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
b
a
Department of Physics, CERN Theory Division, CH-1211, Geneva 23, Switzerland
b
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
ˇ
c
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
6
and E
7
non-abelian
singularities. We briefly comment on a viable E
6
× U (1) effective F-theory model.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
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].
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
4
, over a three-fold base B
3
. The fibration is determined
by the Weierstraß model
y
2
= x
3
+ f (ξ )xz
4
+ g(ξ)z
6
(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
2
:
t
3
: 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
1
On leave from CPHT (UMR CNRS 7644) Ecole Polytechnique, F-91128 Palaiseau,
France.
2
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
3
, their corresponding
degree line bundle has a section that vanishes at these points.
It
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/
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by
SCOAP
3
.