Mass spectrum of type IIB ux compactications comments on AdS vacua and

2025-05-02 0 0 418.4KB 24 页 10玖币

侵权投诉

Mass spectrum of type IIB ﬂux compactiﬁcations

—

comments on AdS vacua and

conformal dimensions

Erik Plauschinn

Institute for Theoretical Physics, Utrecht University

Princetonplein 5, 3584CC Utrecht

The Netherlands

Abstract

In this note we study the mass spectrum of type IIB ﬂux compactiﬁcations.

We ﬁrst give a general discussion of the mass matrix for F-term vacua in

four-dimensional N= 1 supergravity theories and then specialize to type

IIB Calabi-Yau orientifold compactiﬁcations in the presence of geometric

and non-geometric ﬂuxes. F-term vacua in this setting are in general AdS4

vacua for which we compute the conformal dimensions of operators dual to

the scalar ﬁelds. For the mirror-dual of the DGKT construction we ﬁnd

that one-loop corrections to the complex-structure moduli space lead to

real-valued conformal dimensions — only when ignoring these corrections

we recover the integer values previously reported in the literature. For an

example of a ﬂux conﬁgurations more general than the DGKT mirror we

also obtain non-integer conformal dimensions. Furthermore, we argue that

stabilizing moduli in asymptotic regions of moduli space implies that at least

one of the mass eigenvalue diverges.

arXiv:2210.04528v1 [hep-th] 10 Oct 2022

Contents

1 Introduction 2

2 Mass matrix and conformal dimensions 4

3 Type IIB ﬂux compactiﬁcations 6

4 Conformal dimensions 9

5 Asymptotic regions 15

6 Summary and conclusions 18

A Computational details 20

1 Introduction

String theory is a theory of quantum gravity including gauge interactions. Its

supersymmetric version is consistent in ten space-time dimensions and can be

connected to four-dimensional physics through compactiﬁcation. Well-understood

compactiﬁcation spaces are Calabi-Yau three-folds whose resulting lower-dimen-

sional theories have a Minkowski vacuum. Deformations that preserve the Calabi-

Yau condition correspond to massless scalar ﬁelds (moduli) in four dimensions,

however, it is possible to deform the compact space away from being Calabi-Yau

by turning on ﬂuxes. Such ﬂuxes generate mass terms for the moduli and typically

lead to AdS4vacua. For gravity theories in AdS spaces one can then apply the

AdS/CFT dictionary to relate, for instance, the masses of scalar ﬁelds in AdS4to

conformal dimensions of corresponding operators in a putative three-dimensional

CFT. For a certain class of type II ﬂux compactiﬁcations, more concretely for

the construction by DeWolfe, Giryavets, Kachru, and Taylor (DGKT) in type IIA

string theory [1], it was observed in [2–5] that the conformal dimensions of all

scalar ﬁelds in the closed-string sector are integer-valued. In [2, 3] the analysis

was done for toroidal compactiﬁcations, while in [4,5] the authors performed their

computation for a general Calabi-Yau three-fold.

Obtaining integer conformal dimensions irrespective of the compactiﬁcation

space is a rather surprising observation which one would like to understand. In

particular, one would like to know which features of the compactiﬁcation are rel-

evant for this result. This question was addressed in [5], where it was found that

some non-supersymmetric type IIA vacua do not lead to integer-valued conformal

dimensions. However, one can argue that such conﬁgurations are unstable [6] and

therefore are not suitable for the applying the AdS/CFT correspondence [5]. In

this note we approach the question of integer conformal dimensions from the type

IIB side. We study orientifold compactiﬁcations of type IIB string theory with

geometric and non-geometric ﬂuxes that lead to supersymmetric AdS4vacua. In

regard to the DGKT construction in type IIA string theory [1] we note the follow-

ing diﬀerences:

1. On the type IIA side one typically considers the large-volume limit without

corrections. In type IIB string theory this limit corresponds to the large-

complex-structure limit, for which corrections are well-understood. We can

therefore compute conformal dimensions in a setting that includes corrections

to the moduli-space geometry.

2. For DGKT ﬂux compactiﬁcations in type IIA string theory the superpo-

tential splits into the sum of two independent terms [7]. From a type IIB

perspective this is a rather special case that corresponds to turning on only a

speciﬁc component of geometric and non-geometric Neveu-Schwarz–Neveu-

Schwarz ﬂuxes. For a more general ﬂux-choice the superpotential will not

split in this way.

In this work we investigate these two aspects. First, we consider the mirror-

dual of the DGKT setting — including perturbative corrections to the complex-

structure moduli space — and determine masses and conformal dimensions ana-

lytically. Second, we construct a ﬂux conﬁguration for which the superpotential

does not split into two separate terms and determine the masses and conformal

dimensions numerically. More concretely,

•in section 2 we give a general discussion of the mass matrix for F-term vacua

of four-dimensional N= 1 supergravity theories. In section 3 we then spe-

cialize to Calabi-Yau orientifold compactiﬁcations of type IIB string theory

with O3- and O7-planes in the presence of geometric Ramond-Ramond (R-R)

and geometric and non-geometric Neveu-Schwarz–Neveu-Schwarz (NS-NS)

ﬂuxes.

