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JHEP05(2019)105 ) acts on Springer Z May 7, 2019 , May 20, 2019 : : March 21, 2019 : Accepted Published Received , Published for SISSA by ) T-dualities (or subgroups thereof) for https://doi.org/10.1007/JHEP05(2019)105 Z , a = 1 compactifications, whose non-perturbative N [email protected] , and A.M. Uranga a,b . 3 1812.06520 The Authors. L.E. Ibáñez Field Theories in Higher Dimensions, Flux compactifications, Superstring c

Recent string theory tests of swampland ideas like the distance or the dS a,b , [email protected] [email protected] C/ Cabrera Nicolás 13-15, Campus deDepartamento Cantoblanco, Facultad de 28049 Teórica, de F´ısica Madrid, Ciencias, Spain Universidad Autónomade Madrid, 28049 Madrid, Spain E-mail: Instituto IFT-UAM/CSIC, de Teórica F´ısica b a Open Access Article funded by SCOAP explored obey the refined dS conjecture. Keywords: Vacua ArXiv ePrint: to D0-branes orto M-theory potentials KK not modes.the leading asymptotic to The corners weak of non-perturbativeglobal coupling moduli examples symmetries. space, at dynamically explored infinite We forbidding perform point that distance, the a there access but study to are rather of points dS general diverging with maxima modular in and invariant potentials saddle and points find but no dS minima, and that all examples distance in moduli space.agreement with The the divergence distancebased relates on conjecture. to gaugino towers condensation Wedual of in to discuss states compactifications heterotic specific of becoming orbifolds. type examples light,the I’ We of or complex in Horava-Witten show structure this theory, that in of behavior which these an the examples SL(2 underlying are 2-torus, and the tower of light states correspond example we study the case ofeffective heterotic 4d action is knownand to complex be structure invariant moduli, under in4d modular particular SL(2 heterotic symmetries acting or on orbifoldsuperpotentials, the compactifications. the Kähler corresponding Remarkably, duality in invariant models potentials diverge with at non-perturbative points at infinite Abstract: conjectures have been performedcoupling at regime weak remains challenging. coupling. Wemoduli propose Testing effective to these action exploit ideas to the beyond modular check swampland symmetries the constraints of weak beyond the perturbation theory. As an E. Gonzalo, Modular symmetries and the swampland conjectures JHEP05(2019)105 (1.1) ) in a consistent 15 φ ( V. V 2 p 0 . Then in the limit of 12 c M . αt ∈ M ≤ − − 0 e ) p V ' j ∇ m 4 i ]. The main list of such constraints ∇ 7 19 – 26 15 11 10 ]): any scalar potential 10 9 , – 1 – or min( 8 15 , an infinite tower of states appears in the effective V → ∞ ] for a review. These are powerful conditions excluding p t c 20 22 10 M 24 ] (see also [ 7 7 ) = 21 | ≥ , p 0 V ], see [ p 7 ( |∇ – d 1 1 dS conjecture [ Z) modular forms , 3.2.3 The gaugino condensate 3.2.1 From3.2.2 heterotic to type I’ The / 1-loop HW computation of the threshold correction field theory with exponentially decreasing masses Refined theory of quantum gravity must obey either The moduli space of scalars is non-compact. Distance conjecture: consider ainfinite point distance in moduli space • • • 4.3 Two moduli model 4.1 Extrema in4.2 a modular invariant Swampland potential conjectures 3.1 Gaugino condensation3.2 in 4D heterotic The string type I’ or Horava-Witten dual include the following: There has recently beenon effective a theories lot [ ofeffective theories interest which on cannot thethe arise most concept in interesting of swampland consistent the conjecturesscalars theories refer in Swampland of consistent to constraints theories quantum the of gravity. properties quantum of gravity Some [ the of moduli space of B General analysis of minima for two fields 1 Introduction 5 Conclusions and outlook A SL(2 4 Modular invariant potentials and the swampland conjectures 3 String theory examples Contents 1 Introduction 2 Modular symmetries and invariant potentials JHEP05(2019)105 /R 1 → R ]. Modular ]; also, the ) symmetries 15 19 , Z , 11 ], see also [ ]. 7 14 – = 2 setup in [ = 1 supersymmetry, in which 11 N N ]. In this case the Kählerpotential and the – 2 – 87 – 80 ]. 68 – 65 ] and fairly generically in large classes of eliptic CY threefolds see = 1 heterotic orbifolds, for which the modular SL(2 69 ]. ]. Modular symmetries in axion and inflaton potentials have also been N ] and [ 77 79 , – 64 ] and references therein. In this paper we will explore the relations between – 78 73 14 – 20 , duality of tori and orbifolds thereof. This group is generated by ]. These generalized modular symmetries are also generated by some discrete 11 72 T 12 , ) – Z ]. The first two conjectures have been tested in a number of different string theory set- 11 69 , We will focus on compactifications preserving 4d It would certainly be interesting to try and test these conjectures in string settings The string theory tests of the distance conjecture performed so far are either pertur- 11 . The existence of this kind of modular symmetries goes however beyond the toroidal of examples are 4d were well studied(non-perturbative) in superpotential the must transform early suchhence 90’s that the the [ full scalar Kählerpotential (and potential)single is modulus invariant (little under is modular known transformations. about modular In symmetries the for case the of multimoduli a case) the behavior arise also inCY non-perturbative fourfolds superpotentials [ for F-theoryconsidered compactifications in [ on non-perturbative superpotentials for the moduliare may thus arise. constrained The by resulting invariance scalar under potentials modular transformations. A prototypical class folds, like the quintic [ e.g. [ shift symmetry and a transformationshift relating symmetries large and and small theirdistance volumes interplay in (in with moduli fact space the the has discrete tower already of been explored light in states the near points at infinite and are part ofSL(2 the duality symmetries oftransformations string along theory. with discrete The shiftT prototypical symmetries example for is the real the setting, part of since a modular Kählermodulus symmetries arise also in (exact) moduli spaces of Calabi-Yau three- direction by proposing toswampland exploit ideas the beyond modular symmetries weakpactifications of coupling. transforms the in Indeed, effective action general theare to under scalar discrete check moduli modular infinite symmetries space (orgeneral involving of paramodular) the transformations string Kähler,complex symmetries, of structure com- which and the complex moduli. dilaton They of 4d involve in string compactifications, is lacking. Only with theit additional is constraints possible of theories to with e.g. at signal least the 8 emergence supersymmetries ofwith a reduced dual supersymmetry theory and [ This also seems e.g. to including be non-perturbative out induced of potentials. reach. However, in this paper we make substantial progress in this pling. For some recentalso [ papers on the Swampland and thebative Weak or Gravity involve Conjecture at least see eighttests supersymmetries are (or also simple circle often compactifications).light essentially states These kinematic, is in identified but the a sense full that understanding the of their structure effect of on the the towers effective of field theory tion between the lastconnection two between conditions the has distancein been conjecture [ pointed and out global in symmetriestings, see [ has [ been pointedthese out constraints, in the hitherto unexplored regime of low supersymmetry and strong cou- There are presumably interesting interplays among these conjectures. A possible connec- JHEP05(2019)105 – 88 in the 2 T = 2 vacua; N . This observa- = 2 subsectors, 2 T N × = 1 heterotic toroidal N ]. In orbifolds with 84 = 2 supersymmetry. Their properties are – N 82 – 3 – ]. We revisit these results from the perspective of the swampland 87 – 85 ]. We expect that the lessons we present in this paper extend to more general 77 – 73 = 1 string vacua is interesting in its own right, and was already considered in the The fact that many string models lead to theories invariant under modular transfor- A key observation is that the moduli dependence relevant for invariance under modular We describe different specific string contexts in which such modular invariant non- The study of the modular invariant effective actions of a single modulus in general N = 2 sectors. ], which can enter the moduli-dependent non-perturbative superpotential. In this case mations, motivates the study ofpotentials general effective in theories the of context scalarstrivial of with candidate the modular field invariant swampland theories which conjectures. mayconsistency be would They also consistently may coupled require be fully to fledged quantum considered string gravity. as theory This to non- give a physical meaning to their or KK momentumthe states applicability in of our thetransformations results latter to on other pictures. string swampland compactifications. conjecturestentials In from with This particular, invariance modular non-perturbative under discussion behavior superpo- have modular enormously beensee e.g. obtained increases [ in the literaturestring in compactifications. a variety of settings, tion allows us to explore thetype dynamics of IIB moduli orientifolds, using other typetheory. dual descriptions, I’ in These compactifications, particular pictures andnear allow compactifications the a of point clear Horava-Witten at identificationappearance infinity of of in the the moduli divergence tower in of space, the states and scalar becoming which potential, are light and ultimately which responsible correspond for to D0-branes the the singularities appearN due to towers of KK or windingtransformations arises states from subsectors on with 4d thethus controlled fixed by the properties of heterotic compactifications on K3 perturbative potentials appear, and discusspoints. the relevant We towers revisit of states modular atorbifolds, invariant infinite gaugino already distance condensation discussed in in 4d there the early are 90’s moduli-dependent [ 90 one-loop threshold corrections to the gauge couplings [ existence of the states tothis provide a is physical a understanding novel ofrather the behavior, than singularities. as the Notice emergence compared that ofpoints with a are the dual dynamically previous description, (exponentially) discussions what censored. we in find 4d is that the infinite distance the existence of essentiallarge/small singularities volume). at We display points explicitlyarises at that infinite in from concrete distance infinite string in examplesswampland towers moduli the distance of divergence space conjecture (e.g. states expectations.swampland becoming Hence, distance light our conjecture in analysismodular with remarkably those transformations. the relates limits, In the behavior addition in of the agreement modular the symmetries with moduli for the effective moduli action require under the restrictive condition on the theory. 4d early 90’s, see e.g. [ conjectures. One finds in particular that modular invariant potentials necessarily imply superpotential must be an automorphic form with definite modular weight, which is a very JHEP05(2019)105 , /T we 1 (2.1) 5 ]. This 85 → − + 1, which T T ), see [ → = 1 supergravity we review super- Z , T N 3.1 we review the struc- = 1 supergravity. We ), which is ubiquitous 2 Z , , N we discuss effective field 4 = 1 bc − we discuss concrete string theory 3), implying the appearance of sin- 3 − , ad Z = 0. )-invariant scalar potentials, and explore , making it axion-like; and ∈ θ δ Z , for )-invariant scalar potential, and study the va- . The two generators are Z it – 4 – /t , 1 + , a, b, c, d θ → b d collects some useful properties of modular functions t = + + A T cT aT invariant under the modular group SL(2 , e.g. )-invariant scalar potentials in 4d t Z T , −→ T an overview of the extrema in models with two complex scalars. B we turn to other dual realizations, including models in type I’ or Horava-Witten 3.2 The structure of the rest of this paper is as follows. In section We emphasize that in these effective theories the invariant potentials always diverge acting on the compleximplies modulus a discrete periodicitywhich relates for small the and real large part compactifications, and how theseprototypical constraints example fit of with such swamplandin symmetry, ideas. the string modular theory. We group In focustheories SL(2 this on of section the a we single review modulusmodular the symmetry scalar is potential generated for by 4d the transformations and appendix 2 Modular symmetries andWe invariant want potentials to study how modular symmetries can constraint the effective action of string theory, which nicely isolate the towerscorrespond of to states dominating D0-branes at or infinitetheories KK distances with momentum (and general modes). which single-modulus In SL(2 lidity section of the (refined)offer swampland our conjectures final remarks. to their Appendix extrema. Finally, in section setups with non-perturbativetransformations, superpotentials and satisfying describe these thestates properties UV inducing the degrees under divergent of contributions modular to freedompotentials the providing from potential. the gaugino In infinite section condensationsection towers in of heterotic toroidal orbifolds, and subsequently in are recovered. ture of single-modulus SL(2 show the superpotential must havegularities modular at weight the ( boundary of moduli space. In section to e.g. possible quintessence dynamics. exponentially at points at infinity,them. so that In the their modulidistance UV are conjecture completion, dynamically forbidden the should to infinite bepotential, access tower and responsible of thus for states for the demanded the dynamical impossibility by to generation the access swampland of the the regions divergent in which global symmetries modulus effective actions with non-trivialtheir SL(2 extrema. We findfind AdS dS and maxima Minkowski and vacua,analysis saddle show both points, that SUSY but the and we refinedsecond have non-SUSY. dS condition, been conjecture for We unable applies also dS in to maxima). all find examples a We studied find dS (including in minimum. the our Our examples no runaway behavior leading behavior and singularities. In this paper, we perform a detailed study of different single- JHEP05(2019)105 . 