Marginal deformations of heterotic interpolating models and exponential suppression of the cosmological constant
aa r X i v : . [ h e p - t h ] J a n NITEP 89OCU-PHYS 531Jan 26, 2021
Marginal deformations of heterotic interpolating modelsand exponential suppression of the cosmologicalconstant
H. Itoyama a,b,c ∗ , Sota Nakajima b † a Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),Osaka City University b Department of Mathematics and Physics, Graduate School of Science,Osaka City University c Osaka City University Advanced Mathematical Institute (OCAMI)3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
AbstractFollowing our previous work of 1905.10745 [hep-th], 2003.11217 [hep-th], we study heteroticinterpolating models D dimensionally compactified with constant background fields that in-clude the full set of Wilson lines and radii. Focusing on the phenomenoloically viable super-symmetry restoring parameter region, we analyze the pattern of gauge symmetry enhance-ment and the representation of massless fermions. We obtain the set of cases with the expo-nentially small cosmological constant. Our analysis does not depend on non-supersymmetricendpoint models of interpolations. A part of the moduli space of interpolating models is inone-to-one correspondence with the counterpart of toroidal compactification of heteroticsuperstrings. ∗ e-mail: [email protected] † e-mail: [email protected] Introduction
Quest of unification of electroweak and strong forces and the experimental discovery of brokensupersymmetry have been major driving forces of some of our high energy theoretical andexperimental activities for more than thirty years. Currently, the presence of dark energy inour universe obtained from the latest observation presents a provocative question of how togenerate a small and non-vanishing cosmological constant from physics at string scale.A few years after the introduction of heterotic superstrings[1], the modular invariant SO (16) × SO (16) string theory in ten spacetime dimensions[2] was constructed, which istachyon free and does not possess spacetime supersymmetry. Following this work, a possi-bility was suggested in [3] that, upon compactification to lower dimensions (see [4] for relatedwork), one can construct models nowadays called interpolating models which connect theheterotic superstring and this non-supersymmetric heterotic cousin by the Z twist operationas well as by a radius parameter and that, in the supersymmetry restoring region, one cansuppress the value of the cosmological constant. This compactification with twist generalizesthe role played by the T duality, which is originally self-dual for the bosonic string[5], andthe original picture of the heterotic vacua upon toroidal compactification[6].Twisted compactifications[7] of non-supersymmetric heterotic string theory have nowbeen developed in a variety of directions including S duality[8, 9] and more phenomenologicalstudies[10, 11, 12, 13, 14] and many of these cases are characterized as those where the one-loop cosmological constant obeys the following formula[3]:Λ (10 − d ) = ( n F − n B ) ξa − d + O ( e − /a ) , a ≈ . (1.1)Here ξ is a positive constant computable from the knowledge of the spectrum of Kaluza-Kleintowers alone, and therefore from higher-dimensional quantum field theory alone and n F and n B represent the degrees of freedom of massless fermions and massless bosons respectively .The modes in which n F = n B , that is, the cosmological constant is exponentially suppressedat one-loop level are called super no - scale models [14]. Not only interpolating models butalso string models with broken supersymmetry in general have been attracting attention inthe context of string phenomenology and swampland program[15]. The clean separation of (1.1) into two terms and related field theoretic limits are currently under studyby us [16].
