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arXiv:1309.7297v2 [hep-th] 15 Nov 2013
On the Standard Model Group in F-theory
Kang-Sin Choi
∗
Scranton Honors Program, Ewha Womans University, Seoul 120-750, Korea
Abstract
We analyze the Standard Model gauge group SU (3)×SU (2)× U (1) constructed
in F-theory. The non-Abelian part SU(3) × SU(2) is described by singularities of
Kodaira ty pe. It is d istin gu ished to na¨ıve produ ct of SU(3) and SU(2), revealed
by blow-up analysis, since the resolution pr ocedures cannot be done separately to
each group. The Abelian part U (1) is constructed by obtaining a desirable global
two-form harboring it, u sing ‘factorization method’ similar to the decomposition
method of the spectral cover; It makes use of an extra section in the elliptic fiber of
the Calabi–Yau manifold, on which F-theory is compactified. Conventional gauge
coupling unification of SU (5) is achieved, without threshold correction from the flux
along hypercharge direction.
∗
email: kangsin@ewha.ac.kr
1
1 Introduction
It has been well-known wisdom that the gauge group SU(3)×SU(2)×U(1) of the Standard
Model (SM) is far from arbit r ary collection of simple and Abelian factors, because the
matter fields transforming under this have very particular charge assignments. Grand
Unified Theory (GUT) [1] suggests that it is best understood by embedding the group to
a series of the exceptional groups E
n
, including SU(5) ≡ E
4
, SO(10) ≡ E
5
, and E
6
[2]. In
this sense the SM group may be expressed as the unique, maximal E
3
× U(1) subgroup
of E
4
. These E
n
groups naturally occur in heterotic string and F-theory [3–14].
In this work, we analyze the singularities and the two-forms describing the Standard
Model gauge group SU(3) × SU(2) × U(1) in F-theory. The non-Abelian part can be
described by conventional singularities of Kodaira type for the elliptic fiber in an internal
manifold [4, 15–17]. Besides the saga city that the GUT structure suggests, also in F-
theory the desired light matter fields o f the SM—not only the (3, 2) representation but
also (
3, 1) and (1, 2)— emerge on so-called matter curves [6 –8, 18], only if the SU(3) ×
SU(2) is embedded in E
6
at least, because essentially the matter fields can only arise by
branching the gauge multiplets in the adjoint represent ations of (local) unified groups.
With the clue of various gauge symmetry enhancement directions, we can o btain the
desired singularities for SU(3) × SU(2) by deforming the singularity of SU(5), verified
by matter curve structure, etc. [19 , 20]. This will be reviewed and further analyzed by
resolution process in Section 2. Although the shape of the singularity was consistent, it
has been obtained by applying various necessary conditions. The analyses given in Section
2 now provides a complete proof of it.
In constructing the gauge theory, the real problem has been an Abelian U(1) group
that is not obtained from a Kodaira-type singularity. We obtain the Abelian gauge field
by expanding the three-form tensor of M/F-theory along a two-cycle `a la Kaluza–Klein
reduction. For the components of Cartan subalgebra of non-Abelian groups, we can
automatically obtain such cycles by blowing-up the corresponding singularity. On the
other hand it was quite difficult to find such a curve for U(1) which is glo bally valid and
has intersection numbers giving the desired charges of matter fields. Again, one hint can
be embedding all the groups in a unified g r oup, which would lead a two- form for Abelian
group in the similar fashion for non-Abelian group. So-called U(1)-restricted model was
the fir st successful method in describing such global U(1) for SU(5) × U(1) by embedding
the U(1) group into SU(2) [21–23]. However this heavily depends on a clever choice of
ansatz and extending this to general gauge group, following the E
n
series was difficult.
There have been an indirect derivation from spectral cover [24] via heterotic–F-theory
duality [25], which gives a globally valid two-form in so-called stable degeneration limit,
but it is useful in the case admitting the dua lity. Recently, Refs. [27, 28], introduced
2
a ‘multiple section’ method to introduce U(1)’s, essentially by finding a cousin of the
element from Cartan subalgebra from a certain unified group (see also [22,25, 26]). Such
two-form naturally comes from more careful comparison between the elliptic equation and
the spectral cover. In this paper, and esp ecially in Section 3, we employ this method to
obtain the U( 1) correctly describing hypercharge. As a byproduct, this method provides
another proof to the expression of the singularity SU(3)×SU(2) in relation t o the spectral
cover. We employed this method because this is extended to any number of U(1) sections
and although the exceptional divisors may not be directly embedded into a larger group
like E
8
, the intersection relation can be embedded into and traced from such group.
Another interesting direction is to find different sections making use of Mordell-Weil
group generated by a single group element in the elliptic fiber and/or ‘tops’ in to ric
geometry [19,29, 30].
At the moment, there is no clue whether we have the Standard Model at the unification
or string/F-theory scale without an intermediate GUT. Besides such a priori reason, the
direct construction of the SM also has following practical merits. First, we do not need
to t ur n on a flux in the hypercharge direction which gives rise to a threshold correction
to the corresponding gauge coupling, ruining the coupling unification relation. Second,
some sector related to electroweak symmetry breaking is b etter understand if we have
the unbroken group as the SM group, since some fields being footprints of GUT have
nontrivial coupling to the Higgs sector [31, 3 2]. Even we have the SM at t he unification
scale, still we have footprint s of GUT such as the number of generation, since the string
theory itself makes use of the unification relatio n [19, 20, 33]. These two features have
no analogy in heterotic string theory, since in F-theory we have two-step construction
of gauge theory: constructing smaller group tha n E
8
and further symmetry breaking by
G-flux. It controls the number of generation obeying a certain unification relation while
the actual gauge symmetry is the smaller SM gro up.
2 Non-Abel ian factor SU(3) × SU(2)
We first construct a singularity for the non-Abelian algebra SU(3)×SU(2) of the Standard
Model. As discussed in the introduction, it is not a mere product of simple alg ebras SU(3)
and SU(2), but it should be along the chain of E
n
algebra
1
.
1
In this paper, we do not need to distinguish b e tween the product of the group and the sum of the
algebra, wher e the latter is more appropria te to our purpose, but we follow the traditional desc ription.
3
2.1 Description by singularity
We consider F-theory compactified on Calabi–Yau fourfold Y , which is an elliptic fiber-
ation over a three-base B. The elliptic fiber is given as an hypersurface by an elliptic
equation in the ‘Tate form’
P ≡ −y
2
+ x
3
+ a
1
xyz + a
2
x
2
z
2
+ a
3
y + a
4
xz
4
+ a
6
z
6
= 0 (1)
in P
2
[2,3,1]
fiber over B, having homo geneous coordinates (x, y, z) with the respective weights
indicated as subscripts, and all the parameters are also appro pr iate holomorphic sections
over B [3, 17]. This can be regarded as a definition for our Cala bi–Yau fourfold Y as a
hypersurface.
Tuning parameters a
i
in some base coordinate, say w defined with respect to a divisor
of B
W : w = 0, (2)
which should be already present from the construction of B, gives rise to singularities re-
lated to gauge symmetry on the worldvolume W × R
4
. For instance, an SU(5) singularity,
split I
5
, is obtained from the table by Kodaira [16].
a
1
= b
5
, a
2
= b
4
w, a
3
= b
2
w
2
, a
4
= b
2
w
3
, a
6
= b
0
w
5
, (3)
up to higher order terms in w. We have the discriminant of the elliptic equation (1) up
to fifth order in w,
∆ = b
4
5
(b
0
b
2
5
− b
2
b
3
b
5
+ b
2
3
b
4
)w
5
+ O(w
6
),
responsible for the gauge group SU(5) [3,4, 16].
Deforming the singularity by adding lower order terms in w to a
i
’s, we have less severe
singularity with smaller algebra. The claim in Refs. [19,20 ] is that, the singularity for the
SU(3) × SU(2) is given as,
a
1
= b
5
+ O(w), (4)
a
2
= b
4
w + O(w
2
), (5)
a
3
= b
3
(b
6
+ w)w + O(w
3
), (6)
a
4
= b
2
(b
6
+ w)w
2
+ O(w
4
), (7)
a
6
= b
0
(b
6
+ w)
2
w
3
+ O(w
6
). (8)
The discriminant takes the form
∆ = b
3
5
P
2
(3,2)
P
(
3,1)
w
3
+ P
(3,2)
P
30
w
4
+ O(w
5
) (9)
where the parameters are displayed in Table 1 and P
30
is a quite lengthy, non-factorizable
polynomial in b
i
, e.g. containing a term 2b
2
3
b
4
b
4
5
.
