LMU-ASC-21/07
AEI-2007-020
Jumping Through Loops:
On Soft Terms from Large Volume
Compactifications
Marcus Berg+ , Michael Haack† and Enrico Pajer†
+ Max Planck Institute for Gravitational Physics
Albert Einstein Institute
Mühlenberg 1, D-14476 Potsdam, Germany
† Ludwig-Maximilians-University
Department of Physics
Theresienstr. 37, D-80333 München, Germany
Abstract
We subject the phenomenologically successful large volume scenario of hep-th/0502058
to a first consistency check in string theory. In particular, we consider whether the
expansion of the string effective action is consistent in the presence of D-branes and
O-planes. Due to the no-scale structure at tree-level, the scenario is surprisingly ro-
bust. We compute the modification of soft supersymmetry breaking terms, and find
only subleading corrections. We also comment that for large-volume limits of toroidal
orientifolds and fibered Calabi-Yau manifolds the corrections can be more important,
and we discuss further checks that need to be performed.
http://arxiv.org/abs/0704.0737v2
http://arxiv.org/abs/hep-th/0502058
Contents
1 Introduction 3
2 Review 5
2.1 KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Consistency of KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Large volume scenario (LVS) . . . . . . . . . . . . . . . . . . . . . . . . 7
3 String loop corrections to LVS 10
3.1 From toroidal orientifolds to Calabi-Yau manifolds . . . . . . . . . . . . 11
3.2 LVS with loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 The P4[1,1,1,6,9] model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Gaugino masses 20
4.1 Including loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Other soft terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 LVS for other classes of Calabi-Yau manifolds? 25
5.1 Abundance of “Swiss cheese” Calabi-Yau manifolds . . . . . . . . . . . 25
5.2 Toroidal orientifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 Fibered Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . 28
6 Further corrections 29
7 Conclusions 29
A Some details on LVS 31
A.1 LVS for P4
[1,1,1,6,9]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A.2 Many Kähler moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
B Loop corrected inverse Kähler metric for P4[1,1,1,6,9] 34
C No-scale Kähler potential in type II string theory 35
C.1 No-scale structure in type IIA . . . . . . . . . . . . . . . . . . . . . . . 36
C.2 No-scale structure in type IIB . . . . . . . . . . . . . . . . . . . . . . . 37
C.3 Cancellation with just the volume modulus . . . . . . . . . . . . . . . . 39
C.4 Cancellation with many Kähler moduli . . . . . . . . . . . . . . . . . . 39
C.5 Perturbative corrections to Vnp1 and Vnp2 . . . . . . . . . . . . . . . . . 40
D KK spectrum with fluxes 42
E The orientifold calculation 44
F Factorized approximation 48
F.1 Factorized approximation of the scalar potential . . . . . . . . . . . . . 50
1 Introduction
The KKLT strategy [1,2] for producing stabilized string vacua that can serve as a
starting point for phenomenology has been a source of great interest for the last few
years. The “large-volume scenario” (LVS) [3,4] is an extension of KKLT where string
corrections to the tree-level supergravity effective action computed in [5] play a sig-
nificant role, and where the compactification volume can be as large as 1015 in string
units. In LVS, work has been done on soft supersymmetry breaking [4,6,7], the QCD
axion [8,9], neutrino masses [10], inflation [11–14], and even first attempts at LHC
phenomenology [15].
Although tantalizing, the models discussed in the aforementioned papers (nominally
“string compactifications”) raise many questions. It remains an open problem to con-
struct complete KKLT models in string theory, as opposed to supergravity. Problems
one faces include things like the description of RR fluxes in string theory, showing that
the necessary nonperturbative effects actually can and do appear in a way consistent
with other contributions to the potential (for progress in this direction, see [16–32]),
and verifying that one can uplift to a Minkowski or deSitter vacuum without ruining
stabilization [26,33–36]. In LVS, since string corrections play a crucial role, striving
for actual string constructions seems quite important. In the end, the restrictiveness
this entails may greatly improve predictivity, or kill the models completely as string
compactifications.
In this paper, we will not improve on the consistency of KKLT or LVS in general,
but rather assume the existence of LVS models in string theory, and then perform self-
consistency checks. This is a modest step on the way towards reconciling phenomeno-
logically promising scenarios with underlying string models. We will see that although
a priori the situation looks very bleak, and one might have hastily concluded that even
our modest consistency check would put very strong constraints on LVS, things are
more interesting. It turns out that LVS jumps through every hoop we present it with,
and instead of broad qualitative changes, we find only small quantitative changes.
The main difference between KKLT and LVS is that LVS includes a specific string
α′ correction ∆Kα′ in the Kähler potential K of the four-dimensional N = 1 effective
supergravity. Naturally, the four-dimensional string effective action also contains other
string corrections. Here, we will focus on gs corrections due to sources (D-branes
and O-planes). For some N = 1 and N = 2 toroidal orientifolds, these corrections
were computed in [37] (see also [38]; for a comprehensive introduction to orientifolds,
see [39]). Compared to the α′ correction ∆Kα′ considered in LVS, the gs corrections to
the Kähler potential ∆Kgs will scale as
∆Kα′ : ∆Kgs
O(α′3) : O(g2sα′2)
(string frame) . (1)
By naive dimensional analysis, one would expect that in a 1/V expansion, where V is
the overall volume in the Einstein frame, eq. (1) implies
∆Kα′ ∼ O(g−3/2s V−1) , ∆Kgs ∼ O(gsV−2/3) (Einstein frame) . (2)
If there is more than one Kähler modulus, as is usually the case, various combinations
of Kähler moduli may appear in ∆Kgs in eq.(2), and a priori this could lead to even
weaker suppression in 1/V than that shown. However, we will argue that (2) is actually
correct as far as the suppression factors in the 1/V expansion go. Nevertheless, even the
suppression displayed in (2) seems to be a challenge for LVS, if indeed V ∼ 1015. For V
this large, ∆Kgs would dominate ∆Kα′ , since we do not expect the string coupling gs
to be stabilized extremely small. On the other hand, if we are interested in the effects
gs corrections may have on the existence of the large volume minima, the relevant
quantity to look at is the scalar potential V , rather than the Kähler potential K. It
turns out that certain cancellations in the expression for the scalar potential leave us
with leading correction terms to V that scale as
∆Vα′ ∼ O(g−1/2s V−3) , ∆Vgs ∼ O(gsV−3) . (3)
This is already much better news for LVS. However, restoring numerical factors in (3),
and with gs typically not stabilized extremely small, it would seem that ∆Kgs could still
have a significant effect both on stabilization and on the resultant phenomenology (like
soft supersymmetry breaking terms, which also depend on the Kähler potential). We
will see that although this is indeed so in principle, in practice the models we consider
are surprisingly robust against the inclusion of ∆Kgs . The clearest example of this is
the calculation of gaugino masses in sec. 4. The result is that for the “11169 model”
(analyzed in [6]), the correction to the gaugino masses due to ∆Kgs is negligible. Thus,
for the most part, LVS survives our onslaught unscathed.
We consider this a sign that scenarios such as LVS deserve to be taken seriously as
goals to be studied in detail in string theory, even as the caveats above (that apply to
any KKLT-like setup) serve to remind us that there is much work left to be done to
really understand phenomenologically viable stabilized flux compactifications in string
theory.
2 Review
Let us begin by a quick review of the KKLT and large volume scenarios. For reasons
that will become clear, we will want to allow for more than a single Kähler modulus.
2.1 KKLT
The KKLT setup [1,2] is a warped Type IIB flux compactification on a Calabi-Yau (or
more generally, F-theory) orientifold, with all moduli stabilized. In this paper, we will
neglect warping. For progress towards taking warping into account in phenomenological
contexts, see [40,41].
In the four-dimensional N = 1 effective supergravity, the Kähler potential and
superpotential read
K = −ln(S + S̄)− 2 ln(V) +Kcs(U, Ū) ,
W = Wtree +Wnp =W (S, U) +
Ai(S, U)e
−aiTi , (4)
where the volume V is a function of the Kähler moduli Ti = τi + ibi whose real parts
are 4-cycle volumes and whose imaginary parts are axions bi, arising from the integral
of the RR 4-form over the corresponding 4-cycles. In particular, the volume V depends
on the Ti only through the real parts τi,
V = V(Ti + T̄i) = V(τi) , (5)
and the nonperturbative superpotential Wnp a priori depends on the complexified dila-
ton S and the complex structure moduli U . After stabilization of S and U by demand-
ing DUW = 0 = DSW , we have
K = −ln(S + S̄)− 2 ln(V) +Kcs(U, Ū) ,
W = Wtree +Wnp =W0 +
−aiTi . (6)
We keep the dependence on the complexified dilaton S and the complex structure
moduli U in the Kähler potential for now, since the Kähler metric in the F-term
potential
V = eK
GJ̄IDJ̄W̄DIW − 3|W |2
is to be calculated with the full Kähler potential K, including the dependence on S and
U . (In eq. (7), the index I a priori runs over all moduli, but after fixing the complex
structure moduli and the dilaton, only the sum over the Kähler moduli remains). The
scalar potential V has a supersymmetric AdS minimum at a radius that is barely large
enough to make the use of a large-radius effective supergravity self-consistent, typically
τ ∼ 100 (recall that τ has units of (length)4).1 In addition, to obtain a supersymmetric
minimum at all, one needs to tune the flux superpotential W0 to very small values.
That is, the stabilization only works for a small parameter range. This is easy to
understand, since we are balancing a nonperturbative term against a tree-level term.
Let us briefly digress on the reasons for and implications of this balancing.
2.2 Consistency of KKLT
In the previous section we only considered the lowest-order supergravity effective action.
As was already noted in the original KKLT paper, α′ corrections and gs corrections
(string loops) that appear in addition to the tree-level effective action could in principle
1This minimum then has to be uplifted to dS or Minkowski by an additional contribution to the
potential. Various mechanisms were suggested in [2,36,42–45].
affect stabilization. Oftentimes, the logic of string effective actions is that if one such
correction matters, they all do, so no reliable physics can be learned from considering
the first few corrections. If this is true, one can only consider regimes in which all
corrections are suppressed. This is not necessarily so if some symmetry prevents the
tree-level contribution to the effective action from appearing, so that the first correc-
tion (be it α′ or gs) constitutes lowest order. This indeed happens for type IIB flux
compactifications; given the tree level Kähler potential (6), if we were to set Wnp = 0,
the remaining K andW in (6) produce a no-scale potential, i.e. the scalar potential for
the Kähler moduli then vanishes [46]. In KKLT, this no-scale structure is only broken
by the nonperturbative contribution to the superpotential Wnp. Since each term in
Wnp is exponentially suppressed in some Kähler modulus, the resulting terms in the
potential are also exponentially suppressed. For instance, for the simpler example of a
single modulus τ , the potential (after already fixing the axionic partner along the lines
of appendix A.2) reads
4|A|2aτe−aτ
aτ + 1
− 4aτ |AW0|
e−aτ , (8)
meaning that even for moderate values of the Kähler modulus τ , all these terms are
numerically very small. Corrections in α′ and gs, however, are expected to go as powers
of Kähler moduli τ , so will dominate the scalar potential for most of parameter space. In
particular, it was argued in [3,4] that only for very small values of W0 can perturbative
corrections to the Kähler potential be neglected. It was the insight of [3] that even if
W0 is O(1) (which is more generic than the tiny value forW0 required in KKLT), there
can still be a competition between the perturbative and nonperturbative corrections to
the potential in regions of the Kähler cone where large hierarchies between the Kähler
moduli are present. We now review this scenario.
2.3 Large volume scenario (LVS)
As was shown in [5], the no-scale structure (and factorization of moduli space) is broken
by perturbative α′ corrections to the Kähler potential, such as
K = −ln(2S1)− 2 ln(V + 12ξS
1 ) +Kcs(U, Ū) , (9)
where2 ξ = −ζ(3)χ/2(2π)3 and S1 = ReS. For large volume V, we see that the
perturbative correction goes as a power in the volume,
− 2 ln(V + 1
1 ) = −2 lnV −
V + . . . , (10)
2Here ξ differs by a factor (2π)−3 from [5] because we use the string length ls = 2π
which by the discussion in the previous subsection will dominate in the scalar potential
if all Kähler moduli are even moderately large. Using the superpotential
W = W0 +Wnp = W0 +
−aiTi , (11)
the scalar potential has the structure
V = Vnp1 + Vnp2 + V3 (12)
G̄i∂̄W̄np∂iWnp +
G̄iK̄ (W̄0 + W̄np)∂iWnp + c.c.
G̄iK̄Ki − 3
|W |2
For concrete calculations we will use the model based on the hypersurface of degree 18 in
[1,1,1,6,9]
(see [16,47,48] for background information on its topology. Some comments
about generalizations to other models with arbitrary numbers of Kähler moduli are
given in appendix A.2). The defining equation is
z181 + z
2 + z
3 + z
4 + z
5 − 18ψz1z2z3z4z5 − 3φz61z62z63 = 0 (13)
and it has the Hodge numbers h1,1 = 2 and h2,1 = 272 (only two of the complex
structure moduli ψ and φ have been made explicit in (13); moreover, not all of the
272 survive orientifolding). We denote the two Kähler moduli by Tb = τb + ibb and
Ts = τs + ibs, where τb and τs are the volumes of 4-cycles, and the subscripts “b” and
“s” are chosen in anticipation of the fact that one of the Kähler moduli (τb) will be
stabilized big, and the other one (τs) will be stabilized small. An interesting property
of this model is that it allows expressing the 2-cycle volumes ti explicitly as functions of
the 4-cycle volumes τj , so that the total volume of the manifold can be written directly
in terms of 4-cycle volumes, yielding
V = 1
b − τ 3/2s
, (14)
(ts + 6tb)
, τs =
Following [4], we are interested in minima of the potential with the peculiar property
that one Kähler modulus τb ∼ V2/3 is stabilized large and the rest are relatively small
(but still large compared to the string scale),
aτs ∼ lnV ∼
ln τb (15)
in the case at hand. Thus, we expand the potential around large volume, treating
e−aτs as being of the same order as V−1. In the end one has to check that the resulting
potential indeed leads to a minimum consistent with the exponential hierarchy aτs ∼
lnV, so that the procedure is self-consistent. Applying this strategy, the scalar potential
at leading order in 1/V becomes3
VO(1/V3) =
2|A|2a2√τse−2aτs
− 2a|AW0|τse
3|W0|2
eKcs . (16)
From here one can see the existence of the large volume minima rather generally. By
the Dine-Seiberg argument [49], the scalar potential goes to zero asymptotically in
every direction. Along the direction (15), for large volume the leading term in (16) is
V ∼ Vnp2 ∝ −
V3 , (17)
which is negative, so the potential V approaches zero from below. For moderately
small values of the volume, V is positive (this is guaranteed if the Euler number χ
is negative, hence ξ positive), so in between there is a minimum. This minimum
is typically nonsupersymmetric, and because we are no longer balancing a tree-level
versus a nonperturbative term, we can find minima at large volume — hence the name
large volume scenario (LVS).4 To be precise, in flux compactifications we move in
parameter space by the choice of discrete fluxes, but since V is exponentially sensitive
to parameters like S1, large volume minima appear easy to achieve also by small changes
in flux parameters. If we allow for very small values ofW0 (so that KKLT minima exist
at all), the above minimum can coexist with the KKLT minimum [4,50]. Here, we will
allow W0 to take generic values of order one.
The astute reader will have noticed that this argument for the existence of the
LVS minimum is “one dimensional”, as it only takes into account the behavior of the
potential along the direction (15). One must of course check minimization with respect
to all Kähler moduli. In [3] a plausibility argument to this effect was given, and the
existence of the minimum was explicitly checked in the case of the P4
[1,1,1,6,9]
model by
explicitly minimizing the potential (16) with respect to the Kähler moduli. In doing
so, it is convenient to trade the two independent variables {τb,τs} for {V,τs} so that
3Here we have already stabilized the axion bs, i.e. solved ∂V/∂bs = 0, which produces the minus
sign in the second term; this is also true with many small moduli τi. See appendix A.2 for details.
Also note that solving DUW = 0 = DSW causes the values of U and S at the minimum to depend
on the Kähler moduli. However, this dependence arises either from the nonperturbative terms in
the superpotential or from the α′-correction to the Kähler potential. Thus it would only modify the
potential at subleading order in the 1/V expansion.
4By “tree-level” we intend “tree-level supergravity”, i.e. for the purposes of this paper we call both
α′ and gs corrections “quantum corrections”.
∂τsV = 0, as then the last term in (16) is independent of τs (this will be different when
we include loop corrections). Extremizing with respect to τs, and defining
X ≡ Ae−aτs , (18)
one obtains a quadratic equation for X ,
τsS1V
(4aτs − 1)X2 +
2a|W0|
(aτs − 1)X
eKcs . (19)
In (18), we chose A to be real as a potential phase can be absorbed into a shift of the
axion b and disappears after minimization with respect to b (see section A.2). Two
comments are in order. The quadratic equation (19) has just one meaningful solution
(X = 0 corresponds to τs = ∞). Moreover, when expanding (19) in 1/(aτs), the leading
terms arise from derivatives of the exponential.
Formula (19) is an implicit equation determining τs. However, one can easily solve
(19) for X and obtains
X = Ae−aτs =
2|W0|
(aτs)2
. (20)
The hierarchy (15) is obvious in this solution, rendering the procedure self-consistent.
One also notices that reasonably large values of τs (e.g. 35) are not difficult to obtain,
if V is stabilized large enough; for example, simply set a ∼ 1, A ∼ 1, W0 ∼ 1. We
fill in the numerical details, following [3], in appendix A.1 (including some further
observations).
3 String loop corrections to LVS
As already emphasized, the α′ correction proportional to ξ is only one among many
corrections in the string effective action. We now consider the effect of string loop
corrections on this scenario and what the regime of validity is for including or neglect-
ing those corrections. Volume stabilization with string loop corrections but without
nonperturbative effects was considered in [51].
To be precise, the corrections considered in [51] were those of [37], that were com-
puted for toroidal N = 1 and N = 2 orientifolds. Here, we would need the analogous
corrections for smooth Calabi-Yau orientifolds. Needless to say, these are not known.
Faced with the fact that the string coupling gs is stabilized at a finite (and typically
not terribly small) value, we propose that attempting to estimate the corrections based
on experience with the toroidal case is better than arbitrarily discarding them. As we
will see, if our estimates are correct, typically the loop corrections can be neglected,
though there may at least be some regions of parameter space where they must be
taken into account (see figure 4). (In section 5, we will briefly consider “cousins” of
LVS where they cannot be neglected anywhere in parameter space.) Improvement on
our guesswork would of course be very desirable.
3.1 From toroidal orientifolds to Calabi-Yau manifolds
We would like to make an educated guess for the possible form of one-loop corrections
in a general Calabi-Yau orientifold. All we can hope to guess is the scaling of these
corrections with the Kähler moduli T and the dilaton S. The dependence on other
moduli, like the complex structure moduli U , cannot be determined by the following
arguments (even in the toroidal orientifolds this dependence was quite complicated).
In order to generalize the results of [37] to the case of smooth Calabi-Yau mani-
folds, we should first review them and in particular remind ourselves where the various
corrections come from in the case of toroidal orientifolds. There, the Kähler potential
looks as follows (we will explain the notation as we go along):
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
V (21)
E (K)i (U, Ū)
4τiS1
i 6=j 6=k
E (W )k (U, Ū)
4τiτj
There are two kinds of corrections. One comes from the exchange of Kaluza-Klein
(KK) modes between D7-branes (or O7-planes) and D3-branes (or O3-planes, both
localized in the internal space), which are usually needed for tadpole cancellation, cf.
fig. 1. This leads to the first kind of corrections in (21), proportional to E (K)i where
the superscript (K) reminds us that these terms originate from KK modes. In the
toroidal orientifold case, this type of correction is suppressed by the dilaton and a
single Kähler modulus τi, related to the volume of the 4-cycle wrapped by the D7-
branes (or O7-planes, respectively).5 We expect an analog of these terms to arise more
5We should mention that there was no additional correction of this kind coming from KK exchange
between (parallel) D7-branes in [37] (actually that paper considered the T-dual version with D5-branes,
but here we directly translate the result to the D7-brane language). This was due to the fact that
in [37] the D7-brane scalars were set to zero. In general we would also expect a correction coming
PSfrag replacements τb
Figure 1: The loop correction E(K) comes from the exchange of closed strings, or
equivalently an open-string one-loop diagram, between the D3-brane and D7-branes
(or O7-planes) wrapped on either the small 4-cycle τs (as in a) or the large 4-cycle
τb (as in b). The exchanged closed strings carry Kaluza-Klein momentum.
generally, given that they originate from the exchange of KK states which are present
in all compactifications.
The second type of correction comes from the exchange of winding strings between
intersecting stacks of D7-branes (or between intersecting D7-branes and O7-planes).
The exchanged strings are wound around non-contractible 1-cycles within the intersec-
tion locus of the D7-branes (and O7-planes, respectively), cf. fig. 2. This leads to the
PSfrag replacements
Figure 2: The loop correction E(W ) comes from the exchange of winding strings
on the intersection between the small 4-cycle τs and the large 4-cycle τb. If this
intersection is empty, there are no terms with E(W ).
second kind of correction in (21) proportional to E (W )i . The superscript (W ) reminds
us that these terms arise from the exchange of winding strings. In toroidal orientifolds,
this type of correction is suppressed by the two Kähler moduli measuring the volumes
from parallel (or more generally, non-intersecting) D7-branes by exchange of KK-states. These should
scale in the same way with the Kähler moduli as those arising from the KK exchange between D3-
and D7-branes.
of the 4-cycles wrapped by the D7-branes (and O7-planes). One might a priori think
that this kind of correction does not generalize easily to a smooth Calabi-Yau which
has vanishing first Betti number (and therefore at most torsional 1-cycles). However,
the exchanged winding strings are, from the open string point of view, Dirichlet strings
with their endpoints stuck on the D7-branes. Thus, the topological condition is on the
cycle over which the two D7-brane stacks (or one D7-brane stack and an O7-plane)
intersect, as in figure 3. Thus, it depends on the topology of specific cycles within
cycles whether winding open strings exist in a given model.6
PSfrag replacements
no D-brane
no D-
brane
Figure 3: A D7-brane is wrapped on a 4-cycle A, which intersects the 4-cycle B on
a 2-cycle C. For Dirichlet strings, the relevant topological condition (the existence
of nontrivial 1-cycles) is on the intersection locus C, not on cycle B or on the whole
Calabi-Yau. In other words, without the D-brane, the string on cycle C could have
been unwound by sliding it along cycle B (as shown in the figure). With the D-brane,
the string on cycle C is stuck.
Given the expressions in [37] and the subset reproduced in (21) above, it is tempting
to conjecture that some terms at one loop might be suppressed only by powers of single
Kähler moduli like the τi (and the dilaton):
Calabi-Yau: ∆Kgs
for some function E of the complex structure and open string moduli. If this were
the case, the one-loop corrections would typically dominate the α′ correction in (21)
6The toroidal orientifold case seems to be a bit degenerate. Two stacks of D7-branes intersect along
a 2-cycle with the topology of P1. However, there are point-like curvature singularities along the P1
at the orbifold point and strings winding around these singular points cannot be contracted without
crossing the singularities. This seems to allow for stability of winding strings (at least classically).