1n two of our recent publication[10, 11], we investigated the 9D interpolating modelswhich connect the two of the heterotic superstrings with SO (16) × SO (16) heterotic stringby the full set of Wilson lines. The presence of Wilson lines at a generic point of themoduli space breaks the nonabelian gauge group set in 10-dimensional string theory into theproduct that includes the abelian gauge groups, spoiling the hope for the unification. Thisis, however, retrieved in string theory itself: gauge symmetry enhancement takes place atthe extrema of the one-loop potential on the moduli space. We have managed to determinea set of points in the moduli space where this in fact occurs and the nonabelian gauge groupsassociated with these points.In this letter, we extend our previous works in one-dimensional compactification to the D -dimensional counterpart. We generalize the analysis of gauge symmetry enhancementin interpolating models by focusing on the splitting of Narain lattice. In Sect. 2, by D -dimensional twisted toroidal compactification, we construct heterotic interpolating modelsthat are marginally deformed by the full set of the constant moduli. In Sect. 3, we studythe moduli space of the interpolating models in order to identify the massless spectra. Weshow that the Wilson lines in the interpolating heterotic models can be represented bythose in untwisted toroidal heterotic models. Namely, the massless spectra of interpolatingmodels can be specified by using the information of heterotic superstrings with maximalsupersymmetry. Finally, we apply this analysis to two concrete examples of supersymmetricendpoint models and obtain the same results as in our previous works[10, 11], including thepattern of symmetry enhancement with the exponentially suppressed cosmological constants. D -dimensional toroidal compactification of generalheterotic interpolating models In heterotic string models D -dimensional toroidal compactified and deformed by the constantmoduli, the dimensionless internal momenta are[6] ℓ IL = m I − w i A Ii ,p Li = 1 √ (cid:18) m · A i + n i + w j (cid:18) G ij + B ij − A i · A j (cid:19)(cid:19) ,p Ri = 1 √ (cid:18) m · A i + n i − w j (cid:18) G ij − B ij + 12 A i · A j (cid:19)(cid:19) , (2.1)2here i = 10 − D, · · · , I = 1 , · · · ,
16. We shall consider the heterotic models with D compactified dimensions, in which only X -direction is twisted. We should here introduce amomentum lattice defined asΛ [Γ; α, β ] ≡ ( η ¯ η ) − D η − X m I ∈ Γ X w ∈ Z + α X n ∈ Z + β ) X w i =9 ,n i =9 ∈ Z q ( | ℓ L | + p L ) ¯ q p R , (2.2)where Γ is a 16-dimensional Euclidean lattice.In order to construct an interpolating model, we start from a heterotic string modelcompactified on T D with maximal supersymmetry: Z ++ = Z (8 − D ) B (cid:0) ¯ V − ¯ S (cid:1) X β =0 , / Λ [Γ ; 0 , β ] , (2.3)where Γ is a 16-dimensional Euclidean even self-dual lattice and Z (8 − D ) B = τ − D ( η ¯ η ) − (8 − D ) is the contribution from the bosonic part propagating in (10 − D )-dimensional spacetime.Let T (9) and Q be operators representing half translations around X -direction and X I -directions respectively. By using a shift vector δ I ∈ Γ , the operator Q can be representedby exp (2 