Multiple Unfoldings of Orbifold Singularities:
Engineering Geometric Analogies to Unification
Jacob L. Bourjaily∗
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(Dated: 2nd April 2007)
Katz and Vafa [1] showed how charged matter can arise geometrically by the deformation of ADE-
type orbifold singularities in type IIa, M-theory, and F-theory compactifications. In this paper we use
those same basic ingredients, used there to geometrically engineer specific matter representations,
here to deform the compactification manifold itself in a way which naturally compliments many
features of unified model building. We realize this idea explicitly by deforming a manifold engineered
to give rise to an SU5 grand unified model into a one giving rise to the Standard Model. In this
framework, the relative local positions of the singularities giving rise to Standard Model fields are
specified in terms of the values of a small number of complex structure moduli which deform the
original manifold, greatly reducing the arbitrariness of their relative positions.
I. INTRODUCTION
One of the ways in which a gauge theory with
massless charged matter can arise in type IIa, M-
theory, and F-theory is known as geometrical engi-
neering. In this framework, gauge theory at low en-
ergy arises from co-dimension four singular surfaces
in the compactification manifold [2] and charged
matter arises as isolated points (curves in F-theory)
on these surfaces over which the singularity is en-
hanced. Katz and Vafa [1] constructed explicit ex-
amples of local geometry which would give rise to dif-
ferent representations of various gauge groups. Their
work was presented explicitly in the language of type
IIa or F-theory, but the general results have been
shown to apply much more broadly to M-theory as
well [3, 4, 5, 6, 7].
The picture of matter and gauge theory arising
from pure geometry via singular structures has been
used very fruitfully in much of the progress of M-
theory phenomenology. In [8] Witten engineered
an interesting phenomenological model in M-theory
which could possibly solve the Higgs doublet-triplet
splitting problem; this model was explored in great
detail together with Friedmann in [9]. There, the ex-
plicit topology of the ADE-singular surface and the
relative locations of all the isolated conical singulari-
ties was motivated by phenomenology—the descrip-
tion of the geometry of the singularities themselves
was taken for granted.
Unlike model building with D-branes, for exam-
ple, geometrical engineering as it has been under-
stood provides little information about the number,
type, and relative locations of the many different sin-
gularities needed for any phenomenological model.
This information must either come a posteriori from
phenomenological success or via duality to a con-
crete string model. But recent successes in M-theory
model building (for example, [10, 11]) motivate a
new look at how to describe the relative structure
∗Electronic address: jbourjai@princeton.edu
of singularities—at least locally—within the frame-
work of geometrical engineering itself.
In this paper, we reduce the apparent arbitrari-
ness of the number and relative positions of the sin-
gularities required by low-energy phenomenology by
showing how they can be obtained from deforming
a smaller number of singularities in a more unified
model. In section II we review the ingredients of
geometrical engineering as described in [1]. The ba-
sic framework is then interpreted in a novel way in
section III to relate manifolds with matter singular-
ities to those with more or less symmetry. The idea
is used explicitly to deform an SU5 grand unified
model into the Standard model.
To be clear, as in [1] our results apply only strictly
to N = 2 models from type IIa compactifications
or N = 1 models from F-theory compactifications1;
but we suspect that this framework has an M-theory
analogue in the spirit of [7].
II. GEOMETRICAL ENGINEERING
In the framework of geometrical engineering the
compactification manifold is described as a fibration
of (singular) K3 surfaces over a base space of appro-
priate dimension. The collection of point-like (co-
dimension four) singularities of the K3 fibres would
then be a co-dimension four surface in the compact-
ified manifold, giving rise to gauge theory of type
corresponding to the singularities on each K3 fibre.
Table I lists polynomials in C3 whose solutions can
be (locally) taken to be the fibres for each corre-
sponding gauge group.
One of the strengths of this framework is its gen-
erality: the local geometry is specified in terms of
1 We essentially describe non-compact Calabi-Yau three-
folds which are K3-fibrations over C1. If the C1-base is
fibred over CP1 as an O(−2) bundle, for example, then
the total space will be a Calabi-Yau four-fold upon which
F-theory can be compactified, giving rise to an N = 1
theory.
http://arxiv.org/abs/0704.0444v1
mailto:jbourjai@princeton.edu
TABLE I: Hypersurfaces in C3 giving rise to the desired
orbifold singularities.
Gauge group Polynomial
SUn (≡ An−1) xy = z
SO2n (≡ Dn) x
2 + y2z = zn−1
2 = y3 + z4
2 + y3 = yz3
2 + y3 = z5
the K3 fibres, so that the description applies equally
well to compactifications in type IIa, M-theory, and
F-theory—the difference being the dimension of the
space over which the surfaces in Table I are fibred.
To obtain massless charged matter, however, addi-
tional structure is necessary. Specifically, at isolated
points (in type IIa or M-theory) on the co-dimension
four singular surface, the type of singularity of the
K3 fibres must be enhanced by one rank. Mathe-
matically, this requires that one can describe how
the various polynomials in Table I can be deformed
into each other; and the possible two-dimensional
deformations have been classified [12].
For example, to describe the embedding of a mass-
less 5 of SU5 in type IIa, you would need to start
with aK3-fibred Calabi-Yau where each of the fibres
are of the type giving rise to SU5 gauge theory. From
Table I we see that these four-dimensional fibres are
locally the set of solutions to the equation
xy = z5, (1)
in C3. Now, to obtain matter in the 5 representa-
tion, there would need to be an isolated point some-
where on the two-dimensional base space where the
fibre is enhanced to SU6, [1]. A description of the
local geometry can be given by
xy = (z + 5t)(z − t)5, (2)
where t is a complex coordinate on the base over
which the K3’s are fibred. Notice that when t = 0
the equation describes precisely the fibre which
would have given rise to SU6 gauge theory if it were
fibred over the entire base manifold. However, be-
cause it is the fibre only over the origin in the com-
plex t-plane, there is no SU6 gauge theory. Equation
(2) is said to describe the ‘resolution’ SU6 → SU5,
which is found to give rise to SU5 gauge theory at
low energy with a single massless 5 at t = 0. This
and many other explicit examples of such resolutions
and the matter representations obtained are given in
One subtlety which makes the description above
not automatically apply to M-theory constructions,
however, is that in equation (2) the complex pa-
rameter t is two-dimensional: taken as a coor-
dinate over which the K3 surfaces are fibred, it
gives rise to a six-dimensional compactification man-
ifold. In M-theory, co-dimension four singularities
are three-dimensional and chiral matter would live
at isolated points on these three dimensional orb-
ifold singularities. So in M-theory the resolution
SU6 → SU5 would need a three-dimensional defor-
mation. Morally, the structure is identical to that
described in equation (2), but the parameter t must
be upgraded to describe three-dimensional deforma-
tions. This can be done in terms of hyper-Kähler
quotients. We suspect that all the resolutions de-
scribed explicitly for type IIa here and in [1] can be
upgraded to three-dimensional deformations needed
in M-theory, and in many cases these generalizations
have already been given [4, 5, 7].