3 π/ 2 (2.4) (2.6) (2.2) (2.3) i . i, e A In coming = ). 1 T A ]. A general and , so that modular ) transformation. ). Hence we can 2 85 | Z T for examples. For ( , d η 3 + 2 / 1 cT , ) | . )) (2.5) d / )) ) 2 T | ∗ T . Without loss of generality + ( ( , ) (see appendix j T j j W ) self-dual points ( 3 T | ( − ) cT ( Z P ) ( j P d , T 3 T ( 0. Such a behavior is in principle n ( + n/ , → ) being a The Kählermodulus effective → ) W ∞ ) T ) + log 12 cT i T ) ( ∗ T ) ( T ∗ j ( ) 6 ( T ) 2 T 4 ) T η → ( T ( G T η − − m/ ( T 2, H T η T / a,b,c,d ( ) must be of the form m – 5 – = iδ T 1728) e is a polynomial on ( ) W 12 3 log ( − H T ) P ) ( − T 6 −→ T ( ( G η j ) ) = , if one insists in avoiding singularities within the funda- ∗ T ( A ) = ( W ) = T,T T ( T ( = 1. It also corresponds to the Kählerpotential for the overall ( G H H 11 h may correspond to a string modulus like Kähler,complex structure or ) may be chosen different from the SL(2 T j ( ), which has modular weight 1 P are positive integers and are Eisenstein functions of weight 4 and 6 respectively, see appendix T ( 6 ) being a modular invariant holomorphic function. This can always be expressed , η 4 is a moduli-independent phase, which can depend on the SL(2 T ( m, n G δ H Note that these superpotentials with no additional singularities in the fundamental As explained in appendix We now revisit the general study of T-duality invariant potentials in [ Under modular transformations one has ( In general This does not imply the absence of tree level superpotential couplings. These are indeed present, but . Thus this peculiar behavior, if present, requires non-perturbative physics. 1 /t 1 sections we will provide explicit string models with thisnecessarily divergent involve charged behavior, matter fields, including whichfor the transform the non-trivial matching under of the the modular total group and modular account weight for the superpotential. where domain, necessarily diverge exponentiallyrather as surprising, since in perturbation theory potentials are known to scale like a power of where the zeros of Another equivalent way to write the same expression is as a function of the absolute modular invariant function mental domain, it can be proven that useful way to parametrizefunction functions with given modularalways write weight is through the Dedekind with weight (-3), i.e. where This corresponds to e.g. thecompactification large Kählermodulus with dependence of a Calabi-YauKählermodulus (CY) in string a toroidal/orbifold compactifications. invariance of the Kählerpotential dictates that the superpotential should have modular complex dilaton, depending onconcreteness, the we carry examples out the considered, discussionaction in see terms will section of be determined by the full supergravity Kählerpotential JHEP05(2019)105 ) = T ( T (2.7) (2.8) , and H T , which thus ]. for large Im 12 1 in Planck units, → ∞ πt e T , T ]. In the non-perturbative 3 − 14 , 2 ) = 1, which is indeed realized | 2 . 13 T ˆ ( G T | H π 2 ) Im T 2 − π ) T 3(Im ( 3. Barring singularities at finite distance in ]. These constructions show that the expo- 2 − G 82 – 6 – 12 | ) ) = ∗ T ( η | 1 3 T,T ) ( 2 T ˆ . As we discuss in the next section, this type of potentials G 4.1 8(Im = = 2 supersymmetry on which they are based. In those examples, V N ) is explicitly modular invariant and diverges like 2.7 is the non-holomorphic weight-2 Eisenstein function 2 ], the towers of states near those points typically induce corrections to the metric ˆ . Extrema of this potential as well as more general potentials with arbitrary G 11 We would like to emphasize that, nevertheless, this behavior is in stark contrast with The existence of this singular behavior as one approaches points at infinite distance in One may worry that the above computations can be trusted only for large Im It is interesting to consider the simplest case with → ∞ admit very explicit descriptions inframework terms here of described, a dual access theoryin to [ moduli these space, points but istrapped also in excluded a from not non-perturbative regime the merely farfor dynamics from from the any kinematics inducing weak SM coupling a should expansion. potential. then Perturbativity appear The as models an infrared are effect, thus as suggested e.g.in [ states, in agreement with the distance swampland conjecture. the behavior at largedue moduli to found the extended ine.g. other [ examples studied inin the moduli literature, space, possibly possibly making the infinite distance emergent; these states sometimes seems unavoidable. moduli space, resonates withthe the potential swampland dynamically conjectures. censorsrecovered. the For Furthermore, limits Im the in divergences in which those the limits stem continuous from shift towers of symmetries almost are massless the section, but cannotin change the the modular non-trivial propertiesbe topology we an of are holomorphic the highlighting. form bundle,moduli of Hence, space which modular in (which is weight would short, encoded in have the no coming superpotential clear sections) must physical this origin, implies as will the be existence clear of in singularities the at examples Im that there maystatement be is that corrections modular invariance in strongly restrictsthe the the structure superpotential deep of the and interior Kählerpotential, thuspotential of and the moduli superpotential scalar are space. potential notof mere for non-trivial functions However, moduli. over the bundles moduli space, key over More but moduli formally, rather the sections space. Kähler Corrections may change the specific form of This potential ( t will be discussed inarise section in gaugino condensation in heterotic orbifolds, and other related examples. where nential singularities arise from towersswampland of distance states conjecture. becoming light, in nice agreement with the in concrete string constructions,given as by we the will simple see expression in coming sections. The scalar potential is gaugino condensation example studied in [ JHEP05(2019)105 ], 90 = 2 = 1 ]. ] the – (3.1) , and 2 77 N 87 N 88 , which – T 2 × 74 T in heterotic S = 2 subsectors, fixed. In those of the ). In ref. [ 2 N S U T , = 1 heterotic Abelian 0, the heterotic string T for the resulting (8 Im N 0 would be dynamically → 2 , / 2 g p T ∞ , i M 1 = → , and are given by 2 − s 2 S ) M T ¯ τ τ, ( = 1 F-theory compactifications [ torus Z N , corresponding to S 0 would be censored, in agreement with absence – 7 – τ 2 2 τ → ], in the first paper introducing S-duality. Although d g Γ 86 = 1 superpotentials of the type discussed in the previ- Z a N b 0. However in such a case, as discussed above, there are = 2 theories, the gauge kinetic functions can only depend = , a N ∞ i ∆ → on these moduli. S U ) ) S-duality could be expected to be a symmetry only in finite theories, it Z Z , , SL(2 × T ) ) vanishing at Z S , The thresholds corrections depending on the moduli of the From general results in Threshold corrections to 4d gauge kinetic functions arise from a perturbative 1-loop Following the same arguments as above the limits As explained, similar analysis applies when the relevant modulus correspond to other ( H SL(2 theories arise from momentum and winding modes on on moduli inthey vector cannot multiplets. appear Thus,corrections in depending since the on the the threshold K3 Kählerandare complex correction. in moduli structure vector moduli belong multiplets. Hence, We to we will hypermultiplets, pay are special interested attention to in the threshold action of the modular groups i.e. those associated tosectors the elements structure of is thethe essentially orbifold that discussion of can group a be leavingcan compactification phrased be some of obtained in heterotic by this on furtheralso more K3 orbifold for general twists), comparison following setting with the (eventual the classical section breaking computation to to in follow. [ Upon gaugino condensation, suchmoduli-dependent threshold superpotentials corrections which can are give modular rise forms of to the non-perturbative appropriatecomputation. weight. We eventuallyparticular focus are on interested Abelian in the orbifolds, moduli-dependent which piece, are which exact arise in CFTs, and in 3.1 Gaugino condensation inIn 4D this section heterotic we string will discussous how section arise in string theory.orbifolds. In We particular, first we focus will on consider the 4d moduli-dependent threshold corrections for gauge couplings. a necessarily singularities in the fundamental region, with no obvious physical interpretation. 3 String theory examples coupling have appeared in the context of 4d forbidden, and the weak coupling limit of continuous global symmetries. Forbecomes large tension-less since onedivergent has limits for in the the string S-dual scale potentials were avoided by canceling the divergence choosing string moduli. Thecompactifications was case put forward in in [ whichin it principle SL(2 corresponds towas the argued that complex it dilaton mayno apply supersymmetry. to Indeed, some modular extent invariant in superpotentials some involving complexified subsector string of theories with reduced or JHEP05(2019)105 th M i Z (3.5) (3.2) (3.3) (3.4) × , N orbifolds -function a Z c β The proto- 0 N is subject to Z + 2 2 and ) leaving the T i is the 4 N | P is invariant under . Performing the Z 2 )) . , =2 i 1 2 M are frozen and hence | U N a,i ( , , n b 2 = 1. This leads us to , i η . 2 i , 2 | U m 1 U 4 ) . Using the above results, | ) N M i 2 ) m + 2 Z U T U , given by πτ U ( 1 − ). Also η P e | (Im = 2 gauge theory. Note that, as is the mass of KK and winding N ) m ) 2 is a moduli-independent constant. SL(2 Z n U − N a 2 . This includes standard embedding × M c 8 m T for T 1 E + (Im ) + log 1 n 4 Z | n N 1 , ) + 4 | m T ( ( ] for a review. )) i η TU | T 93 2 πi τ ( ) 2 – 8 – n η T e | | Z ) ) i , and thus act crystallographically only for definite ∈ U 2 2 T (Im , 1 Z h ,n X = 2 sub-theory, and = 1 orbifolds, some or all the 2 with the orbifold group (Im , )(Im 1 log 1 T N m N a , which is present if the embedding of the orbifold group in b /P 8 log labeling the three possibly fixed 6 = E − i T 4 (Im = = 1 theory contains gauge sectors which may experience strong =2 torus a by suitable relabeling of the integers = Z N N a,i ∆ 2 U b , with ) i | ] M Z ) U | i , ( , complex structure. For instance, that is the case e.g. for the 88 P i | P 2 ) one obtains for the moduli dependent threshold corrections T | 12. , -function coefficient of the corresponding T SL(2 8 are winding numbers and momenta and = 1 orbifolds i β 3.1 , X i ], which in general transform non-trivially under the modular groups. In particular, this also is the order of the orbifold group (e.g. × N | − T the 92 = 6 , m , ) P i | = a Z n 91 b N , a The resulting 4d Let us consider including additional orbifolds to yield 4d fixed. For simplicity we focus on factorizable orbifolds, and consider the dependence In cases with charged chiral multiplets and possible non-perturbative superpotentials involving them, ∆ 2 2 the gauge degrees of freedom involvesmodels, only the but other also many other orbifolds, see [ the modular properties offields the [ superpotential require includingoccurs the for effect the of tree-level the superpotential modular couplings weights involving of untwisted matter or twisted fields. dynamics effects at lowcontains energies. no Consider charged the matter,confines simplest and so situation produces that that a some it gauginotypical gauge case condensation describes factor is superpotential a an in unbroken pure the infrared. super-Yang-Mills sector, which Also, we note thatonly in contribute a certain constant toextra the orbifold thresholds. action which Thisvalues are occurs of not when the the fixed with where coefficient of the corresponding Recall that the orbifold group is constrained to have a crystallographic action on the torus. orbifolds, with some elements ofT the orbifold group (forming aon subgroup the moduli this is given by [ with expected, this expression is invariant under SL(2 consider excitations of vector