等几何高等几何高等几何高等几何高等几何高等几何高等几何

论文研究-基于时空几何流的Bandelet稀疏正则化在视频修复中的应用.pdf

计算几何 算法几何 程序

Qt 5实现串口调试助手 （源工程文件、0积分下载）

【SystemVerilog】路科验证V2学习笔记（全600页）.pdf

AutoSAR标准协议4.2.2

光伏-储能并网系统仿真.rar

NPPJSONViewer.zip

GD32替换STM32注意事项.pdf

XCP协议的规范文档

VS2015安装证书，JavaScript_ProjectSystem.msi，JavaScript_LanguageService.msi

专题二巧用图形的翻折解决几何问题 2020年中考数冲刺几何难点突破专题汇编(原卷版).docx

黎曼几何引论上册

计算几何算法几何程序

Qt 5实现串口调试助手（源工程文件、0积分下载）