#### On the 3-form formulation of axion potentials from D-brane instantons

Received: December
formulation of axion potentials from
Eduardo Garc a-Valdecasas 0 1 2 3 4
Angel Uranga 0 1 3 4
up the 0 1 4
-form 0 1 4
Open Access 0 1 4
c The Authors. 0 1 4
0 Campus de Cantoblanco , 28049 Madrid , Spain
1 C/Nicolas Cabrera 13-15, Campus de Cantoblanco , 28049 Madrid , Spain
2 Departamento de F sica Teorica, Universidad Autonoma de Madrid
3 Instituto de F sica Teorica UAM-CSIC
4 eld does not arise in the
The study of axion models and quantum corrections to their potential has experienced great progress by phrasing the axion potential in terms of a 3-form eld dual to the axion. Such reformulation of the axion potential has been described for axion monodromy models and for axion potentials from non-perturbative gauge dynamics. In this paper we propose a 3-form description of the axion potentials from non-gauge D-brane instantons. Interestingly, the required 3-form underlying geometry, but rather shows up in the KK compacti cation in the generalized geometry obtained when the backreaction of the D-brane instanton is taken into account.
D-branes; Flux compacti cations; Gauge-gravity correspondence
1 Introduction and main results 2 Review of 3-forms and monodromy. 3 3-forms from D-brane instanton backreaction
D-brane instanton backreaction
The 3-form and its coupling
Some toroidal examples
Gauge non-perturbative e ects
Introduction and main results
Axions have become an essential template to describe physics of scalar elds whose
potential enjoys special protection properties due to an underlying symmetry principle. Naively,
the symmetry corresponds to the perturbative global symmetry shifting the value of the
scalar eld, which is violated by non-perturbative e ects, as originally proposed for the
QCD axion [1]. However, it has recently become clear that the most fundamental
symmetry structure is that of the dual 2-form. Contributions to the axion potential which spoil
the shift symmetry must arise from the existence of a 3-form which eats up the dual 2-form
to make it (and so the dual axion) massive. The gauge symmetry of the 3-form constrains
the form of these contributions in an advantageous way for many phenomenological
applications. The description of the axions in terms of forms and their duals has also been
key to the use of the weak gravity conjecture [2] to constrain transplanckian axion model
This formulation has been well understood:
For the QCD axion, in [21{23], where the 3-form is actually the Chern-Simons
composite 3-form built out of the QCD gauge elds.
In string compacti cations producing axion monodromy [24, 25], as described in [26]
connecting it to the earlier description in [27, 28].1 In these cases, the 3-form is a
fundamental eld, and its couplings arise from di erent sources, ranging from
ChernSimons couplings to
uxes in the 10d action [26, 47] (see also [48]), torsion
homology [26] (see also [49]) or topological brane-bulk couplings [43].
1For other works related to axion monodromy, see [15, 29{46].
The two above phenomena, in particular the presence of uxes and non-perturbative
e ects on D-brane gauge sectors, play an important role in several scenarios of moduli
stabilization (and thus of their axion components), along the lines in [50]. Actually, the gauge
non-perturbative e ects can be described in string theory as particular cases of D-brane
instanton e ects wrapping the same cycle as the gauge D-branes in the compact space. In
general, in string theory there are other non-perturbative e ects from D-brane instantons
not wrapping such cycles (sometimes dubbed stringy or exotic D-brane instantons [51{53],
see [54, 55] for reviews), and contributing to the stabilization of axions as well. It is
therefore natural to wonder about the 3-form description of these latter e ects. Interestingly,
there is no known description of this kind: since there is no gauge group associated to the
cycles, we cannot use any composite Chern-Simons 3-form; on the other hand, for e.g. a
D3-brane instanton on a 4-cycle, there is not any obvious corresponding harmonic form
able to produce a 3-form in the 4d theory upon compacti cation.
In this paper we solve this question and provide the 3-form description for the
stabilization of an axion by non-gauge D-brane instanton e ects. The key idea is to notice
that the stabilization occurs when the non-perturbative e ect is included in the theory, so
it is only then that we can hope to
nd a suitable 3-form. Therefore, the internal form
supporting the 4d 3-form must arise only in the geometry backreacted by the presence of
the D-brane instanton, in the sense discussed in [56, 57]. In general, these correspond to
generalized geometries, so the corresponding form need not be harmonic with respect to
the underlying CY metric, rather it corresponds to (a piece of) a generalized calibration.
