A new twist on heterotic string compactifications

A rich pattern of gauge symmetries is found in the moduli space of heterotic string toroidal compactifications, at fixed points of the T-duality transformations. We analyze this pattern for generic tori, and scrutinize in full detail compactifications on a circle, where we find all the maximal gauge symmetry groups and the points where they arise. We show the gauge symmetry groups that arise at special points, in figures of slices of the 17-dimensional moduli space of Wilson lines and circle radii. We then study the target space realization of the duality symmetry. Although the global continuous duality symmetries of dimensionally reduced heterotic supergravity are completely broken by the structure constants of the maximally enhanced gauge groups, the low energy effective action can be written in a manifestly duality covariant form using heterotic double field theory. As a byproduct, we show that a unique deformation of the generalized diffeomorphisms accounts for both $SO(32)$ and $E_8\times E_8$ heterotic effective field theories, which can thus be considered two different backgrounds of the same double field theory even before compactification. Finally we discuss the spontaneous gauge symmetry breaking and Higgs mechanism that occurs when slightly perturbing the background fields, both from the string and the field theory perspectives.

The distinct backgrounds of heterotic string theory on a k dimensional torus with constant metric, antisymmetric tensor field and Wilson lines are characterized by the points of the O(k,k+16;R) O(k;R)×O(k+16;R)×O(k,k+16;Z) coset manifold, where O(k, k + 16; Z) is the T-duality group [1,2]. At self-dual points of this manifold, some massive modes become massless and the U (1) 2k+16 gauge symmetry becomes non-abelian. In particular, for zero Wilson lines, the massless fields give rise to SO(32) × U (1) 2k or E 8 × E 8 × U (1) 2k at generic values of the metric and B-field. By introducing Wilson lines, not only is it possible to totally or partially break the non-abelian gauge symmetry of the uncompactified theory, but it is also possible to enhance these groups. The construction of [2] further allowed to continuously interpolate between the SO(32) and E 8 × E 8 heterotic theories after compactification [3], and even suggested that these superstrings are two different vacuum states in the same theory before compactification.
Enhancement of the gauge symmetry occurs at fixed points of the T-duality transformations [4]. Massless fields become massive at the neighborhood of such points and the T-duality group mixes massless modes with massive ones [5]. Moreover, by identifying different string backgrounds that provide identical theories, T-duality gives rise to stringy features that are rather surprising from the viewpoint of particle field theories. Nevertheless, some of these ingredients have a correspondence in toroidal compactifications of heterotic supergravity. In particular, although the field theoretical reduction of heterotic supergravity cannot describe the non-abelian fields that give rise to maximally enhanced gauge symmetry 1 , being a gauged supergravity, the reduced theory is completely determined by the gauge group, which can be chosen to be one of maximal enhancement. Likewise, the global symmetries of heterotic supergravities are linked to T-duality. While the theory with the full set of SO(32) or E 8 × E 8 gauge fields has a global continuous O(k, k; R) symmetry, when introducing Wilson lines, the symmetry enlarges to O(k, k + 16; R) [6,7,8], which is related to the discrete T-duality symmetry of the parent string theory.
The global duality symmetries are not manifest in heterotic supergravity. To manifestly display these symmetries, as well as to account for the maximally enhanced gauge groups in a field theoretical setting, one appeals to the double field theory/generalized geometric reformulation of the string effective actions [9,10] (for reviews and more references see [11]). Specifically, these frameworks not only describe the enhancement of gauge symmetry [12]- [15], but also give a geometric description of the non-geometric backgrounds that are obtained from T-duality [16] and provide a gauge principle that requires and fixes the α -corrections of the string effective actions [17]. Dependence of the fields on double internal coordinates and an extension of the tangent space are some of the elements that allow to go beyond the standard dimensional reductions of supergravity.
Motivated by deepening our understanding of heterotic string toroidal compactifications, in section 2 we review the main features of heterotic string propagation on a (10 − k)-dimensional Minkowski space-time times an internal k-torus with constant background metric, antisymmetric tensor field and Wilson lines, and recall their O(k, k + 16) covariant formulation. We focus on the phenomenon of symmetry enhancement arising at special points in moduli space.
In section 3, we concentrate on the simplest case, namely circle compactifications (k = 1). To explore the moduli space, we split the discussion into the situations in which the Wilson line A preserves the E 8 × E 8 or SO(32) gauge symmetry, and those where it breaks it. In the former case, the circle direction can give a further enhancement of symmetry to E 8 × E 8 × SU (2) at radius R = 1, and either to SO(32) × SU (2) at R = 1 or to SO(34) at R = 1 √ 2 . When the Wilson line breaks the E 8 × E 8 or SO(32) gauge symmetry, the pattern of gauge symmetries is very interesting. Not only is it possible to restore the original E 8 × E 8 or SO(32) gauge symmetry for specific values of R and A, but also larger groups of rank 17 can be obtained. We explicitly work out enhancements to SO (18) 4 and U (17) at R 2 = 1 4 . We depict slices of the moduli space for different values of R and A in several figures, which clarify the analysis and neatly exhibit the regions and points with special properties.
Examining the action of T-duality, we can see that all points in moduli space where there is maximal symmetry enhancement, namely enhancement to groups that do not have U (1) factors, are fixed points of T-duality, or more general O(1, 17, Z) dualities that involve some exchange of momentum and winding number on the circle. In the simplest cases, such as those listed above, the enhanced symmetry arises at the self-dual radius given by R 2 sd = 1 − 1 2 |A| 2 . We explore the action of T-duality and its fixed points in section 3.5. One can have other points of symmetry enhancement, which are fixed points of duality symmetries that involve shifts of Wilson lines on top of the exchange of momentum and winding. This is studied in detail in section 3.6, where we obtain the most general duality symmetries that change the sign of the right-moving momenta and rotate the left-moving momenta, leaving the circle direction invariant. Concentrating on the case where the Wilson lines have only one non-zero component, we find a rich pattern of fixed points that correspond to SU (2) × SO(32) or SU (2) × E 8 × E 8 enhanced gauge symmetry, arising at R −1 sd = C, with C an integer number with prime divisors congruent to 1 or 3 (mod 8), and SO(34) or SO (18) × E 8 at R −1 sd = √ 2 C with C a Pythagorean prime number or a product of them.
We then turn to the target space realization of the theory. In section 4, we construct the low energy effective actions of (toroidally compactified) heterotic strings from the three and four point functions of string states. We first consider only the massless states and compare the effective action obtained from the string amplitudes with the dimensional reduction of heterotic supergravity performed in [8]. As expected, we get a gauged supergravity which only differs from the effective action of [8] in the cases of maximal enhancement, in which all the (left-moving) U (1) k Kaluza-Klein (KK) gauge fields of the compactification become part of the Cartan subgroup of the enhanced gauge symmetry.
The higher dimensional origin of the low energy theory with maximally enhanced gauge symmetry cannot be found in supergravity, and one has to refer to DFT. Although the structure constants of the gauge group completely break the global duality symmetry of dimensionally reduced supergravity, the action can still be written in terms of O(k, N ) multiplets, with N the dimension of the gauge group. We show in section 5 that the low energy effective action of the toroidally compactified heterotic string at self-dual points of the moduli space can be reproduced through a generalized Scherk-Schwarz reduction of heterotic DFT. Furthermore, extending the construction of [13], we find the generalized vielbein that reproduces the structure constants of the enhanced gauge groups through a deformation of the generalized diffeomorphisms. An important output of the construction is that a unique deformation is required for the SO(32) and E 8 × E 8 groups, and hence the SO(32) and E 8 × E 8 theories can be considered two different solutions of the same heterotic DFT, even before compactification.
When perturbing the background fields away from the enhancement points, some massless string states become massive. The vertex operators of the massive vector bosons develop a cubic pole in their OPE with the energy-momentum tensor, and it is necessary to combine them with the vertex operators of the massive scalars in order to cancel the anomaly. This fact had been already noticed in [12], but unlike the case of the bosonic string, in the heterotic string all the massive scalars are "eaten" by the massive vectors. We compute the three point functions involving massless and slightly massive states 2 and construct the corresponding effective massive gauge theory coupled to gravity. Comparing the string theory results with the spontaneous gauge symmetry breaking and Higgs mechanism in DFT, we see that the masses acquired by the sligthly massive string states fully agree with those of the DFT fields, provided there is a specific relation between the vacuum expectation value of the scalars along the Cartan directions of the gauge group and the deviation of the metric, B-field and Wilson lines from the point of enhancement.
We have included five appendices. Appendix A collects some known facts about lattices that are used in the main body of the paper. Details of the procedures leading to find the enhancement points and the fixed points of the duality transformations are contained in Appendix B and C, respectively. The three and four point amplitudes of the massless and slightly massive string states are reviewed in Appendix D. Finally we count the number of non-vanishing structure constants of SO(32) and E 8 × E 8 in Appendix E.

Toroidal compactification of the heterotic string
In this section we recall the main features of heterotic string compactifications on T k . We first discuss the generic k case and then we concentrate on the k = 1 example. For a more complete review see [5].

