Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Universality of gauge thresholds in non-supersymmetric heterotic vacua

Carlo Angelantonja, Ioannis Florakisb *, Mirian Tsulaiac

a Dipartimento di Fisica, Universita di Torino, and INFN Sezione di Torino, Via P. Giuria 1,10125 Torino, Italy b Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany c Faculty of Education Science Technology and Mathematics, University of Canberra, Bruce ACT 2617, Australia

CrossMark

A R T I C L E I N F 0

A B S T R A C T

Article history: Received 31 July 2014 Accepted 1 August 2014 Available online 7 August 2014 Editor: L. Alvarez-Gaume

We compute one-loop threshold corrections to non-abelian gauge couplings in four-dimensional heterotic vacua with spontaneously broken N = 2 ^ N = 0 supersymmetry, obtained as Scherk-Schwarz reductions of six-dimensional K3 compactifications. As expected, the gauge thresholds are no-longer BPS protected, and receive contributions also from the excitations of the RNS sector. Remarkably, the difference of thresholds for non-abelian gauge couplings is BPS saturated and exhibits a universal behaviour independently of the orbifold realisation of K3. Moreover, the thresholds and their difference develop infra-red logarithmic singularities whenever charged BPS-like states, originating from the twisted RNS sector, become massless at special loci in the classical moduli space.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

In the last decades we have witnessed a tremendous progress in understanding the structure of supersymmetric vacua in String Theory and M/F-theory. Several semi-realistic vacua that incorporate the salient features of the MSSM have been constructed and analysed to a remarkable extent. Their low-energy effective action with N = 1 supersymmetry has been fully reconstructed at tree-level, and the incorporation of quantum and a' corrections is still a subject of intense study. Despite these successful endeavours, su-persymmetry breaking in String Theory remains a compelling open problem that string phenomenology aspires to address.

A fully-fledged approach to spontaneous supersymmetry breaking in String Theory, that admits an exactly solvable world-sheet description, is the stringy realisation [1-4] of the Scherk-Schwarz mechanism [5,6], via special freely-acting orbifolds. In this class of vacua, the supersymmetry breaking scale is tied to the size of compact dimensions, while the exponential growth of string states may destabilise the classical vacuum due to the emergence of tachy-onic excitations. This is closely related to the Hagedorn problem of String Thermodynamics [7] and can be circumvented in special constructions [8-10]. Moreover, it has been recently argued that

* Corresponding author.

E-mail addresses: carlo.angelantonj@unito.it (C. Angelantonj), florakis@mppmu.mpg.de (I. Florakis), mirian.tsulaia@canberra.edu.au (M. Tsulaia).

closed string tachyons emerging from twisted orbifold sectors of a class of heterotic vacua with explicitly broken supersymmetry can actually acquire a mass by blowing-up the orbifold singularities [11].

In all those cases where supersymmetry is (spontaneously) broken but the vacuum is classically stable, it is meaningful and important to study one-loop radiative corrections to couplings in the low-energy effective action. The emergence of one-loop tadpoles for massless states does not impinge on the validity of the one-loop analysis, although it makes the incorporation of higher loops problematic, unless the back-reaction on the classical vacuum is properly taken into account [12,13].

For this reason, we address in this letter the problem of computing one-loop threshold corrections to gauge couplings in a class of four-dimensional heterotic vacua with spontaneously broken supersymmetry, that can be built as K3 reductions of the SO(16) x SO(16) construction of [14] in terms of freely-acting orbifolds. In contrast to heterotic vacua with unbroken supersymmetry, where the moduli dependence of the one-loop corrected gauge couplings arises from the BPS sector, in the case of spontaneously broken supersymmetry the amplitude receives contributions from the full tower of charged string states, and is no-longer topological. Nevertheless, we find that the difference between gauge thresholds exhibits a remarkable universal structure akin to the N = 2 super-symmetric case, due to highly non-trivial cancellations induced by an MSDS spectral flow [15-17] in the bosonic right-moving sector of the heterotic string.

http://dx.doi.org/10.1016/j.physletb.2014.08.001

0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

The striking signature of spontaneous supersymmetry breaking is the emergence of logarithmic singularities at special points of the classical moduli space. These are ascribed to charged BPS-like states that become massless at points of gauge symmetry enhancement, and survive in the difference of gauge thresholds.