We study this in the particular example of D3-brane instantons on 4-cycles, but
the lesson is general (as expected from T-duality / mirror symmetry). Also, we show
that the picture is compatible with D-brane instantons corresponding to gauge
nonperturbative e ects.
The paper is organized as follows. In section 2 we review the 3-form description of
axion stabilization and its interplay with axion monodromy and non-perturbative gauge
dynamics. In section 3 we provide the 3-form description of axion potentials induced by
non-gauge D-brane instantons: after posing the question in section 3.1, we review the
D3brane instanton backreacted geometry in section 3.2, and obtain the 4d 3-form and its
couplings in section 3.3. A simple example is displayed in section 3.4. Section 3.5 describes
the generalization, in particular the mirror picture of D2-brane instantons in type IIA
compacti cations. In section 4 we discuss the case of gauge D-brane instantons. Finally,
section 5 contains our nal remarks.
Review of 3-forms and monodromy.
Consider an axion , regarded just as a scalar taking values in a circle (i.e. with discrete
periodicity2 2 ) and with an (approximate) shift symmetry. In many applications one is
interested in generating a non-trivial potential for this axion, violating precisely this shift
symmetry. For simplicity we consider the potential expanded at quadratic order around
2For simplicity we set the axion decay constant to f = 1.
a minimum, as for instance arises in moduli stabilization; the general picture is however
more general. Hence we have the lagrangian
A potential of this kind is naively not compatible with the axion periodicity, but may be
made so by including multivalued branches for the potential, thus inducing axion
monodromy. The description of the periodicity properties are automatically implemented by
using a dual formulation, in terms of a 3-form c3 eating up the 4d dual 2-form
db2 = d , in 4d). This is described by the following lagrangian3
where F4 = dc3. This theory has the gauge invariance
As emphasized in [26], it is this gauge symmetry that protects the atness of the axion
potential against uncontrolled corrections even in transplanckian
eld ranges. Dualizing
the 2-form back into the axion , we obtain the Kaloper-Sorbo description of axion
monodromy [27, 28, 35, 59]
Integrating out the non-dynamical 4-form
eld strength F4 one recovers a potential with
the structure (2.1).
The idea can be generalized to other potentials by considering corrections, which due
to the gauge invariance, (2.3) must be in terms of higher powers in F4. This leads for
instance to the attening in [60].
At this point we would like to emphasize an important clari cation: the description of
the massive axion in terms of a 3-form eating up the dual 2-form, or in terms of a coupling
between (a function of) the axion and the 4-form
eld strength, is not necessarily a signature
of axion monodromy, but rather of the existence of a non-trivial axion potential. Axion
monodromy arises in cases when the potential does not naively satisfy the axion periodicity.
On the other hand, the 3-form description should also hold for non-monodromic potentials
(i.e. single-valued and consistent with the axion periodicity). Our main interest in this
note is indeed the 3-form description of single instanton non-perturbative potentials, which
indeed are periodic in the axion.
Before entering this discussion, we conclude this review with a brief recap of the 3-form
description of axion potentials induced by non-perturbative gauge dynamics. This has been
SYM, which will be useful later on. The axion belongs to a chiral multiplet
= Im T , and
we have a coupling
SSYM =
3We allow for a Zn discrete gauge symmetry, see [26, 48, 58] for discussions of such system and the
corresponding Zn charged domain walls.
The non-perturbative dynamics produces a gaugino condensate superpotential
W = !
is the SYM dynamical scale, regarded as a
where ! = e2 i=n and
= exp
The theory has n gapped vacua, which di er by the value of h i (the phase of the
gaugino condensate), in the sense that a shift
(k + 1)th. Therefore the potential has a periodicity of
+ 2 changes the kth vacuum to the
+ 2 n. On the other hand,
the axion should actually have a periodicity of 2 ; in fact, this is ensured because there
is a 3-form description of the above axion potential. This follows from realizing that the
coupling (2.5) describes a coupling to a 4-form
F4 structure, with F4 = tr F 2. Namely,
In other words, although the potential is periodic with a period 2 n, the periodicity of
2 is achieved via a monodromic structure, but with a nite number n of branches. This is
structure of domain walls among vacua is naturally described in terms of the (composite)
3-form description [21{23].