Compactifications on T k
Consider the heterotic string propagating on a background manifold that is a product of a d = 10 − k dimensional flat space-time times an internal torus T k with constant background metric G = e t e ⇒ G mn = e a m δ ab e b n ), antisymmetric two-form field B mn and U (1) 16 gauge field A A m , where m, n, a, b = 1, ..., k and A = 1, . . . , 16. For simplicity we take the background dilaton to be zero. The set of vectors e m define a basis in the compactification lattice Λ k such that the internal part of the target space is the kdimensional torus T k = R k /πΛ k . The vectorsê a constitute the canonical basis for the dual lattice Λ k * , i.e.ê a m e a n = δ m n , and thus they obeyê tê = G −1 ⇒ê a m δ abê b n = G mn .
The contribution from the internal sector to the world-sheet action (we consider only the bosonic sector here) is where we take α = 1, Y A are chiral bosons and the currents ∂Y A form a maximal commuting set of the SO(32) or E 8 × E 8 current algebra. The world-sheet metric has been gauge fixed to δ αβ (α, β = τ, σ) and 01 = 1. The internal string coordinate fields satisfy where w m ∈ Z are the winding numbers. It is convenient to define holomorphic Y m L (z) and antiholomorphic Y m R (z) fields as the dots standing for the oscillators contribution. Then the periodicity condition is The canonical momentum has components 3 The chirality constraint on Y A and the condition of vanishing Dirac brackets between momentum components require the redefinitions Π A →Π A = 2Π A and Π m →Π m = 3 The unusual i factors are due to the use of Euclidean world-sheet metric.
Integrating over σ, we get the center of mass momenta where we used univaluedness of the wave function in the first line. Modular invariance requires π A ∈ Γ 16 or Γ 8 × Γ 8 , corresponding to the SO(32) or E 8 × E 8 heterotic theory, respectively. Whenever we do not need to make the distinction, we use Γ to refer to one of these two lattices. In Appendix A we give all the relevant explanations and details about these lattices.
From these equations we get because π A is on an even lattice, and therefore p forms an even (k, k + 16) Lorentzian lattice. In addition, self-duality Γ (k,k+16) = Γ (k,k+16) * follows from modular invariance [1,20]. Note that p L , p R depend on 2k + 16 integer parameters n m , w m and π A , and on the background fields G, B and A.
The space of inequivalent lattices and inequivalent backgrounds reduces to where O(k, k + 16; Z) is the T-duality group (we give more details about it in the next section).
The mass of the states and the level matching condition are respectively given by where κ is the Killing metric for the Cartan subgroup of SO(32) or E 8 × E 8 , and the "generalized metric" of the k-dimensional torus, given by the (2k + 16) × (2k + 16) scalar matrix, is This is a symmetric element of O(k, k + 16), accounting for the degrees of freedom of the Combining the momentum and winding numbers in an O(k, k + 16)-vector the mass formula (2.11a) and level matching condition (2.11b) read The group O(k, k + 16; Z) is generated by: -Integer Θ-parameter shifts, associated with the addition of an antisymmetric integer matrix Θ mn to the antisymmetric B-field, -Factorized dualities, which are generalizations of the R → 1/R circle duality, of the form where D i is a k × k matrix with all zeros except for a one at the ii component.
-Shifts by a bivector The transformation of the charges under the action of O Θ O Λ , which will be useful later, (2.26) 4 Note that this adds a shift to B of the form Notice the particular role played by the element η viewed as a sequence of factorized dualities in all tori directions, i.e. (2.27) Its action on the generalized metric is where A ≡ A m I and, together with the transformation Z → η −1 O D ηZ which accounts for the exchange w m ↔ n m , it generalizes the R ↔ 1/R duality of the circle compactification. These transformations determine the dual coordinate fields 5 A vielbein E for the generalized metric with M, N = a, b = 1, . . . , 2k + 16, can be constructed from the vielbein for the internal metric and inverse internal metric as follows whereẽ is the vielbein for κ. In the basis of right and left movers, that we denote "RL", where the O(k, k + 16; R) metric η takes the diagonal form Then the momenta (p aR , p aL , p A ) in (2.8b) are (2.34) 5 The transformations also determine a dual coordinateỸ , but this is not actually independent of Y m (z,z) and Y A (z).

Massless spectrum
The massless bosonic spectrum of the heterotic string in ten external dimensions is given, in terms of bosonic and fermionic creation operators α µ −1 ,ψ µ −1/2 , respectively, by where the symmetric traceless, antisymmetric and trace pieces are respectively the graviton, antisymmetric tensor and dilaton.
In compactifications on T k , the spectrum depends on the background fields. In sector 1 there are the same number of massless states at any point in moduli space. In sector 2, we see from (2.8b) that there are no massless states for generic values of the metric, B-field and Wilson lines A I m , while for certain values of these fields the momenta can lie in the weight lattice of a rank 2k + 16 group G L × G R . In this case, there is a subgroup with |(p R , p L )| 2 = 2 which can give rise to massless states. Subtracting (2.11a) and (2.11b) we see that massless states have p R = 0, and thus (unlike in the bosonic string theory), the non-abelian gauge symmetry comes from the left sector only. The group G L × U (1) k R in which the massless states transform defines the gauge group of the theory, with G L a simply-laced group of rank 16 + k and dimension N , that depends on the point in moduli space (which is spanned by G mn , B mn , A I n ). Specifically, the 10 − k dimensional massless bosonic spectrum and the corresponding vertex operators (in the −1 and 0 pictures) are given by (µ, ν = 0, . . . , 9 − k; m, n = 1, . . . , k; I = 1, . . . , 16): with φ the scalar from the bosonization of the superconformal ghost system, and k µ µν = µν k ν = 0.
• k KK left abelian gauge vectors: g mµ + b mµ ≡ a mµ and 16 Cartan generators of SO(32) or where the indexÎ = (I, m) includes both the chiral "heterotic" directions and the compact toroidal ones, labeling the Cartan sector of the gauge group G L .
• k KK right abelian gauge vectors: with k µ A µ = 0 and currents where α are the roots of G L (or equivalently the left momenta) and the cocycles c α verify c α c β = ε(α, β)c α+β , with ε(α, β) = ±1 the structure constants of G L in the Cartan-Weyl basis.
It is convenient to define the index Ω = (Î, α) = 1, ..., N and condense the vertex operators for left vectors and scalars as The massive states are obtained increasing the oscillation numbers N and N or choosing |(p R , p L )| 2 ≥ 4.
Let us see what groups arise. Using that p R = 0, we get from (2.8b) that the massless states have left-moving momentum while their momentum number on the torus is given by Note that quantization of momentum number on the torus is a further condition to be imposed on top of p L 2 = 2.
In the absence of Wilson lines A A m = 0, the k torus directions decouple from the 16 chiral "heterotic directions" Y A ; p A = π A is a vector of the weight lattice of SO(32) or E 8 ×E 8 and then |p A | 2 ∈ 2N. The only possible massless states then have either momenta p L = (0, π A ) with |π| 2 = 2, or p L = ( √ 2 e an w n , 0) with w m g mn w n = 1 (and additionally n m w m = 1). The former are the root vectors of SO(32) or E 8 × E 8 , while the latter have solutions only for certain values of the metric and B-field on the torus and lead to the same groups as in the (left sector of) bosonic string theory, namely all simply-laced groups H of rank k. The total gauge group is then Turning on Wilson lines, the pattern of gauge symmetries is more complicated, and also richer. In the sector with zero winding numbers, w m = 0, we have p A = π A as before, but now requiring a quantized momentum number imposes π A A A m ∈ Z (see (2.47)) which, for a generic Wilson line breaks all the gauge symmetry leaving only π A = 0, which corresponds to the U (1) 16 Cartan subgroup. The opposite situation corresponds (2.21) and thus the pattern of gauge symmetries is as for no Wilson line 7 . In the SO(32) theory, the same conclusions hold if A ∈ Γ 16 , but one has the more interesting possibility A ∈ Γ v or A ∈ Γ c , where the SO(32) symmetry is not broken, and the 16 chiral heterotic directions can be combined with the torus ones, giving larger groups which are not products.
Let us discuss the different groups that can arise in points of moduli space where the enhancement is maximal. In that case, the matrices that embed the internal sector of the heterotic theory on T k into a 16 + k-dimensional bosonic theory are related to the 6 We denote Γ * g the dual of the root lattice, and one has Γ * g = Γ 8 × Γ 8 for E 8 × E 8 and Γ * g = Γ w = Γ 16 + Γ v + Γ c for SO(32) (see Appendix A for more details). 7 The only difference is that the massless states have shifted momenta π A and a shifted momentum number along the circle compared to the ones without Wilson lines, see Eq.(2.26).
Cartan matrix C by [5] (G + 1 2 A I A I ) mn One can then view the possible maximal enhancements from Dynkin diagrams. Let us first consider Wilson lines that do not break the original gauge group, i.e A ∈ Γ * g . We start with the SO(32) heterotic theory. The Dynkin diagram of SO(32) is The Dynkin diagrams of the gauge symmetry groups arising at points of maximal enhancement in the compactification of the SO(32) theory on T k have k extra nodes, with or without lines in between. Since the resulting groups have to be in the ADE class (they are all simply laced), one cannot add nodes with lines on the left side. Therefore, the nodes should be added on the right side, and linked or not linked to the last node or not, and additionally add lines linking them to ech other, or not. For one dimensional compactifications (k = 1), the only possibilities are corresponding respectively to SO(32) × SU (2) and SO(34). Since a line in the Dynkin diagram means that the new simple root is not orthogonal to the former one, then the Cartan matrix for this situation should have an off-diagonal term in the row corresponding to the new node and the column of the previous node, which according to (2.48) means that there is a non-zero Wilson line. Thus, no Wilson line (or a line in Γ 16 , which is equivalent to no Wilson line) gives the enhancement group SO(32) × SU (2) and, as explained above, this enhancement works as in the bosonic theory, at R = 1. The enhancement symmetry group SO(34) is obtained with a Wilson line in the vector or negative-chirality spinor conjugacy classes, and will be presented in detail in section 3.3.1. For the E 8 × E 8 heterotic theory, the situation is less rich in the cases in which the dimension of the resulting group is larger than that of E 8 × E 8 . As we explained above, where we see immediately that the extra nodes cannot be linked to any of the E 8 's, as any extra line would get us away from ADE. Then the possible enhancements are groups which are products of the form E 8 × E 8 × H, where H is any semi-simple group of rank k, and each H arises at the same point in moduli space as in the compactifications of the bosonic theory on T k [13]. However, maximal enhancement can still be obtained by breaknig one of the E 8 to SO (16), and then the richness of the SO(32) case is recovered (e.g. enhancement to SO(18) × E 8 ).
If A / ∈ Γ * g , part or all of the SO(32) or E 8 × E 8 symmetry is broken, and one can still see groups that arise from the Dynkin diagrams. For compactifications on T k , a priori any group of rank 16 + k in the ADE class can arise. However, we need to take into account that there are only k linearly independent Wilson lines that can be turned on, so not any ADE group is actually achievable. For compactifications on a circle, in the SO(32) theory one can reach for example the enhancement groups SU (2) × SO(2p) × SO(32 − 2p) or SO(2p+2)×SO(32−2p), and in the E 8 ×E 8 theory the groups SO(2p)×SO(18−2p)×E 8 for p < 8. A very interesting case is that of p = 8, where we conjecture that there is no Wilson line that achieves the enhancement to SU (2) × SO(16) × SO (16).
Points of enhancement are fixed points of some O(k, k + 16; Z) symmetry. Enhancement groups that are not semi-simple, i.e. that contain U (1) factors, arise at lines, planes or hyper-planes in moduli space. On the contrary, maximal enhancement occurs at isolated points in moduli space. These are fixed points (up to discrete transformations) of the O D duality symmetry, or more general duality symmetries involving O D . This is developed in detail in sections 3.5 and 3.6 for compactifications on a circle, to which we now turn.