4
name parameter representation symm. enhancement
P
(3,2)
b
6
(3, 2) SU(5)
P
(
3,1)
b
2
3
b
4
b
5
− b
2
b
3
b
2
5
+ b
0
b
3
5
− b
3
3
b
6
(
3, 1) SU(4) × SU(2)
P
(1,2)
b
2
3
b
4
− b
2
b
3
b
5
+ b
0
b
2
5
+ b
2
2
b
6
− 4b
0
b
4
b
6
(1, 2) SU(3) × SU(3)
Table 1: Paramters of gauge symmetry enhancements.
We have a singularity so called Kodaira split I
3
for SU(3) located at (x, y, w) =
(0, 0, 0), which has orders ord(a
1
, a
2
, a
3
, a
4
, a
6
, ∆) = (0, 0, 1, 2, 3, 3) in w [16, 17]. On the
discriminant locus W in (2), F-theory interprets it that we have the SU(3) gauge theory
[3, 4]. Setting P
(3,2)
= 0 enhances t he discriminant to degree five, whereas P
(
3,1)
= 0
does to degree four. The subscripts indicate the corresponding quantum numbers of
unhiggsed fields in the branching, since (3, 2) is regarded as off- diagonal component of
the adjoint 24 under the breaking SU(5) → SU(3) × SU(2), while (
3, 1) is that of
15 under SU(4) → SU(3). The former symmetry enhancement shows that the actual
group from the parameters (4)-(8) is larger than SU(3). It is because t he parameters are
specially tuned up to ord(a
1
, a
2
, a
3
, a
4
, a
6
) = (0, 1, 2, 3 , 5), as the deformations of the SU(5)
singularity in (3).
To see the other part, we change the reference as
w
′
≡ w + b
6
, (10)
defining a new divisor W
′
: w
′
= 0 of B. The parameters become
a
1
= b
5
+ O(w
′
),
a
2
= b
4
(w
′
− b
6
) + O(w
′2
),
a
3
= b
3
(w
′
− b
6
)w
′
+ O(w
′3
),
a
4
= b
2
(w
′
− b
6
)
2
w
′
+ O(w
′4
),
a
6
= b
0
(w
′
− b
6
)
3
w
′2
+ O(w
′6
).
(11)
The discriminant has the form
∆ =
b
2
5
− 4b
4
b
6
2
P
3
(3,2)
P
(1,2)
w
′2
+ P
2
(3,2)
P
′
30
w
′3
+ P
(3,2)
P
36
w
′4
+ O(w
′5
) (12)
where the parameters are shown in Ta ble 1 and P
′
30
, P
36
are non-factorizable polynomials
containing respectively 3b
2
3
b
4
b
4
5
, −3b
2
3
b
4
b
5
5
b
6
. From the observation that ord(a
1
, a
2
, a
3
, a
4
, a
6
, ∆) =
(0, 0, 1, 1, 2, 2) in w
′
we see at (x, y, w
′
) = (0, 0, 0) there is the Kodaira singularity, split
I
2
for SU(2). Again we have the SU(2) gauge theory localized on the locus W
′
. The
parameter w
′
is distinguished to w by the relation (10) via the parameter b
6
, which is the
5
section depending on the base coordinate. We see the parameter P
(3,2)
= b
6
again since
(3, 2) is also charged under t his and vanishing of which enhance the gauge symmetry to
SU(5), in which limit we do not distinguish between w and w
′
. Also P
(1,2)
= 0 enhances
the symmetry SU(2) → SU(3). It is clear that our singularity describes the maximal
semisimple algebra SU(3) × SU(2) embedded in SU(5).
2.2 Resolution
We resolve the SU(3) × SU(2) singularities following the Tate algorithm [16,23,34]. This
resolution shall reveal nontrivial algebraic structure. Neglecting higher order terms in w,
the elliptic equation is
P = −y
2
+x
3
+b
5
xyz+b
4
wx
2
z
2
+b
3
(b
6
+w)wyz
3
+b
2
(b
6
+w)w
2
xz
4
+b
0
(b
6
+w)
2
w
3
z
6
. (13)
First we resolve the I
3
part located at (x, y, w) = (0, 0, 0). We introduce another affine
coordinate e
1
of a P
1
curve such that
(x, y, w) = (x
1
e
1
, y
1
e
1
, w
1
e
1
),
and forbid the simultaneous vanishing x
1
= y
1
= w
1
= 0. Then the original singularity
is only accessed by e
1
= 0. The lowest order terms in e
1
have common factor y
1
, so still
the point (e
1
, y
1
) = (0, 0) is again singular. To have smooth resolution of Y , we blow-up
again
(e
1
, y
1
) = (e
′
1
e
2
, y
2
e
2
), (14)
or equivalently (x, y, w) = (x
2
e
′
1
e
2
, y
2
e
′
1
e
2
2
, w
2
e
′
1
e
2
), and remove e
′
1
= y
2
= 0. Then the
lowest order terms now in e
2
have no common factor and the resolution procedure termi-
nates. From now on we drop the subscripts in x, y, w and the prime in e
′
1
, and so on, if
there is no confusion. The resulting polynomial is
˜
P =e
2
1
e
3
2
x
3
e
1
− y
2
e
2
+ b
5
xyz + b
4
e
1
wx
2
z
2
+ b
3
(b
6
+ e
1
e
2
w)wyz
3
+ b
2
(b
6
+ e
1
e
2
w)e
1
w
2
xz
4
+ b
0
(b
6
+ e
1
e
2
w)
2
e
1
w
3
z
6
.
(15)
Next, we go to the SU(2) part, by going to the primed coordinates using the relation
(10), however, now posessing the form
b
6
+ e
1
e
2
w ≡ w
′
, (16)
(Here w means w
2
in the above (14)). We obtain
˜
P
′
=x
3
e
3
1
e
3
2
− y
2
e
2
1
e
4
2
+ b
5
e
2
1
e
3
2
xyz + b
4
e
2
1
e
2
2
(w
′
− b
6
)x
2
z
2
+ b
3
w
′
(w
′
− b
6
)e
1
e
2
2
yz
3
+ b
2
w
′
(w
′
− b
6
)
2
e
1
e
2
xz
4
+ b
0
w
′2
(w
′
− b
6
)
3
z
6
.
(17)
6
x y z e
1
e
2
e e
0
w
′
Z 2 3 1 0 0 0 0 0
E
1
1 1 0 −1 0 0 1 0
E
2
1 2 0 0 −1 0 1 0
E 1 1 0 0 0 −1 0 1
Table 2: Scaling relations from the definition of the exceptional divisors e
1
, e
2
and e. We
do not remove w
′
by scaling, although it scales covar iantly, but just constrained by ( 19).
As before this describes I
2
singularity at (x, y, w
′
) = ( 0, 0, 0). We want blow up there by
introducing another coordinat e e such that w
′
→ w
′
e.
(x, y, w
′
) → (xe, ye, w
′
e).
ˆ
P
′
=e
2
x
3
e
3
1
e
3
2
e − y
2
e
2
1
e
4
2
+ b
5
e
2
1
e
3
2
xyz + b
4
e
2
1
e
2
2
(w
′
e − b
6
)x
2
z
2
+ b
3
w
′
(w
′
e − b
6
)e
1
e
2
2
yz
3
+ b
2
w
′
(w
′
e − b
6
)
2
e
1
e
2
xz
4
+ b
0
w
′2
(w
′
e − b
6
)
3
z
6
.
(18)
This is the standa r d r esolution of I
2
singularity, found in e.g. Ref [27]. It seems in this
primed coordinates, we would have more singularities such a s (e
2
, w
′
) = (0, 0). Shortly
we see, it turns out we have no more. We come back to the original coordinates, now by
the modified relation
w
′
e − b
6
= e
1
e
2
e
0
. (19)
Then the equation becomes
ˆ
P =e
2
1
e
3
2
x
3
e
3
e
1
− y
2
e
2
e
2
+ b
5
e
2
xyz + b
4
e
0
e
1
e
2
x
2
z
2
+ b
3
(e
0
e
1
e
2
+ b
6
)e
0
eyz
3
+ b
2
(e
0
e
1
e
2
+ b
6
)e
2
0
e
1
exz
4
+ b
0
(e
0
e
1
e
2
+ b
6
)
2
e
3
0
e
1
z
6
]
(20)
We have changed the name w to e
0
, since t he divisor e
0
= 0 plays the role of the extended
root of SU(3) below.
