Geometrically Engineering the Standard Model:
Locally Unfolding Three Families out of E8
Jacob L. Bourjaily∗
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(Dated: 3rd April 2007)
This paper extends and builds upon the results of [1], in which we described how to use the tools
of geometrical engineering to deform geometrically-engineered grand unified models into ones with
lower symmetry. This top-down unfolding has the advantage that the relative positions of singular-
ities giving rise to the many ‘low energy’ matter fields are related by only a few parameters which
deform the geometry of the unified model. And because the relative positions of singularities are
necessary to compute the superpotential, for example, this is a framework in which the arbitrariness
of geometrically engineered models can be greatly reduced.
In [1], this picture was made concrete for the case of deforming the representations of an SU5
model into their Standard Model content. In this paper we continue that discussion to show how
a geometrically engineered 16 of SO10 can be unfolded into the Standard Model, and how the
three families of the Standard Model uniquely emerge from the unfolding of a single, isolated E8
singularity.
I. INTRODUCTION
In [2], Katz and Vafa showed how to geometrically
engineer matter representations in terms of the local
singularity structure of type IIa, M-theory, and F-
theory compactifications. In that framework matter
and gauge theory both have purely geometrical ori-
gins: SUn, SO2n and En gauge theories arise from
the existence of co-dimension four singular curves of
certain types in the compactification manifold [3];
and massless matter representations arise from iso-
lated points (in type IIa or M-theory) or curves (in
F-theory) along the singular surface over which the
type of singularity is enhanced by one rank.
Despite the extraordinary generality of this frame-
work, it has not been widely used phenomenologi-
cally. This is largely because the description of the
isolated enhancements of singularities giving rise to
various matter representations is inherently local:
although the geometry near any particular enhance-
ment could be described concretely, the framework
had nothing to say about numbers, types, and rela-
tive locations of different matter fields. This global
data was either to be determined by duality to a
concrete, global string theory model1, or suggested
via the a posteriori success of a given set of relative
positions (as in e.g. [5, 6]).
Another way to relate the number and relative
positions of (enhanced singularities giving rise to)
matter fields was given in [1]: in that paper, we
described for example how a local description of the
geometry giving rise to a massless 5 of SU5 could
be smoothly deformed into a local description of a
(3,1) and a (1,2) of SU3×SU2 which live at distinct
∗Electronic address: jbourjai@princeton.edu
1 Geometrically engineered models in M-theory are dual to
intersecting brane models in type II, (see e.g. [4]).
points—related by a single deformation parameter.
A cartoon of what was described in that paper is
shown in Figure 1.
In this paper we describe pedagogically how to
extend that idea to engineer analogies to SO10 and
E6 × SU2 grand unified models
2. Although in [1]
we were able to analyze explicit unfoldings of SO10
and SU6 singularities sufficiently well by sight, this
will not be possible for our present examples. All
of the examples in this paper involve the unfolding
of isolated En singularities; and although algebraic
descriptions of these are known and classified [7], it
would be unnecessarily cumbersome and unenlight-
ening to analyze them explicitly as we did in [1].
Therefore, in section II we describe a much more
powerful and elegant language in which to study
these resolutions.
In section III we describe in detail how the un-
folding of a 16 of SO10 into the Standard Model is
derived in the language of section II. This is achieved
in two stages: in the first stage, we unfold the 16 into
10⊕5⊕1 of SU5; we then unfold the resulting SU5
model into a single ‘family’ of the Standard Model.
At the end, all the relative positions of the singu-
larities of the family are set by the non-zero values
of two complex structure moduli, thereby greatly re-
ducing the arbitrariness of their relative positions.
The next most obvious example would be a de-
scription of how a 27 of E6 geometrically unfolds
into the Standard Model. However, there are two
reasons to leave this example to the reader: first, it is
a most natural extension of the results of section III;
secondly, it is a consequence of the E6 × SU2 grand
unified model which we describe in section IV. Al-
2 As described in section IV, the resolution E8 → E6 × SU2
naturally starts as a theory with three 27’s of E6 related
by an SU2 family symmetry.
http://arxiv.org/abs/0704.0445v2
mailto:jbourjai@princeton.edu
FIG. 1: A cartoon of the geometric deformation of a 5 and 10 of SU5 into the Standard Model as described in [1].
The surface over which the singularities are enhanced is coordinatized by a complex parameter t and the geometry
is deformed by changing the value of a complex parameter s. The relative locations of the ‘resolved’ singularities are
given in terms of s.
though not given as an example in [2], it is not hard
to see3 that a single isolated E8 singularity at the
intersection of a co-dimension four surfaces of types
E6 and SU2 gives rise to matter in the representa-
tion (27,2)⊕ (27,1) ⊕ (1,2). It is easy to see how
this would unfold into the matter content of three
families—one coming from each of the 27’s. That
three families emerge from E8 is a general conse-
quence of group theory and can be understood from
the fact that E6 ×SU3 is a maximal subgroup of E8
into which the adjoint of E8 partially branches into
an SU3 triplet of 27’s.
As in the preceding paper [1], this work is pre-
sented concretely in the language of Calabi-Yau
compactifications of type IIa string theory, which
can also be naturally extended to F-theory mod-
els. Here, we engineer the explicit local geometry
of (non-compact) Calabi-Yau three-folds which are
K3-fibrations over C1. If type IIa string theory is
compactified on this three-fold, a four-dimensional
N = 2 theory with various massless hypermulti-
plets will result. But if, for example, the C1 base of
this three-fold were fibred as an O(−2) bundle over
1, the resulting total space would be a Calabi-Yau
four-fold4 upon which F-theory would compactify to
an N = 1 theory with chiral multiplets. However,
because the manifold over which the singular K3’s
are fibred in M-theory is a real, three-dimensional
space, our fibrations over C1 do not have a direct
application to M-theory.
It would of course be desirable to have a similar
description of geometric unfolding explicitly in the
language of G2-manifolds so that this picture could
be realized concretely in M-theory as well. This is
particularly important in light of the recent advances
in M-theory phenomenology (e.g. [8, 9]). By exten-
3 This is described in section II.
4 This is just one example of the ways in which these Calabi-
Yau three-folds could be fibred over CP1 to result in a
Calabi-Yau four-fold.
sion of the work of Berglund and Brandhuber in [10],
such a generalization should be relatively straight-
forward, but we will not attempt to do this here.
II. RESOLVING En-TYPE SINGULARITIES
Recall that a gauge theory in type IIa string the-
ory can arise from compactification to six dimen-
sions over a singular K3 surface (similar statements
apply to M-theory and F-theory) [3]. The complex
structures of the singular compactification manifolds
giving rise to SUn(≡ An−1), SO2n(≡ Dn), and En
gauge theory are given in Table I—where the sur-
faces are labelled conveniently by the name of the
resulting gauge theory5.
We can generalize this discussion by considering
a complex, one-dimensional space B over which a
smooth family of singular K3 surfaces are fibred. If
almost everywhere over B the K3-fibres have singu-
larities of a single type, then compactification of type
IIa string theory over the total space will give rise
Gauge group Polynomial
SUn (≡ An−1) xy = z
SO2n (≡ Dn) x
2 + y2z = zn−1
2 = y3 + z4
2 + y3 = 16yz3
2 + y3 = z5
TABLE I: Hypersurfaces in C3 giving rise to the desired
orbifold singularities.
5 It is of curious historical interest that the equations listed
in Table I were first identified by Fleix Klein in 1884 [11].
The reader may also be amused that the full resolutions of
these surfaces were almost completely classified—up to a
few computational errors—by Bramble in 1918 [12].
to gauge theory in four-dimensions of the type corre-
sponding to the typical fibre. Massless charged mat-
ter will arise if over isolated points in B the type of
fibre is enhanced by one rank. The geometry about
a single such isolated point where the singularity is
enhanced was described in detail by Katz and Vafa
in [2].
The representation of matter living at these ‘more-
singular’ points was also given in [2]: suppose that
G ⊃ H ×U1 and that the rank of H is one less than
G; then, if there is an isolated G-type singularity
over a surface of H-type singularities, the resulting
massless representation is given by those parts of the
decomposition of the adjoint of G into H×U1 which
are charged under the U1.