Compactifications on a circle
Let us explicitly work out the circle compactification at radius R, with a Wilson line A A . The momentum components (2.8b) are 8 where |A| 2 = A A A A = AκA t . 9 The massless states, which satisfy p R = 0, have leftmoving momenta and momentum number on the circle In the sector p L = 0 one has n = w = π A = 0, and the massless spectrum corresponds to the common gravitational sector and 18 abelian gauge bosons: 16 from the Cartan sector of E 8 × E 8 or SO(32) and 2 KK vectors, forming the U (1) 18 gauge group.
The condition p L 2 = 2 can be achieved in three possible ways: From (3.2) we see that sector 1 has w = 0 and then (3.1) implies p A = π A . The condition on the norm says that these are the roots of SO(32) or E 8 × E 8 . But as explained in the previous section, one has to impose further that n ∈ Z and thus from (3.3), π · A ∈ Z. We divide the discussion into two cases, one in which this condition does not break the SO(32) or E 8 × E 8 symmetry, and the second one in which it does.

Enhancement of SO(32) or E 8 × E 8 symmetry
If we want the condition π · A ∈ Z not to select a subset of the possible π A in the root lattice, or in other words not to break the SO(32) or E 8 × E 8 gauge symmetry, we have to impose We restrict to this case now, and leave the discussion of the possible symmetry breakings to the next section.
Sector 2 contributes states only at certain radii R 2 = 1/w 2 , and have π = −wA. The momentum number of these states given in (3.3) becomes If A ∈ Γ, 10 one has 1 2 |A| 2 ∈ Z, and thus the only way to satisfy the quantization condition is with w = ±1, which gives two extra states at R = 1, with momentum number n = 9 We are abusing notation, as |A| 2 = AκA is not a scalar under reparameterizations of the circle coordinate, i.e. our definition is |A| 2 = A A m A A m where m here is just the circle coordinate. The scalar quantity is A 2 = |A| 2 /R 2 (see (3.32) below). 10 By Γ we mean Γ 16 or Γ 8 × Γ 8 , according to which heterotic theory one is looking at.
±(1 − 1 2 |A| 2 ). For SO(32), the condition A ∈ Γ w leaves the additional possibilities A ∈ Γ v or A ∈ Γ c . Since π = −wA ∈ Γ 16 , we have to require w to be even. But then one cannot satisfy the quantization of momentum (3.5) since 1 2 |A| 2 w ∈ Z. Thus, there are no extra states from this sector for A ∈ Γ v or A ∈ Γ c . Sector 3 contributes states only at radii R 2 = s 2 /(2w 2 ). The condition |p A | 2 < 2 is only possible if A is not in the root lattice. And as it is required to be in the weight lattice, this possibility arises in the SO(32) heterotic theory only, for A ∈ Γ v or A ∈ Γ c . The quantization of momentum implies n = 1 2 where in the last equality we have used (3.2) and |p L | 2 = 2. Note that the last equality is the same as (3.5). For A ∈ Γ v , π · A ∈ Z for π ∈ Γ g and 1 2 |A| 2 = 1 2 (mod 1), so the only option is s = 1, giving extra states with w = ±1 at R = 1/ √ 2. These states enhance SO(32) × U (1) to SO(34). We present an explicit example of this case in section 3.3.1. For A ∈ Γ c , π · A ∈ Z for π ∈ Γ g but now 1 2 |A| 2 ∈ Z and thus we cannot satisfy the quantization condition (3.6) this way. However π · A = 1 2 (mod 1) for π ∈ Γ s and thus we recover that for these Wilson lines there is an enhancement to SO(34) at R = 1/ √ 2 as well, by states with w = ±1. Note that A ∈ Γ c is equivalent by a Λ shift with Λ ∈ Γ s to A ∈ Γ v . As we can see from (2.26), by this shift the winding number remains invariant, while π ∈ Γ s gets shifted to π ∈ Γ g .
We conclude that in circle compactifications with Wilson lines the pattern of gauge symmetry enhancement is (we give here only the groups on the left-moving side): In figures 1, 2 and 3 we show a slice of the moduli space of the heterotic theory compactified on a circle at a generic radius greater than one, at R 2 = 1 and at R 2 = 1 2 respectively 11 . In figure 4 we show another slice of moduli slice of moduli space that includes the radial direction. The first item above corresponds to the red points in figures 2b and 4b, while the second and third ones correspond, respectively to the red and green points in figures 2a, 3a and 4a. In the next section we will show how the enhancement at some of the other special points in the figures arise.

Enhancement-breaking of gauge symmetry
Whenever the Wilson line is not in the dual root lattice, part or all of the SO(32) or E 8 × E 8 symmetry is broken. However, this does not imply that no symmetry enhancement from the circle direction is possible. The pattern of gauge symmetries can still be rich. We denote these cases enhancement-breaking of gauge symmetry. This nomenclature can be confusing however: for specific values of R and A, there is the possibility that the symmetry enhancement is so large that it restores the original SO(32) or E 8 × E 8 symmetry, or even leads to a larger group of rank 17. This means that we can have a maximal enhancement even if the Wilson line is not in the dual root lattice, either to the groups listed at the end of the previous section, or to any other simply-laced, semi-simple group of rank 17, such as for example SO(18) × E 8 .
The massless states for an arbitrary Wilson line are the following: Sector 1 has w = 0 (and thus p A = π A ) and consists of the roots of SO(32) or in the following section.
Sector 2 contains states only at radii R 2 = 1/w 2 , and these states should have π A = −wA A . Since by definition A / ∈ Γ, there are states in this sector if there is some integer w ( = 1) such that wA ∈ Γ. One should also impose the quantization condition (3.5). If these two conditions are satisfied for a given w, then there are two extra states in this sector, giving an enhacement of Sector 3 contains states only at radii R 2 = s 2 /(2w 2 ). Quantization of momentum gives the condition (3.6), but now |A| 2 is not necessarily integer. If there are states in this sector, there is an enhacement of the group H × U (1). This enhancement can be to H × SU (2) (where the SU (2) is along some direction mixing the circle with the heterotic directions) in the simplest case, but one can actually have enhancement of the group H ×U (1) to a group that is not a product, like for example enhancement of SO(16)×U (1) to SO(18), as we will show in detail.
In sections 3.3.2, 3.3.3, 3.3.4, 3.3.5 and 3.3.6 we show explicitly how the groups mentioned in sector 1 get enhanced respectively to SO (18)

Explicit examples
Here we present some examples of symmetry enhancement-breaking. The roots of SO(32) are given by where underline means all possible permutations of the entries. The roots of E 8 × E 8 are , with even number of + signs Consider the SO(32) heterotic theory compactified on a circle of radius where the first entry corresponds to the circle direction. In sector 1, with w = 0, one has all momenta satisfying |π A | 2 = 2 and π · A ∈ Z. The last condition is true for any π A ∈ Γ g , and thus in this sector we have all the root vectors of SO(32) given in (3.7). There are no states in sector 2, as w ∈ Z. In sector 3 we have s = 1 and w = ±1. Here we get massless states coming from three different sectors of the SO(32) weight lattice, namely 3.a) |π| 2 = 2, with π 1 = ±1 (where the signs are not correlated). These are 60 states with n = 0.
These are 2 states, which have n = w.
Another 2 states with n = −w.
We thus get 64 extra states, which together with the Cartan direction of the circle, enhance the SO(32) to SO(34). This point in moduli space is illustrated in green in figures 3a and 4a. In the figure 3a the other green points differ from this by a Λshift, while the other green points in figure 4a, that appear at a different radius, will be explained in section 3.4.
Consider the E 8 × E 8 heterotic string compactified on a circle of radius R = 1 √ 2 , with Wilson line A = (1, 0 7 , 0 8 ), which is of the form (v0) according to the notation of Appendix A (see (A.10) in particular). This Wilson line leaves the second E 8 unbroken, while from the first E 8 , the surviving states in sector 1 are the ones with integer entries, i.e. those in the first line of (3.8). The group H from sector 1 is then SO(16) × E 8 and the corresponding points in moduli space are illustrated by the grey dots in figure 1b.
There are no states in sector 2, while in sector 3 we have states with w = ±1 such that s = 1, |p A | 2 = 1. The surviving states have the following momenta where the first entry corresponds to the circle and the subsequent ones to the 8 directions along the Cartan of the first E 8 factor. The first line contains the states of sector 1. These are the 144 roots of SO (18). This point in moduli space, together with its equivalent ones, are illustrated by the green dots in figure 3b.
Consider the SO(32) heterotic string compactified on a circle of radius R, with a Wilson line A = 1 2 p , 0 16−p , with 2 ≤ p ≤ 8. 13 . The massless states that survive in sector 1 (w = 0) are those with momentum π A satisfying 1 2 Then the surviving states have momenta For p = 2, these points are illustrated by the violet dots in figures 1a, 2a and 3a.
Additionally, by states in sector 3, the U (1) of the circle can be enhanced to SU (2) at R 2 = 1 − p 8 for p < 8. These states have n = w = ±1, π A = 0 and An interesting phenomenon occurs for p = 2 and p = 6, where there are also massless states in sector 2 at R 2 = 1 4 (note this is equal to 1 − p 8 for p = 6). For p = 2, these have w = ±2, n = 0, π = ∓(1, 1, 0 14 ), and thus one gets enhancement to SO(4) × SO(28) × SU (2) either at R 2 = 1 4 from states in sector 2, or R 2 = 3 4 from states in sector 3. For p = 6, besides the states in (3.15), there are massless states in sector 3 (always at R 2 = 1 4 ) with w = ±1 and momentum 14 Together with the states in (3.15), there are 32 extra states, corresponding to the (s) class of SO (12), which together with the elements in the adjoint displayed in (3.14) give an enhancement to Spin(12)/Z 2 . On top of this, there are two extra states in sector 2, with w = ±2 and π = ∓(1 6 , 0 10 ), giving an additional SU (2). The total enhancement group for p = 6 at R = 1 2 is then SU (2) × Spin(12)/Z 2 × SO (20). A special case is p = 8, where according to the formula for the radius R 2 = 1 − p 8 , one would get enhancement to SO(16) × SO(16) × SU (2) at R = 0, which is not part of the moduli space. We show in more detail in section 3.6.3 that the Wilson line (16). We conjecture that the enhancement group Consider the SO(32) heterotic string compactified on a circle of radius R, with a Wilson line A = 1 4 p , 0 16−p with 2 ≤ p ≤ 14. The massless states that survive in sector 1 are those satisfying 1 4 and thus we get the following states in sector 1 Additionally, by states in sector 3, the U (1) of the circle can be enhanced to SU (2) at R 2 = 1 − p 32 . These states have n = w = ±1, π A = 0 and We thus have enhancement to For p = 8, there are additional massless states, and a larger enhancement group, coming from states with π = ∓(1, 1, 0 6 ) and w = ±1. These have These states, together with those in (3.19) promote the U (8) to an SO (16), and thus at p = 8 the enhancement is to U (1) × SO(16) × SO(16) (as we mentioned at the end of the previous section, the U (1) does not get enhanced to SU (2), at this radius either).
This is an interesting example of enhancement-breaking in the E 8 × E 8 heterotic theory, where first the E 8 is broken to SU (2) × E 7 by the Wilson line A = 1 4 8 , 0 8 and then enhanced by the circle direction to SU (2) × E 8 .
The Wilson line leaves the second E 8 unbroken, while the surviving roots from the first E 8 have 9-momenta This, gives 128 roots, which together with the 8 Cartan directions, gives an unbroken Additionally at R = 1 2 there are two states in sector 2 with w = ±2 and 112 states in sector 3 with w = ±1 and momentum These states give a total of 114 extra states that add up to the previous 136 states, plus the circle direction, adding up to the 251 states of SU (2) × E 8 . So at R = 1 2 we get enhancement to SU (2) × E 8 × E 8 , which works very differently than the enhancement occurring at R = 1, mentioned in section 3.1.
Consider the SO(32) heterotic theory with Wilson line A = ( 1 4 16 ). This Wilson line breaks the SO(32) gauge symmetry leaving the states with weight π = ±(1, −1, 0 14 ), corresponding to U (16). Additionally, at R = 1 2 there are extra states in sector 3 that have momenta These are 32 additional roots, which enhance the U (1) × U (16) occurring for that Wilson line at any radius, to U (17).