The two resolution procedures should commute or should not prefer the order. Indeed,
we see it is, since we can write the overall result as
(x, y, w, w
′
) → (xee
1
e
2
, yee
1
e
2
2
, e
0
e
1
e
2
, w
′
e)
where the coordinate w is cha nged to e
0
for later convenience. Nevertheless two resolutions
affect each other due to the constraint (19). The resolution procedures can be equivalently
re-expresed in terms of the scaling in Table 2. Besides the definition Z for the P
2
2,3,1
, we
have introduced three new coordinates e
1
, e
2
, e and three scaling relations E
1
, E
2
, E.
7
Once the scalings are established, for instance E
1
in Table 2 means (x, y, z, e
1
, e
2
, e, e
0
) →
(λx, λy, z, λ
−1
e
1
, e
2
, e, λe
0
), we should exclude some points x = y = e
0
= 0 as explained
above. A combination of two scalings always gives a new scaling in a different guise and
we see it has the structure of ideal. So we introduce the Stanrey–Reisner (SR) ideal,
containing such data, generated by
{xyz,xye
0
, ye
1
, xe
0
e
2
, yze, xze
2
, ze
1
e
2
, xyw
′
} ∪ {(ze
1
, ze) xor xe
0
} ∪ {ze
2
xor ye
0
}
{e
1
w
′
xor ee
0
} ∪ {e
2
w
′
xor yee
0
} ∪ {zee
2
xor e
0
w
′
} ∪ {xe
2
w
′
xor (ee
0
, e
1
e)} ∪ {xe
1
w
′
xor e
2
e}
(21)
where in each curly parenthesis, we can choose o ne of the elements (xor means exclusive
or), corresponding to a particular triangulation of the toric diagram [36]. We must choose
ze
1
and ze
2
to have four-dimensional Lorentz vector components fo r the Cartan subal-
gebras that will be related to e
1
and e
2
, which we see below. What we choose here are
ee
0
, e
2
w
′
, e
0
w
′
, e
1
e, and xe
1
w
′
, some of which generated by others. Finally we have
{xyz, xye
0
, ye
1
, xe
0
e
2
, ze
1
, ze
2
, ze, xyw
′
, e
0
e, e
1
e, e
0
w
′
, e
2
w
′
}. (22)
2.3 Intersections
We hereafter consider divisors of the Calabi–Yau manifold
ˆ
Y defined by
ˆ
P = 0. Vanishing
loci of the blow-up coordinates e
i
define exceptional divisors E
i
. Explicitly we have
E
1
: e
1
= 0 = −e
2
+ b
5
x + b
3
b
6
e
0
, {ye
1
, ze
1
, ee
1
} (23)
E
2
: e
2
= 0 = x
3
e
3
e
1
+ b
5
e
2
xy + b
4
e
0
e
1
e
2
x
2
(24)
+ b
3
b
6
ee
0
y + b
2
b
6
ee
2
0
e
1
x + b
0
e
3
0
e
1
b
2
6
, {ze
2
}
E
0
: e
0
= 0 = x
3
e
1
− y
2
e
2
+ b
5
xyz = 0, {e
0
e} (25)
E : e = 0 = −y
2
e
4
2
+ b
5
e
3
2
xy − b
4
b
6
e
2
2
x
2
− b
3
b
6
w
′
e
2
2
y + b
2
b
2
6
e
2
w
′
x − b
0
b
3
6
w
′2
, {ee
1
, ze} (26)
W
′
: w
′
= 0 = x
3
e
1
e − y
2
+ b
5
xyz − b
4
b
6
x
2
z
2
. {e
0
w
′
, e
2
w
′
} (27)
In the every second line, we simplified the relation using the SR ideal (22). The divisors
E
1
, E
2
, E
0
are the obj ects in the SU(3) part, so we obtain them f r om
ˆ
P
T
after perfor ming
proper transformation e
2
1
e
3
2
ˆ
P
T
=
ˆ
P in (20) and we obtain E, W
′
of the SU(2) from e
2
ˆ
P
′
T
=
ˆ
P
′
in (18). In particular the divisor E has dependence on the coordinate w
′
that cannot be
eliminated by the constraint (19), so it is not able to be compared with the E
1
, E
2
, E
0
in the
SU(3) part. However, due to the constraint (19), E does not have common intersection
with them. For t he same reason, we simply decouple the f actor e
2
1
e
2
2
in the equation
ˆ
P
′
T
|
w
′
=0
= 0 to define W
′
, since the constraint (19) forbids vanishing of the either factor.
8
Using the constraint e
2
= −b
6
, we may massage the equation (26) to a fancier form, which
in fact becomes important later.
A complete intersection of E
i
with two arbitrary divisors D
a
and D
b
in
ˆ
Y gives a P
1
curve. Since D
a
and D
b
are arbitrary, an intersection number of two such P
1
curves is
given by the number of common solutions to the equations E
i
= E
j
=
ˆ
P = 0:
E
1
· E
2
= 1 : e
1
= e
2
= b
5
x + b
3
b
6
e
0
= 0, (28)
E
1
· E
0
= 1 : e
1
= e
0
= −e
2
+ b
5
x = 0, (29)
E
2
· E
0
= 1 : e
2
= e
0
= e
1
+ b
5
y = 0. {xe
0
e
2
, e
0
e, e
2
e}, (30)
where the dot product notation is understood. Each equation has one solution in x, y,
and/or e
i
, assuming that b
i
’s are all nonzero. This completes the SU(3) root relations via
McKay correspondence that the intersection numbers corresponding to the minus of the
Cartan matrix of the algebra. The above int ersections are also expressed as [37]
Z
ˆ
Y
E
i
∧ E
j
∧ D
a
∧ D
b
= −A
ij
Z
B
W ∧ D
a
∧ D
b
, E
i
∈ Cartan subalgebra, (31)
where A
ij
is the Cartan matrix and D
a
, D
b
are divisors in B ( whose pullback to
ˆ
Y is
omitted without confusion).
These exceptional divisors E
1
, E
2
, E
0
, thus the cor responding roots in the SU(3) al-
gebra, are disconnected to the rest of divisors E and W
′
of SU(2), since the constraint
does not allow simultaneous vanishing of e
i
and e, or of e
i
and w
′
for each i = 1, 2, 0.
For E and W
′
, we have two solutions in y/x to
E · W
′
= 2 : e = w
′
= y
2
− b
5
xy + b
4
b
6
x
2
= 0, {ez, e
0
e, e
1
e, e
2
w
′
} (32)
consistent with the affine (roughly, extended) Dynkin diagram of SU(2). If there is only
one solution, the discriminant of (32) becomes
b
2
5
− 4b
4
b
6
= 0, (33)
which destroy the O(w
′2
) term in the discriminant (12) of the elliptic equation. We will
assume otherwise in what follows.
2.4 Matter curves and sy mmetry enhancement
In Section 2.1, we have studied various gauge symmetry enhancements by analyzing the
discriminant. In Table 1, each equation P
f
= 0 defines a codimension one curve of the
SU(3) surface e
0
= 0 and/or the SU(2) surface w
′
= 0. Since we can interpret it as that
there are light matter fields f localized on the P
f
= 0, we call it as the matter curve. Here
we further analyze the matter curves from properties of t he exceptional divisors resulting
from the resolution.
9
Matter curves for (3, 2) We first analyze t he matter curve P
(3,2)
= 0 for (3, 2). On
this locus, there is local gauge symmetry enhancement to SU(5). Here we will see this is
reflected by further degeneration and rearrangement of the exceptional divisors.
The equations for exceptional divisors become
E
1
→ E
1A
: −e
2
+ b
5
x = 0 = e
1
, (34)
E
2
→ E
2x
∪ E
2E
∪ E
2B
: xe
2
(x
2
ee
1
+ b
5
y + b
4
e
0
e
1
x) = 0 = e
2
, (35)
E
0
→ E
0C
: x
3
e
1
− y
2
e
2
+ b
5
xyz = 0 = e
0
, (36)
E → E
E2
: e
3
2
y(−ye
2
+ b
5
x) = 0 = e, (37)
where we have degeneration of E
2
and we renamed the divisors accordingly. We may
find E
2E
: e
2
= e = 0 from E as well, and now the previously disconnected pa rt can
communicate via this. It satisfies the modified constraint (19) now read as e = e
2
using
the SR elements e
0
e and e
1
e. This locks only possible divisor from E to E
2E
, a mo ng
seemingly possible elements e.g. E
y
: e = y = 0.