III. ENGINEERING GEOMETRIC
ANALOGIES TO UNIFICATION
The main result of this paper is that distinct con-
ical singularities on a surface with some gauge sym-
metry can be deformed into each other in ways anal-
ogous to unification; and conversely, that a descrip-
tion of a single matter field in a unified theory can
be ‘unfolded’ into distinct matter fields in a theory
of lower gauge symmetry. Because the tools used to
perform these unification-like deformations are pre-
cisely the same as those used to describe the singu-
larities themselves, some care must be taken to avoid
unnecessary confusion.
We will start by reinterpreting the tools used
above to engineer charged matter, and then we will
use both interpretations simultaneously to construct
explicit examples of the geometric analogue to uni-
fied model building.
Consider again the resolution SU6 → SU5 de-
scribed by
xy = (z + 5s)(z − s)5, (3)
where we have replaced t 7→ s from equation (2)
to make a interpretative distinction that will soon
become clear. We propose to momentarily discuss
pure gauge theory and ignore any description of mat-
ter. With this in mind, take a fixed (real) two-
dimensional neighborhood over which every point is
fibred by the solutions to equation (3) for any fixed
value of s. Because the fibres are the same every-
where on the manifold, there is no matter: for any
s the geometry would give rise to pure gauge theory
at low energy. For s 6= 0 solutions to equation (3)
are SU5 fibres and so the compactification manifold
would give rise to pure SU5. However, when s = 0
the fibres are all SU6 and so the low-energy theory
would be pure SU6. Therefore s is a ‘global’ param-
eter which deforms the gauge content of the theory
such that for arbitrary values of s 6= 0 the theory is
pure SU5 and for s = 0 it is pure SU6. That this
deformation is ‘smooth’ is apparent at least when
s 6= 0.
An obvious question to ask is how this framework
applies when conical singularities are present. We
(1, 2)(3, 1)
SU6 SU5
SU3×SU3SU4×SU2
FIG. 1: The plane describing the deformation of a
theory with a single of SU into one of SU SU
gauge theory with one ( and one ( as a
function of as described by equation (4). For a fixed
value of , the base space over which solutions to (4) are
fibred are indicated by the black line. Notice that the
relative positions of the two isolated (conical) singulari
ties are fixed by
will show that when the ADE-surface singularity
changes because of some complex structure modu-
lus such as above, the conical singularities giving
rise to charged matter (often) behave as one would
expect from unified model building intuition. This
is best demonstrated with explicit examples.
Suppose that the singular 3 surfaces are fibred
over a two-dimensional base space with local com-
plex coordinate . And say the four-dimensional fi-
bre over the point is given by the solutions to
xy = ( + 5 )( + 3 (4)
for a given value of , which is now to be interpreted
as a complex structure modulus deforming the entire
local geometry near = 0. When = 0 the geometry
is of course identical to our previous description of
SU SU and so the theory would be SU with a
single massless located at = 0.
Consider now to be fixed at some non-zero value.
The gauge theory is then SU SU : for generic
values of , the fibres given by equation (4) have
two singular points, at + 3 = 0 and
= 0, and so the union of these
points over the base manifold coordinatized by will
be two distinct, two-dimensional singular surfaces:
one giving rise to SU and the other SU . These
surfaces become coincident as a single SU surface
when = 0.
Along the complex -plane, there are two iso-
lated points over which the singularities are en-
hanced: at s/2 the fibre is visibly SU SU
and at s/3 the fibre is SU SU . There-
fore the theory has two two charged, massless fields,
in the ( and ( representations of
SU SU at s/2 and s/3, respec-
tively. Figure 1 indicates the singularity structure
as a function of
Notice how this description parallels unified model
building: the = 0 theory of one of SU de-
forms smoothly into one with (
of SU SU
Similarly, we may ask how a 10 of SU would de-
form into distinct singularities supporting Standard
Model matter fields. The fibre structure giving rise
to a massless 10 of SU is given as follows. Let
be a local coordinate on the base space over which
fibres are given by solutions to
+ 2yt 10 ; (5)
at = 0, equation (5) describes an SO10 fibre, while
for = 0 the fibres are SU —although in this case
the result is harder to read off. This resolution,
SO10 SU , gives rise to a 10 of SU [1].
Following the same idea as before, the deformation
of this singularity into SU SU is given by
+ 2 + ( + ( (6)
where is again interpreted as a complex structure
modulus deforming the geometry near the singular-
ity. Notice as before that = 0 describes an SU the-
ory with one massless 10 located at = 0. However,
for = 0 there are again two orbifold singularities
corresponding to SU SU gauge theory. At
three distinct points on the complex plane the rank
of the fibre is enhanced: = 0, and
give rise to matter in the ( , ( , and
representations of SU SU , respec-
tively. The structure of the deformation achieved by
varying is shown in Figure 2.
Again, our intuition from unified model building
is realized naturally in this framework.
FIG. 1: The t-s plane describing the deformation of a
theory with a single 5 of SU5 into one of SU3 × SU2 × U1
gauge theory with one (3,1)1/3 and one (1, 2)−1/2 as a
function of s as described by equation (4). For a fixed
value of s, the base space over which solutions to (4) are
fibred are indicated by the black line. Notice that the
relative positions of the two isolated (conical) singulari-
ties are fixed by s.
will show that when the ADE-surface singularity
changes because of some complex structure modu-
lus such as s above, the conical singularities giving
rise to charged matter (often) behave as one would
expect from unified model building intuition. This
is best demonstrated with explicit examples.
Suppose that the singular K3 surfaces are fibred
over a two-dimensional base space with local com-
plex coordinate t. And say the four-dimensional fi-
bre over the point t is given by the solutions to
xy = (z + 5t)(z − t+ 3s)2(z − t− 2s)3, (4)
for a given value of s, which is now to be interpreted
as a complex structure modulus deforming the entire
local geometry near t = 0. When s = 0 the geometry
is of course identical to our previous description of
SU6 → SU5 and so the theory would be SU5 with a
single massless 5 located at t = 0.
Consider now s to be fixed at some non-zero value.
