On Large Volume Moduli Stabilization in IIB Orientifolds
I present a brief introduction to the construction of explicit type IIB orientifold compactifications and summarize the ‘Large Volume Scenario’ on compact four-modulus Calabi–Yau manifolds. I discuss the relevance of this kind of setups for the physical MSSMlike model building and gravitational cos...
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irk-123456789-561892014-02-14T03:10:19Z On Large Volume Moduli Stabilization in IIB Orientifolds Walliser, N.-O. Поля та елементарні частинки I present a brief introduction to the construction of explicit type IIB orientifold compactifications and summarize the ‘Large Volume Scenario’ on compact four-modulus Calabi–Yau manifolds. I discuss the relevance of this kind of setups for the physical MSSMlike model building and gravitational cosmology. These notes are based on my talk at the ‘Bogolyubov Kyiv Conference 2009’ on ‘Modern Problems of Theoretical and Mathematical Physics’. Подано короткий вступ до побудови явних компактифiкацiй IIБ орiєнтованих множин та розглянуто “сценарiй великих об’ємiв” для компактних чотиримодульних Калабi–Яу многовидiв. Обговорено доречнiсть таких схем для побудови фiзичної моделi МССМ-типу та для гравiтацiйної космологiї. Основою роботи є доповiдь на “Боголюбовськiй київськiй конференцiї 2009” з “Сучасних проблем теоретичної та математичної фiзики”. 2010 Article On Large Volume Moduli Stabilization in IIB Orientifolds / N.-O. Walliser // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 481-486. — Бібліогр.: 10 назв. — англ. 2071-0194 PACS 11.25.M http://dspace.nbuv.gov.ua/handle/123456789/56189 en Український фізичний журнал Відділення фізики і астрономії НАН України |
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Поля та елементарні частинки Поля та елементарні частинки Walliser, N.-O. On Large Volume Moduli Stabilization in IIB Orientifolds Український фізичний журнал |
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I present a brief introduction to the construction of explicit type IIB orientifold compactifications and summarize the ‘Large Volume Scenario’ on compact four-modulus Calabi–Yau manifolds. I discuss the relevance of this kind of setups for the physical MSSMlike model building and gravitational cosmology. These notes are based on my talk at the ‘Bogolyubov Kyiv Conference 2009’ on ‘Modern Problems of Theoretical and Mathematical Physics’. |
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Walliser, N.-O. |
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Walliser, N.-O. |
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Walliser, N.-O. |
| title |
On Large Volume Moduli Stabilization in IIB Orientifolds |
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On Large Volume Moduli Stabilization in IIB Orientifolds |
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On Large Volume Moduli Stabilization in IIB Orientifolds |
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On Large Volume Moduli Stabilization in IIB Orientifolds |
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On Large Volume Moduli Stabilization in IIB Orientifolds |
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on large volume moduli stabilization in iib orientifolds |
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Відділення фізики і астрономії НАН України |
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2010 |
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Поля та елементарні частинки |
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http://dspace.nbuv.gov.ua/handle/123456789/56189 |
| citation_txt |
On Large Volume Moduli Stabilization in IIB Orientifolds / N.-O. Walliser // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 481-486. — Бібліогр.: 10 назв. — англ. |
| series |
Український фізичний журнал |
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AT walliserno onlargevolumemodulistabilizationiniiborientifolds |
| first_indexed |
2025-07-05T07:25:18Z |
| last_indexed |
2025-07-05T07:25:18Z |
| _version_ |
1836790913350762496 |
| fulltext |
ON LARGE VOLUME MODULI STABILIZATION
ON LARGE VOLUME MODULI STABILIZATION
IN IIB ORIENTIFOLDS
N.-O. WALLISER
Institute for Theoretical Physics, Vienna University of Technology
(Wiedner Hauptstr. 8-10, 1040 Vienna, Austria; e-mail: walliser@ hep. itp. tuwien. ac. at )
PACS 11.25.M
c©2010
I present a brief introduction to the construction of explicit type
IIB orientifold compactifications and summarize the ‘Large Vol-
ume Scenario’ on compact four-modulus Calabi–Yau manifolds. I
discuss the relevance of this kind of setups for the physical MSSM-
like model building and gravitational cosmology. These notes are
based on my talk at the ‘Bogolyubov Kyiv Conference 2009’ on
‘Modern Problems of Theoretical and Mathematical Physics’.
