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Threshold Corrections and Symmetry Enhancement in String Compactifications

We present the computation of threshold functions for Abelian orbifold compactifications. Specifically, starting from the massive, moduli-dependent string spectrum after compactification, we derive the threshold functions as target space duality invariant free energies (sum over massive string states). In particular we work out the dependence on the continuous Wilson line moduli fields. In addition we concentrate on the physically interesting effect that at certain critical points in the orbifold moduli spaces additional massless states appear in the string spectrum leading to logarithmic singularities in the threshold functions. We discuss this effect for the gauge coupling threshold corrections; here the appearance of additional massless states is directly related to the Higgs effect in string theory. In addition the singularities in the threshold functions are relevant for the loop corrections to the gravitational coupling constants.


Introduction
Four-dimensional string models possibly provide a consistent description of all known interactions including gravity. At energy scales small compared to the typical string scale M string , which is directly related to the Planck mass M Planck = O (10 19 GeV), it is convenient to use an effective Lagrangian for the light string modes φ L where the effect of the heavy string modes φ H is integrated out. Based on several motivations, the main area of research in this context is on four-dimensional string vacua with N = 1 space-time supersymmetry, implying that the effective string interactions are of the form of an N = 1 supergravity-matter action. Studying four-dimensional string vacua with space-time supersymmetry, it turns out that infinitely many of them are connected by continuous deformations of the underlying two-dimensional conformal field theory, parametrized by coupling constants called moduli, collectively denoted by T i . At the level of the effective supergravity action, and neglecting non-perturbative effects, moduli are described by massless scalar fields with flat potential, whose vacuum expectation values parametrize the continuous deformations. The moduli play a very important role in the effective action, since various coupling constants among the matter fields like the tree-level Yukawa couplings [1,2] or the loop corrections to the gauge and gravitational coupling constants [3]- [19] are moduli dependent functions. Specifically, the one-loop moduli dependence of the gauge and gravitational coupling constants arises from σmodel and Kähler anomalies [5,7,8,10] and from integrating out the infinite tower of massive string modes with moduli dependent masses.
The moduli spaces M of four-dimensional string theories have a very interesting and rich structure. First, as it is true for many known compactification schemes of the ten-dimensional heterotic string, the underlying superconformal field theory is invariant under the target space duality symmetries (see [20]) which act on the the moduli as discrete reparametrizations. These target space duality transformations act in general non-trivially on the infinite massive spectrum φ H , in the sense that states with different quantum numbers (e.g. discrete internal momentum and winding numbers) are mapped onto each other. Consequently, integrating out the massive spectrum φ H implies that certain low energy couplings are given in terms of automorphic functions of the corresponding duality group. This observation can provide very useful informations about the structure of the effective low-energy supergravity action, in particular when combined with some analyticity arguments of holomorphic coupling functions [21].
A second very interesting feature of the string moduli spaces comes from the fact that at certain critical points P c in the moduli spaces a finite number of additional massless states may appear in the moduli-dependent string spectrum, which are otherwise massive at generic points in M. We will call these states φ ′ H . Very often, at P c these fields correspond to additional holomorphic spin one currents on the world sheet. Then the gauge symmetry of the four-dimensional string is enlarged which is nothing else than the stringy version of the well known Higgs effect. The role of the Higgs fields is now taken by moduli fields. The field-theoretical formulation of the stringy Higgs effect, i.e. the correct coupling of the relevant moduli to the gauge bosons, was investigated in [22,23] for the case of the standard Z 3 orbifold [24,25].
The appearance of additional massless fields at some critical points P c in M implies that the description of the string compactification by an low-energy effective action contains discontinuities, since near the critical points the fields φ ′ H should be kept as light degrees of freedom and should not be integrated out from the spectrum. In other words, when integrating over all massive fields including φ ′ H the effective action may acquire singularities at the critical points P c . Let us make this more clear by considering as an example the one loop running of a gauge coupling constant in the low-energy field theory.
The one-loop running coupling constant at a scale p is given by (neglecting Kähler and σ-model anomalies) where b is the one-loop β-function coefficient of the light modes φ L and ∆(T i ) is the moduli-dependent threshold correction of the heavy string modes. ∆(T i ) can be regarded as the suitably regularized free energy [26,27] of all massive modes, ∆(T i ) ∝ log det M 2 φ H (T i ), where the M φ H (T i )'s are the moduli-dependent masses of the heavy modes. Clearly, if this sum contains also states φ ′ H which become massless at P c , ∆(T i ) possesses a singularity at P c . As we will discuss in detail, the masses of φ ′ H are generically of the form M φ ′ H (T i ) ∝ (T i − P c ) (here, T i is one specific 'critical' modulus). Thus we see that ∆(T i ) exhibits a logarithmic singularity of the form ∆ → n log(T i − P c ), (1.2) where n accounts for the degeneracy of states which become massless at P c . In order to get finite threshold corrections∆(T i ) over the whole moduli space it is useful to separate in eq.(1.1) the contribution of the states φ ′ H from the states φ H which are always massive. Then eq.(1.1) can be written as 1 g 2 (p 2 ) = 1 g 2 (M 2 string ) Here b ′ is the contribution of the states φ ′ H to the β-function coefficient, and∆(T i ) does not contain the states φ ′ H . Thus the logarithmic singularity in the moduli T i is nothing else than the threshold effect of φ ′ H with (intermediate) mass scale M φ ′ H (T i ) ∝ (T i − P c ). This discussion was entirely based on field theoretical arguments. As already discussed in [28], in string theory these threshold functions are given in terms of automorphic functions of the underlying target space duality group with the appropriate singularity structure. In fact, since in string theory there is generically an infinite number of states which may become massless at duality equivalent points in M (at one particular point in M only a finite number of states can become massless) the relevant automorphic functions possess singularities at an infinite number of points being related by discrete duality transformations. In this paper we will calculate explicitly these divergent threshold functions which are related to the discontinuities in the string spectrum and to the stringy Higgs effect. To be specific we concentrate on Abelian orbifold compactifications of the ten-dimensional heterotic string. We will discuss the dependence of the threshold corrections as functions of the moduli associated with the six-dimensional orbifold, denoted by T and U, as well as of the socalled Wilson line moduli [29] which take values in the heterotic gauge group. The orbifold moduli T , U (metric and antisymmetric tensor) will be relevant for the discussion of the Higgs effect in the compactification sector; here the relevant automorphic functions will be given in terms of the absolute modular invariant function j. On the other hand, the Wilson line moduli are responsible for the Higgs effect in the heterotic gauge group. This is of rather phenomenological importance since the Wilson line Higgs field may break some GUT gauge group to the gauge group of the standard model. Moreover it may be even possible to identify some of the Wilson line moduli with the supersymmetric standard model Higgs fields H 1 and H 2 .
Our paper is organized as follows. In sections 2 and 3 we discuss the structure of the gauge groups in orbifold compactifications as a function of the various moduli fields. In section 4 we compute the masses of the generically massive fields. Some of these masses become zero at certain fixed points in the orbifold moduli spaces. Here we will use the results of our previous paper [30] where we have determined the moduli spaces plus the target space duality transformations in the presence of Wilson line moduli. In sections 5, 6 and 7 we will apply the results of section 4 to compute the target space free energies as (infinite) sums over the massive string states. Finally we will explain the relation of these free energies to the string threshold corrections. We will display the dependence of the threshold corrections in terms of T and U as well as in terms of the generic Wilson line moduli. A discussion of this is provided in section 8.

Gauge groups in orbifold compactifications with continuous Wilson lines
In this section we will start to discuss the moduli dependence of the gauge group of an orbifold compactification. We will concentrate on the so called gauge sector here, whereas the compactification sector will be studied in the next section. Our aim is to determine the unbroken gauge group in the presence of the most general continuous Wilson lines compatible with a given twist. For a certain class of twists the minimal and therefore generic gauge groups are easily determined, and we describe the method for determining them. The results are listed in tables given at the end of this paper. We also point out under which circumstances this method fails to give the minimal gauge groups, in which case it only yields a lower bound on these minimal gauge groups, that is not saturated.
For these cases, in which a more detailed analysis is necessary, we outline how one has to proceed in order to determine the generic gauge groups.

Wilson lines in the Narain model
Since many properties of an orbifold model can be understood easily in terms of the underlying Narain model [31,32], let us first recall how the gauge group of a toroidal compactification depends on the Wilson line moduli [33]. We will concentrate on the so called gauge sector which is generated by the sixteen extra left-moving worldsheet bosons X(z) := X I L (z), I = 1, . . . , 16. There are sixteen chiral conserved currents ∂X I L (z) on the worldsheet. These can be combined with the right-moving ground state, which is an N = 4 space time vector supermultiplet, to give a U(1) 16  where G (l) is semi-simple and has rank l.
As discussed in [33] these extended symmetries can be described in the following way.
Every vector P ∈ Γ of the Narain lattice can be written as where the integers q I , n i and m i are the charge, winding and momentum quantum numbers, I = 1, . . . , 16, i = 1, . . . , 6. The standard basis vectors are [34] l I = e I , − 1 2 (e I · A i )e i ; − 1 2 (e I · A i )e i , (2.4) (2.5) and where This basis is a function of the moduli where v = q I e I . Then is a Narain vector with v 2 = q T C (16)  and SO(32) were constructed in [33].
Finally note that a small deformation δA i of the Wilson lines, if it destroys some of the conditions v · A i ∈ Z, acts on the lattice as a deformation (v, 0 6 ; 0 6 ) → (v, w; w) (2.11) with w = − 1 2 (v · δA i )e i , which makes the corresponding state acquire a mass α ′ 2 M 2 = w 2 in a smooth way. This is a version of the stringy Higgs effect [23]. 3 More generally all possible extra massless states correspond to Narain vectors with quantum numbers satisfying q T C (16) q + 2n T m = 2. This is a consequence of the mass formula as we will recall in section 5. A second subclass of this set will be the subject of the next section.