Exploring a slice of moduli space
In this section we present a detailed analysis of the slice of moduli space for compactifications of the heterotic theory on a circle at any radius and Wilson line given by The results of this section are displayed in figure 4. Here we present the main ingredients in the calculations, and leave further details to Appendix B.
For this type of Wilson line, the states with w = 0 (sector 1) that survive, are those satisfying This preserves all the roots only if A 1 ∈ Z for the Γ 16 case, or A 1 ∈ 2Z for the Γ 8 × Γ 8 case. These correspond to the horizontal orange lines in figure 4, where at any generic radius, the gauge symmetry is If A 1 / ∈ Z, then we have just the roots with π 1 = 0. That is, the 420 roots of SO(30) or the 324 roots of SO(14) × E 8 . This corresponds to the white regions in figure 4. Now, depending on the value of R, we can have additional states in sectors 2 and 3, i.e. states with non-zero winding 15 which momenta satisfy |p L | 2 = 2 and have a quantized momentum number on the circle. Then, according to (3.2) and (3.5,3.6), they should obey (3.26) The first equation implies R −1 ≥ w, and the simplest solution is But π is in an even lattice, which implies π 1 = −2q, q ∈ Z. The quantization condition for n yields so we have only the winding numbers that are divisors of the numbers that can be written as 2q 2 − 1, for some integer q. In terms of q, the Wilson lines are of the form If the radius also satisfies R < 1 √ 2w < 1 w , we have additional solutions where some of the other components of π are non-zero, such that The quantization conditions are the same as before, but now the Wilson lines have the following behavior as a function of the radius we have yet other possible solutions, but only for the The lines and quantization conditions are: where we used (π 1 ) 2 = |π| 2 − 7 4 and π 1 = − q + 1 2 . For a given q and w, whenever the Wilson line is of the form a w,q in (3.27), we get 2 massless states (one for w > 0 and another one for w < 0). If there are no more states, These correspond to the blue lines in figure 4, where for example in figure 4a, the long blue line going from (R, For Wilson lines of the form b w,q in (3.28), we get 60 extra states for the Γ 16 , and 28 for Γ 8 × Γ 8 . The former promote the enhancement to U (1) × SO(32), while the latter to U (1) × SO(16) × E 8 , and they correspond respectively to the orange lines in figure 4a and the black lines in figure 4b. The largest curved orange line in the former and black line in the latter going from (0, 0) to (0, 2) corresponds to b 0,1 = 1 ± √ 1 − 2R 2 , where the plus sign is for the upper half of the curve, and the minus sign for the lower half.
Finally, Wilson lines of the form c w,q in (3.29) give in the E 8 × E 8 heterotic theory, 2 × 2 6 = 128 states (the sign of one of the seven (± 1 2 ) is determined by the sign of the other 6 and the sign chosen for the Wilson line). Note that c w,q (R) = b 2w,q (R). It is not hard to show that a Wilson line that can be written as c w,q (R) can always be written as b 2w,q (R), but the function b can also have an odd w. Wilson lines b that can also be written as c bring then a total of 28 + 128 = 156 states, which corresponds to the enhancement to U (1) × E 8 × E 8 in the orange lines of figure 4b.
There are only two kinds of intersections between lines, and the points of intersection correspond to points of maximal enhancement (see Appendix B for details): • between a blue curve a(R) with w 1 and an orange curve theory. These are the red dots of figure 4, and arise at for some integer k, with C = 1, 3, 9, 11, ... are all the integers whose prime divisors are 1 or 3 (mod 8) (see Table 1).
• between two blue a(R) with w 1 and w 2 and two orange (black) curves b(R) with w 3 and w 4 , where the enhancement group is SO(34) (SO(18) × E 8 ) for the SO(32) (E 8 × E 8 ) theory. These are the green dots of figure 4, and arise at 16 for some integer k, with C = 1, 5, 13, 17, ... are all the integers whose prime divisors are Pythagorean primes (see Table 1) In Appendix B we give the details of the calculations and also prove that these are the only possible intersections for this type of Wilson lines. In section 3.6 we show how these points arise as fixed points of a duality symmetry.

T-duality in circle compactifications
In this section we discuss the action of T-duality in the heterotic string compactified on a circle. By T-duality we mean the action of certain type of transformations in O(1, 17, Z) that relate a given heterotic theory with 16-dimensional lattice Γ, compactified on a circle of radius R and Wilson line A, to another heterotic theory with lattice Γ , compactified on a circle of radius R and Wilson line A . In this section we discuss the usual T-duality exchanging momentum and winding numbers, while in the next section we discuss more general dualities, and their fixed points.
The duality generated by the matrix O D is the usual T-duality transformation exchanging momentum and winding numbers (w , n , π ) = (n, w, π) . (3.30) Since π stays untouched, this duality is possible if Γ = Γ. Its action on the background fields can be worked out from the generalized metric (2.13), which for the circle is 17 where we have defined the scalar The action of O D transforms this into 33) 16 We get additionally R = . 17 Here we choose the Cartan-Weyl basis where the Killing metric for the Cartan subgroup κ IJ is diagonal. and thus we get in agreement with the heterotic Buscher rules for scalars [21]. We get that a background has R = R for Additionally, if 2A ∈ Γ , then A = −A ∼ A, and therefore the background is fully selfdual, satisfying M = M −1 up to discrete transformations (these are of the form (2.19), (2.20) or (2.21), but for the circle the only non-trivial one is a Λ-shift (2.21)).
In all these examples one has 2 A ∈ Γ, hence A ∼ A and then the backgrounds are fully self-dual, namely R = R, A ∼ A.
For Wilson lines with only one non-zero component, we have that the fixed "points" of this symmetry correspond actually to lines of non-maximal enhancement symmetry where the Wilson lines are functions of the radius (A = A(R sd )), and are such that . We now discuss the differences between fixed points of duality symmetries further, exploring more general dualities and their fixed points.

More general dualities and fixed points
The transformation O D discussed before is a particular type of transformation that changes the sign of p R while it rotates p L , preserving its norm (in compactifications of the bosonic theory on a circle, p L has a single component and O D just leaves it invariant, but in the heterotic theory O D rotates the 17-dimensional vector p L ). It would be very interesting to understand what are all the possible transformations that do this, and obtain their fixed points. Here we do something more modest, namely we work out the set of transformations that change the sign of p R and rotate p L , leaving its circle direction component invariant. We thus require with U ∈ O(16, Z). These transformations generically link a given heterotic theory with lattice Γ, in a background defined by (A, R) to another heterotic theory with lattice Γ in a dual background with (A , R ). The duality transformation depends on the matrix U and we use a convenient parameterization to relate the radii R and R , namely we define a positive number r such that The duality transformation that achieves (3.35) should have the form Requiring this to be in O(1, 17; Z), we get a set of quantization conditions like for example 19 (the full set of quantization conditions is given in (C.2)) r , We divide the discussion into the dualities where Γ = Γ , and those where the dual lattice is not the original one. To denote the different sublattices that will play a role, it is useful to use the (0) We give these in (A.14). Note that a lattice Γ + is equivalent to a lattice Γ − , the choice (s) versus (c) conjugacy class is a convention with no physical relevance. Here it is important however to make the distinction whether a given duality maps, say, Γ + to Γ + , or Γ + to Γ − .
In the following we write the main results, leaving the details to Appendix C. The results for generic Wilson lines, assuming that r is a prime number, are summarized in Table 2. We later concentrate on the situation where the Wilson lines are of the form (3.24), i.e. with only one non-zero component, as we did in section 3.4, to see what happens when the assumption that r is prime is relaxed. For Wilson lines of this form, the O(16) symmetry is broken to O (15), and there are four inequivalent choices of U that we will analyze in detail The dualities for which the lattice does not change involve those where π is invariant, such as the one discussed in the previous section. But as explained above, one can have more general dualities even when Γ = Γ, and thus more general fixed points. Fixed points of a duality are those for which R = R and A = A. 20 To make the analysis tractable for generic Wilson lines, we restrict to the situation where r is a prime number and U = I, and relax this assumption only in the setup where the Wilson lines have just one non-zero component. Under the assumption that r is a prime number, the full set of quantization conditions (C.2) are satisfied if and only if (see and thus the fixed points of these transformations are at R = 1 and A any point in the lattice Γ. They correspond to enhancements to SU (2) × SO(32) and SU (2) × E 8 × E 8 discussed in section 3.1. These points appear in the diagonal entries in Table 2.
Let us now analyze in more detail the fixed points of the dualities for the subset of Wilson lines of the form (3.24), i.e. with only one non-zero component. The quantization conditions evaluated at the fixed points turn into (see Appendix C for details of the calculation) n , m , where now n = 1 Table 1   These points in moduli space are points of maximal enhancement symmetry. Those in the first column give rise to SU (2) × SO(32) for Γ 16