The paper is structured as follows: in Section 2 we define the freely-acting orbifold responsible for the spontaneous breaking of supersymmetry and present the corresponding one-loop partition function. Section 3 is devoted to the evaluation of gauge threshold corrections for the non-abelian gauge couplings and contains the main results of our investigation. Finally, in Section 4 we discuss the relevant decompactification limits and comment on their physical interpretation.

2. Heterotic vacuum with spontaneous supersymmetry breaking

The class of non-supersymmetric vacua that we shall focus on is obtained as a Scherk-Schwarz reduction of six-dimensional K3 compactifications of the E8 x E8 heterotic string. They can also be viewed as K3 reductions of the Itoyama-Taylor vacuum [14], that corresponds to a lower-dimensional freely-acting implementation of the non-supersymmetric, non-tachyonic, SO(16) x SO(16) construction [18,19].

For concreteness, we shall consider the T6/ZN x Z2 compacti-fication of the ten-dimensional E8 x E8 heterotic string, with fac-torised T6 = T4 x T2. The ZN, with N = 2, 3, 4, 6 rotates chrystal-lographycally the complexified T4 coordinates as

v: zi ^ e2iK/Nzi,

z2 ^ e-2in/NZ2,

and realises the singular limit of the K3 surface, preserving 8 supercharges. The Z2 is instead freely acting and is generated by

' = (—1)Fst + F1+F2 ^

Here, Fst is the space-time fermion number, responsible for the breaking of supersymmetry, F1 and F2 are the "fermion numbers" of the two original E8's, whereas 8 acts as an order-two shift along the remaining T2. The combined action of 8 and (—1)Fst is responsible for the spontaneous breaking of the N = 2 supersymmetry down to N = 0, while the presence of (— 1)F 1+F2 guarantees the classical stability of the vacuum.1

The one-loop partition function reads

* =1 E ^

,G=0 h,g= 1 è (-)

a,b=0 i

'[ b/2]

„ [ a/2+h/N ] „ [ a/2-h/N ]

^L b/2+g/Nj ^L b/2-g/N J

Y^ ¿Tk/2 16Ä F k/2+h/N [ k/2-h/N 1 V\_i/2\ U\_i/2+g/N\ \_l/2-g/N J

L r,s=0

k,i=0 1

j12fj 24

H(b+i+s)+G(a+k+r)+HG

ö a-[ si

Here, n is the Dedekind function and ^ ^ j are the standard Ja-cobi theta constants with characteristics. The sum over of the spin

1 This is no-longer true when Wilson lines are turned on, whereby all non-supersymmetric heterotic vacua can be continuously connected [20,21]. In this note we shall always assume a trivial Wilson-line background.

structures a, b, k, t, r and s yield the ten-dimensional E8 x E8 heterotic-string spectrum, while (h, g) and (H, G) correspond to the ZN and Z2 orbifolds. The two-dimensional Narain lattice with characteristics is defined as

M H] = ^2 E e'nG(^2n)rA+ H

H I2,n+ if Ii (T '

rA,n(T, U)

= 4TJUJ \m2-Um 1+T (n1 +Un2)|2 - 4^ | m2-Um 1+T (n'+Un2)|2 (2 5)

and depends on the Kahler and complex structure moduli T and U. As usual, momenta and windings are labeled by m and n, while the integral vectors ii and X2 encode the freely-acting shift of U1. Without loss of generality, we shall focus on the case i 1 = (1, 0) and i2 = (0, 0) corresponding to a momentum shift along the first T2 direction. All other cases can be related to the former by suitable redefinitions of the T and U moduli. Finally,

for (h, g) = (0, 0),

k[£]i"12

r 1/2+h/Ng1gr 1/2-h/N 112 for (h, g) = (0, 0), 1 L 1/2+g/Nj L 1/2-g/NJ'

with r4,4 being the conventional Narain lattice associated to the T4, ^0] = 16 sin4(ng/N) counting the number of twisted

sectors of the ZN orbifold, and the remaining ^gj's with h = 0 being determined by modular invariance.

As a consequence of the Scherk-Schwarz mechanism, the two gravitini acquire a mass m 3/2 = |U ^V T 2 U 2, and supersymmetry is spontaneously broken at a generic point in the classical moduli space. The Z2 also breaks the E8 x E7 gauge group of the N = 2 theory down to SO(16) x SO(12), up to abelian factors. The full spectrum can be derived from (2.3) using standard techniques.