πim · δ ) for m I ∈ Γ , and we can split Γ into Γ +16 and Γ − which are respectivelyeven and odd under Q :Γ +16 = (cid:8) m I ∈ Γ (cid:12)(cid:12) δ · m ∈ Z (cid:9) , Γ − = (cid:26) m I ∈ Γ (cid:12)(cid:12) δ · m ∈ Z + 12 (cid:27) . (2.4)An interpolating model is constructed by orbifolding Z ++ by the Z action α = ( − F Q T (9) ,where F is the spacetime fermion number. The action of α on Z ++ gives Z ++ α −→ Z + − = Z (8 − D ) B (cid:0) ¯ V + ¯ S (cid:1) X β =0 , / e πiβ (cid:0) Λ (cid:2) Γ +16 ; 0 , β (cid:3) − Λ (cid:2) Γ − ; 0 , β (cid:3)(cid:1) . (2.5)For the partition function to be modular invariant, we must add the twisted sectors: Z + − ( − /τ ) = Z − + ( τ ) = Z (8 − D ) B (cid:0) ¯ O − ¯ C (cid:1) X β =0 , / Λ [Γ + δ ; 0 , β ] . (2.6) Z − + α −→ Z −− = Z (8 − D ) B (cid:0) ¯ O + ¯ C (cid:1) X β =0 , / e πiβ (cid:0) Λ (cid:2) Γ +16 + δ ; 0 , β (cid:3) − Λ (cid:2) Γ − + δ ; 0 , β (cid:3)(cid:1) , (2.7)3here we assume that δ is an integer. As a result, the total partition function is Z (10 − D ) int = 12 (cid:0) Z ++ + Z + − + Z − + + Z −− (cid:1) = Z (8 − D ) B (cid:8) ¯ V (cid:0) Λ (cid:2) Γ +16 ; 0 , (cid:3) + Λ (cid:2) Γ − ; 0 , / (cid:3)(cid:1) − ¯ S (cid:0) Λ (cid:2) Γ +16 ; 0 , / (cid:3) + Λ (cid:2) Γ − ; 0 , (cid:3)(cid:1) + ¯ O (cid:0) Λ (cid:2) Γ +16 + δ ; 1 / , (cid:3) + Λ (cid:2) Γ − + δ ; 1 / , / (cid:3)(cid:1) − ¯ C (cid:0) Λ (cid:2) Γ +16 + δ ; 1 / , / (cid:3) + Λ (cid:2) Γ − + δ ; 1 / , (cid:3)(cid:1)(cid:9) . (2.8)In the limit where the volume of the compact space goes to zero, we can check that Z (10 − D ) int provides a 10D non-supersymmetric endpoint model whose partition function is Z (10) M = Z (8) B (cid:8) ¯ V Λ (16) (cid:2) Γ +16 (cid:3) − ¯ S Λ (16) (cid:2) Γ − (cid:3) + ¯ O Λ (16) (cid:2) Γ +16 + δ (cid:3) − ¯ C Λ (16) (cid:2) Γ − + δ (cid:3)(cid:9) , (2.9)where Λ (16) [Γ] = η − P m I ∈ Γ q | m | . As the partition function (2.8) reproduces that of anoriginal 10D supersymmetric model in the large volume limit, we can check that the modelconstructed above interpolates from a 10D supersymmetric endpoint model to a 10D non-supersymmetric one. There are various interpolations, which depend on the choice of theshift vector δ I [2, 4].In this letter, we are interested in a part of the moduli space where supersymmetry isasymptotically restoring because the cosmological constant can be exponentially suppressedin this region. The states with w = 0 acquire huge mass in this region, so the twisted sectors,in which α = 1 /
2, are suppressed. We shall henceforth focus only on the contributions fromthe untwisted sectors.
In this section, we study the massless spectra of interpolating models constructed in theprevious section. The spectra of string theories with some compactified dimensions can beclassified into two sectors; sector 1 does not depend on the moduli and sector 2 does. Insector 1, the massless states are at the origin of the momentum lattice, i.e. ℓ IL = p iL = p iR = 0,and their degree of freedom is 8 × (8 + 16), which corresponds to a 10D gravity multiplet and10D gauge bosons of the Cartan subalgebra of a gauge symmetry whose rank is 16. Since4he origin of Γ is even under Q , there is no massless fermion in sector 1 in