1. Introduction
In addition to the perturbative closed string sector, type
II theories admit non-perturbative objects, the so-called
Dirichlet branes (D-branes, in brief). These are objects,
on which open strings can end. Strings can a) have
both ends on the same D-brane; b) they can stretch be-
tween two different D-branes, or c) propagate (as closed
strings) from one D-brane to another one. Due to their
intrinsic tension, stretched strings give rise to massive
excitation modes. When the distance between two or
more branes is reduced, the strings allow for massless
modes.
A careful analysis of the string spectrum shows that
the massless modes give rise to the gauge theory on a
D-brane. Consider a bunch of N D-branes on the top of
each other, this configuration being often referred to as
a stack (of N D-branes). Then the massless modes in-
duce a non-Abelian group, more precisely a dimensional
reduced Super Yang–Mills gauge theory.
D-branes are classified according to their spatial di-
mension. For example, a D7-brane fills the four-
dimensional spacetime entirely – this accounts for the
T a b l e 1. Spacetime extension of D-branes. Spacetime
directions filled by the branes are denoted by crosses,
whereas the transverse ones are left blank
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
D7-brane × × × × × × × ×
D5-brane × × × × × ×
D3-brane × × × ×
E3-brane × × × ×
three spatial dimensions x1, x2, and x3 (see Table 1) –
and it wraps a four-dimensional subspace of the CY man-
ifold (a holomorphic four-cycle extended, for instance,
along the directions x4, x5, x6, and x7). The transverse
directions x8 and x9 account for the degrees of movement
of the brane.
Stable configurations of D-branes underlie certain su-
persymmetric calibration conditions (BPS conditions).
Whereby, in type IIB, D-branes have to wrap complex
subspaces of the CY manifold. Configurations of other
kinds turn out to be unstable. Therefore, type IIB com-
pactifications naturally come with D3-, D5-, and D7-
branes that wrap, respectively, no cycles, two-cycles, and
four-cycles of the internal space.
The rest of this section presents a rough sketch of
how the matter content of low-energy supergravity arises
from intersecting stacks of branes. Two four-cycles
generically intersect each other in a one-dimensional
complex subspace, i.e. a Riemann surface. Let D1 be
a four-cycle, on which a stack of D7-branes is wrapped.
Consider the second stack on a four-cycle D2 such that
D1 andD2 intersect each other. It can be shown that the
chiral matter is induced at the intersection by bifunda-
mental open strings ‘stretched’ between D1 and D2 with
fluxes. Furthermore, Yukawa couplings arise in the case
where three stacks of D7-branes intersect one another.
In conclusion, the type IIB compactification (with D-
branes and orientifold planes) provides an efficient way
to encode the entire matter content of MSSM-like the-
ories in terms of the geometry of complex subspaces.
Beside the massless modes arising from the perturbative
sector in ten dimensions, we have: a) a Yang–Mills gauge
theory on an eight-dimensional manifold (e.g., on a stack
D1), b) chiral matter on a six-dimensional manifold (e.g.,
D1∩D2), and c) Yukawa coupling in our spacetime (e.g.,
D1 ∩D2 ∩D3).
2. The ‘Large Volume Scenario’
Calabi–Yau manifolds come in families smoothly related
to one another by deformation parameters called mod-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 481
N.-O. WALLISER
uli. These parameters control the shape and the size of
CY manifolds. An important property of CY three-folds
is that their moduli space is the product of two disjoint
parts M = M2,1
CS ×M
1,1
K . The complex-structure mod-
uli account for deformations of the shape. The Kähler
moduli, instead, control the sizes of the three-fold and of
its subspaces.
These moduli give rise to massless fields in the four-
dimensional supergravity via a process similar to the
Kaluza–Klein (KK) reduction. The dimension of the
moduli space is determined by the Hodge stucture of the
CY three-fold: dimM2,1
CS = h2,1 and dimM1,1
K = h1,1.