Notice that, as in the parent ten-dimensional SO(16) x SO(16) non-supersymmetric theory [18,19], the spectrum is free of tachy-onic excitations at a generic point of the (T, U) moduli space. This can be verified by looking at the H = 0, a = 0 contributions to (2.3).

3. One-loop thresholds for non-abelian gauge couplings

Although the vacuum configuration presented in the previous section is not supersymmetric, the absence of physical tachyons in the perturbative spectrum implies that it is classically stable. As a result, it is fully justified and important to study one-loop radiative corrections to couplings in the low-energy effective action, in contrast to higher-loop diagrams that diverge due to the emergence of one-loop tadpoles back-reacting on the vacuum [12,13]. This is still an open problem in String Theory, and has recently triggered a growing interest [22-24].

To this end, we shall address here the question of quantum corrections to the couplings of the non-abelian SO(16) x SO(12) gauge factors, extending the analysis of [25] to non-supersymmetric vacua.

Threshold corrections Aq associated to the group factor G appear in the relation between the running gauge coupling gg (jx.) of the low-energy theory and the string coupling gs

16n 2 Sq W

^^ + ßg log 2 + Ag, Ss2 W2

where, in the case at hand, the Kac-Moody algebra is realised at level one, and Ms sets the string scale. They encode the contribution of the infinite tower of massive string states to the one-loop diagram, and can be organised as

Ад -

- 1 N_1

¿s R-N7* £ £ M "1]

F H,G=0 h,g=0

Jsr-n7 E

H ,G=0 h,g=0

4^] 4 H:î] я 4 H]

_ G : g

кзГ h

In this expression, d¡x denotes the SL(2; R) invariant measure, while R.N. stands for the modular-invariant prescription of [28,29] for regularising the infra-red divergences of the integral.

The quantity H' encodes the spin-structure sum over the integrated world-sheet correlators for the four-dimensional spacetime fields, whereas H'^j encodes the contribution of the gauge sector with the relevant trace insertion. They are defined as

(_)a(1+G)+b(1+H)9i_

M -1 E

Я Г а/2Л ч Я Г a/2 ] Я Г a/2+h/N] Я Г a/2 _h/N ]

Ч Ь/2\ \ Ч Ь/2\ Ч b/2+g/N] Vl b/2_g/N J

(a, b)=(1 : 1)

H:H - 4(

4\ (2ni)2 zg 4пт2

kG+lH,

Lk : l=0

1Г k/2] 6Я Г k/2+h/N ] я Г k/2_h/N ]

\_l/2\ ^|_l/2+g/sJ \_l/2_g/N J

Г: S=0

rG+sH *[ 1/2]

(zG )\zg =0 •

In the latter equation, it is implied that the VEV zg is only inserted along the particular theta function corresponding to the Cartan charge whose group trace we are considering.

It is convenient to arrange the 4N2 sectors of the orbifold so as to distinguish the origin of the various contributions to the thresholds. The (h, g) = (0 , 0) sector corresponds to the Itoyama-Taylor construction [14] reduced to four-dimensions, and is proportional to the T4 lattice ^K3[0], depending on the invariant T4 moduli. Furthermore, since h = g = 0, one is effectively dealing with the SO(16) x SO(16) lattice. Hence, the group traces are independent of the choice of gauge group G, implying that the difference of thresholds is independent of the T4 moduli.

An explicit calculation yields

Ал = _

4N x 122 n12n24

[Г2:2[î] К _ Я34¿44((E2 _ Я4)^ + 8П12)

Г : 2 Г1 ](Я| _ Я2>34Я24((Ё2 + Я34)Я24Я34 _ 8П12)

Г : 2 Г 1 ](Я| _ Я28)Я44Я24((Е2 + Я44)Я24Я44 _ 8П12)]:

where E2 is the weight-two quasi-holomorphic Eisenstein series.2 Notice that the second and third lines can be obtained from the first one upon acting with the SL(2; Z) generators S and TS, as demanded by modularity.