Definition of the orbifold
We can now proceed to extend this to the untwisted sector of an orbifold compactification.
First of all one has to select a Narain lattice with some symmetry that can be modded out.
Our reference lattice will be the one with vanishing Wilson lines, which therefore factorises as Γ 22;6 = Γ 16 ⊕ Γ 6;6 . For definiteness, Γ 16 will be the E 8 ⊗ E 8 lattice. Then, we have to specify the twist action on Γ 16 and Γ 6;6 . Whereas the twist action on Γ 6;6 is defined by choosing one of the 18 twists θ of the compactification lattice Λ that lead to N = 1 space time supersymmetry [35], the twist on Γ 16 will be a Weyl twist (inner automorphism) θ ′ of The total twist Θ = (θ ′ , θ, θ) of the Narain lattice is constrained by world sheet modular invariance [36]. The level matching conditions worked out in [36] restrict the eigenvalues of Θ. In this paper we will not present a classification of Z N Weyl orbifolds.
Instead we will take one of the E 8 as a hidden sector and assume that the Weyl twist in this sector has been chosen in such a way that it cancels the WS modular anomalies of the internal twist and of the Weyl twist in the first E 8 . The orbifold model defined this way still has some Wilson line moduli left. In order to be compatible with the twist the Wilson lines must satisfy [29,37] where A Ij is a matrix containing the components of the Wilson lines and θ J I , θ i j are the matrices of the gauge and of the internal twist with respect to the lattice bases e I and e i of Γ 16 and Λ * respectively. Wilson line moduli do exist if an eigenvalue appears both in the gauge twist and in the internal twist. More precisely, if a complex conjugated pair of eigenvalues (a real eigenvalue) appears d times in the gauge twist and d ′ times in the internal twist, this then leads to 2dd ′ (dd ′ ) real moduli [30,37].

Minimal gauge groups in the presence of Wilson lines
Let us now work out the gauge groups for Weyl twists of a E 8 in the presence of generic continuous Wilson lines. The basic idea is the following. All Weyl twists of E 8 are induced by twists that have a nontrivial action on some sublattice N . This sublattice can be chosen to be the root lattice of a regular semi-simple subalgebra. The twist action on the sublattice can then be described by a so called Carter diagram, which can be thought of as a generalization of the well known Dynkin diagram [38]. In fact most Weyl twists of E 8 are induced by Coxeter twists of regular subalgebras and in this case the Carter diagram is identical with the Dynkin diagram of this subalgebra. To get all inequivalent Weyl twist one has to add a few twists of subalgebras, which are not Coxeter twists. These are then described by Carter diagrams that are not Dynkin diagrams. For more details on Carter diagrams and their relation to Weyl twists see [39] and [40].
Given the sublattice N on which the twist acts non-trivially one has to look for the largest complementary sublattice I on which it acts trivially. That means that I is defined by together with (E 8 ⊃ N ⊕ I and I ′ ⊃ I) =⇒ I ′ = I. (2.14) If we now decompose E 8 into conjugacy classes with respect to N ⊕I, a familiar procedure used in covariant lattice models, we get schematically that This means that there are three types of lattice vectors, namely those belonging to the sublattices N and I and those which have a non-vanishing projection onto both N and I. For a review of lattice techniques see [41].
Using this decomposition the effect of switching on continuous Wilson lines becomes quite obvious. If we assume for the moment that that all eigenvalues of the gauge twist also appear in the internal twist, which is true for all Z 3 , Z 4 , Z ′ 6 and Z 7 orbifolds, then the Wilson lines will take arbitrary values in N R . Thus for any generic choice of the Wilson lines only states corresponding to lattice vectors in (0, I) are massless. Note that these states are automatically twist invariant. Therefore the gauge group of the orbifold contains at least a semi-simple group corresponding to the roots of the lattice I.
However, the rank of this group is not a priori guaranteed to be 8 − dim(N ). This follows from the fact that the subgroup, to which E 8 is broken, need not to be semi-simple but only reductive, that is there can be U(1) factors around. On the other hand, when considering the action of the twist on the Cartan subalgebra, one knows that dim(N ) Cartan generators are not invariant under the twist and are therefore projected out, whereas 8 − dim(N ) are invariant under the twist. Therefore, the rank of the unbroken gauge group is 8 − dim(N ) and the gauge group itself is given by where G I is the semi-simple Lie group associated to the roots of the lattice I.
Decompositions of the form N ⊕ I can easily be found using the formalism of extended Dynkin diagrams [42]. There are, however, cases where the decomposition is not unique. This is not in contradiction with I being maximal, because ⊃ is a partial ordering relation, only. It is well known that it happens in a few number of cases that the twist of the full algebra does not only depend on the isomorphic type of the subalgebra that is twisted, but also on the precise embedding [38]. This is easily illustrated by the following example, namely by decomposing E 8 into cosets with respect to A 8 1 and then taking the A 4 1 Coxeter twist. From the decompostion one sees that, depending on the choice of the A 1 's, one has that I = D 4 or I = A 4 1 . These twists are known as A 4 I 1 and A 4 II 1 , respectively.
Fortunately, we can use the results of the classification of conjugacy classes of the Weyl group, which is equivalent to the classification of Weyl twist modulo conjugation, for determining the minimal gauge groups. In his work [38] Carter gives, for all twists, the decompositions of the root system, which specify the group G I . Then, according to (2.16), all one has to do is to add some U(1) factors, if necessary. We have listed all the minimal gauge groups that can appear in the context of N = 1 supersymmetric Z N orbifold compactifications.
Note, however, that we have so far assumed that the Wilson lines are really allowed to take values in all of N R . But this is not the case if an eigenvalue of the gauge twist does not appear in the internal twist. The simplest example for this is provided by the A 3 Coxeter twist which has, as a lattice twist, the eigenvalues ω i , i = 1, 2, 3 where ω = exp(2πi/4). This can be combined with a Z 8 twist of the internal space with eigenvalues Ω k , k = 1, 2, 3, 5, 6, 7, where Ω = exp(2πi/8). Since θ does not have the eigenvalue −1, there are no Wilson lines taking values in the −1 eigenspace of θ ′ . Therefore, the minimal gauge group is larger then the expected SO (10). A detailed analysis shows that the unbroken gauge group is the non-simply laced group SO (11).
Note that this is not in contradiction with the twist being defined through a Weyl twist of E 8 , because an inner automorphism of the full group may be an outer one of a subgroup.
Therefore, breaking to non-regular subgroups is possible, if Wilson lines are turned on.
In these cases, which include many of the Z 6 and Z 8 , and most of the Z 12 and Z ′ 12 orbifolds, our list only provides a lower bound on the gauge group that is not saturated.
Note that the analysis to be performed in order to get the minimal gauge group is, in these cases, the same as the one one has to use in order to get intermediate gauge groups, that is gauge groups that are neither minimal nor maximal. A combination of counting and embedding arguments as used in [37] is in many cases sufficent to determine the gauge group.
The maximal gauge groups which appear for vanishing Wilson lines have been described in [40]. Here the gauge group of the torus modes is E 8 and therefore all the three classes of vectors in the decomposition are present. In order to determine the gauge group of the orbifold one has to form twist invariant combinations of the states corresponding to lattice vectors in (N , 0) and (W i (N ), W i (I)), because these vectors transform nontrivially under the twist. For convenience we have included the results of [40] in our table.
The table is organized as follows. We list all twists that have order 2, 3, 4, 6, 7, 8 or 12 and can therefore appear in the context of N = 1 orbifolds. We quote the conjugacy class of the twist and its name from [38]. So, a Coxeter twist in the subalgebra X is called X, whereas the non-Coxeter twists, if they are needed, are called X(a i ). If the twist depends on the embedding of the subalgebra, then the inequivalente choices are labeled by X I , X II , . . .. Next, we list the order of the twist. An asteriks is put on those numbers, where the order of the Lie algebra twist is twice the order of the corresponding lattice twist, following [40]. We also list the non-trivial eigenvalues of the twist, because they specify the structure of the untwisted moduli space [30]. Actualy, we list the corresponding powers of the N-th primitive root of unity where N is the order of the twist. The eigenvalues of the Coxeter twists have been calculated using their relation to the ranks of the inequivalent Casimir operators, whereas the eigenvalues of non-Coxeter twists are quoted from [39]. Then we give the minimal gauge groups, by taking the semi-simple part from [38] and adding U(1)s if needed. Finally, we also quote the maximal gauge groups from [40].