Γ ↔ Γ
Note that unless Γ = Γ ± 16 and Γ = Γ ± 8 × Γ ± 8 (or the other way around, and using any combination of signs) − situations that we analyze separately in the next section − there exists some U 1 ∈ O(16, Z) such that Γ = U 1 Γ. In that case, the duality with Γ = Γ, U 2 and A is equivalent to one between Γ and Γ = Γ, U = U 1 U 2 and has A = A U 1 . Restricting to diagonal matrices U , we see that the dualities with U and Γ = Γ are equivalent to the dualities with U = I but where Γ is Γ = Γ ± 16 for Γ = Γ 16 and det(U ) = ±1 (3.44) where det 1 (det 2 ) is the product of the 8 first (last) diagonal elements and the lattices Γ ± are defined in Appendix A. If additionally the Wilson line A is invariant under the action of U (up to a Λ-shift) we get exactly the same fixed points that one gets for a duality with Γ = Γ . Since Wilson lines of the type (3.24) are invariant under the action of a diagonal U such that the first component is +1, we get the same fixed points of section 3.6.1 that correspond to enhancement to SO(34) or SO(18) × E 8 .
Under the assumption that r is a prime number, the quantization conditions are satisfied if and only if A ∈ (Γ∩Γ ) * \Γ , A ∈ (Γ ∩ Γ ) * \Γ , r = 2 (3.46) and thus the fixed points of these transformations are at R = 1 √ 2 , and correspond to the enhancements SO(34) and SO(18) × E 8 . The possible Wilson lines for the different choices of Γ and Γ are given in Table 2.
There is no U ∈ O(16, Z) that transforms the lattices Γ 16 and Γ 8 ×Γ 8 into each other, and thus the case Γ = Γ 16 and Γ = Γ 8 × Γ 8 is different from the ones considered previously.
Here, for simplicity, we restrict to U = 1, namely we analyze dualities such that (p L , p R ) = (p L , −p R ). The quantization conditions under the assumption that r is a prime number, are given in (3.46). For Γ = Γ 16 = (1, 0 7 , 1, 0 7 ) breaks the E 8 × E 8 symmetry also to SO(16) × SO (16). There are also states which are neutral under SO(16) × SO (16), of the same form as before, i.e. with momenta where w = 2m andñ = n + w .
In the following table we write the fixed points of the dualities between a theory with lattice Γ (row) and another one with Γ (column) for the smallest value of the parameter r defined in (3.36). We indicate the conjugation classes of the possible Wilson lines (for a given row and column, any A given can be dualized to any A ), and the enhancement group arising at the fixed point of the duality.

Effective action and Higgs mechanism
Now that we saw the rich structure of duality symmetry, we turn to its explicit target space realization. The global duality symmetry of the dimensionally reduced heterotic supergravity action has been deeply investigated in the seminal papers by J. Maharana and J. Schwarz [6] and N. Kaloper and R. Myers [7], and more recently in [8]. If the gauge fields are truncated to the Cartan subsector of the E 8 × E 8 or SO(32) gauge group, the dimensional reduction of heterotic supergravity from 10 to 10 − k dimensions produces a theory with U (1) 2k+16 abelian gauge symmetry and a continuous global O(k, k + 16; R) symmetry. If the reduction includes the full set of E 8 × E 8 or SO(32) gauge fields and no Wilson lines, the global symmetry reduces to O(k, k; R), while a compactification with Wilson lines for the Cartan gauge fields of a rank 16 − r subgroup of the rank 16 gauge group G L , gives an effective field theory with global O(k, k + 16 − r; R) duality symmetry [8]. The analysis of [8] is based on string-theoretic arguments and holds to any order in the α expansion of the heterotic string effective field theory action involving all the massless string states, except those that become massless at self-dual points of the moduli space.
Including the massless states with nonzero winding or momentum number on T k in the effective field theory of the toroidally compactified heterotic string is not difficult, as it is a gauged supergravity. The action with at most two derivatives of the massless fields is then completely determined by the gauge group. Therefore, although the field theoretical Kaluza-Klein reduction of heterotic supergravity cannot describe the string modes that give rise to maximally enhanced gauge symmetry, the action is entirely fixed.
Nevertheless, we will see in the forthcoming sections that the explicit construction of the (toroidally compactified) heterotic string effective action from the scattering amplitudes of massless string modes at self dual points of the moduli space, and its manifestly duality-covariant reformulation, give important information. In particular, we will obtain novel relations between the SO(32) and E 8 × E 8 theories. We will also consider the light states that acquire mass when slightly perturbing the background fields and revisit the gauge symmetry breaking and Higgs mechanism, both from the field theory and the string theory viewpoints.

Effective action of massless states
The three-point functions of all the (toroidally compactified) heterotic string massless vertex operators are reviewed in Appendix D, where we also compute the four point function of the massless scalars. These amplitudes are reproduced from the S-matrix of the following effective action which also contains terms from higher point functions that we have not computed but need to be included on the basis of gauge symmetry. Here κ d is the effective Planck coupling constant (related to the gauge coupling g d as g d = √ 2κ d ) and 22 22 We have rescaled the polarizations introduced in Section 2 as At the specific points in moduli space where the gauge symmetry is enhanced, it is convenient to split the index Γ = (Î, α = α, α), whereÎ = 1, ..., 16 + k denotes the Cartan generators and α (α) are the positive (negative) roots of G L . The vectors AÎ µ andĀ m µ correspond to the left and right Cartan generators in sector 1, respectively, while A α µ correspond to the vectors of sector 2, as defined in section 2.3. The scalars SÎ m correspond to the (16 + k) × k scalars in sector 1, while the S αm correspond to the scalars in sector 2. In this case, G mn = G sd mn + S (mn) (x), B mn = B sd mn + S [mn] (x), A Im = A sd Im + S Im (x), A αm = S αm (x), and the superindex sd refers to the self-dual values of the background fields. The algebra in the Cartan-Weyl basis is where ε(α, β) = ±1 for simply-laced algebras. Note that it is completely determined by the vertex operators of the vector states: the roots αÎ are the momenta of the string states and ε(α, β) is given by the cocycle factors in the currents (2.42) c α c β = ε(α, β)c α+β . When the gauge group G L is a product, the structure constants (and the indices Γ,Î, α) split into those of each factor, e. For gauge groups of the form G L × U (1) k L × U (1) k R , the action (4.1) agrees with the dimensionally reduced heterotic supergravity action obtained in [8], including the scalar potential (although the reduction of [8] contains an additional term with six scalars that we have not computed) 23 . It possesses O(k, k; R) global symmetry.
In the case of enhanced gauge groups of the form G L × U (1) k R , in which the k leftmoving Cartan generators are absorbed by the Cartan subgroups of the non-abelian group G L , the structure constants completely break the global symmetry. However, (4.1) can be rewritten in O(k, n) covariant form, where n equals the dimension of the full gauge group. We review this rewriting in the next section, where we also present an alternative reformulation of (4.1) from a generalized Scherk-Schwarz compactification of double field theory. This will allow us to obtain novel relations between the E 8 × E 8 and SO(32) heterotic theories.
From (4.1) one can see some of the features of the spontaneous breaking of gauge symmetry that occurs away from the enhancement points. An effective stringy Higgs mechanism is already encoded in the string theory computation, which can be interpreted as triggered by the vacuum expectation values of the scalar fields in the Cartan sector SÎ m , which give mass to the vectors in the non-Cartan sector from the covariant derivatives in the kinetic terms, while the scalars without legs in the Cartan sector acquire mass from the scalar potential. We present the relevant details in the forthcoming sections.

Higgs mechanism in string theory
When moving away from the points in moduli space where the gauge symmetry is enhanced, p R = 0 and the extra massless vectors and scalars in sector 2 acquire mass. The dependence of the vertex operators on the background fields is contained in the exponential factors of the internal coordinates, which become where c α = c α , as we will see later. In particular, the [U (1) L ] k+16 × [U (1) R ] k charges of these states, (qÎ,q m ) = (pÎ L , p m R ), are generated by JÎ ⊗ J m . The OPE of the energy-momentum tensor with the massive vector boson vertex operators develop a cubic pole, and it is necessary to combine these operators with those of the massive scalars in order to cancel the anomaly. As discussed in [12], the vertex operators of the massless vectors "eat" the scalars S αm and the conformal anomalies can be canceled when redefining and B µν = −b µν , B mn = −b mn are necessary to compare with [8]. Note that the KK reductions of the metric and B field, A (1)m µ and A (2) mµ , having the internal indices up and down repectively, cannot couple through one scalar field, unlike the left and right vector fields A Γ µ andĀ m µ in (4.1). See the next section and the equivalent discussion in [12].
where ξ is some coefficient. In terms of fields, this is corresponding to the R ξ t'Hooft gauge condition where p R can be identified with a non vanishing vev. Then the physical massive vector boson vertices are actually A , and the scalars S αm disappear from the spectrum.
Note that the fields associated to A have well defined charges (p L , p R ), and since m 2 = −k 2 , the gauge condition can be written as implying an effective polarization This leads to a massive vector of the form where p 2 R = 0 is related to the vevs. This is the usual massive vector field incorporating the would-be Goldstone bosons p m R S αm that provide the longitudinal polarization. Unlike the case of the toroidally compactified bosonic string, in the heterotic string all the massive scalars are Goldstone bosons. Since the gauge group in the supersymmetric right sector is abelian, there are no other massive scalars from the compactification of the massless states.
The non-vanishing three point functions involving massless and light states, i.e. states that are massless at the self-dual points and become massive when perturbing the background fields, are listed in Appendix C, and they lead to the following effective action The S-matrix of this massive gauge field theory coupled to gravity reproduces the string theory three-point amplitudes. The non-Abelian pieces in the field strength of the massive gauge fields and in the Chern-Simons terms in H µνρ correctly appear in terms of the charges of the corresponding fields (qÎ, q m ) = (pÎ L , p m R ). These charges determine the coefficients of the vector boson three-point functions, which can be identified with structure constants reflecting the fact that the gauge interactions in string theory are a manifestation of an underlying affine Lie algebra. This algebra is isomorphic to that of the enhanced G L group [15], which justifies the identification c α = c α used in (4.4) (we will comment further on this result in the next section).
Not all the terms in the action can be obtained from the three-point functions, but we have completed the expressions so that they correctly reproduce the massless case when p R = 0 and p L ∈ Γ.
All the terms of the scalar potential of the massless theory (4.1) are absorbed by the field strengths of the massive vectors or by interaction terms containing massive vectors.