Because the question of how to (smoothly) deform
the surfaces of Table I into ones of lower rank has in-
trinsic mathematical interest, it is not too surprising
that all possible two-dimensional deformations have
been classified. Our discussion below will make use
of the notation and results presented in [7].
In our present work, we are interested in defor-
mations of En singularities into ones of lower rank.
Unlike SUn singularities, the resolutions of which are
easy enough to read off by sight, the algebraic com-
plexity of En singularities is formidable. To appre-
ciate what is meant by this, consider the resolution
of E7. From Table I we know that an E7 singularity
is locally isomorphic to the surface x2 + y3 = 16yz3
in C3. Its full resolution in terms of the seven defor-
mation parameters ~t = (t1, t2, . . . , t7) is given by
− x2 − y3 + 16yz3 + ǫ2(~t)y
2z + ǫ6(~t)y
2 + ǫ8(~t)yz + ǫ10(~t)z
2 + ǫ12(~t)y + ǫ14(~t)z + ǫ18(~t) = 0, (1)
where the ǫn(~t) are n
th order symmetric polynomi-
als in the components of ~t which are tabulated over
several pages of the appendix of [7].
A näıve way to determine the type of singularity
found by resolving E7 “in the direction ~t” would be
to expand equation (1) completely using the explicit
functions ǫn(~t), find each of its singular points, and
expand locally about each until an isomorphism with
a singularity of lower rank in Table I was clear. This
is the way, for example, that [2] demonstrated that
the resolution of E7 in the direction (0, 0, 0, 0, 0, t, 0)
gives rise to E6 for t 6= 0. All of the results in this
paper could be verified in this way. Luckily, however,
Katz and Morrison described a much more powerful
and direct way to analyze the deformations of En
singularities [7].
We would like a pragmatic answer to the follow-
ing question: what is the type of fibre found by re-
solving an En singularity in the direction ~t? That
there is an easy answer to this question makes our
work much simpler. Although an adequate treat-
ment would take us well beyond the scope of our
present discussion, the answer given in [7] is at least
very easy to make use of6: for each of the equations
in Table II satisfied by the components of ~t, the sin-
gularity has the corresponding root. Given the list of
roots, it is then a straight-forward exercise to con-
struct the Dynkin diagram corresponding to the sin-
gularity7.
6 Of course, this answer does depend on the parameterization
used. As stated before, we are working with the conventions
of [7].
7 Misusing the notation of [7] in a way applicable only to
In an admittedly bad notation, we consider each of
the n deformation parameters ti(t) to be functions
of t, the local coordinate on the base space B. A
(non-Abelian) gauge theory will be present if there
are roots implied by Table II which are preserved
for generic values of t. And charged massless matter
will exist if at isolated points {t∗} an additional root
is added—or, in terms of Dynkin diagrams, if an
additional node is added. At each isolated point we
can therefore identify the resolution G → H and
thereby determine the resulting representation.
Equation Root
ti − tj = 0 =⇒ ei − ej
ti + tj + tk = 0 =⇒ e0 − ei − ej − ek
tij = 0 =⇒ 2e0 −
2ti1 +
tij = 0 =⇒ 3e0 − 2ei1 −
TABLE II: The roots of the singularity resulting from the
resolution of En in the direction~t. This is a reproduction
of Table 4 of Ref. [7].
SUn, SO2n and En, one can think of the vectors ei as an or-
thonormal basis in Minkowski space which is equipped with
a mostly-plus metric. Then roots are vectors in this space
of norm +2. Each (positive) root gives rise to a node in the
resulting Dynkin diagram, and two nodes are connected by
a line if their inner product is −1 and disconnected if they
are orthogonal.
III. GEOMETRIC ANALOGUE OF SO10
GRAND UNIFICATION
A. The Description of a 16 of SO10
A necessary starting point to describe the unfold-
ing of a 16 of SO10 into the Standard Model is a
description of the initial geometry as was done in [2].
We will briefly review that construction in the lan-
guage described above before we unfold it, first into
an SU5 model, and later all the way into SU3×SU2.
Let t be a local complex coordinate on the space
B over which is fibred the resolution of E6 parame-
terized by ~t = (t, t, t, t, t,−2t). To be clear, for each
value of t, the vector~t describes an explicit surface in
3 given in reference [7] analogous to that of equa-
tion (1) above.
Considering the rules of Table II, we see that for
an arbitrary value of t 6= 0 the root lattice of the
fibre is
(e0−e1−e2−e6)
(e1 − e2) (e2 − e3) (e3 − e4) (e4 − e5)
where we have displayed the roots suggestively so as
to reproduce the SO10 Dynkin diagram. At t = 0,
however, E6 is restored. So we have an isolated E6
fibre over the point t = 0, while for any t 6= 0 the
fibre is SO10. This gives rise to SO10 gauge theory
with a single massless 16 located at the origin in the
t-plane.
B. Unfolding the 16 of SO10 into SU5
We would like to unfold the manifold described
above into one with SU5 gauge theory. It is not
hard to guess in what ‘directions’ ~t we may deform
the the geometry so that the fibre over a generic
point is SU5. Let a denote a parameter independent
of t which adjusts the whole geometry over the re-
gion which is coordinatized by t. Then let the fibre
over t be given by the resolution of E6 in the direc-
tion (t, t, t, t, t+ a,−2t− a). Obviously when a = 0
the situation is the same as above and results in a
single massless 16 of SO10. However, when a 6= 0
the situation is different: for generic values of t it is
easy to see that the simple roots are
(e0−e1−e5−e6) (e1−e2) (e2−e3) (e3−e4)
which means that the generic fibre over t is just
SU5—and so the resulting gauge theory is SU5.
To find what matter representations exist, we
must determine over which locations t the rank of
the fibre is enhanced. This means we are seeking
special values of t (determined by a) at which an ad-
ditional equation in Table II is satisfied. For each of
these points, we can draw the resulting Dynkin dia-
TABLE III: The locations on the complex t-plane over
which the singularity of the fibre is enhanced, and the
representations of SU5 × U1 that result.
Location Fibre
Representation
of SU5 × U1
3t + 2a = 0 SU5 × SU2 1−5
3t+ a = 0 SO10 10−1
t = 0 SU6 53
gram to determine the fibre over that point, thereby
determining the representation which arises there.
It is not hard to exhaustively find all these ‘more
singular’ points. They are give in Table III. Notice
that we have included the U1-charge assignments
that result; these are normalized as in the appendix
of [13].
C. Unfolding a 16 of SO10 into the Standard
Model
To complete our task and unfold the 16 of SO10
all the way to the Standard Model, we must deform
the fibres by another ‘global’ parameter, which we
will denote b. It is not hard to guess a direction over
which the generic fibre will be SU3 × SU2: try for
example (t, t, t, t + b, t + a,−2t− a − b). Again, we
notice that for a general location t and generic fixed
values a, b 6= 0, the singularity has the root structure
(e1 − e2) (e2 − e3)
(e0 − e4 − e5 − e6)
, (4)
which is visibly SU3 × SU2.
Like above, it is a straight-forward exercise to de-
termine all the locations over which the singularity
is enhanced, and the resulting representation which
arises. These points including their resulting rep-
resentations (with U1-charges as normalized in [13])
are listed in Table IV. The entire unfolding is repro-
duced graphically in Figure 2.
Location Fibre Representationof SU3×SU2×U1 Name
3t + 2a+ b = 0 SU3 × SU2 × SU2 (1,1)0 ν
3t + a+ 2b = 0 SU3 × SU2 × SU2 (1,1)6 e
3t+ a+ b = 0 SU5 (3,2)1 Q
3t + a = 0 SU4 × SU2 (3,1)−4 u
3t+ b = 0 SU4 × SU2 (3,1)2 d
t = 0 SU3 × SU3 (1,2)−3 L
TABLE IV: The locations on the complex t-plane over
which the singularity of the fibre is enhanced and the
representations of SU3 × SU2 × U1 that result.