The gauge theory is then SU3×SU2×U1: for generic
values of t, the fibres given by equation (4) have
two singular points, at x = y = z − t+ 3s = 0 and
x = y = z − t− 2s = 0, and so the union of these
points over the base manifold coordinatized by t will
be two distinct, two-dimensional singular surfaces:
one giving rise to SU2 and the other SU3. These
surfaces become coincident as a single SU5 surface
when s = 0.
Along the complex t-plane, there are two iso-
lated points over which the singularities are en-
hanced: at t = s/2 the fibre is visibly SU3 × SU3,
and at t = −s/3 the fibre is SU4 × SU2. There-
fore the theory has two two charged, massless fields,
in the (1,2)
−1/2 and (3,1)1/3 representations of
SU3 × SU2 × U1 at t = s/2 and t = −s/3, respec-
tively. Figure 1 indicates the singularity structure
as a function of s.
Notice how this description parallels unified model
building: the s = 0 theory of one 5 of SU5 de-
forms smoothly into one with (3,1)1/3 ⊕ (1,2)−1/2
of SU3 × SU2 × U1.
Similarly, we may ask how a 10 of SU5 would de-
form into distinct singularities supporting Standard
Model matter fields. The fibre structure giving rise
to a massless 10 of SU5 is given as follows. Let t
be a local coordinate on the base space over which
fibres are given by solutions to
x2 + y2z + 2yt5 =
z + t2
− t10
; (5)
at t = 0, equation (5) describes an SO10 fibre, while
for t 6= 0 the fibres are SU5—although in this case
the result is harder to read off. This resolution,
SO10 → SU5, gives rise to a 10 of SU5 [1].
Following the same idea as before, the deformation
of this singularity into SU3 × SU2 is given by
x2 + y2z + 2y(t+ s)3(t− s)2 =
z + (t− s)2
z + (t+ s)2
− (t− s)4(t+ s)6
, (6)
where s is again interpreted as a complex structure
modulus deforming the geometry near the singular-
ity. Notice as before that s = 0 describes an SU5 the-
ory with one massless 10 located at t = 0. However,
for s 6= 0 there are again two orbifold singularities
corresponding to SU3 × SU2 × U1 gauge theory. At
three distinct points on the complex t plane the rank
of the fibre is enhanced: t = −s, t = 0, and t = s
give rise to matter in the (3,1)
−2/3, (3,2)1/6, and
(1,1)1 representations of SU3 × SU2 × U1, respec-
tively. The structure of the deformation achieved by
varying s is shown in Figure 2.
Again, our intuition from unified model building
is realized naturally in this framework.
(3, 2)
(3, 1) (1, 1)
SO10 SU5
SU3×SU2×SU2SU4×SU2
FIG. 2: The plane describing the deformation of a
theory with a single 10 of SU into one of SU SU
gauge theory, with matter content (
, as a function of as described by equation (6).
For a fixed value of , the base space over which solutions
to (6) are fibred are indicated by the black line. Notice
that the relative positions of the three isolated (conical)
singularities are fixed by
IV. DISCUSSION
One of the primary reasons why geometrical engi-
neering had not been more widely used phenomeno-
logically is because the number, type, and relative lo-
cations of the singularities giving rise to various mat-
ter fields were explicitly ad hoc: the inherent local
framework prevented relationships between distinct
singularities from being discussed. In this paper, we
have shown a framework in which these questions
can be addressed concretely, systematically reduc-
ing the arbitrariness of these models.
Of course, the local nature of geometrical engi-
neering is still inherent in this framework, and con-
tinues to prevent us from addressing questions about
the global structure such as stability, quantum grav-
ity, and the quantization of seemingly continuous
parameters like . However, in the spirit of [13], we
think that local engineering is a good step toward re-
alistic string phenomenology, and may perhaps offer
new insights.
In this paper we explicitly illustrated the geomet-
ric unfolding of the matter content of an SU grand
unified model into the Standard Model. But the pro-
cedure can easily be generalized. It is not difficult to
see how this will work for a more unified theory. For
example, one can envision how an entire family could
unfold out of a single SO10 resolution (which
starts as a 16 of SO10), or how all three families of
the Standard Mode could be unfolded out of a sin-
gle SO10 SU or SU resolution.
However, these examples require more sophisticated
tools of analysis, and so we have chosen to describe
these in a forthcoming work.
V. ACKNOWLEDGEMENTS
This work originated from discussions with Mal-
colm Perry whose insights drove this work forward in
its earliest steps. The author also appreciates helpful
discussions, comments, and suggestions from Her-
man Verlinde, Sergei Gukov, Gordon Kane, Edward
Witten, Paul Langacker, Bobby Acharya, Dmitry
Malyshev, Matthew Buican, Piyush Kumar, and
Konstantin Bobkov.
This research was supported in part by the Michi-
gan Center for Theoretical Physics and a Gradu-
ate Research Fellowship from the National Science
Foundation.
[1] S. Katz and C. Vafa, “Matter from Geometry,”
Nucl. Phys., vol. B497, pp. 146–154, 1997, hep-
th/9606086.
[2] A. Klemm, W. Lerche, and P. Mayr, “K3 Fibrations
and Heterotic Type II String Duality,” Phys. Lett.
vol. B357, pp. 313–322, 1995, hep-th/9506112.
[3] M. Atiyah and E. Witten, “M-theory Dynamics on
a Manifold of Holonomy,” Adv. Theor. Math.
Phys., vol. 6, pp. 1–106, 2003, hep-th/0107177.
[4] E. Witten, “Anomaly Cancellation on Mani-
folds,” 2001, hep-th/0108165.
[5] B. Acharya and E. Witten, “Chiral Fermions from
Manifolds of Holonomy,” 2001, hep-th/0109152.
[6] B. S. Acharya and S. Gukov, “M-theory and Singu-
larities of Exceptional Holonomy Manifolds,” Phys.
Rept., vol. 392, pp. 121–189, 2004, hep-th/0409191.
[7] P. Berglund and A. Brandhuber, “Matter from
Manifolds,” Nucl. Phys., vol. B641, pp. 351–375,
2002, hep-th/0205184.
[8] E. Witten, “Deconstruction, Holonomy, and
Doublet-Triplet Splitting,” 2001, hep-ph/0201018.
[9] T. Friedmann and E. Witten, “Unification Scale,
Proton Decay, and Manifolds of G(2) Holonomy,”
Adv. Theor. Math. Phys., vol. 7, pp. 577–617, 2003,
hep-th/0211269.