A generic CY three-fold comes with many moduli that,
after the KK reduction, lead to unwanted massless scalar
fields in the four-dimensional theory. A possible way to
overcome this problem is the introduction of scalar po-
tentials that stabilize the fields at energies beyond the
characteristic compactification scale. The realization of
this strategy is the main issue of type IIB compactifica-
tions.
The full superpotential for the type IIB superstring
theory compactified on the CY manifold X is
W =
∫
X
G3 ∧ Ω +
∑
i
Ai (S,U) e−aiTi . (1)
The first term is the so-called Gukov–Vafa–Witten
(GVW) flux superpotential [2]. It is a purely tree-
level potential generated by the background fluxes
G3 = F3 + iSH3 supported on three-cycles of the CY
manifold X. The quantity Ω is the holomorphic three-
form of X. We stabilize the complex structure moduli U
and the axion-dilaton field S = e−φ + iC0, by requiring
supersymmetry to hold: DSW = 0 = DUW . Note that
no Kähler moduli appear in the GVW potential.
Unlike the superpotential, the Kähler potential gets
corrections order-by-order in both α′ and the string-
loop expansion. On the top of that, there are non-
perturbative effects from either the world sheet or the
brane instantons. But these corrections are subdomi-
nant, and we neglect them: K = Ktree + Kp (+Knp).
The Kähler potential takes the form
K = −2 ln
(
V e−3φ/2 +
ξ
2g3/2
s
)
−
− ln
(
S + S̄
)
− ln
−i∫
X
Ω ∧ Ω̄
. (2)
Here, gs is the string coupling, and ξ = − ζ(3)χ(X)
16π3 en-
codes the leading term of perturbative corrections in
terms of the Euler characteristic of the CY three-fold,
whose volume is given by
V =
1
3!
∫
X
J ∧ J ∧ J =
1
6
κijk t
itjtk .
The coefficients κijk account for the intersection of
three four-cycles Di, Dj , and Dk. They are numbers
that depend on the basis of integral two-forms {ηi} ∈
H1,1 (X,Z), in which we decide to expand the Kähler
form:
J =
∑
i
tiηi , ti ∈ R . (3)
The ‘Large Volume Scenario’ (LVS), developed in [3],
is a new strategy meant to stabilize Kähler moduli via
non-perturbative effects due to Euclidean D3-branes (see
Table 1).1 These effects are accounted for by the second
term of Eq. (1).
Consider an E3-brane wrapped on a cycle Di of vol-
ume τi. In this case, the coefficient Ai depends only
on the complex structure moduli and the axion-dilaton
field. We assume these moduli to be already stabilized
via the flux superpotential, whereby we can think of Ai
as a constant. The Kähler moduli, instead, enter explic-
itly in the exponentials of expression (1). These moduli
are related to the volume of the cycle, on which the E3-
brane is wrapped in the following way:
Ti = τi e
−φ + iρi . (4)
Here, ρi is the axion field originating from the Ramond–
Ramond four-form C4 supported on the four-cycle Di of
volume τi:
τi =
1
2
∫
Di
J ∧ J =
1
2
κijk t
jtk , and ρi =
∫
Di
C4 .
From Eq. (4), we see how the volume of the E3-brane
cycle enters superpotential (1). The instantonic contri-
bution grows exponentially as the cycle decreases in size.
The key idea of the LVS lies in finding a CY three-
fold such that the volume of the manifold is driven by
the volume τl of a single large four-cycle. The remaining
‘small’ four-cycles contribute negatively to the overall
volume:
V ∼ τ3/2
l −
h1,1−1∑
s=1
τ3/2
s . (5)
1 Euclidean D-branes are instantonic objects extended only in the
spatial directions of the internal manifold. For example, an Eu-
clidean D3-brane wraps a four-cycle of the CY manifold, and it
is often denoted by E3.
482 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
ON LARGE VOLUME MODULI STABILIZATION
Fig. 1. ln (V ) for P4
1,1,2,2,6 [12] /Z2 : 1 0 0 0 1 in the large volume
limit, as a function of the CY volume and the size of the E3-cycle.