Heterotic double field theory
Although the action (4.1) can be generically obtained by dimensional reduction of heterotic supergravity from 10 to 10 − k dimensions, not all the effective actions of massless fields obtained from toroidally compactified heterotic string theory can be uplifted to higher dimensional supergravities. In particular, the states with nonzero winding or momentum number on T k cannot be captured by field theoretical Kaluza Klein compactifications. To find the higher dimensional description of these string modes, one has to refer to gauged double field theory (DFT) [18,19,22], an O(D, D + N ; R) covariant rewriting of heterotic supergravity, with D the dimension of space-time and N the dimension of the gauge group.
In this section we review this construction and show that the effective action (4.1) can be rewritten in terms of O(k, N ) multiplets. The reformulation is achieved essentially assembling the N +k gauge fields as a vector, the N k moduli scalars as part of a symmetric tensor and the structure constants of the non-abelian gauge groups as an antisymmetric three-index tensor under O(k, N ) transformations. The procedure generalizes the analysis of [8] by including all the massless string modes at self-dual points of the moduli space, in which the k left Kaluza-Klein vector fields become part of the Cartan subgroup of the maximally enhanced gauge group.
Furthermore, using the equivalence between gauged DFT and generalized Scherk-Schwarz (gSS) compactifications [19], we present an explicit realization of the internal generalized vielbein which reproduces the structure constants of all the enhanced gauge groups under generalized diffeomorphisms. In particular, we show that the structure constants of the E 8 × E 8 and SO(32) groups can be obtained from the same deformation of the generalized diffeomorphisms and then the E 8 × E 8 and SO(32) theories can be described as different solutions of the same heterotic DFT.

Gauged double field theory
The frame-like DFT action reproducing heterotic supergravity was originally introduced in [9] and further developed in [18]. The theory has a global G = O(D, D + N ; R) symmetry, a local double-Lorentz H = O(D − 1, 1; R) × O(1, D − 1 + N ; R) symmetry, and a gauge symmetry generated by a generalized Lie derivative The constant symmetric and invertible metrics η MN and η AB raise and lower the indices that are rotated by G and H, respectively. In addition there is a constant symmetric and invertible H-invariant metric H AB constrained to satisfy The three metrics η MN , η AB and H AB are invariant under the action of L, G and H.
The fields of the theory are a generalized vielbein E A M and a generalized dilaton d. The former is constrained to relate the metrics η AB and η MN , and allows to define a generalized metric H MN from H AB The theory is defined on a 2D + N dimensional space but the coordinate dependence of fields and gauge parameters is restricted by a strong constraint the derivatives ∂ M transforming in the fundamental representation of G and the dots representing arbitrary products of fields.
DFT can be deformed in terms of so-called fluxes or gaugings f MN P [18], a set of constants that satisfy linear and quadratic constraints 5) and the following additional constraint is required to further restrict the coordinate dependence of fields and gauge parameters The generalized dilaton and frame transform under generalized diffeomorphisms and H-transformations as follows The DFT action can be expressed in terms of the generalized fluxes (5.13) and it is fixed by demanding H-invariance, since the generalized fluxes are not Hcovariant.

Parameterization and choice of section
Choosing specific global and local groups and parameterizing the fields in terms of metric, two-form, vector and scalar fields one can make contact with the (toroidally compactified) heterotic string modes and effective actions of the previous sections. To this aim, we first consider the theory at points of the moduli space in which the gauge group is where N ; R) .
Under this decomposition, the degrees of freedom can be decomposed as where E M a parameterizes the coset G i /H i . The G and H invariant metrics are H AB = diag(g AB , g AB , δ ab , δ ab , δ F G ) .

(5.15)
We can parameterize the generalized frame in terms of the d-dimensional fields as where the vielbeins e µ A and e µ A for the right and left sectors define the same space-time

The internal part of the generalized vielbein E a M can be written in terms of the background fields and perturbations as
where e a m and e a m are two different frames for the same background metric G mn ,ê a m ,ê a m are the inverse frames and C mn = B mn + 1 2 A I m A nI . Then the generalized metric is and the symmetric and where the fields depend on the external coordinates.
With this parameterization in (5.11), taking e −2d = √ −Ge −2ϕ in (5.12) and resolving the strong constraint (5.4) in the supergravity frame, after integrating (5.13) along the internal coordinates one gets an action of the form of (the electric bosonic sector of) half-maximal gauged supergravity [10] 5.20) and the scalar potential is and are taken to be the structure constants of G L , satisfying the linear and quadratic constraints (5.5) Identifying A Γ µ = A Γ µ , B µν = −B µν , G µν = G µν one gets the ten dimensional heterotic string low energy effective action (4.1).
For k = 0 and generic values of the background fields, the gauge group is U (1) 2k+16 and then there are no gaugings. In this case, (5.19) reproduces (4.1) when identifying the generalized gauge fields with the string theory fields as  (5.19) and taking for f M N P the structure constants of G L , one recovers (4.1).
In the cases of maximal enhancement, we can take G i = O(k, N ), with N being the dimension of a simply-laced group of rank 16 + k. The k left internal dimensions become part of the dimensions associated to the Cartan subgroup of the enhanced gauge group, the left KK gauge fields A m µ become Cartan components of the non-abelian gauge fields A Γ µ and the gaugings are the structure constants of the gauge group. In the next section we deal with these cases in full detail and we also show that the action (5.19) reproduces the right patterns of symmetry breaking when moving away from a point of enhancement.

Generalized Scherk-Schwarz reductions
We have seen that appropriately choosing the global and local symmetry groups and the gaugings deforming the generalized Lie derivative (5.9), one can account for both the un-compactified and the toroidally compactified versions of the heterotic string effective low energy theory with gauge group G L ×U (1) k L ×U (1) k R . To describe the effective theory with maximally enhanced gauge group G L × U (1) k R , we perform a generalized Scherk-Schwarz (gSS) compactification of DFT. Recall that the result of gauging the theory and parameterizing the generalized fields in terms of the degrees of freedom of the lower dimensional theory is effectively equivalent to a gSS reduction of DFT [19], which has the advantage of providing an explicit realization of the generalized vielbein E A M giving rise to the enhanced gauge algebra under the generalized diffeomorphisms (5.11) [12,13]. In this section we extend the construction to the heterotic case, and in particular, we will show that the formulation of [13] allows to describe the E 8 × E 8 and SO(32) theories as two solutions of the same heterotic DFT, even before compactification.
The generalized vielbein in gSS reductions is a product of two pieces, one depending on the d external coordinates x µ and the other one depending on the internal ones, y L , y R : The matrix Φ parameterizes the scalar, vector and tensor fields of the reduced d-dimensional action and the twist E characterizes the background.
Let us concentrate on the internal part of the vielbein The scalar matrix can be written as We now expand on the explicit parameterization of Φ(x) in terms of fluctuations that can be identified with the string theory fields and on the twist E a M (y L ) realizing the enhanced gauge algebra.

Fluctuations around generic points in moduli space
In order to identify the massless vector and scalar fields of the reduced theory with the corresponding string states at a generic point in moduli space, we first consider a reduction on an ordinary 2k+16 torus (i.e. no twist). There are no gaugings and therefore we get an ungauged action with 2k + 16 abelian U (1) k+16 The internal part of the generalized vielbein in the left-right basis reads, at first order, where now we denote e 0 andê 0 the frames and inverse frames for G to lighten the notation. Note we are not varying the frame for the Killing metricẽ. Performing this expansion and accommodating the terms so that it has the form of a gSS reduction E = Φ(x)E, where now the twist E is constant, one gets The matrix of Φ is an element of SO + (k, k + 16; R), the component of O(k, k + 16; R) connected to the identity. Inserting this into (5.28) we get, up to second order 25 ,  25 We actually get a second order piece in the off-diagonal terms, namely instead of M , one gets M +Q, where Q contains terms of the form δe t δê, δB δB , etc., but this second order piece is not needed for our purpose of computing the action up to quartic order.
The fluxes f ab c computed from (5.11) vanish as the twist E is constant and the theory is not deformed. Then, taking the abelian field strengths F a µν = (F a µν , F a µν , F A µν ) for the U (1) k R and U (1) k+16 L vector fields, we get (up to first order in fluctuations) Plugging these terms in (5.19), the effective action (4.1) derived from toroidally compactified string theory is reproduced if we identify, as in the previous section, F a µν = e a 0 mF m µν , F a µν = e a 0 m F m µν , F A µν =ẽ A I F I µν , δG mn = S (mn) , δB mn = S [mn] , δA Im = S Im , where the vector and scalar fields correspond to the string theory statesā m µ , a m µ , a I µ , SÎ m in sector 1 of section 2.3.