(which is suppressed by the overall volume V) in the Kähler potential, if there are large
hierarchies among the Kähler moduli. However, one should keep in mind that toroidal
orientifolds are rather special in that they have very simple intersection numbers. In
particular, the overall volume can be written as V ∼ τiti, where there is no summation
over i implied. Thus, it is not obvious whether a generalization to the case of a general
Calabi-Yau really contains terms suppressed by single Kähler moduli instead of the
overall volume. Even though we cannot exclude the presence of such terms, we deem it
more likely that the scaling of one-loop corrections to the Kähler potential is not (22)
Calabi-Yau: ∆Kgs
gaK(t, S1)E
W(t, S1)E
V , (23)
where the sums run over KK and winding states, respectively. Also, E (K) and E (W ) are
again unknown functions of the complex structure and open string moduli, t stands
for the 2-cycle volumes (in the Einstein frame; see appendix C.1) and the functions
gK(t, S1) and gW(t, S1) determine the scaling of the KK and winding mode masses
with the Kähler moduli and the dilaton.7 As we review in appendix E, in the toroidal
orientifold case the suppression by the overall volume arises naturally through the Weyl
rescaling to the 4-dimensional Einstein frame.
Starting with the ansatz (23) for smooth Calabi-Yau manifolds, the known form
(22) for toroidal orientifolds follows simply by substituting gK, gW and the intersec-
tion numbers for the toroidal orientifold case. In particular, gK ∼ ti for the 2-cycle
transverse to the relevant D7-brane, while gW ∼ t−1i for the 2-cycle along which the
two D7-branes intersect. Then, the first of the terms in (23) reduces to E (K)i /(S1τi) for
toroidal orientifolds, the second to E (W )i /(τjτk) with j 6= i 6= k, cf (50). Our strategy
in the following chapters will therefore be to assume a scaling like (23) for the 1-loop
corrections to the Kähler potential for general Calabi-Yau spaces.
As already mentioned, the dependence on the complex structure and open string
moduli cannot be inferred by analogy to the orientifold case. We parameterize our igno-
7In rewriting the sums over KK and winding states in terms of the functions g and E , we assume that
the dependence of the corresponding spectra on the complex structure and Kähler moduli factorizes.
In the known examples of toroidal orientifolds (with or without world-volume fluxes), this is always
the case, cf. [52]. Moreover, in general there can appear several contributions (denoted by a and q)
depending on which tower of KK or winding states are exchanged in a given process. We will see
explicit examples of this in the following.
rance by keeping the expressions E in (23) as unknown functions of the corresponding
moduli. Then we investigate the consequences of the one-loop terms, depending on
the size of these unknown functions at the minimum of the potential for the complex
structure and open string moduli. Some further comments on the form of ∆Kgs will
appear in section 5.2.
3.2 LVS with loop corrections
Thus, allowing for string loop corrections of the form (23) in (9), and expanding the
α′ correction as in (10), we can write
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
W = W0 +
−aiTi , (24)
where as explained in the previous section, we have not specified the explicit form of
the loop corrections E , that are allowed to be functions of U (and in general of the open
string moduli, that we neglect in our analysis, assuming that they can be stabilized
by fluxes). The Kähler potential for the complex structure moduli Kcs(U, Ū) is left
unspecified in (24), indeed we will not need its explicit form. For consistency, we have
also included loop corrections to the α′ correction.8 This changes ξ to ξ̃, which is a
small change; for S1 = 10, numerically ξ̃ ≈ 1.02 ξ.
Neglecting fluxes, the functions gaK and g
W are proportional and inversely propor-
tional to some 2-cycle volume, respectively. (We will come back to corrections from
fluxes in appendix D.) When using a particular basis of 2-cycles (with volumes ti as
in appendix C.1), the 2-cycle volume appearing in gaK or g
W might be given by a linear
combination ta =
i citi of the basis cycles ti (and similarly for tq). Depending on
which 2-cycle is the relevant one, this linear combination might or might not contain
the large 2-cycle tb ∼ V1/3, which always exists in LVS. If it is present in the linear
combination, one can neglect the contribution of the small 2-cycles to leading order in
a large volume expansion and obtains possible terms proportional to E (K)b S−11 V−2/3 or
E (W )b V−4/3, where the subscript b refers to the large 4-cycle τb.
8We remind the reader that the α′ correction arises from the R4 term in 10 dimensions whose
coefficient receives corrections at 1-loop (and from D-instantons). The 1-loop correction amounts to
a shift of the prefactor from ξ to ξ̃ = ξ
1 + π
3ζ(3)S2
, see for instance [53] for a review.
Before getting into the details, it is hard to resist trying to anticipate what might
happen. For those terms that are more suppressed in volume than the ξ̃ term (e.g
E (W )b ), one would expect the loop corrections to have little effect on stabilization. They
could still represent a small but interesting correction to physical quantities in LVS. For
those that are less suppressed in volume than the ξ̃ term (e.g. E (K)b ), one would expect
the loop correction to have a huge effect on stabilization, and severely constrain the
allowed values for the complex structure moduli and the dilaton in LVS (in particular,
constrain them to a region in moduli space where the function E (K)b takes very small
values). We will find, however, that this expectation is sometimes too naive. For
example, there can be cancellations in the scalar potential that are not obvious from
just looking at the Kähler potential.
Let us now get into more detail on what happens in the LVS model with loop
corrections.
3.3 The P4
[1,1,1,6,9]
model
We would now like to specify the general form of the Kähler- and superpotential (24) to
the case of the P4[1,1,1,6,9] model. In this space, the divisors that produce nonperturbative
superpotentials when D7- (or D3-) branes are wrapped around them do not intersect,
as reviewed for instance in [48]. Therefore, we do not expect any correction of the E (W )
type in this model (for the generalization to models where there are such intersections,
see appendix D). Moreover, we neglect flux corrections to the KK mass spectrum in
the main text. It is shown in appendix D that, for small fluxes, this correctly captures
all the qualitative features we are interested in, and it leads to much clearer formulas.
Thus, we now consider the scalar potential resulting from
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
τbE (K)b
τsE (K)s
W = W0 + Ae
−aTs . (25)
As τb is very large the corresponding non-perturbative term in the superpotential of
(24) can be neglected, which allowed us to simplify the notation by setting As = A
and as = a.
The general structure of the scalar potential was already given in (12). The three
contributions at leading order (O(V−3)) in the large volume expansion are
Vnp1 = e
24a2|A|2τ 3/2s e−2aτs
V∆ , (26)
Vnp2 = −eKcs
2a|AW0|τse−aτs
6E (K)s
, (27)
3eKcs |W0|2
1 ξ̃ +
4(E (K)s )2
, (28)
where the axion has already been minimized for, as discussed in section A.2, and
2S1τs − 3E (K)s . (29)
The leading α′-correction is the ξ̃ term in V3 above. We now see that it scales with the
volume and the string coupling gs = 1/S1 as claimed in the Introduction, in eq. (3).
Also the volume dependence of the loop correction (E (K)s term) in V3 is as announced
in (3). The gs factors seem to differ from (3); we see g
s , g
s and g
s for Vnp1, Vnp2 and
V3, respectively. This is because the gs dependence advertised in (3) arises in models
where, unlike in P4[1,1,1,6,9], the E (W ) correction is present as well, cf. appendix D.9 It is
also worth mentioning that the loop correction proportional to E (K)s modifies Vnp1 and
Vnp2 at leading order in the V-expansion. whereas the α′ correction does not; it only
appears in V3. This is so even though both corrections are equally suppressed in the
Kähler potential (i.e. ∼ V−1). The reason for this can be traced back to the fact that
the loop-correction explicitly depends on τs and not only on the overall volume, cf. the
discussion in appendix C.4 and C.5.
As anticipated, E (K)b and its first derivatives appear only at the next order, O(V−10/3):
V10/3 = 2
61/3|W0|2eKcs
S31V10/3
(E (K)b )2 +
∂αE (K)b ∂ᾱE
, (30)
where ∂α = ∂/∂U
α and ∂ᾱ = ∂/∂Ū
ᾱ and α enumerates the complex structure moduli.
For E (K)b = E
s = 0, the potential terms at leading order coincide with the original
case discussed in (56), cf. appendix A.1. The singularity from zeros of the denominator
is an artifact of the expansion as discussed in appendix B. The range of validity is
9There, it is shown that including the effect of fluxes on the KK spectrum might also produce this
behavior.
limited to the range in moduli space where the denominator ∆ does not become too
small. It is also apparent that the loop terms are subleading in a large τs, large S1
expansion. However, depending on the relative values of the parameters {E (K)s , τs, S1},
a truncation to the first terms in such an expansion may or may not be valid. We
perform a numerical comparison of the two contributions to V3 in figure 4.
Figure 4: The top surface is the α′ correction, the second is the gs correction, and
the “red carpet” is 10/∆ (we used the values A = 1,W0 = 1, a = 2π/8). We see
that for most of the parameter range, the α′ correction dominates, and only for large
E(K)s , with the string coupling gs = 1/S1 not too small, do the contributions become
comparable.
We can understand the volume dependence of the terms (26)-(30) as follows. The
common prefactor eK gives an overall suppression τ−3b ≃ V−2. The quantum corrections
obey the rule that a term proportional to 1/τλb in K appears in V3 at order 1/τ
(where the +3 comes from the overall eK factor) for all values of λ except for λ = 1.
When it does appear, it is generated by the term (KiKi − 3) and breaks the no-scale
structure. For λ = 1 there is a cancellation at leading order, so it appears only at
order 1/τ 2+3b (see appendix C.3 and C.4). This rule can explicitly be verified in our
calculation: the α′ and the E (K)s corrections are suppressed by 1/τ 3/2b inK, and therefore
they appear with the suppression 1/τ
b in V3. On the other hand, for the E
b term a
cancellation takes place to leading order (λ = 1). It appears neither in Vnp1 nor in Vnp2
at leading order (which can be understood more generally, cf. appendix C.5). Thus, it
only appears subleading in the potential, at O(V−10/3).10
10This cancellation for λ = 1 was already noticed in [51], albeit in the case without nonperturbative
superpotential. In [54] it was argued that this cancellation can be understood from a field redefinition
We now proceed to minimize the potential (26)-(28), using the same strategy as
in the case without loop corrections, cf. section 2.3 and appendix A.1. The equations
∂VV = 0 = ∂τsV are of course more complicated now, but it is easy to solve them
numerically. Doing so we find that the volume V and the small 4-cycle volume τs,
viewed as functions of S1 and E (K)s , are well fit by linear functions when restricted to
a sufficiently limited range in parameter space. For example,
range: S1 = [8, 11], E (K)s = [20, 40]
log10 V = 1.720S1 − 0.1208 E (K)s − 3.437 , (31)
τs = 5.000S1 − 0.3581 E (K)s − 8.638 .
The fits are quite good; the error is no greater than ±0.3 for τs and ±0.1 for log10 V in
this range, for an {S1, E (K)s } grid of 402 points.
From (31) we see an interesting difference to the case without loop corrections. The
value of τs at the minimum depends on the complex structure moduli U , through E (K)s .
This is in contrast to the case without loop corrections, where the value of τs is only
determined by the value of the Euler number ξ and the dilaton S1, cf. (57) below. It is
analogous to the perturbative stabilization in [51] where the volume at the minimum
of the potential also depends on U .
The result (26)-(28) was derived in a particular model, but we expect the appearance
of loop corrections in V to be more general. This opens up the possibility that in
principle, one might obtain large volume minima even for manifolds of vanishing (or
even positive) Euler number, where LVS is not applicable, as LVS-style stabilization
only holds for one sign of ξ. In practice it might be difficult to get large enough values
for the 1-loop corrections to stabilize τs at a value sufficiently bigger than the string
scale. This deserves further study.
We also note that the special structure of (28) and (30), i.e. the appearance of Es
only in (28) and of Eb only in (30), offers additional flexibility in tuning the relative
size of these terms in a purely perturbative stabilization of the Kähler moduli along
the lines of [51,55]. Also this point deserves further study.
argument combined with the no-scale structure of the tree-level Kähler potential. That argument
holds for the case of a single Kähler modulus T with tree-level Kähler potential −3 ln(T + T̄ ) and
under the assumption that the coefficient of the loop correction to the Kähler potential ∼ (T + T̄ )−1
is independent of the complex structure moduli and the dilaton. Here, these assumptions do not
hold, but we showed that the term ∼ (Tb + T̄b)−1 in the Kähler potential nevertheless only appears
at subleading order in the potential in LVS, cf. (26)-(30).
4 Gaugino masses
Now that we know how the stabilization of the (Kähler) moduli is modified by loop
corrections, it is natural to extend our analysis to the soft supersymmetry breaking
Lagrangian (For a review see for instance [56,57].) In LVS, supersymmetry breaking
is mostly due to F -terms: Fs 6= 0, Fb 6= 0. These determine the soft supersymmetry
breaking terms which can be present in the low energy effective action without spoiling
the hierarchy between the electroweak and the Planck scale,
Leff = LMSSM + Lsoft . (32)
The soft Lagrangian contains gaugino masses M , scalar masses m, further scalar bilin-
ear terms B and trilinear terms A. (For explicit expressions, see the aforementioned
reviews, or e.g. [6].)
Let us start considering gaugino masses. In [6] it was shown that in LVS, gaugino
masses Ma are generically suppressed with respect to the gravitino mass m3/2:
|Ma| ≃
ln(1/m3/2)
ln(1/m3/2)
(we use units in which MPl = 1). This suppression results from a cancellation of the
leading order F -term contribution to gaugino masses. We briefly review this calcula-
tion. Given the F -terms
F I = eK/2GJ̄IDJ̄W̄ , (34)
gaugino masses are given by [56]
F I∂Ifa , (35)
where fa are the gauge kinetic functions and a labels the different gauge group fac-
tors. In LVS the Standard Model (SM) gauge groups arise from D7-branes wrapped
around small 4-cycles. We do not try to go into the details of how to embed the SM
concretely, but we mention that different gauge group factors might arise from brane
stacks wrapping the same 4-cycle if world volume fluxes are present on the branes. In
that case the gauge kinetic functions are given by11
+ ha(F)S + f (1)a (U) , (36)
11We use the “phenomenology” normalization of the gauge generators, in the language of [58]; that
explains the relative factor of 4π in (36).
where ha depends on the world volume fluxes and we also included a possible 1-loop
correction to the gauge kinetic function which depends on the complex structure (and
possibly open string) moduli. If several gauge groups arise from branes wrapped around
the same cycle, the same Kähler modulus T would appear in all of them. From (36) it
is clear why the D7-branes of the SM have to wrap small 4-cycles, because otherwise
the gauge coupling would come out too small (unless there is an unnatural cancellation
between the different contributions to fa).
As is also apparent from (36), the gauge kinetic function in general depends not
only on the Kähler moduli but also on the dilaton and the complex structure. Thus,
according to (35) we need to know FU , F S and F i for the small Kähler moduli.12
From the definition (34), it is clear that FU and F S might be non-vanishing even
though we assume DUW = 0 = DSW , provided the inverse metric G
J̄I contains mixed
components between Kähler moduli on the one hand and complex structure moduli
and dilaton on the other hand. Without loop corrections (i.e. considering only the
leading α′ correction) there is no mixing between the Kähler and complex structure
moduli, and one finds
FU = 0 , F S ∼ O(V−2) and F i ∼ O(V−1) (without loop corrections) . (37)
Thus, at leading order in the large volume expansion, the sum in (35) only runs over
the Kähler moduli. Moreover, taking into account the linear dependence of the gauge
kinetic functions (36) on the (small) Kähler moduli, the sum effectively only involves
a single term, i.e.
F a +O(V−2) , (38)
where F a is the F-term of the (small) Kähler modulus appearing in fa.
As a concrete example we consider again the P4[1,1,1,6,9] model with only one small
Kähler modulus τs. The corresponding F-term is given by
F s = eK/2
Gs̄s∂s̄W̄ + (G
s̄sKs̄ +G
b̄sKb̄)W̄
= 2τse
K/2W̄0
− 1 +O((aτs)−2)
+O(V−2) , (39)
where we used (20) and (61) for the first term and (64) for the second.
12With a slight abuse of notation, we denote the F -terms of the Kähler moduli by the index i, but
the F -terms of the other moduli are identified by the symbol for the corresponding modulus, like FS .
This is to avoid introducing too many indices.
Now the leading order cancellation is obvious in (39). Determining the gaugino
masses requires dividing by Refs, cf. (35). In order to further evaluate this, [6,7] as-
sumed that the dilute flux approximation fs = (4π)
−1Ts is valid, i.e. they neglected the
contributions from world-volume fluxes and one-loop terms compared to the tree-level
term. This puts some constraints on the allowed discrete flux values determining hs.
We want to stress that the cancellation appearing in (39) is independent of this approx-
imation. We are mainly interested in the fate of this cancellation when including loop
corrections, and do not have anything to add concerning phenomenological constraints
that may arise from imposing the dilute flux approximation. Using it, the gaugino
masses simplify to
|Ms| =
= eK/2|W0|
+O((aτs)−2)
ln(1/m3/2)
ln(1/m3/2)
which is the announced result. In (40) we used
m3/2 ∼ |W0|/V and aτs ∼ ln(V/|W0|) , (41)
where the second relation holds in LVS due to (20).
4.1 Including loop corrections
The previous section was a review of the results found in [6]. Now we ask what changes
if one considers the loop corrected Kähler potential (25). A priori, as (40) results from
a leading order cancellation, one might wonder whether loop corrections might spoil
this small hierarchy between the gaugino and gravitino masses. To address this concern
we start by observing that the gaugino masses are still determined by the F-terms of
the small Kähler moduli (in the large volume limit). More precisely, the scaling of the
F-terms (37) now becomes
FU = O(V−2), F S ∼ O(V−2) and F i ∼ O(V−1) (with loop corrections) , (42)
i.e. FU no longer vanishes, but it is just as suppressed as F S.
We again focus on the P4[1,1,1,6,9] model and ask how (39) is modified by loop cor-
rections. Amongst other things, we need to generalize equation (20) to include loop
corrections, since we need it to calculate the first term in (39). Thus, we need to
extremize the potential again with respect to τs by setting
∂τsV =
4aτs − 1
∆+ 6E (K)s
2a|W0|
V2S1∆2
aτs − 1
∆2 − 18(E (K)s )2
2aS1τ
s E (K)s
X (43)
− 3|W0|
2(E (K)s )2
4S21V3∆
to zero. Obviously, X = 0 is no longer a solution. Instead, there are now two non-trivial
solutions, one of which goes to zero in the limit E (K)s → 0. This solution corresponds
to a maximum of the potential, so it is of no use to us here. We can expand the other
solution for large aτs, as in the case without loop corrections, yielding
X = Ae−aτs =
2|W0|
2aE (K)s
(aτs)2
. (44)
Another ingredient we need is the quantity Gı̄sKı̄, in order to evaluate the second term
in (39). Using equation (65) we obtain
Gı̄sKı̄ = −2τs
6E (K)s
+ . . .
= −2τs −
2E (K)s
− 18(E
S21τs
2(E (K)s )3
τ 2s S
+ . . . , (45)
where the ellipsis represents terms that are more suppressed in V−1.
Now we see from (44), (45) and (65) that at leading order in an expansion in aτs, the
quantities relevant to evaluate (39) are not affected by the loop corrections. Thus, the
leading order cancellation in the gaugino masses survives the inclusion of loop effects.13
At first glance, though, equations (44), (45) and (65) seem to suggest a correction to the
subleading term, that could potentially give a significant contribution to the gaugino
masses after the leading-order cancellation, cf. (39).
In the actual calculation, this contribution drops out. Putting all the ingredients
together (and employing the dilute flux approximation again), the gaugino mass turns
13One might argue that this result was to be expected, because the main assumption of [6] is
that the stabilization is due to nonperturbative effects, i.e. the dominant effect in ∂τsV should arise
from the nonperturbative superpotential. However, in view of (43), it is no longer obvious that the
nonperturbative terms dominate when loop corrections are included.
out to be
|Ms| =
= 3eK/2|W0|
16a2τ 2s
S1 − 12
2aE (K)s
64S1a3τ 3s
+ . . .
ln(1/m3/2)
ln(1/m3/2)
. (46)
The result of [6] is therefore very robust. Unexpectedly, the correction to (40) due to
E (K)s only appears at sub-sub-leading order in the 1/ ln(1/m3/2) expansion.
4.2 Other soft terms
In [7] all other soft terms were calculated for LVS. The main result (see p. 15 of [7]) is
that roughly speaking, all the soft parameters are determined by F s and by the power
with which the chiral matter metrics scale with τs. As we saw in the previous section,
F s gets modified by loop corrections only at sub-sub-leading order in a 1/τs expansion
(see (46)). Therefore, the calculation of all the soft terms in [7] appears to be quite
robust against including loop effects.
One of the key assumptions in [7] is that all the Yukawa couplings Y are already
present in perturbation theory, i.e. they have the schematic form Y = Y pert(U) +
Y np(e−T ). This requirement featured prominently already in the derivation of the
volume dependence of the chiral matter metrics in [59] by scaling arguments. In [7] the
same schematic form is essential for simplifying the trilinear soft terms A. In general
these terms receive a contribution of the schematic form
F T∂T logY = F
T ∂T (Y
pert(U) + Y np(e−T ))
Y pert(U) + Y np(e−T )
∼ O(e
O(T 0) +O(e−T ) , (47)
which is exponentially suppressed if and only if Y pert(U) is non-vanishing. However, in
many examples the Yukawa couplings are actually only generated nonperturbatively,
see for instance the discussion in [60], and [61] for some examples. This poses a con-
straint on the way the Standard Model is realized in LVS, if one wants to ensure flavor
universality of the soft breaking terms as advertised in [7].
One more comment about the important issue of flavor universality. In [7], section
3.4., it was argued that in LVS, approximate flavor universality is a natural consequence
of the zeroth-order factorization of Kähler and complex structure moduli spaces. We
provide some more details on the factorized approximation in appendix F.
5 LVS for other classes of Calabi-Yau manifolds?
In section 3.3 and 4 we saw that the 1-loop corrections to the moduli Kähler potential
only have relatively small effects on the large volume scenario based on the P4
[1,1,1,6,9]
model of [3]. In this chapter, we would like to ask the question how generic the “Swiss
cheese” form is for a Calabi-Yau manifold and if there are other models in which the
one-loop corrections discussed above might become more important. This is indeed
to be expected if the Calabi-Yau under consideration has a fibered structure, as we
explain in the following. If gs corrections do dominate α
′ corrections, they could ruin
the volume expansion of LVS.
5.1 Abundance of “Swiss cheese” Calabi-Yau manifolds
In the LVS examples discussed in [4] the volume in terms of the Kähler moduli takes
the “Swiss cheese” form
− . . .−
, (48)
where the coefficients ai, ..., ci are only non-vanishing for the small Kähler moduli. The
LVS limit consists in scaling the overall volume of the Calabi-Yau more or less isotrop-
ically while having small holes inside the manifold. The τ ’s are linear combinations of
∂tiV, where now V is considered as a (cubic) function of the 2-cycle volumes ti. For
the effective field theory analysis to be valid one should not only demand that the
4-cycle volumes τi are large compared to the string scale, but also that the 2-cycle
volumes ti are large. In the cases discussed in [4], the linear combinations ∂tiV are
indeed such that one can have one of them exponentially large and the others small
(but still sufficiently larger than the string scale), without taking any of the ti to be
exponentially small. This is obvious for the P4
[1,1,1,6,9]
example where the 2-cycle volume
tb only appears in the definition of one of the τ ’s, cf. (14), but it is also true for the
second example of [4], cf. their formulas (84).
However, the F18 model of [16] does not seem to allow its volume to be written in
the form (48) with one Kähler modulus τb that can become large while keeping all the
others small (again, demanding that the ti stay larger than 1 in string units). Thus,
it is an interesting question how generic or non-generic the “Swiss cheese” Calabi-Yau
manifolds are. We do not attempt to give a general answer; instead, we turn to two
examples in which the form of the volume differs from (48).