The remaining contributions can be organised as

H: G=0 h, g=0

(К g) = (0: 0)

д GH gh

e mîg]+а40:я+Ад[Щ+Ад[î:g])

(h : g) = (0: 0)

- a(u+)

+ А(д _) + А(д+) + a

according to the sectors of the freely-acting orbifold. The first contribution Ag+), corresponding to (H, G) = (0 , 0), computes the gauge thresholds of the N = 2 heterotic string on the orbifold limit of K3. It is thus expected to be BPS saturated and the difference

(u + )

(u + ) G'

to be universal and to depend only on the moduli of the T2 torus3 [25]. The remaining terms, connected among each other by S and TS modular transformations, are inherently non-BPS since the freely-acting orbifold acts non-trivially and breaks supersymmetry. This is reflected by the fact that the modular integral now involves genuinely non-holomorphic contributions.

For concreteness, we shall present explicitly the various contributions in the case N = 2, where one finds

(u+) SO(16)

SO(16)

48 МЯ *[ 0 ]

E2 E4 E6 _ E6

Г0] Я^Я + Я^Ш2 _ + 8n

96 Г2:2Ы n24

г Г 0 ] Я24(Я38 _ Я48) (E 2 _ Я34 )Я34Я4

144 2:2Ы n12 П12

SO(12)

48 Г240]

E 2 E 4 E 6 _ E 4

(u_) SO(12)

"96 Г240]

0] Я38Я48[е 2 (Я3 + я4) + я 2 _ 2гя.4]

г2 2 01

0]/Я24(Я38 _ Я48) E2Я34Я44

TiV^UJV n12 n1

Лл4\л4 _ л4|2 _л4|2

я 2 я44 \ я 24 _ я44\2 _ я 2 я3 \ я 24 + я34\2 _ 4 _ 4

+ пЦ12 я3 я4

(3.10)

In these expressions E4 (E6) is the weight-four (-six) holomorphic Eisenstein series. Again, the remaining terms can be computed by the action of the generators S and TS of SL(2; Z) on the corresponding A(U-) contributions.

As anticipated, Eqs. (3.7) and (3.9) compute the thresholds to the N = 2 supersymmetric E8 and E7 gauge factors. Eqs. (3.8) and

2 Whenever the characteristics of the theta constants equal 0 , 1 /2 we employ the light notation in terms of the §a's.

3 The difference of gauge thresholds is indeed universal for the T4/ZN orbifolds,

though in more general constructions they may exhibit a non-universal structure

[26,27].

(3.10) involve contributions from BPS states whose masses are now deformed by the free action of the Z^ orbifold.

The BPS contributions to these amplitudes can be integrated over the SL(2; Z) fundamental domain F or, after partial unfolding, over the fundamental domain F0[2] of the r0(2) congruence subgroup, following the procedure developed in [28-30]. The non-BPS contributions can be shown to be exponentially suppressed [31] in the large T2 volume limit, and are thus negligible at low-energies. We shall not indulge here in the full computation of the thresholds, but rather focus on their difference. One finds

(u + ) SO(16)

(u + )

SO(12)

that reproduces the result of [25], and

A(u-) _A(U-) ASO(16) aso(12)

-6 ЛиД

(3.11)

(3.12)

Surprisingly, the non-holomorphic contributions to the thresholds cancel when taking their difference, and reduce to a purely holomorphic BPS-like term. As we shall show, the difference of gauge thresholds exhibits a remarkable universal behaviour, independently of the details of the T4/ZN orbifold. Indeed, the non-holomorphic contribution to the difference of thresholds reads

^34 + Û^û4

n12n12

^4>24 - 3^4

n12n12

û24 û34|û24 + û34|2 û34û4

цПц 12 8)

= 12( O 2 V 8 + 3V 83) (O 8 V 8 - V 8)

(3.13)

where in the right-hand side we have introduced the SO(8) characters. Although, this term looks completely non-holomorphic it actually possesses a BPS-like structure due to a remarkable MSDS identity [16,10]

O8 V8 - V83 = 8,

(3.14)

which reflects a hidden MSDS spectral flow in the bosonic sector of the global N = (2, 2) superconformal symmetry on the world-sheet [17,31]. As a result, Eq. (3.13) reduces to the purely holomor-phic contribution (3.12).

To evaluate the integrals, we first notice that the combination #22/n12 corresponds to an automorphic function of the Hecke congruence subgroup r0(2). Moreover, it is regular at the cusp at

t = im while it has a simple pole at the cusp t = 0.4 This is sufficient to identify [30]

= F0(1,1, 0) - 16 = Mr) - 24,

(3.15)

where F0(1, 1, 0) is the meromorphic weight-zero Niebur-Poincare series attached to the cusp at t = 0 of r0(2), and J2(t) is the Fricke transform [30] of the r0(2) Hauptmodul

j2(T) =

П24(т) П24 (2 т)

(3.16)

The modular integrals can be straightforwardly computed using the results of5 [28-30,32] to yield

4 We remind here that the compactification of the fundamental domain F0[2] of r0(2) requires adding two points, i.e. the two cusps, t = im and t = 0. See, for instance, [30].