3-forms from D-brane instanton backreaction
In string theory, there are non-perturbative contributions to the superpotential beyond
the above non-perturbative gauge dynamics contributions. These come from, for instance,
euclidean D-brane instantons which do not correspond to gauge theory instantons. A
prototypical example is provided by D3-branes instantons wrapped on 4-cycles in CY
compacti cations,4 such as those used in moduli stabilization in [50]. In our discussions we will
assume the instantons to indeed produce superpotential terms; it would be interesting, but
beyond our scope, to develop an understanding of a dual description for instanton e ects
generating higher F-terms due to extra fermion zero modes [61, 62]. Denoting by T the
complex modulus associated to the wrapped 4-cycle, we have a superpotential
Wnp = A e T
where A is a prefactor depending on complex structure moduli (and possibly also on open
string moduli), whose detailed structure is not essential for our present purposes.
We expect the axion potential induced by these instantons to admit a description in
terms of a 3-form eating up the dual 2-form. On the other hand, there is no obvious
candidate for such a 3-form in the 4d CY compacti cation: the only available RR
are even-degree gauge potentials, so the 3-form should arise from the RR 6-form integrated
over a 3-cycle. However, there is no natural pairing between a 3-cycle and a 4-cycle in
a CY so as to support the topological coupling
F4 ultimately responsible for the axion
4In general, these may carry world-volume uxes, but for simplicity we will restrict to the case of trivial
potential. Therefore there is no natural candidate for the 3-form coupling to the axion to
reproduce its potential.
The solution to this problem is to follow the intuition gained in the discussion of
the SYM superpotential. In that case, the 3-form arises only when the existence of
nonperturbative sectors in the gauge theory are taken into account; namely, the fact that
implementation of a similar concept for non-gauge D-brane instantons requires proposing
that the 3-form describing the axion stabilization should be looked for not in the original
CY geometry, but rather in the geometry perturbed by the gravitational backreaction of the
D-brane instantons. This perturbation of the geometry has been studied in the literature
in [56, 57] by exploiting the technology of generalized geometry.
D-brane instanton backreaction
In this section we review ideas in [56, 57] to describe the backreaction of D-brane instantons
in terms of generalized geometry. The uninterested reader may wish to jump to section 3.3.
The e ect of D-brane instantons can be encoded in the underlying CY geometry by
means of a deformation turning the SU(3) holonomy into (in general) an SU(3)
structure, associated to the existence of two spinors (not covariantly constant due to the
already in the type IIB case, the two spinors are written
1 = +
here + and + are complex conjugate of
N = 1 supersymmetry, and satisfying r
, and + is the 4d spinor specifying the
+, where W0 is the superpotential
The spinors (1;2) can be used to de ne two polyforms,
2 = +
Noticing the chirality of the spinors in the sandwich, the polyform
+ contains even degree
contains odd degree forms. A common alternative notation is (for type IIB)
The familiar case of SU(3) structure corresponds to (2)
. For SU(3) holonomy the spinors are covariantly constant and the polyforms
1 =
2 =
The compacti cation ansatz is
ds2 = e2A(y)g (x) dx dx
The 10d elds can be organized in complex quantities, in agreement with the 4d susy
structure. One holomorphic quantity is
is the 10d dilaton (not to be confused with the 4d axion). This is motivated
because it provides the calibration for BPS domain wall D-branes (notice that in IIB,
calibrates odd-dimensional cycles). In other words, the tension of a D-brane BPS domain
wall is obtained by integrating the above form over the wrapped cycle. For instance, for
standard CY compacti cations, a D5-brane on a supersymmetric 3-cycle
BPS domain wall, whose tension is given by R
the 3-form part of the polyform
The second quantity is
, where the subindex (3) denotes
C describes the RR backgrounds not encoded in the background
the generalized calibration of BPS D-brane instantons (notice that in IIB,
even-dimensional cycles). For instance, for standard CY compacti cations, a D3-brane
wrapped on a holomorphic 4-cycle
provides a 4d BPS instanton whose action is given
The 3-form and its coupling
From the supersymmetry conditions recast in terms of the 10d version of the 4d elds Z,
T , one can show that, in a weak coupling expansion, we have
dH Z =
backreaction of the instanton e ect on the 10d geometry in terms of the appearance of a
1-form component Z(1) of Z, which was absent in the CY geometry.