Extended gauge groups from the compactification sector
In this section we will discuss the enhancement of the gauge group occuring in the compactification (also called internal) sector of Narain compactifications and Narain orbifolds. We will also take into account the effect of continuous Wilson lines on this enhancement.
In the toroidal case, the generic gauge symmetries come from the twelve conserved chiral world sheet currents ∂X i L (z) and ∂X i R (z) corresponding to the left-and right-moving parts of the six internal coordinates. Thus, the generic gauge group coming from the compactification sector is U(1) 6 L ⊗U(1) 6 R . The six gauge bosons of the U(1) 6 R are graviphotons, that is they are part of an N = 4 gravitational supermultiplet, whereas the other six gauge bosons belong to N = 4 vector multiplets.
In order to break the extended N = 4 supersymmetry to N = 1 one must break the U(1) 6 R completely. This is done by every twist that doesn't have 1 as an eigenvalue. To preserve the minimal N = 1 supersymmetry, the twist must be in the subgroup SU(3) ⊂ SO(6) [25]. This leads to the classification [35] of N = 1 twists.
Since the twist acts in a left -right symmetric way the U(1) 6 L is automatically also broken completely. There are, however, similarly to the gauge sector, special values of the moduli for which one has extra conserved currents. Let us discuss this for the untwisted model first. The additional massless gauge bosons are related to Narain vectors P = q I l I + n i k i + m i k i with quantum numbers q I = 0, n i m i := n T m = 1. In order to extend the generic gauge group U(1) 6 L to G (l) ⊗ U(1) 6−l , where G (l) is a rank l ≤ 6 semisimple simply laced Lie group with Cartan matrix C ij , i, j = 1, . . . , l, the moduli must satisfy the relations [33] This implies that D ij ∈ Z and moreover the vectors are in the Narain lattice. The vectors e i = G ij e j are basis vectors of the compactification lattice Λ. Since C ij is a Cartan matrix, (p For vanishing Wilson lines the conditions (3.1), (3.2) reduce to the more special conditions for points of extended gauge symmetry that are known from [34]. Note that an extended gauge group is compatible with, at least, small deformations of the Wilson lines, because their effect can be compensated by tuning the metric.
In the orbifold case, the moduli are further restricted by the condition of compatibility with a given twist. These conditions are, in the absence of discrete background fields, given by (2.12) and and θ j i , θ i j are the matrices of the internal twist with respect to lattice bases of the compactification lattice Λ and its dual [37].
As an example, let us discuss the compactification on a 2-torus T 2 . There are two rank two semi-simple simply laced Lie algebras, namely A 1 ⊕ A 1 and A 2 with corresponding gauge groups SU(2) 2 and SU(3). These enhanced gauge groups occur at special points in the moduli space. Let us in the following first consider the case of vanishing Wilson lines. The special points of enhanced gauge symmetries are determined by equations (3.1) and (3.2). For the maximal enhancement of the gauge group to SU(2) 2 or SU(3), these equations do completely fix the moduli G ij and B ij , up to discrete transformations.
Introducing complex moduli T and U, where G ij , B ij , i, j = 1, 2 are the real moduli of the 2 -torus, these conditions can be expressed as and U = T = e iπ/6 (3.8) for a maximal enhancement to A 1 ⊕ A 1 and A 2 , respectively [23]. Note that in these equations we have restricted ourselves to the critical points of the standard fundamental domain. If one allows for an abelian factor, then one can also have the gauge group SU(2) ⊗ U(1). Inspection of equations (3.1) and (3.2) shows that, in this case, not all of the moduli are fixed by these equations, but that two real moduli parameters are left unfixed. These two parameters can be taken to be one of the two radii of the T 2 as well as the angle between these two radii. In terms of the complex moduli T and U, this means that the extended gauge group SU(2) ⊗ U(1) occurs for points in moduli space for which U = T . Note that this complex subspace U = T contains the two points (3.7) and (3.8) of maximally extended symmetry. For generic values U = T , the toroidal gauge group is given by U(1) 2 .
Let us now discuss the enhancement of the gauge group in the context of orbifold compactifications for which the underlying internal 6-torus decomposes into a T 6 = T 2 ⊕ T 4 .
For concreteness, let us study the effect on this enhancement of a Z 2 acting on the 2-torus T 2 . This is the situation encountered in a Z 4 -orbifold, for instance. The Z 2 twist, given by the reflection −I 2 , doesn't put any additional constraints on the four real moduli G ij and B ij and on their complex version U and T . Therefore, the moduli space associated with the T 2 is the same as in the toroidal case. As is well known from one-dimensional compactifications, the enhanced (SU(2)) and the generic (U(1)) gauge groups get broken to SU(2) → U(1) and U(1) → ∅ by the Z 2 -twist, respectively. Thus, in the case of the Z 2 -twist acting on the internal T 2 , the gauge groups for the points U = T = 1, exp(iπ/6) = U = T = 1 and U = T in moduli space are given by U(1) k with k = 2, 1, 0, respectively.
Something more interesting happens at the SU(3) symmetric point U = T = exp(iπ/6) in the orbifold case. The Z 2 -twist on T 2 can be decomposed into where W 1 , C and D are the first Weyl reflection, the Coxeter twist and the diagram automorphism of A 2 , respectively. Therefore the twist acts as an outer automorphism and breaks the SU(3) to the maximal non-regular subgroup SU (2). Thus, we have found a bosonic realization of the conformal embedding SU(2) k=4 ⊂ SU(3) k=1 and expect that the SU(2) is realized at the higher level k = 4 in order to have central charge c = 2.
Indeed, a direct calculation of the OPE of the twist-invariant combinations of conserved currents shows that there is a SU(2) current algebra at level k = 4.
Note that this phenomenon of rank reduction and simultaneous increase of the level is also quite generically present in the gauge sector because, as mentioned in the previous section, a Weyl twist of the E 8 will often act as an outer automorphism of a subalgebra left unbroken by Wilson lines. Take as an example the SU(3) 3 model obtained in [43] through switching on Wilson line moduli. By inspection of the vertex operators given in [43] one easily sees that the algebra of the corresponding deformed untwisted model is , like in some of the models described in [37]. Therefore all these models should We will present here the case of the two-dimensional torus and its Z 2 orbifold. The Z 3 orbifold will be discussed at the end of this section.
The modular group of both the torus compactification and its Z 2 orbifold is, in the absence of Wilson lines, the group SO(2, 2, Z). This group has four generators, which we can take to be S, T , D 2 , R as defined in [20]. S and T generate the subgroup SL(2, Z) ⊂ SO(2, 2, Z), which is the subgroup of orientation preserving basis changes in Λ, whereas R is the reflection of the first coordinate and D 2 is the factorized duality in the second coordinate. Note that there is a second SL(2, Z) subgroup which commutes with the first one. It is generated by S ′ = D 2 SD 2 and T ′ = D 2 T D 2 . Whereas T ′ is the axionic shift symmetry, S ′ is almost the full duality D = D 1 D 2 , namely S ′ = SD. Note that D 1 , which is the factorized duality transformation of the first coordinate, is not an independent generator of the group because P := RS permutes the two coordinates and therefore D 1 = PD 2 P. See [20] for a detailed discussion. The explicit matrices given there specify the linear action of the modular group SO(2, 2, Z) on the quantum numbers.
The group SO(2, 2, Z) acts non-faithfully and fractionally linear on the moduli. The faithfull transformation group P SO(2, 2, Z) is known to be generated by the transformations [50] S : In the Z 2 orbifold the critical points are the same, but since one must form twist invariant combinations the numbers of extra massless gauge bosons is divided by the order of the twist. Therefore equation (3.10) holds as well. The case of the Z 3 orbifold will be discussed below.
Let us now switch on Wilson lines in the Z 2 orbifold model discussed above. As already argued above in terms of the real moduli, any extended gauge group can be preserved by an appropriate tuning of the moduli of the compactification sector. All we have to add here is the prescription of this tuning in terms of the complex moduli. We will do this for the case of two complex Wilson line moduli B, C associated with the internal 2-torus The T modulus now reads whereas the U modulus is not modified. Here A µi denotes the µ-th component of the i-th Wilson line with respect to an orthonormal frame, µ, i = 1, 2.
States which become massless for generic U = T (where the orbifold gauge group is (1)) stay massless when Wilson lines are turned on. Thus, no tuning of the complex moduli U and T is necessary. In order to have the maximally extended Wilson lines, replaced by whereas in order to have SU(3) → SU (2), the new condition is given by This follows from the discussion of the zeros of the mass formula, which will be presented in section 6.
Consider, as another example, a Z 3 orbifold defined by the Coxeter twist of the A 2 root lattice. Here, the moduli have to be restricted in order to be compatible with the twist.
In terms of complex moduli one has to set U = 1 2 ( √ 3 + i) = e iπ/6 and C = 0, whereas Since the U and the C modulus are frozen to discrete values, the modular group of the Z 3 orbifold is smaller than the one of the untwisted model. In the case of vanishing Wilson lines the modular group is obviously SU(1, 1, Z) ≃ SL(2, Z) T with generators S ′ and T ′ as defined above. The point T = e iπ/6 of extended U(1) 2 symmetry is a fixed point under which is a transformation of order 3. This reduction of the order, as compared to the modular group of the untwisted model, is due to the fact that the transformation M, which is of order 2, is not in the reduced modular group SU(1, 1, Z). 4 Mass formulae for SO(p + 2, 2) and SU(m + 1, 1) cosets In this chapter we show that in the case of a factorizing 2-torus T 2 , the moduli dependent part of the mass formula for the untwisted sector of an N = 1 orbifold can be written as |M| 2 /Y . M is a holomorphic function of the moduli and depends on the quantum numbers. Y is a real analytic function of the moduli, only, and is related to the Kähler

Torus compactifications and the SO(22, 6) coset
Let us first recall that the mass formula for the heterotic string compactified on a torus and that physical states must also satisfy the level matching condition Here, we have absorbed the normal ordering constant of the NS sector into the definition of the right moving number operator and restricted ourselves to states surviving the GSO projection. Thus N R has an integer valued spectrum in both the NS and the R sector.
Substituting the second equation into the mass formula yields Our aim is to make the moduli dependence of the mass explicit.
The first step is to express p R in terms of the quantum numbers q I , n i , m j and the real moduli G ij , B ij , A i . This can be done by expanding the Narain vector P = (p L ; p R ) ∈ Γ in terms of the lattice basis l I , k i , k j (2.4) -(2.6), and then projecting onto the right handed part by multiplying with the vectors e (R) where the e µ are an orthonormal basis of R 6 The norm squared takes then the form where v is the vector of quantum numbers, The matrix Φ contains the moduli are the moduli matrices and is a 6-bein whose appearence reflects the fact that the e (R) µ can be rotated by an SO (6) transformation. e i are a basis of the dual Λ * of the compactification lattice Λ.
The matrix Φ satisfies the equation where H 22,6 is the standard pseudo-euclidean lattice metric of Γ = Γ 22;6 : where C (16) is the Cartan matrix of E 8 ⊗ E 8 . By a coordinate transformation Φ → Φ equation (4.12) can be brought to the form with the standard pseudo-euclidean metric of type (+) 22 This is the standard form of the constraint equation which defines the coset space (6)) in terms of homogeneous coordinates Φ [44]. Therefore Φ is a modified homogenous coset coordinate. It has the advantage that not only the deformation group SO(22, 6) acts linearly on it, as usual for homogenous coordinates, but that the subgroup of modular transformations acts by integer valued matrices [30].
In the following the pseudo-euclidean lattice metric of an integer lattice Γ m;n and the standard metric of type (+) m (−) n will be denoted by H m,n and η m,n respectively.

Orbifold compactifications and symmetric Kähler spaces
In the following subsections we will derive mass formulae and parametrizations for the untwisted moduli of orbifold compactifications. This will be done in four steps. In the first step we will recall how one can derive an explicit real parametrization of orbifold moduli spaces by solving the constraint equations, imposed by the compatibility requirement with a given twist, on the moduli G ij , B ij , A i of the Narain model. This solution can then be used in a second step to locally factorize the moduli space into spaces corresponding to distinct eigenvalues of the twist. The third step is then to find appropriate complex coordinates on each factor space which make explicit its Kähler structure. This can be done by using the relations between the the real moduli and homogenous coset coordinates. Finally, one can solve the constraint equations for an independent set of complex moduli. Plugging these results into the mass formula allows one to write its moduli dependent part as the ratio of a holomorphic and a real analytic piece, where the latter one is related to the Kähler potential.