multipletsSL(2 involved in the loop.integral in We ( note that and Here with JHEP05(2019)105 , 2 0 3 Z Z = 0 → × (3.6) (3.7) 2 2 T n Z = 1 . The origin , the sector m T U = 0. A similar 2 Im , 1 6= 0, the only effect T, m , i GS , δ 2 4 ) i ) i 1 U U ( ( η η 2 4 ) i ) ], which implies the replacement i 1 T , the opposite regime of Im in this example. This has precisely T ( 96 ( T η – ) η 8 =2 ] Z 3 94 =1 , N i Y i,E b 92 log , 8 S E 91 3 β = 3 ], the mass decreases exponentially with the =2 2 8 e N a,i E – 9 – 3 b 19 , β = 0 leads to an infinite tower of KK states with 16 3 = Λ =1 – i X 2 , 8 8 1 E E 2 10 ), there is an infinite tower of states with f β are gauge group independent constants. However, this n , π 3 2 1 T 3 e )’s is smaller [ 16 3 i i GS δ T − ( . η = Λ S B 8 = E = 0. , which is one of the simplest examples whose extrema are studied a W 6 (and fixed f , where ) i GS T δ i ( GS δ ) for fixed complex structures: we see that at large Im /η → ∞ . As is familiar [ ) − 1 3.3 S U − ) =2 and appendix T N in ( a,i 4 b is the heterotic 4D complex dilaton. Actually, cancellation of duality anomalies 2 ) = Ω( (Im S → M ' The description in terms of the heterotic degrees of freedom certainly resonates with As we advanced, these potentials diverge exponentially for large Im The result can be easily adapted to other orbifold groups. For instance, in orbifolds Finally, recall that in orbifolds with no fixed planes, like the prime order cases The gaugino condensation superpotential then reads Including the effect of the threshold correction leads to holomorphic gauge functions S, T =2 , there are no moduli dependent threshold corrections. The modular invariant scalar ( 2 7 N a,i becoming light. In any ofparticles these modify limits, the the gauge physical theory interpretationcondensate, is dynamics hence that by the leading increasing towers the to of effective a light scale potential of growing the at gaugino infinity. the swampland distance conjecture. In the following section, we consider a dual type I’ / with vanishing winding numbers M distance, if phrased inleads the to proper an frame. infinite Bybehavior tower SL(2 of can winding be modes recoveredspace: with at vanishing as the momenta Im points at infinite distance in complex structure moduli in section of this divergence canlight, be as traced expected to frommass contributions from the infinite swampland towers distance of conjecture. states becoming In particular, consider the Z potentials arising fromstudy. orbifolds with For gaugino instance,on condensation the considering dependence can examples on beW with the subject fixed overall Kählermodulus, to complex we explicit structure, have a and superpotential focusing of the form where, as mentionedsimply earlier, not the present complex inis the structure that superpotential. moduli the Also, exponent are for of projected orbifolds the with out, they are where we have made usethe of modular the properties described fact in that theof previous how section, such and superpotentials it is can a arise most in simple illustration very explicit heterotic compactifications. b point is not essential for ourorbifold, purposes, for so which we rather ignore it by considering e.g. the where requires in general a Green-Schwarz mechanism [ JHEP05(2019)105 ). U with 4 ) (there ], either 2 . Again, 2 2 heterotic, T R T 8 109 – E × ) depends on )(tr × 2 3.3 102 8 F E compactification are 2 T and (tr , and exploit the rich web 2 2 ) 2 compactification down to 8d, T = 1 compactification on K3, 2 F × T N , (tr 4 = 2. In this description, the modular F , whose mass formula ( N 2 moduli independently. , possibly with Wilson lines; clearly, this 2 T U T = 2 corrections to the gauge couplings, thus N (as well as of its complex structure modulus and = 2. As is well known, in the ], while for the SO(32) heterotic, it corresponds – 10 – T T 97 N )[ n corrections, not involving the gauge groups, and which − 2 ) 12 2 ; they are hence not the natural objects to disentangle the two R n, 2 ] for a review. We are thus left with a large class of 6d models T = 1 can be obtained by additional orbifold actions. In this section 101 modular group. However, the relevant heterotic degrees of freedom N and (tr U ) modular properties, or the behavior at independent infinite distance ) 4 Z Z R , moduli independently. , moduli of compactification to 4d ], see [ 2 factors, (12 + U U T 8 100 SL(2 , E = 2 subsectors in terms of a compactification of heterotic on of K3 and × 99 T T N ) and the Z = 4 [ , the The structure of the quartic corrections to the 8d theory, for diverse choices of Wilson It is thus natural to regard the configuration as first a The compactification can be constructed as a 6d In order to do that, we return to the description of moduli dependent threshold correc- T n lines, has been studieddirectly from as different a dual descriptions. perturbativelatter computation For is in instance, closely related heterotic in (in [ string,O7-planes a and or perturbative 8 from limit) D7-branes to F-theory on the on top orientifold (as K3. of counted type in The IIB the on double cover). This corresponds to a are in addition tr we will hence skip).functions of the heterotic As Kählermodulus weUpon show dimensional below, reduction the oninstanton K3, coefficients backgrounds, one including of recovers the the such 4d correspondingwith couplings the gauge are precisely and (combinations modular gravitational of) the same modular functions. eventually followed by a K3 compactificationproperties to 4d should be manifest8d already theory at contains the acorresponding level precursor to of of corrections the the to toroidal corrections the 8d to 8d theory. couplings the Indeed, tr 4d the gauge couplings corrections, on the two to which should subsequently be compactifieddescription on is disadvantageous since theobscured modular by properties the in complications the of the previous stage of K3 compactification. we would like toof focus string on dualities the to structure understand of the heterotic one-loop on threshold K3 correction. followed by a these are determined by the distribution of instanton numbers for the gauge backgrounds description, which explicitly displaysthe the degrees of freedom controlling the dynamicstions of in eventual reduction to SL(2 correspond to momentum and windingboth states in independent SL(2 points. In this section we provide a picture in terms of a dual type I’ or Horava-Witten freedom controlling the dynamics of the 3.2 The type I’3.2.1 or Horava-Witten dual From heterotic toIn type the I’ above / HW heterotic description, the piece of moduli space under discussion has an Horava-Witten description, which allows for a more precise identification of the degrees of JHEP05(2019)105 , 4 τ ]). 1)- 120 − , and 112 gauge spinor τ 8 . Hence, E , or with 2 2 is the type 128 T . These are 2 τ Z ] the spectrum 1 S ] (see also [ compactification 113 d 8 1)-brane instantons 111 arise from D0-brane , as follows. Starting − 4 ) of SO(16) τ → d 16 , compactification of type I’ 1 16 S compactification with Wilson 2 T ). orientifold with two O8-planes, each symmetry on the boundaries down to D0-branes, giving particles in the 2 8 120 Z k , E / 1 1 is not manifest. Indeed, since × S bringing us down from 9d to 8d. Similarly, 8 τ . In the type IIB picture, the corrections were 1 E 2 ) + ( S 1 T – 11 – , + 1) D0-branes, giving particles in the k 120 , given by one circle associated to the lift from type I’ 2 and transforming in the ( -dependent part arises from perturbative fundamental T Z arise from 9d particles arising from open string stretching U ∈ U modulus maps to the type IIB complexified coupling -dependent piece of the quartic corrections arise from D( τ w T to 8d, e.g. with suitable Wilson lines if we wish to relate to the SO(8) 1 ]; here the S modulus remains the complex structure modulus of the type IIB modular group corresponds to S-duality on the type IIB complex coupling 110 U and transforming in the ( 2 ) w Z Namely, we consider type IIA on and , . In this picture the D0-branes lift to KK momentum modes of the 10d 2 3 In this picture, the modular behaviour in In this type I’ picture, the threshold correction is associated to the tower of particles In order to make the connection with towers of particles as in the standard Swampland The original heterotic As will become clear later on, this T-duality is similar to those relating the Weak Gravity Conjecture 3 SO(16) multiplets, which propagate on a to 11d, and the second circle being that already presentfor in charged the particles and 9 instantons. winding 2 IIB complex coupling, theM-theory. corresponding S-duality The group lift isparticular of manifest choice type of only wilson I’ if lines theory we breaking the corresponds lift up to to Horava-Witten (HW) theory, in the representation of SO(16). The 8din correction which arising they as are a allowedthe 1-loop to corrections diagram run depending of along on these the among particles, the D8-branes, and thus9d having particles non-trivial with winding in winding the type I’ in the 9d typeparticles bound I’ to theory, each of concretely theof two the O8/D8 such configurations. corrections BPS As particles depending discussed is in on of given [ each by SO(16), bound and states bound of states 2 of (2 theory. with 16 D8-branes (ason counted an in additional the doubletheory. cover). In This fact, the 9d discussion theorylater can on. is be then easily compactified extended to general Wilson lines, as we will do a tower of chargerather states, than light albeit particles. in this case theyDistance correspond Conjecture, to we will D( exploit yetIt another corresponds picture, to developed T-dualizing in the [ type IIB orientifold into an and the modular group of thecomputed orientifolded in [ brane instantons, whereasstring the corrections. Hence, this pictureterms succeeds of in two kinds disentangling of the instanton twoConjecture, effects. modular since groups This moving in is towards very points reminiscent of at the infinite Swampland distance Distance in a modulus has an effect in from the heterotic, one performsline, an and S-duality two to T-dualities a to type the I type IIB orientifold. the heterotic the SL(2 heterotic model with Wilson lines breaking the gauge group to SO(8) JHEP05(2019)105 2 th T I (3.8) of this 1) τ − D0 KK KK case. D( Mix of Objects 8 F1 winding Wrapped F1 wrapped M2 Wrapped D1 E ) . wrapped F1 / KK J e ` 1 I e ` the Wilson line in the IJ 2 | . It is also straightforward G 2 I 11 + e ` 2 T A τ p − ( 9 e ` πt | and ], albeit for the 2 − τ e 1 8 (2) S-duality p 111 E V 8 Hidden HW -dependent threshold correction for the Duality group d πt τ KählerT-duality KählerT-duality KählerT-duality KählerT-duality − Complex structure Complex structure Complex structure Complex structure e Z I 11 ` X ,` 11 – 12 – s 9 X 4 ` 10 R t s /g 10 HW vector multiplets, described from the perspec- . The above expression for vanishing Wilson lines t t R 10 dt dt 10 /g 9 R 8 s Z R in 8d. Cases with more general Wilson lines can be 9 9 iθ iθ iθ iθ R E 9 ∞ ∞ e e e e /g 9 ∈ 8 i R 0 0 9 9 9 1 i R i R Z Z iR 10 E i + i R 11 + + /R /R 1 1 /R ` 2 + 4! 4! + /R 1 2 + 10 10 3 11 9 B geometry. 0 Modulus = C 2 B = = I C R R R R 2 C 1 2 C × I S -depended contribution in type IIB theory arises from M2-branes 11 T 1 T 2 × R e ` S R 2 T U , possibly with Wilson lines, is the natural framework to understand , with Λ a root vector of R × , R 2 T I gauge 2 R Z T A A Z · / ∈ 1 9 Λ S ` − = I ` 9 e ` = Type I Picture Type I’ . Dual pictures and some of their ingredients. Despite similar notation for the compactifi- ). Similarly, the Heterotic Type IIB I orientifold orientifold orientifold e Z ` , The 1-loop computation of the The dual pictures and relevant ingredients are shown in table Horava-Witten Here direction. We focus on thesuch case that without Wilson lines,is and drop manifestly the modular tildes for invariantto the under analyze momenta transformations the of surviving subgroups in the presence of non-trivial Wilson lines, like the configuration with an unbroken discussed similarly. We follow closely the steps in [ tive of the worldline, reads 3.2.2 The 1-loopThus, computation HW of theory on the thresholdthe correction modular properties offollowing the we threshold illustrate corrections. the computation To of flesh the out the discussion, in the of type I’. Asmaps in to standard M-theory/type the IIB typeSL(2 duality, IIB the complex complex coupling, structure wrapped and on the there is a geometric interpretation for the Table 1 cation radii, they in general have meaning adapted to the corresponding picture. JHEP05(2019)105 2 F and (3.9) (3.11) (3.12) (3.10) (3.13) ]. 11 ` 114 0) and anti-D0- . . + cc. (3.14) ) bringing us down 1 > τ   1 24 1 ) 11 q ` S 11 11 9 ` ` represents the winding . q 9 πiw 11 − ` w 2 + log 9 e . w (1 t 1 1 24 ! 0 , 2 11 q > ` 2 πiτ 2 2 Y cc. 2 11 τ ` 72 e πτ − + (log |   2 9 . A useful way to eventually extract 11 = w 1 t 11 t ` 12 2 ` 9 π log 9 reads 2 11 − w accompanied by non-trivial actions on w 1 | − t 2 1 q w e 12 2 = 9 , of D0-branes in type I’ language. With πτ Z 1 − T 2 w 6=0 ∈ 11 − 0 ` 11 0 11 τ = X e > > w ,` | X )] 9 9 24 πi 9 is to perform a Poisson resummation to get 11 X – 13 – 1 w ` π 2 w 2 = 0 term (discussed below). cc. R w τ τ | 2 , 4! /