Consequently, the only nontrivial intersections are
E
0C
· E
1A
= 1 : e
0
= e
1
= −e
2
+ b
5
x = 0, (38)
E
1A
· E
2x
= 1 : e
2
= e
1
= x = 0, (39)
E
2E
· E
2x
= 1 : e
2
= e = x = 0, (40)
E
2E
· E
2B
= 1 : e
2
= e = b
5
y + b
4
x = 0, {e
0
e} (41)
E
2B
· E
0C
= 1 : e
2
= e
0
= e
1
+ b
5
y = 0. (42)
The rest intersection numbers are zero. For instance, E
2x
· E
0C
= 0 : e
0
= e
2
= x = 0
forbidden by the SR element xe
0
e
2
. In particular there is no intersection between E
2x
and E
2B
since the required equations e
2
= x = y = 0 are forbidden by the SR element xy
which is inherited by xyw
′
with w
′
= 0.
Altogether these relations give rise to the extended Dynkin diag r am the locally en-
hanced gauge symmetry SU(5). However globally the unbroken gauge symmetry on
ˆ
Y
still remains SU(3) × SU(2). Although the divisor E had no intersections to E
2
before
the symmetry enhancement, now their degenerated da ughters do have nonzero intersec-
tions. This can be tracked to general factors e appearing in the divisors, resulted from
the r esolution of the I
2
part tha t cannot be separately done with respect to the I
3
part.
Also we can calculate the SU(3) × SU(2) weight of the divisors (34)-(37) in Dynkin
basis, as [E
i
· E
1
, E
i
· E
2
; E
i
· E], shown in Table 2.4. At the intersection or matter curve
b
6
= 0, we have local gauge symmetry enhancement which explains t he emergence of a
light field with quantum number (3, 2). Their six components and weights are displayed
in Table 4. As expected two roots of SU(3) are played by E
1A
and E
2x
+ E
2E
+ E
2B
, and
10
divisor weight
E
1A
[−2, 1; 0]
E
2x
[1, −1; 1]
E
2E
[0, 0; −2]
E
2B
[0, −1; 1]
E
0C
[1, 1; 0]
Table 3: SU(3) × SU( 2) weights of the exceptional divisors in Dynkin basis.
divisor weight
E
2B
+ E
0C
[1, 0; 1]
E
2B
+ E
0C
+ E
2E
[1, 0; −1]
E
2B
+ E
0C
+ E
1A
[−1, 1; 1]
E
2B
+ E
0C
+ E
1A
+ E
2E
[−1, 1; −1]
E
2B
[0, −1; 1]
E
2B
+ E
2E
[0, −1; −1]
Table 4: The components of the matter representations (3, 2), at the matt er curve b
6
= 0.
the root of SU(2) is E
2E
. One can easily see that this is the only representation whose
components are only effective divisors.
Matter curves for (
3, 1) There are other gauge symmetry enhancement directions,
according to Table 1. We have SU(3) → SU(4 ) symmetry enhancement, yielding light
matter (
3, 1) at the matter curve P
(
3,1)
= 0. For example solving in b
6
and restoring
appropriately b
6
again, we have a splitting of E
2
.
E
2
→ E
2D
∪ E
2F
: e
2
= 0 = b
−2
3
b
−1
5
b
0
(b
3
b
6
e
0
+ b
5
ex)
×
b
3
b
5
b
6
e
2
0
e
1
+ b
5
(b
2
b
3
− b
0
b
5
)ee
0
e
1
x + b
2
3
e(b
5
y + ee
1
x
2
)
.
(43)
We have E
0
· E
1
= 1 as before and in addition we have intersections of new divisors
E
1
· E
2D
= 1 : e
1
= e
2
= b
5
x + b
3
b
6
e
0
= 0, (44)
E
1
· E
2F
= 0 : e
1
= e
2
= b
4
3
b
5
= b
5
x + b
3
b
6
= 0, {e
1
e, e
1
y} (45)
E
0
· E
2D
= 0 : e
0
= e
2
= x = 0, { xe
0
e
2
} (46)
E
0
· E
2F
= 1 : e
0
= e
2
= e
1
+ b
5
y = 0. {xe
0
e
2
} (47)
The representations are calculated in Table 5, where we must take the highest weight
[−1, 1; 0], not [0, 1; 0].
11
divisor weight
E
0
+ E
1
+ E
2F
[−1, 1; 0]
E
2F
[0, −1; 0]
E
1
+ E
2F
[−2, 0; 0]
divisor weight
E
G
[0, 0; 1]
E
G
+ E [0, 0; −1]
Table 5: The components of the matter representations (3, 1) at the matter curve P
(
3,1)
=
0, and (1, 2) at the matter curve P
(1,2)
= 0.
Matter curves for (1, 2) Another symmetry enhancement direction is P
(1,2)
= 0, shown
in Table 6. We use the equation for the divisor E after applying the constraint (19), now
reads as e
2
= −b
6
,
E : e = 0 = b
6
y
2
+ b
5
xy + b
4
x
2
+ b
3
w
′
y + b
2
w
′
x + b
0
w
′2
, (48)
after dropping −b
3
6
. It may degenerate into two divisors
E
G
∪ E
H
: e = 0 = (pw
′
+ qx + ry)(sw
′
+ ux + vy), (49)
where
b
0
= ps, b
2
= pu + qs, b
3
= sr + pv, b
4
= qu, b
5
= ru + qv, b
6
= rv,
satisfying the relatio n P
(1,2)
= 0 in a highly nontrivial way. We can a lways solve six
parameters p, q, r, s, u, v for as many b
i
’s. In fact, setting w
′
= 1 makes (48) to be identical
to spectral cover equation, in which the condition for local factorization (49) is precisely
the requirement for the existence o f the (1, 2, 20) representation of SU(3) × SU(2) ×
SU(6) ⊂ E
8
[10]. This is another evidence that the divisor equation for E (48) should be
obtained from
ˆ
P
′
T
not from
ˆ
P
T
.
Accordingly, each E
G
or E
H
provides a r epresentation for (1, 2), shown in Table 5.
We may check the intersection relations in the same way
E
G
· E
H
= 1 : e = pw
′
+ qx + ry = sw
′
+ ux + vy = 0 (50)
where we have nontrivial solution to the last two equations in x and y if
(ur − qv)
2
= b
2
5
− 4b
4
b
6
6= 0, (51)
which we have already met in (33) and assumed nonvanishing. As is
E
G
· E = E
G
· (E
G
+ E
H
) = −2 + 1 = −1, (52)
so holds the same relation f or E
H
, by the symmetry that we have no qualitative difference
between E
G
and E
H
. We see also there is no distinction between E
G
and E
H
group
theoretically, since (1, 2) is a self-conjugate representation. But this may be subject to
Freed–Witten global anomaly. This factorization and intersection structure of E indirectly
suggests that we should obta in E fro m the primed coordinates
ˆ
P
′
not
ˆ
P .
12
3 Abelian factor U(1)
The work by Mayrhofer, Palti and Weigand [27], following that o f Esole and Yau [28],
introduced a systematic way to obtain arbitrary number of U(1) gauge fields with desired
gauge quant um numbers, which we follow here (see also Ref. [25], which has similar
structure making use of heterotic/F-theory dua lity). The idea is to introduce a new
section than zero section in the elliptic fiber, by tuning the coefficients of the elliptic
equation. This is very similar and indeed related t o the factorization of the spectral
cover to which we can relate the group elements. Then we have a conifold singularity at
the new section. Resolving this new singularity will give rise to a new section S having
wished intersection structures with the existing resolution divisors for the desirable gauge
quantum numbers.
3.1 More global sections from factorization
We need one-forms or gauge fields A
1
of Abelian symmetries for either Cartan subalgebra
of non-Abelian groups or just U(1) groups. To realize them, we require harmonic two-
forms w
2
∈ H
1,1
(
ˆ
Y ) by which the M/F-theory three-form tensor is expanded as
C
3
=
X
A
1
∧ w
2
= A
e
1
1
∧ w
e
1
2
+ A
e
2
1
∧ w
e
2
2
+ A
e
1
∧ w
e
2
+ A
Y
1
∧ w
Y
2
+ · · · .
(53)
Here, w
e
i
2
are the dual two-forms to the divisors E
1
, E
2
or E of
ˆ
Y obtained from the
blowing- ups in the previous section.