interpolatingmodels. Thus, the special points in the moduli space, where some states in sector 2 aremassless, can realize n F = n B . In sector 2, the massless states must satisfy the following twoconditions: | ℓ L | + p L = 2 , p R = 0 . (3.1)Using eq. (2.1) and the invariant Lorentzian inner product | ℓ L | + p L − p R = | m | + 2 n i w i ,these conditions are expressed as | m | + 2 n i w i = 2 , m · A i − w j (cid:18) G ij − B ij + 12 A i · A j (cid:19) = − n i . (3.2)At generic points in the moduli space, sector 2 contains no massless states and the gaugesymmetry is U (1) D . However, there are special points in the moduli space where addi-tional massless states appear if conditions (3.2) are satisfied.Before we study the pattern of symmetry enhancement, it is worth to comment on theshift symmetry in the moduli space. Let us consider a shift of a Wilson line A Ik by a vector a Ik accompanied with a shift of B ki by a k · A i : A Ik → A Ik + a Ik , B ki → B ki + 12 a k · A i (cid:18) B ik → B ik − a k · A i (cid:19) , (3.3)If k = 9 and a Ik ∈ Γ +16 , then we can check from eq. (2.1) that the partition function (2.8) isinvariant with the replacements m I → m I − w k a Ik , n k → n k − w k | a k | + m · a k , n i = k → n i = k , w i → w i . (3.4)In fact, the shift (3.3) is a part of the duality symmetry of heterotic strings on T D [17] .On the other hand, for k = 9, which is the direction with a twist, the partition function isinvariant if a Ik ∈ rather than a Ik ∈ Γ +16 , as w ∈ Z + α and n ∈ Z + β ). By thedefinition (2.4) of Γ +16 , it is clear that v ∈ Γ +16 if v ∈ .In this letter, for simplicity, we focus on the states in sector 2 with w i = 0 for all directionsnot only for i = 9. Namely, the massless spectra depend only on the value of the Wilson In non-supersymmetric heterotic models, the shift vector a Ik has to be an element of Γ +16 , not Γ , inorder to maintain the splitting Γ = Γ +16 + Γ − . A Ii , and G ij and B ij take generic values. Then, the conditions (3.2) are | m | = 2 , m · A i = − n i . (3.5)Note that if an even self-dual lattice Γ is the root lattice of Spin (32) / Z (i.e. Γ =Γ (16) g + Γ (16) s ), the elements with | m | = 2 are only in Γ (16) g . We briefly review the conjugacyclasses and the characters of SO (2 n ) in Appendix A.First, we consider conditions (3.5) in maximally supersymmetric models compactified onuntwisted tori. Then, we investigate the moduli space of interpolating models compacti-fied on twisted tori. We shall provide the relation between the Wilson lines in maximallysupersymmetric models and those in interpolating models.(1) Maximally supersymmetric heterotic modelsThe partition function takes the form (2.3), and m I ∈ Γ and n i ∈ Z for bothspacetime vectors and spinors. Denoting as A ( g ′ ) i the Wilson lines realizing a semisimplesubalgebra g ′ ⊂ g , where g is SO (32) or E × E , as a nonabelian part of a gaugesymmetry, A ( g ′ ) i should satisfy m · A ( g ′ ) i ∈ Z for m I ∈ Γ g ′ and | m | = 2 m · A ( g ′ ) i / ∈ Z for m I ∈ Γ g \ Γ g ′ and | m | = 2 , (3.6)where Γ g ′ and Γ g are the root lattices of g ′ and g . For example, the condition thatthe gauge symmetry is maximally enhanced, i.e. g ′ = g , is A ( g ) Ii ∈ Γ ∗ g , where Γ ∗ g is theweight lattice of g . From the replacements (3.4), we find that the interesting sector,in which w i = 0 and m I ∈ Γ g , is invariant under the shift (3.3) with a Ik ∈ Γ ∗ g , so thecondition A ( g ) Ii ∈ Γ ∗ g means A Ii = (0 ) up to the shift symmetry .