− 11 Z 1 ) + ∈ 2 w 6=0 = in a rescaling of 4! − (18 9 t 11 11 X in the integral is associated to the four external legs in ` w ` O 24 πiτ t = q (2) dt 2 4 =0) 1 t 4! V × 9 − ∞ w ) = 0 piece, which corresponds to no D0-branes, and corre- 6=0) gauge ( 0 = 9 R Z 1 A w 12 11 gauge ( ` π 6=0) − log(1 (2; A 4! 9 √ 0 O w = [ > gauge ( / = 11 X ) A , and gathering the contributions from D0’s ( ` = 0 contribution, and performing a Poisson resummation in =0) R points. Moreover, for general Wilson lines, there are combined iden- 9 9 1 πiτ 12 w 2 2 w ( gauge gauge e 18; − A , A , we have = t = 0), we have (2 q 6= 0 contribution, after integration in O < \ 9 6=0) ) term. Note that the same result is obtained by taking two external legs i.e. tr 9 11 w w Z ` limit of an infinite tower labeled by or SO(16) 4 gauge ( F 4 τ A 18; The As will be clear later on, this contribution extracts the dominant contribution in the Isolating the In the following we reabsorb In the above expression, the , (2 Here have substracted the sponds to a perturbative piece mentioned later on. The above expression can be recast as Introducing branes ( from D0-branes in the Gopalumar-Vafa interpretation of the topological string [ large hindsight we write Incidentally, this zero-winding contribution from 11d KK modes is similar to a contribution where we have exclude the divergent In the 10d picture upon reductionof on the the direction 9d 11, objects thefrom integer (eventually 9d D0-branes) to running 8d. in the loop inintegrating the in in the 4d theory. the dominant contribution at infinity in tifications due to modularthe transformations Wilson line of moduli.O This is just the HW description ofthe the tr dualities in the heterotic SO(8) JHEP05(2019)105 ) 2 2 R ; in = 2, HW. term Z (3.15) (3.17) (3.18) (3.19) (3.16) 2 on the N ) )(tr → ∈ 2 (namely, 2 k k F τ F -dependent U . 2 = 1 theory obtained F 2 = 0 part) to achieve a N F ) tr in the lift type I’ . 2 4 11 1 | ` R ) S τ tr 12) tr ( η | − − corresponds to the zero winding 2 n τ . log F ] ]( . ] 4 4 1 | | 4 24 ) | ] in the type I’ picture for the SO(16) ) ) ] ( tr − U U τ 4 ( ( | ( 111 = η ) η η | | | | U ) ) ) ) ( τ τ U η U ( | – 14 – η ) | U (Im log 4 | ) 1 6 log [ (Im (Im τ log [ ( Im − ( 4 | η ) | = ) τ ) we have ( τ η | ) ) piece (arising from the dropped gauge 3.11 τ τ A has no quartic Casimir). One can similarly get the (tr log [ (Im 8 . The above computation produces the coefficient of the (tr 2 E log [ (Im T × theories, to which we refer the reader for details. Here it suffices to note that, 4 ), and taking into account the Euler characteristic of K3, we have a threshold n An equivalent way to understand this picture is to indeed consider first the compact- Let us conclude with a final remark bringing us back to the original discussion of A very similar computation involving fundamental string winding states in type I’ As mentioned before, the dominant term at large will produce a tower of 6d (massive) D0-brane states, in diverse representations of the = 0 contribution of the whole tower of D0-branes (or HW KK modes in the 11 direction). in the heterotic description). 8 9 equivalently the coefficient offrom the just anomaly the polynomial K3 compactification. of the 6d ification on K3 toE 6d. In this compactification, the tower of D0-branes in the adjoint of (12 + correction to the 4d gauge coupling constants given by which, as announced at the beginning, is controlled by the beta function of the 4d in 8d (recall that term, resulting in a quartic correction of the form Upon compactification on K3, with a gauge instanton background with instanton number This multiplicity of bound states implies the result heterotic in K3 and SO(8) from the perspective ofas the it 9d simply theory, reliesthe the the present computation existence case, is of the identicaltwo-dimensional space one to relevant defined 9d the 9d by states previous BPS the are one, state HW M2-branes interval for times with each the wrapping integer number charge It is thus these states thatT dominate in the point at infinity in the moduli space of theory (i.e. wrapped M2-branethreshold states corrections. in These HW) were can discussed be in shown [ to produce the amplitude in the HW picture. Hence we get w Here the exponent ismatches with chosen by the adjusting log(Im modular the invariant coefficient, quantity, as but expected it from can the be original checked manifestly that modular it invariant So putting together with ( JHEP05(2019)105 8 E , and produce, 2 T . These of course T ), keeping invariant the Z , = 2 context this gives rise to a tower N = 1 context produces the non-perturbative N – 15 – = 1 supergravity models with one chiral field and . In the 4d N 2 T = 1 setup, these instanton would have too many fermion zero N . 2 R tr − 2 ], but with the additional difficulty that the M5-brane is bound to the F ], namely the effect of instantons from D0-brane loops on the action of an 117 116 , In the HW lift, we have just an instanton effect from an M5-brane on K3 times the 11d 115 = 1 superpotentials which are modular forms under SL(2 or could perhaps representthe some other subsector hand of onethemselves a as could more explicit ignore complicated examples itsadmitting compactification. of possible a consistent string On coupling theorymodels to origin the quantum and connection gravity. consider between The them duality idea by symmetries here is in to string test compactifications in and specific the N moduli scalar potential. Ingeneral this section class we of exploreare in modular some simplified invariant detail models, potentials the since extremamultiple for a of moduli, fully a the gauge realistic most single bosons stringsome modulus and compactification particular matter is compactification fields. expected after to So the involve at rest best, of these the models spectrum could has arise been in integrated out, 4 Modular invariant potentials and4.1 the swampland conjectures Extrema in aIn modular the invariant previous potential sections we have seen how non-perturbative effects are able to generate circle, with arbitrary momentumto excitations that on in the [ boundary, latter. which thus This renders the configuration quantitative is description similar beyond present knowledge. picture is not soshows useful. that the In effect anyis of event, nothing the it threshold but is correctionsin interesting the on to [ sum the point non-perturbative of outinstanton superpotential the that from this a contributions D4-brane description from loop. polyinstantons like those introduced superpotential. The relevant objectK3, is and a 6d which BPS runsof particle in given spacetime a by instantons, loop a with D4-brane in the (in wrapped D8-branes. their on In Higgs themodes branch) 4d correspond to contribute to to gaugenon-trivial the instantons contribution, superpotential, thanks on and to the only extra a orbifold few projections. (fractional) Thus instantons the provide 1-loop diagram integral of tr 3.2.3 The gauginoLet condensate us conclude by mentioningthe non-perturbative that effects, the which type in I’ the 4d or HW picture also includes the source of termined by the indexbackgrounds. theorem These on massive 6d K3via states for a are states one-loop subsequently diagram coupled compactified whose on torections computation the to is identical the gravitational to 4d and thedue gauge above gauge to one, couplings. the the familiar The threshold fact cor- matching that with the the index theorem above relates description the is multiplicities precisely of 6d states to the surviving 6d gauge group. The ground states in the dimensional reduction in K3 are de- JHEP05(2019)105 . π 3 I 2 i iT = 0 e (4.4) (4.3) (4.1) (4.2) + m = ). The R ρ T 2.5 = = ] it was also so for it )). The proof T 85 | T + ( . j θ ( and ) = P 2 i are never dS minima. (1728) | T = 0 and , which type of extrema H P = |P | 2 H 3 , dT T dH ) given by eq. ( 2 . − m/ | . T } 2 (