5.2 Toroidal orientifolds
The reason loop corrections may be more important in toroidal orientifolds than in
compactifications on “Swiss cheese” Calabi-Yau manifolds is the following. As we al-
ready discussed in section 3.1, the conjectured form of 1-loop corrections (23) simplifies
in the case of toroidal orientifolds, because they have very special and simple intersec-
tion numbers. More concretely, using the definition τi = ∂tiV, together with the special
form of the intersection numbers in the toroidal case, i.e. V = t1t2t3, the volume can
alternatively be expressed as
V = √τ1τ2τ3 = tiτi (no summation; i = 1, 2 or 3) . (49)
Thus, formula (23) simplifies and the 1-loop corrections proportional to E (K)i are only
suppressed by single Kähler moduli instead of by the overall volume. Also the terms
proportional to E (W )i can be rewritten in the toroidal orientifold case and the Kähler
potential takes the form (for the T 6/(Z2 × Z2) example)
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
V (50)
E (K)i (U, Ū)
4τiS1
i 6=j 6=k
E (W )k (U, Ū)
4τiτj
where the functions E are non-holomorphic Eisenstein series in this case [37]. It is easy
to see that the origin of this simplification is the fact that there is just a single non-
vanishing intersection number in the toroidal orientifold case and all Kähler moduli
appear linearly in the cubic expression for the volume.
The difference of the toroidal orientifold to the “Swiss cheese” case of LVS can also
be seen in the different forms of the functional dependence of the volume on the Kähler
moduli. In the toroidal orientifold case one has the relations
∂t1V = t2t3 , ∂t2V = t1t3 , ∂t3V = t1t2 , (51)
so that two of them will always become large if one takes one of the ti to be large
and demands that the other two stay larger than 1. This also holds for any linear
combinations of them. The difference is also obvious from the fact that the 2-cycle
volume tb that is responsible for τb becoming large in the LVS examples of [4] always
appears cubically in the volume. This is related to the fact that the term (τb +
aiτi)
should be the square of a linear combination of the ti, in order for (48) to be expressible
as a cubic polynomial in the ti. In contrast, any (untwisted) 2-cycle volume in the
toroidal orientifold case only appears linearly in the cubic volume polynomial.
To illustrate the effect of terms in the Kähler potential that are suppressed only by
single Kähler moduli instead of the overall volume, we take the Kähler potential (50)
and expand V3 in the region of the Kähler cone where τ1 = τ2 = τb ≫ τ3 = τs (as we
explained above, at least two of the Kähler moduli have to become large simultaneously,
if one wants to avoid any of the 2-cycle volumes becoming very small). This leads to
(for simplicity setting all Ui = U , all E (K)i = E (K) and all E
i = E (W )):
|W0|2eKcs
2S1V2
(E (K))2 + 1
(∂UŪKcs)
−1∂UE (K)∂ŪE (K)
8τ 2s S
3 ξ̃ S
E (W )
(E (K))2 + (∂UŪKcs)−1∂UE (K)∂ŪE (K)
4S21τs
. (52)
Obviously, the leading term in the large τb expansion now comes from the loop correc-
tion and not from the α′ term (which term really dominates depends on the values of
S1 and U as well, of course). Thus, an expansion of the potential as in LVS, cf. (16),
would not be realized in this case, even if one found a way to lift enough zero modes
by fluxes for τs to appear in a nonperturbative superpotential.
This toy example was meant to show that for a consistent large volume expansion
in models with large hierarchies in the Kähler moduli, it is important to make sure that
there are no correction terms in the Kähler potential (from loop or α′-corrections) that
are suppressed only by some of the small Kähler moduli. We should stress again that
also terms suppressed by the large volume can be dangerous if the suppression is less
than for the α′ term, i.e. if it is τ−λb with λ < 3/2. The only exception to this rule is
the case λ = 1 as we showed above (and as is shown more generally in appendices C.4
and C.5). In this respect it would be important to know if the conjecture (25) really
bears out. If it turns out that the actual form of the 1-loop corrections also contains
terms like
tλ1b t
, (53)
with λ1 + λ2 = 1 but 0 < λ1 6= 1 or 0, such a 1-loop correction would spoil the large
volume expansion performed in (16).14
14In principle, one would also need an argument that no such terms arise at higher loop order, which
5.3 Fibered Calabi-Yau manifolds
The feature of orientifolds that all Kähler moduli appear linearly in the cubic expression
for the volume shows that a similar simplification can occur in the case of (K3) fibered
Calabi-Yau manifolds, which also have the property that one Kähler modulus (the one
corresponding to the volume of the base) only appears linearly in the cubic expression
for the volume. This takes the form
V = tbηijtitj + dijktitjtk , (54)
where ηij are the intersection numbers of the (K3) fiber, and neither they nor the triple
intersection numbers dijk contain the index b, which is chosen to denote “base”, but it
is also suggestively the same index as the one we used for the large Kähler modulus in
the P4
[1,1,1,6,9]
model. Two-parameter examples of this type appear in e.g. [47,62]. In a
region of the Kähler moduli space where the base tb is rather large but all the other
ti stay relatively small, the volume is approximately V = tbηijtitj . Thus, if the Kähler
potential has a 1-loop correction ∼ E (K)b tb/V, it could be approximated in this region
E (K)b tb
E (K)b
, (55)
where τf = ηijtitj is the volume of the (K3) fiber (which is small compared to t
Obviously, this would lead to a correction to the Kähler potential that is only sup-
pressed by a single (small) 4-cycle volume, similar to the toroidal orientifold example
we discussed in the last section.
We should note that this limit (large base and small fiber for (K3) fibered Calabi-
Yau manifolds), is quite different from the one performed in the usual LVS of [3], even
though both cases involve hierarchical limits of the Kähler moduli. As explained in
section 5.1, the LVS limit consists in scaling the overall volume of the Calabi-Yau more
or less isotropically while keeping holes in the bulk of the manifold small. In contrast,
the limit of large base and small fiber is anisotropic. At the moment we have nothing to
add about whether such anisotropic configurations with all moduli stabilized actually
exist. We merely wanted to point out that if they do exist, that would be an example
of smooth Calabi-Yau compactifications where the gs corrections we consider dominate
over the α′ corrections considered in the large volume limit, as in the toroidal orientifold
case.
would, however, have to be further suppressed in the dilaton S1.
6 Further corrections
In [4], further α′ corrections to the string effective action beyond the one in (9) were
considered. In the case of bulk α′ corrections (i.e. those already present in type IIB
bulk theory without D-branes, arising from sphere level) scaling arguments were given
as to why they are suppressed in the large volume limit. Although that discussion was
surprisingly powerful in its simplicity, we do not consider it completely conclusive, if
large hierarchies between the Kähler moduli exist. After all, dimensional analysis alone
does not guarantee that the other α′-corrections are always suppressed by additional
powers in the overall volume, instead of powers of some of the small Kähler moduli.
Moreover, in addition to the bulk α′ corrections that appear at order O(α′3), in the
models of interest for LVS further α′-corrections arise on the worldvolume of D-branes
and O-planes, cf. [63–70]. These corrections begin already at order O(α′) and scaling
arguments of the kind used for the bulk corrections do not seem to be sufficient to
neglect them.
Indeed, there are correction terms involving two powers of the Riemann tensor which
do modify the effective D3-brane charge and tension, if the D7-branes are wrapped
over 4-cycles with non-vanishing Euler number. These terms were already taken into
account in [1]. However, there are further contributions to the DBI action at the same
order in α′, like F 23R or F
3 , where F3 stands for the RR 3-form field strength, R for
the Riemann tensor and we left index contractions unspecified. If the D7-branes do
not break supersymmetry and remain BPS, it seems unlikely that these terms could
contribute to the potential for the closed string moduli, i.e. induce some effective D3-
brane tension. The reason is that there does not seem to be a corresponding term in
the Chern-Simons action that could lead to the necessary modification of the effective
D3-brane charge at the same time. This could be checked in more detail.
In general, we think that the question of additional corrections to the moduli
(Kähler) potential deserves further attention. Here we only outlined some steps in
that direction.
7 Conclusions
In this paper, we have investigated whether string loop corrections may impact a) stabi-
lization in the large volume compactification scenario (LVS), and b) the phenomenology
of those scenarios, as manifested in the soft supersymmetry breaking terms. The result
is that for the specific class of compactification manifolds considered in LVS, so-called
“Swiss cheese” Calabi-Yau manifolds, changes are minuscule. Only if the loop cor-
rections become abnormally large (in the toroidal orientifold case, this can happen if
the complex structure is stabilized very large) do they affect LVS. For other classes
of manifolds, the corrections may be important. We hasten to add that the detailed
expressions for the loop corrections in LVS remain unknown; we have merely tried to
infer their scaling with the Kähler moduli from experience in the toroidal orientifold
limit. We think it is important to attempt to address this issue, as the string coupling
is stabilized at a nonzero value, so the corrections cannot be turned off.
We also stress the (to some readers obvious) fact that there remain a host of is-
sues that must ultimately be dealt with if one wishes to claim that these are “string
compactifications”.
• We cannot be sure that fluxes do not alter the corrections, since backgrounds
with RR and NSNS fluxes are not well understood in string perturbation theory.
• Additional corrections may appear (see section 6) that could be equally threat-
ening to LVS as the loop corrections, or worse.
• In [37] only the corrections to the Kähler potential coming from N = 2 sectors
were determined and we based our generalization on those results. However,
there might be interesting corrections coming from the N = 1 sectors as well.
• It has not yet been shown that a local Standard Model-like construction can
be embedded in the simplest examples like the P4
[1,1,1,6,9]
model. If more general
models turn out to be needed, one needs to reconsider whether the requisite
nonperturbative superpotentials are generated.
• We have largely ignored open string moduli, under the proviso that they are
stabilized heavy, as are the dilaton and complex structure moduli.
• The coefficient A(S, U) in the nonperturbative superpotential is generally as-
sumed to be of order 1. It is not known how generic this is.
• All string computations we have discussed were performed in a supersymmetric
context. In LVS supersymmetry is broken already before uplift, in the AdS
minimum. Supersymmetry breaking directly in string theory is not very well
understood [39,71].
Faced with all these caveats, a pessimist might be inclined to give up. We think we
have shown that it is worth considering these issues in detail. Sometimes, an effect one
would have thought to be devastating turns out to be as gentle as a summer breeze.
Acknowledgments
It is a pleasure to thank Carlo Angelantonj, Vijay Balasubramanian, Massimo Bianchi,
Joe Conlon, Gottfried Curio, Robbert Dijkgraaf, Michael Douglas, Bogdan Florea,
Elias Kiritsis, Max Kreuzer, Fernando Marchesano, Peter Mayr, Thomas Mohaupt,
Hans-Peter Nilles, Gabi Pfuff, Fernando Quevedo, Waldemar Schulgin, Mike Schulz,
Stephan Stieberger, Angel Uranga, and Alexander Westphal for helpful discussions
and comments and Boris Körs for initial collaboration. This work is supported in part
by the European Community’s Human Potential Program under contract MRTN-CT-
2004-005104 ’Constituents, fundamental forces and symmetries of the universe’. The
work of M. B. is supported by European Community’s Human Potential Program un-
der contract MRTN-CT-2004-512194, ‘The European Superstring Theory Network’.
He would like to thank the Galileo Galilei Institute in Florence for hospitality. M. H.
would like to thank the university of Nis for hospitality. The work of M. H. and E. P.
is supported by the German Research Foundation (DFG) within the Emmy Noether-
Program (grant number: HA 3448/3-1). Both M. B. and M. H. would like to thank
the KITP in Santa Barbara for hospitality during the program “String Phenomenol-
ogy” and the university of Hamburg for hospitality during the workshop “Generalized
Geometry and Flux Compactifications”.
A Some details on LVS
In this appendix we collect some details on the minimization of the potential in LVS,
mainly reviewing the results of [3,8], but filling in some details. The minimization with
respect to the axions (i.e. the imaginary parts of the Kähler moduli) is performed for
an arbitrary number of Kähler moduli, while for the minimization of the real parts, we
restrict to the example of the hypersurface in P4[1,1,1,6,9] discussed throughout the main
text.
A.1 LVS for P4
[1,1,1,6,9]
Here we give some more numerical details on large-volume stabilization in the P4
[1,1,1,6,9]
orientifold. The relevant features of this Calabi-Yau have been described in chapter
2.3. The leading terms of the scalar potential are
V e−Kcs =
τe−2aτ
V2 τe
−aτ +
V3 , (56)
where we use τ = τs and V as the independent variables and for the expansion we
have in mind the limit (15). The minimum of this potential under the assumption that
aτ ≫ 1 is given by
V = µ
eaτ . (57)
In the P4[1,1,1,6,9] orientifold the coefficients λ, µ and ν can be calculated explicitly,
yielding
2a2|A|2
, µ =
2a|AW0|
and ν = ξ
S1|W0|2 . (58)
We notice that the value of τ at the minimum is determined only by the Euler number
τ ∝ χ2/3 and the value of the dilaton S1 at its minimum. An example of a set of
possible parameters (using a = 2π/10, A = 1, S1 = 10 and W0 = 10) is
ξ = − ζ(3)χ
2(2π)3
≃ 1.31 −→ ν ≃ 155 ,
λ ≃ 0.67 , µ = 4π
. (59)
There is an unknown overall factor eKcs that does not change the shape of the potential
and so leaves the position of the minima unchanged. For the parameters given in
equation (59), the minimum is at τ ≃ 41.1 and V ≃ 9.96 · 1011. These values come
from equation (57) which is just approximated using the assumption aτ ≫ 1. This
solution has the shortcoming that, if one is interested in the value of the potential at
the minimum, after substitution of (57) into (56), one finds V = 0. If instead one
solves the exact equation for the minimum of the potential numerically, the result is
V ≃ −6.6 · 10−37 at the point τ ≃ 41.7 and V ≃ 1.38 1012. From this one checks that,
apart from the shortcoming that V = 0, the approximate solution gives the position of
the minimum with a good precision.
A.2 Many Kähler moduli
The simple picture of P4
[1,1,1,6,9]
, gets slightly more involved in models with more than
two Kähler moduli, but some general statements can still be made. For a single small
Kähler modulus, among the leading contributions to the potential only the one from
Vnp2 is axion dependent, while the leading terms in V3 and Vnp1 are axion independent.
For several small Kähler moduli, all three terms are axion dependent. However, the
argument that the leading term in Vnp2 only receives a sign change due to axion stabi-
lization generalizes (and holds also for the regular KKLT scenario with relatively small
volume, see e.g. [8], section 3.2).
Indeed, with the superpotential (11) one obtains
Vnp1 = e
K G̄i
aiaj |AiAj |e−aiτi−ajτj cos(−aibi + ajbj + βi − βj)
Vnp2 = −2eK aiGk̄iKk̄
|AiW0|e−aiτi cos(−aibi + βi − βW0)
+|AiAj |e−aiτi−ajτj cos(−aibi + ajbj + βi − βj)
, (60)
V3 = e
K (Gk̄lKk̄Kl − 3)
|W0|2 + 2 |W0Ai|e−aiτi cos(−aibi + βi − βW0)
+|AiAj |e−aiτi−ajτj cos(−aibi + ajbj + βi − βj)
where Ai = |Ai|eiβi, W0 = |W0|eiβW0 and a sum over repeated indices is understood
throughout. As the only dependence on the axions is in form of cosines, one can easily
see that this potential has a minimum for
aibi = −βW0 + βi + niπ , ni ∈ 2Z+ 1 . (61)
We notice that the minimum of the bi depends on the (already fixed) complex structure
moduli, but it is independent of the Kähler moduli.
In the regime (15) the scalar potential again contains three terms at leading order,
Vnp1 ∼ 2eKcs
aiaj |AiAj |e−aiτi−ajτjM ljMki (−VVlk + VlVk)
+ . . . ,
Vnp2 ∼ −2eKcs
ai|AiW0|e−aiτiτi
+ . . . , (62)
V3 ∼ eKcs
8V3 |W0|
2 + . . . ,
where the sum over i and j effectively only picks up terms from the small moduli
because of the exponential suppression of Vnp1 and Vnp2. Moreover, for Vnp1 we used
the form (89) for the inverse of the moduli metric with respect to the basis (83). The
Kähler moduli appearing in the nonperturbative superpotential are linear combinations
of these, which we account for by a basis-changing matrix Mki , i.e.
Ti =M
i T̃k , (63)
where T̃k are the fields defined in (83) and Ti are the Kähler moduli appearing in the
nonperturbative superpotential. (Another way of saying this is that the real parts of Ti
measure the volumes of a basis of divisors that have the right properties to contribute
to the nonperturbative superpotential.) In the second term we used
Gk̄iKk̄ = −2τi + . . . = −2ReTi + . . . . (64)
In the basis (83), this would follow straightforwardly from (87), (89) and the relations
(80), but it holds equally well after a change of basis, because both sides of (64)
transform linearly under a change of basis (63).
Note finally that the ellipsis in (62) and (64) stand for subleading corrections in the
large volume limit (assuming also (15)).
B Loop corrected inverse Kähler metric for P4
[1,1,1,6,9]
We now have a closer look at the inverse metric from the Kähler potential in equation
(25). We invert the 4× 4 matrix and focus on the four terms that appear in the scalar
potential for the Kähler moduli,
Gb̄b =
τ 2b +O (τb) ,
Gb̄s = Gs̄b = 4τbτs
6E (K)s
, (65)
Gs̄s =
2S1τs
where we have performed an expansion in τb ≃ V2/3 and the quantity ∆ was introduced
in (29). We notice that only Gb̄b is not corrected at leading order. The apparent
divergence from zeros of the denominator ∆ is an artifact of the expansion. In fact,
the determinant of the (entire) Kähler metric behaves as
detG ∼ Aτ−7/2b +Bτ
b + . . . (66)
for some expressions15 A and B, which depend on the moduli τs, U and S1. In par-
ticular, one finds A ∼ ∆, but B does not vanish at a zero of ∆. Thus, in general
the expansion in large τb picks up the factor A, which is responsible for the apparent
divergence in (65). However, this is fictitious because when ∆ = 0, the next term
proportional to B is non-vanishing and the determinant stays away from zero. Indeed,
we do not expect to find any zero of the determinant in the range of validity of the
parameters.
If E (K)s ≪ (S1τs), one can further expand (65) with respect to E (K)s /(S1τs), yielding
Gb̄b =
τ 2b +O (τb) ,
Gb̄s = Gs̄b = 4τbτs
2E (K)s
((E (K)s )2
Gs̄s =
3E (K)s√
2S1τs
((E (K)s )2
Depending on the values of the moduli (τs and S1), this expansion may or may not be
useful. In general, only the expansion in τb makes sense and one has to deal with the
full expressions (65). That is what we did in section 3.3.
C No-scale Kähler potential in type II string the-
In this appendix we review why compactification of type IIA and type IIB theory
on general Calabi-Yau manifolds, or orientifolds thereof, lead to no-scale (F-term)
potentials if
i) the superpotential does not depend on the Kähler moduli (68)
and if
ii) one uses the tree-level form of the Kähler potential. (69)
(Of course, in LVS neither i) nor ii) holds, but one can think of jointly imposing i)
and ii) as a zeroth-order approximation, that we will successively move away from in
later subsections of this appendix.)
15This A has nothing to do with the A in Wnp.
If the moduli spaces of Kähler and complex structure moduli factorize (see appendix
F for more details on this), and under assumption i), the F-term potential takes the
V = eK
GĪJDĪW̄DJW − 3|W |2
GābDāW̄DbW + (G
ı̄jKı̄Kj − 3)|W |2
. (71)
The indices a and b run over the complex structure moduli and the dilaton, i, j over
the Kähler moduli and I and J refer to all moduli.
The condition for a no-scale potential (V = 0 for the Kähler moduli) is then
Gı̄jKı̄Kj = 3 , (72)
and we will verify in turn that this is fulfilled in both type IIA and type IIB Calabi-
Yau compactifications, if one uses the tree-level Kähler potential, as in assumption ii).
In that case, the moduli spaces of Kähler and complex structure moduli do factorize
exactly.
C.1 No-scale structure in type IIA
The tree level Kähler potential for the Kähler moduli is
K = − ln
dijk(σ + σ̄)i(σ + σ̄)j(σ + σ̄)k
= − ln
dijktitjtk
= − ln(V) , (73)
where dijk are the intersection numbers of the Calabi-Yau,
dijk =
ωi ∧ ωj ∧ ωk , (74)
σi = ti + ici (75)
are the complexified Kähler moduli whose real parts ti represent the volumes of 2-cycles
and whose imaginary parts originate from the expansion of the NSNS 2-form. Using
the Kähler form
J = tiωi (76)
of the Calabi-Yau, it is useful to introduce the notation
V = 1
J ∧ J ∧ J = 1
dijktitjtk ,
ωi ∧ J ∧ J =
dijktjtk ,
Vij =
ωi ∧ ωj ∧ J = dijktk . (77)
Note that here the index i does not denote a derivative with respect to the Kähler
variables (in contrast to subscripts on the Kähler potential K). Instead, one has the
relations Vi = 2∂σiV and Vij = 4∂σi∂σ̄̄V. It is straightforward to calculate
Ki = −
2V = Kı̄ Gi̄ = Ki̄ = −
. (78)
Then one can show that the inverse Kähler metric is
G̄i = −4VjiV + 2tjti . (79)
To verify this, one has to use
V ijVj =
ti , Vijtj = 2Vi , Viti = 3V . (80)
Putting everything together, one arrives at
Gı̄jKı̄Kj =
−4V ijV + 2titj
V = 3 , (81)
i.e. (72) is fulfilled under assumption (69).
C.2 No-scale structure in type IIB
In the type IIB case, the tree-level Kähler potential for the Kähler moduli is
K = −2 ln
dijktitjtk
= −2 ln(V) . (82)
The difference to the IIA case is that, even if K in (82) is expressed in terms of the
2-cycle volumes ti, the real parts of the good Kähler moduli, T̃i, are now the 4-cycle
volumes τ̃i (the imaginary parts, on the other hand, arise from the RR 4-form). The
relation between them depends on the particular Calabi-Yau:
Re T̃i = τ̃i =
dijktjtk = Vi , (83)
which cannot be inverted in general.16 In order to calculate Ki = ∂T̃iK we note that
∂ti = (∂ti T̃j)∂T̃j + (∂ti
T̄)∂ ¯̃T̄
= Vij
∂T̃j + ∂ ¯̃T̄
. (84)
If acting on a function F that only depends on T̃ +
T , as is the case for K, (84)
simplifies to
∂tiF (T̃ +
T ) = 2Vij∂T̃jF (T̃ +
T ) , (85)
where on the left hand side T̃ is understood as a function of t. Alternatively, one has
∂T̃iF (T̃ +
T ) =
V ij∂tjF (T̃ +
T ) = ∂ ¯̃
F (T̃ +
T ) . (86)
Using this, one can calculate
Ki = −
V ∂T̃iV = −
V ijVj
V = −
2V = Kı̄ , (87)
where in the last step we used (80). In the same way one can calculate
Gi̄ =
. (88)
Using this formula one can check that the inverse Kähler metric is given by
Gı̄j = 4 (−VVij + ViVj) . (89)
Putting everything together, no-scale structure holds also for type IIB:
Gı̄jKı̄Kj = (−VVij + ViVj)
V = 3 , (90)
again under the assumption (69).
16Note that the Kähler moduli appearing in the non-perturbative superpotentials in the examples
of [16] are related to the ones in (83) by a linear field redefinition. However, this does not play any role
in verifying the no-scale structure at leading order, as (90) below is invariant under field redefinitions.
We chose to make the distinction clear by using tildes for the Kähler moduli defined by (83).
C.3 Cancellation with just the volume modulus
Now we relax assumption (69). For simplicity, let us first consider the Kähler potential
K = −3 ln(T + T̄ ) + Ξ
(T + T̄ )λ
, (91)
which corresponds to the case of a single Kähler modulus and the complex structure
moduli and the dilaton are neglected. A generic quantum correction was added to the
tree level term, which could be an α′ or a loop correction, depending on the value of
λ. Focusing on V3, i.e.
eK |W |2 = G
̄iK̄Ki − 3 , (92)
one calculates
eK |W |2 =
(3(2τ)λ + ξλ)2
3(2τ)2λ + Ξ(2τ)λλ(λ+ 1)
= 3− 3 + (λ− λ
(2τ)λ
3(2τ)2λ
. (93)
This simplified calculation gives an intuition of why the E (K)b -term does not appear
in V3 of (28) whereas the α
′- and E (K)s -terms contribute. When the exponent of the
quantum correction is exactly 1, there is a cancellation at leading order in the scalar
potential (compare also the discussion in footnote 10). Note that since we focused on
V3 in this subsection, it did not matter whether assumption (68) holds or not.