(2) Interpolating heterotic modelsIn order to distinguish from the maximally supersymmetric case, we express theWilson lines in interpolating models as ˜ A Ii . From the partition function (2.8), vectorsand spinors have the values of n i and m I following Table 1. Conditions (3.5) for the The rank of g ′ is not always 16. If it is smaller than 16, there is the product of U (1) in the gaugesymmetry, in addition to the semisimple algebra g ′ . Including the w i = 0 sector, the models are invariant under the shift (3.3) with a Ik ∈ Γ . If a Ik ∈ Γ v or a Ik ∈ Γ c , the Wilson lines can not vanish by the shift symmetry and there are possibilities to give the gaugesymmetries larger than SO (32), e.g. SO (34). i =9 n m I vector Z Z Γ +16 Z Z + 1 Γ − spinor Z Z Γ − Z Z + 1 Γ +16 Table 1: Assignment of n i and m I in interpolating models X i =9 -directions are the same as in the maximally supersymmetric case since n i =9 ∈ Z for any m I . On the other hand, for the X -direction, the value of n depends on thesplitting of Γ . So, the second condition in (3.5) for massless vectors is˜ A · m ∈ Z for m I ∈ Γ +16 and/or ˜ A · m ∈ Z + 1 for m I ∈ Γ − , (3.7)while for massless spinors,˜ A · m ∈ Z + 1 for m I ∈ Γ +16 and/or ˜ A · m ∈ Z for m I ∈ Γ − . (3.8)From (3.7), we find that the Wilson lines ˜ A ( g ′ ) i which realize a semisimple Lie algebra g ′ in interpolating models are represented by using A ( g ′ ) i as follows:˜ A ( g ′ ) i =9 = A ( g ′ ) i =9 , ˜ A ( g ′ )9 = 2 (cid:16) A ( g ′ )9 + δ (cid:17) . (3.9)For instance, the Wilson lines with maximally gauge symmetry enhancement are˜ A Ii =9 = (0 ), ˜ A I = 2 δ I . We can check from the replacemants (3.4) that the sec-tor with w i = 0 and m I ∈ Γ g in interpolating models is invariant under the shift (3.3)of ˜ A I by a I ∈ ∗ g , which is consistent with the relations (3.9) and the invariance of A I under the shift (3.3) with a I ∈ Γ ∗ g .The massless spinors transform in a different representation of g ′ than the masslessvectors if they exist. Conditions (3.8) and the relations (3.9) imply that spinors canbe massless if there is a set of m I ∈ Γ g \ Γ g ′ satisfying A ( g ′ )9 · m ∈ Z + 1 / δ I , which determines the non-supersymmetric endpoint model of the interpola-tion, is not required in order to clarify the massless spectra of interpolating models.7 .1 Example 1: supersymmetric SO (32) endpoint model As a example, let us consider the interpolation in which one of the endpoint model isthe supersymmetric SO (32) heterotic model. The choice of the model, which is non-supersymmetric, at the other endpoint depends on the shift vector δ I : for example, thechoice δ I = (cid:16)(cid:0) (cid:1) , (cid:17) gives the SO (16) × SO (16) model, which was investigated in [10, 11].In the 10D supersymmetric SO (32) model, Γ is the root lattice of Spin (32) / Z and thecontribution of X I to the partition function is η − X m I ∈ Γ q | m | = O + S = O O + V V + S S + C C . (3.10)As