We need two kinds o f requirement s fo r w
2
’s. (i) Each A
1
should have desired gauge
quantum number, so the Poincar´e-dual divisor of the paired w
2
in
ˆ
Y should have appro-
priate int ersections with other divisors. (ii) Every A
1
should be seven-dimensional vector
field, which restricts the index structure of the components of the pared w
2
[7]. For a
resolved Kodaira singularity, the blown-up cycles w
e
i
2
automatically possess Requirement
(i), seen from the intersection structure, seen in Section 2.3. However there is no Kodaira
singularity for an Abelian symmetry, or would-be-related I
1
singularity is actually smooth.
So we cannot do the resolution as in Section 2.2. We will see shortly that the desired
two-form is obtained from ano ther kind of singularity under a more special condition.
The goal of this section is to find w
Y
2
harboring the hypercharge satisfying Requirement
(ii).
An important hint is the relation between the elliptic equation and spectral cover
equation. The latter in a sense shows the algebraic relat ion in more suggestive form, since
the coefficients are directly related to combinations of weight vectors. The relation is best
13
seen by restricting the former on the hypersurface
X ≡ e
0
e
2
1
ex
3
− (e
0
e
1
e
2
+ b
6
)y
2
= e(e
0
e
2
1
x
3
− w
′
y
2
) = 0. (54)
It will be convenient to define t as
xe
1
t ≡ y, {e
1
y, xy} (55)
serving as a well-behaved holo mo r phic coordinate. It is because the condition t = 0 means
y = 0, and the opposite also holds because the SR elements indicated in (5 5) forbid x = 0
and e
1
= 0. So from now on we use t = ye
−1
1
x
−1
, under which (54) becomes
e
0
ex − (e
0
e
1
e
2
+ b
6
)t
2
. (56)
Putting this to
ˆ
P , we obtain
ˆ
P
X=0
= e
−3
0
e
1
e
2
w
′2
(b
0
e
6
0
z
6
+ b
2
t
2
e
4
0
z
4
+ b
3
t
3
e
3
0
z
3
+ b
4
t
4
e
2
0
z
2
+ b
5
t
5
e
0
z + b
6
t
6
). (57)
Unless b
6
= 0, E
0
has no intersection to X so we do not care about the factor e
−3
0
,
otherwise there is a cancellation and no overall factor in e
0
remains. Besides the prefactor,
the polynomial in z, in the parenthesis, is nothing but the spectral cover equation, whose
relation is discussed in Ref. [10, 27]. The F-theory compactification requires a global
section in the elliptic fiber, which is at (x, y, z) = (1, 1, 0) in our case and usually called
zero section Z. In the spectral cover description, the vanishing sum of five distinguished
points, for the unimodular g r oup SU, should be translated to the absence of z
5
term,
with our choice of the zero section.
In the spectral cover description, we have obtained a globally valid U(1) by ‘decompo-
sition’ [2 5]. Its first procedure is tuning of parameters (4)- ( 8) as follows [7,10,12,19,27,35].
b
0
= a
0
d
1
, b
2
= a
0
d
2
+ a
1
d
1
, b
3
= a
0
d
3
+ a
1
d
2
,
b
4
= a
0
d
4
+ a
1
d
3
, b
5
= a
0
d
5
+ a
1
d
4
, b
6
= a
1
d
5
.
(58)
In other words,
a
1
= (a
0
d
5
+ a
1
d
4
), (59)
a
2
= (a
0
d
4
+ a
1
d
3
)w, (60)
a
3
= (a
0
d
3
+ a
1
d
2
)(a
1
d
5
+ w)w, (61)
a
4
= (a
0
d
2
+ a
1
d
1
)(a
1
d
5
+ w)w
2
, (62)
a
6
= a
0
d
0
(a
1
d
5
+ w)
2
w
3
. (63)
And t he absence of a
5
or b
1
should be translated as a constraint
0 = a
0
d
1
+ a
1
d
0
.
14
matter parameters
P
X
a
1
P
q
◦
d
5
P
u
c
◦
a
3
1
d
2
+ a
0
a
2
1
d
3
+ a
2
0
a
1
d
4
+ a
3
0
d
5
P
d
c
◦
a
0
a
1
d
2
d
3
d
4
+ a
2
0
d
2
3
d
4
+ a
2
1
d
0
d
2
4
− a
0
a
1
d
2
2
d
5
−a
2
0
d
2
d
3
d
5
+ 2a
0
a
1
d
0
d
4
d
5
+ a
2
0
d
0
d
5
P
l
◦
a
5
1
d
5
d
2
0
− a
3
1
d
5
a
2
0
d
2
d
0
− 2a
1
d
5
a
4
0
d
0
d
4
− 3a
2
1
d
5
a
3
0
d
0
d
3
+a
3
0
d
0
a
2
1
d
2
4
+a
5
0
d
0
d
2
5
+a
4
0
d
2
a
1
d
4
d
3
−a
5
0
d
2
d
5
d
3
+a
3
1
d
0
d
4
a
2
0
d
3
+a
0
a
4
1
d
0
d
4
d
2
+ a
5
0
d
4
d
2
3
+ a
4
0
a
1
d
3
3
+ 2a
3
0
a
2
1
d
2
3
d
2
+ a
2
0
a
3
1
d
3
d
2
2
P
e
c
◦
−2d
1
a
4
1
+ d
2
a
0
a
3
1
− d
3
a
2
0
a
2
1
+ d
4
a
3
0
a
2
1
− d
5
a
4
0
Table 6: Defining equation for the matter curves.
Then the Tate p olynomial becomes factorized form
ˆ
P
X=0
= e
−3
0
e
1
e
2
w
′2
(a
0
e
0
z + a
1
t)(d
0
e
5
0
z
5
+ d
1
e
4
0
z
4
t + d
2
e
3
0
z
3
t
2
+ d
3
e
2
0
z
2
t
3
+ d
4
e
0
zt
4
+ d
5
t
5
).
≡ Y
1
Y
2
.
(64)
This will become the S[U(1)×U(5)] spectral cover equation in the heterotic side. However
to ensure this U(1) be global, we require another global section other than the zero
section [19,27]. In our case, indeed we have the new section is at X = Y
1
= 0. Since this
section is related to the ‘U(1) part’ Y
1
or the parameter a
1
/a
0
, we may expect this section
has appropriate structure for the hyperchar ge.
The factorization structure (64) may be expressed in another way [27]
ˆ
P
T
= XQ − Y
1
Y
2
= 0, (65)
by introducing a polynomial Q, holomorphic in z and t. This leads to a conifold singularity
at
X = Q = Y
1
= Y
2
= 0, (66)
which is o f higher codimension t han one. We blow up Y
1
= Q = 0 by introducing a P
1
with homogeneous coordinates (λ
1
, λ
2
) such that [28]
Y
1
λ
2
= Qλ
1
, Y
2
λ
1
= Xλ
2
. (67)
The origina l singularity (66) gives unconstrained λ
1
and λ
2
, which means it is replaced
by the P
1
. Away from the singularity, we recover the original equation (65) by solving λ
1
15
and λ
2
. In effect, the equations in (67) have redefined the Calabi–Yau space, which we
denote as
ˆ
Y again by abusing the nota tion, as a hypersurface in the new ambient space
including the P
1
. Then, we obtain an an extra section than zero section as the following
divisor in
ˆ
Y
S : λ
1
= 0, (68)
which f orces λ
2
be nonzero and gives Y
1
= X = 0 from (67). The lesson f rom the
spectral cover [25] tells us that Y
1
= 0 is related to the hypercharge U(1) as a subset of
the commutant S[U(1) × U(5)] to t he SM group in E
8
. The desired candidate for the
hypercharge (1, 1)- form is therefore Poincar´e-Hodge dua l to the threefold S in
ˆ
Y , up to
some correction we should consider below.
Next, we consider Condition (ii) below Eq. (53), for A
1
being a Lorentz vector. A
natural method is given in Ref. [27], which we follow here with similar notations. Such
forms w should satisfy following constraints
Z
ˆ
Y
w ∧ D
a
∧ D
b
∧ D
c
= 0, (69)
Z
ˆ
Y
w ∧ Z ∧ D
a
∧ D
b
= 0, (70)
from which the two indices of w have one leg on the elliptic fiber and the other leg on the
base B. Here the divisors D
a
, D
b
, D
c
of
ˆ
Y are pullbacks of arbitrary divisors in B. It is
far from trivial for the Cartan subalgebra elements E
1
and E
2
to satisfy the relation (70),
without choosing the SR ideal (21). For w we find the desired linear combination
w
Y
= S − Z −
¯
K + a
1
+
X
t
i
E
i
(71)
where
¯
K is the canonical class of the base B, a
1
is the divisor defined by a
1
= 0, and the
coefficients t
i
will be determined later. The general such pr ocess is called Shioda map [38 ].