C.4 Cancellation with many Kähler moduli
We would now like to see how the previous result is changed when we have an arbitrary
number of moduli. We do not make any assumption on the dependence of the volume
on the Kähler moduli (“Swiss cheese” or fibered manifolds are special cases). Due to
its relevance for LVS, we consider a single correction to the Kähler potential which only
depends on the large Kähler modulus Tb (an example would be the α
′-correction or the
loop term proportional to E (K)b , considering the moduli other than the Kähler moduli
as fixed; this is allowed at leading order in a τb-expansion, as we argue in appendix F).
Thus, we take the Kähler potential to be of the form
K = K(0) + δK = −2ln(V) + δK(Tb, T̄b) ≡ −2ln(V) +
(Tb + T̄b)λ
. (94)
Again focusing on V3, we obtain
eK |W |2 = G
̄iK̄Ki − 3 = (G̄i0 + δG̄i)
̄ + δK̄
i + δKi
− 3 ,
where δKi ≡ ∂TiδK and G
0 is the inverse metric of appendix C.2; finally δG
̄i is
the modification of the inverse metric coming from considering the modified Kähler
potential (94). Explicitly one has
G̄i = (G0i̄ + δKi̄)
−1 ≃ G̄i0 −G
0 δKhk̄G
0 + . . . ,
δKi = −
(2τb)λ+1
δib , δKi̄ =
(λ2 + λ) Ξb
(2τb)λ+2
δibδ̄b . (96)
We now put everything together and use the results of appendix C.2 and formula (64)
(which, for the unperturbed metric and Kähler potential, is an exact equality) to arrive
eK |W |2 =
i − 3
̄ δKi
0 δKkh̄G
i + . . .
= 0 +
(2τb)λ+1
4(λ2 + λ)Ξb
(2τb)λ+2
τbτb + . . .
(λ− λ2)Ξb
(2τb)λ
+ . . . . (97)
We notice that the term 1/τλb vanishes exactly for λ = 1, independently of the explicit
form of the volume in terms of the Kähler moduli. In particular, the loop correction
proportional to E (K)b experiences a cancellation at leading order in V3 (and it is not
difficult to see that the subleading order is suppressed by τ−2λb ). Therefore, the loop
correction is subleading in the potential compared to the α′ correction, even though it
is leading in the Kähler potential. Next, we would like to extend this analysis to the
other parts of the potential, i.e. Vnp1 and Vnp2.
C.5 Perturbative corrections to Vnp1 and Vnp2
We now introduce the nonpertubative superpotential into the game, i.e. relax assump-
tion (68), and look at the other terms of the scalar potential, Vnp1 and Vnp2 (see eq.
(12)). For this, we restrict to the P4[1,1,1,6,9] model again. The contribution Vnp1 is pro-
portional to Gs̄sW̄,s̄W,s. From (65) we see that no E (K)b appears at leading order. This
can be understood as follows. Consider the Kähler potential (94) where now V is the
volume of P4
[1,1,1,6,9]
, given in (14). Then the scaling with the large Kähler modulus τb
is schematically given by
Gs̄s ≃ Gs̄s0 −Gs̄b0 δKbb̄Gb̄s0 + . . .
∼ τ 3/2b + τ 2b
τλ+2b
∼ τ 3/2b +
+ . . . , (98)
which shows that any loop correction to the Kähler potential of the form Ξ/τλb leads to
a subleading contribution to Vnp1 in the large volume expansion. As usual, the ellipsis
stands for terms that are even more subleading in the τb expansion.
To understand the E (K)s correction to Gs̄s we need to consider
K = −2 ln(V) + δ̃K(T, T̄ ) ≡ −2 ln(V) + Ξb g(Ts, T̄s)
(Tb + T̄b)λ
for some function g(Ts, T̄s) of the small Kähler modulus and we assume λ ≥ 3/2 in the
following. Then, again very schematically, the scaling behavior is given by17
Gs̄s ≃ Gs̄s0 −Gs̄i0 δ̃Ki̄G
0 + . . .
∼ τ 3/2b + Ξb (τb, τ
τ−λ−2b τ
τ−λ−1b τ
+ . . . (100)
∼ τ 3/2b + τ−λb + τ
−λ+3/2
b + τ
b + . . . .
One sees that λ = 3/2 indeed contributes at the same order as Gs̄s0 . This is confirmed
by the dependence of Gs̄s in (65) on E (K)s through ∆, cf. (29).
We now consider Vnp2. This is proportional to G
̄sK̄. Again we start by considering
a correction to the Kähler potential whose only dependence on the Kähler moduli is
via τb, as in (94). Schematically, we find
G̄sK̄ ≃ (G̄s0 −G
0 δKbb̄G
̄ + δK̄
+ . . .
∼ τs +
Gs̄b0 Ξb
τλ+1b
+ . . . ∼ τ 0b +
+ . . . . (101)
This result is confirmed by the absence of E (K)b in the leading term of Vnp2. A calculation
very similar to the one in (100) shows, however, that Vnp2 is modified by a correction
to the Kähler potential of the form (99) for λ = 3/2. It is straightforward to generalize
this analysis to a more general form of the “Swiss cheese” volume, with more than one
small Kähler modulus.
17For λ = 3/2 we can still use the expansion of the inverse metric (96), because the correction term
would also be further suppressed e.g. in the dilaton.
D KK spectrum with fluxes
In this section we would like to develop some intuition on how the analysis of sections
3.3 and 4 might change in the presence of fluxes. We will restrict the discussion to
one possible effect of the fluxes, namely their influence on the KK spectrum. It is not
known explicitly how closed string fluxes, which are present in LVS, would change the
mass spectrum. We will consider a toy example, using an analogy to the correction
arising from world volume fluxes (cf. [52]), in order to get a feeling for what kind of
effects one might expect. In particular, for the purposes of this appendix we assume a
modified KK mass spectrum of the form
m2KK ∼
1 + F
1 + F
) , (102)
where F represents any of the fluxes that may be present, and in the second equality
the factors of S1 appeared when expressing the 2-cycle volumes in Einstein frame as
compared to the string frame (t ∼ e−Φ/2tstr). Note that expanding (102) for large
values of t would lead to a correction ∆m2 ∼ F 2/t3, whose scaling with the flux and
with t is reminiscent of the moduli masses induced by closed string 3-form flux [72,73].
In that case, the suppression would be by the overall volume (which would lead to only
mild effects in LVS), but in (102) we allow for a suppression by single 2-cycle volumes
(which might be the small 2-cycle in the P4[1,1,1,6,9] model).
Substituting (102) in (23), we now consider the scalar potential resulting from
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
τbE (K)b
τsE (K)s
F 2S1
W = W0 + Ae
−aTs . (103)
We have not included any flux correction to the term proportional to E (K)b because we
expect such corrections to be subleading in a large volume expansion.18 Note that the
F -dependent correction term we did include is of the same form as the winding string
18Even though we think it is unlikely, we cannot exclude that the correction to KK masses that
scale like t−1b without fluxes is only suppressed by F
2/τs instead of F
2/τb. In that case, one would
have to redo the analysis of appendix C.5, using (99) with λ = 1. This would prohibit the use of the
expansion (96), because in the large volume limit the leading contribution to Gss̄ would arise from the
loop correction (it would scale as τ−1b as opposed to the scaling of the tree level contribution ∼ τ
In that case the leading terms in Vnp1 and Vnp2 would be suppressed compared to V3 and only arise
at order V−10/3, thus invalidating the volume expansion of LVS.
correction ∼ E (W )s , when one neglects any potential flux corrections to the winding
string spectrum, cf. (24) (remember that gsW would just be proportional to 1/
without fluxes). Thus, by considering (103) we implicitly also analyze in the following
the effect of corrections from winding strings (recall from section 3.3 that this correction
is not present in the P4[1,1,1,6,9] model, but may be present in general).
We now give the generalization of (26)-(28) when using the modified Kähler po-
tential (103). The three contributions at leading order (O(V−3)) in the large volume
expansion are
Vnp1 = e
24a2|A|2τ 3/2s e−2aτs
V∆ , (104)
Vnp2 = −eKcs
2a|AW0|τse−aτs
6E (K)s
1− 2F
, (105)
3eKcs |W0|2
1 ξ̃ + (106)
4(E (K)s )2 − 8(E (K)s )2F 2S1τ−1s (1 + F 2S1τ−1s )− 8
F 2S21E
where the axion has already been minimized for, as discussed in section A.2, and now
∆ is generalized to
2S1τs − 3E (K)s
1− 3F
. (107)
Plots for F = 1 and F = 3 are given in fig. 5, and they look quite similar to the
plot without flux, fig. 4. Qualitatively, the conclusion is the same; only for nongeneric
values of the gs corrections do they compete with the α
′ correction. Note, however,
that the amount of fine-tuning seems to depend on the value of the flux, cf. fig. 5.
The same is true for the dependence of the values of V and τs at the minimum on S1
and E (K)s . For F = 3, for instance, this dependence becomes more complicated than
what we found in (31). For the parameter range shown in fig. 5, the values of τs and
V in the minimum vary in the ranges τs ∈ [14.6, 46.3] and log10 V ∈ [3.7, 15.5], where
the smallest value for both of them is reached in the corner where the two corrections
become comparable.
Also the cancellation that we found for the gaugino masses survives the inclusion
of the flux factor in (103). The correction still only appears at sub-sub-leading order
0.0 9
PSfrag replacements
0.0 9
PSfrag replacements
Figure 5: Similarly to figure 4, the top surface is the α′ correction, the second is
the gs correction (with F = 1 in the left graph and F = 3 in the right), and the “red
carpet” is 10/∆, with ∆ from (107), using the same values as in fig. 4. The result
is qualitatively the same as before. Note, however, that the range for E(K)s differs.
For larger values of F one does not need to fine-tune E(K)s as much in order for the
two corrections to become of similar order.
in an expansion in ln(1/m3/2) and we find (again using the dilute flux approximation
for the prefactor (Ref)−1):
|Ms| =
= 3eK/2|W0|
16a2τ 2s
S1 − 12
2a(1− 2F 2S1τ−1s )E
64S1a3τ 3s
+ . . .
ln(1/m3/2)
ln(1/m3/2)
. (108)
This concludes our brief study of the direct effects of fluxes on the loop corrections.
E The orientifold calculation
In the main text, we are interested in how ∆Kgs , the one-loop correction to the Kähler
potential, scales with the Kähler moduli Ti. Our argument in section 3.1 is based on the
known result for ∆Kgs in the case of N = 2 supersymmetric K3× T 2 orientifolds and
N = 1 supersymmetric T 6/ZN (or T 6/(Z2 ×Z2)) orientifolds, from [37] (see also [74]).
Here we review this computation for the case of K3 × T 2, and take this opportunity
to adapt it to our case of D3-branes and D7-branes from the beginning. (One can also
obtain them by T-duality on the final D9/D5 results of [37], e.g. as in the appendix
of [75], but as we shall see, the direct computation is enlightening in its own right.)
We will leave out details that are essentially identical to [37], and only emphasize the
differences.
As shown in [37] using “Kähler adapted” vertex operators, the easiest way to com-
pute ∆Kgs is by considering the 2-point function of the complex structure modulus U
of T 2, with vanishing Wilson line moduli, i.e.
〈VUVŪ〉 = −
4g2cα
(U − Ū)2
〈V (0,0)ZZ V
(0,0)
〉σ . (109)
Here, we use the notation of [37],
(0,0)
U = −gcα′−2
U − Ū
(0,0)
(0,0)
= gcα
′−2 2
U − Ū
(0,0)
(110)
(0,0)
ZZ = −
i∂Z +
α′(p · ψ)Ψ
i∂̄Z +
α′(p · ψ̃)Ψ̃
eipX ,
(0,0)
= − 2
i∂Z̄ +
α′(p · ψ)Ψ̄
i∂̄Z̄ +
α′(p · ψ̃) ¯̃Ψ
eipX . (111)
As in [37], and [76] before that, we find these complex worldsheet variables particularly
convenient:
(X4 + ŪX5) , Z̄ =
(X4 + UX5) ,
(ψ4 + Ūψ5) , Ψ̄ =
(ψ4 + Uψ5) , (112)
where
Gstr is the volume of T
2 measured in string frame. The 2-point function (109)
can be expanded for small momenta, p1 · p2 ≪ 1, and we obtain
〈V (0,0)ZZ V
(0,0)
〉σ = −V4
(p1 · p2)
16(4π2α′)2
d2ν1d
k=0,1
~n=(n,m)T
e−π~n
~nt−1
](0, τ)
η3(τ)
γσ,kZ intσ,k[
〈∂̄Z(ν̄1)∂̄Z̄(ν̄2)〉σ〈Ψ(ν1)Ψ̄(ν2)〉α,βσ 〈ψ(ν1)ψ(ν2)〉α,βσ (113)
+〈∂̄Z(ν̄1)∂Z̄(ν2)〉σ〈Ψ(ν1) ¯̃Ψ(ν̄2)〉α,βσ 〈ψ(ν1)ψ̃(ν̄2)〉α,βσ + c.c.
+O((p1 · p2)2) .
For the details we refer to [37]. The main difference to the corresponding formula
(C.3) in [37] is the appearance of the inverse metric G−1str in the exponent arising from
the zero mode sum, and in the prefactor. This is due to the fact that the D3 and
D7 branes are localized along the T 2, and so the closed string channel involves a
Kaluza-Klein momentum sum instead of a winding sum. The sum over bosonic zero
modes has been made explicit, since there is also an implicit dependence on m,n in
the bosonic correlators: this arises from the classical piece in the split into zero modes
and fluctuations. That is, Z(ν) = Zclass(ν) + Zqu(ν), where the classical part is given
Zclass =
n+mŪ
Re(ν) c̃σ , c̃σ =
1 for K
2 for A ,M . (114)
These zero modes have the right periodicity under Re(ν) → Re(ν) + π (for A,M) or
Re(ν) → Re(ν) + 2π (for K), i.e. X4 → X4 + 2πn
α′ and X5 → X5 + 2πm
α′. In
contrast to [37] they involve the real part of ν. The reason is again that in the D3/D7
case the branes are localized along T 2 and thus the winding appears in the open string
channel as opposed to the closed string channel (as was the case for D9/D5 branes).
The sum over spin structures is performed using Riemann identities. This leaves
the correlators of the bosonic fields as the only piece that depends on the positions
νi of the vertex operators. The νi integral can then be evaluated. As the zero modes
(114) involve the real part of ν in the case of D3/D7-branes, in contrast to the D9/D5-
case studied in [37], the zero mode contribution in the Z-correlators drops out. The
quantum part is evaluated using the method of images on the worldsheet [38,77,78].
To evaluate the KK sum in (113), it is useful to regularize the integral over t by a UV
cutoff Λ. With this we obtain
1/(eσΛ2)
~n=(n,m)T
π3c2σtα
′e−π~n
~nt−1
π3α′c2σe
4 + πα′c2σ
E2(0, U) + . . . , (115)
where the prime at the sum indicates that the (n,m) = (0, 0) term is left out, and cσ,
eσ are constants whose precise values will not be important in the following (but can
be found in [37]). Terms that go to zero in the limit Λ → ∞ have been dropped, as
indicated by the ellipsis. The nonholomorphic Eisenstein series E2(0, U) is the s = 2
special case of
Es(0, U) =
~n=(n,m)T
|n+mU |2s . (116)
The terms involving the UV cutoff Λ drop out after summing over all diagrams, due
to tadpole cancellation. We have then reduced (113) to
〈V (0,0)ZZ V
(0,0)
〉σ = −(p1 · p2)α′
(4π2α′)2
E2(0, U)γσ,kQσ,k
+O((p1 · p2)2) . (117)
The quantities Qσ,k come from the sum over spin structures and are defined in [37].
Introducing the notation
E2(0, U) =
k=0,1
E2(0, U)γσ,kQσ,k
, (118)
we end up with (neglecting some irrelevant factors of gc, α
′, terms subleading in the
low-energy expansion, and constants of order 1)
〈VUVŪ〉 ∼ −i(p1 · p2)
(4π2α′)2
(U − Ū)2
E2(0, U) . (119)
To read off the one-loop correction to the kinetic term of U we need to perform a Weyl
rescaling to the Einstein frame. In the one-loop term (119) this just leads to
Weyl rescaling: × e
Vstr , (120)
where
Vstr = VstrK3
Gstr (121)
is the overall volume in string frame. The Kähler potential can then be read off from
the kinetic term by use of the identity
∂U∂ŪE2(0, U) = −
(U − Ū)2
E2(0, U) , (122)
producing the final result
∆Kgs ∼
Gstre
Vstr(S + S̄)
E2(0, U) , (123)
where
Gstre
Φ/Vstr is to be interpreted as a function of the Kähler variables. In the
K3× T 2 orientifold case, using (121), this is just proportional to eΦ/VstrK3 ∼ (T + T̄ )−1
(with Re T the volume of K3 measured in Einstein frame), giving a result T-dual
to [37] (note that we switched the real and imaginary parts in the definition of T and
S as compared to [37], to conform with the rest of this paper). As we argue in section
3.1, in general the dependence on the Kähler moduli will be more complicated than
this, because there is no analog to the relation (121). It is still clear that the inverse
suppression in the overall volume will appear as in (123), given that it is a direct
consequence of the Weyl rescaling.
F Factorized approximation
As mentioned in section 4.2, it is an important issue to what extent the moduli spaces
of Kähler and complex structure moduli factorize. In this appendix, we give further
details on the factorized approximation.
A common starting point in the analysis of the potential arising in type IIB theory
with 3-form fluxes is to assume that all complex structure moduli Uα and the dilaton
S are stabilized by demanding
DUαW = 0 = DSW . (124)
In this case the F-term potential for the moduli (7) reduces to
V = eK
G̄iD̄W̄DiW − 3|W |2
, (125)
where as in the main text, the indices i and j refer only to the Kähler moduli and
thus run from 1 to h1,1. Note that even though the complex structure moduli and
the dilaton are assumed to be stabilized by (124), the inverse metric G̄i is part of
the inverse of the whole moduli space metric. More precisely, if we denote the Kähler
moduli by Ti, as before, and all other moduli (i.e. the complex structure moduli and
the dilaton) collectively as Za, the moduli space metric is given by
GIJ̄ ∼
Ki̄ Kib̄
Ka̄ Kab̄
. (126)
We denote the inverse of this (whole) metric by GJ̄I . In general
G̄i 6= (Ki̄)−1 . (127)
Equality only holds if Gib̄ = 0, i.e. if the moduli space of the Kähler moduli is factorized
from the rest, as it is the case without loop and α′ corrections.
In this appendix, we would like to investigate at which order in a large volume
expansion the two matrices in (127) start to deviate from each other. For this analysis
we assume a volume of the “Swiss cheese” form as in (48) and a Kähler potential of
the form (24) (without taking possible effects of fluxes on the KK and winding mode
spectra into account as was done in appendix D; thus, gaK ∼ ta and g
K ∼ t−1q for some
2-cycle volumes). To avoid cumbersome notation we will indicate all the small moduli
collectively as τs. We then use the formula
A−1(1 +BP−1CA−1) −A−1BP−1
−P−1CA−1 P−1
, (128)
where P is the Schur complement of A, defined as
P = D − CA−1B . (129)
In our case P is the Schur complement of Ki̄. From (24) we read off that
GIJ̄ ∼
τ−2b τ
τ−2b τ
τ−2b τ
, (130)
where we only indicate the τb dependence and the indices run over I, J = {Tb, Ts, U, S}.
Here β = −2 for those τi with a nonvanishing ai in (48) (so β has an implicit index
i), otherwise β = −5/2 (which is in particular the value in the P[1,1,1,6,9] case). We
decompose GIJ̄ as in (128)
τ−2b τ
, A−1 ∼
τ 2b τ
7/2+β
7/2+β
B = CT ∼
τ−2b τ
, D ∼
τ 0b τ
τ−1b τ
τ 0b τ
τ−1b τ
, P−1 ∼
τ 0b τ
τ−1b τ
. (131)
Using (128) one easily obtains the scaling of the inverse:
GJ̄I ∼
τ 2b τ
7/2+β
7/2+β
τ 0b τ
τ 0b τ
. (132)
Now, from (128), G̄i receives two contributions. The first is K ̄i, that would be the
only term in the case of a factorized metric; the second is K−1h̄ Khb̄P
Kal̄K
, that
breaks factorization. Let us compare their τb scaling:
G̄i = A−1 + A−1BP−1CA−1 (133)
τ 2b τ
7/2+β
7/2+β
τ 0b τ
τ 0b τ
Thus the corrections coming from non-vanishing off-diagonal metric elements in (126)
set in with a suppression by τ−2b , τ
−7/2−β
b and τ
b in G
b̄b, Gb̄s and Gs̄s, respectively.
In the explicit example based on P[1,1,1,6,9], β = −5/2, and we checked this result by
comparing to the subleading terms in (65).
F.1 Factorized approximation of the scalar potential
What we are really interested in is not the (inverse) metric itself, but the scalar po-
tential, to which we now turn. For the nonperturbative terms Vnp1 and Vnp2, the
suppression of the off-diagonal terms in (133) is inherited by the scalar potential, as
they are proportional to Gs̄sW̄,s̄W,s and G
̄sK̄, respectively. For V3 things are not as
simple, due to the no-scale structure at leading order. Let us neglect for a moment all
the quantum corrections, then the no-scale structure implies
Gı̄jKı̄Kj − 3
no−scale ∼ (τ
b , τ
τ 2b τ
7/2+β
7/2+β
∼ τ 0b + τ
2β+7/2
b = 0 . (134)
The two terms have to vanish independently. Now let us add corrections that break
no-scale structure. Because of the cancellation described in appendix C.3, the leading
contribution can be seen to come at order τ
b (from the α
′, E (K)s and E (W )s corrections).
On the other hand, the off-diagonal terms appear at order
Gı̄jKı̄Kj
off−diagonal ∼ (τ
b , τ
τ 0b τ
τ 0b τ
∼ τ−2b + τ
b + τ
∼ τ−2b + . . . , (135)
for both β = −2 and β = −5/2. Therefore, the off-diagonal terms of the moduli space
metric appear in the scalar potential with a suppression of at least τ
b (as is confirmed
by the explicit example of section 3.3, cf. formulas (26)-(30)). The suppression can be
even stronger if some corrections are absent and the leading term in (135) vanishes.
To summarize: if one is only interested in the leading term of the scalar potential in
the large volume (i.e. large τb) expansion, then one can use the factorized approximation,
G̄i = K ̄i +O
. (136)
This provides a useful tool to simplify the calculations.
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	Introduction
	Review
	KKLT
	Consistency of KKLT
	Large volume scenario (LVS)
	String loop corrections to LVS
	From toroidal orientifolds to Calabi-Yau manifolds
	LVS with loop corrections
	The P[1,1,1,6,9]4 model
	Gaugino masses
	Including loop corrections
	Other soft terms
	LVS for other classes of Calabi-Yau manifolds?