simple configurations, we consider the following Wilson lines A i : A Ii =9 = (cid:0) (cid:1) , A I = (cid:18) p , (cid:18) (cid:19) q , (cid:18) (cid:19) r (cid:19) , p + q + r = 16 . (3.11)Note that the above configuration is in the maximally supersymmetric model and we canobtain those in the interpolating model by using the relations (3.9). Obviously A i =9 · m ∈ Z for any m I ∈ Γ , so the massless spectrum of the interpolating model is determined by A I .Following the above discussion about the massless spectrum in sector 2, massless vectorsand massless spinors have the following values of m I :vectors: m I = (cid:0) ± , ± , p − , q + r (cid:1) , (cid:0) p , ± , ± , q − , r (cid:1) , (cid:0) p + q , +1 , − , r − (cid:1) , (3.12)spinors: m I = (cid:0) ± , p − , ± , q − , r (cid:1) , ± (cid:0) p + q , +1 , +1 , r − (cid:1) , (3.13)where the underline indicates the permutations of the components. Thus, the gauge sym-metry is SO (2 p ) × SO (2 q ) × U ( r ) and the massless spinors transform in a bi-fundamentalrepresentation of SO (2 p ) × SO (2 q ) and an antisymmetric representation and its conjugateof SU ( r ). At the point (3.11) in the moduli space, n B = 8 { p ( p −
1) + 2 q ( q −
1) + r ( r −
1) + 24 } , (3.14) n F = 8 { pq + r ( r − } , (3.15) We have omitted the abelian factors which come from the vectors G iµ and B iµ . ×
24 is the degree of freedom in sector 1. With p + q + r = 16, the solutions of n F = n B are ( p, q, r ) = (6 , , , (6 , , , (7 , , SO (12) × SO (12) × U (4)or SO (14) × SO (12) × U (3) or SO (18) × SO (14). E × E endpoint model The other example is constructed from the supersymmetric E × E heterotic model. In thissupersymmetric model, an even self-dual lattice Γ is the root lattice of E × E , and thecontribution from X I is represented by the SO (16) characters as follows: η − X m I ∈ Γ q | m | = ( O + S ) ( O + S ) . (3.16)One of the simple and non-trivial configurations of the Wilson lines is A Ii =9 = (cid:0) (cid:1) , A I = p , (cid:18) (cid:19) q ; 0 p ′ , (cid:18) (cid:19) q ′ ! , p + q = p ′ + q ′ = 8 . (3.17)At the point (3.17) in the moduli space, n B = 8 { p ( p −
1) + 2 q ( q −
1) + 2 p ′ ( p ′ −
1) + 2 q ′ ( q ′ −
1) + S i ( p, q ) + S i ( p ′ , q ′ ) + 24 } , (3.18) n F = 8 { pq + 4 p ′ q ′ + S h ( p, q ) + S h ( p ′ , q ′ ) } , (3.19)where S i ( p, q ) and S h ( p, q ) are the degrees of freedom from the spinor conjugacy class of SO (16) and defined as follows: S i ( p, q ) = p, q ∈ Z + 164 ( p, q ) = (2 , , (4 , , (6 , p, q ) = (0 , , (8 , , (3.20) S h ( p, q ) = p, q ∈ Z + 1 or ( p, q ) = (0 , , (8 , p, q ) = (2 , , (4 , , (6 , . (3.21)9he solutions of n F = n B are ( p, q, p ′ , q ′ ) = (3 , , , , ), which realize SO (6) × SO (10) × SO (8) × SO (8) gauge symmetry.Let us comment on the w i =9 = 0 sector. As mentioned in the footnote 1, including the w i =9 = 0 sector, we have the possibility of a richer pattern of gauge symmetry enhancement.In particular, the gauge symmetry can be enhanced to a semisimple Lie algebra whose rankis greater than 16 when G ij and/or B ij take special values, which correspond to the fixedpoints of the duality symmetry. In [18], such a pattern of symmetry enhancement has beenexplored in maximally supersymmetric heterotic strings compactified on T D . Acknowledgments