Untwisted orbifold moduli
We will now implement the first step of the four described above. Let us recall that it was shown in [37], based on earlier results of [29,45], that the continuous parts of the background fields G ij , B ij , A i must satisfy the equations where θ j i , θ j k and θ J I are the matrices of the internal twist θ and of the gauge twist θ ′ with respect to the lattice bases e i , e i and e I of the lattices Λ, Λ * and Γ 16 . Although θ and θ ′ are orthogonal maps their matrices with respect to non-orthonormal lattice bases are not. Thus, it is convenient to express equations (4.16) in terms of orthonormal bases e µ and e M of R 6 and R 16 , before solving them [45,37]. The transformations between the lattice and orthonormal frames is given by n-bein matrices A useful relation to be used later is E * = E T,−1 . One complication is that the metric moduli drop out of the D-matrix when written with respect to an orthonormal frame, because e µ ·e ν = δ µν and therefore (see below for details of the transformation) Instead, they are now contained in the 6-bein E which is sensitive to deformations of Λ.
To study the effect of lattice deformations let us fix a reference lattice Λ and introduce a deformation map S (or better a family of deformation maps) which maps it to Λ This is compatible with the twist θ : In order to transform equations (4.16) and (4.19) into the orthonormal frame we will have to work out some formulae. Consider therefore the deformation described in terms Thus, a compactification on Λ with background metric δ µν can be reinterpreted as a compactification on a fixed lattice Λ in a deformed background G µν . Noting that S ν µ = e ′ µ · e ν = e ′ µ · e ν =: S µν and S µ ν := e ′µ · e ν = G µν S µν (4.21) and defining S * = (S µ ν ), S = (S ν µ ) we have that S * = S T,−1 . Next we introduce a fixed (moduli independent) coordinate transformation which connects the orthonormal basis e µ to the lattice basis e i of the reference lattice Λ: This matrix and its inverse T i µ = e µ · e i connect the deformation matrices by Therefore, the matrix T µ i also connects the deformed orthonormal basis e ′ µ to the deformed lattice basis e i : The relations between the four n-bein matrices E ν i , T µ i , S j i , S ν µ and the four bases e i , e i , e µ , e ′ µ are summarized by which shows that S j i and S ν µ relate the undeformed lattice basis e i and the orthonormal basis e µ to their images e i , e µ under the deformation S, whereas T describes a fixed coordinate transformation relating e i to e µ and e i to e ′ µ . E is a family of coordinate transformations relating the moving frame e i to the fixed orthonormal frame e µ . Equation (4.25) shows that E can be factorised into a moduli-dependent piece and a moduliindependent one in two different ways.
We can now use the n-bein matrices and transform the equations (4.16), (4.19) into the orthonormal bases. In order to do this we have to introduce the transformed moduli and the transformed twist matrices Note that these relations are consistent thanks to (4.19). Since e µ , e M are orthonormal bases it follows that are orthogonal matrices.
Using the formulae derived above we find that equations (4.16), (4.19) are equivalent to The orthonormal bases can be chosen such that the twists θ, θ ′ take their real standard forms, with nonvanishing 2 × 2 blocks along the diagonal. A slight modification will turn out to be useful in order to display the coset structure. Namely, we will group together degenerate eigenvalues into bigger blocks. For definiteness let us consider the gauge twist θ ′ and assume that it has complex eigenvalues e ±iψ j and real eigenvalues −1 and 1 with multiplicities m j , p and q. Then there exists a basis e M such that The internal twist θ can be brought into the same form with multiplicities n j , r and s.
In these coordinates the equations (4.29) can be easily solved. The result for the Wilson lines matrix is and for the D-matrix the result is with S has the same structure as D. Note, however, that only the symmetric positive part (in the polar decomposition) of S is physically relevant, because the orthogonal part describes a pure rotation of Λ. We use here that, when referring to an orthonormal frame, the matrix of the positive symmetric (orthogonal) part of an invertible map is a positive symmetric (orthogonal) matrix. The irrelevance of rotations will also be manifest by the fact that the masses only depend on the metric through

Factorization of the moduli space
We can now perform the second step, namely use the constrained form (4.32) -(4. 35) of the moduli in order to factorize the untwisted orbifold moduli space in factors corresponding to the various distinct eigenvalues. In order to display its coset structure we must work with the homogenous coset coordinate. Therefore we start by replacing the lattice indices appearing in Φ by orthonormal indices. This defines a new homogenous coordinate Φ on the SO(22, 6) coset. In order to keep the metric moduli inside Φ we do not use the 6-bein E i ν but only its moduli-independent part T i ν to define the new coset coordinate, namely The new coset coordinate satisfies the coset equation in which the pseudo-euclidean lattice metric H 22,6 is replaced by because of the coordinate transformation. Inside the mass formula we absorbe the trans- Now we can use the block-diagonal form of the A, D and S matrices and permute the rows and columns of Φ such that it becomes block-diagonal and a similar expression for φ (−1) ∈ M(p + r, r, R) and φ (+1) ∈ M(q + s, s, R).
In the coset equation (4.39) this permutation results in replacing which makes manifest the factorization into factors corresponding to the distinct eigenvalues of the twist.
Again we can keep the mass formula form-invariant by absorbing this permutation into the components v. Note, however, that the underlying lattice will not have a corresponding decomposition, because a generic lattice vector has nonvanishing projections onto more than one eigenspace. Therefore the mass formula will not factorize for all states, but only for those having quantum numbers which live in only one of the eigenspaces.
Before we proceed to discuss the irreducible factors in the decomposition, let us recall from [25] that N = 1 space time supersymmetry requires θ to have no eigenvalue +1 and the eigenvalue −1 can only have multiplicities 0 or 2. Therefore s = 0 and either r = 0 or r = 2. This implies that Φ (+1) does not appear. More generally, if some eigenvalue does only appear in θ (θ ′ ) but not in θ ′ (θ) this gives rise to vanishing rows in the Wilson line matrix and therefore also in the rearranged Φ. They correspond to directions of the Narain lattice which possess no deformations.

Real eigenvalues and the SO(p + 2, 2) coset
We will now study the subspace which corresponds to the real twist eigenvalue −1 and is parametrized by φ (−1) . For the later discussion of modular symmetries it is convenient to go one step back and to introduce a lattice basis in this sector. More precisely (see [30]) we can find a sublattice Γ p+2,2 of the Narain lattice Γ 22,6 on which the twist Θ acts as −I p+4 . Note, however, that this lattice is only a sublattice but not a direct factor, which means that there is no decomposition Γ 22,6 = Γ p+2,2 ⊕ · · ·, but only a sublattice Γ p+2,2 ⊕ · · · ⊂ Γ 22,6 . We will now consider those states that have non-vanishing quantum numbers lying in Γ p+2,2 only.
The moduli dependent part of the mass formula for such states now takes the form which are lattice coordinates for Γ p+2,2 . For vanishing Wilson lines Γ p+2,2 factorizes as where Γ p is a sublattice but not a factor of the E 8 ⊗E 8 or SO(32) lattice Γ 16 . Γ 2,2 denotes the momentum/winding lattice corresponding to a two-dimensional sublattice Λ (2) of the compactification lattice Λ, which we assume to have a decomposition Λ = Λ (2) ⊕ Λ (4) , as this happens for example for the Z 4 twist. If Wilson lines are switched on, Γ p+2,2 does not factorize any more but can still be described in terms of Γ p and Γ 2,2 .
is the analog of Φ, because it satisfies the constraint equation of a SO(p + 2, 2) coset with the standard metric replaced by the lattice metric H p+2,2 of Γ p+2,2 where and C (p) is the lattice metric of Γ p .
In order to introduce complex coordinates and to make explicit the Kähler structure of the SO(p + 2, 2) coset, we will now transform equation (4.51) into its standard form.
This can be done in two steps. The first step consists in converting the lattice indices I, J, . . . = 1, . . . , p which refer to a lattice basis of Γ p , into orthonormal indices M, N, . . .
as well as converting the lattice indices i, j, . . . = 1, 2, which refer to lattice bases of Λ 2 (as lower indices) and of Λ * 2 (as upper indices), into orthonormal indices µ, ν. This is done in a similar way to the one discussed in the last subsection, and one arrives at the The second step for bringing equation (4.51) into its canonical form consists in replacing the metric H p+2,2 by the standard metric η p+2,2 . This is achieved by In terms of the real moduli we have that The new homogenous coset coordinate φ satisfies the standard coset relation Recall that η p+2,2 is the standard metric of type (+) p+2 (−) 2 .
The complex coordinate φ c satisfies the equations and is therefore the standard complex homogenous coordinate on the SO(p + 2, 2) coset [44]. Note that we can replace v Tφ by v T φ c in the mass formula, because Thus the moduli dependent part of the mass is proportional to the norm squared of a complex number.
Our next step towards the derivation of the mass formula is to solve the complex constraint equations in terms of unconstrained complex coordinates and to make explicit the Kähler structure of the moduli space and the Kähler potential. This procedure is well known both in the mathematical [44] and in the physics [27] literature. We will use first the physicists approach and then explain the relation to the results in the mathematical literature.

A bounded realization
One way of solving the constraint equations is to first extract a scale factor √ Y from the coordinates, where Y is a positive, real analytic functions of the moduli. It turns out that this function is closely related to the Kähler potential [27]. In terms of the new One can now take the first p + 2 variables as the independent ones and express the other two in terms of them [27] y i = z i , i = 1, . . . , p + 2, y p+3 = 1 2 This solves equation (4.61). Defining z = (z i ) ∈ C p+2 we see that equation (4.60) yields that The domain defined by Y > 0 has two connected components, which is readily seen from |z T z| = 1 ⇒ Y = 0. Choosing |z T z| < 1 defines a bounded open domain in C p+2 , called a complex polydisc, which provides the standard bounded realization of the SO(p + 2, 2) coset [44]. The domain possesses a Kähler metric, with Kähler potential [46]
Taking for definiteness y p+3 + y p+3 > 0 we get the unbounded open domain which differs by a factor of i from the one used in the mathematical literature [44]. Note that the imaginary part of y is unconstrained whereas the real part lives in the forward light cone of a p + 2 dimensional Minkowski space.
Using the holomorphic transformation between the two parametrizations given in [48] we know that also is a Kähler potential.