[10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and
D. Vaman, “An M Theory Solution to the Hierarchy
Problem,” Phys. Rev. Lett., vol. 97, p. 191601, 2006,
hep-th/0606262.
[11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar,
and J. Shao, “Explaining the Electroweak Scale
and Stabilizing Moduli in M-Theory,” 2007, hep-
th/0701034.
[12] S. Katz and D. Morrison, “Gorenstein Threefold
Singularities with Small Resolutions via Invariant
Theory for Weyl Groups,” J. Algebraic Geometry
vol. 1, pp. 449–530, 1992.
[13] H. Verlinde and M. Wijnholt, “Building the Stan-
dard Model on a D3-Brane,” JHEP, vol. 01, p. 106,
2007, hep-th/0508089.
FIG. 2: The t-s plane describing the deformation of a
theory with a single 10 of SU5 into one of SU3×SU2×U1
gauge theory, with matter content (3,1)
−2/3⊕(3, 2)1/6⊕
(1,1)1, as a function of s as described by equation (6).
For a fixed value of s, the base space over which solutions
to (6) are fibred are indicated by the black line. Notice
that the relative positions of the three isolated (conical)
singularities are fixed by s.
IV. DISCUSSION
One of the primary reasons why geometrical engi-
neering had not been more widely used phenomeno-
logically is because the number, type, and relative lo-
cations of the singularities giving rise to various mat-
ter fields were explicitly ad hoc: the inherent local
framework prevented relationships between distinct
singularities from being discussed. In this paper, we
have shown a framework in which these questions
can be addressed concretely, systematically reduc-
ing the arbitrariness of these models.
Of course, the local nature of geometrical engi-
neering is still inherent in this framework, and con-
tinues to prevent us from addressing questions about
the global structure such as stability, quantum grav-
ity, and the quantization of seemingly continuous
parameters like s. However, in the spirit of [13], we
think that local engineering is a good step toward re-
alistic string phenomenology, and may perhaps offer
new insights.
In this paper we explicitly illustrated the geomet-
ric unfolding of the matter content of an SU5 grand
unified model into the Standard Model. But the pro-
cedure can easily be generalized. It is not difficult to
see how this will work for a more unified theory. For
example, one can envision how an entire family could
unfold out of a single E6 → SO10 resolution (which
starts as a 16 of SO10), or how all three families of
the Standard Mode could be unfolded out of a sin-
gle E8 → SO10×SU3 or E8 → E6 ×SU2 resolution.
However, these examples require more sophisticated
tools of analysis, and so we have chosen to describe
these in a forthcoming work.
V. ACKNOWLEDGEMENTS
This work originated from discussions with Mal-
colm Perry whose insights drove this work forward in
its earliest steps. The author also appreciates helpful
discussions, comments, and suggestions from Her-
man Verlinde, Sergei Gukov, Gordon Kane, Edward
Witten, Paul Langacker, Bobby Acharya, Dmitry
Malyshev, Matthew Buican, Piyush Kumar, and
Konstantin Bobkov.
This research was supported in part by the Michi-
gan Center for Theoretical Physics and a Gradu-
ate Research Fellowship from the National Science
Foundation.
[1] S. Katz and C. Vafa, “Matter from Geome-
try,” Nucl. Phys., vol. B497, pp. 146–154, 1997,
hep-th/9606086.
[2] A. Klemm, W. Lerche, and P. Mayr, “K3 Fibrations
and Heterotic Type II String Duality,” Phys. Lett.,
vol. B357, pp. 313–322, 1995, hep-th/9506112.
[3] M. Atiyah and E. Witten, “M-theory Dynamics on
a Manifold of G2 Holonomy,” Adv. Theor. Math.
Phys., vol. 6, pp. 1–106, 2003, hep-th/0107177.
[4] E. Witten, “Anomaly Cancellation on G2 Mani-
folds,” 2001, hep-th/0108165.
[5] B. Acharya and E. Witten, “Chiral Fermions from
Manifolds of G2 Holonomy,” 2001, hep-th/0109152.
[6] B. S. Acharya and S. Gukov, “M-theory and Singu-
larities of Exceptional Holonomy Manifolds,” Phys.
Rept., vol. 392, pp. 121–189, 2004, hep-th/0409191.
[7] P. Berglund and A. Brandhuber, “Matter from G2
Manifolds,” Nucl. Phys., vol. B641, pp. 351–375,
2002, hep-th/0205184.
[8] E. Witten, “Deconstruction, G2 Holonomy, and
Doublet-Triplet Splitting,” 2001, hep-ph/0201018.
[9] T. Friedmann and E. Witten, “Unification Scale,
Proton Decay, and Manifolds of G(2) Holonomy,”
Adv. Theor. Math. Phys., vol. 7, pp. 577–617, 2003,
hep-th/0211269.
[10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and
D. Vaman, “An M Theory Solution to the Hierarchy
Problem,” Phys. Rev. Lett., vol. 97, p. 191601, 2006,
hep-th/0606262.
[11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Ku-
mar, and J. Shao, “Explaining the Electroweak
Scale and Stabilizing Moduli in M-Theory,” 2007,
hep-th/0701034.
[12] S. Katz and D. Morrison, “Gorenstein Threefold
Singularities with Small Resolutions via Invariant
Theory for Weyl Groups,” J. Algebraic Geometry,
vol. 1, pp. 449–530, 1992.
[13] H. Verlinde and M. Wijnholt, “Building the Stan-
dard Model on a D3-Brane,” JHEP, vol. 01, p. 106,
2007, hep-th/0508089.
http://arxiv.org/abs/hep-th/9606086
http://arxiv.org/abs/hep-th/9506112
http://arxiv.org/abs/hep-th/0107177
http://arxiv.org/abs/hep-th/0108165
http://arxiv.org/abs/hep-th/0109152
http://arxiv.org/abs/hep-th/0409191
http://arxiv.org/abs/hep-th/0205184
http://arxiv.org/abs/hep-ph/0201018
http://arxiv.org/abs/hep-th/0211269
http://arxiv.org/abs/hep-th/0606262
http://arxiv.org/abs/hep-th/0701034
http://arxiv.org/abs/hep-th/0508089

  Katz and Vafa showed how charged matter can arise geometrically by the
deformation of ADE-type orbifold singularities in type IIa, M-theory, and
F-theory compactifications. In this paper we use those same basic ingredients,
used there to geometrically engineer specific matter representations, here to
deform the compactification manifold itself in a way which naturally
compliments many features of unified model building. We realize this idea
explicitly by deforming a manifold engineered to give rise to an $SU_5$ grand
unified model into a one giving rise to the Standard Model. In this framework,
the relative local positions of the singularities giving rise to Standard Model
fields are specified in terms of the values of a small number of complex
structure moduli which deform the original manifold, greatly reducing the
arbitrariness of their relative positions.