This equation de nes an special 1-form
Z(1). It is associated to the globally
de ned supersymmetric spinor, in the presence of the non-perturbative correction to the
geometry. Its structure can be obtained by integrating the above equation. For the
particf = 0, one obtains [56]
where the tilded superpotential has the dependence on open string degrees of freedom
One intuitive way to understand the above expression is to notice that, in a theory
containing gauge D3-branes (i.e. D3-branes sitting at a point in the internal space), the
4d superpotential as a function of the D3-brane position is obtained by considering a
1chain L joining two di erent points in the CY and integrating Z(1) over it. This follows
because Z(1) is the calibrating form for a D3-brane wrapped on the 1-chain L, which de nes
a domain wall interpolating between the two con gurations of the D3-branes at the two
(end)points. We thus have
Z(1) = f W~ np
where we regard f as the D3-brane position in the direction normal to the instanton
4cycle. The result therefore reproduces the familiar dependence on open string moduli,
microscopically associated to Ganor strings [63] (see also [64]).
In the following we recast (3.7) as d 1 =
2, and we use both
1 and 2 as internal
pro les for the KK reduction of higher-dimensional form elds in the backreacted geometry.
This is similar to the non-harmonic forms used in KK reduction of massive U(1)'s, and
studied in [49] in compacti cation spaces with torsion (co)homology.
We may use these forms to perform the KK reduction of the 10d RR 4-form C4 as
C4 =
This produces a 3-form in 4d spacetime, naturally associated to the non-perturbative e ect,
and a 2-form, dual of . Moreover, is it clear that the 3-form is eating up the 2-form, by
noticing that the eld strength F5 has a term
F5 = (1 + 10d) ( 2 ^ (c3 + db2)
where 10d is added to take the self-duality of F5 into account. This clearly has the gauge
This implies that the 3-form is eating up the 2-form to become massive, and correspondingly
provides a dual description of the axion
becoming massive, as in section 2.
It is also straightforward to show that this 3-form has a Kaloper-Sorbo coupling to the
axion. We focus just on the leading term
= R C4, and F4 = dc3. We simply
massage the kinetic term of the (self-dual) 4-form and focus on the components in (3.11)
F5 ^ F5 =
C4 ^ 2 ^ F4 =
where we used 2
Some toroidal examples
To esh out this somewhat abstract description, let us now consider a toroidal
compact
T6, where for simplicity we take a factorizable, T6 = T2
local complex coordinates be z1; z2; z3. Let us study the backreaction caused by a
instantonic D3-brane wrapping the 4-cycle
in [56, 65], the complex structure Z =
structure with a 1-form piece
4 de ned by z3 = 0. Using the general formulas
gets corrected, becoming a generalized complex
This is the 1-form to be used to produce the 4d 3-form upon compacti cation of the 10d
Notice that it actually corresponds to dz3, a harmonic 1-form already present in the
underlying toroidal geometry. Therefore there seems to be essentially no new geometric
structure associated to the backreacted geometry, namely, no axion potential due to the
instanton e ect. This feature is clearly related to the existence of extra harmonic forms in
the T6 geometry, which are not present in generic CYs. However, it nicely dovetails the
expectation that D3-brane instantons in toroidal geometries have additional fermion zero
modes, and do not produce non-perturbative superpotentials for the corresponding moduli.
In order to actually get non-trivial structure, we can consider orbifolds which
remove the extra harmonic forms, and produce genuine CY geometries. Consider for
instance T6=(Z2
Z2), where the generators of the orbifold group act as
( z1; z2; z3), and ! : (z1; z2; z3) ! (z1; z2; z3).
To describe the quotient, we
inu3 = 0, so f = u3, and we have
It is now clear that the 1-form supporting the 3-form in the compacti cation of the 10d
4-form is non-harmonic with respect to the original CY geometry.
Although we have focused on the case of axions with potential arising from D3-brane
instantons on 4-cycles, the ideas hold for general RR axions associated to other cycles, and
with potentials arising from the corresponding wrapped D-brane instantons. In order to
illustrate this, we consider the mirror con guration of type IIA with axions arising from
the RR 3-form on a 3-cycle, stabilized by D2-brane instantons.