	Abundance of ``Swiss cheese'' Calabi-Yau manifolds
	Toroidal orientifolds
	Fibered Calabi-Yau manifolds
	Further corrections
	Conclusions
	Some details on LVS
	LVS for P[1,1,1,6,9]4
	Many Kähler moduli
	Loop corrected inverse Kähler metric for P[1,1,1,6,9]4
	No-scale Kähler potential in type II string theory
	No-scale structure in type IIA
	No-scale structure in type IIB
	Cancellation with just the volume modulus
	Cancellation with many Kähler moduli
	Perturbative corrections to Vnp1 and Vnp2
	KK spectrum with fluxes
	The orientifold calculation
	Factorized approximation
	Factorized approximation of the scalar potential

  We subject the phenomenologically successful large volume scenario of
hep-th/0502058 to a first consistency check in string theory. In particular, we
consider whether the expansion of the string effective action is consistent in
the presence of D-branes and O-planes. Due to the no-scale structure at
tree-level, the scenario is surprisingly robust. We compute the modification of
soft supersymmetry breaking terms, and find only subleading corrections. We
also comment that for large-volume limits of toroidal orientifolds and fibered
Calabi-Yau manifolds the corrections can be more important, and we discuss
further checks that need to be performed.

Introduction 3
2 Review 5
2.1 KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Consistency of KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Large volume scenario (LVS) . . . . . . . . . . . . . . . . . . . . . . . . 7
3 String loop corrections to LVS 10
3.1 From toroidal orientifolds to Calabi-Yau manifolds . . . . . . . . . . . . 11
3.2 LVS with loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 The P4[1,1,1,6,9] model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Gaugino masses 20
4.1 Including loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Other soft terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 LVS for other classes of Calabi-Yau manifolds? 25
5.1 Abundance of “Swiss cheese” Calabi-Yau manifolds . . . . . . . . . . . 25
5.2 Toroidal orientifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 Fibered Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . 28
6 Further corrections 29
7 Conclusions 29
A Some details on LVS 31
A.1 LVS for P4
[1,1,1,6,9]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A.2 Many Kähler moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
B Loop corrected inverse Kähler metric for P4[1,1,1,6,9] 34
C No-scale Kähler potential in type II string theory 35
C.1 No-scale structure in type IIA . . . . . . . . . . . . . . . . . . . . . . . 36
C.2 No-scale structure in type IIB . . . . . . . . . . . . . . . . . . . . . . . 37
C.3 Cancellation with just the volume modulus . . . . . . . . . . . . . . . . 39
C.4 Cancellation with many Kähler moduli . . . . . . . . . . . . . . . . . . 39
C.5 Perturbative corrections to Vnp1 and Vnp2 . . . . . . . . . . . . . . . . . 40
D KK spectrum with fluxes 42
E The orientifold calculation 44
F Factorized approximation 48
F.1 Factorized approximation of the scalar potential . . . . . . . . . . . . . 50
1 Introduction
The KKLT strategy [1,2] for producing stabilized string vacua that can serve as a
starting point for phenomenology has been a source of great interest for the last few
years. The “large-volume scenario” (LVS) [3,4] is an extension of KKLT where string
corrections to the tree-level supergravity effective action computed in [5] play a sig-
nificant role, and where the compactification volume can be as large as 1015 in string
units. In LVS, work has been done on soft supersymmetry breaking [4,6,7], the QCD
axion [8,9], neutrino masses [10], inflation [11–14], and even first attempts at LHC
phenomenology [15].
Although tantalizing, the models discussed in the aforementioned papers (nominally
“string compactifications”) raise many questions. It remains an open problem to con-
struct complete KKLT models in string theory, as opposed to supergravity. Problems
one faces include things like the description of RR fluxes in string theory, showing that
the necessary nonperturbative effects actually can and do appear in a way consistent
with other contributions to the potential (for progress in this direction, see [16–32]),
and verifying that one can uplift to a Minkowski or deSitter vacuum without ruining
stabilization [26,33–36]. In LVS, since string corrections play a crucial role, striving
for actual string constructions seems quite important. In the end, the restrictiveness
this entails may greatly improve predictivity, or kill the models completely as string
compactifications.
In this paper, we will not improve on the consistency of KKLT or LVS in general,
but rather assume the existence of LVS models in string theory, and then perform self-
consistency checks. This is a modest step on the way towards reconciling phenomeno-
logically promising scenarios with underlying string models. We will see that although
a priori the situation looks very bleak, and one might have hastily concluded that even
our modest consistency check would put very strong constraints on LVS, things are
more interesting. It turns out that LVS jumps through every hoop we present it with,
and instead of broad qualitative changes, we find only small quantitative changes.
The main difference between KKLT and LVS is that LVS includes a specific string
α′ correction ∆Kα′ in the Kähler potential K of the four-dimensional N = 1 effective
supergravity. Naturally, the four-dimensional string effective action also contains other
string corrections. Here, we will focus on gs corrections due to sources (D-branes
and O-planes). For some N = 1 and N = 2 toroidal orientifolds, these corrections
were computed in [37] (see also [38]; for a comprehensive introduction to orientifolds,
see [39]). Compared to the α′ correction ∆Kα′ considered in LVS, the gs corrections to
the Kähler potential ∆Kgs will scale as
∆Kα′ : ∆Kgs
O(α′3) : O(g2sα′2)
(string frame) . (1)
By naive dimensional analysis, one would expect that in a 1/V expansion, where V is
the overall volume in the Einstein frame, eq. (1) implies
∆Kα′ ∼ O(g−3/2s V−1) , ∆Kgs ∼ O(gsV−2/3) (Einstein frame) . (2)
If there is more than one Kähler modulus, as is usually the case, various combinations
of Kähler moduli may appear in ∆Kgs in eq.(2), and a priori this could lead to even
weaker suppression in 1/V than that shown. However, we will argue that (2) is actually
correct as far as the suppression factors in the 1/V expansion go. Nevertheless, even the
suppression displayed in (2) seems to be a challenge for LVS, if indeed V ∼ 1015. For V
this large, ∆Kgs would dominate ∆Kα′ , since we do not expect the string coupling gs
to be stabilized extremely small. On the other hand, if we are interested in the effects
gs corrections may have on the existence of the large volume minima, the relevant
quantity to look at is the scalar potential V , rather than the Kähler potential K. It
turns out that certain cancellations in the expression for the scalar potential leave us
with leading correction terms to V that scale as
∆Vα′ ∼ O(g−1/2s V−3) , ∆Vgs ∼ O(gsV−3) . (3)
This is already much better news for LVS. However, restoring numerical factors in (3),
and with gs typically not stabilized extremely small, it would seem that ∆Kgs could still
have a significant effect both on stabilization and on the resultant phenomenology (like
soft supersymmetry breaking terms, which also depend on the Kähler potential). We
will see that although this is indeed so in principle, in practice the models we consider
are surprisingly robust against the inclusion of ∆Kgs . The clearest example of this is
the calculation of gaugino masses in sec. 4. The result is that for the “11169 model”
(analyzed in [6]), the correction to the gaugino masses due to ∆Kgs is negligible. Thus,
for the most part, LVS survives our onslaught unscathed.
We consider this a sign that scenarios such as LVS deserve to be taken seriously as
goals to be studied in detail in string theory, even as the caveats above (that apply to
any KKLT-like setup) serve to remind us that there is much work left to be done to
really understand phenomenologically viable stabilized flux compactifications in string
theory.
2 Review
Let us begin by a quick review of the KKLT and large volume scenarios. For reasons
that will become clear, we will want to allow for more than a single Kähler modulus.
2.1 KKLT
The KKLT setup [1,2] is a warped Type IIB flux compactification on a Calabi-Yau (or
more generally, F-theory) orientifold, with all moduli stabilized. In this paper, we will
neglect warping. For progress towards taking warping into account in phenomenological
contexts, see [40,41].
In the four-dimensional N = 1 effective supergravity, the Kähler potential and
superpotential read
K = −ln(S + S̄)− 2 ln(V) +Kcs(U, Ū) ,
W = Wtree +Wnp =W (S, U) +
Ai(S, U)e
−aiTi , (4)
where the volume V is a function of the Kähler moduli Ti = τi + ibi whose real parts
are 4-cycle volumes and whose imaginary parts are axions bi, arising from the integral
of the RR 4-form over the corresponding 4-cycles. In particular, the volume V depends
on the Ti only through the real parts τi,
V = V(Ti + T̄i) = V(τi) , (5)
and the nonperturbative superpotential Wnp a priori depends on the complexified dila-
ton S and the complex structure moduli U . After stabilization of S and U by demand-
ing DUW = 0 = DSW , we have
K = −ln(S + S̄)− 2 ln(V) +Kcs(U, Ū) ,
W = Wtree +Wnp =W0 +
−aiTi . (6)
We keep the dependence on the complexified dilaton S and the complex structure
moduli U in the Kähler potential for now, since the Kähler metric in the F-term
potential
V = eK
GJ̄IDJ̄W̄DIW − 3|W |2
is to be calculated with the full Kähler potential K, including the dependence on S and
U . (In eq. (7), the index I a priori runs over all moduli, but after fixing the complex
structure moduli and the dilaton, only the sum over the Kähler moduli remains). The
scalar potential V has a supersymmetric AdS minimum at a radius that is barely large
enough to make the use of a large-radius effective supergravity self-consistent, typically
τ ∼ 100 (recall that τ has units of (length)4).1 In addition, to obtain a supersymmetric
minimum at all, one needs to tune the flux superpotential W0 to very small values.
That is, the stabilization only works for a small parameter range. This is easy to
understand, since we are balancing a nonperturbative term against a tree-level term.
Let us briefly digress on the reasons for and implications of this balancing.
2.2 Consistency of KKLT
In the previous section we only considered the lowest-order supergravity effective action.
As was already noted in the original KKLT paper, α′ corrections and gs corrections
(string loops) that appear in addition to the tree-level effective action could in principle
1This minimum then has to be uplifted to dS or Minkowski by an additional contribution to the
potential. Various mechanisms were suggested in [2,36,42–45].
affect stabilization. Oftentimes, the logic of string effective actions is that if one such
correction matters, they all do, so no reliable physics can be learned from considering
the first few corrections. If this is true, one can only consider regimes in which all
corrections are suppressed. This is not necessarily so if some symmetry prevents the
tree-level contribution to the effective action from appearing, so that the first correc-
tion (be it α′ or gs) constitutes lowest order. This indeed happens for type IIB flux
compactifications; given the tree level Kähler potential (6), if we were to set Wnp = 0,
the remaining K andW in (6) produce a no-scale potential, i.e. the scalar potential for
the Kähler moduli then vanishes [46]. In KKLT, this no-scale structure is only broken
by the nonperturbative contribution to the superpotential Wnp. Since each term in
Wnp is exponentially suppressed in some Kähler modulus, the resulting terms in the
potential are also exponentially suppressed. For instance, for the simpler example of a
single modulus τ , the potential (after already fixing the axionic partner along the lines
of appendix A.2) reads
4|A|2aτe−aτ
aτ + 1
− 4aτ |AW0|
e−aτ , (8)
meaning that even for moderate values of the Kähler modulus τ , all these terms are
numerically very small. Corrections in α′ and gs, however, are expected to go as powers
of Kähler moduli τ , so will dominate the scalar potential for most of parameter space. In
particular, it was argued in [3,4] that only for very small values of W0 can perturbative
corrections to the Kähler potential be neglected. It was the insight of [3] that even if
W0 is O(1) (which is more generic than the tiny value forW0 required in KKLT), there
can still be a competition between the perturbative and nonperturbative corrections to
the potential in regions of the Kähler cone where large hierarchies between the Kähler
moduli are present. We now review this scenario.
2.3 Large volume scenario (LVS)
As was shown in [5], the no-scale structure (and factorization of moduli space) is broken
by perturbative α′ corrections to the Kähler potential, such as
K = −ln(2S1)− 2 ln(V + 12ξS
1 ) +Kcs(U, Ū) , (9)
where2 ξ = −ζ(3)χ/2(2π)3 and S1 = ReS. For large volume V, we see that the
perturbative correction goes as a power in the volume,
− 2 ln(V + 1
1 ) = −2 lnV −
V + . . . , (10)
2Here ξ differs by a factor (2π)−3 from [5] because we use the string length ls = 2π
which by the discussion in the previous subsection will dominate in the scalar potential
if all Kähler moduli are even moderately large. Using the superpotential
W = W0 +Wnp = W0 +
−aiTi , (11)
the scalar potential has the structure
V = Vnp1 + Vnp2 + V3 (12)
G̄i∂̄W̄np∂iWnp +
G̄iK̄ (W̄0 + W̄np)∂iWnp + c.c.
G̄iK̄Ki − 3
|W |2
For concrete calculations we will use the model based on the hypersurface of degree 18 in
[1,1,1,6,9]
(see [16,47,48] for background information on its topology. Some comments
about generalizations to other models with arbitrary numbers of Kähler moduli are
given in appendix A.2). The defining equation is
z181 + z
2 + z
3 + z
4 + z
5 − 18ψz1z2z3z4z5 − 3φz61z62z63 = 0 (13)
and it has the Hodge numbers h1,1 = 2 and h2,1 = 272 (only two of the complex
structure moduli ψ and φ have been made explicit in (13); moreover, not all of the
272 survive orientifolding). We denote the two Kähler moduli by Tb = τb + ibb and
Ts = τs + ibs, where τb and τs are the volumes of 4-cycles, and the subscripts “b” and
“s” are chosen in anticipation of the fact that one of the Kähler moduli (τb) will be
stabilized big, and the other one (τs) will be stabilized small. An interesting property
of this model is that it allows expressing the 2-cycle volumes ti explicitly as functions of
the 4-cycle volumes τj , so that the total volume of the manifold can be written directly
in terms of 4-cycle volumes, yielding
V = 1
b − τ 3/2s
, (14)
(ts + 6tb)
, τs =
Following [4], we are interested in minima of the potential with the peculiar property
that one Kähler modulus τb ∼ V2/3 is stabilized large and the rest are relatively small
(but still large compared to the string scale),
aτs ∼ lnV ∼
ln τb (15)
in the case at hand. Thus, we expand the potential around large volume, treating
e−aτs as being of the same order as V−1. In the end one has to check that the resulting
potential indeed leads to a minimum consistent with the exponential hierarchy aτs ∼
lnV, so that the procedure is self-consistent. Applying this strategy, the scalar potential
at leading order in 1/V becomes3
VO(1/V3) =
2|A|2a2√τse−2aτs
− 2a|AW0|τse
3|W0|2
eKcs . (16)
From here one can see the existence of the large volume minima rather generally. By
the Dine-Seiberg argument [49], the scalar potential goes to zero asymptotically in
every direction. Along the direction (15), for large volume the leading term in (16) is
V ∼ Vnp2 ∝ −
V3 , (17)
which is negative, so the potential V approaches zero from below. For moderately
small values of the volume, V is positive (this is guaranteed if the Euler number χ
is negative, hence ξ positive), so in between there is a minimum. This minimum
is typically nonsupersymmetric, and because we are no longer balancing a tree-level
versus a nonperturbative term, we can find minima at large volume — hence the name
large volume scenario (LVS).4 To be precise, in flux compactifications we move in
parameter space by the choice of discrete fluxes, but since V is exponentially sensitive
to parameters like S1, large volume minima appear easy to achieve also by small changes
in flux parameters. If we allow for very small values ofW0 (so that KKLT minima exist
at all), the above minimum can coexist with the KKLT minimum [4,50]. Here, we will
allow W0 to take generic values of order one.
The astute reader will have noticed that this argument for the existence of the
LVS minimum is “one dimensional”, as it only takes into account the behavior of the
potential along the direction (15). One must of course check minimization with respect
to all Kähler moduli. In [3] a plausibility argument to this effect was given, and the
existence of the minimum was explicitly checked in the case of the P4
[1,1,1,6,9]
model by
explicitly minimizing the potential (16) with respect to the Kähler moduli. In doing
so, it is convenient to trade the two independent variables {τb,τs} for {V,τs} so that
3Here we have already stabilized the axion bs, i.e. solved ∂V/∂bs = 0, which produces the minus
sign in the second term; this is also true with many small moduli τi. See appendix A.2 for details.
Also note that solving DUW = 0 = DSW causes the values of U and S at the minimum to depend
on the Kähler moduli. However, this dependence arises either from the nonperturbative terms in
the superpotential or from the α′-correction to the Kähler potential. Thus it would only modify the
potential at subleading order in the 1/V expansion.
4By “tree-level” we intend “tree-level supergravity”, i.e. for the purposes of this paper we call both
α′ and gs corrections “quantum corrections”.
∂τsV = 0, as then the last term in (16) is independent of τs (this will be different when
we include loop corrections). Extremizing with respect to τs, and defining
X ≡ Ae−aτs , (18)
one obtains a quadratic equation for X ,
τsS1V
(4aτs − 1)X2 +
2a|W0|
(aτs − 1)X
eKcs . (19)
In (18), we chose A to be real as a potential phase can be absorbed into a shift of the
axion b and disappears after minimization with respect to b (see section A.2). Two
comments are in order. The quadratic equation (19) has just one meaningful solution
(X = 0 corresponds to τs = ∞). Moreover, when expanding (19) in 1/(aτs), the leading
terms arise from derivatives of the exponential.
Formula (19) is an implicit equation determining τs. However, one can easily solve
(19) for X and obtains
X = Ae−aτs =
2|W0|
(aτs)2
. (20)
The hierarchy (15) is obvious in this solution, rendering the procedure self-consistent.
One also notices that reasonably large values of τs (e.g. 35) are not difficult to obtain,
if V is stabilized large enough; for example, simply set a ∼ 1, A ∼ 1, W0 ∼ 1. We
fill in the numerical details, following [3], in appendix A.1 (including some further
observations).
3 String loop corrections to LVS
As already emphasized, the α′ correction proportional to ξ is only one among many
corrections in the string effective action. We now consider the effect of string loop
corrections on this scenario and what the regime of validity is for including or neglect-
ing those corrections. Volume stabilization with string loop corrections but without
nonperturbative effects was considered in [51].
To be precise, the corrections considered in [51] were those of [37], that were com-
puted for toroidal N = 1 and N = 2 orientifolds. Here, we would need the analogous
corrections for smooth Calabi-Yau orientifolds. Needless to say, these are not known.
Faced with the fact that the string coupling gs is stabilized at a finite (and typically
not terribly small) value, we propose that attempting to estimate the corrections based
on experience with the toroidal case is better than arbitrarily discarding them. As we
will see, if our estimates are correct, typically the loop corrections can be neglected,
though there may at least be some regions of parameter space where they must be
taken into account (see figure 4). (In section 5, we will briefly consider “cousins” of
LVS where they cannot be neglected anywhere in parameter space.) Improvement on
our guesswork would of course be very desirable.
3.1 From toroidal orientifolds to Calabi-Yau manifolds
We would like to make an educated guess for the possible form of one-loop corrections
in a general Calabi-Yau orientifold. All we can hope to guess is the scaling of these
corrections with the Kähler moduli T and the dilaton S. The dependence on other
moduli, like the complex structure moduli U , cannot be determined by the following
arguments (even in the toroidal orientifolds this dependence was quite complicated).
In order to generalize the results of [37] to the case of smooth Calabi-Yau mani-
folds, we should first review them and in particular remind ourselves where the various
corrections come from in the case of toroidal orientifolds. There, the Kähler potential
looks as follows (we will explain the notation as we go along):
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
V (21)
E (K)i (U, Ū)
4τiS1
i 6=j 6=k
E (W )k (U, Ū)
4τiτj
There are two kinds of corrections. One comes from the exchange of Kaluza-Klein
(KK) modes between D7-branes (or O7-planes) and D3-branes (or O3-planes, both
localized in the internal space), which are usually needed for tadpole cancellation, cf.
fig. 1. This leads to the first kind of corrections in (21), proportional to E (K)i where
the superscript (K) reminds us that these terms originate from KK modes. In the
toroidal orientifold case, this type of correction is suppressed by the dilaton and a
single Kähler modulus τi, related to the volume of the 4-cycle wrapped by the D7-
branes (or O7-planes, respectively).5 We expect an analog of these terms to arise more
5We should mention that there was no additional correction of this kind coming from KK exchange
between (parallel) D7-branes in [37] (actually that paper considered the T-dual version with D5-branes,
but here we directly translate the result to the D7-brane language). This was due to the fact that
in [37] the D7-brane scalars were set to zero. In general we would also expect a correction coming
PSfrag replacements τb
Figure 1: The loop correction E(K) comes from the exchange of closed strings, or
equivalently an open-string one-loop diagram, between the D3-brane and D7-branes
(or O7-planes) wrapped on either the small 4-cycle τs (as in a) or the large 4-cycle
τb (as in b). The exchanged closed strings carry Kaluza-Klein momentum.
generally, given that they originate from the exchange of KK states which are present
in all compactifications.
The second type of correction comes from the exchange of winding strings between
intersecting stacks of D7-branes (or between intersecting D7-branes and O7-planes).
The exchanged strings are wound around non-contractible 1-cycles within the intersec-
tion locus of the D7-branes (and O7-planes, respectively), cf. fig. 2. This leads to the
PSfrag replacements
Figure 2: The loop correction E(W ) comes from the exchange of winding strings
on the intersection between the small 4-cycle τs and the large 4-cycle τb. If this
intersection is empty, there are no terms with E(W ).
second kind of correction in (21) proportional to E (W )i . The superscript (W ) reminds
us that these terms arise from the exchange of winding strings. In toroidal orientifolds,
this type of correction is suppressed by the two Kähler moduli measuring the volumes
from parallel (or more generally, non-intersecting) D7-branes by exchange of KK-states. These should
scale in the same way with the Kähler moduli as those arising from the KK exchange between D3-
and D7-branes.
of the 4-cycles wrapped by the D7-branes (and O7-planes). One might a priori think
that this kind of correction does not generalize easily to a smooth Calabi-Yau which
has vanishing first Betti number (and therefore at most torsional 1-cycles). However,
the exchanged winding strings are, from the open string point of view, Dirichlet strings
with their endpoints stuck on the D7-branes. Thus, the topological condition is on the
cycle over which the two D7-brane stacks (or one D7-brane stack and an O7-plane)
intersect, as in figure 3. Thus, it depends on the topology of specific cycles within
cycles whether winding open strings exist in a given model.6
PSfrag replacements
no D-brane
no D-
brane
Figure 3: A D7-brane is wrapped on a 4-cycle A, which intersects the 4-cycle B on
a 2-cycle C. For Dirichlet strings, the relevant topological condition (the existence
of nontrivial 1-cycles) is on the intersection locus C, not on cycle B or on the whole
Calabi-Yau. In other words, without the D-brane, the string on cycle C could have
been unwound by sliding it along cycle B (as shown in the figure). With the D-brane,
the string on cycle C is stuck.
Given the expressions in [37] and the subset reproduced in (21) above, it is tempting
to conjecture that some terms at one loop might be suppressed only by powers of single
Kähler moduli like the τi (and the dilaton):
Calabi-Yau: ∆Kgs
for some function E of the complex structure and open string moduli. If this were
the case, the one-loop corrections would typically dominate the α′ correction in (21)
6The toroidal orientifold case seems to be a bit degenerate. Two stacks of D7-branes intersect along
a 2-cycle with the topology of P1. However, there are point-like curvature singularities along the P1
at the orbifold point and strings winding around these singular points cannot be contracted without
crossing the singularities. This seems to allow for stability of winding strings (at least classically).
(which is suppressed by the overall volume V) in the Kähler potential, if there are large
hierarchies among the Kähler moduli. However, one should keep in mind that toroidal
orientifolds are rather special in that they have very simple intersection numbers. In
particular, the overall volume can be written as V ∼ τiti, where there is no summation
over i implied. Thus, it is not obvious whether a generalization to the case of a general
Calabi-Yau really contains terms suppressed by single Kähler moduli instead of the
overall volume. Even though we cannot exclude the presence of such terms, we deem it
more likely that the scaling of one-loop corrections to the Kähler potential is not (22)
Calabi-Yau: ∆Kgs
gaK(t, S1)E
W(t, S1)E
V , (23)
where the sums run over KK and winding states, respectively. Also, E (K) and E (W ) are
again unknown functions of the complex structure and open string moduli, t stands
for the 2-cycle volumes (in the Einstein frame; see appendix C.1) and the functions
gK(t, S1) and gW(t, S1) determine the scaling of the KK and winding mode masses
with the Kähler moduli and the dilaton.7 As we review in appendix E, in the toroidal
orientifold case the suppression by the overall volume arises naturally through the Weyl
rescaling to the 4-dimensional Einstein frame.