Multiple Unfoldings of Orbifold Singularities:
Engineering Geometric Analogies to Unification
Jacob L. Bourjaily∗
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(Dated: 2nd April 2007)
Katz and Vafa [1] showed how charged matter can arise geometrically by the deformation of ADE-
type orbifold singularities in type IIa, M-theory, and F-theory compactifications. In this paper we use
those same basic ingredients, used there to geometrically engineer specific matter representations,
here to deform the compactification manifold itself in a way which naturally compliments many
features of unified model building. We realize this idea explicitly by deforming a manifold engineered
to give rise to an SU5 grand unified model into a one giving rise to the Standard Model. In this
framework, the relative local positions of the singularities giving rise to Standard Model fields are
specified in terms of the values of a small number of complex structure moduli which deform the
original manifold, greatly reducing the arbitrariness of their relative positions.
I. INTRODUCTION
One of the ways in which a gauge theory with
massless charged matter can arise in type IIa, M-
theory, and F-theory is known as geometrical engi-
neering. In this framework, gauge theory at low en-
ergy arises from co-dimension four singular surfaces
in the compactification manifold [2] and charged
matter arises as isolated points (curves in F-theory)
on these surfaces over which the singularity is en-
hanced. Katz and Vafa [1] constructed explicit ex-
amples of local geometry which would give rise to dif-
ferent representations of various gauge groups. Their
work was presented explicitly in the language of type
IIa or F-theory, but the general results have been
shown to apply much more broadly to M-theory as
well [3, 4, 5, 6, 7].
The picture of matter and gauge theory arising
from pure geometry via singular structures has been
used very fruitfully in much of the progress of M-
theory phenomenology. In [8] Witten engineered
an interesting phenomenological model in M-theory
which could possibly solve the Higgs doublet-triplet
splitting problem; this model was explored in great
detail together with Friedmann in [9]. There, the ex-
plicit topology of the ADE-singular surface and the
relative locations of all the isolated conical singulari-
ties was motivated by phenomenology—the descrip-
tion of the geometry of the singularities themselves
was taken for granted.
Unlike model building with D-branes, for exam-
ple, geometrical engineering as it has been under-
stood provides little information about the number,
type, and relative locations of the many different sin-
gularities needed for any phenomenological model.
This information must either come a posteriori from
phenomenological success or via duality to a con-
crete string model. But recent successes in M-theory
model building (for example, [10, 11]) motivate a
new look at how to describe the relative structure
∗Electronic address: jbourjai@princeton.edu
of singularities—at least locally—within the frame-
work of geometrical engineering itself.
In this paper, we reduce the apparent arbitrari-
ness of the number and relative positions of the sin-
gularities required by low-energy phenomenology by
showing how they can be obtained from deforming
a smaller number of singularities in a more unified
model. In section II we review the ingredients of
geometrical engineering as described in [1]. The ba-
sic framework is then interpreted in a novel way in
section III to relate manifolds with matter singular-
ities to those with more or less symmetry. The idea
is used explicitly to deform an SU5 grand unified
model into the Standard model.
To be clear, as in [1] our results apply only strictly
to N = 2 models from type IIa compactifications
or N = 1 models from F-theory compactifications1;
but we suspect that this framework has an M-theory
analogue in the spirit of [7].
II. GEOMETRICAL ENGINEERING
In the framework of geometrical engineering the
compactification manifold is described as a fibration
of (singular) K3 surfaces over a base space of appro-
priate dimension. The collection of point-like (co-
dimension four) singularities of the K3 fibres would
then be a co-dimension four surface in the compact-
ified manifold, giving rise to gauge theory of type
corresponding to the singularities on each K3 fibre.
Table I lists polynomials in C3 whose solutions can
be (locally) taken to be the fibres for each corre-
sponding gauge group.
One of the strengths of this framework is its gen-
erality: the local geometry is specified in terms of
1 We essentially describe non-compact Calabi-Yau three-
folds which are K3-fibrations over C1. If the C1-base is
fibred over CP1 as an O(−2) bundle, for example, then
the total space will be a Calabi-Yau four-fold upon which
F-theory can be compactified, giving rise to an N = 1
theory.
http://arxiv.org/abs/0704.0444v1
mailto:jbourjai@princeton.edu
TABLE I: Hypersurfaces in C3 giving rise to the desired
orbifold singularities.
Gauge group Polynomial
SUn (≡ An−1) xy = z
SO2n (≡ Dn) x
2 + y2z = zn−1
2 = y3 + z4
2 + y3 = yz3
2 + y3 = z5
the K3 fibres, so that the description applies equally
well to compactifications in type IIa, M-theory, and
F-theory—the difference being the dimension of the
space over which the surfaces in Table I are fibred.
To obtain massless charged matter, however, addi-
tional structure is necessary. Specifically, at isolated
points (in type IIa or M-theory) on the co-dimension
four singular surface, the type of singularity of the
K3 fibres must be enhanced by one rank. Mathe-
matically, this requires that one can describe how
the various polynomials in Table I can be deformed
into each other; and the possible two-dimensional
deformations have been classified [12].
For example, to describe the embedding of a mass-
less 5 of SU5 in type IIa, you would need to start
with aK3-fibred Calabi-Yau where each of the fibres
are of the type giving rise to SU5 gauge theory. From
Table I we see that these four-dimensional fibres are
locally the set of solutions to the equation
xy = z5, (1)
in C3. Now, to obtain matter in the 5 representa-
tion, there would need to be an isolated point some-
where on the two-dimensional base space where the
fibre is enhanced to SU6, [1]. A description of the
local geometry can be given by
xy = (z + 5t)(z − t)5, (2)
where t is a complex coordinate on the base over
which the K3’s are fibred. Notice that when t = 0
the equation describes precisely the fibre which
would have given rise to SU6 gauge theory if it were
fibred over the entire base manifold. However, be-
cause it is the fibre only over the origin in the com-
plex t-plane, there is no SU6 gauge theory. Equation
(2) is said to describe the ‘resolution’ SU6 → SU5,
which is found to give rise to SU5 gauge theory at
low energy with a single massless 5 at t = 0. This
and many other explicit examples of such resolutions
and the matter representations obtained are given in
One subtlety which makes the description above
not automatically apply to M-theory constructions,
however, is that in equation (2) the complex pa-
rameter t is two-dimensional: taken as a coor-
dinate over which the K3 surfaces are fibred, it
gives rise to a six-dimensional compactification man-
ifold. In M-theory, co-dimension four singularities
are three-dimensional and chiral matter would live
at isolated points on these three dimensional orb-
ifold singularities. So in M-theory the resolution
SU6 → SU5 would need a three-dimensional defor-
mation. Morally, the structure is identical to that
described in equation (2), but the parameter t must
be upgraded to describe three-dimensional deforma-
tions. This can be done in terms of hyper-Kähler
quotients. We suspect that all the resolutions de-
scribed explicitly for type IIa here and in [1] can be
upgraded to three-dimensional deformations needed
in M-theory, and in many cases these generalizations
have already been given [4, 5, 7].