Let us thus study the mirror dual to the con guration of D3-branes on a 4-cycle, in
the setup of a general CY (alternatively the main ideas can already be illustrated in the
toroidal examples in section 3.4). Consider the CY in the large complex structure limit,
where it can be regarded as a T3 (parametrized by coordinates yi, 1; 2; 3;) bered over a 3d
The mirror dual can be obtained by applying three T-dualities [66], along the
coordinates yi. The D3-brane instanton thus turns into a D2-brane wrapped on the 3-cycles
locally spanned by x1, x2, y30 (with the prime denoting the T-dual coordinate). One further
sees that the complex structure deformation Z(1)
df = dx3 + idy3 gives rise to a polyform
T = T(2) + T(4);
This follows from the fact that Z(1) is eventually use to expand the RR forms and obtain
the 4d 3-form. Hence, these are the 2- and 4-form components of (3.6) produced by the
backreaction of the D2-instanton, as we argue later on. Before that, let us conclude that
5A global construction is easily produced by using Weierstrass equations for the 2-tori, but we will not
need this extra complication.
C =
T ^ c3 ! C7 = T(4) ^ c3 ; C5 = T(2) ^ c3
where the c3 in the last two expansions is understood to be the same 4d 3-form.
Let us nish by arguing further that the above T is indeed the backreaction
corresponding to the D2-brane instanton. In the original picture in section 3.3 we considered
the superpotential in the theory in the presence of a gauge D3-brane, given by the integral
of the calibrating form over a 1-chain. Under the mirror transformation we must consider
the theory in the presence of a gauge D6-brane wrapped on the 3-cycle
3 (i.e. the
mirthe calibrating form T over a (generalized) 4-chain
generalized) 3-cycles (see e.g. [67])
interpolating between two (possibly
WD6 =
We are interested in the superpotential as a function of one D6-brane complex modulus,
given by one deformation of the special lagrangian 3-cycle, and the corresponding Wilson
line along one 1-cycle in
3. The actual components turned on in this T are T(4), which
will be integrated along the 4-chain produced by the deformation of the 3-cycle
component T2 integrated over the 2-chain spanned by the 1-cycle in such deformation. The
latter accounts for the contribution to the superpotential of the induced D4-brane charge
arising from possible D6-brane worldvolume uxes on the 2-forms Poincare dual to the
The resulting variation of the superpotential is
WD6 =
T(2) ^ F =
where F is the magnetic eld induced in the D6-brane.
Therefore, the only way to reproduce the D6-brane open string moduli dependence
described by the Ganor zeroes is that the instanton backreaction indeed produces the
deformation T with components T(2) and T(4).
Gauge non-perturbative e ects
There are D3-brane instantons which admit the interpretation of gauge theory instantons.
This happens when the D3-brane instanton wraps precisely the same 4-cycle as a stack of
4d spacetime- lling D7-branes.
The description of stabilization of axions coupling to non-abelian gauge interactions
has been recast in terms of coupling to the composite Chern-Simons 3-form in [21{23] cf.
section 2. It is natural to ask about any possible interplay between this and the 3-form
discussed in earlier sections.
Actually, the 3-form in earlier sections arises only when the D-brane instantons
backreact, in other words when the gauge dynamics is geometrized. This implies that we must
instantons) is de ned by
condensate hSi = h
To this order there is no deformation of Z, and moreover no 1-form that can support
the 3-form. The latter is however generated when the non-perturbative gaugino
condensate (which is described by (fractional) euclidean D3-branes) is included. The gaugino
i = e2 ik=n 3 is a non-zero vev for the gaugino bilinear. In [65] it
Z =
T = e
exp(ie =2J )
dZ = i`s4hSi 2( )
consider the geometry that results when the D7-branes, together with the euclidean
D3brane instantons, are backreacted on the geometry. The resulting con guration no longer
contains open string degrees of freedom, as everything is encoded in the backreacted
geometry. Therefore the relation between the 3-forms is essentially holographic: on one side there
are open string degrees of freedom, and the axion stabilization mechanism is described as
in [21{23] in terms of a 3-form constructed out of the open string sector gauge elds; on
the other side, there is a backreacted geometry, and no open string degrees of freedom, and
the axion stabilization arises from a 3-form supported by the distorted geometry.
esh out the latter claim, let us consider the description of the backreaction of
D7-branes and their euclidean D3-brane instanton e ects.
We concentrate in the case
To describe the backreacted geometry, we borrow results from [65]. The
backreaction of the D7-branes, at the perturbative level (i.e not including the euclidean D3-brane
This is exactly as in (3.7), thereby con rming the anticipation that the backreacted D7/D3
system can support a 3-form in agreement with the mechanism in earlier sections.