} 3 W 2 | 3 H ˆ G {− {− 2 H = 1728 | | π 2 12 12 3 | | | 2 ) + log H η η ∗ = 0 one has | , with | | (0) + 6 3 3 T I I n ) ). The summary of our results is that we (1728) T T T |P − 8 8 ( dT dH = 0, |P 2.5 T

/η and choice of polynomial 2 ) ) with the fundamental region in the complex I dj ) = dT – 16 – 3 T T ∗ Z ) = ( 4 , ∗ = 3 log ( H n, m ( in eq. ( 2 − T,T ( ˆ T,T in turn. 12 G = P ( | V η ) = = 0 so that for | V 1 ∗ = 1 supergravity Kählerpotential 3 W I n, m dj dT T ]. First we study, for each choice of N 8 T,T = 85 ( = 1728, 2 G ) = j ˆ . One can show then that the points G ∗ 1 T,T ( = 0 0 and polynomial V one has one has m i ≥ ρ = = = 0 and one gets T m, n T = 0 or dT dH It is always a maximum andAt it is always in AdS. n At Although we do not have a general proof for the absence of dS minima, one can We have performed a general study of minima for the different models obtained for Let us consider then the • borrows some results from [ is generated at these points. Oncewe we show have that, found a for set these ofconsidering parameters, parameters different it generating a values can minimum, of never be in dS. Let us see how his comes about, saddle points, but wethis have structure been is unable a tothe consequence models. find of dS the minima. modular It invariance (duality is in naturalexplicitly a to prove that string believe the setting) that extremaThis at of the may self-dual be points or proved zeros for of any value of explored the potential numericallyare and not we aware find of a that general this proof. is indeeddifferent the case, although we find Minkowski and AdS extrema with and without SUSY. We also find dS maxima and The potential is invariantplane under shown PSL(2 in figure are always extrema, sincenon-trivially on they derivatives are of fixedconjectured the points that potential, under all which finite must extrema thus order lie vanish. subgroups on which In the act [ boundary of the fundamental region. We have corresponding potential is given by Here we have renamed the real and imaginary parts of the field as: imply or are related to any of the proposed swampland constraints. with a general superpotential swampland ideas. Thus e.g. one may ask whether the properties of modular invariance JHEP05(2019)105 : 1 ρ . ) = (4.6) (4.5) 2