FIG. 2: An illustration of the resolution of a geometrically engineered 16 of SO10 into the Standard Model as a
function of two complex structure moduli a and b as described in section III. The coordinate along the base space is t
and runs vertically in the diagram. When a = b = 0, along the left-hand side, there is just one isolated E6 singularity
at t = 0. When b = 0 but a is allowed to vary, this single singularity splits into three, and any a =constant slice will
have three isolated singularities in the complex t-plane as shown above. Moving rightward in the diagram, at the
dashed line a is held fixed and b is allowed to grow, causing the three enhancements of SU5 to break apart into six
total isolated singularities over SU3 ×SU2, which is shown on the right-hand-side. Also shown are the (appropriately
normalized) U1 charges of fields obtained via this multiple unfolding.
IV. GEOMETRIC ANALOGUE OF E6 × SU2
GRAND UNIFICATION
After having completed the unfolding of a 16 of
SO10 into the Standard Model, it is natural to ask if
this idea can be extended to relate all the singulari-
ties of the Standard Model as perhaps the unfolding
of a single isolated singularity of higher-rank. The
answer is in fact yes—and there is a sense in which
precisely three families arise if the notion of ‘geomet-
ric unification’ is saturated.
Because a 16 of SO10 arises from the resolution
E6 → SO10, it can only be unfolded out an ex-
ceptional singularity. Clearly the highest level of
unification one can achieve along this line would
be to start with a resolution E8 → H where H is
a rank-seven subgroup of E8 which contains SO10.
The possible ‘top-level’ gauge groups are then E7,
E6 × SU2, and SO10 × SU3. We choose to study
E8 → E6 × SU2 as our example because it will nat-
urally include a description of the unfolding of 27 of
E6 into the Standard Model, which is interesting in
its own right, and because it follows quite directly
from our work in section III.
The initial geometry which we will deform into the
Standard Model is given as follows. Let t be a com-
plex coordinate on the base space B over which is fi-
bred the resolution (t, t, 0, 0, 0, 0, 0, 0) of E8. Clearly,
when t = 0 we recover E8; when t 6= 0 we see that
the roots of the fibre are
(e0−e3−e4−e5)
(e3 − e4) (e4 − e5) (e5 − e6) (e6 − e7) (e7 − e8)
(e1 − e2)
, (5)
which is visibly E6 × SU2. Following the general
rule to determine the representation resulting from
a given resolution [2], we find that at t = 0 lives
massless matter charged in the (27,2)1⊕(27,1)−2⊕
(1,2)3 representation of E6 × SU2 × U1ϕ .
To avoid pedantic redundancy, in Figure 3 we have
summarized in great detail the entire unfolding into
SU3×SU2×U1Y ×U1χ×U1Ψ×U1ζ×U1ϕ . An outline
of the steps involved in deriving this unfolding is
given presently.
First, the unfolding of the E6 × SU2 gauge the-
ory into E6 gauge theory is obtained by defining
the fibre over t to be given by the resolution of E8
in the direction (t + a, t − a, 0, 0, 0, 0, 0, 0) for some
a 6= 0. This clearly kills the SU2 node of the fibre
in equation (5). There are five locations at which
the singularity is enhanced by one rank, giving rise
to three 27’s and two singlets as shown in the left-
most section of Figure 3.
FIG. 3: An illustration of the resolution of a single isolated E8 into the Standard Model in terms of four deformation
parameters a, b, c, d. Along the left hand side, for a = b = c = d = 0, the generic E6 × SU2 fibre is enhanced to E8
at t = 0. Moving from left to right, a, b, c, d are sequentially allowed to grow to some non-zero value—and between
dashed lines all but one of the moduli are held fixed. Solid lines indicate the locations of enhanced singularities
relative to the plane for as functions of a, b, c, d. The complete list of isolated singularities, their locations, and charge
assignments are given on the right hand side of the diagram.
The rest of the unfolding is a natural appli-
cation of the work in section III. Let us now
set the fibre over t to be given by the resolution
(t+ a, t− a, b, b, b, b, b,−2b) of E8 for arbitrary com-
plex deformation parameters a, b 6= 0. From section
III we see immediately that the generic fibre is SO10.
A thorough scanning for possible solutions to equa-
tions in Table II shows that there are 11 isolated
points on the complex t-plane over which the singu-
larity is enhanced. These correspond to the ‘break-
ing’ of each 27 of E6 into 16 ⊕ 10 ⊕ 1 of SO10,
while the singlets remain singlets. This is seen in
the second vertical strip (from the left) in Figure 3.
Again, following our discussion above, it is easy
to guess possibilities for the next two resolution di-
rections. First, we set the fibre over t to be given by
(t+ a, t− a, b, b, b, b, b+ c,−2b− c) which will result
SU5 gauge theory with matter content correspond-
ing to the ‘canonical’ decomposition of three 27’s of
E6 with two singlets. And finally, the full resolution
of the E6×SU2 grand unified model into SU3×SU2
can be given by letting the fibre over t be given by
the (t+a, t−a, t, t, t, t+d, t+c,−2b−c−d) resolution
of E8 for (generic) arbitrary fixed complex structure
moduli a, b, c, d 6= 0.
V. IMPLICATIONS
Let us clarify what we have done. For a given set
of fixed, nonzero complex structure moduli, the reso-
lution given above describes the explicit, local geom-
etry of a non-compact Calabi-Yau three-fold, which
is a K3-fibration over C1. If type IIa string theory
is compactified on this three-fold, the resulting four-
dimensional theory will have SU3 × SU2 gauge the-
ory with hypermultiplets at isolated points as given
in Figure 3 which reproduce the spectrum of three
families of the Standard Model with an extended
Higgs sector and some exotics. Alternatively, if one
takes this (non-compact) Calabi-Yau three-fold and
fibres it over CP1 as described in section I so that
the total space is Calabi-Yau, then F-theory on this
space will give rise to N = 1 supersymmetry with
SU3×SU2 gauge theory and chiral multiplets in the
representations given in Figure 3. And although it
does not follow directly from our construction above,
considering the close similarities between two- and
three-dimensional resolutions of the singularK3 sur-
faces we have every reason to suspect an analogous
geometry can be engineered for M-theory in terms
of hyper-Kähler quotients by extension of the results
in [10, 14, 15]. We are currently working on building
this geometry in M-theory, and we expect to report
on this work soon.
Given these four complex structure moduli, all
the relative positions of the 35 disparate singulari-
ties giving rise to all three families of the (extended)
Standard Model are then known8. Beyond the usual
three families of the Standard Model, the manifold
also gives rise to two Higgs doublets for each fam-
ily, six Higgs colour triplets, three right-handed neu-
trinos and five other Standard Model singlets. We
should point out that this matter content (and their
U1-charge assignments) is a consequence of group
theory and algebraic geometry alone—it is simply
what is found when unfolding E8 all the way to the
Standard Model.
And given the relative positions and local ge-
ometry of the singularities together with the U1-
structure, one can in principle compute the full su-
perpotential coming from instantons wrapping dif-
ferent singularities. Because these are fixed by the
values of the complex structure moduli, there is a
(complex) four-dimensional landscape9 of different,
explicit SU3 × SU2 embeddings at the compactifi-
cation scale. Although this large landscape may
appear to have too much freedom, we remind the
reader that in the traditional understanding of ge-
ometrical engineering there would be hundreds of
parameters describing the (independent) relative lo-
cations of each of the isolated singularities.
There are a few things to notice about the form of
the superpotential that will emerge. First, because
of the U1-charge assignments, each term in the su-
perpotential must combine exactly one term arising
from each of the 27’s. This greatly limits the form
of the superpotential. And in particular, it implies
that neither mass nor flavour eigenstates will arise
from any single 27—that is, the ‘families’ in the col-
loquial sense are necessarily linear combinations of
fields resulting from different 27’s.
Also notice that in general the terms in the su-
perpotential will be proportional to e−
dVol where
dVol is the volume form of some cycle wrapping sin-
gularities (the details of which depends on whether
we are talking about type IIa, M-theory, or F-theory
realizations), and are in principle calculable in terms
of the deformation moldui. And because these coef-
ficients are exponentially related to the volumes of
cycles, we expect the high-scale Lagrangian will be
generically hierarchical. This structure could be im-
portant for solving problems in phenomenology—for
example the µ problem in the Higgs potential, the
Higgs doublet-triplet splitting problem, or avoiding
8 A subtlety, however, is that because our language has been
explicitly that of N = 2 theory from type IIa, we are unable
to distinguish the 5 from the 5 in the splitting of the 10’s
of SO10. In Figure 3, a consistent choice was made—and
although we do not justify this claim here, it is the choice
that will be correct for the M-theory generalization of this
work.
