Mordell-Weil U(1)约束下的F理论GUT：E6和E7非阿贝尔奇点的探讨

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本文主要探讨的是F-theory (弦理论的一种实现，特别关注于四维空间时间中的超对称场论) 中的Grand Unified Theory (GUT) 的一种特殊情况，即带有Mordell-Weil U(1) 的模型。Mordell-Weil群是椭圆曲线上的有理点集合，这里提到的U(1) 是额外的阿贝尔因素，其在理论中的引入对于理解高维度几何结构及其与物理粒子的关联至关重要。作者们研究了一个具体情境，即在具有椭圆纤维和有理截面的纤维丛中，特别关注Mordell-Weil排名为一的简单情况。这种U(1) 的存在对于椭圆纤维的Tate形式的系数有着显著的影响。Tate形式是一种用于描述椭圆曲线的标准形式，它有助于揭示曲线的局部特征，特别是那些与代数几何中的特殊点相对应的性质。通过将代表通用超曲面P^1_12的方程转化为Tate形式，研究者发现这个额外的U(1) 与理论中出现的E6和E7这两个著名的非阿贝尔规范群的例外奇异点相匹配。E6和E7是非阿贝尔规范群，它们在F-theory中扮演着重要的角色，尤其是在弦理论的GUT模型构建中，因为它们可能对应于某些基本粒子的对称性。文章进一步讨论了一个可行的E6×U(1) 结构的有效F-theory模型，这意味着这个模型能够将E6规范群与额外的U(1) 稳定器相结合，从而可能提供了一种解释粒子物理标准模型之外的额外对称性的途径。这个模型不仅受到数学约束的检验，还必须满足物理学上的自然性和一致性。这篇论文的核心内容是探究了在F-theory GUT框架下，Mordell-Weil U(1) 对模型的结构和规范对称性产生的影响，以及如何通过数学手段如Tate形式来刻画这种关系。研究者提出的E6×U(1) 模型提供了一种可能的途径，用于结合额外的U(1) 效应与已知的E6规范对称性，这对于深入理解弦理论与粒子物理的连接具有重要意义。

Physics Letters B 735 (2014) 226–230

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

F-GUTs with Mordell–Weil U (1)’s

I. Antoniadis

a,1

, G.K. Leontaris

Department of Physics, CERN Theory Division, CH-1211, Geneva 23, Switzerland

Physics Department, Theory Division, Ioannina University, GR-45110 Ioannina, Greece

a r t i c l e i n f o a b s t r a c t

Article history:

Received

12 May 2014

Received

in revised form 14 June 2014

Accepted

16 June 2014

Available

online 19 June 2014

Editor:

M. Cveti

In this note we study the constraints on F-theory GUTs with extra U (1)’s in the context of elliptic

ﬁbrations with rational sections. We consider the simplest case of one abelian factor (Mordell–Weil rank

one) and investigate the conditions that are induced on the coeﬃcients of its Tate form. Converting

the equation representing the generic hypersurface P

112

to this Tate’s form we ﬁnd that the presence

of a U (1), already in this local description, is consistent with the exceptional E

and E

non-abelian

singularities. We brieﬂy comment on a viable E

× U (1) effective F-theory model.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

It has been by now widely accepted that additional U (1) or dis-

crete

symmetries constitute an important ingredient in GUT model

building. Such symmetries are useful to prevent dangerous super-

potential

couplings of the effective ﬁeld theory model, in particular

those inducing proton decay operators and lepton number violat-

ing

interactions at unacceptable rates. Model building in the con-

text

of string theory has shown that such symmetries are naturally

incorporated in the emerging effective ﬁeld theory model. In the

context of F-theory [1] in particular, the last few years several GUT

symmetries have been analysed with the presence of additional

U (1) factors [2].

In F-theory models the non-abelian part of the gauge group is

determined by speciﬁc geometric singularities of the internal man-

ifold.

The internal space is an elliptically ﬁbred Calabi–Yau (CY)

fourfold Y

, over a three-fold base B

. The ﬁbration is determined

by the Weierstraß model

= x

+ f (ξ )xz

+ g(ξ)z

(1)

where the base of the ﬁbration corresponds to the point of the

torus z → 0 and as such it deﬁnes a zero section at [x : y : z] =[t

: 0]. For particularly restricted f , g functions the ﬁber degener-

ates

over certain points of the base. The non-abelian singularities

of the ﬁber are well known and have been systematically classi-

ﬁed

with respect to the vanishing order of the functions f , g and

On leave from CPHT (UMR CNRS 7644) Ecole Polytechnique, F-91128 Palaiseau,

France.

For an incomplete list see [3–10], the reviews [11–14] and references therein.

the roots of the discriminant of (1), by Kodaira [15]. An equiva-

lent

description useful for local model building is also given by

Tate [16,17]. There are U (1) symmetries however which do not

emerge from a non-abelian singularity and as such they do not fall

into the category of a Cartan subalgebra. There is no classiﬁcation

for such U (1) symmetries analogous to the non-abelian case and

up to now they have not been fully explored. Abelian factors corre-

spond

to extra rational sections and as such they imply additional

restrictions on the form of the functions f , g. Because sections are

given in terms of divisors whose intersection points with the ﬁber

should be distinct and not identiﬁable by any monodromy action,

this can occur only for rational intersection points. Therefore, for

such points of an elliptic curve ﬁbred over B

, their corresponding

degree line bundle has a section that vanishes at these points.

is known that rational points on elliptic curves constitute a

group, the so called Mordell–Weil group. The Mordell Weil group

is ﬁnitely generated in the sense that there exists a ﬁnite basis

which generates all its elements [18]. A ﬁnitely generated group

can be written as

Z ⊕ Z ⊕···⊕ Z ⊕ G

where G is the torsion subgroup, which in principle could be a

source for useful discrete symmetries in the effective Lagrangian.

Recent developments in F-theory have analysed some properties

of the latter and its implications on effective ﬁeld theory mod-

els.

The rank of the abelian group is the rank of the Mordell–Weil

group [19,20], however, the latter in not known. Up to now, stud-

ies

with one, two and three extra sections have appeared and

some general implications on the low energy models have been

accounted for [21–33].

http://dx.doi.org/10.1016/j.physletb.2014.06.044

0370-2693/

SCOAP

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