m > . Other T dj does not dT i, ρ ) j = ∂j ∂H ( = 1. We can T P ρ P 3 = n/ T ) = 0 and if . Again it is always 57, a saddle point if j . ρ 3 2 = 1 then, at 2 ( m 1 is always a minimum, is ﬁnite and non-zero. : maximum, minimum i n − > ρ H 1+

= 0 with = − dT . Additionally, there is a dH < i = 19 T m . 0 000 T = H 1 then at H = �� 1728) , + 1 1 then T n < − 00 o j H n > 2 H �� | ( 57

n > . 1 then C 2 m 2 | n − �� + . If 2 3 n n > | i 12 = 0. If | 1+ �� η = (0) | − dj 2, in the boundary of the fundamental region, dT I j . T 2 |P T – 17 – 3 = ). The self-dual points are located at -�� 2 m/ = 1 Z 2 , = ˆ G T -�� 1 and V 1728) and a saddle point at diverges in such a way that . − ρ -�� i 57 and a minimum if j . m > ∂j ∂H ( 1 = = 3 all types of extrema can occur at n �� − T T P

one obtains > 2 I we consider the particular case . ρ 0 3 T 000 6 1 at 2 4 H ) H = T ( = 0. Thus indeed it is always a Minkowski minimum, and the same ( 1 T 12 m > | /η = 1 V η 57 or | . 1 = 0. However, m > m 3 2 I = 1 (1) = 0. In both points 2 T − = 1, at ˆ 8 G H W 1 or < n = . Fundamental domain for PSL(2 = 0 = 1 or 000 V H H For Some of these giveH rise to dSThen maxima, the but potential, at no both minima. fixed points, If simplifies to: and the same appliesthen for diverge, so happens for n but not at a fixedwith point. This is the simplest example superpotentialn discussed > above These always yield Minkowski minima. If or saddle point. Namely, it is a maximum if in AdS. In figure see the maximum at non-SUSY AdS minimum at Im By choosing different • • Figure 1 extrema are found on the border of the modular domain. JHEP05(2019)105 (4.7) (4.8) = 0, so 2, which . = 1. We dT dH P = 1 T = = 0, H we are still unable m P = 1, and = 0. the auxiliary field n = 1). There is a SUSY AdS m H V : i , P dT dH = ). They verify o . The potential is always positive 2 j | ( T dj = 0, dT D P → | 1 2 . Note that in the model considered m n in the zoom in around this point. In . This is always a maxima and the − 3 2 2 ρ | = dj ˆ dT G 3 2 n 12 = π | 3 − 1728) and 2 η | j T 2 1 3 I − (1728) + (i.e. T ρ j – 18 – 6 3 ( |P ) → and a non-SUSY AdS minimum at Im 1 2 dT dH 2 T T i i ( ) we plot the potential for 1 → = H 3 = /η T T V | = lim = 1 H = lim C W there is an AdS minimum, since ∝ | i D T = h T = 1, we evaluate the potential at m , a saddle point at = 0. ρ 2 ˆ = G T . Scalar potential for where we have defined at the extrema. By changinga H minimum. we can make it a saddle point or a maximum but not the same plot, at Moving on to where we havepotential defined is positive. Incan figure ( see the predicted dS maximum at This completes the study ofIt extrema is on quite interesting self-dual that points, modular which functions are conspire generic. to produce all kinds of extrema The remaining extrema are the multiple zeros of has a non-zero vacuumpoints, expectation where value. In the last step we specified toexcept dS the minima. self-dual One may wonder why, given the freedom in they preserve supersymmetry andminima are at always the Minkowski. self-dualsupersymmetry points We is in summarize spontaneously the broken the if tables results at for the minimum of maximum at is also on the boundary of the fundamental domain. Figure 2 JHEP05(2019)105 No Yes Yes SUSY No Yes Yes SUSY 0 0 dT dH 6= 0 0 0 dT dH 6= 0 0 0 H 6= 0 . . i ρ 0 0 = = H 6= 0 = 1): right: a zoom around 57 T T . 1 3 4 P 57 . − 1 3 2 < > − = 0, > 19 0 . 000

< m H H 0 19 000 . + 1 H H = 1, 00 H + 1 H n < Minimum Maximum Maximum Min 57 or . 00 < 2 H 57 H . Type of Extrema

(i.e. 3 4 − 2 6 − − < Type of Extrema 0 – 19 – /η 0 3 Min 0 000 / > H 1 H SP (Saddle Point) if else j Max < Max o 2 } = | 3 SP D | W ) {− 2 ρ 0 | 2 | 0 = > = 0 . Classification of the extrema found at . Classification of the extrema found at < T o 12 ( | 2 V } (1728) | (1728) η | 3 V |P 3 C I | T |P = 1) m {− 2 = 0 | 4 3 Table 2 2 T Table 3 a | 12 ( | n | V 1728 η n | (0) V 12 3 I | 1 ∝ P | η 12 T | | η 1 | ∝ 3 I T 1 . Left: scalar potential for = 1 = 0 1 n > n n = 0 = 1 m m m > Figure 3 its dS maximum. JHEP05(2019)105 , ] 7 ρ and = ), as (4.9) 1728) 2 (4.11) (4.10) T 3 T π ( − η j → ) or ( T ( j 0 or instanton 2 ˆ D G . According to [ c , πT 2 i − e , → j is fixed and we have only some + 3 η r V V j 0 + 2 ) and the Dedekind function c 6 T n V ∂ ( i . Since there is a dS maximum at j ] we find: c ∂ V ≤ − j + 4 i 85 ) m V K j 6 q ]. However, in all examples analyzed the refined . It is easy to see why using only this freedom – 20 – ∇ 5 i j π and for other points after fixing = ∇ I c | T 3 V 2 V √ min ( |∇ = | in eq. (2.20) of [ V V |∇ πT 2 i − e P T ) r + 3 in which we plot an example with a dS maximum. This quantity should be n ] including the third condition in the beginning of this paper, seems to be 7 + 4 2 m ( π 2 e are never dS minima. → One can also check whethermaxima, the dS which condition in is fact obeyed can forH large be moduli, studied away analytically. from any Taking dS for those points we have to make sure that the smallest eigenvalue of the Hessian satisfies: In all examples analyzed with a dS maximum, the third condition on the Hessian is satisfied. as in figure either negative or larger thanit a will certain not constant be true at this point for any modes depending on theoriginal dS specific swampland model. conjecture ofconjecture ref. We [ have [ found dSobeyed. maxima, We in have checked violation it of numerically for the each case by plotting the non-perturbative level. Inmodular particular invariance we have dictates seen thewhich how existence for would of large allow values a of forstring dynamical models the global barrier studied moduli, correspond symmetries. censoring tothe large towers of moduli distance The states conjecture origin becoming massless, ideas. of in agreement such with The a towers barrier of in states the are specific KK, winding, H 4.2 Swampland conjectures We have already seen thatswampland modular ideas invariant and potentials how have such features potentials consistent appear with in the models of reduced supersymmetry at freedom in choosing thethe dependence Minkowski or on AdS minimumare cannot be zero “perturbed” at to the athat minimum dS something minima. nothing similar changes If happens bymodular at adding region, the them but other as wechecked points perturbations. that have in the been We the other unable believe minima boundary to in of prove the the it boundary, fundamental which analytically. appear for Numerically some we specific have choices of that, starting from a Minkowskithere vacuum in should a be self-dual point someadding and nearby adding (modular dS some minima, invariant) perturbation perturbations notconstructed one only around AdS. never the lands Klein This in Invarianta is function consequence dS. in of fact modular The not invariance. whole the The potential dependence case, on is and to uplift one of the Minkowski vacua that we have found to dS. One would naively say JHEP05(2019)105 . T = 1 P = 0, ]. It would be m 118 = n ] in these one-modulus 4 . This quantity is always positive and P – 21 – . The ratio is always bigger than one and grows linearly with Im 6 , so it is guaranteed that in this limit the dS conjecture holds. T /η 3 / 1 j = . Such vacua should be unstable if the conjecture is correct, perhaps W 2 for | V V |∇ . is the highest exponent of the polynomial r It is interesting to note that this is an specific exampleWe of have a not large much to class say of about scalar the AdS conjecture of ref. [ gauge groups andmultiple charged contributions particles. and amay In subsector be the which perhaps by overwhelmed enda by itself the other dS would minima. sectors total yield of This vacuum e.g.subsector the is an energy of theory, something the yielding AdS depends that theory. positive vacuum, we on energy cannot and predict on the basis of the potential of a interesting to check if such an instability exists. 4.3 Two moduliThe model above single-modulus classcompactifications of potentials yield is effective already field very rich. theories However which generic contain string typically multiple moduli, models. This conjecture statesconsistent that with there quantum are gravity. no Inthe non-SUSY, the self-dual AdS points class stable are of vacua SUSY. in modelsboundary a There described of are theory above the non-SUSY all AdS fundamental AdSdepicted vacua region, vacua only in in in as figure other in pointsdecaying the at by the simple a mechanism example analogous with to that of bubbles of nothing, as in [ potentials which obey the refinedmonotonous swampland conjecture decreasing but potential, do not astype correspond of considered to potentials. a up simple to now, and leading to quintessence Figure 4 where it goes to infinity forAgain, large we resorted to numerical methods to check that it holds everywhere in all examples. JHEP05(2019)105 B = 1 . (4.12) (4.13) N 2 vanishes | Ω S . A simple | F 2 S | , 2 | S ) H | . In appendix 3 S 6= 0 there are new S, T − ( S or 2 | F W | Ω T | 2