9 That it is continuous is a consequence of the fact that we are
engineering non-compact Calabi-Yaus. If one matched this
local geometry to a compact global structure, the landscape
would of course be discrete.
proton decay.
We are in the process of studying the phenomenol-
ogy of models on this landscape. At first glance,
the U1-structure combined with high-scale hierar-
chies could possibly be complex enough to be able to
avoid some of the typical problems of E6-like grand
unified models. We should point out that if there
were no high-scale hierarchies, however, then the al-
lowed terms in the superpotential would generically
give rise to low-energy lepton and baryon number vi-
olation, similar to any ‘generic’ E6 model—i.e. one
which includes all types of terms allowed by the E6-
mandated U1-structure [16]. We could always im-
pose additional symmetries and add fields by hand
to solve these problems, but this would not be very
compelling. However, if viable models already ex-
ist in the landscape which do not require additional
fields or symmetries, these would be compelling even
if we do not yet understand how they are selected.
One of the most important phenomenological
questions about these models is the fate of the
additional U1 symmetries. Although we suspect
that one can determine which of the U1 symmetries
are dynamical below the compactification scale by
studying the normalizability of their corresponding
vector multiplets, we do not presently have have
a complete understanding of this situation. Of
course, if any additional U1’s survive to low en-
ergy they could have very interesting—or damning—
phenomenological consequences.
VI. DISCUSSION
An important point to bear in mind when consid-
ering geometrically engineered models is that there
generically exist10 moduli which can deform the ge-
ometry into one which gives rise to a theory with
less gauge symmetry. For example, if you are given a
geometrically-engineered SO10 grand unified model,
then our results show explicitly that the model can
be locally deformed into an SU5 model, and this
can be deformed further into the Standard Model;
the original SO10 theory is seen to be a single
point in a (complex) two-dimensional landscape of
SU3×SU2 theories. And because larger symmetries
always lie in lower dimensional surfaces of moduli
space, it is very relevant to ask what physics pre-
vents this unfolding from taking place. Indeed, this
question applies to the Standard Model as well—
our analysis could easily go further to unfold away
SU3×SU2. We are not presently able to answer why
this does not happen11; although this observation
10 There could be global obstructions which prevent such a
deformation from taking place. But these are invisible to
the non-compact, local constructions considered here.
11 Although, perhaps the unfolding of SU2 may provide an al-
ternative to tuning in the usual Higgs sector [17]. It would
be interesting to understand in greater detail the relation-
suggests that perhaps theories with less symmetry,
like SU3 × SU2, could be much more natural than
grand unified theories.
More generally, it is not presently understood
what physics controls the values of the geometric
moduli which deform the manifold—the parameters
which deform the E8 → E6×SU2 complex structure,
for example. We do not yet have a general mecha-
nism which would fix these parameters; we simply
observe that any non-zero values of the moduli will
give rise to a geometrically engineered manifold with
SU3×SU2 gauge theory ‘peppered’ with all the nec-
essary singularities of the three families of the Stan-
dard Model together with the usual E6-like exotics.
And importantly, for any point in the complex four-
dimensional ‘landscape,’ the relative locations of all
the relevant singularities are known—and hence in
principle so is the superpotential.
This relationship between moduli-fixing and
gauge symmetry breaking could be a novel feature
of geometrically-unfolded models. It may allow one
to apply the results in [8], for example, to single out
theories on the landscape. However, a prerequisite
to this type of analysis would be an identification
of which moduli should be identified with the ones
which deform the geometry as described here.
Although the motivation in this paper and in
[1] appears to be a top-down realization of grand
unification, there is a sense in which we are re-
ally engineering from the bottom-up. Specifically,
because the local geometry we have described is
non-compact, the resulting theory is decoupled from
quantum gravity, and the parameters along the land-
scape of deformations are continuous. This is not
unlike the situation in [18]. But what we lose in
global constraints we perhaps gain by concrete local
structure. Not only do we have a framework which
naturally predicts three families with a rather de-
tailed phenomenological structure, but we have done
so in a way that preserves all the information about
the local geometry. And because this framework re-
alizes the ‘physics from pure geometry’ paradigm in
a potentially powerful way, it could prove impor-
tant to concrete phenomenological constructions in
M-theory, for example.
Of course we envision these local geometries to
be embedded within compact Calabi-Yau manifolds.
It is an assumption of the framework that the pre-
cise global topology of the compactification mani-
fold can be ignored at least as a good first approx-
imation. One may ask the extent to which these
constructions can be glued into compact manifolds.
Concretely: under what circumstances can a non-
compact Calabi-Yau three-fold which is a fibration of
K3 surfaces with asymptotically uniform ADE-type
ship between unfolding and the Higgs mechanism.
singularities be compactified? This is an important
question for mathematicians, the answer to which
would likely lead to important physical insight—e.g.
quantization of the moduli space of deformations.
A possible objection to this framework is that
our constructions appear to depend on several seem-
ingly arbitrary choices (the specific chain from E8 to
the Standard Model, which roots were eliminated at
each step, etc.). However, it is likely that the particle
content, for example, which results is completely in-
dependent of these choices. Furthermore, we suspect
that different realizations of the unfolding merely re-
sult in different parameterizations of the landscape,
and do not reflect true additional arbitrariness. But
this is still an area that deserves attention.
Lastly, because in this picture the Standard Model
is seen to unfold at the compactification scale, one
may ask what has become of gauge coupling unifi-
cation. Because the gauge coupling constants are
functions of the volumes of their corresponding co-
dimension four singular surfaces12 which depend on
the deformation moduli, the traditional meaning of
grand unification is more subtle here—as is typical
in string phenomenology. For example, although we
chose to unfold the Standard Model sequentially as
a series of less unified models, there is no reason to
suspect that that order has any physical importance.
Surely, if as we parameterized the unfolding in sec-
tion IV, setting d → 0 (or c → 0) would result in an
SU5 grand unified theory; but setting a → 0 instead
would result in a restoration of family symmetry.
The four complex structure moduli tune different
types of unification separately—and should simulta-
neously be at play in the question of gauge coupling
unification.
It is interesting to note, however, that if one were
to simultaneously scale the values of all the moduli
to be very small, the spectrum would be more and
more unified: the relative distances between singu-
larities shrink, unifying the coefficients in the su-
perpotential; and the volumes of the co-dimension
four singularities (if realized in a compact manifold)
would approach one another, resulting in a unifica-
tion of their gauge couplings. What this may mean
phenomenologically remains to be understood.
In this paper we have described a local, purely ge-
ometric framework in which gauge symmetry ‘break-
ing’ can be re-cast as a problem of moduli fixing—
and in which the same moduli which describe this
geometric ‘unfolding’ also determine the physics of
massless matter. And although we still do not un-
derstand the mechanisms by which these moduli are
fixed, the landscape of possibilities is already enor-
mously reduced: what would have been the hun-
dreds of parameters describing the relative positions
on the compactification manifold of the Standard
Model’s three families worth of matter fields, we
specify them all in terms of only four complex struc-
ture moduli which describe the unfolding of an iso-
lated E8 singularity. And the fact that three families
emerges is group-theoretic and not added by hand.
VII. ACKNOWLEDGEMENTS
It is a pleasure to thank helpful discussions with
and insightful comments of Herman Verlinde, Sergei
Gukov, Gordy Kane, Paul Langacker, Edward Wit-
ten, Cumrun Vafa, Brent Nelson, Malcolm Perry,
Dmitry Malyshev, Matthew Buican, Piyush Kumar,
and Konstantin Bobkov.
This work was funded in part by a Graduate Re-
search Fellowship from the National Science Foun-
dation.
12 Of course, this can only be discussed concretely when the
compact manifold is known.
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http://arxiv.org/abs/hep-th/0508089

  This paper extends and builds upon the results of an earlier paper, in which
we described how to use the tools of geometrical engineering to deform
geometrically-engineered grand unified models into ones with lower symmetry.
This top-down unfolding has the advantage that the relative positions of
singularities giving rise to the many `low energy' matter fields are related by
only a few parameters which deform the geometry of the unified model. And
because the relative positions of singularities are necessary to compute the
superpotential, for example, this is a framework in which the arbitrariness of
geometrically engineered models can be greatly reduced.