Here, we have set W0 = 5, A = 1, and gs = 1/10
For the obvious reason, these kinds of three-folds have
been dubbed the ‘Swiss-cheese’ CY. This allows us to
make the cycles small, while keeping the CY manifolds
large. The advantage is double. First, the gauge the-
ory, that takes place on the small cycles, decouples from
the Planck scale dynamics. Second, if we choose the E3-
brane to wrap one of the ‘small’ cycles, then we obtain
instanton contributions that grow exponentially with de-
crease in τs. These give rise to an anti de Sitter (AdS)
vacuum at a large CY volume. We devote the rest of
this section to the explanation of this result.
The four-dimensional potential for the scalar fields
gets contributions from two terms: the F- and D-term
potentials. At this stage of the analysis, only the F-term
potential is relevant for the LVS:2
V = eK
∑
i=T,S,U
Kij̄DiW Dj̄W̄ − 3|W |2
. (6)
When V � 1, V can be expanded in inverse powers of the
three-fold volume. The potential decomposes into three
parts: V = Vnp1 +Vnp2 +Vα′ . The two non-perturbative
terms depend explicitly on the Kähler moduli:
Vnp1 ∼
1
V
(
−κssj tj
)
e−2as τs eK +O
(
e−2asτs
V2
)
, (7)
Vnp2 ∼ −
asτse
−asτs
V2
eK +O
(
e−asτs
V3
)
. (8)
The first term is proportional to the self-intersection of
the E3-brane and is positive; Vnp2, instead, is negative.
2 The D-term potential depends on the choice of the gauge bun-
dles, which we equip the D-branes with.
Fig. 2. V for P4
1,1,2,2,6 [12] /Z2 : 1 0 0 0 1 in the large-volume limit
as a function of the volume of the E3-cycle. Each curve corre-
sponds to a different value of the CY volume. The minimum flat-
tens and shifts to the right as the volume increases
There is the third contribution that depends on the α′-
corrections:
Vα′ ∼ ξ
V3
eK +O
(
1
V4
)
. (9)
We want this term to be positive. This is the case for
negative Euler characteristics. Therefore, a necessary
condition for CY three-folds to realize LVS is h2,1 > h1,1
(since χ = 2
(
h1,1 − h1,2
)
).
In the general case, the right-hand side of (7) domi-
nates the power expansion in V−1. But let us consider
the special limit, in which the E3-cycle scales with the
logarithm of the CY volume, i.e. τs ∼ lnV. In (7)
and (8), the Kähler moduli appear in the exponentials.
Therefore, in this limit, Vnp1 and Vnp2 become propor-
tional to V−3 and compete with Vα′ on the same footing.
In Fig. 1, the logarithm of the scalar potential is plot-
ted as a function of the volume of E3-cycle τ and the CY
volume. The ‘void channel’ corresponds to the region,
where the potential becomes negative.
Figure 2 shows the potential for the same model as
a function of the ‘small’ cycle, on which the E3-brane
is wrapped. Each curve corresponds to a fixed value
of the CY volume. We can choose volumes of the CY,
for which the potential is minimum. But increasing V
drives the potential to a negative minimum: this is the
AdS vacuum solution. For larger volumes, the minimum
flattens and shifts to the right.
The general behavior of V in the LVS can be sum-
marized as follows. Due to the positive contributions of
Vnp1 and Vα′ , the potential starts positive at small CY
volumes and then, driven by Vnp2, reaches a negative
minimum. Afterward, it approaches zero asymptotically
for large values of the volume.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 483
N.-O. WALLISER
Remark: This behavior is of main relevance for cos-
mology. Such a flat potential accommodates slow-roll
inflationary scenarios. The Kähler modulus plays the
role of the inflaton slowly rolling down the potential.
This is one of the few known setups that allow, at least
in principle, a stringy inflation [4].
3. Four-Modulus ‘Swiss-Cheese’ Calabi–Yau
Manifolds
The following section is based on my work in collabora-
tion with Kreuzer, Collinucci, and Mayrhofer [1].
In [5], Blumenhagen et al. stress that it is too naive
to treat the stabilization of the complex structure mod-
uli via instanton effects and the configuration of chiral
matter D7-branes, as two independent tasks. In the gen-
eral case, indeed, D7-cycles intersect the E3-cycle. This
gives rise to charged zero modes that could spoil the
non-pertubative stabilization. This means that the mod-
uli stabilization and the vanishing chiral intersection be-
tween D7- and E3-branes must be accounted for at the
same time.