Another unbounded realization
For applications one prefers to rearrange the complex moduli y in terms of a T and an U modulus and additional complex Wilson line moduli B k , C k . Then, for vanishing Wilson lines, one is left with an SO(2, 2) coset which factorizes into two SU(1, 1) cosets parametrised by the T and the U modulus. To do this one has to set (generalizing the treatment of SO(4, 2) cosets [27,30]) with k = 1, . . . , r. If p is even, then r = p 2 . If p is odd, r = p−1 2 and then there is one additional unpaired complex coordinate In terms of the new moduli, the Y u function now reads For completeness, let us display how the y i look in terms of the new moduli . . , and y p = A, if p is odd Let us now comment on several special cases. First, by setting all of the Wilson moduli to zero, B k = C k = A = 0, one obtains the Kähler potential for a SO(2, 2) coset, which factorizes into two SU(1, 1) cosets parametrised by the U and the T modulus, respectively. Again, Y > 0 has two connected components, and U and T have been defined in such a way, that the condition y p+3 + y p+3 > 0 implies that they both have a positive real part, as usual.
Next, by inspection of the Kähler potential, one sees that, for a fixed value of k, T, U, B k , C k parametrize a subspace which is a SO(4, 2) coset. Likewise T, U, A parametrize a SO(3, 2) coset. Note also that by setting B k = C k one can truncate a SO(4, 2) coset to a SO(3, 2) coset. Moreover, for even p = 2m we can eliminate half of the moduli by setting U + U = r 1 and (B k + C k )(B k + C k ) = r 2 A k A k , with real positive constants r 1,2 and thus truncate the SO(p + 2, 2) coset to a SU(m + 1, 1) coset. This is obvious from the fact that Y then takes the form which leads to the Kähler potential of a SU(m + 1, 1) coset.

The mass formula
Having found various parametrizations y = √ Y φ c with y = y(z), y(y) or y = y(T, U, B k , C K , A) of the coset, we can substitute each of them into the mass formula with the result that Therefore the mass squared can be written as the ratio of the square of the norm of a chiral mass M = v T y, which is a holomorphic function of the moduli and contains the dependence on the quantum numbers, and of a real analytic positive function Y , which is related to a Kähler potential by K = − log Y .
Let us recall that this expression is invariant under the group of target space modular transformations. To be precise we will consider here the modular group of the sublattice Γ p+2,2 only. The connection of this group with the full modular group was discussed in the extended version of our paper [30]. A general lattice vector P ∈ Γ p+2,2 can be decomposed as Here v = (v A ) denotes the set of all components with respect to a lattice basis of Γ p+2,2 and e M an orthonormal basis. The p + 4 bein E M A (m) connecting the two bases contains the dependence on the moduli m = (T, U, B k , C k , A). Note that in [30] the full p + 4 bein (E M A ) was denoted by φ, a symbol that we now use for the right moving part (E i A ) of it. Note also that we discussed the full Narain lattice there, but these considerations also apply to sublattices. Now a modular transformation consists of acting with a matrix while simultanously transforming the moduli m → m ′ such that P is invariant [49,30].
As we discussed in some detail in [30] the modular group can be interpreted as a discrete subgroup of the deformation group SO(p + 2, 2) and therefore, the transformation of the moduli is given by the left action of this discrete subgroup on the SO(p + 2, 2) coset.
Starting from this observation we then showed how the concrete transformation laws of the moduli can be worked out.
We can now combine this with the results of this section to deduce the general form of the transformation of Y and M. First we know that Y only depends on the moduli, but not on the quantum numbers, and that it is related to the Kähler potential by K = − log Y .
Since the moduli are holomorphic coset coordinates, and modular transformations result from the left action of SO(p + 2, 2) on its coset, which is a symmetric Kähler manifold, K must transform by a Kähler transformation in order to convert the lattice basis of our reference lattice to an orthonormal one (with respect to the Euclidean scalar product) Note that we have written all of the indices as lower ones for simplicity. In a second step, we have to switch to a basis which is also orthonormal with respect to the pseudoeuclidean metric. This gives v T = ( q 1 , q 2 , n 1 , n 2 , m 1 , m 2 ) = √ 2q 1 , √ 2q 2 , n 1 + 1 2 m 1 , n 2 + 1 2 Next, let us use the solution (4.73) of the coset equations for the case p = 2 with U, T, B, C as independent variables. Then the chiral mass is given by Using the mass formula (4.75), with the Kähler potential for p = 2 inserted in it, then gives the following mass formula for an SO(4, 2) coset The mass formula for a SO(2, 2)-coset is obtained by setting B = C = 0, yielding [27] i M = m 2 − im 1 U + in 1 T − n 2 UT (4.86)

Complex eigenvalues and the SU(m + 1, 1) coset
We will now discuss the moduli subspace corresponding to complex eigenvalues. We will, however, restrict ourselves to the case of a non-degenerate rightmoving complex eigenvalue, n i = 1. In geometrical terms this means that we again focus on a two torus with Wilson line moduli switched on, but this time with the twist on this subsector acting as a rotation and not as a reflection. As in the case of the SO(p + 2, 2) coset, one can step by step repeat the procedure given there for introducing a new homogenous coset coordinateφ satisfying φ T η 2m+2,2φ = −I 2 . (4.87) In this case, however, one has to take into account that only half of the components ofφ are independent. This can be done in the following way. After a suitable reordering of the components one finds that φ ′ has the form This effectively leads to the replacement η 2m+2,2 → η m+1,1 ⊕ η m+1,1 in (4.87). Then, by introducing the complex coordinate one finds that φ c satisfies the relation characterizing the coset SU(m + 1, 1)/ (SU(m + 1) ⊗ U(1)) [44]. Finally, one also has to introduce complex quantum numbers by v i (c) = v ′ i + iv ′ m+2+i , i = 1, . . . , m + 2 (4.93) in order to be able to rewrite the mass formula as

A bounded realization
Since the homogenous coset coordinate φ c is again a complex vector (and not a matrix) we can proceed as in the last subsection. First we introduce rescaled coordinates y i = √ Y φ i c , i = 1, . . . , m + 2 and obtain the equation  This solves (4.92) with and the standard Kähler potential for this realization is [46] Comparing to standard projective coordinates (which are also known to provide a solution to the constraints [44]) we see that those are identical to the z i , z.

An unbounded realization
Again, we would like to have another, unbounded representation in terms of a T Clearly we have found an unbounded realization, because the imaginary part of T , for example, is not constrained at all, whereas the real part can be arbitrarily large.
There is a second way of connecting the bounded and the unbounded realization. Namely one can start with the bounded realization and then introduce T and A i by the map For vanishing Wilson lines this reduces to the standard map from the open unit disc onto the right half plane Substituting the transformation (4.103) into K ′ = − log Y u yields which differs from K = − log Y b by a Kähler transformation. Note that, when relating z, z i and T, A i by equating (4.101) and (4.96), this is equivalent to relating them by (4.103) modulo this Kähler transformation.