  In our earlier paper, this picture was made concrete for the case of
deforming the representations of an SU(5) model into their Standard Model
content. In this paper we continue that discussion to show how a geometrically
engineered 16 of SO(10) can be unfolded into the Standard Model, and how the
three families of the Standard Model uniquely emerge from the unfolding of a
single, isolated E8 singularity.

Geometrically Engineering the Standard Model:
Locally Unfolding Three Families out of E8
Jacob L. Bourjaily∗
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(Dated: 3rd April 2007)
This paper extends and builds upon the results of [1], in which we described how to use the tools
of geometrical engineering to deform geometrically-engineered grand unified models into ones with
lower symmetry. This top-down unfolding has the advantage that the relative positions of singular-
ities giving rise to the many ‘low energy’ matter fields are related by only a few parameters which
deform the geometry of the unified model. And because the relative positions of singularities are
necessary to compute the superpotential, for example, this is a framework in which the arbitrariness
of geometrically engineered models can be greatly reduced.
In [1], this picture was made concrete for the case of deforming the representations of an SU5
model into their Standard Model content. In this paper we continue that discussion to show how
a geometrically engineered 16 of SO10 can be unfolded into the Standard Model, and how the
three families of the Standard Model uniquely emerge from the unfolding of a single, isolated E8
singularity.
I. INTRODUCTION
In [2], Katz and Vafa showed how to geometrically
engineer matter representations in terms of the local
singularity structure of type IIa, M-theory, and F-
theory compactifications. In that framework matter
and gauge theory both have purely geometrical ori-
gins: SUn, SO2n and En gauge theories arise from
the existence of co-dimension four singular curves of
certain types in the compactification manifold [3];
and massless matter representations arise from iso-
lated points (in type IIa or M-theory) or curves (in
F-theory) along the singular surface over which the
type of singularity is enhanced by one rank.
Despite the extraordinary generality of this frame-
work, it has not been widely used phenomenologi-
cally. This is largely because the description of the
isolated enhancements of singularities giving rise to
various matter representations is inherently local:
although the geometry near any particular enhance-
ment could be described concretely, the framework
had nothing to say about numbers, types, and rela-
tive locations of different matter fields. This global
data was either to be determined by duality to a
concrete, global string theory model1, or suggested
via the a posteriori success of a given set of relative
positions (as in e.g. [5, 6]).
Another way to relate the number and relative
positions of (enhanced singularities giving rise to)
matter fields was given in [1]: in that paper, we
described for example how a local description of the
geometry giving rise to a massless 5 of SU5 could
be smoothly deformed into a local description of a
(3,1) and a (1,2) of SU3×SU2 which live at distinct
∗Electronic address: jbourjai@princeton.edu
1 Geometrically engineered models in M-theory are dual to
intersecting brane models in type II, (see e.g. [4]).
points—related by a single deformation parameter.
A cartoon of what was described in that paper is
shown in Figure 1.
In this paper we describe pedagogically how to
extend that idea to engineer analogies to SO10 and
E6 × SU2 grand unified models
2. Although in [1]
we were able to analyze explicit unfoldings of SO10
and SU6 singularities sufficiently well by sight, this
will not be possible for our present examples. All
of the examples in this paper involve the unfolding
of isolated En singularities; and although algebraic
descriptions of these are known and classified [7], it
would be unnecessarily cumbersome and unenlight-
ening to analyze them explicitly as we did in [1].
Therefore, in section II we describe a much more
powerful and elegant language in which to study
these resolutions.
In section III we describe in detail how the un-
folding of a 16 of SO10 into the Standard Model is
derived in the language of section II. This is achieved
in two stages: in the first stage, we unfold the 16 into
10⊕5⊕1 of SU5; we then unfold the resulting SU5
model into a single ‘family’ of the Standard Model.
At the end, all the relative positions of the singu-
larities of the family are set by the non-zero values
of two complex structure moduli, thereby greatly re-
ducing the arbitrariness of their relative positions.
The next most obvious example would be a de-
scription of how a 27 of E6 geometrically unfolds
into the Standard Model. However, there are two
reasons to leave this example to the reader: first, it is
a most natural extension of the results of section III;
secondly, it is a consequence of the E6 × SU2 grand
unified model which we describe in section IV. Al-
2 As described in section IV, the resolution E8 → E6 × SU2
naturally starts as a theory with three 27’s of E6 related
by an SU2 family symmetry.
http://arxiv.org/abs/0704.0445v2
mailto:jbourjai@princeton.edu
FIG. 1: A cartoon of the geometric deformation of a 5 and 10 of SU5 into the Standard Model as described in [1].
The surface over which the singularities are enhanced is coordinatized by a complex parameter t and the geometry
is deformed by changing the value of a complex parameter s. The relative locations of the ‘resolved’ singularities are
given in terms of s.
though not given as an example in [2], it is not hard
to see3 that a single isolated E8 singularity at the
intersection of a co-dimension four surfaces of types
E6 and SU2 gives rise to matter in the representa-
tion (27,2)⊕ (27,1) ⊕ (1,2). It is easy to see how
this would unfold into the matter content of three
families—one coming from each of the 27’s. That
three families emerge from E8 is a general conse-
quence of group theory and can be understood from
the fact that E6 ×SU3 is a maximal subgroup of E8
into which the adjoint of E8 partially branches into
an SU3 triplet of 27’s.
As in the preceding paper [1], this work is pre-
sented concretely in the language of Calabi-Yau
compactifications of type IIa string theory, which
can also be naturally extended to F-theory mod-
els. Here, we engineer the explicit local geometry
of (non-compact) Calabi-Yau three-folds which are
K3-fibrations over C1. If type IIa string theory is
compactified on this three-fold, a four-dimensional
N = 2 theory with various massless hypermulti-
plets will result. But if, for example, the C1 base of
this three-fold were fibred as an O(−2) bundle over
1, the resulting total space would be a Calabi-Yau
four-fold4 upon which F-theory would compactify to
an N = 1 theory with chiral multiplets. However,
because the manifold over which the singular K3’s
are fibred in M-theory is a real, three-dimensional
space, our fibrations over C1 do not have a direct
application to M-theory.
It would of course be desirable to have a similar
description of geometric unfolding explicitly in the
language of G2-manifolds so that this picture could
be realized concretely in M-theory as well. This is
particularly important in light of the recent advances
in M-theory phenomenology (e.g. [8, 9]). By exten-
3 This is described in section II.
4 This is just one example of the ways in which these Calabi-
Yau three-folds could be fibred over CP1 to result in a
Calabi-Yau four-fold.
sion of the work of Berglund and Brandhuber in [10],
such a generalization should be relatively straight-
forward, but we will not attempt to do this here.
II. RESOLVING En-TYPE SINGULARITIES
Recall that a gauge theory in type IIa string the-
ory can arise from compactification to six dimen-
sions over a singular K3 surface (similar statements
apply to M-theory and F-theory) [3]. The complex
structures of the singular compactification manifolds
giving rise to SUn(≡ An−1), SO2n(≡ Dn), and En
gauge theory are given in Table I—where the sur-
faces are labelled conveniently by the name of the
resulting gauge theory5.
We can generalize this discussion by considering
a complex, one-dimensional space B over which a
smooth family of singular K3 surfaces are fibred. If
almost everywhere over B the K3-fibres have singu-
larities of a single type, then compactification of type
IIa string theory over the total space will give rise
Gauge group Polynomial
SUn (≡ An−1) xy = z
SO2n (≡ Dn) x
2 + y2z = zn−1
2 = y3 + z4
2 + y3 = 16yz3
2 + y3 = z5
TABLE I: Hypersurfaces in C3 giving rise to the desired
orbifold singularities.
5 It is of curious historical interest that the equations listed
in Table I were first identified by Fleix Klein in 1884 [11].
The reader may also be amused that the full resolutions of
these surfaces were almost completely classified—up to a
few computational errors—by Bramble in 1918 [12].
to gauge theory in four-dimensions of the type corre-
sponding to the typical fibre. Massless charged mat-
ter will arise if over isolated points in B the type of
fibre is enhanced by one rank. The geometry about
a single such isolated point where the singularity is
enhanced was described in detail by Katz and Vafa
in [2].