5 The first integral was actually originally computed in [25] by unfolding the fundamental domain against the Narain lattice.

R.N. J й/лГ2,2 (T, U ) = -log T 2 U 2|n(T )n(U )|4, (3.17)

j /2,2 [0] (T, U ) = log T 2 U 2 |$4(T )Û2(U )|4, (3.18)

and R.N.

I d/r2,^0] (T, U)

^2 (т)

Пи(т)

= -2log[| j2(T/2) - j2(U)|41 j2(U) - 24|4]. Combining the various contributions, one finds

Aso(16) - Aso(12) = 36log[T2U21n(T)n(U)|4] - 4 log[T2U2|Û4(T)Û2(U)|4]

(3.19)

+ 3 lo^| j2(T/2) - J2(U)|4| j2(U) - 24|4].

(3.20)

Again, the various terms have a clear physical interpretation. The first line generalises the celebrated result of [25]. The presence of second term is ascribed to the modified Kaluza-Klein masses of BPS states, that are indeed affected by the free action of the orbifold. In fact, since the Z2 orbifold corresponds to a spontaneous breaking of supersymmetry, the model in (2.3) contains precisely the same excitations as the E8 x E8 heterotic string on T2 x T4/ZN, whose masses are continuously deformed by the scale of super-symmetry breaking. As a result, the duality group is broken down to the subgroup Г0(2)t x Го(2)и of SL(2; Z)t x SL(2; Z)u.

While the first two contributions are regular at any point in the classical moduli space, the term in the second line, particular to this vacuum with broken supersymmetry, possesses logarithmic singularities at the locus T/2 = U and its Го(2) images. The origin of these singularities is ascribed to massive charged BPS-like states that become massless at special points in moduli space. To manifest their origin in the perturbative spectrum, it is convenient to express their contribution to (2.3) in terms of the SO(2n) characters

1 (O4 O4 x V12 O4 V 16)(Г2,2[0] + Г2,2[}]) • (3.21)

These states include the left-moving NS vacuum and its stringy excitations, while the right-moving sector is massless and belongs to the bi-fundamental representation (16, 12) of the SO(16) x SO(12) gauge group. They always carry non-trivial momentum and winding quantum numbers, and the lightest states have mass

lo 4 O 4

|T/2 - U|2 T 2 U 2

(3.22)

Indeed, these states become massless at the point T/2 = U, where pR = 0, and are responsible for the logarithmic divergence in (3.20).

Notice that the fact that extra massless states emerge from the Z2 twisted sector is compatible with the fact that the term $12/n12, originating from the un-twisted sector, has a pole at the cusp t = 0 but is regular at t = im. In fact, the two cusps are related by an S modular transformation that also relates the untwisted and twisted sectors. As a result, the singularity of the u-sector at t = 0 is to be understood as the map under S of the physical infra-red singularity of the twisted sector.

One can compute the gauge thresholds also in the case of the other singular limits of K3, namely N = 3, 4, 6. Although the result of the thresholds depends on the particular value of N, their

difference exhibits a remarkable universal behaviour. In fact, one finds

^SO(16) — ^SO(12)

= a log[T 2 U 2\n(T )n(U )|4] + P log[T 2 U 2^ )#2 (U )|4 ]

+ Y log[\ jj2(T/2) — h(U)\4\ j2(U) — 24\4]

(3.23)

with (a, P, y) = (36, — 3, 1) for the Z2 and Z3 orbifolds, for the Z4 orbifold and (a, P, y) =

(a, P, y) = 8(36, — 4, )

8w"> 3, 15;

H(36, — 3, 1) for the Z6 orbifold.

This universality structure is a direct consequence of the universal behaviour of the N = 2 thresholds [25,33], which is preserved by the free action of the Z^.