The mass formula
Again, the mass formula is given by the ratio of the square of the chiral mass and the function Y where y and Y are functions of the complex moduli T, A i , i = 1, . . . , m.
Again this expression is invariant under the group of modular transformations. The relevant group SU(m + 1, 1, Z) is a subgroup of the group SO(p + 2, 2, Z), namely the normalizer with respect to the Z N group generated by the twist [49]. Recall that these groups explictly depend on the reference lattice, so we did not make this explicit in our notation. For the same reasons as discussed before in the case of SO(p + 2, 2) cosets, Y and the chiral mass M transform as where F is a holomorphic function of the moduli.
Example: The SU(2, 1) and SU(1, 1) mass formulae We will again illustrate the general procedure with a concrete example, a two dimensional Z 3 orbifold with one independent two-component Wilson line. This time we take both the reference compactification lattice and the sublattice of Γ 16 to be A 2 root lattices.
Both the internal and the gauge twist are taken to be the A 2 Coxeter twist.
Now the transformation matrices from the lattice to the orthonormal basis are given by The transformed quantum numbers are given by v T = ( q 1 , q 2 , n 1 , n 2 , m 1 , Diagonalizing the pseudo-euclidean lattice metric yields Next, we have to reorder the components v → v ′ and finally complexify them, v ′ → v c .
Introducing the complex quantum numbers yields that v T (c) = (q c , n c , m c ) (4.112) Therefore, the chiral mass (setting y 1 = y) is given by giving raise to the mass formula We now proceed to show that one can get the mass formula (4.114) of an SU(2, 1) coset by a suitable truncation of the one for the SO(4, 2) coset given in (4.84). To do this truncation correctly, one has to take two things into account. First, the lattice Λ must be proportional to the A 2 root lattice in order to have the A 2 Coxeter twist as a lattice automorphism. This freezes the U modulus to the value U = 1 2 ( √ 3 + i), while T is still arbitrary. Secondly, one has to choose inside the E 8 ⊕ E 8 lattice a sublattice with the appropriate symmetry. Taking an A 2 sublattice amounts to setting in (4.81). Also note that the Wilson line moduli have to be fixed to B = √ 3A, C = 0 [30].
These states give rise to moduli dependent threshold corrections, which are given in terms of automorphic functions of the modular group under consideration [5,11,26,27]. We will thus focus on massive untwisted states in the following.
We will assume that the internal 6-torus factorises into T 6 = T 2 ⊕ T 4 and that the lattice twist θ acts on the 2-torus T 2 as a Z 2 -twist. We will then focus on the SO(p + 2, 2)-coset space associated with the T 2 and discuss its mass formula. That is, we will consider those massive untwisted states which have non-vanishing quantum numbers v T = (q 1 , ..., q p , n 1 , n 2 , m 1 , m 2 ) in the Narain sublattice Γ p+2,2 ⊂ Γ 22,6 , as discussed in section 4.2.3. We will, in addition, also allow these massive untwisted states to carry nonvanishing quantum numbers in an orthogonal sublattice Γ 20−p,4 with Γ p+2,2 ⊕ Γ 20−p,4 ⊂ Γ 22,6 .
Recall that the level matching condition for physical states in the heterotic string reads where (p L ; p R ) ∈ Γ p+2,2 and (P L ; P R ) ∈ Γ 20−p,4 . The mass formula for physical states can then be written as The untwisted states associated with the Narain sublattice Γ p+2,2 do, on the other hand, where C denotes the lattice metric of the sublattice Γ p of the We have shown in section 4.2.3 that for a SO(p + 2, 2)-coset p 2 R can be written as where Y is related to the Kähler potential by K = − log Y , and where M is a holomorphic function of the complex coordinates for the SO(p + 2, 2)-coset. From the study of a few examples in the past [11,26,27], it is expected that a prominent role in threshold corrections to gauge and gravitational couplings is going to be played by those massive untwisted states which satisfy 2N R + P 2 R = 0, so that Thus, we will in the following only consider untwisted states for which 2N R + P 2 R = 0. Then, (5.4) turns into Note that, for any given 2N L + P 2 L , the orbit 2n T m+ q T Cq = 2(1 −N L − 1 2 P 2 L ) is invariant under modular SO(p + 2, 2, Z)-transformations. Let us now consider the following cases: a) first, consider untwisted states for which N L = 0, P 2 L = 0. This defines an orbit for which 2n T m + q T Cq = 2 (5.8) This is the orbit which is relevant for the discussion of the stringy Higgs effect. It can namely happen [23] that at certain finite points in the fundamental region of the moduli space certain (otherwise) massive states precisely become massless, M 2 = 0.
Then, at these special points in moduli space one has that p 2 L = 2, and, hence, one has additional massless gauge bosons in the spectrum as well as additional massless scalar fields. These additional massless gauge bosons enhance the gauge group of the underlying two-dimensional Z 2 -orbifold. This enhancement can give rise to an additional SU(2), as discussed in section 3. Thus, it it crucial to take into account orbit (5.8) when discussing 1-loop corrections to gauge couplings associated with the enhanced gauge group of the compactification sector of the orbifold. Similarly, orbit (5.8) must also be taken into account when discussing threshold corrections to gravitational couplings. b) now consider untwisted states for which 2N L + P 2 L = 2. This defines an orbit for which p 2 L = p 2 R and, hence, An example is provided by setting N L = 0, P 2 L = 2. Then, take P 2 L = 2 to be a vector in the root lattice of the gauge group G ′ which is not affected by the moduli associated with Γ p+2,2 . For instance, G ′ can be taken to be the E ′ 8 in the hidden sector left unbroken when turning on Wilson lines. Then, the orbit given in (5.9) is the relevant one for the discussion of threshold corrections for this type of gauge couplings in the presence of Wilson lines.
For untwisted states with quantum numbers q = 0, (5.9) obviously reduces down to n T m = 0. This reduced orbit is not invariant under general modular SO(p + 2, 2, Z)transformations anymore, but it is still invariant under modular transformations belonging to the subgroup SL(2, Z) T ⊗ SL(2, Z) U ⊂ SO(p + 2, 2, Z). This reduced orbit is the one which has been discussed quite extensively in the literature [26,27,11] in the context of (2, 2)Z N -orbifold theories. In this context, it is well known that there are no finite points in the fundamental region of the (T, U)-moduli space where (otherwise) massive states might become massless. Indeed, by setting [27] n 1 = r 1 s 2 , n 2 = s 1 s 2 , m 1 = −r 2 s 1 and m 2 = r 1 r 2 , it follows that p 2 . This shows that it is only in the large radius limit, namely at T = ∞ with U fixed (and vice-versa), that states (Kaluza-Klein states) become massless. This is in manifest contrast to what happens for the orbit discussed in a).
c) finally, consider untwisted states for which 2N L + P 2 L ≥ 4. Inspection of (5.7) shows that there aren't any points, finite or otherwise, in moduli space for which M 2 = 0, since then p 2 L < 0, which isn't allowed. Hence, it is only for the orbits a) and b) that it can happen that massive states become massless at some special points in the fundamental region of moduli space. Threshold corrections to couplings should then exhibit a singular behaviour at precisely those points in moduli space. Thus, it appears that the interesting physics contained in threshold corrections is associated with orbits a) and b). This is then why we will be focussing on orbits a) and b) in the following.
It was shown [11,27] in the context of threshold corrections to the E 6 and E ′ 8 gauge couplings in (2, 2)Z N -orbifold theories, that it is of relevance to consider the quantity orbit log M. There, the relevant orbit is the reduced orbit discussed in case b), given by n T m = 0, q = 0. Indeed, when suitably regularised [27], n T m=0,q=0 log M| reg = log (η −2 (T )η −2 (U)), this quantity is precisely the object appearing in these threshold corrections. In the context of threshold corrections to gravitational couplings, on the other hand, one should apriori consider both the orbits a) and b). We will thus introduce the following quantities for cases a) and b), respectively It is implied in (5.10) that a regularisation procedure should exist for turning these for- Now consider the case when all Wilson lines have been turned off. As will be discussed in detail in the next section, ∆ 0 should contain a term of the form ∆ 1 , on the other hand, will contain a term of the form As shown in [27], this term arises when summing over all massive states for which n T m = 0, q = 0.
As discussed above, all the physically interesting information relevant to the threshold corrections to gravitational couplings should be contained in ∆ 0 and in ∆ 1 . These gravitational threshold corrections should then, for the case of vanishing Wilson moduli, be proportional to where the constants α and β have to be determined from an appropriate string scattering amplitude calculation. We will, in the next sections, study ∆ 0 and ∆ 1 in detail. We will, in particular, also study the effect of non-vanishing Wilson moduli on (5.13). The last section will be devoted to a discussion of threshold corrections to gauge and gravitational couplings.
Note that the analogous result to (5.13) for a SU(m + 1, 1)-coset can be obtained by an appropriate truncation of the mass formula (5.6) for a SO(2m + 2, 2)-coset, as discussed in section 4.2.4.
6 Automorphic functions for the orbit 2n T m + q T Cq = 2 In this section we will consider massive untwisted states possessing N L = 0, P 2 L = 0. Then the associated modular invariant orbit is defined by It can happen that certain massive states, characterised by a set of quantum numbers satisfying (6.1), become massless at special points in moduli space. Then, at these special points one has that p 2 L = 2, and, hence, there are some additional massless states corresponding to massless gauge bosons. Thus, orbit (6.1) is of big relevance for the stringy Higgs-effect. We will in the following try to construct automorphic functions for the orbit (6.1). We will, for concretenesss, first consider a SO(4, 2)-coset associated with an internal 2-torus T 2 . At the end of this section, by making use of the truncation procedure given in section 4.2.4, we will also discuss automorphic functions for a SU(2, 1)-coset.
Consider the quantity ∆ 0 introduced in (5.10). Taking into account that the Cartan matrix C for the Narain sublattice Γ 4;2 under consideration reads C = 2diag(1, 1), yields (5.10) as where we have rewritten the sum over all the states laying on the SO(4, 2, Z)-invariant orbit n T m + q 2 = 1 into a sum over SL(2, Z) T,U -invariant orbits n T m = constant. Then, (6.2) can be conveniently written as We begin by studying the case where the Wilson lines B and C have been switched off.
We will later generalise our results to the case of non-vanishing Wilson lines.
As already mentioned, there is an enhancement of the gauge group of the model at some Let us consider case i) first. Setting T = U turns (6.5) into Then, the holomorphic mass (6.6) vanishes for the following 2 untwisted states Note that these states satisfy n T m = 1. Thus, they are precisely the 2 states corresponding to the 2 additional gauge bosons which enhance the gauge group to G = SU(2)⊗U (1) at the points where T = U.
Next, consider case ii). Setting T = U = 1 turns (6.6) into In addition to the 2 states (6.7) there are now 2 additional ones Finally, consider case iii). Inserting T = U = e iπ 6 into (6.6) yields The holomorphic mass (6.10) vanishes not only for the states (6.7), but also for the As shown above, this is precisely the orbit for which some of the massive states become massless at the special points i)-iii) in the fundamental region of the (T, U)-moduli space.
Thus, at each of these special points ∆ 0,0 has to exhibit a logarithmic singularity. The order of this singularity should be determined by the number of states which become massless at these special points. As shown above, the ratio of additional massless states at the special points i)-iii) is 1:2:3. Thus, the order of the zeros of exp ∆ 0,0 should be 1:2:3 for the points i)-iii). According to well-known theorems of modular functions [51], the order of the zeros and of the poles, together with the modular weights (possibly leading to non-trivial multiplier systems), determine a modular function in a unique way.
Applying this to the above case yields that exp ∆ 0,0 ∝ (j(T ) − j(U)) r η(T ) −2 η(U) −2 (6.12) where r denotes some non-zero integer. j(T ) denotes the absolute modular invariant function. 5 The formal definition (6.4) of ∆ 0,0 implies that ∆ 0,0 has to carry a modular  Comparing (6.4) with (6.12), on the other hand, gives that for some non-zero integer r. In (6.13) a regularisation of the infinite sum is implied. Thus, Let us now switch on the Wilson lines B and C and discuss the resulting modifications to (6.13). We will, unfortunately, have to restrict ourselves to the case where B and C are small, that is, we will work to lowest non-trivial order in B and C, only. When switching on Wilson lines, ∆ 0,0 becomes equal to as can be seen from (4.83). Then, a suitable ansatz for the modification to (6.13) is, to lowest order in BC, given by where X(T, U) and Y (T, U) will be determined in the following. Let us first discuss the modification to the (j(T ) − j(U))-term given by X(T, U). Under SL(2, Z) Utransformations [17,30] it follows that, to lowest order in BC, Then, if one requires that j(T ) − j(U) + 1 2 BCX(T, U) should be invariant under SL(2, Z) T,U -transformations (6.16), one obtains that X(T, U) has to transform as Similarly it follows that, under SL(2, Z) T -transformations, X(T, U) has to transform to lowest order in BC as Note that, in addition, X(T, U) has to vanish at T = U. Only then does ∆ 0,0 again become singular at T = U. This has to be the case because, as discussed in section 3, the enhancement of the gauge group still occurs at T = U when Wilson lines are switched on.
Similarly, when inserting (6.9) into (6.14) shows that another singularity of ∆ 0,0 occurs at T = U = 1 + BC 2 . And finally, when inserting (6.11) into (6.14) yields yet another singularity of ∆ 0,0 at T = U = i 2 + 3 4 + BC 2 , thus substanciating our claims in section 2. All these requirements do not uniquely specify X(T, U), however. The following X(T, U) satisfies all the above requirements where a is an arbitrary constant which cannot be determined alone by symmetry arguments. Note that the first term in (6.20) transforms inhomogenously as in (6.19), whereas the second term in (6.20) transforms homogenously with modular weight 2 under SL(2, Z) T,U -transformations. 6 One can now try to determine the function Y (T, U) in a similar manner [17] by demanding that η(T ) −2 η(U) −2 + 1 2 BC Y (T, U) should transform with modular weight of −1 under SL(2, Z) T,U -transformations. Then, one has to require Y (T, U) to transform as [17] Y (T, U) → (iγU + δ)Y (T, U) + iγη −2 (U)∂ T η −2 (T ) (6.21) under SL(2, Z) U -transformations as well as [17] Y (T, under SL(2, Z) T -transformations. These transformation laws do, however, not uniquely specify Y (T, U). The following Y (T, U) transforms appropriately up to corrections of where b denotes an arbitrary constant which cannot be determined alone by symmetry arguments. Note that the first term [17] in (6.23) transforms inhomogenously under SL(2, Z) T,U -transformations, whereas the second term in (6.23) transforms homogenously with modular weight 1. 7 Hence, it follows from (6.15) that the BC-corrected exp ∆ 0,0 is given by with X(T, U) and Y (T, U) given by (6.20) and (6.23), respectively.
with appropriate constants p and t. Note that each of the factors in exp ∆ 0,1 transforms with modular weight of −1 under SL(2, Z) T,U -transformations.
Inserting (6.15) and (6.32) into (6.2) yields that +b BC η 4 (T ) η 4 (U) + p log(B + C) + p log(B − C) + . . . (6.33) where, again, p, r,t andb denote some appropriate constants. Note that each of the terms in (6.33) is either invariant or the logarithm of a term transforming with weight of −1 under modular SL(2, Z)-transformations. Also note that, up to quadratic order in B and C, the terms given in (6.33) are the only ones one can write down which can exhibit singular behaviour at finite points in the moduli space. At these points, otherwise massive states become massless, thus inducing this singular behaviour. The terms in (6.33) represent possible threshold corrections terms which one should find in calculations of the running of gauge and gravitational couplings in orbifolds with Z 2 -sectors. In that context, the coefficients p, r,t andb are related to beta-function coefficients as well as to Kähler and σ-model anomaly coefficients. This will be discussed in the last section of this paper. Finally, note that the terms given in (6.33) transform appropriately under modular SL(2, Z) T,U -transformations, but not under additional transformations belonging to the full modular SO(4, 2, Z)-group. The dots in (6.33) stand for additional contributions, which we haven't computed, whose role is to restore the proper transformation behaviour of ∆ 0 under full SO(4, 2, Z)-transformations.
Finally, let as look at the SU(2, 1)-coset discussed in section 4.2.4, which was based on an A 2 -root lattice. Applying the truncation U = 1 2 ( √ 3 + i), C = 0, B = √ 3A to (6.33) yields that where we have used that j(U = ρ) = 0. The dots in (6.34) stand for additional terms whose role is to restore the proper transformation behaviour of (6.34) under transformations belonging to the full modular SU(2, 1, Z)-group.
7 Automorphic functions for the orbit 2n T m + q T Cq = 0 In this section we will consider massive untwisted states possessing 2N L + P 2 L = 2. Then, the associated modular invariant orbit is defined by Let us again look at the example of an SO(4, 2)-coset, for concreteness. As in the previous section, by taking into account that the Cartan matrix C for the Narain sublattice Γ 4;2 under consideration reads C = 2diag(1, 1), then yields (5.10) as log M + . . .
Using that upon regularisation, (r 1 ,s 1 ) =(0,0) log(r 1 + is 1 U) = log η −2 (U) [26,27], allows one to rewrite (7.7) as This can also be written as Inserting (7.9) into (7.2) yields that This result agrees with the one given in [17], which was obtained by requiring exp ∆ 1 to transform with weight −1 under SL(2, Z) T,U -transformations [17]. Note that ∆ 1 also has to transform appropriately under additional SO(4, 2, Z)-transformations. The dots in (7.10) stand for additional contributions, which we haven't computed, whose role is to restore the proper transformation behaviour of ∆ 1 under transformations belonging to the full modular SO(4, 2, Z)-group.