The representation of matter living at these ‘more-
singular’ points was also given in [2]: suppose that
G ⊃ H ×U1 and that the rank of H is one less than
G; then, if there is an isolated G-type singularity
over a surface of H-type singularities, the resulting
massless representation is given by those parts of the
decomposition of the adjoint of G into H×U1 which
are charged under the U1.
Because the question of how to (smoothly) deform
the surfaces of Table I into ones of lower rank has in-
trinsic mathematical interest, it is not too surprising
that all possible two-dimensional deformations have
been classified. Our discussion below will make use
of the notation and results presented in [7].
In our present work, we are interested in defor-
mations of En singularities into ones of lower rank.
Unlike SUn singularities, the resolutions of which are
easy enough to read off by sight, the algebraic com-
plexity of En singularities is formidable. To appre-
ciate what is meant by this, consider the resolution
of E7. From Table I we know that an E7 singularity
is locally isomorphic to the surface x2 + y3 = 16yz3
in C3. Its full resolution in terms of the seven defor-
mation parameters ~t = (t1, t2, . . . , t7) is given by
− x2 − y3 + 16yz3 + ǫ2(~t)y
2z + ǫ6(~t)y
2 + ǫ8(~t)yz + ǫ10(~t)z
2 + ǫ12(~t)y + ǫ14(~t)z + ǫ18(~t) = 0, (1)
where the ǫn(~t) are n
th order symmetric polynomi-
als in the components of ~t which are tabulated over
several pages of the appendix of [7].
A näıve way to determine the type of singularity
found by resolving E7 “in the direction ~t” would be
to expand equation (1) completely using the explicit
functions ǫn(~t), find each of its singular points, and
expand locally about each until an isomorphism with
a singularity of lower rank in Table I was clear. This
is the way, for example, that [2] demonstrated that
the resolution of E7 in the direction (0, 0, 0, 0, 0, t, 0)
gives rise to E6 for t 6= 0. All of the results in this
paper could be verified in this way. Luckily, however,
Katz and Morrison described a much more powerful
and direct way to analyze the deformations of En
singularities [7].
We would like a pragmatic answer to the follow-
ing question: what is the type of fibre found by re-
solving an En singularity in the direction ~t? That
there is an easy answer to this question makes our
work much simpler. Although an adequate treat-
ment would take us well beyond the scope of our
present discussion, the answer given in [7] is at least
very easy to make use of6: for each of the equations
in Table II satisfied by the components of ~t, the sin-
gularity has the corresponding root. Given the list of
roots, it is then a straight-forward exercise to con-
struct the Dynkin diagram corresponding to the sin-
gularity7.
6 Of course, this answer does depend on the parameterization
used. As stated before, we are working with the conventions
of [7].
7 Misusing the notation of [7] in a way applicable only to
In an admittedly bad notation, we consider each of
the n deformation parameters ti(t) to be functions
of t, the local coordinate on the base space B. A
(non-Abelian) gauge theory will be present if there
are roots implied by Table II which are preserved
for generic values of t. And charged massless matter
will exist if at isolated points {t∗} an additional root
is added—or, in terms of Dynkin diagrams, if an
additional node is added. At each isolated point we
can therefore identify the resolution G → H and
thereby determine the resulting representation.
Equation Root
ti − tj = 0 =⇒ ei − ej
ti + tj + tk = 0 =⇒ e0 − ei − ej − ek
tij = 0 =⇒ 2e0 −
2ti1 +
tij = 0 =⇒ 3e0 − 2ei1 −
TABLE II: The roots of the singularity resulting from the
resolution of En in the direction~t. This is a reproduction
of Table 4 of Ref. [7].
SUn, SO2n and En, one can think of the vectors ei as an or-
thonormal basis in Minkowski space which is equipped with
a mostly-plus metric. Then roots are vectors in this space
of norm +2. Each (positive) root gives rise to a node in the
resulting Dynkin diagram, and two nodes are connected by
a line if their inner product is −1 and disconnected if they
are orthogonal.
III. GEOMETRIC ANALOGUE OF SO10
GRAND UNIFICATION
A. The Description of a 16 of SO10
A necessary starting point to describe the unfold-
ing of a 16 of SO10 into the Standard Model is a
description of the initial geometry as was done in [2].
We will briefly review that construction in the lan-
guage described above before we unfold it, first into
an SU5 model, and later all the way into SU3×SU2.
Let t be a local complex coordinate on the space
B over which is fibred the resolution of E6 parame-
terized by ~t = (t, t, t, t, t,−2t). To be clear, for each
value of t, the vector~t describes an explicit surface in
3 given in reference [7] analogous to that of equa-
tion (1) above.
Considering the rules of Table II, we see that for
an arbitrary value of t 6= 0 the root lattice of the
fibre is
(e0−e1−e2−e6)
(e1 − e2) (e2 − e3) (e3 − e4) (e4 − e5)
where we have displayed the roots suggestively so as
to reproduce the SO10 Dynkin diagram. At t = 0,
however, E6 is restored. So we have an isolated E6
fibre over the point t = 0, while for any t 6= 0 the
fibre is SO10. This gives rise to SO10 gauge theory
with a single massless 16 located at the origin in the
t-plane.
B. Unfolding the 16 of SO10 into SU5
We would like to unfold the manifold described
above into one with SU5 gauge theory. It is not
hard to guess in what ‘directions’ ~t we may deform
the the geometry so that the fibre over a generic
point is SU5. Let a denote a parameter independent
of t which adjusts the whole geometry over the re-
gion which is coordinatized by t. Then let the fibre
over t be given by the resolution of E6 in the direc-
tion (t, t, t, t, t+ a,−2t− a). Obviously when a = 0
the situation is the same as above and results in a
single massless 16 of SO10. However, when a 6= 0
the situation is different: for generic values of t it is
easy to see that the simple roots are
(e0−e1−e5−e6) (e1−e2) (e2−e3) (e3−e4)
which means that the generic fibre over t is just
SU5—and so the resulting gauge theory is SU5.
To find what matter representations exist, we
must determine over which locations t the rank of
the fibre is enhanced. This means we are seeking
special values of t (determined by a) at which an ad-
ditional equation in Table II is satisfied. For each of
these points, we can draw the resulting Dynkin dia-
TABLE III: The locations on the complex t-plane over
which the singularity of the fibre is enhanced, and the
representations of SU5 × U1 that result.
Location Fibre
Representation
of SU5 × U1
3t + 2a = 0 SU5 × SU2 1−5
3t+ a = 0 SO10 10−1
t = 0 SU6 53
gram to determine the fibre over that point, thereby
determining the representation which arises there.
It is not hard to exhaustively find all these ‘more
singular’ points. They are give in Table III. Notice
that we have included the U1-charge assignments
that result; these are normalized as in the appendix
of [13].
C. Unfolding a 16 of SO10 into the Standard
Model
To complete our task and unfold the 16 of SO10
all the way to the Standard Model, we must deform
the fibres by another ‘global’ parameter, which we
will denote b. It is not hard to guess a direction over
which the generic fibre will be SU3 × SU2: try for
example (t, t, t, t + b, t + a,−2t− a − b). Again, we
notice that for a general location t and generic fixed
values a, b 6= 0, the singularity has the root structure
(e1 − e2) (e2 − e3)
(e0 − e4 − e5 − e6)
, (4)
which is visibly SU3 × SU2.
Like above, it is a straight-forward exercise to de-
termine all the locations over which the singularity
is enhanced, and the resulting representation which
arises. These points including their resulting rep-
resentations (with U1-charges as normalized in [13])
are listed in Table IV. The entire unfolding is repro-
duced graphically in Figure 2.
Location Fibre Representationof SU3×SU2×U1 Name
3t + 2a+ b = 0 SU3 × SU2 × SU2 (1,1)0 ν
3t + a+ 2b = 0 SU3 × SU2 × SU2 (1,1)6 e
3t+ a+ b = 0 SU5 (3,2)1 Q
3t + a = 0 SU4 × SU2 (3,1)−4 u
3t+ b = 0 SU4 × SU2 (3,1)2 d
t = 0 SU3 × SU3 (1,2)−3 L
TABLE IV: The locations on the complex t-plane over
which the singularity of the fibre is enhanced and the
representations of SU3 × SU2 × U1 that result.