2 ˆ . Essentially the same classes G ) + log B ∗ H π T 3 . 2 ) − + T T ( 6 ) H T dT dH ) (

S η 3 log ( 2 I 3 T − Ω( 4 ) ∗ – 22 – + S ) = 2 | − = 2 subsectors from fixed tori have non-trivial dy- H | S S, T 2 ( N | Ω W log ( − − S Ω I ) = ∗ iS 2 = 1 supersymmetry. On the other hand, we have discussed fairly | ,T,T N ∗ (4.14) 12 | η | ) = S, S 3 ∗ I ( 1 T I G S , S, S . So, inspired by the gaugino condensate model one can consider an ∗ 16 3 = T,T The string constructions we have considered are explicit toroidal orbifolds of the het- In this sense, it is interesting to check whether the results found above for a single ( V forbidden by exponential growth ofarises the from (non-perturbative) infinite scalar towers potential. ofinterestingly This linked states potential to becoming the light, swampland distance soanism. conjecture the in This absence a very new of explicit mechanism global andsupersymmetry, may novel symmetries of mech- well is the be mechanisms discussed the in counterpart, the in literature in the cases context with of 8 lower supercharges. (or no) extent to which the latter imply satisfying the swamplanderotic constraints. string, for whichnamics moduli due in to theirparticular, appearance we in have threshold described correctionsuli of examples space, 4d in where gauge which a kinetic the global functions. access shift In to symmetry points for at the infinity axion in would mod- be restored, is dynamically points. On one hand, wesuch have modular constructed explicit symmetries string are theoryswampland compactifications ingrained in constraints in which arise the from UVin construction, the models and dynamics with derived of 4d general how e.g. effective several non-perturbative theories for superpotentials scalars enjoying duality modular symmetry and explored the 5 Conclusions and outlook In this paperaction we for moduli have to put test swampland conjectures forward in the regimes beyond weak use coupling expansion of the modular symmetries in the effective essentially two classes of vacua,or depending on not whether at the the auxiliarymodulus field minimum. case of discussed If above itpossibilities and vanishes, which the are the same summarized structure in conclusions theof of apply. tables minima vacua in For as appendix are in identical the to single modulus the case single are obtained and again no dS minima are found. We denote derivatives with respectwe to study the the fields extrema with of a sub-index this general potential in some detail. For general Ω there are with The resulting scalar potential is then given by: modulus is very muchexample modified by is the provided additionsection by of an the extra heterotic chiralsupergravity gaugino multiplets model condensation superpotential discussed in JHEP05(2019)105 would 2 T ) acting on and enlarged moduli space Z , the presence of Wilson lines on 20; 2 , T – 23 – × ) symmetries, since the Wilson lines are not invariant. In fact, Z , A final remark concerns the fact that, naively, modular symmetries are manifest in the We have performed a careful numerical analysis of the properties of scalar potentials We have also considered several dual pictures, in particular type I’ / Horava-Witten discussions. This work has been supported by the ERC Advanced Grant SPLE under discriminating it from the swampland. Acknowledgments We thank A. Font, A. Herráez,F. Marchesano, M. Montero and I. Valenzuela for useful act consistently on the setthe of fluxes. all flux Restricting vacua to byhave axion acting an scalars, non-trivially axion this on monodromy is bothmuch structure. just the richer a moduli The set re-statement and extension that ofconstraints to the fluxes, of the flux invariance necessarily potentials full under including modular dualityvacua transformations non-geometric group bring of brings ones. new moduli in insights in into We a the both expect landscape the of that structure flux of the the landscape and the set of conditions e.g. in heterotic compactificationsseem on to K3 break thethe SL(2 correct interpretation islines that and the define duality a symmetrieswhich larger includes act duality non-trivially the group on Wilson SO(4 the lines. Wilson Similarly, in flux compactifications, the duality groups effective actions of compactifications inintroduction which of further fluxes ingredients typically are induces absent.modular scalar For transformations, potentials instance, thus which seemingly arethe breaking not effective invariant the under action. symmetry the In explicitlysense fact, at this in the interpretation which level is the of far symmetry from is complete, present. and The there situation is is a precise analogous to the statement that gravity. This is certainlythat a we do promising not research claim that directionapply the to fact to pursue. that any we have Let consistent beenmodular us theory unable invariant to however effective of find insist actions any quantum dS we gravity. minimum have should been Only unable that to in find the any dS simplest minimum. classes of already mentioned distanceto conjectures. (supersymmetric) Minkowski The andno potentials AdS dS have minima, minima. extrema and MoreoverThese the dS that remarkable potentials maxima facts correspond satisfy suggest or the the saddle tantalizing refinedingrained proposal points, de in that Sitter but modular the conjecture properties conditions. conditions are deeply to guarantee consistency of effective theories with quantum quantum gravity may applythe in future. a far more general context, whichin we general effective hope theories to forinvariance explore scalars, under with modular in essentially symmetries. the Surprisingly, onlyto this constraint requirement render that alone seems they the to enjoy suffice theories consistent with a number of swampland conjecture, besides the theory compactifications, in which theor degrees Kählermoduli of can freedom associated bethe to studied swampland complex structure in distance isolation.ubiquitous conjecture appearance in Such of the statesalong modular present are the groups context. lines key discussed, in to the We CY strategy the of expect moduli relating derivation that, modular spaces, of properties given and with consistency the of with string dualities JHEP05(2019)105 F 0 (as (A.3) (A.4) (A.5) (A.6) (A.1) (A.2) which r < /T 1 . → − n 1 t T n c by the expression r =0 ∞ X n ) has self-dual points at , +1 and 2 Z . / . A meromorphic function , T ρ 3 2 ν ρ = 1 t Z → . . ) / of the form bc ) p ) ∗ ) πT T ρ ν 2 ρ T i ) = Z − , ρ ( e , ∞ − T + ( F ,ρ, = r T T , ad = 1 F ) ( X ( ) 6=1 d Z ρ p T . The lowest order of the expansion of , q ∈ + the uniformizing variables are has an expansion in terms of uniformizing n D + + = q ) = SL(2 ρ F n T ρ cT ]. The modular group is generated by the ν Z ( i, ρ a , . The group PSL(2 , 85 + , t = ( = =0 ∞ n 1 – 24 – 1 1 X n t 2 ν T n , a, b, c, d if ∞ b b d + ν ) ) b d r i i q + + ∞ =0 one expands + ∞ + − + ν X n cT aT ) = T T = 2 cT aT ( ( T / ( 1 r 1 ν → ∞ F 12 F t = T −→ 1 at infinity. At t ) = is shown in figure . Since changing the sign of all the parameters leaves invariant the T T , as well as at infinity. A modular form admits a Laurent expansion F t ( 3 ) admits expansions around D F T π/ ) ( 2 i i F e . Around the self-dual points + p = ν T . This is generated by a shift transformation ( ρ it Z) modular forms + , θ with is the order of = ∞ T i, ρ ν ) is said to have modular weight is denoted The symmetry divides the complex plane in equivalent regions and the conventional T p = ( One interesting consequence of thisin formula the is physics that examples for negative in modular the weight main tex) there must necessarily be a singularity in the In terms of them It may be proven that the orders are related to the modular weight and at each point in the interiorin of the closed domain variables. In particular, as fundamental domain T with relates small and large transformation, the group isF actually PSL(2 Here we just list sometext. well known We properties follow fortransformations the closely modular the forms appendix discussed in in the [ main 65480-P from the MINECOSevero and Ochoa” the Programme. grantnumber SEV-2016-0597 The FPU16/03985. of work of the “Centro E. de Gonzalo Excelencia is supportedA by a FPU SL(2 fellowship contract ERC-2012-ADG-20120216-320421, by the grants FPA2016-78645-P and FPA2015- JHEP05(2019)105 (A.7) (A.8) (A.9) rather (A.11) (A.12) (A.14) (A.15) (A.16) (A.17) (A.10) 2 G = 1 k , )) .... . For of weight 12 given by T ( )) (A.13) k + j T 2 ( ( q . . j ). Then the non-holomorphic P is implied. ( may be written as ... d k n , 2 6 ... 2 T P . ) − D + 3 T G ) 2 ) . + 8 3 π n/ T n ) q cusp form T 3 3 ). Some useful expansions in powers 49 ) ( cT . ) ( ) Im q T 1 2 ( T + 4 T ) may be written in terms of the cusp 96 ( − T T ( ( 24 G η ( − G j T 3 4 πc j / T ) − 4 + 3 + 21493760 ( 1 ) 2 1 2 ∆( ) G j T 2 2 q G n T ( − q q ( T m/ ( m 2 η 20 2 Z 72 π G 12 1 G ∈ ) + 2 2 4 ) for large Im 2 π 12 12 ) − – 25 – q T ) T 91125 − d ,n 1728) π X ( q ( 1 T ) = ∆( 6 ) = 675 n ( F + = − ∗ 24 T G η 1 + 256 ) ) = ( which is regular in ) is at the origin of the superpotential divergences for − η T T cT T r t ) = ( ( ( r T,T ( 1 24 a polynomial of j 2 j ( T dT / ) = ) ( 1 2 + 744 + 196884 dη 2 T k → P ˆ − T 3 G 2 q π 1 ( q 2 η ∆( G ) = ( G = T at large ( ) = ) = 1 ) = 1 − T T H ) ( ( − T 2 j ) ( T ( G T F η ( η are positive integers and 1 these are holomorphic modular forms of weight 2 πT 2 i e 0 and the exponential growth of n, m k > = Of particular interest are the modular forms constructed from the Eisenstein series < ). Any modular form of weight modular form q T 2 ∞ ( ˆ The divergence of large modulus. with of and equivalently as form as It may be shownj that any holomorphic modular invariant form is a rational function of One can check the identity, used in the main text The holomorphic Klein modular invariant form The Dedekind function is given by For transforms as a connection G transforms as a modular form of weight 2. There is a ν fundamental domain. Thus if, on physics grounds, there are no such singularities, then JHEP05(2019)105 i ) 4 S = (B.4) (B.1) (B.2) (B.3) T arise as a can also be 4 P . 2