III. ENGINEERING GEOMETRIC
ANALOGIES TO UNIFICATION
The main result of this paper is that distinct con-
ical singularities on a surface with some gauge sym-
metry can be deformed into each other in ways anal-
ogous to unification; and conversely, that a descrip-
tion of a single matter field in a unified theory can
be ‘unfolded’ into distinct matter fields in a theory
of lower gauge symmetry. Because the tools used to
perform these unification-like deformations are pre-
cisely the same as those used to describe the singu-
larities themselves, some care must be taken to avoid
unnecessary confusion.
We will start by reinterpreting the tools used
above to engineer charged matter, and then we will
use both interpretations simultaneously to construct
explicit examples of the geometric analogue to uni-
fied model building.
Consider again the resolution SU6 → SU5 de-
scribed by
xy = (z + 5s)(z − s)5, (3)
where we have replaced t 7→ s from equation (2)
to make a interpretative distinction that will soon
become clear. We propose to momentarily discuss
pure gauge theory and ignore any description of mat-
ter. With this in mind, take a fixed (real) two-
dimensional neighborhood over which every point is
fibred by the solutions to equation (3) for any fixed
value of s. Because the fibres are the same every-
where on the manifold, there is no matter: for any
s the geometry would give rise to pure gauge theory
at low energy. For s 6= 0 solutions to equation (3)
are SU5 fibres and so the compactification manifold
would give rise to pure SU5. However, when s = 0
the fibres are all SU6 and so the low-energy theory
would be pure SU6. Therefore s is a ‘global’ param-
eter which deforms the gauge content of the theory
such that for arbitrary values of s 6= 0 the theory is
pure SU5 and for s = 0 it is pure SU6. That this
deformation is ‘smooth’ is apparent at least when
s 6= 0.
An obvious question to ask is how this framework
applies when conical singularities are present. We
(1, 2)(3, 1)
SU6 SU5
SU3×SU3SU4×SU2
FIG. 1: The plane describing the deformation of a
theory with a single of SU into one of SU SU
gauge theory with one ( and one ( as a
function of as described by equation (4). For a fixed
value of , the base space over which solutions to (4) are
fibred are indicated by the black line. Notice that the
relative positions of the two isolated (conical) singulari
ties are fixed by
will show that when the ADE-surface singularity
changes because of some complex structure modu-
lus such as above, the conical singularities giving
rise to charged matter (often) behave as one would
expect from unified model building intuition. This
is best demonstrated with explicit examples.
Suppose that the singular 3 surfaces are fibred
over a two-dimensional base space with local com-
plex coordinate . And say the four-dimensional fi-
bre over the point is given by the solutions to
xy = ( + 5 )( + 3 (4)
for a given value of , which is now to be interpreted
as a complex structure modulus deforming the entire
local geometry near = 0. When = 0 the geometry
is of course identical to our previous description of
SU SU and so the theory would be SU with a
single massless located at = 0.
Consider now to be fixed at some non-zero value.
The gauge theory is then SU SU : for generic
values of , the fibres given by equation (4) have
two singular points, at + 3 = 0 and
= 0, and so the union of these
points over the base manifold coordinatized by will
be two distinct, two-dimensional singular surfaces:
one giving rise to SU and the other SU . These
surfaces become coincident as a single SU surface
when = 0.
Along the complex -plane, there are two iso-
lated points over which the singularities are en-
hanced: at s/2 the fibre is visibly SU SU
and at s/3 the fibre is SU SU . There-
fore the theory has two two charged, massless fields,
in the ( and ( representations of
SU SU at s/2 and s/3, respec-
tively. Figure 1 indicates the singularity structure
as a function of
Notice how this description parallels unified model
building: the = 0 theory of one of SU de-
forms smoothly into one with (
of SU SU
Similarly, we may ask how a 10 of SU would de-
form into distinct singularities supporting Standard
Model matter fields. The fibre structure giving rise
to a massless 10 of SU is given as follows. Let
be a local coordinate on the base space over which
fibres are given by solutions to
+ 2yt 10 ; (5)
at = 0, equation (5) describes an SO10 fibre, while
for = 0 the fibres are SU —although in this case
the result is harder to read off. This resolution,
SO10 SU , gives rise to a 10 of SU [1].
Following the same idea as before, the deformation
of this singularity into SU SU is given by
+ 2 + ( + ( (6)
where is again interpreted as a complex structure
modulus deforming the geometry near the singular-
ity. Notice as before that = 0 describes an SU the-
ory with one massless 10 located at = 0. However,
for = 0 there are again two orbifold singularities
corresponding to SU SU gauge theory. At
three distinct points on the complex plane the rank
of the fibre is enhanced: = 0, and
give rise to matter in the ( , ( , and
representations of SU SU , respec-
tively. The structure of the deformation achieved by
varying is shown in Figure 2.
Again, our intuition from unified model building
is realized naturally in this framework.
FIG. 1: The t-s plane describing the deformation of a
theory with a single 5 of SU5 into one of SU3 × SU2 × U1
gauge theory with one (3,1)1/3 and one (1, 2)−1/2 as a
function of s as described by equation (4). For a fixed
value of s, the base space over which solutions to (4) are
fibred are indicated by the black line. Notice that the
relative positions of the two isolated (conical) singulari-
ties are fixed by s.
will show that when the ADE-surface singularity
changes because of some complex structure modu-
lus such as s above, the conical singularities giving
rise to charged matter (often) behave as one would
expect from unified model building intuition. This
is best demonstrated with explicit examples.
Suppose that the singular K3 surfaces are fibred
over a two-dimensional base space with local com-
plex coordinate t. And say the four-dimensional fi-
bre over the point t is given by the solutions to
xy = (z + 5t)(z − t+ 3s)2(z − t− 2s)3, (4)
for a given value of s, which is now to be interpreted
as a complex structure modulus deforming the entire
local geometry near t = 0. When s = 0 the geometry
is of course identical to our previous description of
SU6 → SU5 and so the theory would be SU5 with a
single massless 5 located at t = 0.