Symmetry enhancement
In order to incorporate the massless degrees of freedom that enhance the U (1) 16+k gauge symmetry to a N -dimensional group G L of rank 16 + k, we identify the 16 + k torus with the maximal torus of the enhanced symmetry group, so that the O(k, k + 16; R) covariance of the abelian theory is promoted to O(k, N ; R).
The effective action is formally as (5.19), where now the non-abelian left vector fields A G µ and scalars S Gm absorb the KK left vector and scalar fields, yielding The structure constants in the field strengths, covariant derivatives and scalar potential can be explicitly computed from the twist E a M , generalizing the procedure introduced for the bosonic string in [12,13] (see also [23]). Namely, the extra massless vectors with non-trivial momentum and winding can be thought of as coming from a metric, a B-field and a Wilson line defined in an extended tangent space, with extra dimensions. The fields in this fictitious manifold depend on a set of coordinates dual to the components of momentum and winding along the compact directions. Promoting the internal piece of the vielbein E a M to an element in O(k, N ; R), the fluxes computed from the deformed generalized Lie derivative by reproduce the structure constants of the enhanced gauge algebra, with the deformation Ω abc defined below. A dependence on the left internal coordinates is therefore mandatory, but we restrict it to dependence only on the Cartan subsector, namely on the k + 16 To be specific, start with the generalized vielbein where e,ê,ẽ, A and C are the fields on the torus at the point of enhancement. Then, identify ∂ y m ↔ dỹ m , rotate to the left-right basis on the spacetime indices and bring the generalized vielbein to a block-diagonal form rotating the flat indices, which leads to Finally, we extend this (2k + 16) × (2k + 16) matrix so that it becomes an element of O(k, N ) of the form where the index M = 1, · · · , k + N and the (N − (k + 16)) × (N − (k + 16)) diagonal block J contains the left-moving ladder currents associated to the α i roots of the enhanced gauge group, J α i (y 1 L , ..., y k+16 L ) = δ α i e i √ 2α i ·y L . Note that the (N + k) × (N + k) matrix (5.37) depends only on the coordinates associated to the Cartan directions of the algebra. In case the gauge symmetry is enhanced to a product of groups, J contains the currents of all the factors, each set of currents depending on the corresponding Cartan directions.
Taking for the deformation Ω abc = ε(α, β) δ α+β+γ if two roots are positive, −ε(α, β) δ α+β+γ if two roots are negative, if a, b, c are associated with roots, and zero if one or more indices correspond to Cartan generators, all the structure constants can be obtained replacing (5.37) in (5.34). The deformation accounts for the cocycle factors that were excluded from the CFT current operators in (5.37) but are necessary in order to compensate for the minus sign in the OPE J α (z)J β (w) when exchanging the two currents and their insertion points z ↔ w (c α c β = ε(α, β)c α+β ). It was conjectured in [24] that such factors would also appear in the gauge and duality transformations of double field theory, and actually, they can be included without spoiling the local covariance of the theory. Indeed, the cocycle tensor Ω abc satisfies the consistency constraints of gauged DFT, (5.5) and (5.6), and it breaks the O(k, N ) covariance of (5.19) to O(k, k + 16). In this way, all the structure constants can be obtained from (5.34) using the expression (5.37) for the generalized vielbein with the appropriate currents corresponding to the enhanced gauge groups. All the gaugings obtained in this way satisfy the quadratic constraints (5.23), and therefore the construction is consistent.
It is interesting to note that the deformation Ω abc can be chosen to be the same one for the E 8 × E 8 and SO(32) groups. Indeed, we show in Appendix E that both groups have 26880 non-vanishing structure constants of the form f αβ α+β , half of which can be chosen to be +1 and the other half −1, so that one unique deformation accounts for both heterotic theories. The generalized vielbeins giving the remaining structure constants which involve one Cartan index can be obtained from (5.37) with J containing the E 8 × E 8 or the SO(32) currents. Choosing the former or the latter amounts to choosing a background, and in this sense the two heterotic theories can be considered as two solutions of the same gauged DFT, even before compactification.
Plugging all this in (5.19), we get precisely the effective action (4.1) derived from the string amplitudes, where the potential is, to lowest order in M , Note that unlike in the bosonic theory, there is neither a cosmological constant nor a cubic piece in the potential, which is now bounded from below. Additionally, the quadratic piece cancels. There is also a sixth-order potential, but in order to get its explicit form we would need to expand M in (5.30) to quartic order in the fields. 27

Away from the self-dual points
In this section we show that moving away from a point of enhancement corresponds to giving a vacuum expectation value to MÂā, the piece in the matrix of scalar fields that belongs to the Cartan subsector, corresponding to the KK scalars for the metric, B-field and Wilson lines. In the next section we show that the mass acquired by the vectors and scalars that are not in the Cartan directions agree with the string theory masses.
In the neighborhood of a given point of enhancement, the scalars in the Cartan subsector acquire a vacuum expectation value vÂā. Then we redefine so that M Gā = 0 for all indices G,ā. These vevs spontaneously break the enhanced symmetry: some or all of the left-moving vectors in non-Cartan directions A α µ get a mass from the covariant derivative of the scalars, given by Note that, as expected, this is always positive, unlike in the bosonic theory.
We discuss now in more detail the process of spontaneous symmetry breaking. It is simpler for this to use the Chevalley basis for the Cartan generators, where the Killing form is equal to the Cartan matrix κÎĴ = CÎĴ , and the components of a simple root αÎ (where the subscriptÎ labels the root) are (αÎ)Ĵ = δÎĴ .
We thus have for simple roots αÎ and non-simple roots β = (n β )ÎαÎ, We see that by giving arbitrary vevs to all scalars in the Cartan subsector, all the gauge vectors corresponding to ladder generators acquire mass and the gauge symmetry is spontaneously broken to U (1) k+16 Similarly, if v has a row with all zeros, let's say the rowÎ 0 , then the corresponding (complex) vector A αÎ 0 remains massless, and there is at least an SU (2) subgroup of G L that remains unbroken. The converse is also true, namely For the vectors associated to non-simple roots β the situation is more tricky as it depends on which integers n β I are non-zero. A β remains massless if vÎā = 0 for allÎ such that n β I = 0 and for allā. Note that one cannot give masses only to the vectors corresponding to non-simple roots: if all the vectors corresponding to simple roots are massless, then necessarily v = 0 and there is no symmetry breaking at all. This implies that the spontaneous breaking of symmetry always involves at least one U (1) factor, corresponding to the Cartan of the SU (2) associated to the simple root whose vector becomes massive. Thus we cannot go from one point of maximal enhancement in moduli space (given by a semi-simple group) to another point of maximal enhancement by a spontaneous breaking of symmetry.
Regarding the scalars, introducing the vevs for those in the Cartan subsector in the potential (5.38), we get at quadratic order in the scalar fields 1 16 f The first term gives and M α = b mbM αb is the Goldstone boson contribution which is eaten by the vectors to become massive. This agrees with the results in [15] on which we expand in the next section.
We thus get that for arbitrary vevs, all vectors and scalars except those along Cartan directions acquire masses, and the symmetry is broken to U (1) k+16 L × U (1) k R . If vÎ 0ā = 0 for a givenÎ 0 and for allā, while all other vevs are non-zero, then the remaining symmetry is at least (SU (2) × U (1) k+15 ) L × U (1) k R where the SU (2) L factor corresponds to the root αÎ 0 , and the massless scalars are, besides those purely along Cartan directions, at least all those of the form M αÎ 0ā .