We thank Shun’ya Mizoguchi and Yuji Sugawara for helpful discussion on this subject. Thework of HI is supported in part by JSPS KAKENHI Grant Number 19K03828 and by theOsaka City University (OCU) Strategic Research Grant 2020 for priority area.
A Lattices and characters
Irreducible representations of SO (2 n ) can be classified into four conjugacy classes: • The trivial conjugacy class (the root lattice):Γ ( n ) g = ( ( n , · · · , n n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n i ∈ Z , n X i =1 n i ∈ Z ) . (A.1) • The vector conjugacy class:Γ ( n ) v = ( ( n , · · · , n n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n i ∈ Z , n X i =1 n i ∈ Z + 1 ) . (A.2) • The spinor conjugacy class:Γ ( n ) s = ((cid:18) n + 12 , · · · , n n + 12 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n i ∈ Z , n X i =1 n i ∈ Z ) . (A.3)10 The conjugate spinor conjugacy class:Γ ( n ) c = ((cid:18) n + 12 , · · · , n n + 12 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n i ∈ Z , n X i =1 n i ∈ Z + 1 ) . (A.4)The dual lattice of Γ ( n ) g is the weight lattice of SO (2 n ), which is the set of weights of allconjugacy classes: Γ ( n ) ∗ g = Γ ( n ) w = Γ ( n ) g + Γ ( n ) v + Γ ( n ) s + Γ ( n ) c . (A.5)Modular invariance of the partition function of the supersymmetric heterotic string theoryrequires that X I be compactified on an even self-dual Euclidean lattice. In 16-dimensions,only two such lattices exist, one of which is the root lattice of E × E :Γ = (cid:0) Γ (8) g + Γ (8) s (cid:1) × (cid:0) Γ (8) g + Γ (8) s (cid:1) . (A.6)The other is the root lattice of Spin (32) / Z which is the sum of the trivial and spinorconjugacy classes of SO (32): Γ = Γ (16) g + Γ (16) s . (A.7)The SO (2 n ) characters of the corresponding conjugacy classes are defined as O n = 1 η n X m I ∈ Γ ( n ) g q | m | = 12 η n ϑ n " (0 , τ ) + ϑ n " / (0 , τ ) ! , (A.8) V n = 1 η n X m I ∈ Γ ( n ) v q | m | = 12 η n ϑ n " (0 , τ ) − ϑ n " / (0 , τ ) ! , (A.9) S n = 1 η n X m I ∈ Γ ( n ) s q | m | = 12 η n ϑ n " / (0 , τ ) + ϑ n " / / (0 , τ ) ! , (A.10) C n = 1 η n X m I ∈ Γ ( n ) c q | m | = 12 η n ϑ n " / (0 , τ ) − ϑ n " / / (0 , τ ) ! , (A.11)11here the Dedekind eta function and the theta function with characteristics are defind as η ( τ ) = q / ∞ Y n =1 (1 − q n ) , (A.12) ϑ " αβ ( z, τ ) = ∞ X n = −∞ exp (cid:0) πi ( n + α ) τ + 2 πi ( n + α )( z + β ) (cid:1) . (A.13) References [1] D. J. Gross, J. A. Harvey, E. J. Martinec and R. Rohm, Phys. Rev. Lett. , 502 (1985).[2] L. J. Dixon and J. A. Harvey, Nucl. Phys. B , 93 (1986);L. Alvarez-Gaume, P. H. Ginsparg, G. W. Moore and C. Vafa, Phys. Lett. B , 155(1986).[3] H. Itoyama and T. R. Taylor, Phys. Lett. B , 129 (1987); FERMILAB-CONF-87-129-T, Proceedings of International Europhysics Conference on High-energy Physics, 25June-1 July 1987. Uppsala, Sweden (C87-06-25).[4] V. P. Nair, A. D. Shapere, A. Strominger and F. Wilczek, Nucl. Phys. B , 402(1987);P. H. Ginsparg and C. Vafa, Nucl. Phys. B , 414 (1987).[5] K. Kikkawa and M. Yamasaki, Phys. Lett. B , 357-360 (1984) ;N. Sakai and I. Senda, Prog. Theor. Phys. , 692 (1986) [erratum: Prog. Theor. Phys. , 773 (1987)] .