In this note we have provided the description of the axion potential from non-gauge D-brane
instanton e ects in terms of a 3-form eating up the 2-form dual to the axion. The 3-form
arises from the KK reduction of higher-dimensional RR
elds in the generalized geometry
arising when the D-brane instanton backreaction is taken into account. The mechanism
also holds for D-brane instantons corresponding to gauge instantons, in which case the
generalized geometry description of the axion couplings can be regarded as holographically
related to earlier 3-form descriptions of stabilization of QCD-like axions.
Our works puts axion potentials from non-perturbative e ects in a similar footing to
other stabilization mechanisms, like
ux compacti cations.
We hope this can improve
the study of the interplay between di erent stabilization mechanisms in string theory. In
another line, given recent results in applying the Weak Gravity Conjecture to axion models
in terms of their dual 3-forms [15], we expect our analysis to allow similar analysis for
nonperturbative axion potentials from instantons.
We hope to come back to these questions in the near future.
Acknowledgments
We would like to thank L. Iban~ez, F. Marchesano, A. Retolaza and G. Zoccarato for
useful discussions. E. G. and A. U. are partially supported by the grants
FPA2015-65480P from the MINECO, the ERC Advanced Grant SPLE under contract
ERC-2012-ADG20120216-320421 and the grant SEV-2012-0249 of the \Centro de Excelencia Severo Ochoa"
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
38 (1977) 1440 [INSPIRE].
(2015) 020 [arXiv:1503.00795] [INSPIRE].
[arXiv:1503.03886] [INSPIRE].
gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
[3] A. de la Fuente, P. Saraswat and R. Sundrum, Natural in ation and quantum gravity, Phys.
Rev. Lett. 114 (2015) 151303 [arXiv:1412.3457] [INSPIRE].
[4] T. Rudelius, Constraints on axion in ation from the weak gravity conjecture, JCAP 09
constraints on large eld in ation, JHEP 10 (2015) 023 [arXiv:1503.04783] [INSPIRE].
[7] T.C. Bachlechner, C. Long and L. McAllister, Planckian axions and the weak gravity
conjecture, JHEP 01 (2016) 091 [arXiv:1503.07853] [INSPIRE].
evading the weak gravity conjecture with F-term winding in ation?, Phys. Lett. B 748 (2015)
455 [arXiv:1503.07912] [INSPIRE].
[9] J. Brown, W. Cottrell, G. Shiu and P. Soler, On axionic eld ranges, loopholes and the weak
gravity conjecture, JHEP 04 (2016) 017 [arXiv:1504.00659] [INSPIRE].
[10] D. Junghans, Large- eld in ation with multiple axions and the weak gravity conjecture,
JHEP 02 (2016) 128 [arXiv:1504.03566] [INSPIRE].
in ation, JHEP 12 (2015) 108 [arXiv:1506.03447] [INSPIRE].
[12] S. Bielleman, L.E. Iban~ez and I. Valenzuela, Minkowski 3-forms, ux string vacua, axion
stability and naturalness, JHEP 12 (2015) 119 [arXiv:1507.06793] [INSPIRE].
[arXiv:1508.00009] [INSPIRE].
[14] B. Heidenreich, M. Reece and T. Rudelius, Sharpening the weak gravity conjecture with
dimensional reduction, JHEP 02 (2016) 140 [arXiv:1509.06374] [INSPIRE].
(2016) 085 [arXiv:1601.00647] [INSPIRE].
[arXiv:1602.06517] [INSPIRE].
[arXiv:1605.05311] [INSPIRE].
gravity conjecture, JHEP 04 (2016) 020 [arXiv:1512.00025] [INSPIRE].
[16] A. Hebecker, F. Rompineve and A. Westphal, Axion monodromy and the weak gravity
conjecture, JHEP 04 (2016) 157 [arXiv:1512.03768] [INSPIRE].
in ation with warped KK-modes, Phys. Lett. B 754 (2016) 328 [arXiv:1512.04463]
[18] J.P. Conlon and S. Krippendorf, Axion decay constants away from the lamppost, JHEP 04
quantum gravity via the weak gravity conjecture, Int. J. Mod. Phys. D 25 (2016) 1643005
Lett. 96 (2006) 081602 [hep-th/0511175] [INSPIRE].
(2014) 105025 [arXiv:1312.7273] [INSPIRE].
Phys. Rev. D 78 (2008) 106003 [arXiv:0803.3085] [INSPIRE].
monodromy, Phys. Rev. D 82 (2010) 046003 [arXiv:0808.0706] [INSPIRE].
[26] F. Marchesano, G. Shiu and A.M. Uranga, F-term axion monodromy in ation, JHEP 09
(2014) 184 [arXiv:1404.3040] [INSPIRE].
(2009) 121301 [arXiv:0811.1989] [INSPIRE].
03 (2011) 023 [arXiv:1101.0026] [INSPIRE].
[arXiv:0912.1341] [INSPIRE].
[28] N. Kaloper, A. Lawrence and L. Sorbo, An ignoble approach to large eld in ation, JCAP
[30] L.E. Iban~ez and I. Valenzuela, BICEP2, the Higgs mass and the SUSY-breaking scale, Phys.
Lett. B 734 (2014) 354 [arXiv:1403.6081] [INSPIRE].
[arXiv:1403.7507] [INSPIRE].
[32] R. Blumenhagen and E. Plauschinn, Towards universal axion in ation and reheating in
string theory, Phys. Lett. B 736 (2014) 482 [arXiv:1404.3542] [INSPIRE].
[33] L.E. Iban~ez and I. Valenzuela, The in aton as an MSSM Higgs and open string modulus
monodromy in ation, Phys. Lett. B 736 (2014) 226 [arXiv:1404.5235] [INSPIRE].
(2014) 16 [arXiv:1404.3711] [INSPIRE].
Fortsch. Phys. 62 (2014) 647 [arXiv:1405.0283] [INSPIRE].
throats, JHEP 02 (2015) 086 [arXiv:1405.7044] [INSPIRE].
01 (2015) 128 [arXiv:1411.5380] [INSPIRE].
compacti cations, Nucl. Phys. B 899 (2015) 414 [arXiv:1412.5537] [INSPIRE].
ux-scaling scenario for high-scale moduli stabilization in string
theory, Nucl. Phys. B 897 (2015) 500 [arXiv:1503.07634] [INSPIRE].
JHEP 07 (2015) 099 [arXiv:1504.02103] [INSPIRE].
[45] S. Bielleman, L.E. Iban~ez, F.G. Pedro and I. Valenzuela, Multi eld dynamics in Higgs-otic
in ation, JHEP 01 (2016) 128 [arXiv:1505.00221] [INSPIRE].
[arXiv:1510.01522] [INSPIRE].
09 (2014) 123 [arXiv:1405.3652] [INSPIRE].
in ux compacti cations, JHEP 04 (2013) 138 [arXiv:1211.5317] [INSPIRE].
[50] S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, de Sitter vacua in string theory, Phys.
Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].
[hep-th/0609191] [INSPIRE].
[52] L.E. Iban~ez and A.M. Uranga, Neutrino Majorana masses from string theory instanton
e ects, JHEP 03 (2007) 052 [hep-th/0609213] [INSPIRE].
theories, JHEP 05 (2007) 024 [hep-th/0610003] [INSPIRE].
056 [hep-th/0409149] [INSPIRE].
JHEP 04 (2011) 061 [arXiv:1012.4018] [INSPIRE].
[5] M. Montero , A.M. Uranga and I. Valenzuela , Transplanckian axions!?, JHEP 08 ( 2015 ) 032 [6] J. Brown , W. Cottrell , G. Shiu and P. Soler , Fencing in the swampland: quantum gravity [19] F. Baume and E. Palti , Backreacted axion eld ranges in string theory , JHEP 08 ( 2016 ) 043 [20] B. Heidenreich , M. Reece and T. Rudelius , Axion experiments to algebraic geometry: testing [21] G. Dvali , Three-form gauging of axion symmetries and gravity , hep-th/ 0507215 [INSPIRE].
[22] G. Dvali , R. Jackiw and S.-Y. Pi , Topological mass generation in four dimensions , Phys. Rev.
[23] G. Dvali , S. Folkerts and A. Franca , How neutrino protects the axion , Phys. Rev. D 89 [24] E. Silverstein and A. Westphal , Monodromy in the CMB: gravity waves and string in ation , [31] E. Palti and T. Weigand , Towards large r from [p; q] -in ation, JHEP 04 ( 2014 ) 155 [35] N. Kaloper and A. Lawrence , Natural chaotic in ation and ultraviolet sensitivity , Phys. Rev.
[36] M. Arends et al., D7-brane moduli space in axion monodromy and uxbrane in ation , [37] S. Franco , D. Galloni , A. Retolaza and A. Uranga , On axion monodromy in ation in warped [38] R. Blumenhagen , D. Herschmann and E. Plauschinn , The challenge of realizing F-term axion monodromy in ation in string theory , JHEP 01 ( 2015 ) 007 [arXiv:1409.7075] [INSPIRE].
[39] A. Hebecker , P. Mangat , F. Rompineve and L.T. Witkowski , Tuning and backreaction in F-term axion monodromy in ation, Nucl . Phys . B 894 ( 2015 ) 456 [arXiv:1411. 2032 ] [40] L.E. Iban~ez , F. Marchesano and I. Valenzuela , Higgs-otic in ation and string theory , JHEP [41] I. Garc a-Etxebarria , T.W. Grimm and I. Valenzuela , Special points of in ation in ux [42] R. Blumenhagen et al., A [43] A. Retolaza , A.M. Uranga and A. Westphal , Bi d throats for axion monodromy in ation , [44] D. Escobar , A. Landete , F. Marchesano and D. Regalado , Large eld in ation from [46] R. Blumenhagen , C. Damian , A. Font , D. Herschmann and R. Sun , The ux-scaling scenario: de Sitter uplift and axion in ation, Fortsch . Phys. 64 ( 2016 ) 536 [47] L. McAllister , E. Silverstein , A. Westphal and T. Wrase , The powers of monodromy , JHEP [48] M. Berasaluce-Gonzalez , P.G. Camara , F. Marchesano and A.M. Uranga , Zp charged branes [49] P.G. Camara , L.E. Iban~ez and F. Marchesano , RR photons, JHEP 09 ( 2011 ) 110 [51] R. Blumenhagen , M. Cvetic and T. Weigand , Spacetime instanton corrections in 4D string vacua: the seesaw mechanism for D-brane models, Nucl . Phys . B 771 ( 2007 ) 113 [53] B. Florea , S. Kachru , J. McGreevy and N. Saulina , Stringy instantons and quiver gauge [54] R. Blumenhagen , M. Cvetic , S. Kachru and T. Weigand , D-brane instantons in Type II orientifolds , Ann. Rev. Nucl. Part. Sci . 59 ( 2009 ) 269 [arXiv:0902.3251] [INSPIRE].
[55] L.E. Ibanez and A.M. Uranga , String theory and particle physics: an introduction to string phenomenology , Cambridge University Press, Cambridge U.K. ( 2012 ).
[56] P. Koerber and L. Martucci , From ten to four and back again: how to generalize the geometry , JHEP 08 ( 2007 ) 059 [arXiv:0707.1038] [INSPIRE].
[57] P. Koerber and L. Martucci , Warped generalized geometry compacti cations, e ective theories and non-perturbative e ects, Fortsch . Phys. 56 ( 2008 ) 862 [arXiv:0803.3149] [INSPIRE].
[58] M. Berasaluce-Gonzalez , G. Ram rez and A.M. Uranga , Antisymmetric tensor Zp gauge symmetries in eld theory and string theory , JHEP 01 ( 2014 ) 059 [arXiv:1310.5582] [59] S. Dubovsky , A. Lawrence and M.M. Roberts , Axion monodromy in a model of holographic gluodynamics , JHEP 02 ( 2012 ) 053 [arXiv:1105.3740] [INSPIRE].
[60] X. Dong , B. Horn , E. Silverstein and A. Westphal , Simple exercises to atten your potential , Phys. Rev. D 84 ( 2011 ) 026011 [arXiv:1011.4521] [INSPIRE].
[61] C. Beasley and E. Witten , New instanton e ects in supersymmetric QCD , JHEP 01 ( 2005 ) [62] I. Garcia-Etxebarria , F. Marchesano and A.M. Uranga , Non-perturbative F-terms across lines of BPS stability , JHEP 07 ( 2008 ) 028 [arXiv:0805.0713] [INSPIRE].
[63] O.J. Ganor , A note on zeros of superpotentials in F-theory, Nucl . Phys . B 499 ( 1997 ) 55 [64] D. Baumann et al., On D3-brane potentials in compacti cations with uxes and wrapped