The condition (69) is satisfied using t he relation
Z
ˆ
Y
S ∧ D
a
∧ D
b
∧ D
c
=
Z
ˆ
Y
Z ∧ D
a
∧ D
b
∧ D
c
=
Z
B
D
a
∧ D
b
∧ D
c
since both S and Z a r e global section. The next condition (70) is satisfied by (71) thanks
to the following. First, the intersection o f Z : z = 0 means Y
1
= a
1
t among the defining
equation of S. Using that xyz is an element SR ideal, the only nontriviality for intersection
S · Z comes from
Z
ˆ
Y
S ∧ Z ∧ D
a
∧ D
b
=
Z
B
a
1
∧ D
a
∧ D
b
.
Also t he adjunction formula states
Z
ˆ
Y
Z ∧ Z ∧ D
a
∧ D
b
= −
Z
B
K ∧ D
a
∧ D
b
.
16
3.2 More symmetry enhancement
The procedure described in the pr evious subsection introduces two new things: one is the
new section S and the o ther is further factorization of the matter curve. We calculate
the intersection of the divisors with S under various symmetry enhancement conditions
on matter curves. From this, we see that, although S did not originate from the Cartan
subalgebra of the SU(5) singularity or Kodaira I
5
, it plays exactly the same role for the
fourth generator of it, other than existing generators E
1
, E
2
, E.
Matter c urves for (3, 2) Under the parametrization (59)-(63) the matter curve equa-
tion P
(3,2)
= 0 in (9) further factors as
P
(3,2)
= P
q
P
X
= 0,
with different intersection structures of exceptional divisors for each factor shown in Table
6. We can directly compute the int ersections as, omitting λ
1
= D
a
= 0,
S · E
1A,q
= 0 : e
1
= y = −e
2
+ a
1
d
4
x = d
5
= 0, {ye
1
} (72)
S · E
2E,X
= 1 : e
2
= e = x = a
1
= 0, {xe
0
e
2
, e
0
e, e
1
e} (73)
S · E
2E,q
= 1 : e
2
= e = Y
1
= d
5
= 0, (74)
S · E
2B,X
= 1 : e
2
= e
0
= e
1
+ a
0
d
5
y = a
1
= 0, {xe
0
e
2
, e
0
e} (75)
S · E
2B,q
= 0 : e
2
= X = Y
1
= x
2
ee
1
+ a
1
d
4
y + (a
1
d
3
+ a
0
d
4
)e
0
e
1
x = d
5
= 0, (76)
S · E
2x,q
= 0 : e
2
= x = y = d
5
= 0. {xy} (77)
In calculating the intersection of Y
1
= a
0
e
0
+ a
1
t with x = 0 or e
1
= 0, the definition of t
is not valid so we need to restore its original form a
0
e
0
e
1
x + a
1
y.
We should remember that a lt hough we have local gauge symmetry enhancement, still
on
ˆ
Y the gauge symmetry is SU(3)×SU(2)×U(1), whose basis corresponds to E
1
, E
2
, E.
Thus we have a definite product between those with S. We should have a definite inter-
section S · E
1
= S · E
1A,q
= 0 so we should also have
S · E
1A,X
= 0.
As before, we have an invariant
S · E
2
= S · (E
2x
+ E
2E
+ E
2B
) = 1 (78)
calculated from the matter curve q. Since S · E
2
should be independent of the decompo-
sition, therefore we should have the same value for the X and we have
S · E
2x,X
= −1.
17
Also indirectly we can obtain the value. From the definition of the extended root E
0
, we
have linear dependence relation E
0C
+ E
1A
+ E
2x
+ E
2E
+ E
2B
= 0, fixing, for both X and
q,
S · E
0C
= −1.
Matter curves for (
3, 1) Now we go to the case of (3, 1). In t his case our facto r izat ion
is
P
(
3,1)
= P
u
c
P
d
c
= 0.
where each factor is again displayed in Ta ble 6. Also omitting λ
1
= D
a
= 0, we have
S · E
2D,u
c
= 1 : e
2
= b
3
b
6
e
0
+ b
5
ex = X = 0, (79)
S · E
2D,d
c
= 0 : e
2
= b
3
b
6
e
0
+ b
5
ex = X = Y
1
= 0, (80)
S · E
2F ,u
c
= 0 : e
2
= b
3
b
5
b
6
e
2
0
e
1
+ b
5
(b
2
b
3
− b
0
b
5
)ee
0
e
1
x + b
2
3
e(b
5
y + ee
1
x
2
) = X = Y
1
= 0
(81)
S · E
2F ,d
c
= 1 : e
2
= b
3
b
5
b
6
e
2
0
e
1
+ b
5
(b
2
b
3
− b
0
b
5
)ee
0
e
1
x + b
2
3
e(b
5
y + ee
1
x
2
) = X = 0.
(82)
While the constraint P
u
c
= a
2
1
b
3
+ a
2
0
b
5
= 0 makes the conditions in (79) automatically
solve the equation Y
1
= 0, it is not in the case of d
c
curve in (80). The same situation holds
for E
2F
. The rest of intersection is the same as in the previous case S ·E
0
= −1, S ·E
1
= 0.
There are still no ma tter curve for (1, 2) for the factorization (58); For generic a
i
’s
and d
i
’s, we cannot solve the six parameters in (49).
Matter curve for (1, 2) With the factorization at X = 0, one of E
G
and E
H
has a
solution in the form
E
G
: p + qx + ry = (a
0
e
0
z + a
1
t)(g
0
e
2
0
z
2
+ g
1
e
0
zt + g
2
t
2
) = 0, (83)
with t he constraint a
0
g
1
+ a
1
g
0
= 0. This is regarded as the definition of E
G
from now on,
and the other part E
H
is untouched. Thus the modified defining equation of E
G
contains
the factor Y
1
= 0 and the condition is redundant. This is the reason why we have no
further factorization of P
(2,1)
. Therefore we have the intersection structure
S · E
G
= 1 : P
(2,1)
= X = Y
1
= λ
1
= D
a
= 0, (84)
S · E
H
= 0. (85)
At the same time can distinguish the lepton doublet from the down-type Higgs doublet
by extra U(1) quantum numbers than hypercharge.
18
Matter curve for (1, 1 ) After the factorizatio n t o obtain the hypercharge symmetry
U(1)
Y
, we have a new charged singlet e
c
: (1, 1)
1
under the SM group SU(3) × SU(2)
L
×
U(1)
Y
, shown in Table 6. On the matter curve P
e
c
= 0, we have gaug e symmetry en-
hancement U(1)
Y
→ SU(2)
R
so that the resulting SU(3) × SU(2)
L
× SU(2)
R
is still a
subgroup of SO(10) [41].
We note that P
e
c
= 0 is contained in the complete intersection between Y
1
and Y
2
.
Around here, the fiber equation (65) locally has a binomial structure of a deformed Ko-
daira I
2
equation xy = z
1
z
2
, which describes not hing but this SU(2)
R
gauge symme-
try [21, 27, 28]. Therefore the small resolution gives the P
1
fiber (66 ) over the locus
P
e
c
= 0, which we now call S
′
. This S
′
will be related to the weight vector fo r e
c
. Away
from the intersection P
e
c
= 0 in the base B, its fiber described by (66) is already a P
1
,
which we call E
′
. These two have McKay correspondence of the affine SU(2)
R
, namely
S
′
·E
′
= 2. And we can show that S intersects the entire fiber S
′
+E
′
at a single point [27].
Therefore S
′
provides the desirable intersection giving the correct hypercharge of e
c
,
S · S
′
= −1. (86)
3.3 Hypercharge generator from the embedding
In the previous section, we have studied the intersections of the new divisor S in (68) with
various divisors, or, to be more precise, t he intersection numbers between their P
1
fibers
in the sense of (31). Altho ugh on various loci P
f
= 0 each of the divisors E
1
, E
2
, E may
further degenerate into many, we can recollect the results in terms o f the intersections
among E
1
, E
2
, E and S. For example, the relation (78) may be recollected as S · E
2
= 1
since this relation is independent on any specific locus D
a
= D
b
= 0 o n which we calculate.