Comparison with string theory
Let us compare the vector and scalar masses that we got in the previous section from the double field theory effective action, to those of string theory given by (2.16).
We decompose the generalized metric M as in (5.27), where E is the twist containing the information on the background at the point of enhancement and M ab represents the fluctuations from the point, parameterized as in (5.30) in terms of the matrix M in (5.31) 28 . Inserting this in the mass formula (2.16) we get On the other hand, from Eq. (2.34) We thus get The bosonic states that are massless at the point of enhancement (when M = 0) have p R = 0 andN = 1 2 in the N S sector. The left-moving vectors have either N = 1 and p L = 0, or N = 0 and p L = α with α a root of the enhanced gauge algebra (and thus |α| 2 = 2). The former vectors (Cartan) are massless for any M , while, according to (5.48), the latter have mass It is interesting to recall that the combinationsfā αα ≡ fÂ αα vÂā appearing in the vector and scalar masses (5.40) and (5.43) agree with the coefficients of the string theory three-point functions involving one massless right or left vector and two massive left vectors. Then following [15], one could identify the DFT fluxes with the string theory three-point amplitudes and conclude that the fluxes depend on the moduli. Actually, from a gSS DFT point of view, the vevs can be thought of as being encoded either in the twists E a M (y L ) or in the fluctuations Φ a b (x). In this section we have developed the latter identification, i.e. the fluxes fÂ αα are computed from (5.34) with the twist (5.37) containing the currents corresponding to the enhanced gauge group, and the symmetry is broken by the vevs shifting the fluctuations in (5.39). In the former case, i.e. to get moduli dependent fluxes, one can replace the currents in (5.37) by those of the massive vectors in (4.4), and then the twists depend on both the left-and the right-moving internal coordinates, E a M (y L , y R ). In this way, the fluxes computed from the deformed generalized Lie derivative (5.34) get mixed indices from the left and right moving sectors, reproducing the coefficients of the string theory three-point functions which involve massive vectors (4.13). One could then interpret that the fluxesfā αα encode the information about the background through the vertex operators creating the string theory vector and scalar states.
Modular invariance of the one-loop partition function of the heterotic string implies that the 16-dimensional internal momenta must take values in an even self-dual Euclidean lattice, Γ = Γ * , of dimension 16. There are only two of these: Γ 8 × Γ 8 , where Γ 8 is the root lattice of E 8 , and Γ 16 , which is the root lattice of SO(32) in addition to the (s) or (c) conjugacy class In this Appendix we summarize some basic notions on these lattices, which are named Narain lattices.
Given a Lie algebra g of rank n, taking arbitrary integer linear combinations of root vectors, one generates an n-dimensional Euclidean lattice Γ g , called the root lattice. E.g., for the rank n orthogonal groups SO(2n), the n component simple root vectors are  Any Lie group G has infinitely many irreducible representations which are characterized by their weight vectors. Irreducible representations fall into different conjugacy classes, and Γ g can be thought of as the (0) conjugacy class. Two different representations are said to be in the same conjugacy class if the difference between their weight vectors is a vector of the root lattice.
While E 8 has only one conjugacy class, namely (0), the SO(2n) algebras have four inequivalent conjugacy classes: The weight lattice Γ w is formed by all weights of all conjugacy classes including the root lattice itself. Clearly Γ g ⊂ Γ w , and for a simply-laced Lie algebra, which roots have squared modulus 2, it can be shown that Γ g = Γ * w . Therefore, the weight lattice of E 8 contains the weights of the form Γ 8 w : (n 1 , . . . , n 8 ) with n i ∈ Z, is identical to its root lattice, which implies that it is even self-dual. It is also identical to the SO(16) lattice with the (0) and (s) conjugacy classes A necessary condition for a self-dual lattice is that it be unimodular. The SO(2n) Lie algebra lattices are unimodular if they contain two conjugacy classes. The weight lattice of Spin(32)/Z 2 is identical to the SO(32) lattice with the (0) and (s) conjugacy classes. It is even self-dual and it's vectors are: Note that both for SO(32) (or rather Spin(32)/Z 2 ) and for E 8 , one could have chosen the opposite chirality, namely the (c) class instead of (s). We will denote this choice SO(32) − and E − 8 . We can then build the following pairs The maximal enhancement points are those where two or more curves intersect. There are three types of intersections: a w 1 ,q 1 (R) = a w 2 ,q 2 (R), b w 1 ,q 1 (R) = b w 2 ,q 2 (R) and a w 1 ,q 1 (R) = b w 2 ,q 2 (R), that we treat separately. In the case of Γ 8 × Γ 8 , the curves b can in principle have a curve c on top of them.
The case C = 0 is trivial, so we must assume C = 0, which leads to w 2 w 1 + 4q 1 q 2 ∈ Z, we can rewrite (B.3) as Since (1 − 2q 2 i ) and w i are odd, N is even. Also, since C and R are non-zero we get N 2 < 4, which implies N = 0, then R 2 = 2 C 2 . Then the radius where a curve a with winding w 1 intersects another curve a with winding w 2 is The constraint must be a perfect square .
If w 1 = w 2 = w, then q 1 = q 2 ± 1. The winding must be a divisor of both 2q 2 1 − 1 and 2q 2 2 − 1, but these numbers are coprime ∀q 1 . Then the only possible value of w is 1. In conclusion, the only curves a with the same winding number that intersect are a 1,q (R) and a 1,q±1 (R). And the intersection is on In this case, If C = 0, then w 1 = w 2 and q 1 = q 2 . If C = 0 and then R 2 = 1 2C 2 . Replacing in (B.8), C 2 = w 2 1 + w 2 2 , and then the radius where curve b with winding w 1 intersects curve b with winding w 2 is R −2 = 2(w 2 1 + w 2 2 ). The constraint If w 1 = w 2 = w then |2(q 1 + q 2 )| = √ 2w. The l.h.s. is integer and the r.h.s. is irrational, then there is no winding such that b w,q 1 (R) = b w,q 2 (R). B.3 a w 1 ,q 1 (R) = b w 2 ,q 2 (R) Since w 1 is always odd, then C is also odd (in particular it is non-zero). Then w 1 + q 1 (2q 2 + 1) ∈ Z, and then N = 0 or 1, which give Then the radii where a curve a with w 1 intersects another curve b with w 2 intersect are: For each case we have one of these constraints: and then w 2 1 + 2w 2 2 or (w 1 − w 2 ) 2 + w 2 2 must be a perfect square. If w 1 = w 2 = w we get the constraints: leaving only the second case, with q 2 = q 1 or q 1 − 1. The quantization conditions imply that w must be a divisor of both 2q 2 1 − 1 and 2q 1 (q 1 ± 1). But it can be shown that these numbers are coprime, and then w = 1. The only curves with the same windings that intersect are a 1,q (R) and b 1,q (R) or b 1,q−1 (R). The intersections are at R = 1 √ 2 . Summarising, we have: The winding numbers on b can in principle be any positive integer and those on a can only be the divisors of some number of the form 2q 2 − 1, q ∈ Z. B.4 Enhancements to SO(34) or SO(18) × E 8 Here we prove that a w 1 ,q 1 (R) = a w 2 ,q 2 (R) implies that there exist integers w 3 , q 3 , w 4 and q 4 such that a w 1 ,q 1 (R) = b w 3 ,q 3 (R) = b w 4 ,q 4 (R).
We start with R −2 = w 2 1 + w 2 2 . If w 1 > w 2 , there are integers w 3 and w 4 such that w 1 = w 3 + w 4 and w 2 = w 3 − w 4 , because w 1 and w 2 are odd numbers. Then Since R −2 = 2((w 1 − w 4 ) 2 + w 2 4 ) as well, there exist integers w 3 , w 4 , q 3 and q 4 such that a w 1 ,q 1 (R) = b w 3 ,q 3 (R) = b w 4 ,q 4 (R). Note that we can always find q 3 and q 4 because the functions b admit any value of w.
We still have to prove that 2q 3 (q 3 + 1) and 2q 4 (q 4 + 1) are divisible by w 3 and w 4 , respectively, which amounts to proving that w i is a divisor of 2q 2 i − 1 and |q 1 w 2 − q 2 w 1 | = w 2 1 +w 2 2 2 =⇒ w 1 ± w 2 is a divisor of (q 1 ± q 2 ) 2 − 1 (B. 19) We checked that this is satisfied for the first 300 values of q i .
We still have to prove that 2q 2 1 − 1 and 2q 2 2 − 1 are divisible by w 1 and w 2 , respectively. This is the same as proving that q i is a divisor of 2q i (q i + 1) and |(2q 3 + 1)w 4 − (2q 4 + 1) which we checked is satisfied.
In conclusion, we have that, for The Wilson lines that give this enhancement can be written in four different ways , after a few steps, we get ∓ 4 = ± 3 = ± 2 = ∓ 1 and then the Wilson lines are and then, after a few steps, we can prove that Defining integers m = ( √ 2R) −1 and n = A/ √ 2, all this type of enhancement points are given by We want to see if the b lines considered here can be interposed with a c line. q 3 and q 4 are suitable for curves b with w 3 and w 4 . For curves c to coincide with them, we need w i even and q i (q i +1) w i ∈ Z. If one of the two curves b has also a curve c then we have an intersection between an a and a c curve. Analyzing all the possibilities, it can be shown that there are no c curves that intersect with more than one other curve. B.5 Enhancements to SU (2) × SO(32) or SU (2) × E 8 × E 8 The equality a w 1 ,q 1 (R) = b w 2 ,q 2 (R) arises for two type of radius The second type gives R −2 = w 2 1 + w 2 3 if w 2 = w 3 +w 1 2 , which implies that there is an intersection with another curve a of winding w 3 . Then, we restrict to the first type, where R −2 is odd for odd w 2 1 . Thus the even R −2 found in the previous section cannot have additional curves a or b on the intersection.
For R −2 = w 2 1 + 2w 2 2 , the constraints are The Wilson line can be written as and equating them leads to ± 2 = ∓ 1 and implying that R −1 is an odd number. After some algebra, we get It is not hard to prove that all the curves b that intersect just one curve a are superimposed by a curve c (in the Γ 8 × Γ 8 case).

C More on fixed points and dualities
Here we collect the details of the calculations involved in section 3.6. The transformations O U in (3.37) acting on a vector Z of momentum, winding and "heterotic" momenta, result in the transformed momenta (C.1) Requiring these to be quantized leads to the conditions r, We analyze these in more detail, depending whether the duality acts on the same theory or links two theories with different lattices Γ and Γ . , (Q ± 1) Q 2r , Qr 2 ∈ Z , and where U ± are defined in (3.40), and A is defined in (3.32). Here we have used that for the fixed points R = R fp , one has r = R −2 fp since R = 1 rR = R. For U = ±I we define p = (Q±1) 2 r and q = Q/2, then: Quotienting these equations, we see that p, q r can be written as with t, n, m, k ∈ Z. Then which implies k = 1 and t = 2n 2 ±1 m . Taking into account that n = where we have used the fact that as Q, 2p and √ pQ are integers with Q odd, implies that p is also integer. Q+1 2 is integer, then we have the same situation as in the first case.
with t, n, m, k ∈ Z. Then but this implies that k = 1 and t = n 2 ±1 2m . Then, taking into account that n = and m = r 2 = R −1 √ 2 , the only condition is: The possible values of m and n that verify these conditions with the plus signs give the fixed points (R, A 1 ) presented in Table 1.
If A does not satisfy this for any π ∈ Γ, then A ∈ Γ, and viceversa, i.e. either A ∈ Γ or A ∈ Γ. But r|A | 2 2 ∈ Z, A + r|A | 2 2 A ∈ Γ and the reciprocal conditions imply A, A ∈ Γ. We just need to verify 1 ∈ Z. But A · A is integer and |A| 2 , |A | 2 are even, then we get: 1 r ∈ Z, which is only possible for r = 1. Then 1 is the only non-composite possible value for r when the duality does not change the lattice and U = 1.

C.2 Γ ↔ Γ = Γ
The quantization conditions (C.2) for the case where the dual lattice is not the original one become r, where Γ U is the lattice obtained by applying the transformation U to all the elements of Γ and A U = A U . This proves the statements at the beginning of 3.6.2.
Another condition that must hold is 2A ∈ Γ ∩ Γ . But this occurs trivially for the lattices that we consider. 2A will always be in the adjoint conjugacy class of SO(16) × SO (16), which is contained in all Γ ∩ Γ we will study (this could vary with other groups where, for instance, (s) + (s) = (v)...).

D Three and four-point functions
For completeness, in this appendix we list the scattering amplitudes of massless states of the (toroidally compactified) heterotic string that give the effective action (4.1). The results hold for arbitrary points of the moduli space, including enhanced and broken symmetry points, and differ only on the possible values taken by the indices and structure constants. Details of the calculations can be found in [20,25,26].
We use the following expectation values In terms of Mandelstam variables s = −2k 1 · k 2 , t = −2k 1 · k 3 , u = −2k 1 · k 4 and summing over all cyclic orderings of the vertex operators to compensate for the fixing of z 2 , z 3 and z 4 , we get  we finally get which adds up to when using t s + s t + u t + t u + s u + u s = −3 and S Γm S Λ m S Γ n S Λ n f ΓΛΠ f Γ Λ Π = 0.

D.3 Three-point functions involving slightly massive states
It is easy to see that the amplitudes of three massless right vectors or three massless scalars vanish at the enhancement points. However, in the neighborhood of these points, the currents acquire dependence on p R and then the amplitude of three scalars or that of two left and one right vectors get a non-vanishing value and give extra terms in the E Counting structure constants of SO(32) and E 8 ×E 8 In this Appendix we count and compare the number of non-vanishinig structure constants of the SO(32) and E 8 × E 8 algebras, which in the Weyl-Cartan basis are To calculate the number of combinations of α, β indices giving non-vanishing structure constants of SO(32) , it is convenient to denote the 480 roots as The number of pairs of roots (i+, j+), (k−, l−) is: (16 − i)(16 − i − 1) = 1120 if i = k, i < j, i < l = j 560 with j > l 560 with j < l The number of pairs of roots (1; i+, j+), (1; k−, l−) is with i = k, j = l The second line counts the number of pairs j, l such that i < j and i < l = j, and the fourth one, the number of pairs i, l such that i < j and j < l. Then we have  (1, s). That is 26880 non-vanishing structure constants of type f αβ α+β . In addition, there are 2 × (4 × 28 + 128) × 16 = 7680 structure constants of type f αᾱ A . In conclusion, the number of structure constants of type f αβ α+β is 26880 and of type f αα A is 7680, for both the SO(32) and the E 8 × E 8 groups.