Threshold corrections
In this section, we will discuss 1-loop corrections to gauge and gravitational couplings in the context of (0, 2)Z N -orbifold compactifications. We will begin by reviewing some wellknown facts about effective gauge couplings in locally N = 1 supersymmetric effective quantum field theories (EQFT).
Consider first the case of a locally supersymmetric EQFT with gauge group G = ⊗G a where the light charged particles are exactly massless, and where the massive charged fields decouple at some scale, say M X . Then, at energy scales p 2 ≪ M 2 X , the 1-loop corrected low energy gauge couplings are given by [3,4,5,8,57] 1 g 2 a (p 2 ) where b a is the coefficient of the 1-loop N = 1 β-function, β a = bag 3 a 16π 2 , computed from the massless charged spectrum of the theory. b a describes the running between M 2 X and p 2 ≪ M 2 X and is given by b a = −3c(G a ) + C T a (r C ). Here, c(G a ) denotes the quadratic Casimir of the gauge group and the sum is over chiral matter superfields transforming under some representation r of the gauge group G a . Also, T a (r) = T r r (T 2 (a) ), where T (a) denotes a generator of G a .∆ a , on the other hand, determines the boundary conditions for the running gauge couplings at M 2 X and is given bỹ The massive charged fields, which have been integrated out, contribute a finite threshold correction ∆ a to the low energy gauge coupling. There are, however, also contributions tõ ∆ a from the massless modes in the theory. They too need to be taken into account when discussing effective couplings. These massless contributions arise due to non-vanishing Kähler and σ-model anomalies [5,8,9,10,57], present in generic supergravity-matter systems, and they are given by the term c a K − 2 r T a (r) log det g r in (8.2). Here, c a = −c(G a ) + C T a (r C ), and g r denotes the σ-model metric of the massless subsector of the matter fields in the representation r.
Next consider the case where some gauge or matter particles do have small masses of the order M I ≪ M X . In a regime where M 2 I ≪ p 2 ≪ M 2 X , all interactions can be described in terms of a massless EQFT, whereas for p 2 ≪ M 2 I there is another EQFT given in terms of the truly massless fields, only [57]. We will assume that, at the threshold scale M I , supersymmetry remains unbroken whereas the gauge group G = ⊗G a is spontaneoulsy broken down toĜ. Let us consider one such factor G a and assume that it gets spontaneously broken down to G a →Ĝ a = ⊗ iĜa,i . Let us then discuss the running of the couplingĝ a,i (p 2 ) of one such subgroupĜ a,i . In order to simplify the notation, we will, in the following, simply denote this subgroupĜ a,i byĜ a and its associated coupling constantĝ a,i byĝ a . At low energies, p 2 ≪ M 2 I , the effective gauge couplingĝ 2 a (p 2 ) is given as [55,57]  We now turn to string theory and consider orbifolds with N = 2 spacetime sectors. The reason for this is as follows. Explicit string scattering amplitude calculations of threshold corrections to gauge [4,6,11,13,17,19] and gravitational [15] couplings in the context of (2, 2)Z N -orbifold compactifications show that a non-trivial moduli dependence of these thresholds only arises if the orbifold point twist group P contains a subgroupP that, by itself, would produce an orbifold with N = 2 spacetime supersymmetry. Furthermore, if the underlying T 6 torus factorises into T 6 = T 2 ⊕ T 4 , where the T 2 remains untwisted under the action ofP, then the moduli dependent threshold corrections associated with this T 2 are invariant under Γ = SL(2, Z) T,U . We will, in the following, stick to those (0, 2)Z N -orbifolds for which T 6 = T 2 ⊕ T 4 with the untwisted plane lying in T 2 , and we will derive formulae for the gauge and gravitational threshold corrections associated with this untwisted plane T 2 . As an example one can think of a Z 4 -orbifold, for whichP is the Z 2 generated byθ = θ 2 = (Ω 2 , Ω, Ω), where Ω = e πi .
Let us consider the case where the gauge group G = ⊗G a in the observable sector of the (2, 2)Z N -orbifold gets broken to a subgroupĜ by turning on Wilson moduli. Thus, the Wilson moduli act as Higgs fields. Let us assume that this breaking takes place when turning on Wilson moduli B and C associated with the T 2 (in the decomposition T 6 = T 2 ⊕ T 4 ). We will now determine the running of the gauge couplings in both the hidden and the observable sectors from equations (8.1), (8.3) and (8.4). We will throughout this section be working to lowest order in the Wilson moduli B and C.
As a consequence, the results given below will only be invariant under the subgroup SO(2, 2, Z) of the modular group SO(4, 2, Z). Also, for notational simplicity we will not For concreteness, take this to be the case for the E ′ 8 in the hidden sector. Of the orbits discussed in section 5, there is one orbit which is the relevant one for the discussion of threshold corrections to g E ′ 8 , namely the orbit 2n T m + q T Cq = 0. As also discussed in section 5, there are no finite points in the fundamental region of moduli space for which the massive states on this orbit might become massless and, hence, no additional threshold scales. Thus, for this case the only relevant threshold scale is given by M X = M string . Inspection of (7.10) shows that the threshold corrections ∆ E ′ 8 should be given by where α E ′ 8 = −c(E ′ 8 ) [8]. Inserting K = − log (T +T )(U +Ū) − 1 2 (B +C)(C +B) and (8.6) into (8.1) and (8.2) yields that . Note that we have not yet taken into account the Green-Schwarz mechanism. The Green-Schwarz mechanism can remove an amount δ GS from the above [7,8]. It might, in addition, also remove a modular invariant function [56], yielding Next, let us look at the running of the gauge coupling constants in the observable sector.
Consider a non-abelian factor G a and assume that it gets broken down toĜ a = ⊗ iĜa,i when turning on the Wilson moduli B and C. Let us then discuss the running of the couplingĝ a,i (p 2 ) of one such subgroupĜ a,i . In order to simplify the notation, we will, in the following, simply denote this subgroupĜ a,i byĜ a and its associated coupling constant g a,i byĝ a . This time, the relevant orbit is the one for which n T m = 0, q 2 = 1. Inspection of whereα a = −c(Ĝ a ) + ĈT a (rĈ)(1 + 2nĈ). Inspection of (6.32), on the other hand, shows 10 All moduli are taken to be dimensionless.
that the threshold corrections ∆ a should be given by where r denotes an unknown coefficient which cannot be determined by symmetry considerations alone. α a can either be written in terms of the gauge group G a as α a = −c(G a ) + C T a (r C )(1 + 2n C ), or in terms of the unbroken gauge groupĜ a as Here the n C denote the modular weights of the light chiral superfields. Then, it follows from (8.3) and (8.4) that the effective gauge coupling associated with the unbroken subgroupĜ a is given by r denotes a constant which cannot be determined by symmetry. Note that we have not yet taken into account the Green-Schwarz mechanism. The Green-Schwarz mechanism can, again, remove an amount δ GS from the above [7,8] and possibly also a modular invariant function [56], yielding Restrictions of a similar type on the spectrum where already considered in [55].
For completeness, let us consider the case of vanishing Wilson lines B = C = 0. Then, equation (8.12) turns into where Y = S +S − δ GS 8π 2 log (T +T )(U +Ū) + (modular inv. f unction). This describes the running of the gauge coupling associated with the unbroken group G a [4,5,6,8].
An example resembling the above situation is provided by a Z 4 -orbifold of the type Let us now consider a (2, 2)Z N -orbifold model with N = 2 spacetime supersymmetry.
Let us again assume that the internal torus factorises as T 6 = T 2 ⊕ T 4 , and that the T 2 is not twisted under the action of the internal twist, thus leading to a model with N = 2 spacetime supersymmetry. Let us discuss the threshold corrections to the gauge couplings associated with the gauge group of the internal T 2 . As discussed extensively in section 6, the orbit relevant to the discussion of the threshold corrections to these gauge couplings is given by n T m = 1, q 2 = 0. Then, inspection of (6.13) shows that a different threshold scale M I = |j(T ) − j(U)|M string arises in this context besides M X = M string .
For points where T = U in the (T, U)-moduli space, the generic gauge group U(1) ⊗ U(1) gets enlarged to SU(2) ⊗ U(1), because 2 additional N = 2 vector multiplets become massless at these points. At generic values in the (T, U)-moduli space, the running of one such effective U(1)-gauge coupling should, in analogy to (8.11), be given by 16π 2 log(T +T )(U +Ū )|η(T )η(U)| 4 (8.16) Here, the N = 2 β-function coefficientb U (1) vanishes,b U (1) = 0, since there are no massless hyper multiplets in the theory, which are charged under the U(1). The running above the threshold scale M I is determined in terms of the N = 2 β-function coefficient b U (1) associated with the light charged multiplets in the theory. Then, b U (1) is given by vec denotes the N = 2 β-function coefficient for one N = 2 vector multiplet. This is so, because at energies above the threshold M I the 2 additional N = 2 vector multiplets are effectively massless. Finally, the coefficientα U (1) is related to the N = 2 β-function coefficient [4] asα U (1) =b U (1) = 0. Note that (8.16) is manifestly invariant under modular SL(2, Z) T,U -transformations 11 and also that we have ignored a possible removal due to the Green-Schwarz mechanism.
Let us now turn to the discussion of moduli dependent threshold corrections to gravitational couplings in the context of (0, 2)Z N -orbifold theories. As in the gauge case, we will stick to those orbifolds for which the underlying T 6 torus factorises into T 6 = T 2 ⊕ T 4 and we will discuss the moduli dependent threshold corrections associated with this T 2 .
The gravitational coupling we will be considering in the following is the one associated with a C 2 in the low energy effective action of (0, 2)Z N -compactifications of the heterotic string, L = − 1 2 R + 1 Here, C 2 denotes the square of the Weyl tensor C mnpq . The conventional choice [52,53,54] for the tree-level couplings to quadratic gravitational curvature terms is taken to be the one Then, in the gravitational sector, the dilaton only couples to the Gauss-Bonnet combination GB = C 2 − 2R 2 mn + 2 3 R 2 at the tree-level, Let us first consider the running of the gravitational coupling g 2 grav in the context of (2, 2)Z N -orbifold models with gauge group G (B = C = 0). In analogy to the gauge case 11 As discussed in footnote (4), we have ignored the issue of appearance of additional non covariant terms. This will be discussed in [58]. Note that such additional terms do not change the singular behaviour of (8.16). Decomposing this N = 2 vector multiplet into one N = 1 vector multiplet and one N = 1 chiral multiplet, it follows that b grav −b grav = δ V grav + δ C grav , where δ V grav and δ C grav denote the gravitational 1-loop β-function coefficient for an N = 1 vector and chiral multiplet, respectively. b grav −b grav is also proportional to the trace anomaly coefficient of this additional N = 2 vector multiplet. The coefficientα grav denotes the contribution from the truly massless modes due to non-vanishing Kähler and σ-model anomalies, and is given by [9,16]α grav = 1 24 (21 + 1 − dim G + γ M + Ĉ (1 + 2nĈ)), where the sum now runs over all the massless chiral matter/twisted moduli fieldsĈ with modular weight nĈ, see equation (8.5). γ M denotes the contribution from the untwisted modulinos. For the case that there is no Green-Schwarz mechanism for the T 2 under consideration,α grav is also computed [15] to beα grav =b N =2 grav , whereb N =2 grav denotes the trace anomaly contribution to C 2 of all the truly massless N = 2 multiplets in the associated N = 2-orbifold with gauge groupG generated byP, T m m = − 1 (4π) 2b N =2 grav C 2 . This was checked explicitly for the example of a Z 4 -orbifold in [16]. Finally, the coefficient α grav denotes the contribution from all the massive states which decouple at M string and is also given by α grav =b N =2 grav [15].
Inserting all of this into (8.18) yields where now Y = S +S − δ GS 8π 2 log((T +T ) 2 − 1 2 (B +B) 2 ) + (modular inv. f unction). Here, α denotes the contribution to the Kähler and σ-model anomalies due to the additional N = 2 vector multiplet which has become massless at T = U = 1, ρ. Since a N = 2 vector multiplet can be decomposed into a N = 1 vector multiplet and into a N = 1 chiral multiplet C, it follows thatα = 1 24 (−1 + 1 + 2n c ) = − 2 24 , which is precisely twice the trace anomaly of one N = 2 vector multiplet. Note that, in the associated N = 2 orbifold generated byP , the gauge group associated with the internal T 2 gets enhanced from U(1) ⊗ U(1) to SU(2) ⊗ U(1) at T = U, resulting in 2 additional N = 2 vector multiplets. Thus,α is indeed equal to the additional trace anomaly contribution in the associated N = 2 orbifold generated byP . The coefficients b grav andα grav are the same as in the previous case of generic T and U values. Finally, let us discuss the running in the presence of a Wilson line B for the case where T = U = 1, ρ. When taking into account the Green-Schwarz mechanism, the running (8.21) is modified to where Y = S +S − δ GS 8π 2 log((T +T ) 2 − 1 2 (B +B) 2 ) + (modular inv. f unction) [56]. The coefficients are all given in the above discussion.