Therefore we summarize the result as follows. We have McKay correspondence of
intersections of the P
1
fibers
E
1
E
2
S E
E
1
−2 1 0 0
E
2
1 −2 1 0
S 0 1 −2 1
E 0 0 1 −2
= −A
SU (5)
(87)
being the minus of the Cart an ma trix of SU(5). The divisor S provides the ‘fourth root’ of
SU(5). This is a good news, since the hypercharges are correctly given to the fields when
we choose S as the generator with a suitable normalization. The divisors E
1
, E
2
, E may
be blown down to zero size to recover nonabelian singularity SU(3) × SU(2). However
the divisor S cannot be blown down maintaining the fa cto rization ( 64), since we cannot
19
allow the conifold singularity with higher codimension. Therefore, at best we can have
the gauge group SU(3) × SU(2) × U(1), not the full SU(5).
However the Poincar´e–Hodge dual two- form to the divisor S is not exactly what we
want as hypercharge generator, since we need a disconnected ω
Y
≃ S from the other
group SU(3) × SU(2) as
Z
ˆ
Y
w
Y
∧ E
i
∧ D
a
∧ D
b
= 0 for E
1
, E
2
, E. (88)
We may form a linear combination of S with E
i
’s to have the desired property. This is
to find the coefficients t
i
in (71) in the Shioda map [38] done in the following mnemonics.
Take the inverse Cartan matrix A
−1
of the enhanced gauge symmetry of rank r + 1. The
Dynkin ba sis is defined to be and provides a convenient orthogona l relation between roots
α
i
and weights w
i
in a group under consideration
α
i
· w
j
= δ
j
i
, a
i
=
X
j
A
ij
w
j
,
where t he Cartan ma trix provides the product metric and the sum is done over all the
weights in that algebra. The symmetry breaking is described by deleting the jth no de of
the Dynkin diagram and the resulting unbroken symmetry with Carta n matrix being the
one with without the jth row and jth column. From the ortho normality relation, what
we need here is to take jth row of the inverted Cartan matrix as the coefficients t
i
of
linear combinations of root divisors E
i
. In our case
A
−1
SU (5)
=
1
5
4 3 2 1
3 6 4 2
2 4 6 3
1 2 3 4
.
The symmetry breaking SU(5) → SU(3)×SU(2)×U( 1) is done by removing 3rd row ( and
removing the extended root of the SU(5)). Therefore, we take the third row (2, 4, 6, 3),
neglecting the overall normalization 1/5, to obtain 2, 4, 6, 3 fo r coefficients of E
1
, E
2
, S,
and E, respectively. This always guar antee the integral charges under this U(1). We
finally have the hypercharge generator
w
Y
= −
S − Z −
¯
K − a
1
+
1
6
(2E
1
+ 4E
2
+ 3E)
, ( 89)
with the overall norma lization chosen according to the conventional charge. Applying this
to any component E
f
of each field f gives the hypercharge Y
f
=
R
E
f
w
Y
as
Y
q
=
1
6
, Y
X
= −
5
6
, Y
u
c
= −
2
3
, Y
d
c
=
1
3
, Y
l
= −
1
2
, Y
e
c
= 1. (90)
20
It is highly nontrivial that every component has different inner product with 6S and
2E
1
+ 4E
2
+ 3E, but their sum is always the same, as it should be for the components in
a same multiplet.
3.4 Further factorization
The model we have been building so far cannot be realistic f or some reasons below.
1. Even with a ‘1+5’ factorization giving the hypercha r ge U(1), we could not obtain
the massless field with the quantum number (2, 1) for generic parameters a
i
’s and
d
i
’s, since we have no solution to the equation for the desired weights (49). We may
hope that further tuning of these parameters may solve the problem.
2. The sp ectrum so far, listed in Table 6, cannot ta ke int o account the Higgs fields.
By f urt her factorization, we hope we can distinguish up and down Higgses by their
localization on different matter curves.
3. To have four dimensional chiral spectrum, we have to turn on G-flux. If we turn
on the universal G-flux along the entire ‘SU( 5)’ part
2
we have partial unification
relation of SU(5). That is, the SM field belonging to the same represent ation
of SU(5) has the same number of generations among themselves. For example
n
q
= n
u
c
= n
e
c
from 10 where n
f
is the number of generations of a matter field f. If
we want more strong unification relation, we may turn on G-flux along smaller part
than SU(5), for instance SU(4), which gives a larger commutant for the unification
relation, for instance of SO(10).
Note that always t he unbroken group here is the SM group SU(3) × SU(2) × U(1)
Y
,
purely determined by the tuning of t he parameters (58) of the elliptic equation,
regardless the choice of G-flux.
In spectral cover construction, it was shown that factorization with the spectral cover
S[U(3) × U(1) × U(1) × U(1)] is most realistic [31], and the same applies to our F-theory
version.
As before, we seek extra sections as subset of the variety X = 0. So we will require
the factorization of the elliptic equation in the form (we will drop the factor e
0
and z for
simplicity)
3
ˆ
P |
X=0
= (a
0
+ a
1
t)(b
0
+ b
1
t)(d
0
+ d
1
t)(f
0
+ f
1
t + f
2
t
2
+ f
3
t
3
) ≡ Y
1
Y
2
Y
3
Y
4
2
For convenience we call this SU (5) part as a commutant group of the SU(3) × SU(2) × U (1) in E
8
,
just borrowing the nomenclatures of spectral cover construction. The other commuta nt in this case is
the hypercharge U (1)
Y
since the abelian g roup commutes to itself.
3
In what fo llows we have new definitions on Y
i
’s a nd Q, etc. and they are not related to similar ones
in the previous section.
21
with the constraint
a
1
b
0
d
0
f
0
+ a
0
b
1
d
0
f
0
+ a
0
b
0
d
1
f
0
+ a
0
b
0
d
0
f
1
= 0.
Rewriting this again as XQ = Y
1
Y
2
Y
3
Y
4
by introducing a holomorphic polynomial Q, the
above four factors of Y
i
’s give rise to singularit ies. We know we shall have three U(1)’s
so we introduce as many P
1
’s with homogeneous coordina tes (λ
1
, λ
2
), (µ
1
, µ
2
), (ν
1
, ν
2
) and
choose a resolution as
Y
1
λ
2
= Qλ
1
, Y
2
µ
2
= µ
1
ν
2
, Y
3
λ
1
µ
1
ν
1
= λ
2
µ
2
, Y
4
ν
2
= Xν
1
. (91)
The resulting manifold is smooth since the Jacobian has the maximal rank. We are
content to verify that at least locally this reduces to the binomial resolution for three
factors, shown in R ef. [28]. When we have locally Y
2
≃ 1, we have µ
2
= µ
1
ν
2
. Plugging
them, the relations agree as
Y
1
λ
2
= Qλ
1
, Y
3
λ
1
ν
1
= λ
2
ν
2
, Y
4
ν
2
= Xν
1
.
When Y
3
≃ 1, we have µ
2
= λ
1
µ
1
ν
1
/λ
2
on one patch λ
2
6= 0 and we reproduce
Y
1
λ
2
= Qλ
1
, Y
2
λ
1
ν
1
= λ
2
ν
2
, Y
4
ν
2
= Xν
1
.
This also holds good on the other patch λ
1
6= 0.
Consequently, we have new exceptional hypersurfaces containing the sections X =
Y
i
= 0 for i = 1, 2, 3
S : λ
1
= µ
2
= ν
2
= 0 =⇒ X = Y
1
= 0, (92)
S
X
: λ
2
= µ
1
= ν
1
= 0 =⇒ X = Y
2
= Q = 0 , (93)
S
Z
: λ
2
= µ
2
= ν
2
= 0 =⇒ X = Y
3
= Q = 0 . (94)
The two extra U(1) charges we call X and Z. This S divisor has essentially the same
definition as that in the previous factorization (68), having the same g roup a nd Lorentz
properties. As before, the newly found divisors S
X
and S
Z
provide ‘missing’ Cartan
subalgebra of SU(5) or SO(10), respectively. We can recycles the U(1) generators w
Y
,
since the latter satisfies all the requirement of Cartan subalgebra (88) and the whole
generators satisfy desirable conditions (69), (70). Thus we find the new generators with
normalization
w
X
=5(S
X
− Z −
K − b
1
) + 2E
1
+ 4E
2
+ 6E + 3(−6w
Y
), (95)
w
Z
=4(S
Z
− Z −
K − d
1
) + 2E
1
+ 4E
2
+ 6E + 5w
X
+ 3(−6w
Y
), (96)
22
where the factor −6 in f r ont of w
Y
is due to the special fractional convention of hy-
percharge. Since S
X
and S
Z
are going to belong to SO(10) and E
6
, respectively, the
coefficients are also found from the inverted Carta n matrices
A
−1
SO(10)
=
1
4
4 4 4 2 2
4 8 8 4 4
4 8 12 6 6
2 4 6 5 3
2 4 6 3 5
, A
−1
E
6
=
1
3
4 5 6 4 2 3
5 10 12 8 4 6
6 12 18 12 6 9
4 8 12 10 5 6
2 4 6 5 4 3
3 6 9 6 3 6
Further generalization is straightfor ward. The spectrum of the fields and the correspond-
ing charges are shown in Table. 1 in Ref. [31].