4. Decompactification limits

It is instructive to study Eq. (3.23) in the decompactification limits. For convenience, we shall assume a squared T2 with

T = iR1 R2 and U = i —, (4.1)

so that the masses of the two gravitini and of the O 4 O 4 states read

Notice that in the R1 — 0 limit, the freely-acting orbifold degenerates into an explicit breaking of supersymmetry. This implies that the universal behaviour (3.23) should hold also in the case when the ten-dimensional O(16) x O(16) theory of [18,19] is com-pactified on T2 x K3. As a result, a similar universal behaviour of the threshold differences is expected to arise also when T2 x K3 is replaced by a generic Calabi-Yau manifold. It would be interesting to investigate whether Eq. (3.23) also holds when the ten-dimensional heterotic string, whether supersymmetric or not, is compactified on a manifold that does not preserve any supersym-metry.

Acknowledgements

We are grateful to B. Pioline for fruitful discussions. M.T. would like to thank the Department of Physics, the University of Auckland, where part of this work has been performed, for its kind hospitality. The work of C.A. has been supported in part by the European ERC Advanced Grant No. 226455 "Supersymmetry, Quantum Gravity and Gauge Fields" (SUPERFIELDS) and in part by the Com-pagnia di San Paolo contract "Modern Application in String Theory" (MAST) TO-Call3-2012-0088. The work of M.T. has been supported in part by an Australian Research Council grant DP120101340. M.T. would also like to acknowledge grant 31/89 of the Rustaveli National Science Foundation.

m2i/2 = R2 and <4O4 = R1 —

References

In the R1 — < limit, N = 2 supersymmetry is recovered, and the leading behaviour of Eq. (3.23)

lim [^so(16) — ^so(12)] = ^t- R1 R1 — < 3

R 2 + R- 1+ ...

grows linearly with the T2 volume. This is expected from scaling arguments, since in six dimensions the gauge coupling has length dimension —1. The term proportional to P in (3.23) only grows logarithmically with R1 as a result of supersymmetry enhancement since, charged states lighter than the supersymmetry-breaking scale are effectively BPS-like and thus contribute logarithmically to the difference of threshold corrections, whereas the infinite towers of charged states heavier than m3/2 have an effective N = 4 supersymmetry and, thus, do not contribute. Finally, the term proportional to y is exponentially suppressed because the lightest charged states O4O4 have mass mO4O4 ^ m3/2 and effectively decouple.

In the R2 — < limit, the leading behaviour of (3.23) is

lim [^so(16) — ^so(12)]

R2 —X

na „ / 1 = T" R 2{R 1 + R1

np R 2

+ 2nY R 2\R1 — R- —

As expected, the term proportional to a is again linearly divergent with the T2 volume. The term proportional to P now scales as R2/R1, and consistently vanishes as m3/2 — 0. The term proportional to y depends on the scale of supersymmetry breaking. When R1 > 42 it is exponentially suppressed because m3/2 < mO4o4, whereas when R1 < \/2 it scales as R2(2/R1 — R1). This is a consequence of the fact that, in the R1 — 0 limit, supersym-metry is explicitly broken, and this term grows with the volume R2R1 ~ R2/R1 of the T-dual torus.

R. Rohm, Spontaneous supersymmetry breaking in supersymmetric string theories, Nucl. Phys. B 237 (1984) 553.

C. Kounnas, M. Porrati, Spontaneous supersymmetry breaking in string theory, Nucl. Phys. B 310 (1988) 355.

S. Ferrara, C. Kounnas, M. Porrati, F. Zwirner, Superstrings with spontaneously broken supersymmetry and their effective theories, Nucl. Phys. B 318 (1989) 75.

C. Kounnas, B. Rostand, Coordinate dependent compactifications and discrete symmetries, Nucl. Phys. B 341 (1990) 641.

J. Scherk, J.H. Schwarz, Spontaneous breaking of supersymmetry through dimensional reduction, Phys. Lett. B 82 (1979) 60.

J. Scherk, J.H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979) 61.

J.J. Atick, E. Witten, The Hagedorn transition and the number of degrees of freedom of string theory, Nucl. Phys. B 310 (1988) 291. C. Angelantonj, M. Cardella, N. Irges, An alternative for moduli stabilisation, Phys. Lett. B 641 (2006) 474, arXiv:hep-th/0608022.

C. Angelantonj, C. Kounnas, H. Partouche, N. Toumbas, Resolution of Hagedorn singularity in superstrings with gravito-magnetic fluxes, Nucl. Phys. B 809 (2009) 291, arXiv:0808.1357 [hep-th].