FIG. 2: An illustration of the resolution of a geometrically engineered 16 of SO10 into the Standard Model as a
function of two complex structure moduli a and b as described in section III. The coordinate along the base space is t
and runs vertically in the diagram. When a = b = 0, along the left-hand side, there is just one isolated E6 singularity
at t = 0. When b = 0 but a is allowed to vary, this single singularity splits into three, and any a =constant slice will
have three isolated singularities in the complex t-plane as shown above. Moving rightward in the diagram, at the
dashed line a is held fixed and b is allowed to grow, causing the three enhancements of SU5 to break apart into six
total isolated singularities over SU3 ×SU2, which is shown on the right-hand-side. Also shown are the (appropriately
normalized) U1 charges of fields obtained via this multiple unfolding.
IV. GEOMETRIC ANALOGUE OF E6 × SU2
GRAND UNIFICATION
After having completed the unfolding of a 16 of
SO10 into the Standard Model, it is natural to ask if
this idea can be extended to relate all the singulari-
ties of the Standard Model as perhaps the unfolding
of a single isolated singularity of higher-rank. The
answer is in fact yes—and there is a sense in which
precisely three families arise if the notion of ‘geomet-
ric unification’ is saturated.
Because a 16 of SO10 arises from the resolution
E6 → SO10, it can only be unfolded out an ex-
ceptional singularity. Clearly the highest level of
unification one can achieve along this line would
be to start with a resolution E8 → H where H is
a rank-seven subgroup of E8 which contains SO10.
The possible ‘top-level’ gauge groups are then E7,
E6 × SU2, and SO10 × SU3. We choose to study
E8 → E6 × SU2 as our example because it will nat-
urally include a description of the unfolding of 27 of
E6 into the Standard Model, which is interesting in
its own right, and because it follows quite directly
from our work in section III.
The initial geometry which we will deform into the
Standard Model is given as follows. Let t be a com-
plex coordinate on the base space B over which is fi-
bred the resolution (t, t, 0, 0, 0, 0, 0, 0) of E8. Clearly,
when t = 0 we recover E8; when t 6= 0 we see that
the roots of the fibre are
(e0−e3−e4−e5)
(e3 − e4) (e4 − e5) (e5 − e6) (e6 − e7) (e7 − e8)
(e1 − e2)
, (5)
which is visibly E6 × SU2. Following the general
rule to determine the representation resulting from
a given resolution [2], we find that at t = 0 lives
massless matter charged in the (27,2)1⊕(27,1)−2⊕
(1,2)3 representation of E6 × SU2 × U1ϕ .
To avoid pedantic redundancy, in Figure 3 we have
summarized in great detail the entire unfolding into
SU3×SU2×U1Y ×U1χ×U1Ψ×U1ζ×U1ϕ . An outline
of the steps involved in deriving this unfolding is
given presently.
First, the unfolding of the E6 × SU2 gauge the-
ory into E6 gauge theory is obtained by defining
the fibre over t to be given by the resolution of E8
in the direction (t + a, t − a, 0, 0, 0, 0, 0, 0) for some
a 6= 0. This clearly kills the SU2 node of the fibre
in equation (5). There are five locations at which
the singularity is enhanced by one rank, giving rise
to three 27’s and two singlets as shown in the left-
most section of Figure 3.
FIG. 3: An illustration of the resolution of a single isolated E8 into the Standard Model in terms of four deformation
parameters a, b, c, d. Along the left hand side, for a = b = c = d = 0, the generic E6 × SU2 fibre is enhanced to E8
at t = 0. Moving from left to right, a, b, c, d are sequentially allowed to grow to some non-zero value—and between
dashed lines all but one of the moduli are held fixed. Solid lines indicate the locations of enhanced singularities
relative to the plane for as functions of a, b, c, d. The complete list of isolated singularities, their locations, and charge
assignments are given on the right hand side of the diagram.
The rest of the unfolding is a natural appli-
cation of the work in section III. Let us now
set the fibre over t to be given by the resolution
(t+ a, t− a, b, b, b, b, b,−2b) of E8 for arbitrary com-
plex deformation parameters a, b 6= 0. From section
III we see immediately that the generic fibre is SO10.
A thorough scanning for possible solutions to equa-
tions in Table II shows that there are 11 isolated
points on the complex t-plane over which the singu-
larity is enhanced. These correspond to the ‘break-
ing’ of each 27 of E6 into 16 ⊕ 10 ⊕ 1 of SO10,
while the singlets remain singlets. This is seen in
the second vertical strip (from the left) in Figure 3.
Again, following our discussion above, it is easy
to guess possibilities for the next two resolution di-
rections. First, we set the fibre over t to be given by
(t+ a, t− a, b, b, b, b, b+ c,−2b− c) which will result
SU5 gauge theory with matter content correspond-
ing to the ‘canonical’ decomposition of three 27’s of
E6 with two singlets. And finally, the full resolution
of the E6×SU2 grand unified model into SU3×SU2
can be given by letting the fibre over t be given by
the (t+a, t−a, t, t, t, t+d, t+c,−2b−c−d) resolution
of E8 for (generic) arbitrary fixed complex structure
moduli a, b, c, d 6= 0.
V. IMPLICATIONS
Let us clarify what we have done. For a given set
of fixed, nonzero complex structure moduli, the reso-
lution given above describes the explicit, local geom-
etry of a non-compact Calabi-Yau three-fold, which
is a K3-fibration over C1. If type IIa string theory
is compactified on this three-fold, the resulting four-
dimensional theory will have SU3 × SU2 gauge the-
ory with hypermultiplets at isolated points as given
in Figure 3 which reproduce the spectrum of three
families of the Standard Model with an extended
Higgs sector and some exotics. Alternatively, if one
takes this (non-compact) Calabi-Yau three-fold and
fibres it over CP1 as described in section I so that
the total space is Calabi-Yau, then F-theory on this
space will give rise to N = 1 supersymmetry with
SU3×SU2 gauge theory and chiral multiplets in the
representations given in Figure 3. And although it
does not follow directly from our construction above,
considering the close similarities between two- and
three-dimensional resolutions of the singularK3 sur-
faces we have every reason to suspect an analogous
geometry can be engineered for M-theory in terms
of hyper-Kähler quotients by extension of the results
in [10, 14, 15]. We are currently working on building
this geometry in M-theory, and we expect to report
on this work soon.
Given these four complex structure moduli, all
the relative positions of the 35 disparate singulari-
ties giving rise to all three families of the (extended)
Standard Model are then known8. Beyond the usual
three families of the Standard Model, the manifold
also gives rise to two Higgs doublets for each fam-
ily, six Higgs colour triplets, three right-handed neu-
trinos and five other Standard Model singlets. We
should point out that this matter content (and their
U1-charge assignments) is a consequence of group
theory and algebraic geometry alone—it is simply
what is found when unfolding E8 all the way to the
Standard Model.
And given the relative positions and local ge-
ometry of the singularities together with the U1-
structure, one can in principle compute the full su-
perpotential coming from instantons wrapping dif-
ferent singularities. Because these are fixed by the
values of the complex structure moduli, there is a
(complex) four-dimensional landscape9 of different,
explicit SU3 × SU2 embeddings at the compactifi-
cation scale. Although this large landscape may
appear to have too much freedom, we remind the
reader that in the traditional understanding of ge-
ometrical engineering there would be hundreds of
parameters describing the (independent) relative lo-
cations of each of the isolated singularities.
There are a few things to notice about the form of
the superpotential that will emerge. First, because
of the U1-charge assignments, each term in the su-
perpotential must combine exactly one term arising
from each of the 27’s. This greatly limits the form
of the superpotential. And in particular, it implies
that neither mass nor flavour eigenstates will arise
from any single 27—that is, the ‘families’ in the col-
loquial sense are necessarily linear combinations of
fields resulting from different 27’s.