Starting with the ansatz (23) for smooth Calabi-Yau manifolds, the known form
(22) for toroidal orientifolds follows simply by substituting gK, gW and the intersec-
tion numbers for the toroidal orientifold case. In particular, gK ∼ ti for the 2-cycle
transverse to the relevant D7-brane, while gW ∼ t−1i for the 2-cycle along which the
two D7-branes intersect. Then, the first of the terms in (23) reduces to E (K)i /(S1τi) for
toroidal orientifolds, the second to E (W )i /(τjτk) with j 6= i 6= k, cf (50). Our strategy
in the following chapters will therefore be to assume a scaling like (23) for the 1-loop
corrections to the Kähler potential for general Calabi-Yau spaces.
As already mentioned, the dependence on the complex structure and open string
moduli cannot be inferred by analogy to the orientifold case. We parameterize our igno-
7In rewriting the sums over KK and winding states in terms of the functions g and E , we assume that
the dependence of the corresponding spectra on the complex structure and Kähler moduli factorizes.
In the known examples of toroidal orientifolds (with or without world-volume fluxes), this is always
the case, cf. [52]. Moreover, in general there can appear several contributions (denoted by a and q)
depending on which tower of KK or winding states are exchanged in a given process. We will see
explicit examples of this in the following.
rance by keeping the expressions E in (23) as unknown functions of the corresponding
moduli. Then we investigate the consequences of the one-loop terms, depending on
the size of these unknown functions at the minimum of the potential for the complex
structure and open string moduli. Some further comments on the form of ∆Kgs will
appear in section 5.2.
3.2 LVS with loop corrections
Thus, allowing for string loop corrections of the form (23) in (9), and expanding the
α′ correction as in (10), we can write
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
W = W0 +
−aiTi , (24)
where as explained in the previous section, we have not specified the explicit form of
the loop corrections E , that are allowed to be functions of U (and in general of the open
string moduli, that we neglect in our analysis, assuming that they can be stabilized
by fluxes). The Kähler potential for the complex structure moduli Kcs(U, Ū) is left
unspecified in (24), indeed we will not need its explicit form. For consistency, we have
also included loop corrections to the α′ correction.8 This changes ξ to ξ̃, which is a
small change; for S1 = 10, numerically ξ̃ ≈ 1.02 ξ.
Neglecting fluxes, the functions gaK and g
W are proportional and inversely propor-
tional to some 2-cycle volume, respectively. (We will come back to corrections from
fluxes in appendix D.) When using a particular basis of 2-cycles (with volumes ti as
in appendix C.1), the 2-cycle volume appearing in gaK or g
W might be given by a linear
combination ta =
i citi of the basis cycles ti (and similarly for tq). Depending on
which 2-cycle is the relevant one, this linear combination might or might not contain
the large 2-cycle tb ∼ V1/3, which always exists in LVS. If it is present in the linear
combination, one can neglect the contribution of the small 2-cycles to leading order in
a large volume expansion and obtains possible terms proportional to E (K)b S−11 V−2/3 or
E (W )b V−4/3, where the subscript b refers to the large 4-cycle τb.
8We remind the reader that the α′ correction arises from the R4 term in 10 dimensions whose
coefficient receives corrections at 1-loop (and from D-instantons). The 1-loop correction amounts to
a shift of the prefactor from ξ to ξ̃ = ξ
1 + π
3ζ(3)S2
, see for instance [53] for a review.
Before getting into the details, it is hard to resist trying to anticipate what might
happen. For those terms that are more suppressed in volume than the ξ̃ term (e.g
E (W )b ), one would expect the loop corrections to have little effect on stabilization. They
could still represent a small but interesting correction to physical quantities in LVS. For
those that are less suppressed in volume than the ξ̃ term (e.g. E (K)b ), one would expect
the loop correction to have a huge effect on stabilization, and severely constrain the
allowed values for the complex structure moduli and the dilaton in LVS (in particular,
constrain them to a region in moduli space where the function E (K)b takes very small
values). We will find, however, that this expectation is sometimes too naive. For
example, there can be cancellations in the scalar potential that are not obvious from
just looking at the Kähler potential.
Let us now get into more detail on what happens in the LVS model with loop
corrections.
3.3 The P4
[1,1,1,6,9]
model
We would now like to specify the general form of the Kähler- and superpotential (24) to
the case of the P4[1,1,1,6,9] model. In this space, the divisors that produce nonperturbative
superpotentials when D7- (or D3-) branes are wrapped around them do not intersect,
as reviewed for instance in [48]. Therefore, we do not expect any correction of the E (W )
type in this model (for the generalization to models where there are such intersections,
see appendix D). Moreover, we neglect flux corrections to the KK mass spectrum in
the main text. It is shown in appendix D that, for small fluxes, this correctly captures
all the qualitative features we are interested in, and it leads to much clearer formulas.
Thus, we now consider the scalar potential resulting from
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
τbE (K)b
τsE (K)s
W = W0 + Ae
−aTs . (25)
As τb is very large the corresponding non-perturbative term in the superpotential of
(24) can be neglected, which allowed us to simplify the notation by setting As = A
and as = a.
The general structure of the scalar potential was already given in (12). The three
contributions at leading order (O(V−3)) in the large volume expansion are
Vnp1 = e
24a2|A|2τ 3/2s e−2aτs
V∆ , (26)
Vnp2 = −eKcs
2a|AW0|τse−aτs
6E (K)s
, (27)
3eKcs |W0|2
1 ξ̃ +
4(E (K)s )2
, (28)
where the axion has already been minimized for, as discussed in section A.2, and
2S1τs − 3E (K)s . (29)
The leading α′-correction is the ξ̃ term in V3 above. We now see that it scales with the
volume and the string coupling gs = 1/S1 as claimed in the Introduction, in eq. (3).
Also the volume dependence of the loop correction (E (K)s term) in V3 is as announced
in (3). The gs factors seem to differ from (3); we see g
s , g
s and g
s for Vnp1, Vnp2 and
V3, respectively. This is because the gs dependence advertised in (3) arises in models
where, unlike in P4[1,1,1,6,9], the E (W ) correction is present as well, cf. appendix D.9 It is
also worth mentioning that the loop correction proportional to E (K)s modifies Vnp1 and
Vnp2 at leading order in the V-expansion. whereas the α′ correction does not; it only
appears in V3. This is so even though both corrections are equally suppressed in the
Kähler potential (i.e. ∼ V−1). The reason for this can be traced back to the fact that
the loop-correction explicitly depends on τs and not only on the overall volume, cf. the
discussion in appendix C.4 and C.5.
As anticipated, E (K)b and its first derivatives appear only at the next order, O(V−10/3):
V10/3 = 2
61/3|W0|2eKcs
S31V10/3
(E (K)b )2 +
∂αE (K)b ∂ᾱE
, (30)
where ∂α = ∂/∂U
α and ∂ᾱ = ∂/∂Ū
ᾱ and α enumerates the complex structure moduli.
For E (K)b = E
s = 0, the potential terms at leading order coincide with the original
case discussed in (56), cf. appendix A.1. The singularity from zeros of the denominator
is an artifact of the expansion as discussed in appendix B. The range of validity is
9There, it is shown that including the effect of fluxes on the KK spectrum might also produce this
behavior.
limited to the range in moduli space where the denominator ∆ does not become too
small. It is also apparent that the loop terms are subleading in a large τs, large S1
expansion. However, depending on the relative values of the parameters {E (K)s , τs, S1},
a truncation to the first terms in such an expansion may or may not be valid. We
perform a numerical comparison of the two contributions to V3 in figure 4.
Figure 4: The top surface is the α′ correction, the second is the gs correction, and
the “red carpet” is 10/∆ (we used the values A = 1,W0 = 1, a = 2π/8). We see
that for most of the parameter range, the α′ correction dominates, and only for large
E(K)s , with the string coupling gs = 1/S1 not too small, do the contributions become
comparable.
We can understand the volume dependence of the terms (26)-(30) as follows. The
common prefactor eK gives an overall suppression τ−3b ≃ V−2. The quantum corrections
obey the rule that a term proportional to 1/τλb in K appears in V3 at order 1/τ
(where the +3 comes from the overall eK factor) for all values of λ except for λ = 1.
When it does appear, it is generated by the term (KiKi − 3) and breaks the no-scale
structure. For λ = 1 there is a cancellation at leading order, so it appears only at
order 1/τ 2+3b (see appendix C.3 and C.4). This rule can explicitly be verified in our
calculation: the α′ and the E (K)s corrections are suppressed by 1/τ 3/2b inK, and therefore
they appear with the suppression 1/τ
b in V3. On the other hand, for the E
b term a
cancellation takes place to leading order (λ = 1). It appears neither in Vnp1 nor in Vnp2
at leading order (which can be understood more generally, cf. appendix C.5). Thus, it
only appears subleading in the potential, at O(V−10/3).10
10This cancellation for λ = 1 was already noticed in [51], albeit in the case without nonperturbative
superpotential. In [54] it was argued that this cancellation can be understood from a field redefinition
We now proceed to minimize the potential (26)-(28), using the same strategy as
in the case without loop corrections, cf. section 2.3 and appendix A.1. The equations
∂VV = 0 = ∂τsV are of course more complicated now, but it is easy to solve them
numerically. Doing so we find that the volume V and the small 4-cycle volume τs,
viewed as functions of S1 and E (K)s , are well fit by linear functions when restricted to
a sufficiently limited range in parameter space. For example,
range: S1 = [8, 11], E (K)s = [20, 40]
log10 V = 1.720S1 − 0.1208 E (K)s − 3.437 , (31)
τs = 5.000S1 − 0.3581 E (K)s − 8.638 .
The fits are quite good; the error is no greater than ±0.3 for τs and ±0.1 for log10 V in
this range, for an {S1, E (K)s } grid of 402 points.
From (31) we see an interesting difference to the case without loop corrections. The
value of τs at the minimum depends on the complex structure moduli U , through E (K)s .
This is in contrast to the case without loop corrections, where the value of τs is only
determined by the value of the Euler number ξ and the dilaton S1, cf. (57) below. It is
analogous to the perturbative stabilization in [51] where the volume at the minimum
of the potential also depends on U .
The result (26)-(28) was derived in a particular model, but we expect the appearance
of loop corrections in V to be more general. This opens up the possibility that in
principle, one might obtain large volume minima even for manifolds of vanishing (or
even positive) Euler number, where LVS is not applicable, as LVS-style stabilization
only holds for one sign of ξ. In practice it might be difficult to get large enough values
for the 1-loop corrections to stabilize τs at a value sufficiently bigger than the string
scale. This deserves further study.
We also note that the special structure of (28) and (30), i.e. the appearance of Es
only in (28) and of Eb only in (30), offers additional flexibility in tuning the relative
size of these terms in a purely perturbative stabilization of the Kähler moduli along
the lines of [51,55]. Also this point deserves further study.
argument combined with the no-scale structure of the tree-level Kähler potential. That argument
holds for the case of a single Kähler modulus T with tree-level Kähler potential −3 ln(T + T̄ ) and
under the assumption that the coefficient of the loop correction to the Kähler potential ∼ (T + T̄ )−1
is independent of the complex structure moduli and the dilaton. Here, these assumptions do not
hold, but we showed that the term ∼ (Tb + T̄b)−1 in the Kähler potential nevertheless only appears
at subleading order in the potential in LVS, cf. (26)-(30).
4 Gaugino masses
Now that we know how the stabilization of the (Kähler) moduli is modified by loop
corrections, it is natural to extend our analysis to the soft supersymmetry breaking
Lagrangian (For a review see for instance [56,57].) In LVS, supersymmetry breaking
is mostly due to F -terms: Fs 6= 0, Fb 6= 0. These determine the soft supersymmetry
breaking terms which can be present in the low energy effective action without spoiling
the hierarchy between the electroweak and the Planck scale,
Leff = LMSSM + Lsoft . (32)
The soft Lagrangian contains gaugino masses M , scalar masses m, further scalar bilin-
ear terms B and trilinear terms A. (For explicit expressions, see the aforementioned
reviews, or e.g. [6].)
Let us start considering gaugino masses. In [6] it was shown that in LVS, gaugino
masses Ma are generically suppressed with respect to the gravitino mass m3/2:
|Ma| ≃
ln(1/m3/2)
ln(1/m3/2)
(we use units in which MPl = 1). This suppression results from a cancellation of the
leading order F -term contribution to gaugino masses. We briefly review this calcula-
tion. Given the F -terms
F I = eK/2GJ̄IDJ̄W̄ , (34)
gaugino masses are given by [56]
F I∂Ifa , (35)
where fa are the gauge kinetic functions and a labels the different gauge group fac-
tors. In LVS the Standard Model (SM) gauge groups arise from D7-branes wrapped
around small 4-cycles. We do not try to go into the details of how to embed the SM
concretely, but we mention that different gauge group factors might arise from brane
stacks wrapping the same 4-cycle if world volume fluxes are present on the branes. In
that case the gauge kinetic functions are given by11
+ ha(F)S + f (1)a (U) , (36)
11We use the “phenomenology” normalization of the gauge generators, in the language of [58]; that
explains the relative factor of 4π in (36).
where ha depends on the world volume fluxes and we also included a possible 1-loop
correction to the gauge kinetic function which depends on the complex structure (and
possibly open string) moduli. If several gauge groups arise from branes wrapped around
the same cycle, the same Kähler modulus T would appear in all of them. From (36) it
is clear why the D7-branes of the SM have to wrap small 4-cycles, because otherwise
the gauge coupling would come out too small (unless there is an unnatural cancellation
between the different contributions to fa).
As is also apparent from (36), the gauge kinetic function in general depends not
only on the Kähler moduli but also on the dilaton and the complex structure. Thus,
according to (35) we need to know FU , F S and F i for the small Kähler moduli.12
From the definition (34), it is clear that FU and F S might be non-vanishing even
though we assume DUW = 0 = DSW , provided the inverse metric G
J̄I contains mixed
components between Kähler moduli on the one hand and complex structure moduli
and dilaton on the other hand. Without loop corrections (i.e. considering only the
leading α′ correction) there is no mixing between the Kähler and complex structure
moduli, and one finds
FU = 0 , F S ∼ O(V−2) and F i ∼ O(V−1) (without loop corrections) . (37)
Thus, at leading order in the large volume expansion, the sum in (35) only runs over
the Kähler moduli. Moreover, taking into account the linear dependence of the gauge
kinetic functions (36) on the (small) Kähler moduli, the sum effectively only involves
a single term, i.e.
F a +O(V−2) , (38)
where F a is the F-term of the (small) Kähler modulus appearing in fa.
As a concrete example we consider again the P4[1,1,1,6,9] model with only one small
Kähler modulus τs. The corresponding F-term is given by
F s = eK/2
Gs̄s∂s̄W̄ + (G
s̄sKs̄ +G
b̄sKb̄)W̄
= 2τse
K/2W̄0
− 1 +O((aτs)−2)
+O(V−2) , (39)
where we used (20) and (61) for the first term and (64) for the second.
12With a slight abuse of notation, we denote the F -terms of the Kähler moduli by the index i, but
the F -terms of the other moduli are identified by the symbol for the corresponding modulus, like FS .
This is to avoid introducing too many indices.
Now the leading order cancellation is obvious in (39). Determining the gaugino
masses requires dividing by Refs, cf. (35). In order to further evaluate this, [6,7] as-
sumed that the dilute flux approximation fs = (4π)
−1Ts is valid, i.e. they neglected the
contributions from world-volume fluxes and one-loop terms compared to the tree-level
term. This puts some constraints on the allowed discrete flux values determining hs.
We want to stress that the cancellation appearing in (39) is independent of this approx-
imation. We are mainly interested in the fate of this cancellation when including loop
corrections, and do not have anything to add concerning phenomenological constraints
that may arise from imposing the dilute flux approximation. Using it, the gaugino
masses simplify to
|Ms| =
= eK/2|W0|
+O((aτs)−2)
ln(1/m3/2)
ln(1/m3/2)
which is the announced result. In (40) we used
m3/2 ∼ |W0|/V and aτs ∼ ln(V/|W0|) , (41)
where the second relation holds in LVS due to (20).
4.1 Including loop corrections
The previous section was a review of the results found in [6]. Now we ask what changes
if one considers the loop corrected Kähler potential (25). A priori, as (40) results from
a leading order cancellation, one might wonder whether loop corrections might spoil
this small hierarchy between the gaugino and gravitino masses. To address this concern
we start by observing that the gaugino masses are still determined by the F-terms of
the small Kähler moduli (in the large volume limit). More precisely, the scaling of the
F-terms (37) now becomes
FU = O(V−2), F S ∼ O(V−2) and F i ∼ O(V−1) (with loop corrections) , (42)
i.e. FU no longer vanishes, but it is just as suppressed as F S.
We again focus on the P4[1,1,1,6,9] model and ask how (39) is modified by loop cor-
rections. Amongst other things, we need to generalize equation (20) to include loop
corrections, since we need it to calculate the first term in (39). Thus, we need to
extremize the potential again with respect to τs by setting
∂τsV =
4aτs − 1
∆+ 6E (K)s
2a|W0|
V2S1∆2
aτs − 1
∆2 − 18(E (K)s )2
2aS1τ
s E (K)s
X (43)
− 3|W0|
2(E (K)s )2
4S21V3∆
to zero. Obviously, X = 0 is no longer a solution. Instead, there are now two non-trivial
solutions, one of which goes to zero in the limit E (K)s → 0. This solution corresponds
to a maximum of the potential, so it is of no use to us here. We can expand the other
solution for large aτs, as in the case without loop corrections, yielding
X = Ae−aτs =
2|W0|
2aE (K)s
(aτs)2
. (44)
Another ingredient we need is the quantity Gı̄sKı̄, in order to evaluate the second term
in (39). Using equation (65) we obtain
Gı̄sKı̄ = −2τs
6E (K)s
+ . . .
= −2τs −
2E (K)s
− 18(E
S21τs
2(E (K)s )3
τ 2s S
+ . . . , (45)
where the ellipsis represents terms that are more suppressed in V−1.
Now we see from (44), (45) and (65) that at leading order in an expansion in aτs, the
quantities relevant to evaluate (39) are not affected by the loop corrections. Thus, the
leading order cancellation in the gaugino masses survives the inclusion of loop effects.13
At first glance, though, equations (44), (45) and (65) seem to suggest a correction to the
subleading term, that could potentially give a significant contribution to the gaugino
masses after the leading-order cancellation, cf. (39).
In the actual calculation, this contribution drops out. Putting all the ingredients
together (and employing the dilute flux approximation again), the gaugino mass turns
13One might argue that this result was to be expected, because the main assumption of [6] is
that the stabilization is due to nonperturbative effects, i.e. the dominant effect in ∂τsV should arise
from the nonperturbative superpotential. However, in view of (43), it is no longer obvious that the
nonperturbative terms dominate when loop corrections are included.
out to be
|Ms| =
= 3eK/2|W0|
16a2τ 2s
S1 − 12
2aE (K)s
64S1a3τ 3s
+ . . .
ln(1/m3/2)
ln(1/m3/2)
. (46)
The result of [6] is therefore very robust. Unexpectedly, the correction to (40) due to
E (K)s only appears at sub-sub-leading order in the 1/ ln(1/m3/2) expansion.
4.2 Other soft terms
In [7] all other soft terms were calculated for LVS. The main result (see p. 15 of [7]) is
that roughly speaking, all the soft parameters are determined by F s and by the power
with which the chiral matter metrics scale with τs. As we saw in the previous section,
F s gets modified by loop corrections only at sub-sub-leading order in a 1/τs expansion
(see (46)). Therefore, the calculation of all the soft terms in [7] appears to be quite
robust against including loop effects.
One of the key assumptions in [7] is that all the Yukawa couplings Y are already
present in perturbation theory, i.e. they have the schematic form Y = Y pert(U) +
Y np(e−T ). This requirement featured prominently already in the derivation of the
volume dependence of the chiral matter metrics in [59] by scaling arguments. In [7] the
same schematic form is essential for simplifying the trilinear soft terms A. In general
these terms receive a contribution of the schematic form
F T∂T logY = F
T ∂T (Y
pert(U) + Y np(e−T ))
Y pert(U) + Y np(e−T )
∼ O(e
O(T 0) +O(e−T ) , (47)
which is exponentially suppressed if and only if Y pert(U) is non-vanishing. However, in
many examples the Yukawa couplings are actually only generated nonperturbatively,
see for instance the discussion in [60], and [61] for some examples. This poses a con-
straint on the way the Standard Model is realized in LVS, if one wants to ensure flavor
universality of the soft breaking terms as advertised in [7].
One more comment about the important issue of flavor universality. In [7], section
3.4., it was argued that in LVS, approximate flavor universality is a natural consequence
of the zeroth-order factorization of Kähler and complex structure moduli spaces. We
provide some more details on the factorized approximation in appendix F.
5 LVS for other classes of Calabi-Yau manifolds?
In section 3.3 and 4 we saw that the 1-loop corrections to the moduli Kähler potential
only have relatively small effects on the large volume scenario based on the P4
[1,1,1,6,9]
model of [3]. In this chapter, we would like to ask the question how generic the “Swiss
cheese” form is for a Calabi-Yau manifold and if there are other models in which the
one-loop corrections discussed above might become more important. This is indeed
to be expected if the Calabi-Yau under consideration has a fibered structure, as we
explain in the following. If gs corrections do dominate α
′ corrections, they could ruin
the volume expansion of LVS.
5.1 Abundance of “Swiss cheese” Calabi-Yau manifolds
In the LVS examples discussed in [4] the volume in terms of the Kähler moduli takes
the “Swiss cheese” form
− . . .−
, (48)
where the coefficients ai, ..., ci are only non-vanishing for the small Kähler moduli. The
LVS limit consists in scaling the overall volume of the Calabi-Yau more or less isotrop-
ically while having small holes inside the manifold. The τ ’s are linear combinations of
∂tiV, where now V is considered as a (cubic) function of the 2-cycle volumes ti. For
the effective field theory analysis to be valid one should not only demand that the
4-cycle volumes τi are large compared to the string scale, but also that the 2-cycle
volumes ti are large. In the cases discussed in [4], the linear combinations ∂tiV are
indeed such that one can have one of them exponentially large and the others small
(but still sufficiently larger than the string scale), without taking any of the ti to be
exponentially small. This is obvious for the P4
[1,1,1,6,9]
example where the 2-cycle volume
tb only appears in the definition of one of the τ ’s, cf. (14), but it is also true for the
second example of [4], cf. their formulas (84).
However, the F18 model of [16] does not seem to allow its volume to be written in
the form (48) with one Kähler modulus τb that can become large while keeping all the
others small (again, demanding that the ti stay larger than 1 in string units). Thus,
it is an interesting question how generic or non-generic the “Swiss cheese” Calabi-Yau
manifolds are. We do not attempt to give a general answer; instead, we turn to two
examples in which the form of the volume differs from (48).
5.2 Toroidal orientifolds
The reason loop corrections may be more important in toroidal orientifolds than in
compactifications on “Swiss cheese” Calabi-Yau manifolds is the following. As we al-
ready discussed in section 3.1, the conjectured form of 1-loop corrections (23) simplifies
in the case of toroidal orientifolds, because they have very special and simple intersec-
tion numbers. More concretely, using the definition τi = ∂tiV, together with the special
form of the intersection numbers in the toroidal case, i.e. V = t1t2t3, the volume can
alternatively be expressed as
V = √τ1τ2τ3 = tiτi (no summation; i = 1, 2 or 3) . (49)
Thus, formula (23) simplifies and the 1-loop corrections proportional to E (K)i are only
suppressed by single Kähler moduli instead of by the overall volume. Also the terms
proportional to E (W )i can be rewritten in the toroidal orientifold case and the Kähler
potential takes the form (for the T 6/(Z2 × Z2) example)
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
V (50)
E (K)i (U, Ū)
4τiS1
i 6=j 6=k
E (W )k (U, Ū)
4τiτj
where the functions E are non-holomorphic Eisenstein series in this case [37]. It is easy
to see that the origin of this simplification is the fact that there is just a single non-
vanishing intersection number in the toroidal orientifold case and all Kähler moduli
appear linearly in the cubic expression for the volume.