Consider now s to be fixed at some non-zero value.
The gauge theory is then SU3×SU2×U1: for generic
values of t, the fibres given by equation (4) have
two singular points, at x = y = z − t+ 3s = 0 and
x = y = z − t− 2s = 0, and so the union of these
points over the base manifold coordinatized by t will
be two distinct, two-dimensional singular surfaces:
one giving rise to SU2 and the other SU3. These
surfaces become coincident as a single SU5 surface
when s = 0.
Along the complex t-plane, there are two iso-
lated points over which the singularities are en-
hanced: at t = s/2 the fibre is visibly SU3 × SU3,
and at t = −s/3 the fibre is SU4 × SU2. There-
fore the theory has two two charged, massless fields,
in the (1,2)
−1/2 and (3,1)1/3 representations of
SU3 × SU2 × U1 at t = s/2 and t = −s/3, respec-
tively. Figure 1 indicates the singularity structure
as a function of s.
Notice how this description parallels unified model
building: the s = 0 theory of one 5 of SU5 de-
forms smoothly into one with (3,1)1/3 ⊕ (1,2)−1/2
of SU3 × SU2 × U1.
Similarly, we may ask how a 10 of SU5 would de-
form into distinct singularities supporting Standard
Model matter fields. The fibre structure giving rise
to a massless 10 of SU5 is given as follows. Let t
be a local coordinate on the base space over which
fibres are given by solutions to
x2 + y2z + 2yt5 =
z + t2
− t10
; (5)
at t = 0, equation (5) describes an SO10 fibre, while
for t 6= 0 the fibres are SU5—although in this case
the result is harder to read off. This resolution,
SO10 → SU5, gives rise to a 10 of SU5 [1].
Following the same idea as before, the deformation
of this singularity into SU3 × SU2 is given by
x2 + y2z + 2y(t+ s)3(t− s)2 =
z + (t− s)2
z + (t+ s)2
− (t− s)4(t+ s)6
, (6)
where s is again interpreted as a complex structure
modulus deforming the geometry near the singular-
ity. Notice as before that s = 0 describes an SU5 the-
ory with one massless 10 located at t = 0. However,
for s 6= 0 there are again two orbifold singularities
corresponding to SU3 × SU2 × U1 gauge theory. At
three distinct points on the complex t plane the rank
of the fibre is enhanced: t = −s, t = 0, and t = s
give rise to matter in the (3,1)
−2/3, (3,2)1/6, and
(1,1)1 representations of SU3 × SU2 × U1, respec-
tively. The structure of the deformation achieved by
varying s is shown in Figure 2.
Again, our intuition from unified model building
is realized naturally in this framework.
(3, 2)
(3, 1) (1, 1)
SO10 SU5
SU3×SU2×SU2SU4×SU2
FIG. 2: The plane describing the deformation of a
theory with a single 10 of SU into one of SU SU
gauge theory, with matter content (
, as a function of as described by equation (6).
For a fixed value of , the base space over which solutions
to (6) are fibred are indicated by the black line. Notice
that the relative positions of the three isolated (conical)
singularities are fixed by
IV. DISCUSSION
One of the primary reasons why geometrical engi-
neering had not been more widely used phenomeno-
logically is because the number, type, and relative lo-
cations of the singularities giving rise to various mat-
ter fields were explicitly ad hoc: the inherent local
framework prevented relationships between distinct
singularities from being discussed. In this paper, we
have shown a framework in which these questions
can be addressed concretely, systematically reduc-
ing the arbitrariness of these models.
Of course, the local nature of geometrical engi-
neering is still inherent in this framework, and con-
tinues to prevent us from addressing questions about
the global structure such as stability, quantum grav-
ity, and the quantization of seemingly continuous
parameters like . However, in the spirit of [13], we
think that local engineering is a good step toward re-
alistic string phenomenology, and may perhaps offer
new insights.
In this paper we explicitly illustrated the geomet-
ric unfolding of the matter content of an SU grand
unified model into the Standard Model. But the pro-
cedure can easily be generalized. It is not difficult to
see how this will work for a more unified theory. For
example, one can envision how an entire family could
unfold out of a single SO10 resolution (which
starts as a 16 of SO10), or how all three families of
the Standard Mode could be unfolded out of a sin-
gle SO10 SU or SU resolution.
However, these examples require more sophisticated
tools of analysis, and so we have chosen to describe
these in a forthcoming work.
V. ACKNOWLEDGEMENTS
This work originated from discussions with Mal-
colm Perry whose insights drove this work forward in
its earliest steps. The author also appreciates helpful
discussions, comments, and suggestions from Her-
man Verlinde, Sergei Gukov, Gordon Kane, Edward
Witten, Paul Langacker, Bobby Acharya, Dmitry
Malyshev, Matthew Buican, Piyush Kumar, and
Konstantin Bobkov.
This research was supported in part by the Michi-
gan Center for Theoretical Physics and a Gradu-
ate Research Fellowship from the National Science
Foundation.
[1] S. Katz and C. Vafa, “Matter from Geometry,”
Nucl. Phys., vol. B497, pp. 146–154, 1997, hep-
th/9606086.
[2] A. Klemm, W. Lerche, and P. Mayr, “K3 Fibrations
and Heterotic Type II String Duality,” Phys. Lett.
vol. B357, pp. 313–322, 1995, hep-th/9506112.
[3] M. Atiyah and E. Witten, “M-theory Dynamics on
a Manifold of Holonomy,” Adv. Theor. Math.
Phys., vol. 6, pp. 1–106, 2003, hep-th/0107177.
[4] E. Witten, “Anomaly Cancellation on Mani-
folds,” 2001, hep-th/0108165.
[5] B. Acharya and E. Witten, “Chiral Fermions from
Manifolds of Holonomy,” 2001, hep-th/0109152.
[6] B. S. Acharya and S. Gukov, “M-theory and Singu-
larities of Exceptional Holonomy Manifolds,” Phys.
Rept., vol. 392, pp. 121–189, 2004, hep-th/0409191.
[7] P. Berglund and A. Brandhuber, “Matter from
Manifolds,” Nucl. Phys., vol. B641, pp. 351–375,
2002, hep-th/0205184.
[8] E. Witten, “Deconstruction, Holonomy, and
Doublet-Triplet Splitting,” 2001, hep-ph/0201018.