Also notice that in general the terms in the su-
perpotential will be proportional to e−
dVol where
dVol is the volume form of some cycle wrapping sin-
gularities (the details of which depends on whether
we are talking about type IIa, M-theory, or F-theory
realizations), and are in principle calculable in terms
of the deformation moldui. And because these coef-
ficients are exponentially related to the volumes of
cycles, we expect the high-scale Lagrangian will be
generically hierarchical. This structure could be im-
portant for solving problems in phenomenology—for
example the µ problem in the Higgs potential, the
Higgs doublet-triplet splitting problem, or avoiding
8 A subtlety, however, is that because our language has been
explicitly that of N = 2 theory from type IIa, we are unable
to distinguish the 5 from the 5 in the splitting of the 10’s
of SO10. In Figure 3, a consistent choice was made—and
although we do not justify this claim here, it is the choice
that will be correct for the M-theory generalization of this
work.
9 That it is continuous is a consequence of the fact that we are
engineering non-compact Calabi-Yaus. If one matched this
local geometry to a compact global structure, the landscape
would of course be discrete.
proton decay.
We are in the process of studying the phenomenol-
ogy of models on this landscape. At first glance,
the U1-structure combined with high-scale hierar-
chies could possibly be complex enough to be able to
avoid some of the typical problems of E6-like grand
unified models. We should point out that if there
were no high-scale hierarchies, however, then the al-
lowed terms in the superpotential would generically
give rise to low-energy lepton and baryon number vi-
olation, similar to any ‘generic’ E6 model—i.e. one
which includes all types of terms allowed by the E6-
mandated U1-structure [16]. We could always im-
pose additional symmetries and add fields by hand
to solve these problems, but this would not be very
compelling. However, if viable models already ex-
ist in the landscape which do not require additional
fields or symmetries, these would be compelling even
if we do not yet understand how they are selected.
One of the most important phenomenological
questions about these models is the fate of the
additional U1 symmetries. Although we suspect
that one can determine which of the U1 symmetries
are dynamical below the compactification scale by
studying the normalizability of their corresponding
vector multiplets, we do not presently have have
a complete understanding of this situation. Of
course, if any additional U1’s survive to low en-
ergy they could have very interesting—or damning—
phenomenological consequences.
VI. DISCUSSION
An important point to bear in mind when consid-
ering geometrically engineered models is that there
generically exist10 moduli which can deform the ge-
ometry into one which gives rise to a theory with
less gauge symmetry. For example, if you are given a
geometrically-engineered SO10 grand unified model,
then our results show explicitly that the model can
be locally deformed into an SU5 model, and this
can be deformed further into the Standard Model;
the original SO10 theory is seen to be a single
point in a (complex) two-dimensional landscape of
SU3×SU2 theories. And because larger symmetries
always lie in lower dimensional surfaces of moduli
space, it is very relevant to ask what physics pre-
vents this unfolding from taking place. Indeed, this
question applies to the Standard Model as well—
our analysis could easily go further to unfold away
SU3×SU2. We are not presently able to answer why
this does not happen11; although this observation
10 There could be global obstructions which prevent such a
deformation from taking place. But these are invisible to
the non-compact, local constructions considered here.
11 Although, perhaps the unfolding of SU2 may provide an al-
ternative to tuning in the usual Higgs sector [17]. It would
be interesting to understand in greater detail the relation-
suggests that perhaps theories with less symmetry,
like SU3 × SU2, could be much more natural than
grand unified theories.
More generally, it is not presently understood
what physics controls the values of the geometric
moduli which deform the manifold—the parameters
which deform the E8 → E6×SU2 complex structure,
for example. We do not yet have a general mecha-
nism which would fix these parameters; we simply
observe that any non-zero values of the moduli will
give rise to a geometrically engineered manifold with
SU3×SU2 gauge theory ‘peppered’ with all the nec-
essary singularities of the three families of the Stan-
dard Model together with the usual E6-like exotics.
And importantly, for any point in the complex four-
dimensional ‘landscape,’ the relative locations of all
the relevant singularities are known—and hence in
principle so is the superpotential.
This relationship between moduli-fixing and
gauge symmetry breaking could be a novel feature
of geometrically-unfolded models. It may allow one
to apply the results in [8], for example, to single out
theories on the landscape. However, a prerequisite
to this type of analysis would be an identification
of which moduli should be identified with the ones
which deform the geometry as described here.
Although the motivation in this paper and in
[1] appears to be a top-down realization of grand
unification, there is a sense in which we are re-
ally engineering from the bottom-up. Specifically,
because the local geometry we have described is
non-compact, the resulting theory is decoupled from
quantum gravity, and the parameters along the land-
scape of deformations are continuous. This is not
unlike the situation in [18]. But what we lose in
global constraints we perhaps gain by concrete local
structure. Not only do we have a framework which
naturally predicts three families with a rather de-
tailed phenomenological structure, but we have done
so in a way that preserves all the information about
the local geometry. And because this framework re-
alizes the ‘physics from pure geometry’ paradigm in
a potentially powerful way, it could prove impor-
tant to concrete phenomenological constructions in
M-theory, for example.
Of course we envision these local geometries to
be embedded within compact Calabi-Yau manifolds.
It is an assumption of the framework that the pre-
cise global topology of the compactification mani-
fold can be ignored at least as a good first approx-
imation. One may ask the extent to which these
constructions can be glued into compact manifolds.
Concretely: under what circumstances can a non-
compact Calabi-Yau three-fold which is a fibration of
K3 surfaces with asymptotically uniform ADE-type
ship between unfolding and the Higgs mechanism.
singularities be compactified? This is an important
question for mathematicians, the answer to which
would likely lead to important physical insight—e.g.
quantization of the moduli space of deformations.
A possible objection to this framework is that
our constructions appear to depend on several seem-
ingly arbitrary choices (the specific chain from E8 to
the Standard Model, which roots were eliminated at
each step, etc.). However, it is likely that the particle
content, for example, which results is completely in-
dependent of these choices. Furthermore, we suspect
that different realizations of the unfolding merely re-
sult in different parameterizations of the landscape,
and do not reflect true additional arbitrariness. But
this is still an area that deserves attention.
Lastly, because in this picture the Standard Model
is seen to unfold at the compactification scale, one
may ask what has become of gauge coupling unifi-
cation. Because the gauge coupling constants are
functions of the volumes of their corresponding co-
dimension four singular surfaces12 which depend on
the deformation moduli, the traditional meaning of
grand unification is more subtle here—as is typical
in string phenomenology. For example, although we
chose to unfold the Standard Model sequentially as
a series of less unified models, there is no reason to
suspect that that order has any physical importance.
Surely, if as we parameterized the unfolding in sec-
tion IV, setting d → 0 (or c → 0) would result in an
SU5 grand unified theory; but setting a → 0 instead
would result in a restoration of family symmetry.
The four complex structure moduli tune different
types of unification separately—and should simulta-
neously be at play in the question of gauge coupling
unification.
It is interesting to note, however, that if one were
to simultaneously scale the values of all the moduli
to be very small, the spectrum would be more and
more unified: the relative distances between singu-
larities shrink, unifying the coefficients in the su-
perpotential; and the volumes of the co-dimension
four singularities (if realized in a compact manifold)
would approach one another, resulting in a unifica-
tion of their gauge couplings. What this may mean
phenomenologically remains to be understood.
In this paper we have described a local, purely ge-
ometric framework in which gauge symmetry ‘break-
ing’ can be re-cast as a problem of moduli fixing—
and in which the same moduli which describe this
geometric ‘unfolding’ also determine the physics of
massless matter. And although we still do not un-
derstand the mechanisms by which these moduli are
fixed, the landscape of possibilities is already enor-
mously reduced: what would have been the hun-
dreds of parameters describing the relative positions
on the compactification manifold of the Standard
Model’s three families worth of matter fields, we
specify them all in terms of only four complex struc-
ture moduli which describe the unfolding of an iso-
lated E8 singularity. And the fact that three families
emerges is group-theoretic and not added by hand.
VII. ACKNOWLEDGEMENTS
It is a pleasure to thank helpful discussions with
and insightful comments of Herman Verlinde, Sergei
Gukov, Gordy Kane, Paul Langacker, Edward Wit-
ten, Cumrun Vafa, Brent Nelson, Malcolm Perry,
Dmitry Malyshev, Matthew Buican, Piyush Kumar,
and Konstantin Bobkov.
This work was funded in part by a Graduate Re-
search Fellowship from the National Science Foun-
dation.
12 Of course, this can only be discussed concretely when the
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