4 Comment on gauge coupling unification
With localization of each gauge theory on a complex surface S
4
in B, a part of eight
dimensional worldvolume, we have the following field theory limit having dimensional
reduction [6, 7]
−
e
−φ
(2πα
′
)
4
Z
S
4
×R
4
d
8
xF
2
8D
= −
Vol S
4
4g
2
YM
Z
R
4
d
4
xF
2
4D
+ · · · , (97)
with t he vacuum expectation va lue of the dilaton e
φ
becoming string coupling. In the IIB
string theory limit, this Vo l S
4
is interpreted as the effective volume of the cycle wrapped
by dynamical severbranes with both NSNS and RR charges.
The volumes of S
(3)
and S
(2)
respectively spanned by the SU(3 ) locus W ≡ E
0
: e
0
= 0
(in B) and the SU(2) locus W
′
: w
′
= 0 are related, using (87).
VolS
(3)
=
1
2
Z
S
(3)
J ∧ J =
1
2
Z
B
W ∧ J ∧ J = −
1
2
Z
ˆ
Y
E
2
∧ S ∧ J ∧ J
= −
1
2
Z
ˆ
Y
E ∧ S ∧ J ∧ J =
1
2
Z
B
W
′
∧ J ∧ J =
1
2
Z
S
(2)
J ∧ J = VolS
(2)
.
(98)
where J is the K¨ahler form of
ˆ
Y . Therefore we have the same worldvolume for these
two non- Abelian gauge groups. Here the calibrated geometry plays a role: the effec-
tive volumes are g iven by intersection numbers, not depending on scaling factors of the
coordinates.
The gauge coupling of the hypercharge U(1) can be readily determined in the relatio n
to the enhanced group such as SU(5). It should be a global limit b
6
= 0, i.e. not local
23
gauge symmetry enhancement on matter curves
ˆ
P |
b
6
=0
=e
2
1
e
3
2
x
3
e
3
e
1
− y
2
e
2
e
2
+ b
5
e
2
xyz + b
4
e
0
e
1
e
2
x
2
z
2
+ b
3
e
2
0
e
1
e
2
eyz
3
+ b
2
e
3
0
e
2
1
e
2
exz
4
+ b
0
e
5
0
e
3
1
e
2
2
z
6
]
(99)
with the tuned para meters b
i
in (58). It is happy to see that in this limit, the two-
cycles e
0
, e
1
, e
2
are identical to those in the standard resolution of SU(5) singularity I
5
(See, e.g. [27], after renaming e
2
→ e
4
). The woldvolume is provided by the divisor
W
0
: e
0
=
ˆ
P
T
|
b
6
=0
= 0, which is the same as W of the SU(3). Thus
VolS
(5)
=
1
2
Z
S
(5)
J ∧ J =
1
2
Z
B
W
0
∧ J ∧ J =
1
2
Z
B
W ∧ J ∧ J = VolS
(3)
. (100)
The fact that W, W
′
, W
0
have the same volume is obvious since the SU(3) and the SU(2)
are obtained by deforming SU(5) singularity. It is not affected by another deformation
arising from the resolution S of the conifold singularity. The volume of the P
1
fiber of S
cannot be nonzero so there cannot be unbroken SU(5). Nevertheless the gauge couplings
are unaffected by the volume of this P
1
and in the low energy limit, we just have heavy
X , Y gauge multiplets.
In this limit, S provides the Cartan subalgebra element related to hypercharge, as
seen in the relation (87) thus
−
1
4g
2
trF
2
SU (5)
= −
1
4g
2
(trF
2
SU (3)
+ trF
2
SU (2)
+ trF
2
U(1)
),
where we defined the ga uge field as matrix valued A
M
= A
a
M
t
a
, trt
a
t
b
=
1
2
δ
ab
. In particular,
from t he Cartan matrix (87), the generator S is related to the Cartan element with
t =
1
√
60
diag(2, 2 , 2, −3, −3). The two-cycle w
Y
is j ust a modification of that of S, and the
linear transformatio n within the same group SU(5) (otherwise even the definition (89)
does not make sense) is just a transfor ma t ion not affecting the g auge coupling. Thus the
gauge coupling of the hypercharge U(1)
Y
should be related by group theory of the unified
group SU(5) rembedding SU(3) × SU(2) × U(1)
Y
, in the standard way. Normalizing the
U(1) cha r ge of the e
c
to be 1, we fix the coupling as
g
2
= g
2
3
= g
2
2
=
3
5
g
2
Y
,
with the weak mixing angle at this string theory scale
sin θ
0
W
=
g
2
Y
g
2
2
+ g
2
Y
=
3
8
,
consistent with the observation. For any U(1) having an embedding to a certain GUT,
we may use this method, however it is an open question whether every U(1) obtainable
in F-theory has such embedding.
24
To this coupling relation, we have threshold correction, if there is a nontrivial G-flux
along a certain U(1) direction. When we construct the SM group at the string scale,
we should not turn on G-flux alo ng the hypercharg e direction if we want it to be gauge
symmetry. For GUT such as SU(5), we may break it by turning on G-flux without
breaking gauged hypercharge [6, 8]. Other U(1) symmetries constructed for the realistic
model building, it is desirable to broken down by G-flux. Then by St¨uckelberg mechanism,
the corresponding gauge boson acquires mass a nd the symmetry becomes global. There
are also threshold correction to it f r om the flux [8, 39,40 ].
5 Conclusion
We a nalyzed the Standa r d Model gauge group SU(3) × SU(2) × U(1), and also its ac-
companying matter fields, constructed in F-theory, using resolution procedures. The
non-Abelian part SU(3) × SU(2) is described by the singularities of Kodaira type, which
locally looks like I
3
and I
2
, as nontrivial deformation of the SU(5) singularity I
5
. They are
respectively supported at different divisors w = 0 and w
′
= 0, which are related by a co-
ordinate transformation (19), nevertheless described by single elliptic equation (20). The
resolution a nalysis revealed that the SM group should be distinguished to na¨ıve product of
SU(3) and SU( 2), since the two groups are connected by coordinate transformations and
the blowing-ups cannot be done independently. On the matter curves, there are ga uge
the symmetry enhancements to var ious unified groups, and the exceptional divisors from
different simple groups mix in some particular way, yielding matter fields having desired
charges. This desirable feature is present only if the SM group is embedded in E
n
series
group.
The Abelian part U(1) is obtained by ‘factorization method’ making use of an extra
section in the elliptic fiber of an internal manifold. At a particular restriction X = 0,
the factorization of the elliptic equation is related to gauge symmetry enhancement in
certain g roup direction. The resolution at the conifold singular ity originating from this
factorization gives rise to the two-form harboring the desired gauge group, having the
correct assignment of U(1) charges. This new two-form and the corresponding divisor
should be understood in terms of a certain unified group, and from which the conventional
SU(5) gauge coupling unification relation is achieved if no flux is turned on the U(1) part .
We hope that this analysis provides a complete proof to the SM singularity suggested
before: Either by relating to t he spectral cover in the heterotic dual limit and explicit
calculation of the charges of the matter fields. Gauge coupling unification can be anot her
good clue for the model building, which is not shared by other models in the similar
context having an intermediate Grand Unification. We may apply this method to a direct
25
construction of t he Standard Model in the native F-theory context. Mathematically, the
appearance of an extra U(1) as an element of Cartan subalgebra in a larger unified group
is very suggestive, so it would be interesting to extend the wor k to find more systematic
method to find groups involving multiple U(1)’s.
Acknowledgements
The author is grateful to Ralph Blumenhagen, Thomas G r imm, Stefan Groot-Nibbelink,
Hirotaka Hayashi, Seung-Joo Lee, Christopher Mayrhofer, and Timo Weigand for dis-
cussions and correspondences. This work is pa r tly supported by the National Research
Foundation of Korea with grant number 2012-R1A1A1040695.
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