In our attempt to realize a MSSM-like scenario, we
proceed in two steps. Our approach is operational:
Given what we think is the minimal set of constraints
for a MSSM-like configuration, we search for CY three-
folds and explicit brane realizations that satisfy these
conditions. The starting point is the following ‘wishlist.’
We look for
• a CY three-fold with ‘Swiss-cheese’ structure;
• two stacks of intersecting D7-branes such that they
– give rise to a U (Na)× U (Nb) gauge group;
– induce bifundamental chiral fermions at the
intersecting locus;
– do not chirally intersect the E3-brane.
At the same time, we have to take care of consistency
conditions such as
• tadpole cancellation;
• the Freed–Witten (FW) anomaly which is an open
string world-sheet anomaly that arises when D7-
branes are wrapped on non-spin manifolds. These
are the subspaces that do not admit a spin struc-
ture.
The FW anomaly was discovered in [6] and can be com-
pensated by means of half-integral world-volume fluxes.
We start from the complete classification of toric CY
three-fold hypersurfaces in [7]. The hypersurfaces are
given in terms of combinatoric data encoded in four-
dimensional polytopes. Toric CY hypersurfaces are di-
visors, i.e. co-dimension one manifolds embedded in a
toric four-fold ambient space.
The list of CY hypersurfaces with h1,1 = 4 contains
1197 objects. For simplicity, we focus on 11 models that
correspond to simplicial polytopes.
Out of these 11 polytopes, there are only four
inequivalent CY three-folds with the ‘Swiss-cheese’
structure. Each of them allows for two different
D7-/E3-brane setups.
Only one of our ‘Swiss-cheese’ CY hypersurfaces satis-
fies the conditions for a MSSM-like setup and stabilizes
(three out of four) Kähler moduli at the same time. The
other CY three-folds either miss the conditions or fail
to stabilize any Kähler modulus. Let us consider the
(partly) successful model in more details. The following
analysis is assisted by a recently enhanced version of the
PALP package [8].
Consider the CY hypersurface of degree 12 embed-
ded in the weighted projective space P4
1,1,2,2,6 orbifolded
by the Z2-action (1, 0, 0, 0, 1) identifying x1 ∼ −x1 and
x5 ∼ −x5.
The resulting hypersurface is singular. We repair the
singularities, by introducing three toric blow-ups of the
ambient space. These induce three extra Kähler moduli.
The resolution is shown in the following table.
We can recast Table 2 in terms of equivalence rela-
tions: every point (x1, . . . , x8) is equivalent to(
λµx1, λν x2, λ
2µ2ν2ρ x3, λ
2µ2ν2ρ x4,
λ6µ5ν5ρ3x5, µ
2x6, ν
2x7, ρ x8
)
, (10)
where λ, µ, ν, ρ ∈ C∗, and the corresponding exponents
are encoded by the rows of Table 2. On the other hand,
each column four-vector is associated to a global ho-
mogeneous coordinate xi. The corresponding ambient
space divisor Di = {xi = 0}. The divisors D1, . . . , D5
come from the original singular ambient space, whereas
D6, D7, and D8 are the exceptional ones associated to
T a b l e 2. Projective weights for the resolution of
P4
1,1,2,2,6 [12] /Z2 : 1 0 0 0 1
x1 x2 x3 x4 x5 x6 x7 x8 p
1 1 2 2 6 0 0 0 12
1 0 2 2 5 2 0 0 12
0 1 2 2 5 0 2 0 12
0 0 1 1 3 0 0 1 6
484 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
ON LARGE VOLUME MODULI STABILIZATION
the blow-ups. The eight divisors generate four divisor
classes modulo four linear equivalences, which is what
we need for the four-(Kähler)-modulus model. For con-
venience, let us introduce the following basis: η1 = D1,
η2 = D2, η3 = D8, and η4 = D6. Table 3 shows all the
divisors expressed in the η-basis.
The CY divisor is DCY = D1 + · · ·+D8. The smooth
CY three-fold we have constructed has the Euler charac-
teristic χ = −180 and the Hodge numbers h1,1 = 4 and
h1,2 = 94. PALP determines the Stanley–Reisner ideal:
ISR = {x1x2, x1x5, x1x6, x2x5, x2x7, x5x8,
x3x4x6x7, x3x4x6x8, x3x4x7x8} .