. = 0 and 2 ˆ o G 2 | dT dH H Ω | multiplied by the constant π 3 2 3 T − + = 0 for Type A extrema reads: 2 after its minimization is trivial, | . Ω S dT = 0 then ∂S dH ∂V

6= 0 n − 2 = 0, and the same happens at | , S 2 I Ω) Ω H dj T | dT I − 3 respectively are always minima. iS = S 4 ] that the extrema from section Ω = 0 i 2 | Ω 0 (only possible in extrema of Type B) and = 0 so if − 85 I − n = 0 case, for which we rely on numerics. Now = > ρ H S = 0 in both extrema. Just like it happened for iS dj dT 12 – 26 – T to show that if at the self-dual points there is a m | 0. There is no simple way to decide the sign of 2 Ω = 2 (2 2 | = η I V 4 | | | | = or Ω < Ω 6= 0 so the condition for Type B extrema reduces to | 3 T iS 2 I H ρ (0) − 2 2 2 ˆ T G | S I dT dH = = 0 or Ω : ∝ | Ω − |P S | I ρ n 2 T S 2 | iS 16 h 2 = | Ω i − 2 ∗ T 2 − 1 then at − ) = | Ω ∗ e S Ω 1 then = 1 then Ω SS I − n > n = 0) at this stage. The condition Ω T,T This case is the exception, for which analytically we have not been S iS ( 2 I If = 0 at both extrema. By changing H we can make it a saddle point If S 2 m > Ω S | h V I 4 V − iS 1. 2 1 or | = 0. = 1. m n > m > m 1. The the potential is simply . In this way, we recover the single modulus potential and the results of section 1 or 2 | ) I m > S = 0 or S = 1 or 2 Ω( | It is a minimuma if maximum if the vacuum energy in this case. If we consider Ω as a sum of exponentials numerically or a maximum but not a minimum. n able to prove our result. At n > for Type A, if n Ω = 0. Therefore proof that there aredone no analytically, dS except minima for in the we the go self-dual through points the or sameminimum in steps then the of the zeros section potential of cannot have positive minima. If the S field issince in the potential a is Type the A one we extrema, would the find for dependence a on single modulus which we discussed in the mainis text. not trivial Although in anymore, Type we B will extrema see the how dependence the on results Ω( can be easily extended. In particular the while for Type B: Using this equation weis can broken distinguish spontaneously two by types theremains minimization of unbroken ( in extrema. S. In In Type contrast, in B Type extrema A SUSY extrema, SUSY In this appendix weby study gaugino the condensation. extrema Let of usparticular the recall case. potential from The in [ second the field provides two a moduli new model condition for inspired spontaneously susy breaking: B General analysis of minima for two fields JHEP05(2019)105 = | (B.5) H | No No Yes Susy = 0, ]. ]. No No 0 dj 0 0 Yes 0 dT Susy and we have not < < > > 2 SPIRE 2 2 2 | SPIRE | | H | . IN = 0, Ω IN Ω 57, a saddle point if Ω . Ω . | [ | . o | | i ][ ρ 2 1 2 3 : maximum, minimum 2 2 3 | 2 dj i dT = = − Ω − − | − − . Again we do not prove it T T = ). These are SUSY minima 2 2 = 3 2 2 3 2 < | | j | | ( 2 Ω T Ω − Ω Ω 0 > 000 ˆ P G 2 H −

| − Minimum − − H Ω S S S S 19 Minimum . Ω < Ω Ω Ω Type of Extrema − No simple criteria No simple criteria I I I I The string landscape, black holes and S hep-th/0601001 hep-th/0509212 Type of Extrema + 1 iS 0. iS iS 57 iS [ , Ω . 2 2 2 0 2 00 = 1728, | I | | 2 | H H j − o iS

2 V < 2 | : | V < i Min Min Max Max Ω n | = 3 2 12 – 27 – | (2007) 060 | o 0 − η T 2 | ) | 2 i 06 3 | I Ω | T Ω = = 0 and V < I (1728) 3 = 0 = 0 − S T ( − dT dH V V |P ) S ), which permits any use, distribution and reproduction in JHEP 16 2 ρ V , | Ω I Ω = . Classification of Type B extrema at . Classification of Type B extrema at = 0 = 0 = 0 ) = iS − ∗ T 2 57 and a minimum if | . ( all types of extrema can occur at m V V S 1 n Ω V − P T,T I = 0 extrema are the multiple zeros of ∝ ( CC-BY 4.0 Table 5 Table 4 > iS Numerically all min have H V so for 2 | 0 This article is distributed under the terms of the Creative Commons 000 | n H H 1 Numerically all min ∝ The string landscape and the swampland = 0 = 1 (1728) m m m > 1 57 or . |P 2 = 0 = 1 2 − n n n > gravity as the weakest force C. Vafa, N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, m/ The remaining However, we have searched numerically for dS minima at other points on the boundary, < [1] [2] 0 000 H H References supposing Ω is givenfound by any. a sum of exponentials, forOpen various choices Access. of Attribution License ( any medium, provided the original author(s) and source are credited. analytically, but numerically we also find that and are always in Minkowski. Wepoints summarize the in results the for next Type B two extrema tables. at the self-dual By choosing different or saddle point. Namely, it is a maximum if we find that it1728 is always negative. At JHEP05(2019)105 67 ]. , , , SPIRE ]. IN , ][ Fortsch. Phys. ]. SPIRE ]. ]. , ]. IN Adv. Theor. Math. (2018) 570 , ]. ][ (2017) 034 SPIRE SPIRE SPIRE SPIRE IN 08 IN IN ]. IN ][ ]. ]. SPIRE ]. ][ ][ B 785 ][ IN ][ ]. arXiv:1802.08698 JHEP SPIRE ]. SPIRE SPIRE [ , SPIRE IN (2019) 1800097 IN IN IN ][ ][ ][ ][ SPIRE 67 ]. On the Cosmological Implications of the SPIRE IN Phys. 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