[9] T. Friedmann and E. Witten, “Unification Scale,
Proton Decay, and Manifolds of G(2) Holonomy,”
Adv. Theor. Math. Phys., vol. 7, pp. 577–617, 2003,
hep-th/0211269.
[10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and
D. Vaman, “An M Theory Solution to the Hierarchy
Problem,” Phys. Rev. Lett., vol. 97, p. 191601, 2006,
hep-th/0606262.
[11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Kumar,
and J. Shao, “Explaining the Electroweak Scale
and Stabilizing Moduli in M-Theory,” 2007, hep-
th/0701034.
[12] S. Katz and D. Morrison, “Gorenstein Threefold
Singularities with Small Resolutions via Invariant
Theory for Weyl Groups,” J. Algebraic Geometry
vol. 1, pp. 449–530, 1992.
[13] H. Verlinde and M. Wijnholt, “Building the Stan-
dard Model on a D3-Brane,” JHEP, vol. 01, p. 106,
2007, hep-th/0508089.
FIG. 2: The t-s plane describing the deformation of a
theory with a single 10 of SU5 into one of SU3×SU2×U1
gauge theory, with matter content (3,1)
−2/3⊕(3, 2)1/6⊕
(1,1)1, as a function of s as described by equation (6).
For a fixed value of s, the base space over which solutions
to (6) are fibred are indicated by the black line. Notice
that the relative positions of the three isolated (conical)
singularities are fixed by s.
IV. DISCUSSION
One of the primary reasons why geometrical engi-
neering had not been more widely used phenomeno-
logically is because the number, type, and relative lo-
cations of the singularities giving rise to various mat-
ter fields were explicitly ad hoc: the inherent local
framework prevented relationships between distinct
singularities from being discussed. In this paper, we
have shown a framework in which these questions
can be addressed concretely, systematically reduc-
ing the arbitrariness of these models.
Of course, the local nature of geometrical engi-
neering is still inherent in this framework, and con-
tinues to prevent us from addressing questions about
the global structure such as stability, quantum grav-
ity, and the quantization of seemingly continuous
parameters like s. However, in the spirit of [13], we
think that local engineering is a good step toward re-
alistic string phenomenology, and may perhaps offer
new insights.
In this paper we explicitly illustrated the geomet-
ric unfolding of the matter content of an SU5 grand
unified model into the Standard Model. But the pro-
cedure can easily be generalized. It is not difficult to
see how this will work for a more unified theory. For
example, one can envision how an entire family could
unfold out of a single E6 → SO10 resolution (which
starts as a 16 of SO10), or how all three families of
the Standard Mode could be unfolded out of a sin-
gle E8 → SO10×SU3 or E8 → E6 ×SU2 resolution.
However, these examples require more sophisticated
tools of analysis, and so we have chosen to describe
these in a forthcoming work.
V. ACKNOWLEDGEMENTS
This work originated from discussions with Mal-
colm Perry whose insights drove this work forward in
its earliest steps. The author also appreciates helpful
discussions, comments, and suggestions from Her-
man Verlinde, Sergei Gukov, Gordon Kane, Edward
Witten, Paul Langacker, Bobby Acharya, Dmitry
Malyshev, Matthew Buican, Piyush Kumar, and
Konstantin Bobkov.
This research was supported in part by the Michi-
gan Center for Theoretical Physics and a Gradu-
ate Research Fellowship from the National Science
Foundation.
[1] S. Katz and C. Vafa, “Matter from Geome-
try,” Nucl. Phys., vol. B497, pp. 146–154, 1997,
hep-th/9606086.
[2] A. Klemm, W. Lerche, and P. Mayr, “K3 Fibrations
and Heterotic Type II String Duality,” Phys. Lett.,
vol. B357, pp. 313–322, 1995, hep-th/9506112.
[3] M. Atiyah and E. Witten, “M-theory Dynamics on
a Manifold of G2 Holonomy,” Adv. Theor. Math.
Phys., vol. 6, pp. 1–106, 2003, hep-th/0107177.
[4] E. Witten, “Anomaly Cancellation on G2 Mani-
folds,” 2001, hep-th/0108165.
[5] B. Acharya and E. Witten, “Chiral Fermions from
Manifolds of G2 Holonomy,” 2001, hep-th/0109152.
[6] B. S. Acharya and S. Gukov, “M-theory and Singu-
larities of Exceptional Holonomy Manifolds,” Phys.
Rept., vol. 392, pp. 121–189, 2004, hep-th/0409191.
[7] P. Berglund and A. Brandhuber, “Matter from G2
Manifolds,” Nucl. Phys., vol. B641, pp. 351–375,
2002, hep-th/0205184.
[8] E. Witten, “Deconstruction, G2 Holonomy, and
Doublet-Triplet Splitting,” 2001, hep-ph/0201018.
[9] T. Friedmann and E. Witten, “Unification Scale,
Proton Decay, and Manifolds of G(2) Holonomy,”
Adv. Theor. Math. Phys., vol. 7, pp. 577–617, 2003,
hep-th/0211269.
[10] B. Acharya, K. Bobkov, G. Kane, P. Kumar, and
D. Vaman, “An M Theory Solution to the Hierarchy
Problem,” Phys. Rev. Lett., vol. 97, p. 191601, 2006,
hep-th/0606262.
[11] B. S. Acharya, K. Bobkov, G. L. Kane, P. Ku-
mar, and J. Shao, “Explaining the Electroweak
Scale and Stabilizing Moduli in M-Theory,” 2007,
hep-th/0701034.
[12] S. Katz and D. Morrison, “Gorenstein Threefold
Singularities with Small Resolutions via Invariant
Theory for Weyl Groups,” J. Algebraic Geometry,
vol. 1, pp. 449–530, 1992.
[13] H. Verlinde and M. Wijnholt, “Building the Stan-
dard Model on a D3-Brane,” JHEP, vol. 01, p. 106,
2007, hep-th/0508089.
http://arxiv.org/abs/hep-th/9606086
http://arxiv.org/abs/hep-th/9506112
http://arxiv.org/abs/hep-th/0107177
http://arxiv.org/abs/hep-th/0108165
http://arxiv.org/abs/hep-th/0109152
http://arxiv.org/abs/hep-th/0409191
http://arxiv.org/abs/hep-th/0205184
http://arxiv.org/abs/hep-ph/0201018
http://arxiv.org/abs/hep-th/0211269
http://arxiv.org/abs/hep-th/0606262
http://arxiv.org/abs/hep-th/0701034
http://arxiv.org/abs/hep-th/0508089