Conclusions
In this paper we have computed string threshold corrections from the massive string spectrum for Abelian orbifold compactifications. Our discussion contains two main results which were not obtained before in the literature. First, we derive from the massive spectrum the dependence of the threshold functions on the continuous Wilson line moduli at least for small values of these fields. Second, we derive the threshold functions also for those gauge groups whose massless charged spectrum is enlarged at certain points in the orbifold moduli spaces. In terms of target space free energies these thresholds correspond to summation orbits which were neglected in previous considerations. This discussion includes the stringy Higgs effect, i.e. that a gauge group gets spontaneously broken to some subgroup when the moduli take values away from the critical points in the moduli space. Then the corresponding threshold functions possess logarithmic singularities at the critical points in M. This also happens for the gravitational threshold corrections since the whole spectrum contributes to the loop corrections of the gravitational coupling constants. We have shown that if the appearance of the additional massless states is related to the six-dimensional orbifold, the correct singularity structure of the threshold functions is provided by the absolute modular invariant function j. (This result was already anticipated in [28] in quite general terms.) For the case that the enlargement of the massive spectrum is associated with the Wilson line fields, one encounters logarithmic singularities in these fields. (The leading Wilson line singularities in the threshold functions were discussed in [55,59] by using renormalization group arguments.) Our results can be applied in various ways. First, the complete moduli dependence of the threshold functions is relevant for several soft supersymmetry breaking parameters after breaking of local N = 1 supersymmetry. In particular, the Wilson line dependence is important for the µ-problem [17,60,61]. Second one can use the threshold corrections for the non-perturbative gaugino condensation mechanism in the hidden gauge sector.
Specifically, if the hidden gauge group is spontaneosly broken by Wilson line moduli, the effective superpotential depends on these fields; thus one can determine the nonperturbatively fixed vacuum expectation values of these fields. Even more interesting effects may happen if the discontinuities in the spectrum related to the six-dimensional orbifold enter the non-perturbative superpotential. This might be the case when considering non-perturbative superpotential by gravitational instantons [62], since in this case the gravitational threshold corrections are most likely to be relevant for the nonperturbative dynamics. Then the non-perturbative superpotential will contain also the absolute modular invariant function j, and the vacuum structure of the moduli depends on the singularities of the threshold functions in an interesting way [28]. Finally, these types of superpotentials are also important for the formation of domain walls and other topological objects, as discussed in [63,64].