[6] K. S. Narain, Phys. Lett. , 41 (1986);K. S. Narain, M. H. Sarmadi and E. Witten, Nucl. Phys. B , 369 (1987).[7] J. Scherk and J. H. Schwarz, Phys. Lett. , 60 (1979); R. Rohm, Nucl. Phys. B ,553 (1984);C. Kounnas and B. Rostand, Nucl. Phys. B , 641 (1990).[8] J. D. Blum and K. R. Dienes, Nucl. Phys. B , 83 (1998) [hep-th/9707160]; Phys.Lett. B , 260 (1997) [hep-th/9707148].129] S. Abel, E. Dudas, D. Lewis and H. Partouche, JHEP , 226 (2019)[arXiv:1812.09714 [hep-th]];H. Partouche, arXiv:1901.02428 [hep-th];C. Angelantonj, H. Partouche and G. Pradisi, arXiv:1912.12062 [hep-th].S. Abel, T. Coudarchet and H. Partouche, arXiv:2003.02545 [hep-th];T. Coudarchet and H. Partouche, [arXiv:2011.13725 [hep-th]].[10] H. Itoyama and S. Nakajima, PTEP , no. 12, 123B01 (2019) [arXiv:1905.10745[hep-th]].[11] H. Itoyama and S. Nakajima, Nucl. Phys. B , 115111 (2020) [arXiv:2003.11217[hep-th]].[12] S. Abel, K. R. Dienes and E. Mavroudi, Phys. Rev. D , no. 12, 126014 (2015)[arXiv:1502.03087 [hep-th]]; Phys. Rev. D , no. 12, 126017 (2018) [arXiv:1712.06894[hep-ph]];B. Aaronson, S. Abel and E. Mavroudi, Phys. Rev. D , no. 10, 106001 (2017)[arXiv:1612.05742 [hep-th]];S. Abel and R. J. Stewart, Phys. Rev. D , no. 10, 106013 (2017) [arXiv:1701.06629[hep-th]].[13] A. E. Faraggi and M. Tsulaia, Phys. Lett. B , 314-320 (2010) [arXiv:0911.5125 [hep-th]];J. M. Ashfaque, P. Athanasopoulos, A. E. Faraggi and H. Sonmez, Eur. Phys. J. C ,no. 4, 208 (2016) [arXiv:1506.03114 [hep-th]];A. E. Faraggi, Eur. Phys. J. C , no. 8, 703 (2019) [arXiv:1906.09448 [hep-th]];A. E. Faraggi, V. G. Matyas and B. Percival, arXiv:1912.00061 [hep-th]; Nucl.Phys. B , 115231 (2020) [arXiv:2006.11340 [hep-th]]; [arXiv:2010.06637 [hep-th]];[arXiv:2011.04113 [hep-th]]; [arXiv:2011.12630 [hep-th]];A. E. Faraggi, B. Percival, S. Schewe and D. Wojtczak, [arXiv:2101.03227 [hep-th]].[14] C. Kounnas and H. Partouche, PoS PLANCK , 070 (2015) [arXiv:1511.02709 [hep-th]]; Nucl. Phys. B , 593 (2016) [arXiv:1607.01767 [hep-th]]; Nucl. Phys. B , 41(2017) [arXiv:1701.00545 [hep-th]];I. Florakis and J. Rizos, Nucl. Phys. B , 495 (2016) [arXiv:1608.04582 [hep-th]];13. Coudarchet, C. Fleming and H. Partouche, Nucl. Phys. B , 235 (2018)[arXiv:1711.09122 [hep-th]];T. Coudarchet and H. Partouche, Nucl. Phys. B , 134 (2018) [arXiv:1804.00466[hep-th]].H. Partouche, Universe , no. 11, 123 (2018) [arXiv:1809.03572 [hep-th]].[15] Y. Hamada, H. Kawai and K. y. Oda, Phys. Rev. D , 045009 (2015) [arXiv:1501.04455[hep-ph]];M. McGuigan, [arXiv:1907.01944 [hep-th]];C. Angelantonj, Q. Bonnefoy, C. Condeescu and E. Dudas, JHEP , 125 (2020)[arXiv:2007.12722 [hep-th]];B. S. Acharya, JHEP , 128 (2020) [arXiv:1906.06886 [hep-th]];B. S. Acharya, G. Aldazabal, E. Andr´es, A. Font, K. Narain and I. G. Zadeh,[arXiv:2010.02933 [hep-th]].[16] H. Itoyama, Y. Koga and S. Nakajima in progress;See also H. Itoyama and P. Moxhay, Nucl. Phys. B , 685-708 (1987); Y. Arakane,H. Itoyama, H. Kunitomo and A. Tokura, Nucl. Phys. B , 149-163 (1997)[arXiv:hep-th/9609151 [hep-th]].[17] A. Giveon, E. Rabinovici and G. Veneziano, Nucl. Phys. B , 167-184 (1989) ; Phys.Rept. , 77-202 (1994) [arXiv:hep-th/9401139 [hep-th]].[18] B. Fraiman, M. Gra˜na and C. A. N´u˜nez, JHEP09