The difference of the toroidal orientifold to the “Swiss cheese” case of LVS can also
be seen in the different forms of the functional dependence of the volume on the Kähler
moduli. In the toroidal orientifold case one has the relations
∂t1V = t2t3 , ∂t2V = t1t3 , ∂t3V = t1t2 , (51)
so that two of them will always become large if one takes one of the ti to be large
and demands that the other two stay larger than 1. This also holds for any linear
combinations of them. The difference is also obvious from the fact that the 2-cycle
volume tb that is responsible for τb becoming large in the LVS examples of [4] always
appears cubically in the volume. This is related to the fact that the term (τb +
aiτi)
should be the square of a linear combination of the ti, in order for (48) to be expressible
as a cubic polynomial in the ti. In contrast, any (untwisted) 2-cycle volume in the
toroidal orientifold case only appears linearly in the cubic volume polynomial.
To illustrate the effect of terms in the Kähler potential that are suppressed only by
single Kähler moduli instead of the overall volume, we take the Kähler potential (50)
and expand V3 in the region of the Kähler cone where τ1 = τ2 = τb ≫ τ3 = τs (as we
explained above, at least two of the Kähler moduli have to become large simultaneously,
if one wants to avoid any of the 2-cycle volumes becoming very small). This leads to
(for simplicity setting all Ui = U , all E (K)i = E (K) and all E
i = E (W )):
|W0|2eKcs
2S1V2
(E (K))2 + 1
(∂UŪKcs)
−1∂UE (K)∂ŪE (K)
8τ 2s S
3 ξ̃ S
E (W )
(E (K))2 + (∂UŪKcs)−1∂UE (K)∂ŪE (K)
4S21τs
. (52)
Obviously, the leading term in the large τb expansion now comes from the loop correc-
tion and not from the α′ term (which term really dominates depends on the values of
S1 and U as well, of course). Thus, an expansion of the potential as in LVS, cf. (16),
would not be realized in this case, even if one found a way to lift enough zero modes
by fluxes for τs to appear in a nonperturbative superpotential.
This toy example was meant to show that for a consistent large volume expansion
in models with large hierarchies in the Kähler moduli, it is important to make sure that
there are no correction terms in the Kähler potential (from loop or α′-corrections) that
are suppressed only by some of the small Kähler moduli. We should stress again that
also terms suppressed by the large volume can be dangerous if the suppression is less
than for the α′ term, i.e. if it is τ−λb with λ < 3/2. The only exception to this rule is
the case λ = 1 as we showed above (and as is shown more generally in appendices C.4
and C.5). In this respect it would be important to know if the conjecture (25) really
bears out. If it turns out that the actual form of the 1-loop corrections also contains
terms like
tλ1b t
, (53)
with λ1 + λ2 = 1 but 0 < λ1 6= 1 or 0, such a 1-loop correction would spoil the large
volume expansion performed in (16).14
14In principle, one would also need an argument that no such terms arise at higher loop order, which
5.3 Fibered Calabi-Yau manifolds
The feature of orientifolds that all Kähler moduli appear linearly in the cubic expression
for the volume shows that a similar simplification can occur in the case of (K3) fibered
Calabi-Yau manifolds, which also have the property that one Kähler modulus (the one
corresponding to the volume of the base) only appears linearly in the cubic expression
for the volume. This takes the form
V = tbηijtitj + dijktitjtk , (54)
where ηij are the intersection numbers of the (K3) fiber, and neither they nor the triple
intersection numbers dijk contain the index b, which is chosen to denote “base”, but it
is also suggestively the same index as the one we used for the large Kähler modulus in
the P4
[1,1,1,6,9]
model. Two-parameter examples of this type appear in e.g. [47,62]. In a
region of the Kähler moduli space where the base tb is rather large but all the other
ti stay relatively small, the volume is approximately V = tbηijtitj . Thus, if the Kähler
potential has a 1-loop correction ∼ E (K)b tb/V, it could be approximated in this region
E (K)b tb
E (K)b
, (55)
where τf = ηijtitj is the volume of the (K3) fiber (which is small compared to t
Obviously, this would lead to a correction to the Kähler potential that is only sup-
pressed by a single (small) 4-cycle volume, similar to the toroidal orientifold example
we discussed in the last section.
We should note that this limit (large base and small fiber for (K3) fibered Calabi-
Yau manifolds), is quite different from the one performed in the usual LVS of [3], even
though both cases involve hierarchical limits of the Kähler moduli. As explained in
section 5.1, the LVS limit consists in scaling the overall volume of the Calabi-Yau more
or less isotropically while keeping holes in the bulk of the manifold small. In contrast,
the limit of large base and small fiber is anisotropic. At the moment we have nothing to
add about whether such anisotropic configurations with all moduli stabilized actually
exist. We merely wanted to point out that if they do exist, that would be an example
of smooth Calabi-Yau compactifications where the gs corrections we consider dominate
over the α′ corrections considered in the large volume limit, as in the toroidal orientifold
case.
would, however, have to be further suppressed in the dilaton S1.
6 Further corrections
In [4], further α′ corrections to the string effective action beyond the one in (9) were
considered. In the case of bulk α′ corrections (i.e. those already present in type IIB
bulk theory without D-branes, arising from sphere level) scaling arguments were given
as to why they are suppressed in the large volume limit. Although that discussion was
surprisingly powerful in its simplicity, we do not consider it completely conclusive, if
large hierarchies between the Kähler moduli exist. After all, dimensional analysis alone
does not guarantee that the other α′-corrections are always suppressed by additional
powers in the overall volume, instead of powers of some of the small Kähler moduli.
Moreover, in addition to the bulk α′ corrections that appear at order O(α′3), in the
models of interest for LVS further α′-corrections arise on the worldvolume of D-branes
and O-planes, cf. [63–70]. These corrections begin already at order O(α′) and scaling
arguments of the kind used for the bulk corrections do not seem to be sufficient to
neglect them.
Indeed, there are correction terms involving two powers of the Riemann tensor which
do modify the effective D3-brane charge and tension, if the D7-branes are wrapped
over 4-cycles with non-vanishing Euler number. These terms were already taken into
account in [1]. However, there are further contributions to the DBI action at the same
order in α′, like F 23R or F
3 , where F3 stands for the RR 3-form field strength, R for
the Riemann tensor and we left index contractions unspecified. If the D7-branes do
not break supersymmetry and remain BPS, it seems unlikely that these terms could
contribute to the potential for the closed string moduli, i.e. induce some effective D3-
brane tension. The reason is that there does not seem to be a corresponding term in
the Chern-Simons action that could lead to the necessary modification of the effective
D3-brane charge at the same time. This could be checked in more detail.
In general, we think that the question of additional corrections to the moduli
(Kähler) potential deserves further attention. Here we only outlined some steps in
that direction.
7 Conclusions
In this paper, we have investigated whether string loop corrections may impact a) stabi-
lization in the large volume compactification scenario (LVS), and b) the phenomenology
of those scenarios, as manifested in the soft supersymmetry breaking terms. The result
is that for the specific class of compactification manifolds considered in LVS, so-called
“Swiss cheese” Calabi-Yau manifolds, changes are minuscule. Only if the loop cor-
rections become abnormally large (in the toroidal orientifold case, this can happen if
the complex structure is stabilized very large) do they affect LVS. For other classes
of manifolds, the corrections may be important. We hasten to add that the detailed
expressions for the loop corrections in LVS remain unknown; we have merely tried to
infer their scaling with the Kähler moduli from experience in the toroidal orientifold
limit. We think it is important to attempt to address this issue, as the string coupling
is stabilized at a nonzero value, so the corrections cannot be turned off.
We also stress the (to some readers obvious) fact that there remain a host of is-
sues that must ultimately be dealt with if one wishes to claim that these are “string
compactifications”.
• We cannot be sure that fluxes do not alter the corrections, since backgrounds
with RR and NSNS fluxes are not well understood in string perturbation theory.
• Additional corrections may appear (see section 6) that could be equally threat-
ening to LVS as the loop corrections, or worse.
• In [37] only the corrections to the Kähler potential coming from N = 2 sectors
were determined and we based our generalization on those results. However,
there might be interesting corrections coming from the N = 1 sectors as well.
• It has not yet been shown that a local Standard Model-like construction can
be embedded in the simplest examples like the P4
[1,1,1,6,9]
model. If more general
models turn out to be needed, one needs to reconsider whether the requisite
nonperturbative superpotentials are generated.
• We have largely ignored open string moduli, under the proviso that they are
stabilized heavy, as are the dilaton and complex structure moduli.
• The coefficient A(S, U) in the nonperturbative superpotential is generally as-
sumed to be of order 1. It is not known how generic this is.
• All string computations we have discussed were performed in a supersymmetric
context. In LVS supersymmetry is broken already before uplift, in the AdS
minimum. Supersymmetry breaking directly in string theory is not very well
understood [39,71].
Faced with all these caveats, a pessimist might be inclined to give up. We think we
have shown that it is worth considering these issues in detail. Sometimes, an effect one
would have thought to be devastating turns out to be as gentle as a summer breeze.
Acknowledgments
It is a pleasure to thank Carlo Angelantonj, Vijay Balasubramanian, Massimo Bianchi,
Joe Conlon, Gottfried Curio, Robbert Dijkgraaf, Michael Douglas, Bogdan Florea,
Elias Kiritsis, Max Kreuzer, Fernando Marchesano, Peter Mayr, Thomas Mohaupt,
Hans-Peter Nilles, Gabi Pfuff, Fernando Quevedo, Waldemar Schulgin, Mike Schulz,
Stephan Stieberger, Angel Uranga, and Alexander Westphal for helpful discussions
and comments and Boris Körs for initial collaboration. This work is supported in part
by the European Community’s Human Potential Program under contract MRTN-CT-
2004-005104 ’Constituents, fundamental forces and symmetries of the universe’. The
work of M. B. is supported by European Community’s Human Potential Program un-
der contract MRTN-CT-2004-512194, ‘The European Superstring Theory Network’.
He would like to thank the Galileo Galilei Institute in Florence for hospitality. M. H.
would like to thank the university of Nis for hospitality. The work of M. H. and E. P.
is supported by the German Research Foundation (DFG) within the Emmy Noether-
Program (grant number: HA 3448/3-1). Both M. B. and M. H. would like to thank
the KITP in Santa Barbara for hospitality during the program “String Phenomenol-
ogy” and the university of Hamburg for hospitality during the workshop “Generalized
Geometry and Flux Compactifications”.
A Some details on LVS
In this appendix we collect some details on the minimization of the potential in LVS,
mainly reviewing the results of [3,8], but filling in some details. The minimization with
respect to the axions (i.e. the imaginary parts of the Kähler moduli) is performed for
an arbitrary number of Kähler moduli, while for the minimization of the real parts, we
restrict to the example of the hypersurface in P4[1,1,1,6,9] discussed throughout the main
text.
A.1 LVS for P4
[1,1,1,6,9]
Here we give some more numerical details on large-volume stabilization in the P4
[1,1,1,6,9]
orientifold. The relevant features of this Calabi-Yau have been described in chapter
2.3. The leading terms of the scalar potential are
V e−Kcs =
τe−2aτ
V2 τe
−aτ +
V3 , (56)
where we use τ = τs and V as the independent variables and for the expansion we
have in mind the limit (15). The minimum of this potential under the assumption that
aτ ≫ 1 is given by
V = µ
eaτ . (57)
In the P4[1,1,1,6,9] orientifold the coefficients λ, µ and ν can be calculated explicitly,
yielding
2a2|A|2
, µ =
2a|AW0|
and ν = ξ
S1|W0|2 . (58)
We notice that the value of τ at the minimum is determined only by the Euler number
τ ∝ χ2/3 and the value of the dilaton S1 at its minimum. An example of a set of
possible parameters (using a = 2π/10, A = 1, S1 = 10 and W0 = 10) is
ξ = − ζ(3)χ
2(2π)3
≃ 1.31 −→ ν ≃ 155 ,
λ ≃ 0.67 , µ = 4π
. (59)
There is an unknown overall factor eKcs that does not change the shape of the potential
and so leaves the position of the minima unchanged. For the parameters given in
equation (59), the minimum is at τ ≃ 41.1 and V ≃ 9.96 · 1011. These values come
from equation (57) which is just approximated using the assumption aτ ≫ 1. This
solution has the shortcoming that, if one is interested in the value of the potential at
the minimum, after substitution of (57) into (56), one finds V = 0. If instead one
solves the exact equation for the minimum of the potential numerically, the result is
V ≃ −6.6 · 10−37 at the point τ ≃ 41.7 and V ≃ 1.38 1012. From this one checks that,
apart from the shortcoming that V = 0, the approximate solution gives the position of
the minimum with a good precision.
A.2 Many Kähler moduli
The simple picture of P4
[1,1,1,6,9]
, gets slightly more involved in models with more than
two Kähler moduli, but some general statements can still be made. For a single small
Kähler modulus, among the leading contributions to the potential only the one from
Vnp2 is axion dependent, while the leading terms in V3 and Vnp1 are axion independent.
For several small Kähler moduli, all three terms are axion dependent. However, the
argument that the leading term in Vnp2 only receives a sign change due to axion stabi-
lization generalizes (and holds also for the regular KKLT scenario with relatively small
volume, see e.g. [8], section 3.2).
Indeed, with the superpotential (11) one obtains
Vnp1 = e
K G̄i
aiaj |AiAj |e−aiτi−ajτj cos(−aibi + ajbj + βi − βj)
Vnp2 = −2eK aiGk̄iKk̄
|AiW0|e−aiτi cos(−aibi + βi − βW0)
+|AiAj |e−aiτi−ajτj cos(−aibi + ajbj + βi − βj)
, (60)
V3 = e
K (Gk̄lKk̄Kl − 3)
|W0|2 + 2 |W0Ai|e−aiτi cos(−aibi + βi − βW0)
+|AiAj |e−aiτi−ajτj cos(−aibi + ajbj + βi − βj)
where Ai = |Ai|eiβi, W0 = |W0|eiβW0 and a sum over repeated indices is understood
throughout. As the only dependence on the axions is in form of cosines, one can easily
see that this potential has a minimum for
aibi = −βW0 + βi + niπ , ni ∈ 2Z+ 1 . (61)
We notice that the minimum of the bi depends on the (already fixed) complex structure
moduli, but it is independent of the Kähler moduli.
In the regime (15) the scalar potential again contains three terms at leading order,
Vnp1 ∼ 2eKcs
aiaj |AiAj |e−aiτi−ajτjM ljMki (−VVlk + VlVk)
+ . . . ,
Vnp2 ∼ −2eKcs
ai|AiW0|e−aiτiτi
+ . . . , (62)
V3 ∼ eKcs
8V3 |W0|
2 + . . . ,
where the sum over i and j effectively only picks up terms from the small moduli
because of the exponential suppression of Vnp1 and Vnp2. Moreover, for Vnp1 we used
the form (89) for the inverse of the moduli metric with respect to the basis (83). The
Kähler moduli appearing in the nonperturbative superpotential are linear combinations
of these, which we account for by a basis-changing matrix Mki , i.e.
Ti =M
i T̃k , (63)
where T̃k are the fields defined in (83) and Ti are the Kähler moduli appearing in the
nonperturbative superpotential. (Another way of saying this is that the real parts of Ti
measure the volumes of a basis of divisors that have the right properties to contribute
to the nonperturbative superpotential.) In the second term we used
Gk̄iKk̄ = −2τi + . . . = −2ReTi + . . . . (64)
In the basis (83), this would follow straightforwardly from (87), (89) and the relations
(80), but it holds equally well after a change of basis, because both sides of (64)
transform linearly under a change of basis (63).
Note finally that the ellipsis in (62) and (64) stand for subleading corrections in the
large volume limit (assuming also (15)).
B Loop corrected inverse Kähler metric for P4
[1,1,1,6,9]
We now have a closer look at the inverse metric from the Kähler potential in equation
(25). We invert the 4× 4 matrix and focus on the four terms that appear in the scalar
potential for the Kähler moduli,
Gb̄b =
τ 2b +O (τb) ,
Gb̄s = Gs̄b = 4τbτs
6E (K)s
, (65)
Gs̄s =
2S1τs
where we have performed an expansion in τb ≃ V2/3 and the quantity ∆ was introduced
in (29). We notice that only Gb̄b is not corrected at leading order. The apparent
divergence from zeros of the denominator ∆ is an artifact of the expansion. In fact,
the determinant of the (entire) Kähler metric behaves as
detG ∼ Aτ−7/2b +Bτ
b + . . . (66)
for some expressions15 A and B, which depend on the moduli τs, U and S1. In par-
ticular, one finds A ∼ ∆, but B does not vanish at a zero of ∆. Thus, in general
the expansion in large τb picks up the factor A, which is responsible for the apparent
divergence in (65). However, this is fictitious because when ∆ = 0, the next term
proportional to B is non-vanishing and the determinant stays away from zero. Indeed,
we do not expect to find any zero of the determinant in the range of validity of the
parameters.
If E (K)s ≪ (S1τs), one can further expand (65) with respect to E (K)s /(S1τs), yielding
Gb̄b =
τ 2b +O (τb) ,
Gb̄s = Gs̄b = 4τbτs
2E (K)s
((E (K)s )2
Gs̄s =
3E (K)s√
2S1τs
((E (K)s )2
Depending on the values of the moduli (τs and S1), this expansion may or may not be
useful. In general, only the expansion in τb makes sense and one has to deal with the
full expressions (65). That is what we did in section 3.3.
C No-scale Kähler potential in type II string the-
In this appendix we review why compactification of type IIA and type IIB theory
on general Calabi-Yau manifolds, or orientifolds thereof, lead to no-scale (F-term)
potentials if
i) the superpotential does not depend on the Kähler moduli (68)
and if
ii) one uses the tree-level form of the Kähler potential. (69)
(Of course, in LVS neither i) nor ii) holds, but one can think of jointly imposing i)
and ii) as a zeroth-order approximation, that we will successively move away from in
later subsections of this appendix.)
15This A has nothing to do with the A in Wnp.
If the moduli spaces of Kähler and complex structure moduli factorize (see appendix
F for more details on this), and under assumption i), the F-term potential takes the
V = eK
GĪJDĪW̄DJW − 3|W |2
GābDāW̄DbW + (G
ı̄jKı̄Kj − 3)|W |2
. (71)
The indices a and b run over the complex structure moduli and the dilaton, i, j over
the Kähler moduli and I and J refer to all moduli.
The condition for a no-scale potential (V = 0 for the Kähler moduli) is then
Gı̄jKı̄Kj = 3 , (72)
and we will verify in turn that this is fulfilled in both type IIA and type IIB Calabi-
Yau compactifications, if one uses the tree-level Kähler potential, as in assumption ii).
In that case, the moduli spaces of Kähler and complex structure moduli do factorize
exactly.
C.1 No-scale structure in type IIA
The tree level Kähler potential for the Kähler moduli is
K = − ln
dijk(σ + σ̄)i(σ + σ̄)j(σ + σ̄)k
= − ln
dijktitjtk
= − ln(V) , (73)
where dijk are the intersection numbers of the Calabi-Yau,
dijk =
ωi ∧ ωj ∧ ωk , (74)
σi = ti + ici (75)
are the complexified Kähler moduli whose real parts ti represent the volumes of 2-cycles
and whose imaginary parts originate from the expansion of the NSNS 2-form. Using
the Kähler form
J = tiωi (76)
of the Calabi-Yau, it is useful to introduce the notation
V = 1
J ∧ J ∧ J = 1
dijktitjtk ,
ωi ∧ J ∧ J =
dijktjtk ,
Vij =
ωi ∧ ωj ∧ J = dijktk . (77)
Note that here the index i does not denote a derivative with respect to the Kähler
variables (in contrast to subscripts on the Kähler potential K). Instead, one has the
relations Vi = 2∂σiV and Vij = 4∂σi∂σ̄̄V. It is straightforward to calculate
Ki = −
2V = Kı̄ Gi̄ = Ki̄ = −
. (78)
Then one can show that the inverse Kähler metric is
G̄i = −4VjiV + 2tjti . (79)
To verify this, one has to use
V ijVj =
ti , Vijtj = 2Vi , Viti = 3V . (80)
Putting everything together, one arrives at
Gı̄jKı̄Kj =
−4V ijV + 2titj
V = 3 , (81)
i.e. (72) is fulfilled under assumption (69).
C.2 No-scale structure in type IIB
In the type IIB case, the tree-level Kähler potential for the Kähler moduli is
K = −2 ln
dijktitjtk
= −2 ln(V) . (82)
The difference to the IIA case is that, even if K in (82) is expressed in terms of the
2-cycle volumes ti, the real parts of the good Kähler moduli, T̃i, are now the 4-cycle
volumes τ̃i (the imaginary parts, on the other hand, arise from the RR 4-form). The
relation between them depends on the particular Calabi-Yau:
Re T̃i = τ̃i =
dijktjtk = Vi , (83)
which cannot be inverted in general.16 In order to calculate Ki = ∂T̃iK we note that
∂ti = (∂ti T̃j)∂T̃j + (∂ti
T̄)∂ ¯̃T̄
= Vij
∂T̃j + ∂ ¯̃T̄
. (84)
If acting on a function F that only depends on T̃ +
T , as is the case for K, (84)
simplifies to
∂tiF (T̃ +
T ) = 2Vij∂T̃jF (T̃ +
T ) , (85)
where on the left hand side T̃ is understood as a function of t. Alternatively, one has
∂T̃iF (T̃ +
T ) =
V ij∂tjF (T̃ +
T ) = ∂ ¯̃
F (T̃ +
T ) . (86)
Using this, one can calculate
Ki = −
V ∂T̃iV = −
V ijVj
V = −
2V = Kı̄ , (87)
where in the last step we used (80). In the same way one can calculate
Gi̄ =
. (88)
Using this formula one can check that the inverse Kähler metric is given by
Gı̄j = 4 (−VVij + ViVj) . (89)
Putting everything together, no-scale structure holds also for type IIB:
Gı̄jKı̄Kj = (−VVij + ViVj)
V = 3 , (90)
again under the assumption (69).
16Note that the Kähler moduli appearing in the non-perturbative superpotentials in the examples
of [16] are related to the ones in (83) by a linear field redefinition. However, this does not play any role
in verifying the no-scale structure at leading order, as (90) below is invariant under field redefinitions.
We chose to make the distinction clear by using tildes for the Kähler moduli defined by (83).
C.3 Cancellation with just the volume modulus
Now we relax assumption (69). For simplicity, let us first consider the Kähler potential
K = −3 ln(T + T̄ ) + Ξ
(T + T̄ )λ
, (91)
which corresponds to the case of a single Kähler modulus and the complex structure
moduli and the dilaton are neglected. A generic quantum correction was added to the
tree level term, which could be an α′ or a loop correction, depending on the value of
λ. Focusing on V3, i.e.
eK |W |2 = G
̄iK̄Ki − 3 , (92)
one calculates
eK |W |2 =
(3(2τ)λ + ξλ)2
3(2τ)2λ + Ξ(2τ)λλ(λ+ 1)
= 3− 3 + (λ− λ
(2τ)λ
3(2τ)2λ
. (93)
This simplified calculation gives an intuition of why the E (K)b -term does not appear
in V3 of (28) whereas the α
′- and E (K)s -terms contribute. When the exponent of the
quantum correction is exactly 1, there is a cancellation at leading order in the scalar
potential (compare also the discussion in footnote 10). Note that since we focused on
V3 in this subsection, it did not matter whether assumption (68) holds or not.