This set fixes the coordinates that cannot vanish at
the same time. For example, x1 = 0 implies x2 6= 0.
A toric divisor Di intersects the CY hypersurface on a
two-dimensional subspace: dimC (Di ∩DCY) = 2. From
the point of view of the CY hypersurface, the ambient
space divisor Di induces a four-cycle in H4 (DCY) that
we denoted again by Di for simplicity.3
On the CY, three CY divisors intersect one another
at points. In our case, the resulting triple intersection
numbers in the η-basis are encoded as coefficients of a
polynomial as follows:
I3 = −78η3
4 − 6η3η2
4 − 6η2
3η4 + 2η3
3+
+36η2η2
4 + 6η2η3η4 + η2η
2
3 − 18η2
2η4−
−3η2
2η3 + 9η3
2 + η1η
2
3 − 3η2
1η3 + 9η3
1 . (11)
T a b l e 3. Toric divisors of P4
1,1,2,2,6/Z2 : 1 0 0 0 1
Divisor η-basis
D1 η1
D2 η2
D3 2η2 + η3 + η4
D4 2η2 + η3 + η4
D5 η1 + 5η2 + 3η3 + 2η4
D6 η4
D7 −2η1 + 2η2 + η4
D8 η3
DCY 12η2 + 6η3 + 6η4
3 This abuse of notation is justified since the four divisor classes
of the CY three-fold descend from the four divisor classes of
the toric ambient space X. In general, it might happen that a
toric divisor intersects the DCY at two or more loci. On the
CY hypersuface, such a divisor reduces into a sum of disjoint
non-intersecting four-cycles. More precisely, the number of dis-
connected parts of the cycle D is counted by h0,0 (D).
With help of an appropriate recast, the volume of the CY
three-fold exhibits the wanted Swiss-cheese structure of
formula (5):
V =
√
2
9
(
3
2
√
6
τ3/2
a − 1
2
√
2
τ
3/2
b − τ3/2
c − τ3/2
d
)
. (12)
Here, τa, . . . , τd are the volumes of the cycles
Da = η1 + 5η2 + 3η3 + 2η4 ,
Db = η1 + η2 + 3η3 ,
Dc = η1 ,
Dd = η2 .
The convenience of the diagonal basis becomes obvious
by rewriting the triple intersections from expression (11):
I3 = 24D3
a + 72D3
b + 9D3
c + 9D3
d . (13)
The divisorsDb, Dc, andDd contribute negatively to the
overall volume. Indeed, we will check if these cycles are
shrinkable in such a way that the LVS accommodates.
The Kähler cone is a subspace of the space of parame-
ters ti, for which the integral of the Kähler form (3) over
all effective curves is positive:
∫
curve
J > 0. We obtain
the Kähler cone starting from the Mori cone. The Mori
cone is the cone of (numerically) effective curves. It is
determined, by using the Oda–Parks algorithm, as de-
scribed in [9]. The result is a set of constraints for the
volumes of the cycles:
√
τa − 3
√
τb > 0 , 2
√
τb −
√
τd > 0
2
√
τb −
√
τc > 0 ,
√
τc > 0 ,
√
τd > 0 .
We can arbitrarily shrink Db, Dc, and Dd while keeping
Da large. In terms of the η-basis, η1, η2, and η3 are the
small directions, whereas η4 is the large one.
Let us take a closer look at the topology of the ‘small’
cycles. The Euler characteristic χ (Di), the holomorphic
Euler characteristic
χh (Di) =
2∑
p=0
(−1)p h0,p (Di) ,
and the Hodge numbers are listed in Table 4.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 485
N.-O. WALLISER
T a b l e 4. Topological quantities of the divisors
D1, . . . , D8
Divisor D1 D2 D3 D4 D5 D6 D7 D8
χ 3 3 46 46 108 –30 –30 10
χh 1 1 4 4 11 –9 –9 1
h0,0 1 1 1 1 1 1 1 1
h0,1 0 0 0 0 0 10 10 0
h0,2 0 0 3 3 10 0 0 0
h1,1 1 1 38 38 86 8 8 8
The cycles η1 and η2 turn out to be both CP2. These
are Del Pezzo surfaces of the dP0 type. The anticanon-
ical line bundle −KD of a Del Pezzo surface D is am-
ple by definition. This implies that −KD must have
positive intersections with any effective curve C on the
surface:
∫
D
(−KD) ∧ C > 0. The right-hand side of
expression (11) yields η2η
2
3 = +1, whereby, for the
curve C : {x8 = 0} ∩ {x2 = 0}, the following holds:∫
η3
(−η3) ∧ C = −1. This means that η3 is not Del
Pezzo and, hence, is not shrinkable.