•In section 4 we determine masses and conformal dimensions for AdS4ﬂux-

vacua. For the mirror-dual of DGKT we ﬁnd analytically that perturbative

corrections to the prepotential lead to non-integer conformal dimensions.

We also study an example with a ﬂux choice more general than DGKT for

which the superpotential does not split into two separate terms. Here we

ﬁnd numerically that the conformal dimensions are not integer.

•Section 5 is independent of our discussion of AdS vacua and conformal di-

mensions, but uses many results from sections 2 and 3. Here we study ﬂux

compactiﬁcations relevant for the KKLT and large-volume scenarios [8, 9],

where only the F-terms of the complex-structure moduli and the axio-dilaton

are considered. We compute the trace of the canonically-normalized mass

matrix and argue that if moduli are stabilized in an asymptotic regime of

moduli space, then at least one of the mass eigenvalues diverges.

•In section 6 we summarize our ﬁndings and in appendix A we give some

technical details of the computations in the main text.

2 Mass matrix and conformal dimensions

Let us start with a discussion of masses of chiral multiplets in four-dimensio-

nal N= 1 supergravity theories. We consider minima of the scalar potential

corresponding to vanishing F-terms and determine the general form of the mass

matrix of the scalar ﬁelds. In the case of AdS4vacua we furthermore compute the

conformal dimension of operators dual to the scalar ﬁelds.

F-term minima

Let us consider a four-dimensional N= 1 supergravity theory with ncomplex

scalar ﬁelds φMwhere M= 1, . . . , n. The F-term scalar potential can be written

V=eKhFMGMN FN−3|W|2i,(2.1)

where Kdenotes the real K¨ahler potential, Wdenotes the holomorphic superpo-

tential, and GMN =∂M∂NKdenotes the K¨ahler metric. The F-terms are given by

FM=∂MW+KMWwith KM=∂MK. In this work we are interested in F-term

minima of this potential given by

FM= 0 .(2.2)

Mass matrix

The mass matrix for the complex scalar ﬁelds of this theory corresponds to the

second derivatives of the potential (2.1). It can be arranged into the form

m2="m2

MN m2

MN #,(2.3)

where the blocks in the ﬁrst line are related to the ones appearing in the second

line by complex conjugation. For the former we ﬁnd the following expressions at

the minimum (2.2)

MN =eKh∂MFPGP Q ∂NFQ−2GM N |W|2i,

MN =eK−2∂MFNW.

(2.4)

In order to obtain the canonically-normalized mass matrix we note that the K¨ahler

metric GMN is hermitian and positive deﬁnite. We can therefore write Gas the

square of a positive-deﬁnite matrix Γ and we deﬁne a matrix Qas

G= ΓΓ†, Q = Γ−1(∂F )Γ−T.(2.5)

Here and in the following we suppress identity matrices δMN ,δMN , and δMN.

The canonically-normalized mass matrix is obtained by multiplying (2.3) with

appropriate factors of Γ from the left and the right and we ﬁnd

can =eKQQ†−2|W|2−QW

−Q†W Q†Q−2|W|2.(2.6)

Mass eigenvalues

In order to determine the eigenvalues of (2.6) we ﬁrst perform a singular-value

decomposition of Qas

Q=UΣV†,(2.7)

where Uand Vare unitary matrices and Σ is a diagonal matrix that contains only

real entries. We can then write the canonically-normalized mass matrix (2.6) as

can =eKU0

0V Σ2−2|W|2−ΣW

−ΣWΣ2−2|W|2 U†0

0V†,(2.8)

and the corresponding eigenvalue equation for the mass eigenvalues m2is given by

0 = dethm2

can −m2i

= dethe2Kh(Σ2−2|W|2−e−Km2)2−Σ2|W|2ii.

(2.9)

Denoting the entries of the diagonal matrix Σ2by σ2

α, we can solve (2.9) as

α±=eKhσ2

α±σα|W| − 2|W|2i.(2.10)

We ﬁnally note that QQ†=UΣ2U†and Q†Q=VΣ2V†, and hence σ2

αare the real

and positive eigenvalues of the hermitian matrices Q†Qand QQ†.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

MassspectrumoftypeIIBuxcompactications|commentsonAdSvacuaandconformaldimensionsErikPlauschinnInstituteforTheoreticalPhysics,UtrechtUniversityPrincetonplein5,3584CCUtrechtTheNetherlandsAbstractInthisnotewestudythemassspectrumoftypeIIBuxcompactications.WerstgiveageneraldiscussionofthemassmatrixforF...

收起<<

Mass spectrum of type IIB ux compactications comments on AdS vacua and.pdf

共24页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Mass spectrum of type IIB ux compactications comments on AdS vacua and

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: