Modular Symmetries and the Swampland Conjectures
IIFT-UAM/CSIC-18-123
Modular Symmetries and the Swampland Conjectures
E. Gonzalo , , L.E. Ib´a˜nez , and A.M. Uranga Instituto de F´ısica Te´orica IFT-UAM/CSIC,C/ Nicol´as Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain Departamento de F´ısica Te´orica, Facultad de CienciasUniversidad Aut´onoma de Madrid, 28049 Madrid, Spain
Abstract
Recent string theory tests of swampland ideas like the distance or the dS conjectures have been per-formed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging.We propose to exploit the modular symmetries of the moduli effective action to check swamplandconstraints beyond perturbation theory. As an example we study the case of heterotic 4d N = 1compactifications, whose non-perturbative effective action is known to be invariant under modularsymmetries acting on the K¨ahler and complex structure moduli, in particular SL (2 , Z ) T-dualities(or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models withnon-perturbative superpotentials, the corresponding duality invariant potentials diverge at pointsat infinite distance in moduli space. The divergence relates to towers of states becoming light, inagreement with the distance conjecture. We discuss specific examples of this behavior based ongaugino condensation in heterotic orbifolds. We show that these examples are dual to compacti-fications of type I’ or Horava-Witten theory, in which the SL (2 , Z ) acts on the complex structureof an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KKmodes. The non-perturbative examples explored point to potentials not leading to weak couplingat infinite distance, but rather diverging in the asymptotic corners of moduli space, dynamicallyforbidding the access to points with global symmetries. We perform a study of general modularinvariant potentials and find that there are dS maxima and saddle points but no dS minima, andthat all examples explored obey the refined dS conjecture. a r X i v : . [ h e p - t h ] F e b ontents SL (2 , Z ) modular forms 30B General Analysis of Minima for two fields 32 There has recently been a lot of interest on the concept of the Swampland constraintson effective theories [1–7], see [9] for a review. These are powerful conditions excludingeffective theories which cannot arise in consistent theories of quantum gravity. Someof the most interesting swampland conjectures refer to the properties of the modulispace of scalars in consistent theories of quantum gravity [10–18]. The main list ofsuch constraints include the following: • The moduli space of scalars is non-compact. • Distance conjecture: Consider a point in moduli space p ∈ M . Then in thelimit of infinite distance d ( p , p ) = t → ∞ , an infinite tower of states appears inthe effective field theory with exponentially decreasing masses m (cid:39) e − αt .1 Refined dS conjecture [7] (see also [8]): Any scalar potential V ( φ ) in a consistenttheory of quantum gravity must obey either |∇ V | ≥ cM p V or min( ∇ i ∇ j V ) ≤ − c (cid:48) M p V . (1.1)There are presumably interesting interplays among these conjectures. A possible con-nection between the last two conditions has been pointed out in [7], see also [14];also, the connection between the distance conjecture and global symmetries has beenpointed out in [10]. The first two conjectures have been tested in a number of differentstring theory settings, see [10–13] and references therein. In this paper we will explorethe relations between these constraints, in the hitherto unexplored regime of low su-persymmetry and strong coupling. For some recent papers on the Swampland and theWeak Gravity Conjecture see also [19–23] and [24, 25].The string theory tests of the distance conjecture performed so far are either per-turbative or involve at least eight supersymmetries (or simple circle compactifications).These tests are also often essentially kinematic, in the sense that the structure of thetowers of light states is identified but a full understanding of their effect on the effectivefield theory is lacking. Only with the additional constraints of theories with at least 8supersymmetries it is possible to e.g. signal the emergence of a dual theory [10–13].It would certainly be interesting to try and test these conjectures in string set-tings with reduced supersymmetry and also e.g. including non-perturbative inducedpotentials. This seems to be out of reach. However, in this paper we make substantialprogress in this direction by proposing to exploit the modular symmetries of the effec-tive action to check swampland ideas beyond weak coupling. Indeed, the scalar modulispace of string compactifications transforms in general under modular (or paramodu-lar) symmetries, which are discrete infinite symmetries involving transformations of themoduli. They involve in general the K¨ahler, complex structure and complex dilatonof 4d string compactifications, and are part of the duality symmetries of string the-ory. The prototypical example is the SL (2 , Z ) T duality of tori and orbifolds thereof.This group is generated by R → /R transformations along with discrete shift sym-metries for the real part of a K¨ahler modulus T . The existence of this kind of modularsymmetries goes however beyond the toroidal setting, since modular symmetries arisealso in (exact) moduli spaces of Calabi-Yau threefolds, like the quintic [26] and fairlygenerically in large classes of eliptic CY threefolds see e.g. [26–29]. These generalizedmodular symmetries are also generated by some discrete shift symmetry and a trans-formation relating large and small volumes (in fact the discrete shift symmetries and2heir interplay with the tower of light states near points at infinite distance in modulispace has already been explored in the N = 2 setup in [10, 18]. Modular behaviorarise also in non-perturbative superpotentials for F-theory compactifications on CYfourfolds [30–34]. Modular symmetries in axion and inflaton potentials have also beenconsidered in [35, 36]We will focus on compactifications preserving 4d N = 1 supersymmetry, in whichnon-perturbative superpotentials for the moduli may arise. The resulting scalar poten-tials are thus constrained by invariance under modular transformations. A prototypicalclass of examples are 4d N = 1 heterotic orbifolds, for which the modular SL (2 , Z )symmetries were well studied in the early 90’s [37–44]. In this case the K¨ahler poten-tial and the (non-perturbative) superpotential must transform such that the full K¨ahlerpotential (and hence the scalar potential) is invariant under modular transformations.In the case of a single modulus (little is known about modular symmetries for the mul-timoduli case) the superpotential must be an automorphic form with definite modularweight, which is a very restrictive condition on the theory.The study of the modular invariant effective actions of a single modulus in general4d N = 1 string vacua is interesting in its own right, and was already consideredin the early 90’s, see e.g. [42–44]. We revisit these results from the perspective ofthe swampland conjectures. One finds in particular that modular invariant potentialsnecessarily imply the existence of essential singularities at points at infinite distance inmoduli space (e.g. large/small volume). We display explicitly that in concrete stringexamples the divergence arises from infinite towers of states becoming light in thoselimits, in agreement with swampland distance conjecture expectations. Hence, ouranalysis remarkably relates the swampland distance conjecture with the behavior ofthe moduli effective action under modular transformations. In addition the modularsymmetries for the moduli require the existence of the states to provide a physicalunderstanding of the singularities. Notice that this is a novel behavior, as comparedwith the previous discussions in 4d N = 2 vacua; rather than the emergence of adual description, what we find is that the infinite distance points are dynamically(exponentially) censored.We describe different specific string contexts in which such modular invariant non-perturbative potentials appear, and discuss the relevant towers of states at infinitedistance points. We revisit modular invariant gaugino condensation in 4d N = 1heterotic toroidal orbifolds, already discussed in the early 90’s [39–41]. In orbifoldswith N = 2 subsectors, there are moduli-dependent one-loop threshold corrections to3he gauge couplings [45–47], which can enter the moduli-dependent non-perturbativesuperpotential. In this case the singularities appear due to towers of KK or windingstates on the fixed T in the N = 2 sectors.A key observation is that the moduli dependence relevant for invariance under mod-ular transformations arises from subsectors with 4d N = 2 supersymmetry. Their prop-erties are thus controlled by the properties of heterotic compactifications on K3 × T .This observation allows us to explore the dynamics of moduli using other dual descrip-tions, in particular type IIB orientifolds, type I’ compactifications, and compactifica-tions of Horava-Witten theory. These pictures allow a clear identification of the towerof states becoming light near the point at infinity in moduli space, and which are ul-timately responsible for the appearance of the divergence in the scalar potential, andwhich correspond to D0-branes or KK momentum states in the latter pictures. Thisdiscussion enormously increases the applicability of our results on swampland conjec-tures from invariance under modular transformations to other string compactifications.In particular, non-perturbative superpotentials with modular behavior have been ob-tained in the literature in a variety of settings, see e.g. [30–34]. We expect that thelessons we present in this paper extend to more general string compactifications.The fact that many string models lead to theories invariant under modular trans-formations, motivates the study of general effective theories of scalars with modularinvariant potentials in the context of the swampland conjectures. They may be con-sidered as non-trivial candidate field theories which may be consistently coupled toquantum gravity. This consistency would also require fully fledged string theory togive a physical meaning to their behavior and singularities. In this paper, we performa detailed study of different single-modulus effective actions with non-trivial SL (2 , Z )-invariant scalar potentials, and explore their extrema. We find AdS and Minkowskivacua, both SUSY and non-SUSY. We also find dS maxima and saddle points, but wehave been unable to find a dS minimum. Our analysis show that the refined dS conjec-ture applies in all examples studied (including the second condition, for dS maxima).We find in our examples no runaway behavior leading to e.g. possible quintessencedynamics.We emphasize that in these effective theories the invariant potentials always divergeexponentially at points at infinity, so that the moduli are dynamically forbidden toaccess them. In their UV completion, the infinite tower of states demanded by theswampland distance conjecture should be responsible for the dynamical generation ofthe divergent potential, and thus for the impossibility to access the regions in which4lobal symmetries are recovered.The structure of the rest of this paper is as follows. In Section 2 we review the struc-ture of single-modulus SL (2 , Z )-invariant scalar potentials in 4d N = 1 supergravity.We show the superpotential must have modular weight ( − SL (2 , Z )-invariant scalar potential, and study the validity of the (refined) swamplandconjectures to their extrema. Finally, in section 5 we offer our final remarks. AppendixA collects some useful properties of modular functions and Appendix B an overview ofthe extrema in models with two complex scalars. We want to study how modular symmetries can constraint the effective action of stringcompactifications, and how these constraints fit with swampland ideas. We focus onthe prototypical example of such symmetry, the modular group SL (2 , Z ), which is ubiq-uitous in string theory. In this section we review the scalar potential for 4d N = 1 su-pergravity theories of a single modulus T invariant under the modular group SL (2 , Z ),see [42]. This modular symmetry is generated by the transformations T −→ aT + bcT + d , a, b, c, d ∈ Z , ad − bc = 1 . (2.1)acting on the complex modulus T = θ + it . The two generators are T → T + 1, whichimplies a discrete periodicity for the real part θ , making it axion-like; and T → − /T ,which relates small and large t , e.g. t → /t for δ = 0.In general T may correspond to a string modulus like K¨ahler, complex structureor complex dilaton, depending on the examples considered, see Section 3 for examples.For concreteness, we carry out the discussion in terms of T being a K¨ahler modulus5he effective action will be determined by the full supergravity K¨ahler potential G ( T, T ∗ ) = − T − T ∗ ) + log | W | . (2.2)This corresponds to e.g. the large K¨ahler modulus dependence of a Calabi-Yau (CY)string compactification with h = 1. It also corresponds to the K¨ahler potential forthe overall K¨ahler modulus in a toroidal/orbifold compactifications.Under modular transformations one has ( T − T ∗ ) → ( T − T ∗ ) / | cT + d | , so thatmodular invariance of the K¨ahler potential dictates that the superpotential should havemodular weight (-3), i.e. W ( T ) −→ e iδ ( a,b,c,d ) W ( T )( cT + d ) , (2.3)where δ is a moduli-independent phase, which can depend on the SL (2 , Z ) transfor-mation.We now revisit the general study of T-duality invariant potentials in [42]. A generaland useful way to parametrize functions with given modular weight is through theDedekind function η ( T ), which has modular weight 1 / η ( T ) → ( cT + d ) / η ( T ).Hence we can always write W = H ( T ) η ( T ) (2.4)with H ( T ) being a modular invariant holomorphic function. This can always be ex-pressed as a function of the absolute modular invariant function j ( T ) (see AppendixA).As explained in appendix A, if one insists in avoiding singularities within the fun-damental domain, it can be proven that H ( T ) must be of the form H ( T ) = ( j ( T ) − m/ j ( T ) n/ P ( j ( T )) (2.5)where m, n are positive integers and P is a polynomial on j . Without loss of generalitythe zeros of P ( j ) may be chosen different from the SL (2 , Z ) self-dual points T = i, e i π/ . Another equivalent way to write the same expression is H ( T ) = (cid:18) G ( T ) η ( T ) (cid:19) m (cid:18) G ( T ) η ( T ) (cid:19) n P ( j ( T )) , (2.6)where G , are Eisenstein functions of weight 4 and 6 respectively, see Appendix A.Note that these superpotentials with no additional singularities in the fundamentaldomain, necessarily diverge exponentially as T → i ∞ ,
0. Such a behavior is in principlerather surprising, since in perturbation theory potentials are known to scale like a power6f 1 /t . Thus this peculiar behavior, if present, requires non-perturbative physics . Incoming sections we will provide explicit string models with this divergent behavior,including the gaugino condensation example studied in [39]. These constructions showthat the exponential singularities arise from towers of states becoming light, in niceagreement with the swampland distance conjecture.It is interesting to consider the simplest case with H ( T ) = 1, which is indeedrealized in concrete string constructions, as we will see in coming sections. The scalarpotential is given by the simple expression V = 18(Im T ) | η ( T ) | (cid:18) T ) π | ˆ G | − (cid:19) , (2.7)where ˆ G is the non-holomorphic weight-2 Eisenstein functionˆ G ( T, T ∗ ) = G ( T ) − π Im T . (2.8)This potential (2.7) is explicitly modular invariant and diverges like e πt for large Im T = t → ∞ . Extrema of this potential as well as more general potentials with arbitrary H ( T ) will be discussed in section 4.1. As we discuss in the next section, this typeof potentials arise in gaugino condensation in heterotic orbifolds, and other relatedexamples.One may worry that the above computations can be trusted only for large Im T ,and that there may be corrections in the deep interior of moduli space. However,the key statement is that modular invariance strongly restricts the structure of theK¨ahler potential, the superpotential and thus the scalar potential for moduli. Moreformally, the K¨ahler potential and superpotential are not mere functions over modulispace, but rather sections of non-trivial bundles over moduli space. Corrections maychange the specific form of the section, but cannot change the non-trivial topology ofthe bundle, which is encoded in the modular properties we are highlighting. Hence, inshort, the superpotential must be an holomorphic form of modular weight −
3. Barringsingularities at finite distance in moduli space (which would have no clear physicalorigin, as will be clear in the examples in coming sections) this implies the existenceof singularities at Im T → ∞ , which thus seems unavoidable.The existence of this singular behavior as one approaches points at infinite distancein moduli space, resonates with the swampland conjectures. For Im T (cid:29) This does not imply the absence of tree level superpotential couplings. These are indeed present,but necessarily involve charged matter fields, which transform non-trivial under the modular groupand account for the matching of the total modular weight for the superpotential. N = 2 supersymmetry on which they are based. In thoseexamples, e.g. [10], the towers of states near those points typically induce correctionsto the metric in moduli space, possibly making the infinite distance emergent; thesestates sometimes admit very explicit descriptions in terms of a dual theory [12, 13].In the non-perturbative framework here described, access to these points is excludednot merely from kinematics in moduli space, but also from the dynamics inducing apotential. The models are thus trapped in a non-perturbative regime far from anyweak coupling expansion. Perturbativity for the SM should then appear as an infraredeffect, as suggested e.g.in [11].As explained, similar analysis applies when the relevant modulus correspond toother string moduli. The case in which it corresponds to the complex dilaton S inheterotic compactifications was put forward in [43], in the first paper introducing S-duality. Although in principle SL (2 , Z ) S-duality could be expected to be a symmetryonly in finite theories, it was argued that it may apply to some extent in some subsectorof theories with reduced or no supersymmetry. Indeed, modular invariant superpoten-tials involving complexified string coupling have appeared in the context of 4d N = 1F-theory compactifications [31–34].Following the same arguments as above the limits S → i ∞ , g → S , corresponding to g →
0, the het-erotic string becomes tension-less since one has for the string scale M s = M p / (8 Im S ).In ref. [44] the divergent limits in the S-dual potentials were avoided by canceling thedivergence choosing a H ( S ) vanishing at S → i ∞ ,
0. However in such a case, as dis-cussed above, there are necessarily singularities in the fundamental region, with noobvious physical interpretation. 8
String theory examples
In this section we will discuss how N = 1 superpotentials of the type discussed in theprevious section arise in string theory. In particular, we will consider 4d N = 1 het-erotic Abelian orbifolds. We first focus on the moduli-dependent threshold correctionsfor gauge couplings. Upon gaugino condensation, such threshold corrections can giverise to non-perturbative moduli-dependent superpotentials which are modular forms ofthe appropriate weight.Threshold corrections to 4d gauge kinetic functions arise from a perturbative 1-loop computation. We eventually focus on Abelian orbifolds, which are exact CFTs,and in particular are interested in the moduli-dependent piece, which arise in N = 2subsectors, i.e. those associated to elements of the orbifold group leaving some T fixed. In those sectors the structure is essentially that of a compactification of heteroticon K3 × T , and the discussion can be phrased in this more general setting (eventualbreaking to N = 1 can be obtained by further orbifold twists), following the classicalcomputation in [45–47], also for comparison with the section to follow.From general results in N = 2 theories, the gauge kinetic functions can only dependon moduli in vector multiplets. Thus, since the K3 moduli belong to hypermultiplets,they cannot appear in the threshold correction. Hence, we are interested in thresholdcorrections depending on the K¨ahler and complex structure moduli T , U of the T ,which are in vector multiplets. We will pay special attention to the action of themodular groups SL (2 , Z ) T × SL (2 , Z ) U on these moduli.The thresholds corrections depending on the moduli of the T for the resulting N = 2 theories arise from momentum and winding modes on T , and are given by∆ a = b a (cid:90) Γ d ττ (cid:0) Z torus ( τ, ¯ τ ) − (cid:1) , (3.1)with Z torus = (cid:88) m , ,n , ∈ Z e πi τ ( m n + m n ) e − πτ M , (3.2)and M = 14 (Im T )(Im U ) | n T U + n T − m U + m | . (3.3)Here n i , m i are winding numbers and momenta and M is the mass of KK and windingexcitations of vector multiplets involved in the loop. We note that M is invariant under9 L (2 , Z ) T × SL (2 , Z ) U by suitable relabeling of the integers m , , n , . Performing theintegral in (3.1) one obtains for the moduli dependent threshold corrections∆ a = − b a log (cid:104) (Im T ) | η ( T ) | (Im U ) | η ( U ) | (cid:105) , (3.4)with b a the β -function coefficient of the corresponding N = 2 gauge theory. Note that,as expected, this expression is invariant under SL (2 , Z ) T × SL (2 , Z ) U .Let us consider including additional orbifolds to yield 4d N = 1. This leads us toconsider N = 1 orbifolds T /P with the orbifold group P given by Z N and Z N × Z M orbifolds, with some elements of the orbifold group (forming a subgroup P i ) leavingthe i th T fixed. For simplicity we focus on factorizable orbifolds, and consider thedependence on the moduli T i , U i , with i labeling the three possibly fixed T . Usingthe above results, this is given by [45]∆ a = − (cid:88) i | P ( i ) || P | b N =2 a,i (cid:8) log (cid:2) (Im T i ) | η ( T i )) | (cid:3) + log (cid:2) (Im U i ) | η ( U i )) | (cid:3)(cid:9) + c a , (3.5)where | P | is the order of the orbifold group (e.g. N for Z N ). Also b N =2 a,i is the β -functioncoefficient of the corresponding N = 2 sub-theory, and c a is a moduli-independentconstant. Recall that the orbifold group is constrained to have a crystallographicaction on the torus. Also, we note that in certain N = 1 orbifolds, some or all the U i are frozen and hence only contribute a constant to the thresholds. This occurswhen the fixed T is subject to extra orbifold action which are not Z , and thus actcrystallographically only for definite values of the T complex structure. For instance,that is the case e.g. for the Z (cid:48) N orbifolds with N = 6 , , N = 1 theory contains gauge sectors which may experience strongdynamics effects at low energies. Consider the simplest situation that some gaugefactor contains no charged matter, so that it describes a pure super-Yang-Mills sector,which confines and produces a gaugino condensation superpotential in the infrared . The prototypical case is an unbroken E , which is present if the embedding of theorbifold group in the gauge degrees of freedom involves only the other E . This includesstandard embedding models, but also many other orbifolds, see [50] for a review. In cases with charged chiral multiplets and possible non-perturbative superpotentials involvingthem, the modular properties of the superpotential require including the effect of the modular weightsof matter fields [48,49], which in general transform non-trivially under the modular groups. In partic-ular, this also occurs for the tree-level superpotential couplings involving untwisted or twisted fields f a = S − π (cid:88) i =1 b N =2 a,i log (cid:2) η ( T i ) η ( U i ) (cid:3) , (3.6)where S is the heterotic 4D complex dilaton. Actually, cancellation of duality anomaliesrequires in general a Green-Schwarz mechanism [51–53], which implies the replacement b N =2 a,i → b N =2 a,i − δ iGS , where δ iGS are gauge group independent constants. However, thispoint is not essential for our purposes, so we rather ignore it by considering e.g. the Z × Z orbifold, for which δ iGS = 0.The gaugino condensation superpotential then reads W E = Λ e fE βE = Λ e S βE (cid:89) i =1 η ( T i ) η ( U i ) , (3.7)where we have made use of the fact that β E = 3 b N =2 i,E in this example. This has pre-cisely the modular properties described in the previous section, and it is a most simpleillustration of how such superpotentials can arise in very explicit heterotic compactifi-cations.The result can be easily adapted to other orbifold groups. For instance, in orbifoldswhere, as mentioned earlier, the complex structure moduli are projected out, they aresimply not present in the superpotential. Also, for orbifolds with δ i GS (cid:54) = 0, the onlyeffect is that the exponent of the η ( T i )’s is smaller [48, 49]Finally, recall that in orbifolds with no fixed planes, like the prime order cases Z , Z , there are no moduli dependent threshold corrections. The modular invariant scalarpotentials arising from orbifolds with gaugino condensation can be subject to explicitstudy. For instance, considering examples with fixed complex structure, and focusingon the dependence on the overall K¨ahler modulus, we have a superpotential of theform W ( S, T ) = Ω( S ) /η ( T ) , which is one of the simplest examples whose extrema arestudied in section 4 and Appendix B.As we advanced, these potentials diverge exponentially for large Im T, Im U . Theorigin of this divergence can be traced to contributions from infinite towers of statesbecoming light, as expected from the swampland distance conjecture. In particular,consider the mass M in 3.3 for fixed complex structures: we see that at large Im T , thesector with vanishing winding numbers n , = 0 leads to an infinite tower of KK stateswith M (cid:39) (Im T ) − . As is familiar [3, 9–15, 18], the mass decreases exponentiallywith the distance, if phrased in the proper frame. By SL (2 , Z ) T , the opposite regimeof Im T → , = 0. A similar behavior can be recovered at the points at infinite distance incomplex structure moduli space: As Im U → ∞ (and fixed T ), there is an infinitetower of states with m = n = 0 becoming light. In any of these limits, the physicalinterpretation is that the towers of light particles modify the gauge theory dynamicsby increasing the effective scale of the gaugino condensate, hence leading to a potentialgrowing at infinity.The description in terms of the heterotic degrees of freedom certainly resonateswith the swampland distance conjecture. In the following section, we consider a dualtype I’ / Horava-Witten description, which allows for a more precise identification ofthe degrees of freedom controlling the dynamics of the T and U moduli independently. In the above heterotic description, the piece of moduli space under discussion hasan SL (2 , Z ) T × SL (2 , Z ) U modular group. However, the relevant heterotic degrees offreedom correspond to momentum and winding states in T , whose mass formula 3.3depends on both the T and U moduli of T ; they are hence not the natural objectsto disentangle the two independent SL (2 , Z ) modular properties, or the behavior atindependent infinite distance points. In this section we provide a picture in terms ofa dual type I’ or Horava-Witten description, which explicitly displays the degrees offreedom controlling the dynamics of the T and the U moduli independently.In order to do that, we return to the description of moduli dependent threshold cor-rections in N = 2 subsectors in terms of a compactification of heterotic on of K3 × T .Again, eventual reduction to N = 1 can be obtained by additional orbifold actions. Inthis section we would like to focus on the structure of heterotic on K3 × T , and exploitthe rich web of string dualities to understand the one-loop threshold correction.The compactification can be constructed as a 6d N = 1 compactification on K3,followed by a T compactification to 4d N = 2. As is well known, in the E × E heterotic, these are determined by the distribution of instanton numbers for the gaugebackgrounds on the two E factors, (12 + n, − n ) [54], while for the SO (32) heterotic,it corresponds to n = 4 [56, 57], see [58] for a review. We are thus left with a largeclass of 6d models which should subsequently be compactified on T , possibly withWilson lines; clearly, this description is disadvantageous since the modular propertiesin the T compactification are obscured by the complications of the previous stage of123 compactification.It is thus natural to regard the configuration as first a T compactification downto 8d, eventually followed by a K3 compactification to 4d N = 2. In this descrip-tion, the modular properties should be manifest already at the level of the toroidal 8dtheory. Indeed, the 8d theory contains a precursor of the corrections to the 4d gaugecouplings corrections, corresponding to corrections to the 8d couplings tr F , (tr F ) and (tr F )(tr R ) (there are in addition tr R and (tr R ) corrections, not involvingthe gauge groups, and which we will hence skip). As we show below, the coefficientsof such couplings are modular functions of the heterotic K¨ahler modulus T (as well asof its complex structure modulus U ). Upon dimensional reduction on K3, includingthe corresponding gauge and gravitational instanton backgrounds, one recovers the 4d N = 2 corrections to the gauge couplings, thus with the precisely (combinations of)the same modular functions.The structure of the quartic corrections to the 8d theory, for diverse choices ofWilson lines, has been studied from different dual descriptions. For instance, in [59–66],either directly as a perturbative computation in heterotic string, or from F-theory onK3. The latter is closely related (in a perturbative limit) to the orientifold of typeIIB on T with 4 O7-planes and 8 D7-branes on top (as counted in the double cover).This corresponds to a heterotic model with Wilson lines breaking the gauge group to SO (8) , as follows. Starting from the heterotic, one performs an S-duality to a type I T compactification with Wilson line, and two T-dualities to the type IIB orientifold.The original heterotic T modulus maps to the type IIB complexified coupling τ ,and the heterotic U modulus remains the complex structure modulus of the type IIB T . Hence, the SL (2 , Z ) modular group corresponds to S-duality on the type IIBcomplex coupling τ , and the modular group of the orientifolded T . In the typeIIB picture, the corrections were computed in [67]; here the τ -dependent piece of thequartic corrections arise from D( − U -dependent partarises from perturbative fundamental string corrections. Hence, this picture succeedsin disentangling the two modular groups in terms of two kinds of instanton effects.This is very reminiscent of the Swampland Distance Conjecture, since moving towardspoints at infinite distance in a modulus has an effect in a tower of charge states, albeitin this case they correspond to D( − S compact-13fication of type I’ theory . Namely, we consider type IIA on and S / Z orientifoldwith two O8-planes, each with 16 D8-branes (as counted in the double cover). This 9dtheory is then compactified on an additional S to 8d, e.g. with suitable Wilson lines ifwe wish to relate to the SO (8) theory. In fact, the discussion can be easily extendedto general Wilson lines, as we will do later on.In this type I’ picture, the threshold correction is associated to the tower of particlesin the 9d type I’ theory, concretely the corrections depending on τ arise from D0-brane particles bound to each of the two O8/D8 configurations. As discussed in [70]the spectrum of such BPS particles is given by bound states of 2 k D0-branes, givingparticles in the of each SO (16), and bound states of (2 k + 1) D0-branes, givingparticles in the spinor representation of SO (16). The 8d correction arising asa 1-loop diagram of these particles, in which they are allowed to run along the S bringing us down from 9d to 8d. Similarly, the corrections depending on U arise from9d particles arising from open string stretching among the D8-branes, and thus havingnon-trivial winding in the type I’ S Z . These are 9d particles with winding w ∈ Z and transforming in the ( , ) of SO (16) , or with winding 2 w and transforming inthe ( , ) + ( , ).In this picture, the modular behaviour in τ is not manifest. Indeed, since τ is thetype IIB complex coupling, the corresponding S-duality group is manifest only if welift up to M-theory. The lift of type I’ theory corresponds to Horava-Witten (HW)theory, in the particular choice of wilson lines breaking the E × E symmetry on theboundaries down to SO (16) . In this picture the D0-branes lift to KK momentummodes of the 10d E gauge multiplets, which propagate on a T , given by one circleassociated to the lift from type I’ to 11d, and the second circle being that alreadypresent in the 9 d → d compactification of type I’. As in standard M-theory/type IIBduality, the complex structure τ of this T maps to the type IIB complex coupling,and there is a geometric interpretation for the SL (2 , Z ). Similarly, the U -dependedcontribution in type IIB theory arises from M2-branes wrapped on the S / Z × T geometry.The dual pictures and relevant ingredients are shown in Table 1. As will become clear later on, this T-duality is similar to those relating the Weak Gravity Con-jecture for charged particles and instantons. (cid:82) T B + i R R K¨ahler T-duality Mix of R /R e iθ Complex structure wrapped F1 / KKType I (cid:82) T C + i R R /g s K¨ahler T-duality Wrapped D1 R /R e iθ Complex structure KKType IIB C + i /g s S-duality D( − R /R e iθ Complex structure Wrapped F1Type I’ (cid:82) S C + i R /g s Hidden HW D0orientifold (cid:82) S × I B + i R R K¨ahler T-duality F1 windingHorava-Witten R /R e iθ Complex structure KKorientifold (cid:82) T × I C + iR R R K¨ahler T-duality wrapped M2Table 1: Dual pictures and some of their ingredients. Despite similar notation forthe compactification radii, they in general have meaning adapted to the correspondingpicture.
Thus, HW theory on T , possibly with Wilson lines, is the natural framework to under-stand the modular properties of the threshold corrections. To flesh out the discussion,in the following we illustrate the computation of the τ -dependent threshold correctionfor the configuration with an unbroken E in 8d. Cases with more general Wilson linescan be discussed similarly. We follow closely the steps in [68], albeit for the E case.The 1-loop computation of the E HW vector multiplets, described from the per-spective of the worldline, reads A gauge = 14! (cid:90) ∞ dtt t (cid:88) (cid:96) I (cid:90) d p e − πt ( p + G IJ (cid:101) (cid:96) I (cid:101) (cid:96) J )= 14! (cid:90) ∞ dtt (cid:88) (cid:96) ,(cid:96) e − πt V (2) τ | (cid:101) (cid:96) − τ (cid:101) (cid:96) | (3.8)Here (cid:101) (cid:96) I = (cid:96) I − Λ · A I , with Λ a root vector of E and A I the Wilson line in the I th direction. We focus on the case without Wilson lines, and drop the tildes for themomenta such that (cid:101) (cid:96) = (cid:96) ∈ Z , (cid:101) (cid:96) = (cid:96) ∈ Z . The above expression for vanishingWilson lines is manifestly modular invariant under transformations of T . It is alsostraightforward to analyze the surviving subgroups in the presence of non-trivial Wilsonlines, like the SO (8) or SO (16) points. Moreover, for general Wilson lines, there are15ombined identifications due to modular transformations of T accompanied by non-trivial actions on the Wilson line moduli. This is just the HW description of thedualities in the heterotic O (2 , Z ) \ O (2 , R ) / [ O (2; R ) × O (18 , R )]In the above expression, the t in the integral is associated to the four external legsin the tr F term. Note that the same result is obtained by taking two external legsi.e. tr F in the 4d theory.In the following we reabsorb V (2) in a rescaling of t . A useful way to eventuallyextract the dominant contribution at infinity in τ is to perform a Poisson resummationto get A gauge = √ π (cid:90) ∞ dtt t − / (cid:88) w ,(cid:96) ∈ Z e − π w t − τ (cid:96) t e πiw (cid:96) τ . (3.9)In the 10d picture upon reduction on the direction 11, the integer w represents thewinding of the 9d objects (eventually D0-branes) running in the loop in the S bringingus down from 9d to 8d.Isolating the w = 0 contribution, and performing a Poisson resummation in (cid:96) and integrating in t , we have A gauge ( w =0) = τ π (cid:88) w (cid:54) =0 w = πτ
72 (3.10)where we have exclude the divergent w = 0 term (discussed below).As will be clear later on, this contribution extracts the dominant contribution inthe large τ limit of an infinite tower labeled by (cid:96) , of D0-branes in type I’ language.With hindsight we write A gauge ( w =0) = − (cid:18) πiτ − πiτ (cid:19) = −
112 (log q + log q ) (3.11)Incidentally, this zero-winding contribution from 11d KK modes is similar to a contribu-tion from D0-branes in the Gopalumar-Vafa interpretation of the topological string [71].The w (cid:54) = 0 contribution, after integration in t reads A gauge ( w (cid:54) =0) = 14! (cid:88) w (cid:54) =0 (cid:96) ∈ Z | w | e − πτ | w (cid:96) | e πiτ w (cid:96) . (3.12)Introducing q = e πiτ , and gathering the contributions from D0’s ( (cid:96) >
0) and anti-D0-branes ( (cid:96) < A gauge ( w (cid:54) =0) = 24! (cid:32) (cid:88) w > (cid:96) > w q w (cid:96) + cc. (cid:33) (3.13)16ere have substracted the (cid:96) = 0 piece, which corresponds to no D0-branes, andcorresponds to a perturbative piece mentioned later on. The above expression can berecast as A gauge ( w (cid:54) =0) = − (cid:88) (cid:96) > log(1 − q (cid:96) ) + cc. = −
112 log (cid:34) (cid:89) (cid:96) > (1 − q (cid:96) ) (cid:35) + cc. (3.14)So putting together with (3.11) we have A gauge = −
16 log | η ( τ ) | = −
124 log | η ( τ ) | (3.15)Here the exponent is chosen by adjusting the coefficient, but it can be checked that itmatches with the log(Im τ ) piece (arising from the dropped (cid:96) = 0 part) to achieve amodular invariant quantity, as expected from the original manifestly modular invariantamplitude in the HW picture. Hence we getlog [ (Im τ ) | η ( τ ) | ] (3.16)As mentioned before, the dominant term at large τ corresponds to the zero winding w = 0 contribution of the whole tower of D0-branes (or HW KK modes in the 11direction). It is thus these states that dominate in the point at infinity in the modulispace of τ (namely, T in the heterotic description).A very similar computation involving fundamental string winding states in typeI’ theory (i.e. wrapped M2-brane states in HW) can be shown to produce the U -dependent threshold corrections. These were discussed in [68] in the type I’ picturefor the SO (16) and SO (8) theories, to which we refer the reader for details. Here itsuffices to note that, from the perspective of the 9d theory, the computation is identicalto the previous one, as it simply relies the existence of one 9d BPS state for each integercharge k ∈ Z ; in the present case, the relevant 9d states are M2-branes with wrappingnumber k on the two-dimensional space defined by the HW interval times the S inthe lift type I’ → HW. This multiplicity of bound states implies the resultlog [ ( Im U ) | η ( U ) | ] (3.17)Let us conclude with a final remark bringing us back to the original discussion of het-erotic in K3 × T . The above computation produces the coefficient of the (tr F ) termin 8d (recall that E has no quartic Casimir). One can similarly get the (tr F )(tr R )term, resulting in a quartic correction of the formlog [ (Im τ ) | η ( τ ) | (Im U ) | η ( U ) | ] ( tr F − tr R ) tr F (3.18)17pon compactification on K3, with a gauge instanton background with instanton num-ber (12+ n ), and taking into account the Euler characteristic of K3, we have a thresholdcorrection to the 4d gauge coupling constants given bylog [ (Im τ ) | η ( τ ) | (Im U ) | η ( U ) | ] ( n −
12) tr F (3.19)which, as announced at the beginning, is controlled by the beta function of the 4d N = 2, equivalently the coefficient of the anomaly polynomial of the 6d N = 1 theoryobtained from just the K3 compactification.An equivalent way to understand this picture is to indeed consider first the com-pactification on K3 to 6d. In this compactification, the tower of D0-branes in theadjoint of E will produce a tower of 6d (massive) D0-brane states, in diverse rep-resentations of the surviving 6d gauge group. The ground states in the dimensionalreduction in K3 are determined by the index theorem on K3 for states coupled to thegravitational and gauge backgrounds. These massive 6d states are subsequently com-pactified on T , and produce, via a one-loop diagram whose computation is identicalto the above one, the threshold corrections to the 4d gauge couplings. The matchingwith the above description is precisely due to the familiar fact that the index theoremrelates the multiplicities of 6d states to the integral of tr F − tr R . Let us conclude by mentioning that the type I’ or HW picture also includes the sourceof the non-perturbative effects, which in the 4d N = 1 context produces the non-perturbative superpotential. The relevant object is a 6d BPS particle given by aD4-brane wrapped on K3, and which runs in a loop in T . In the 4d N = 2 con-text this gives rise to a tower of spacetime instantons, with (in their Higgs branch)correspond to gauge instantons on the D8-branes. In the 4d N = 1 setup, these in-stanton would have too many fermion zero modes to contribute to the superpotential,and only a few (fractional) instantons provide non-trivial contribution, thanks to theextra orbifold projections. Thus the 1-loop diagram picture is not so useful. In anyevent, it is interesting to point out that this description shows that the effect of thethreshold corrections on the non-perturbative superpotential is nothing but the sum ofthe contributions from polyinstantons like those introduced in [72], namely the effectof instantons from D0-brane loops on the action of an instanton from a D4-brane loop.In the HW lift, we have just an instanton effect from an M5-brane on K3 times the11d circle, with arbitrary momentum excitations on the latter. This configuration is18imilar to that in [73], but with the additional difficulty that the M5-brane is boundto the E boundary, which thus renders the quantitative description beyond presentknowledge. In the previous sections we have seen how non-perturbative effects are able to generate N = 1 superpotentials which are modular forms under SL (2 , Z ), keeping invariant themoduli scalar potential. In this section we explore in some detail the extrema of themost general class of modular invariant potentials for a single modulus T . These ofcourse are simplified models, since a fully realistic string compactification is expectedto involve multiple moduli, gauge bosons and matter fields. So at best, these mod-els could arise in some particular compactification after the rest of the spectrum hasbeen integrated out, or could perhaps represent some subsector of a more complicatedcompactification. On the other hand one could ignore its possible string theory originand consider them by themselves as explicit examples of N = 1 supergravity modelswith one chiral field and admitting a consistent coupling to quantum gravity. The ideahere is to test in specific models the connection between duality symmetries in stringcompactifications and the swampland ideas. Thus e.g. one may ask whether the prop-erties of modular invariance imply or are related to any of the proposed swamplandconstraints.Let us consider then the N = 1 supergravity K¨ahler potential G ( T, T ∗ ) = − T − T ∗ ) + log | W | , (4.1)with a general superpotential W = H ( T ) /η ( T ) , with H ( T ) given by eq.(2.5). Thecorresponding potential is given by V ( T, T ∗ ) = 18 T I | η | (cid:40) T I (cid:12)(cid:12)(cid:12)(cid:12) dHdT + 32 π H ˆ G (cid:12)(cid:12)(cid:12)(cid:12) − | H | (cid:41) . (4.2)Here we have renamed the real and imaginary parts of the field as: T = θ + it = T R + iT I .The potential is invariant under P SL (2 , Z ) with the fundamental region in the complexplane shown in fig.(1). One can show then that the points T = i and T = ρ = e i π are19lways extrema, since they are fixed points under order finite order subgroups whichact non-trivially on derivatives of the potential, which must thus vanish. In [42] itwas also conjectured that all extrema lie on the boundary of the fundamental region.We have explored the potential numerically and we find that this is indeed the case,although we are not aware of a general proof.We have performed a general study of minima for the different models obtainedfor different m, n ≥ P in eq.(2.5). The summary of our results isthat we find Minkowski and AdS extrema with and without SUSY. We also find dSmaxima and saddle points, but we have been unable to find dS minima. It is naturalto believe that this structure is a consequence of the modular invariance (duality in astring setting) of the models.Although we do not have a general proof for the absence of dS minima, one canexplicitly prove that the extrema at the self-dual points or zeros of P are never dSminima. This may be proved for any value of n, m and choice of polynomial P ( j ( T )).The proof borrows some results from [42]. First we study, for each choice of H , whichtype of extrema is generated at these points. Once we have found a set of parametersgenerating a minimum, we show that, for these parameters, it can never be in dS. Letus see how his comes about, considering different values of n, m in turn. - ��� - ��� - ��� ��� ��� ��� ��������������� Figure 1:
Fundamental domain for
P SL (2 , Z ). The self-dual points are located at T = i, ρ . Otherextrema are found on the border of the modular domain. • n = 0 or m = 0At T = ρ one has ˆ G = djdT = 0 so that for n = 0 one has dHdT = 0 and V ( T, T ∗ ) = |P (0) | T I | η | {− } . (4.3)20t is always a maximum and it is always in AdS.At T = i one has j = 1728, ˆ G = djdT = 0, | H | = 1728 m/ |P (1728) | so for m = 0 dHdT = 0 and one gets V ( T, T ∗ ) = |P (1728) | T I | η | {− } (4.4)By choosing different P all types of extrema can occur at T = i : maximum,minimum or saddle point. Namely, it is a maximum if − . < H (cid:48)(cid:48)(cid:48) H (cid:48) < − .
57, asaddle point if H (cid:48)(cid:48)(cid:48) H (cid:48) < − .
57 or H (cid:48)(cid:48)(cid:48) H (cid:48) > − .
57 and a minimum if (cid:12)(cid:12) H (cid:48)(cid:48) H + 1 . (cid:12)(cid:12) > .Again it is always in AdS. In Fig. 4 we consider the particular case n = m = 0with P = 1. We can see the maximum at T = ρ and a saddle point at T = i .Additionally, there is a non-SUSY AdS minimum at Im T = 1 .
2, in the boundaryof the fundamental region, but not at a fixed point. This is the simplest examplesuperpotential discussed above with W = 1 /η ( T ) .Figure 2: Scalar potential for W = 1 /η ( T ) (i.e. n = m = 0, P = 1). There is a SUSY AdSmaximum at T = ρ , a saddle point at T = i and a non-SUSY AdS minimum at Im T = 1 .
2, which isalso on the boundary of the fundamental domain. • n > m > n > T = ρ is always a minimum,and the same applies for m > T = i . If n > H ( ρ ) = 0 and if m > H (1) = 0. In both points ˆ G = djdT = 0. If n > T = ρ ∂H∂j doesnot diverge, so V = 0. Thus indeed it is always a Minkowski minimum, and thesame happens for m > T = i . 21 n = 1 or m = 1Some of these give rise to dS maxima, but no minima. If n = 1 then, at T = ρ : H = ˆ G = 0. However, ∂H∂j diverges in such a way that dHdT is finite and non-zero.Then the potential, at both fixed points, simplifies to: V = 18 T I | η | (cid:40) T I (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) n j − m/ j − n + m j − − m j n/ (cid:17) P ( j ) djdT (cid:12)(cid:12)(cid:12)(cid:12) (cid:41) (4.5)For n = 1, at T = ρ one obtains V = 2 |P (0) | T I | η | (cid:8) | C | (cid:9) , (4.6)where we have defined C = lim T → ρ j − djdT . This is always a maxima and thepotential is positive. In Fig 4.1 we plot the potential for n = 1, m = 0, P = 1.We can see the predicted dS maximum at T = ρ in the zoom in around thispoint. In the same plot, at T = i there is an AdS minimum, since m = 0.Moving on to m = 1, we evaluate the potential at T = i : V = 2 |P (1728) | T I | η | (cid:8) | D | (cid:9) , (4.7)where we have defined D = lim T → i ( j − − djdT . The potential is alwayspositive at the extrema. By changing H we can make it a saddle point or amaximum but not a minimum.The remaining extrema are the multiple zeros of P ( j ). They verify H = dHdT = 0,so they preserve supersymmetry and are always Minkowski. We summarize the resultsfor minima at the self-dual points in the tables (2) and (3). Note that in the modelconsidered supersymmetry is spontaneously broken if at the minimum of V the auxiliaryfield h T ∝ | H | (cid:18) H dHdT + 32 π ˆ G (cid:19) → (cid:18) dHdT (cid:19) (4.8)has a non-zero vacuum expectation value. In the last step we specified to the self-dualpoints, where ˆ G = 0.This completes the study of extrema on self-dual points, which are generic.22igure 3: Left: Scalar potential for W = j / /η (i.e. n = 1, m = 0, P = 1): Right: A zoom aroundits dS maximum. V ( T = 1) Type of Extrema H dHdT SUSY m > V = 0 Min 0 0 Yes m = 1 T I | η | (cid:8) | a | | C | (cid:9) > − . < H (cid:48)(cid:48)(cid:48) H (cid:48) < − .
57 0 (cid:54) = 0 NoSP H (cid:48)(cid:48)(cid:48) H (cid:48) < − .
57 or H (cid:48)(cid:48)(cid:48) H (cid:48) > − . m = 0 ∝ | P (0) | T I | η | {− } < (cid:12)(cid:12) H (cid:48)(cid:48) H + 1 . (cid:12)(cid:12) > (cid:54) = 0 0 YesMax − < H (cid:48)(cid:48) H + 1 . < SP (Saddle Point) if elseTable 2: Classification of the extrema found at T = i . V ( T = ρ ) Type of Extrema H dHdT SUSY n > V = 0 Minimum 0 0 Yes n = 1 | η | (cid:8) |P (1728) | | D | (cid:9) > (cid:54) = 0 No n = 0 ∝ m |P (1728) | T I | η | {− } < (cid:54) = 0 0 YesTable 3: Classification of the extrema found at T = ρ .It is quite interesting that modular functions conspire to produce all kinds of ex-trema except dS minima. One may wonder why, given the freedom in H and P we arestill unable to uplift one of the Minkowski vacua that we have found to dS. One wouldnaively say that, starting from a Minkowski vacuum in a self-dual point and addingsome perturbation there should be some nearby dS minima, not only AdS. This is in23act not the case, and adding (modular invariant) perturbations one never lands in dS.The whole potential is constructed around the Klein Invariant function j ( T ) and theDedekind function η ( T ), as a consequence of modular invariance. The dependence on η is fixed and we have only some freedom in choosing the dependence on j . It is easyto see why using only this freedom the Minkowski or AdS minimum cannot be “per-turbed” to a dS minima. If j or ( j − H are never dS minima. We have already seen that modular invariant potentials have features consistent withthe swampland ideas and how such potentials appear in models of reduced supersym-metry at the non-perturbative level. In particular we have seen how for large values ofthe moduli, modular invariance dictates the existence of a dynamical barrier censoringlarge moduli which would allow for global symmetries. The origin of such a barrier inthe specific string models studied correspond to towers of states becoming massless, inagreement with the distance conjecture ideas. The towers of states are KK, winding, D |∇ V | V = (cid:113) K ij ∂ i V ∂ j VV (4.9)as in Fig. 4 in which we plot an example with a dS maximum. This quantity shouldbe either negative or larger than a certain constant c . Since there is a dS maximumat T = ρ , it will not be true at this point for any c and for other points after fixing c .According to [7] for those points we have to make sure that the smallest eigenvalue ofthe Hessian satisfies: min ( ∇ i ∇ j V ) ≤ − c (cid:48) V (4.10)24igure 4: |∇ V | V for W = j / /η . The ratio is always bigger than one and grows linearly with Im T . In all examples analyzed with a dS maximum, the third condition on the Hessian issatisfied. One can also check whether the dS condition is obeyed for large moduli, awayfrom any dS maxima, which in fact can be studied analytically. Taking j → e − i πT ,ˆ G ( T ) → π and H → e π ( m + n + r ) T P (cid:0) e − i πT (cid:1) in Eq. (2.20) of [42] we find: |∇ V | V = 2 √ T I π (cid:18) m + 4 n + 2 r + 36 (cid:19) , (4.11)where r is the highest exponent of the polynomial P . This quantity is always positiveand it goes to infinity for large T , so it is guaranteed that in this limit the dS conjectureholds. Again, we resorted to numerical methods to check that it holds everywhere inall examples.It is interesting to note that this is an specific example of a large class of scalarpotentials which obey the refined swampland conjecture but do not correspond toa simple monotonous decreasing potential, as considered up to now, and leading toquintessence type of potentials.We have not much to say about the AdS conjecture of ref. [4] in these one-modulusmodels. This conjecture states that there are no non-SUSY, AdS stable vacua in atheory consistent with quantum gravity. In the class of models described above allAdS vacua in the self-dual points are SUSY. There are non-SUSY AdS vacua only inother points at the boundary of the fundamental region, as in the simple example with n = m = 0, P = 1 depicted in fig.(2). Such vacua should be unstable if the conjecture25s correct, perhaps decaying by a mechanism analogous to that of bubbles of nothing,as in [74]. It would be interesting to check if such an instability exists. The above single-modulus class of potentials is already very rich. However genericstring compactifications yield effective field theories which contain typically multiplemoduli, gauge groups and charged particles. In the end the total vacuum energydepends on multiple contributions and a subsector which by itself would yield e.g. anAdS vacuum, may be perhaps overwhelmed by other sectors of the theory, yieldingpositive energy and a dS minima. This is something that we cannot predict on thebasis of the potential of a subsector of the theory.In this sense, it is interesting to check whether the results found above for a singlemodulus is very much modified by the addition of an extra chiral multiplets S . A simpleexample is provided by the heterotic gaugino condensation superpotential discussed insection 3. So, inspired by the gaugino condensate model one can consider an N = 1supergravity model G ( S, S ∗ , T, T ∗ ) = − log ( S − S ∗ ) − T − T ∗ ) + log | W ( S, T ) | (4.12)with W ( S, T ) = Ω ( S ) H ( T ) η ( T ) . (4.13)The resulting scalar potential is then given by: V ( T, T ∗ , S, S ∗ ) = (4.14)= 116 S I T I | η | (cid:26) | iS I Ω S − Ω | | H | + 4 T I (cid:12)(cid:12)(cid:12)(cid:12) dHdT + 32 π H ˆ G (cid:12)(cid:12)(cid:12)(cid:12) | Ω | − | H | | Ω | (cid:27) . We denote derivatives with respect to the fields with a sub-index T or S . In AppendixB we study the extrema of this general potential in some detail. For general Ω thereare essentially two classes of vacua, depending on whether the auxiliary field of S , F S vanishes or not at the minimum. If it vanishes, the structure of vacua are identical to thesingle modulus case discussed above and the same conclusions apply. For F S (cid:54) = 0 thereare new possibilities which are summarized in the tables in Appendix B. Essentiallythe same classes of minima as in the single modulus case are obtained and again no dSminima are found. 26 Conclusions and outlook
In this paper we have put forward the use of the modular symmetries in the effec-tive action for moduli to test swampland conjectures in regimes beyond weak couplingexpansion points. On one hand, we have constructed explicit string theory compact-ifications in which such modular symmetries are ingrained in the UV construction,and derived how several swampland constraints arise from the dynamics of e.g. non-perturbative superpotentials in models with 4d N = 1 supersymmetry. On the otherhand, we have discussed fairly general effective theories for scalars enjoying dualitymodular symmetry and explored the extent to which the latter imply satisfying theswampland constraints.The string constructions we have considered are explicit toroidal orbifolds of theheterotic string, for which moduli in N = 2 subsectors from fixed tori have non-trivial dynamics due to their appearance in threshold corrections of 4d gauge kineticfunctions. In particular, we have described examples in which the access to pointsat infinity in moduli space, where a global shift symmetry for the axion would berestored, is dynamically forbidden by exponential growth of the (non-perturbative)scalar potential. This potential arises from infinite towers of states becoming light,so the absence of global symmetries is interestingly linked to the swampland distanceconjecture in a very explicit and novel mechanism. This new mechanism may well bethe counterpart, in the context of lower (or no) supersymmetry, of the mechanismsdiscussed in the literature in cases with 8 supercharges.We have also considered several dual pictures, in particular type I’ / Horava-Wittentheory compactifications, in which the degrees of freedom associated to complex struc-ture or K¨ahler moduli can be studied in isolation. Such states are key to the derivationof the swampland distance conjecture in the present context. We expect that, given theubiquitous appearance of modular groups in CY moduli spaces, and of string dualitiesalong the lines discussed, the strategy of relating modular properties with consistencywith quantum gravity may apply in a far more general context, which we hope toexplore in the future.We have performed a careful numerical analysis of the properties of scalar poten-tials in general effective theories for scalars, with essentially the only constraint thatthey enjoy invariance under modular symmetries. Surprisingly, this requirement aloneseems to suffice to render the theories consistent with a number of swampland conjec-ture, besides the already mentioned distance conjectures. The potentials have extremathat correspond to (supersymmetric) Minkowski and AdS minima, and dS maxima or27addle points, but no dS minima. Moreover the potentials satisfy the refined de Sitterconjecture conditions. These remarkable facts suggest the tantalizing proposal thatmodular properties are deeply ingrained in the conditions to guarantee consistency ofeffective theories with quantum gravity. This is certainly a promising research direc-tion to pursue. Let us however insist that we do not claim that the fact that we havebeen unable to find any dS minimum should apply to any consistent theory of quantumgravity. Only that in the simplest classes of modular invariant effective actions we havebeen unable to find any dS minimum.A final remark concerns the fact that, naively, modular symmetries are manifestin the effective actions of compactifications in which further ingredients are absent.For instance, introduction of fluxes typically induces scalar potentials which are notinvariant under the modular transformations, thus seemingly breaking the symmetryexplicitly at the level of the effective action. In fact, this interpretation is far fromcomplete, and there is a precise sense in which the symmetry is present. The situationis analogous to the statement that e.g. in heterotic compactifications on K3 × T , thepresence of Wilson lines on T would seem to break the SL (2 , Z ) symmetries, since theWilson lines are not invariant. In fact, the correct interpretation is that the dualitysymmetries act non-trivially on the Wilson lines and define a larger duality group SO (4 , Z ) acting on and enlarged moduli space which includes the Wilson lines.Similarly, in flux compactifications, the duality groups act consistently on the set ofall flux vacua by acting non-trivially on both the moduli and the fluxes. Restrictingto axion scalars, this is just a re-statement that the flux potentials have an axionmonodromy structure. The extension to the full duality group brings in a much richerset of fluxes, necessarily including non-geometric ones. We expect that the constraintsof invariance under modular transformations of moduli in the landscape of flux vacuabring new insights into both the structure of the landscape and the set of conditionsdiscriminating it from the swampland. Acknowledgments
We thank A. Font, A. Herr´aez, F. Marchesano, M. Montero and I. Valenzuela foruseful discussions. This work has been supported by the ERC Advanced Grant SPLEunder contract ERC-2012-ADG-20120216-320421, by the grants FPA2016-78645-P andFPA2015-65480-P from the MINECO and the grant SEV-2016-0597 of the “Centro de28xcelencia Severo Ochoa” Programme. The work of E. Gonzalo is supported by a FPUfellowship number FPU16/03985. 29 SL (2 , Z ) modular forms Here we just list some well known properties for the modular forms discussed in themain text. We follow closely the appendix in [42]. The modular group is generated bythe transformations T −→ aT + bcT + d , a, b, c, d ∈ Z , ad − bc = 1 , (A.1)with T = θ + it . This is is generated by a shift transformation T → T + 1 and T → − /T which relates small and large t . Since changing the sign of all the parametersleaves invariant the transformation, the group is actually P SL (2 , Z ) = SL (2 , Z ) / Z .A meromorphic function F ( T ) is said to have modular weight r if F (cid:18) aT + bcT + d (cid:19) = ( cT + d ) r F ( T ) . (A.2)The symmetry divides the complex plane in equivalent regions and the conventionalfundamental domain D is shown in fig.(1). The group P SL (2 , Z ) has self-dual pointsat T = i, ρ with ρ = e i π/ , as well as at infinity. A modular form admits a Laurentexpansion at each point in the interior of the closed domain D . The lowest order ofthe expansion of F in p is denoted ν p . Around the self-dual points F has an expansionin terms of uniformizing variables. In particular, as T → ∞ one expands F ( T ) = q ν ∞ ∞ (cid:88) n =0 a n q n , q = e i πT (A.3)and ν ∞ is the order of F at infinity. At T = i, ρ the uniformizing variables are t = (cid:18) ( T − i )( T + i ) (cid:19) , t ρ = (cid:18) ( T − ρ )( T + ρ ∗ ) (cid:19) . (A.4)In terms of them F ( T ) admits expansions around T = 1 , ρ of the form( T + i ) F ( T ) = t ν / ∞ (cid:88) n =0 b n t n , ( T + ρ ) F ( T ) = t ν ρ / ρ ∞ (cid:88) n =0 c n t n . (A.5)It may be proven that the orders are related to the modular weight r by the expression r
12 = ν ∞ + ν + ν ρ + (cid:88) p (cid:54) =1 ,ρ, ∞ ν p . (A.6)One interesting consequence of this formula is that for negative modular weight r < ν ∞ < F ( T ) for large Im T is implied.Of particular interest are the modular forms constructed from the Eisenstein series G k ( T ) = (cid:88) n ,n ∈ Z n T + n ) k . (A.7)For k > k . For k = 1 G rathertransforms as a connection G → ( cT + d ) G − πc ( cT + d ). Then the the non-holomorphic ˆ G modular formˆ G ( T, T ∗ ) = G ( T ) − π Im T (A.8)transforms as a modular form of weight 2. There is a cusp form of weight 12 given by∆( T ) = 675256 π (cid:2) G − G (cid:3) , (A.9)The Dedekind function is given by η ( T ) = ∆( T ) / . (A.10)One can check the identity, used in the main text dη ( T ) dT = − π η ( T ) G ( T ) . (A.11)The holomorphic Klein modular invariant form j ( T ) may be written in terms of thecusp form as j ( T ) = 91125 π G ( T ) ∆( T ) . (A.12)It may be shown that any holomorphic modular invariant form is a rational functionof j ( T ). Any modular form of weight r which is regular in D may be written as H ( T ) = ( j ( T ) − m/ j ( T ) n/ P ( j ( T )) (A.13)and equivalently as F ( T ) = η ( T ) r (cid:18) G ( T ) η ( T ) (cid:19) m (cid:18) G ( T ) η ( T ) (cid:19) n P ( j ( T )) , (A.14)with n, m positive integers and P a polynomial of j ( T ). Some useful expansions inpowers of q = e i πT are η ( T ) − = q − / (cid:0) q + 2 q + 3 q + ... (cid:1) (A.15) G ( T ) = π (cid:0) − q − q − q − ... (cid:1) (A.16) j ( T ) = 1 q + 744 + 196884 q + 21493760 q + ... . (A.17)The divergence of η ( T ) − at large t is at the origin of the superpotential divergencesfor large modulus. 31 General Analysis of Minima for two fields
In this Appendix we study the extrema of the potential in the two moduli modelinspired by gaugino condensation. Let us recall from [42] that the extrema from Section4 arise as a particular case. The second field provides a new condition for spontaneouslysusy breaking: h S ∝ | H | (2 iS I Ω S − Ω) (cid:54) = 0 (B.1)Using this equation we can distinguish two types of extrema. In Type B extrema SUSYis broken spontaneously by the minimization in S. In contrast, in Type A extrema,SUSY remains unbroken ( h S = 0) at this stage. The condition ∂V∂S = 0 for Type Aextrema reads: 2 iS I Ω S − Ω = 0 , (B.2)while for Type B: − S I Ω SS e − i | iS I Ω S − Ω | Ω ∗ = 2 − T I | H | (cid:12)(cid:12)(cid:12)(cid:12) dHdT + 32 π H ˆ G (cid:12)(cid:12)(cid:12)(cid:12) . (B.3)If the S field is in a Type A extrema, the dependence on S after its minimization istrivial, since the potential is the one we would find for a single modulus T multiplied bythe constant | Ω( S ) | S I . In this way, we recover the single modulus potential and the resultsof Section 4 which we discussed in the main text. Even tough in Type B extrema thedependence on Ω( S ) is not trivial anymore, we will see how the results can be easilyextended. In particular the proof that there are no dS minima in the self-dual pointsor in the zeros of P can also be done analytically, except for the n = 0 or m = 0 case,for which we rely on numerics. Now we go through the same steps of Section 4 toshow that if at the self-dual points there is a minimum then the potential cannot havepositive minima. n > or m > n > T = ρ H = djdT = 0, and the same happens at T = i for m >
1. Thethe potential is simply V = 0 in both extrema. Just like it happened for Type A, if n > m > T = ρ or T = i respectively are always minima. n = 1 or m = 1If n = 1 then dHdT (cid:54) = 0 so the condition for Type B extrema reduces to Ω = 0. Therefore V = 0 at both extrema. By changing H we can make it a saddle point or a maximumbut not a minimum. 32 = 0 or m = 0This case is the exception, for which analytically we have not been able to prove ourresult. At T = ρ : ˆ G = djdT = 0 so if n = 0 then dHdT = 0 and V ( T, T ∗ ) = |P (0) | S I T I | η | (cid:8) | iS I Ω S − Ω | − | Ω | (cid:9) (B.4)It is a minimum if | iS I Ω S − Ω | − | Ω | > | iS I Ω S − Ω | − | Ω | <
0. There is no simple way to decide thesign of the vacuum energy in this case. If we consider Ω as a sum of exponentialsnumerically we find that it is always negative. At T = i : j = 1728, ˆ G = djdT = 0, djdT = 0, | H | = 1728 m/ |P (1728) | so for m = 0 dHdT = 0 and V ( T, T ∗ ) = |P (1728) | S I T I | η | (cid:8) | iS I Ω S − Ω | − | Ω | (cid:9) (B.5)By choosing different P all types of extrema can occur at T = i : maximum, minimumor saddle point. Namely, it is a maximum if − . < H (cid:48)(cid:48)(cid:48) H (cid:48) < − .
57, a saddle point if H (cid:48)(cid:48)(cid:48) H (cid:48) < − .
57 or H (cid:48)(cid:48)(cid:48) H (cid:48) > − .
57 and a minimum if (cid:12)(cid:12) H (cid:48)(cid:48) H + 1 . (cid:12)(cid:12) > . Again we do notprove it analytically, but numerically we also find that V < H = 0 extrema are the multiple zeros of P ( j ). These are SUSYminima and are always in Minkowski. We summarize the results for Type B extremaat the self-dual points in the next two tables. V ( T = ρ ) Type of Extrema Susy n > V = 0 Minimum Yes n = 1 V = 0 Min | iS I Ω S − Ω | − | Ω | > | iS I Ω S − Ω | − | Ω | < n = 0 Numerically all min V < | iS I Ω S − Ω | − | Ω | > ∝ (cid:8) | iS I Ω S − Ω | − | Ω | (cid:9) Max | iS I Ω S − Ω | − | Ω | < T = ρ .33 ( T = i ) Type of Extrema Susy m > V = 0 Minimum Yes m = 1 V = 0 No simple criteria No m = 0 Numerically all min have V < ∝ (cid:8) | iS I Ω S − Ω | − | Ω | (cid:9) Table 5: Classification of Type B extrema at T = i .However, we have searched numerically for dS minima at other points on the bound-ary, supposing Ω is given by a sum of exponentials, for various choices of H and wehave not found any. 34 eferences [1] C. Vafa, “The String landscape and the swampland,” hep-th/0509212[2] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, “The String landscape, blackholes and gravity as the weakest force,” JHEP (2007) 060 [hep-th/0601001][3] H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and theSwampland,” Nucl. Phys. B , 21 (2007) [hep-th/0605264].[4] H. Ooguri and C. Vafa, “Non-supersymmetric AdS and the Swampland,” Adv.Theor. Math. Phys. (2017) 1787 [arXiv:1610.01533 [hep-th]].[5] G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, “De Sitter Space and the Swamp-land,” arXiv:1806.08362 [hep-th].[6] P. Agrawal, G. Obied, P. J. Steinhardt and C. Vafa, “On the Cosmological Impli-cations of the String Swampland,” Phys. Lett. B (2018) 271 [arXiv:1806.09718[hep-th]].[7] H. Ooguri, E. Palti, G. Shiu and C. Vafa, “Distance and de Sitter Conjectures onthe Swampland,” arXiv:1810.05506 [hep-th].[8] S. K. Garg and C. Krishnan, “Bounds on Slow Roll and the de Sitter Swampland,”arXiv:1807.05193 [hep-th]S. K. Garg, C. Krishnan and M. Zaid, “Bounds on Slow Roll at the Boundary ofthe Landscape,” arXiv:1810.09406 [hep-th].[9] T. D. Brennan, F. Carta and C. Vafa, “The String Landscape, the Swampland,and the Missing Corner,” arXiv:1711.00864 [hep-th].[10] T. W. Grimm, E. Palti and I. Valenzuela, “Infinite Distances in Field Space andMassless Towers of States,” JHEP (2018) 143 [arXiv:1802.08264 [hep-th]].[11] B. Heidenreich, M. Reece and T. Rudelius, “Emergence of Weak Coupling atLarge Distance in Quantum Gravity,” Phys. Rev. Lett. (2018) no.5, 051601[arXiv:1802.08698 [hep-th]].[12] S. J. Lee, W. Lerche and T. Weigand, “Tensionless Strings and the Weak GravityConjecture,” JHEP (2018) 164 [arXiv:1808.05958 [hep-th]].3513] S. J. Lee, W. Lerche and T. Weigand, “A Stringy Test of the Scalar Weak GravityConjecture,” arXiv:1810.05169 [hep-th].[14] A. Hebecker and T. Wrase, “The asymptotic dS Swampland Conjecture - a sim-plified derivation and a potential loophole,” arXiv:1810.08182 [hep-th].[15] E. Palti, “The Weak Gravity Conjecture and Scalar Fields,” JHEP (2017)034 [arXiv:1705.04328 [hep-th]].[16] D. Lust and E. Palti, “Scalar Fields, Hierarchical UV/IR Mixing and The WeakGravity Conjecture,” JHEP (2018) 040 [arXiv:1709.01790 [hep-th]].[17] F. Denef, A. Hebecker and T. Wrase, “de Sitter swampland conjecture and theHiggs potential,” Phys. Rev. D (2018) no.8, 086004 [arXiv:1807.06581 [hep-th]].[18] T. W. Grimm, C. Li and E. Palti, “Infinite Distance Networks in Field Space andCharge Orbits,” arXiv:1811.02571 [hep-th].[19] G. Dvali and C. Gomez, “On Exclusion of Positive Cosmological Constant,”arXiv:1806.10877 [hep-th].G. Dvali, C. Gomez and S. Zell, “Quantum Breaking Bound on de Sitter andSwampland,” arXiv:1810.11002 [hep-th]D. Andriot, “On the de Sitter swampland criterion,” Phys. Lett. B (2018)570 doi:10.1016/j.physletb.2018.09.022 [arXiv:1806.10999 [hep-th]]C. Roupec and T. Wrase, “de Sitter extrema and the swampland,” Fortsch. Phys. (2018) no.29, 1850178doi:10.1142/S0217751X18501786 [arXiv:1808.05040 [hep-th]]Y. Akrami, R. Kallosh, A. Linde and V. Vardanyan, “The landscape, theswampland and the era of precision cosmology,” Fortsch. Phys. (2015) no.09, 020 [arXiv:1503.00795 [hep-th]]M. Montero, A. M. Uranga and I. Valenzuela, “Transplanckian axions!?,” JHEP (2015) 032 [arXiv:1503.03886 [hep-th]]J. Brown, W. Cottrell, G. Shiu and P. Soler, “Fencing in the Swampland: Quan-tum Gravity Constraints on Large Field Inflation,” JHEP (2015) 023[arXiv:1503.04783 [hep-th]]J. Brown, W. Cottrell, G. Shiu and P. Soler, “On Axionic Field Ranges, Loopholesand the Weak Gravity Conjecture,” JHEP , 017 (2016) [arXiv:1504.00659[hep-th]]B. Heidenreich, M. Reece and T. Rudelius, “Weak Gravity Strongly ConstrainsLarge-Field Axion Inflation,” JHEP (2015) 108 [arXiv:1506.03447 [hep-th]]C. Cheung and G. N. Remmen, “Naturalness and the Weak Gravity Conjecture,”Phys. Rev. Lett. (2014) 051601 [arXiv:1402.2287 [hep-ph]]A. de la Fuente, P. Saraswat and R. Sundrum, “Natural Inflation and QuantumGravity,” Phys. Rev. Lett. (2015) no.15, 151303 [arXiv:1412.3457 [hep-th]]A. Hebecker, P. Mangat, F. Rompineve and L. T. Witkowski, “Winding out of theSwamp: Evading the Weak Gravity Conjecture with F-term Winding Inflation?,”Phys. Lett. B (2015) 455 [arXiv:1503.07912 [hep-th]]T. C. Bachlechner, C. Long and L. McAllister, “Planckian Axions and the WeakGravity Conjecture,” JHEP (2016) 091 [arXiv:1503.07853 [hep-th]]T. Rudelius, “On the Possibility of Large Axion Moduli Spaces,” JCAP (2015) no.04, 049 [arXiv:1409.5793 [hep-th]]D. Junghans, “Large-Field Inflation with Multiple Axions and the Weak GravityConjecture,” JHEP (2016) 128 [arXiv:1504.03566 [hep-th]]K. Kooner, S. Parameswaran and I. Zavala, “Warping the Weak Gravity Conjec-ture,” Phys. Lett. B , 402 (2016) [arXiv:1509.07049 [hep-th]]37. Harlow, “Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjec-ture,” JHEP , 122 (2016) [arXiv:1510.07911 [hep-th]]L. E. Ib´a˜nez, M. Montero, A. Uranga and I. Valenzuela, “Relaxion Monodromyand the Weak Gravity Conjecture,” JHEP (2016) 020 [arXiv:1512.00025[hep-th]]A. Hebecker, F. Rompineve and A. Westphal, “Axion Monodromy and the WeakGravity Conjecture,” JHEP (2016) 157 [arXiv:1512.03768 [hep-th]].[21] B. Heidenreich, M. Reece and T. Rudelius, “Evidence for a Lattice Weak GravityConjecture,” arXiv:1606.08437 [hep-th]M. Montero, G. Shiu and P. Soler, “The Weak Gravity Conjecture in three di-mensions,” arXiv:1606.08438 [hep-th]P. Saraswat, “The Weak Gravity Conjecture and Effective Field Theory,”arXiv:1608.06951 [hep-th]D. Klaewer and E. Palti, “Super-Planckian Spatial Field Variations and QuantumGravity,” arXiv:1610.00010 [hep-th]L. McAllister, P. Schwaller, G. Servant, J. Stout and A. Westphal, “Runaway Re-laxion Monodromy,” arXiv:1610.05320 [hep-th]A. Herr´aez and L. E. Ib´a˜nez, “An Axion-induced SM/MSSM Higgs Landscapeand the Weak Gravity Conjecture,” JHEP (2017) 109 [arXiv:1610.08836[hep-th]]M. Montero, “Are tiny gauge couplings out of the Swampland?,” [arXiv:1708.02249[hep-th]]L. E. Ib´a˜nez and M. Montero, “A Note on the WGC, Effective Field Theory andClockwork within String Theory,” JHEP (2018) 057 [arXiv:1709.02392 [hep-th]]G. Aldazabal and L. E. Ib´a˜nez ‘A Note on 4D Heterotic String Vacua, FI-termsand the Swampland,” Phys. Lett. B (2018) 375 [arXiv:1804.07322 [hep-th]].[22] C. Cheung, J. Liu and G. N. Remmen, “Proof of the Weak Gravity Conjecturefrom Black Hole Entropy,” arXiv:1801.08546 [hep-th]T. W. Grimm, E. Palti and I. Valenzuela, `‘Infinite Distances in Field Space andMassless Towers of States,” arXiv:1802.08264 [hep-th]B. Heidenreich, M. Reece and T. Rudelius, “Emergence and the Swampland Con-jectures,” arXiv:1802.08698 [hep-th]S. Andriolo, D. Junghans, T. Noumi and G. Shiu, “A Tower Weak Gravity Con-38ecture from Infrared Consistency,” arXiv:1802.04287 [hep-th]R. Blumenhagen, D. Klaewer, L. Schlechter and F. Wolf, “The Refined SwamplandDistance Conjecture in Calabi-Yau Moduli Spaces,” arXiv:1803.04989 [hep-th]A. Landete and G. Shiu, “Mass Hierarchies and Dynamical Field Range,”arXiv:1806.01874 [hep-th].[23] U. Danielsson and G. Dibitetto, “The fate of stringy AdS vacua and the WGC,”arXiv:1611.01395 [hep-th]B. Freivogel and M. Kleban, “Vacua Morghulis,” arXiv:1610.04564 [hep-th]T. Banks, “Note on a Paper by Ooguri and Vafa,” arXiv:1611.08953 [hep-th]H. Ooguri and L. Spodyneiko, “New Kaluza-Klein Instantons and Decay of AdSVacua,” arXiv:1703.03105 [hep-th].[24] L. E. Ib´a˜nez, V. Martin-Lozano and I. Valenzuela, “Constraining Neutrino Masses,the Cosmological Constant and BSM Physics from the Weak Gravity Conjecture,”JHEP (2017) 066 [arXiv:1706.05392 [hep-th]]E. Gonzalo, A. Herr´aez and L. E. Ib´a˜nez, “AdS-phobia, the WGC, the StandardModel and Supersymmetry,” JHEP (2018) 051 [arXiv:1803.08455 [hep-th]]E. Gonzalo and L. E. Ibez, “The Fundamental Need for a SM Higgs and the WeakGravity Conjecture,” Phys. Lett. B (2018) 272 [arXiv:1806.09647 [hep-th]].[25] Y. Hamada and G. Shiu, “Weak Gravity Conjecture, Multiple Point Principleand the Standard Model Landscape,” JHEP (2017) 043 [arXiv:1707.06326[hep-th]].[26] P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, “A Pair of Calabi-Yaumanifolds as an exactly soluble superconformal theory,” Nucl. Phys. B (1991)21 [AMS/IP Stud. Adv. Math. (1998) 31].[27] P. Candelas, A. Font, S. H. Katz and D. R. Morrison, “Mirror symmetry for twoparameter models. 2.,” Nucl. Phys. B (1994) 626 [hep-th/9403187].[28] B. Haghighat, H. Movasati and S. T. Yau, “Calabi-Yau modular forms in limit: El-liptic Fibrations,” Commun. Num. Theor. Phys. (2017) 879 [arXiv:1511.01310[math.AG]].[29] B. Haghighat, “Mirror Symmetry and Modularity,” arXiv:1712.00601 [hep-th].[30] E. Witten, “Nonperturbative superpotentials in string theory,” Nucl. Phys. B (1996) 343 [hep-th/9604030]. 3931] R. Donagi, A. Grassi and E. Witten, ‘A Nonperturbative superpotential with E(8)symmetry,” Mod. Phys. Lett. A (1996) 2199 [hep-th/9607091].[32] G. Curio and D. Lust, “A Class of N=1 dual string pairs and its modular super-potential,” Int. J. Mod. Phys. A (1997) 5847 [hep-th/9703007].[33] T. W. Grimm, “Non-Perturbative Corrections and Modularity in N=1 Type IIBCompactifications,” JHEP (2007) 004 [arXiv:0705.3253 [hep-th]].[34] L. B. Anderson, F. Apruzzi, X. Gao, J. Gray and S. J. Lee, “Instanton super-potentials, Calabi-Yau geometry, and fibrations,” Phys. Rev. D (2016) no.8,086001 [arXiv:1511.05188 [hep-th]].[35] J. P. Conlon and S. Krippendorf, “Axion decay constants away from the lamp-post,” JHEP (2016) 085 [arXiv:1601.00647 [hep-th]].[36] Y. Olguin-Trejo, S. L. Parameswaran, G. Tasinato and I. Zavala, “Run-away Quintessence, Out of the Swampland,” JCAP (2019) no.01, 031[arXiv:1810.08634 [hep-th]].[37] S. Ferrara, .D. Lust and S. Theisen, “Target Space Modular Invariance and Low-Energy Couplings in Orbifold Compactifications,” Phys. Lett. B (1989) 147.[38] S. Ferrara, D. Lust, A. D. Shapere and S. Theisen, “Modular Invariance in Super-symmetric Field Theories,” Phys. Lett. B (1989) 363.[39] A. Font, L. E. Ib´a˜nez, D. Lust and F. Quevedo, “Supersymmetry Breaking FromDuality Invariant Gaugino Condensation,” Phys. Lett. B (1990) 401.[40] S. Ferrara, N. Magnoli, T. R. Taylor and G. Veneziano, “Duality and supersym-metry breaking in string theory,” Phys. Lett. B (1990) 409.[41] H. P. Nilles and M. Olechowski, “Gaugino Condensation and Duality Invariance,”Phys. Lett. B (1990) 268.[42] M. Cvetic, A. Font, L. E. Ib´a˜nez, D. Lust and F. Quevedo, “Target space duality,supersymmetry breaking and the stability of classical string vacua,” Nucl. Phys.B (1991) 194.[43] A. Font, L. E. Ib´a˜nez, D. Lust and F. Quevedo, “Strong - weak coupling dualityand nonperturbative effects in string theory,” Phys. Lett. B (1990) 35.4044] J. H. Horne and G. W. Moore, “Chaotic coupling constants,” Nucl. Phys. B ,109 (1994) [hep-th/9403058].[45] L. J. Dixon, V. Kaplunovsky and J. Louis, “Moduli dependence of string loopcorrections to gauge coupling constants,” Nucl. Phys. B (1991) 649.[46] I. Antoniadis, K. S. Narain and T. R. Taylor, “Higher genus string corrections togauge couplings,” Phys. Lett. B (1991) 37.[47] I. Antoniadis, E. Gava and K. S. Narain, “Moduli corrections to gauge and gravi-tational couplings in four-dimensional superstrings,” Nucl. Phys. B (1992) 93[hep-th/9204030].[48] D. Lust and T. R. Taylor, “Hidden sectors with hidden matter,” Phys. Lett. B (1991) 335.[49] B. de Carlos, J. A. Casas and C. Munoz, “Supersymmetry breaking and determi-nation of the unification gauge coupling constant in string theories,” Nucl. Phys.B (1993) 623 [hep-th/9204012].[50] L. E. Ib´a˜nez and A. M. Uranga, “String theory and particle physics: An introduc-tion to string phenomenology,” Cambridge Univ. Press 2012.[51] J. P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, “On loop corrections tostring effective field theories: Field dependent gauge couplings and sigma modelanomalies,” Nucl. Phys. B (1992) 145.[52] G. Lopes Cardoso and B. A. Ovrut, “A Green-Schwarz mechanism for D = 4,N=1 supergravity anomalies,” Nucl. Phys. B (1992) 351.[53] L. E. Ib´a˜nez and D. Lust, “Duality anomaly cancellation, minimal string unifica-tion and the effective low-energy Lagrangian of 4-D strings,” Nucl. Phys. B (1992) 305 [hep-th/9202046].[54] N. Seiberg and E. Witten, “Comments on string dynamics in six-dimensions,”Nucl. Phys. B (1996) 121 [hep-th/9603003].[55] G. Aldazabal, A. Font, L. E. Ib´a˜nez and A. M. Uranga, “New branches of stringcompactifications and their F theory duals,” Nucl. Phys. B (1997) 119 [hep-th/9607121]. 4156] D. R. Morrison and C. Vafa, “Compactifications of F theory on Calabi-Yau three-folds. 1,” Nucl. Phys. B (1996) 74 [hep-th/9602114].[57] M. Berkooz, R. G. Leigh, J. Polchinski, J. H. Schwarz, N. Seiberg and E. Witten,“Anomalies, dualities, and topology of D = 6 N=1 superstring vacua,” Nucl. Phys.B (1996) 115 [hep-th/9605184].[58] L. E. Ib´a˜nez and A. M. Uranga, “D = 6, N=1 string vacua and duality,”In *Seoul/Sokcho 1997, Dualities in gauge and string theories* 230-282 [hep-th/9707075].[59] C. Bachas, C. Fabre, E. Kiritsis, N. A. Obers and P. Vanhove, “Heterotic / type Iduality and D-brane instantons,” Nucl. Phys. B (1998) 33 [hep-th/9707126].[60] E. Kiritsis and N. A. Obers, “Heterotic type I duality in D ¡ 10-dimensions, thresh-old corrections and D instantons,” JHEP (1997) 004 [hep-th/9709058].[61] W. Lerche and S. Stieberger, “Prepotential, mirror map and F theory on K3,”Adv. Theor. Math. Phys. (1998) 1105 Erratum: [Adv. Theor. Math. Phys. (1999) 1199] [hep-th/9804176].[62] W. Lerche, S. Stieberger and N. P. Warner, “Quartic gauge couplings from K3geometry,” Adv. Theor. Math. Phys. (1999) 1575 [hep-th/9811228].[63] W. Lerche and S. Stieberger, Fortsch. Phys. (2000) 155 [hep-th/9903232].[64] K. Forger, “On heterotic / type I duality in d = 8,” Lect. Notes Phys. (1999)365 [hep-th/9812154].[65] K. Foerger and S. Stieberger, “Higher derivative couplings and heterotic type Iduality in eight-dimensions,” Nucl. Phys. B (1999) 277 [hep-th/9901020].[66] E. Gava, A. Hammou, J. F. Morales and K. S. Narain, “On the perturbativecorrections around D string instantons,” JHEP (1999) 023 [hep-th/9902202].[67] M. Billo, L. Ferro, M. Frau, L. Gallot, A. Lerda and I. Pesando, “Exotic in-stanton counting and heterotic/type I-prime duality,” JHEP (2009) 092[arXiv:0905.4586 [hep-th]].[68] C. Petersson, P. Soler and A. M. Uranga, “D-instanton and polyinstanton effectsfrom type I’ D0-brane loops,” JHEP (2010) 089 [arXiv:1001.3390 [hep-th]].4269] M. Gutperle, “A Note on heterotic / type I-prime duality and D0-brane quantummechanics,” JHEP (1999) 007 [hep-th/9903010].[70] S. Kachru and E. Silverstein, “On gauge bosons in the matrix model approach toM theory,” Phys. Lett. B (1997) 70 [hep-th/9612162].[71] R. Gopakumar and C. Vafa, “M theory and topological strings. 2.,” hep-th/9812127.[72] R. Blumenhagen, X. Gao, T. Rahn and P. Shukla, “A Note on Poly-InstantonEffects in Type IIB Orientifolds on Calabi-Yau Threefolds,” JHEP , 162(2012) [arXiv:1205.2485 [hep-th]].[73] C. P. Bachas, P. Bain and M. B. Green, “Curvature terms in D-brane actions andtheir M theory origin,” JHEP , 011 (1999) [hep-th/9903210].[74] E. Witten, “Instability of the Kaluza-Klein Vacuum,” Nucl. Phys. B195