C.4 Cancellation with many Kähler moduli
We would now like to see how the previous result is changed when we have an arbitrary
number of moduli. We do not make any assumption on the dependence of the volume
on the Kähler moduli (“Swiss cheese” or fibered manifolds are special cases). Due to
its relevance for LVS, we consider a single correction to the Kähler potential which only
depends on the large Kähler modulus Tb (an example would be the α
′-correction or the
loop term proportional to E (K)b , considering the moduli other than the Kähler moduli
as fixed; this is allowed at leading order in a τb-expansion, as we argue in appendix F).
Thus, we take the Kähler potential to be of the form
K = K(0) + δK = −2ln(V) + δK(Tb, T̄b) ≡ −2ln(V) +
(Tb + T̄b)λ
. (94)
Again focusing on V3, we obtain
eK |W |2 = G
̄iK̄Ki − 3 = (G̄i0 + δG̄i)
̄ + δK̄
i + δKi
− 3 ,
where δKi ≡ ∂TiδK and G
0 is the inverse metric of appendix C.2; finally δG
̄i is
the modification of the inverse metric coming from considering the modified Kähler
potential (94). Explicitly one has
G̄i = (G0i̄ + δKi̄)
−1 ≃ G̄i0 −G
0 δKhk̄G
0 + . . . ,
δKi = −
(2τb)λ+1
δib , δKi̄ =
(λ2 + λ) Ξb
(2τb)λ+2
δibδ̄b . (96)
We now put everything together and use the results of appendix C.2 and formula (64)
(which, for the unperturbed metric and Kähler potential, is an exact equality) to arrive
eK |W |2 =
i − 3
̄ δKi
0 δKkh̄G
i + . . .
= 0 +
(2τb)λ+1
4(λ2 + λ)Ξb
(2τb)λ+2
τbτb + . . .
(λ− λ2)Ξb
(2τb)λ
+ . . . . (97)
We notice that the term 1/τλb vanishes exactly for λ = 1, independently of the explicit
form of the volume in terms of the Kähler moduli. In particular, the loop correction
proportional to E (K)b experiences a cancellation at leading order in V3 (and it is not
difficult to see that the subleading order is suppressed by τ−2λb ). Therefore, the loop
correction is subleading in the potential compared to the α′ correction, even though it
is leading in the Kähler potential. Next, we would like to extend this analysis to the
other parts of the potential, i.e. Vnp1 and Vnp2.
C.5 Perturbative corrections to Vnp1 and Vnp2
We now introduce the nonpertubative superpotential into the game, i.e. relax assump-
tion (68), and look at the other terms of the scalar potential, Vnp1 and Vnp2 (see eq.
(12)). For this, we restrict to the P4[1,1,1,6,9] model again. The contribution Vnp1 is pro-
portional to Gs̄sW̄,s̄W,s. From (65) we see that no E (K)b appears at leading order. This
can be understood as follows. Consider the Kähler potential (94) where now V is the
volume of P4
[1,1,1,6,9]
, given in (14). Then the scaling with the large Kähler modulus τb
is schematically given by
Gs̄s ≃ Gs̄s0 −Gs̄b0 δKbb̄Gb̄s0 + . . .
∼ τ 3/2b + τ 2b
τλ+2b
∼ τ 3/2b +
+ . . . , (98)
which shows that any loop correction to the Kähler potential of the form Ξ/τλb leads to
a subleading contribution to Vnp1 in the large volume expansion. As usual, the ellipsis
stands for terms that are even more subleading in the τb expansion.
To understand the E (K)s correction to Gs̄s we need to consider
K = −2 ln(V) + δ̃K(T, T̄ ) ≡ −2 ln(V) + Ξb g(Ts, T̄s)
(Tb + T̄b)λ
for some function g(Ts, T̄s) of the small Kähler modulus and we assume λ ≥ 3/2 in the
following. Then, again very schematically, the scaling behavior is given by17
Gs̄s ≃ Gs̄s0 −Gs̄i0 δ̃Ki̄G
0 + . . .
∼ τ 3/2b + Ξb (τb, τ
τ−λ−2b τ
τ−λ−1b τ
+ . . . (100)
∼ τ 3/2b + τ−λb + τ
−λ+3/2
b + τ
b + . . . .
One sees that λ = 3/2 indeed contributes at the same order as Gs̄s0 . This is confirmed
by the dependence of Gs̄s in (65) on E (K)s through ∆, cf. (29).
We now consider Vnp2. This is proportional to G
̄sK̄. Again we start by considering
a correction to the Kähler potential whose only dependence on the Kähler moduli is
via τb, as in (94). Schematically, we find
G̄sK̄ ≃ (G̄s0 −G
0 δKbb̄G
̄ + δK̄
+ . . .
∼ τs +
Gs̄b0 Ξb
τλ+1b
+ . . . ∼ τ 0b +
+ . . . . (101)
This result is confirmed by the absence of E (K)b in the leading term of Vnp2. A calculation
very similar to the one in (100) shows, however, that Vnp2 is modified by a correction
to the Kähler potential of the form (99) for λ = 3/2. It is straightforward to generalize
this analysis to a more general form of the “Swiss cheese” volume, with more than one
small Kähler modulus.
17For λ = 3/2 we can still use the expansion of the inverse metric (96), because the correction term
would also be further suppressed e.g. in the dilaton.
D KK spectrum with fluxes
In this section we would like to develop some intuition on how the analysis of sections
3.3 and 4 might change in the presence of fluxes. We will restrict the discussion to
one possible effect of the fluxes, namely their influence on the KK spectrum. It is not
known explicitly how closed string fluxes, which are present in LVS, would change the
mass spectrum. We will consider a toy example, using an analogy to the correction
arising from world volume fluxes (cf. [52]), in order to get a feeling for what kind of
effects one might expect. In particular, for the purposes of this appendix we assume a
modified KK mass spectrum of the form
m2KK ∼
1 + F
1 + F
) , (102)
where F represents any of the fluxes that may be present, and in the second equality
the factors of S1 appeared when expressing the 2-cycle volumes in Einstein frame as
compared to the string frame (t ∼ e−Φ/2tstr). Note that expanding (102) for large
values of t would lead to a correction ∆m2 ∼ F 2/t3, whose scaling with the flux and
with t is reminiscent of the moduli masses induced by closed string 3-form flux [72,73].
In that case, the suppression would be by the overall volume (which would lead to only
mild effects in LVS), but in (102) we allow for a suppression by single 2-cycle volumes
(which might be the small 2-cycle in the P4[1,1,1,6,9] model).
Substituting (102) in (23), we now consider the scalar potential resulting from
K = −ln(2S1)− 2 ln(V) +Kcs(U, Ū)−
τbE (K)b
τsE (K)s
F 2S1
W = W0 + Ae
−aTs . (103)
We have not included any flux correction to the term proportional to E (K)b because we
expect such corrections to be subleading in a large volume expansion.18 Note that the
F -dependent correction term we did include is of the same form as the winding string
18Even though we think it is unlikely, we cannot exclude that the correction to KK masses that
scale like t−1b without fluxes is only suppressed by F
2/τs instead of F
2/τb. In that case, one would
have to redo the analysis of appendix C.5, using (99) with λ = 1. This would prohibit the use of the
expansion (96), because in the large volume limit the leading contribution to Gss̄ would arise from the
loop correction (it would scale as τ−1b as opposed to the scaling of the tree level contribution ∼ τ
In that case the leading terms in Vnp1 and Vnp2 would be suppressed compared to V3 and only arise
at order V−10/3, thus invalidating the volume expansion of LVS.
correction ∼ E (W )s , when one neglects any potential flux corrections to the winding
string spectrum, cf. (24) (remember that gsW would just be proportional to 1/
without fluxes). Thus, by considering (103) we implicitly also analyze in the following
the effect of corrections from winding strings (recall from section 3.3 that this correction
is not present in the P4[1,1,1,6,9] model, but may be present in general).
We now give the generalization of (26)-(28) when using the modified Kähler po-
tential (103). The three contributions at leading order (O(V−3)) in the large volume
expansion are
Vnp1 = e
24a2|A|2τ 3/2s e−2aτs
V∆ , (104)
Vnp2 = −eKcs
2a|AW0|τse−aτs
6E (K)s
1− 2F
, (105)
3eKcs |W0|2
1 ξ̃ + (106)
4(E (K)s )2 − 8(E (K)s )2F 2S1τ−1s (1 + F 2S1τ−1s )− 8
F 2S21E
where the axion has already been minimized for, as discussed in section A.2, and now
∆ is generalized to
2S1τs − 3E (K)s
1− 3F
. (107)
Plots for F = 1 and F = 3 are given in fig. 5, and they look quite similar to the
plot without flux, fig. 4. Qualitatively, the conclusion is the same; only for nongeneric
values of the gs corrections do they compete with the α
′ correction. Note, however,
that the amount of fine-tuning seems to depend on the value of the flux, cf. fig. 5.
The same is true for the dependence of the values of V and τs at the minimum on S1
and E (K)s . For F = 3, for instance, this dependence becomes more complicated than
what we found in (31). For the parameter range shown in fig. 5, the values of τs and
V in the minimum vary in the ranges τs ∈ [14.6, 46.3] and log10 V ∈ [3.7, 15.5], where
the smallest value for both of them is reached in the corner where the two corrections
become comparable.
Also the cancellation that we found for the gaugino masses survives the inclusion
of the flux factor in (103). The correction still only appears at sub-sub-leading order
0.0 9
PSfrag replacements
0.0 9
PSfrag replacements
Figure 5: Similarly to figure 4, the top surface is the α′ correction, the second is
the gs correction (with F = 1 in the left graph and F = 3 in the right), and the “red
carpet” is 10/∆, with ∆ from (107), using the same values as in fig. 4. The result
is qualitatively the same as before. Note, however, that the range for E(K)s differs.
For larger values of F one does not need to fine-tune E(K)s as much in order for the
two corrections to become of similar order.
in an expansion in ln(1/m3/2) and we find (again using the dilute flux approximation
for the prefactor (Ref)−1):
|Ms| =
= 3eK/2|W0|
16a2τ 2s
S1 − 12
2a(1− 2F 2S1τ−1s )E
64S1a3τ 3s
+ . . .
ln(1/m3/2)
ln(1/m3/2)
. (108)
This concludes our brief study of the direct effects of fluxes on the loop corrections.
E The orientifold calculation
In the main text, we are interested in how ∆Kgs , the one-loop correction to the Kähler
potential, scales with the Kähler moduli Ti. Our argument in section 3.1 is based on the
known result for ∆Kgs in the case of N = 2 supersymmetric K3× T 2 orientifolds and
N = 1 supersymmetric T 6/ZN (or T 6/(Z2 ×Z2)) orientifolds, from [37] (see also [74]).
Here we review this computation for the case of K3 × T 2, and take this opportunity
to adapt it to our case of D3-branes and D7-branes from the beginning. (One can also
obtain them by T-duality on the final D9/D5 results of [37], e.g. as in the appendix
of [75], but as we shall see, the direct computation is enlightening in its own right.)
We will leave out details that are essentially identical to [37], and only emphasize the
differences.
As shown in [37] using “Kähler adapted” vertex operators, the easiest way to com-
pute ∆Kgs is by considering the 2-point function of the complex structure modulus U
of T 2, with vanishing Wilson line moduli, i.e.
〈VUVŪ〉 = −
4g2cα
(U − Ū)2
〈V (0,0)ZZ V
(0,0)
〉σ . (109)
Here, we use the notation of [37],
(0,0)
U = −gcα′−2
U − Ū
(0,0)
(0,0)
= gcα
′−2 2
U − Ū
(0,0)
(110)
(0,0)
ZZ = −
i∂Z +
α′(p · ψ)Ψ
i∂̄Z +
α′(p · ψ̃)Ψ̃
eipX ,
(0,0)
= − 2
i∂Z̄ +
α′(p · ψ)Ψ̄
i∂̄Z̄ +
α′(p · ψ̃) ¯̃Ψ
eipX . (111)
As in [37], and [76] before that, we find these complex worldsheet variables particularly
convenient:
(X4 + ŪX5) , Z̄ =
(X4 + UX5) ,
(ψ4 + Ūψ5) , Ψ̄ =
(ψ4 + Uψ5) , (112)
where
Gstr is the volume of T
2 measured in string frame. The 2-point function (109)
can be expanded for small momenta, p1 · p2 ≪ 1, and we obtain
〈V (0,0)ZZ V
(0,0)
〉σ = −V4
(p1 · p2)
16(4π2α′)2
d2ν1d
k=0,1
~n=(n,m)T
e−π~n
~nt−1
](0, τ)
η3(τ)
γσ,kZ intσ,k[
〈∂̄Z(ν̄1)∂̄Z̄(ν̄2)〉σ〈Ψ(ν1)Ψ̄(ν2)〉α,βσ 〈ψ(ν1)ψ(ν2)〉α,βσ (113)
+〈∂̄Z(ν̄1)∂Z̄(ν2)〉σ〈Ψ(ν1) ¯̃Ψ(ν̄2)〉α,βσ 〈ψ(ν1)ψ̃(ν̄2)〉α,βσ + c.c.
+O((p1 · p2)2) .
For the details we refer to [37]. The main difference to the corresponding formula
(C.3) in [37] is the appearance of the inverse metric G−1str in the exponent arising from
the zero mode sum, and in the prefactor. This is due to the fact that the D3 and
D7 branes are localized along the T 2, and so the closed string channel involves a
Kaluza-Klein momentum sum instead of a winding sum. The sum over bosonic zero
modes has been made explicit, since there is also an implicit dependence on m,n in
the bosonic correlators: this arises from the classical piece in the split into zero modes
and fluctuations. That is, Z(ν) = Zclass(ν) + Zqu(ν), where the classical part is given
Zclass =
n+mŪ
Re(ν) c̃σ , c̃σ =
1 for K
2 for A ,M . (114)
These zero modes have the right periodicity under Re(ν) → Re(ν) + π (for A,M) or
Re(ν) → Re(ν) + 2π (for K), i.e. X4 → X4 + 2πn
α′ and X5 → X5 + 2πm
α′. In
contrast to [37] they involve the real part of ν. The reason is again that in the D3/D7
case the branes are localized along T 2 and thus the winding appears in the open string
channel as opposed to the closed string channel (as was the case for D9/D5 branes).
The sum over spin structures is performed using Riemann identities. This leaves
the correlators of the bosonic fields as the only piece that depends on the positions
νi of the vertex operators. The νi integral can then be evaluated. As the zero modes
(114) involve the real part of ν in the case of D3/D7-branes, in contrast to the D9/D5-
case studied in [37], the zero mode contribution in the Z-correlators drops out. The
quantum part is evaluated using the method of images on the worldsheet [38,77,78].
To evaluate the KK sum in (113), it is useful to regularize the integral over t by a UV
cutoff Λ. With this we obtain
1/(eσΛ2)
~n=(n,m)T
π3c2σtα
′e−π~n
~nt−1
π3α′c2σe
4 + πα′c2σ
E2(0, U) + . . . , (115)
where the prime at the sum indicates that the (n,m) = (0, 0) term is left out, and cσ,
eσ are constants whose precise values will not be important in the following (but can
be found in [37]). Terms that go to zero in the limit Λ → ∞ have been dropped, as
indicated by the ellipsis. The nonholomorphic Eisenstein series E2(0, U) is the s = 2
special case of
Es(0, U) =
~n=(n,m)T
|n+mU |2s . (116)
The terms involving the UV cutoff Λ drop out after summing over all diagrams, due
to tadpole cancellation. We have then reduced (113) to
〈V (0,0)ZZ V
(0,0)
〉σ = −(p1 · p2)α′
(4π2α′)2
E2(0, U)γσ,kQσ,k
+O((p1 · p2)2) . (117)
The quantities Qσ,k come from the sum over spin structures and are defined in [37].
Introducing the notation
E2(0, U) =
k=0,1
E2(0, U)γσ,kQσ,k
, (118)
we end up with (neglecting some irrelevant factors of gc, α
′, terms subleading in the
low-energy expansion, and constants of order 1)
〈VUVŪ〉 ∼ −i(p1 · p2)
(4π2α′)2
(U − Ū)2
E2(0, U) . (119)
To read off the one-loop correction to the kinetic term of U we need to perform a Weyl
rescaling to the Einstein frame. In the one-loop term (119) this just leads to
Weyl rescaling: × e
Vstr , (120)
where
Vstr = VstrK3
Gstr (121)
is the overall volume in string frame. The Kähler potential can then be read off from
the kinetic term by use of the identity
∂U∂ŪE2(0, U) = −
(U − Ū)2
E2(0, U) , (122)
producing the final result
∆Kgs ∼
Gstre
Vstr(S + S̄)
E2(0, U) , (123)
where
Gstre
Φ/Vstr is to be interpreted as a function of the Kähler variables. In the
K3× T 2 orientifold case, using (121), this is just proportional to eΦ/VstrK3 ∼ (T + T̄ )−1
(with Re T the volume of K3 measured in Einstein frame), giving a result T-dual
to [37] (note that we switched the real and imaginary parts in the definition of T and
S as compared to [37], to conform with the rest of this paper). As we argue in section
3.1, in general the dependence on the Kähler moduli will be more complicated than
this, because there is no analog to the relation (121). It is still clear that the inverse
suppression in the overall volume will appear as in (123), given that it is a direct
consequence of the Weyl rescaling.
F Factorized approximation
As mentioned in section 4.2, it is an important issue to what extent the moduli spaces
of Kähler and complex structure moduli factorize. In this appendix, we give further
details on the factorized approximation.
A common starting point in the analysis of the potential arising in type IIB theory
with 3-form fluxes is to assume that all complex structure moduli Uα and the dilaton
S are stabilized by demanding
DUαW = 0 = DSW . (124)
In this case the F-term potential for the moduli (7) reduces to
V = eK
G̄iD̄W̄DiW − 3|W |2
, (125)
where as in the main text, the indices i and j refer only to the Kähler moduli and
thus run from 1 to h1,1. Note that even though the complex structure moduli and
the dilaton are assumed to be stabilized by (124), the inverse metric G̄i is part of
the inverse of the whole moduli space metric. More precisely, if we denote the Kähler
moduli by Ti, as before, and all other moduli (i.e. the complex structure moduli and
the dilaton) collectively as Za, the moduli space metric is given by
GIJ̄ ∼
Ki̄ Kib̄
Ka̄ Kab̄
. (126)
We denote the inverse of this (whole) metric by GJ̄I . In general
G̄i 6= (Ki̄)−1 . (127)
Equality only holds if Gib̄ = 0, i.e. if the moduli space of the Kähler moduli is factorized
from the rest, as it is the case without loop and α′ corrections.
In this appendix, we would like to investigate at which order in a large volume
expansion the two matrices in (127) start to deviate from each other. For this analysis
we assume a volume of the “Swiss cheese” form as in (48) and a Kähler potential of
the form (24) (without taking possible effects of fluxes on the KK and winding mode
spectra into account as was done in appendix D; thus, gaK ∼ ta and g
K ∼ t−1q for some
2-cycle volumes). To avoid cumbersome notation we will indicate all the small moduli
collectively as τs. We then use the formula
A−1(1 +BP−1CA−1) −A−1BP−1
−P−1CA−1 P−1
, (128)
where P is the Schur complement of A, defined as
P = D − CA−1B . (129)
In our case P is the Schur complement of Ki̄. From (24) we read off that
GIJ̄ ∼
τ−2b τ
τ−2b τ
τ−2b τ
, (130)
where we only indicate the τb dependence and the indices run over I, J = {Tb, Ts, U, S}.
Here β = −2 for those τi with a nonvanishing ai in (48) (so β has an implicit index
i), otherwise β = −5/2 (which is in particular the value in the P[1,1,1,6,9] case). We
decompose GIJ̄ as in (128)
τ−2b τ
, A−1 ∼
τ 2b τ
7/2+β
7/2+β
B = CT ∼
τ−2b τ
, D ∼
τ 0b τ
τ−1b τ
τ 0b τ
τ−1b τ
, P−1 ∼
τ 0b τ
τ−1b τ
. (131)
Using (128) one easily obtains the scaling of the inverse:
GJ̄I ∼
τ 2b τ
7/2+β
7/2+β
τ 0b τ
τ 0b τ
. (132)
Now, from (128), G̄i receives two contributions. The first is K ̄i, that would be the
only term in the case of a factorized metric; the second is K−1h̄ Khb̄P
Kal̄K
, that
breaks factorization. Let us compare their τb scaling:
G̄i = A−1 + A−1BP−1CA−1 (133)
τ 2b τ
7/2+β
7/2+β
τ 0b τ
τ 0b τ
Thus the corrections coming from non-vanishing off-diagonal metric elements in (126)
set in with a suppression by τ−2b , τ
−7/2−β
b and τ
b in G
b̄b, Gb̄s and Gs̄s, respectively.
In the explicit example based on P[1,1,1,6,9], β = −5/2, and we checked this result by
comparing to the subleading terms in (65).
F.1 Factorized approximation of the scalar potential
What we are really interested in is not the (inverse) metric itself, but the scalar po-
tential, to which we now turn. For the nonperturbative terms Vnp1 and Vnp2, the
suppression of the off-diagonal terms in (133) is inherited by the scalar potential, as
they are proportional to Gs̄sW̄,s̄W,s and G
̄sK̄, respectively. For V3 things are not as
simple, due to the no-scale structure at leading order. Let us neglect for a moment all
the quantum corrections, then the no-scale structure implies
Gı̄jKı̄Kj − 3
no−scale ∼ (τ
b , τ
τ 2b τ
7/2+β
7/2+β
∼ τ 0b + τ
2β+7/2
b = 0 . (134)
The two terms have to vanish independently. Now let us add corrections that break
no-scale structure. Because of the cancellation described in appendix C.3, the leading
contribution can be seen to come at order τ
b (from the α
′, E (K)s and E (W )s corrections).
On the other hand, the off-diagonal terms appear at order
Gı̄jKı̄Kj
off−diagonal ∼ (τ
b , τ
τ 0b τ
τ 0b τ
∼ τ−2b + τ
b + τ
∼ τ−2b + . . . , (135)
for both β = −2 and β = −5/2. Therefore, the off-diagonal terms of the moduli space
metric appear in the scalar potential with a suppression of at least τ
b (as is confirmed
by the explicit example of section 3.3, cf. formulas (26)-(30)). The suppression can be
even stronger if some corrections are absent and the leading term in (135) vanishes.
To summarize: if one is only interested in the leading term of the scalar potential in
the large volume (i.e. large τb) expansion, then one can use the factorized approximation,
G̄i = K ̄i +O
. (136)
This provides a useful tool to simplify the calculations.
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	Introduction
	Review
	KKLT
	Consistency of KKLT
	Large volume scenario (LVS)
	String loop corrections to LVS
	From toroidal orientifolds to Calabi-Yau manifolds
	LVS with loop corrections
	The P[1,1,1,6,9]4 model
	Gaugino masses
	Including loop corrections
	Other soft terms
	LVS for other classes of Calabi-Yau manifolds?
	Abundance of ``Swiss cheese'' Calabi-Yau manifolds
	Toroidal orientifolds
	Fibered Calabi-Yau manifolds
	Further corrections
	Conclusions
	Some details on LVS
	LVS for P[1,1,1,6,9]4
	Many Kähler moduli
	Loop corrected inverse Kähler metric for P[1,1,1,6,9]4
	No-scale Kähler potential in type II string theory
	No-scale structure in type IIA
	No-scale structure in type IIB
	Cancellation with just the volume modulus
	Cancellation with many Kähler moduli
	Perturbative corrections to Vnp1 and Vnp2
	KK spectrum with fluxes
	The orientifold calculation
	Factorized approximation
	Factorized approximation of the scalar potential