I. Florakis, C. Kounnas, N. Toumbas, Marginal deformations of vacua with massive boson-fermion degeneracy symmetry, Nucl. Phys. B 834 (2010) 273, arXiv:1002.2427 [hep-th].

M. Blaszczyk, S. Groot Nibbelink, O. Loukas, S. Ramos-Sanchez, Non-super-symmetric heterotic model building, arXiv:1407.6362 [hep-th]. W. Fischler, L. Susskind, Dilaton tadpoles, string condensates and scale invariance, Phys. Lett. B 171 (1986) 383.

W. Fischler, L. Susskind, Dilaton tadpoles, string condensates and scale invariance. 2, Phys. Lett. B 173 (1986 ) 262.

H. Itoyama, T.R. Taylor, Supersymmetry restoration in the compactified O(16) x O(16)' heterotic string theory, Phys. Lett. B 186 (1987) 129.

C. Kounnas, Massive boson-fermion degeneracy and the early structure of the universe, Fortschr. Phys. 56 (2008) 1143, arXiv:0808.1340 [hep-th].

I. Florakis, C. Kounnas, Orbifold symmetry reductions of massive boson-fermion degeneracy, Nucl. Phys. B 820 (2009) 237, arXiv:0901.3055 [hep-th]. A.E. Faraggi, I. Florakis, T. Mohaupt, M. Tsulaia, Conformal aspects of spinor-vector duality, Nucl. Phys. B 848 (2011) 332, arXiv:1101.4194 [hep-th].

L. Alvarez-Gaume, P.H. Ginsparg, G.W. Moore, C. Vafa, An O(16) x O(16) heterotic string, Phys. Lett. B 171 (1986) 155.

L.J. Dixon, J.A. Harvey, String theories in ten-dimensions without space-time supersymmetry, Nucl. Phys. B 274 (1986) 93.

P.H. Ginsparg, C. Vafa, Toroidal compactification of non-supersymmetric het-erotic strings, Nucl. Phys. B 289 (1987) 414.

V.P. Nair, A.D. Shapere, A. Strominger, F. Wilczek, Compactification of the twisted heterotic string, Nucl. Phys. B 287 (1987) 402.

[22] E. Dudas, G. Pradisi, M. Nicolosi, A. Sagnotti, On tadpoles and vacuum redefinitions in string theory, Nucl. Phys. B 708 (2005) 3, arXiv:hep-th/0410101.

[23] N. Kitazawa, Tadpole resummations in string theory, Phys. Lett. B 660 (2008) 415, arXiv:0801.1702 [hep-th].

[24] R. Pius, A. Rudra, A. Sen, String perturbation theory around dynamically shifted vacuum, arXiv:1404.6254 [hep-th].

[25] L.J. Dixon, V. Kaplunovsky, J. Louis, Moduli dependence of string loop corrections to gauge coupling constants, Nucl. Phys. B 355 (1991) 649.

[26] P. Mayr, S. Stieberger, Threshold corrections to gauge couplings in orbifold com-pactifications, Nucl. Phys. B 407 (1993) 725, arXiv:hep-th/9303017.

[27] E. Kiritsis, C. Kounnas, P.M. Petropoulos, J. Rizos, String threshold corrections in models with spontaneously broken supersymmetry, Nucl. Phys. B 540 (1999) 87, arXiv:hep-th/9807067.

[28] C. Angelantonj, I. Florakis, B. Pioline, A new look at one-loop integrals in string theory, Commun. Number Theory Phys. 6 (2012) 159, arXiv:1110.5318 [hep-th].

[29] C. Angelantonj, I. Florakis, B. Pioline, One-loop BPS amplitudes as BPS-state sums, J. High Energy Phys. 1206 (2012) 070, arXiv:1203.0566 [hep-th].

[30] C. Angelantonj, I. Florakis, B. Pioline, Rankin-Selberg methods for closed strings on orbifolds, J. High Energy Phys. 1307 (2013) 181, arXiv:1304.4271 [hep-th].

[31] C. Angelantonj, I. Florakis, M. Tsulaia, in preparation.

[32] C. Angelantonj, I. Florakis, B. Pioline, in preparation.

[33] E. Kiritsis, C. Kounnas, P.M. Petropoulos, J. Rizos, Universality properties of N = 2 and N = 1 heterotic threshold corrections, Nucl. Phys. B 483 (1997) 141, arXiv:hep-th/9608034.