4. Conclusions and Outlook
In [1], we have analyzed two different configurations of
MSSM D7-branes with chiral matter for the resolved
P4
1,1,2,2,6/Z2[12] : 1 0 0 0 1. The most relevent results are
as follows.
Not every rigid surface Di that contributes negatively
to the CY volume can be stabilized at small values of
τi. Indeed, we find that only Del Pezzo surfaces can be
stabilized at small values. This is a general result. In
our model, three out of the four Kähler moduli can be
fixed via instantonic effects in the LVS. Cicoli suggests
in [10] that, if gs corrections are taken into account in
(2), a full moduli stabilization might be achieved even
if not all the cycles satisfy the topological constraint of
being Del Pezzo.
The physical and consistency conditions we have taken
into account turn out to be very restrictive and signifi-
cantly reduce the list of Swiss-cheese CY hypersurfaces
that can accommodate MSSM-like configurations. This
is true, in particular, if the FW anomaly cancellation is
required. This forces D7- and E3-branes to be magne-
tized in such a way that the condition of zero chiral inter-
sections between the instantonic brane and the matter
branes, on the one hand, and between the hidden D7-
branes – that we eventually need to introduce for tadpole
saturation – and the physical sector, on the other hand,
is hard to satisfy.
While we did not succeed so far in finding a model with
all the Kähler moduli stabilized, it appears likely that
the systematic analysis of a larger list of toric examples
will provide phenomenologically attractive models.
1. A. Collinucci, M. Kreuzer, C. Mayrhofer, and N.O. Wal-
liser, JHEP 0907, 074 (2009); [arXiv:0811.4599 [hep-th]].
2. S. Gukov, C. Vafa, and E. Witten, Nucl. Phys. B 584,
69 (2000); [Erratum-ibid. B 608, 477 (2001)] [arXiv:hep-
th/9906070].
3. V. Balasubramanian, P. Berglund, J.P. Conlon, and
F. Quevedo, JHEP 0503, 007 (2005); [arXiv:hep-
th/0502058].
4. J.P. Conlon and F. Quevedo, JHEP 0601, 146 (2006);
[arXiv:hep-th/0509012].
5. R. Blumenhagen, S. Moster, and E. Plauschinn, JHEP
0801, 058 (2008); [arXiv:0711.3389 [hep-th]].
6. D.S. Freed and E. Witten, [arXiv:hep-th/9907189].
7. M. Kreuzer and H. Skarke, Adv. Theor. Math. Phys. 4,
1209 (2002); [arXiv:hep-th/0002240].
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157, 87 (2004); [arXiv:math/0204356].
9. T. Oda and H. Park, Tohoku Math. J. 43, (1991).
10. M. Cicoli, [arXiv:0907.0665 [hep-th]].
Received 14.10.09
ЩОДО СТАБIЛIЗАЦIЇ МОДУЛIВ ДЛЯ ВЕЛИКИХ
ОБ’ЄМIВ В IIБ ОРIЄНТОВАНИХ МНОЖИНАХ
Н.-О. Уоллизер
Р е з ю м е
Подано короткий вступ до побудови явних компактифiкацiй
IIБ орiєнтованих множин та розглянуто “сценарiй великих
об’ємiв” для компактних чотиримодульних Калабi–Яу много-
видiв. Обговорено доречнiсть таких схем для побудови фiзи-
чної моделi МССМ-типу та для гравiтацiйної космологiї. Осно-
вою роботи є доповiдь на “Боголюбовськiй київськiй конфе-
ренцiї 2009” з “Сучасних проблем теоретичної та математичної
фiзики”.
486 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
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