Symmetry enhancement interpolation, non-commutativity and Double Field Theory

We present a moduli dependent target space effective field theory action for (truncated) heterotic string toroidal compactifications. When moving continuously along moduli space, the stringy gauge symmetry enhancement-breaking effects, which occur at particular points of moduli space, are reproduced. Besides the expected fields, originating in the ten dimensional low energy effective theory, new vector and scalar fields are included. These fields depend on “double periodic coordinates” as usually introduced in Double Field Theory. Their mode expansion encodes information about string states, carrying winding and KK momenta, associated to gauge symmetry enhancements. It is found that a non-commutative product, which introduces an intrinsic non-commutativity on the compact target space, is required in order to make contact with string theory amplitude results.


Introduction
In this article we propose a target space effective field theory description of string theory interactions. Clearly the subject is not new. Indeed, the conventional low energy effective action for given values of moduli fields can be found in string books [1][2][3]. However, several works point towards a richer structure with some intrinsic compact target space non-commutativity. Among them, there are recent analyses [4][5][6][7][8] performed from the perspective of Double Field Theory (DFT) 1 aiming at the inclusion of gauge symmetry enhancement aspects in the field theory description, as well as recent (and not so recent) proposals about non-commutativity of string zero modes [22][23][24].
A key guide in our analysis is the field theory description of gauge symmetry enhancement on toroidal compactifications. Gauge symmetry enhancement is a very stringy JHEP03(2019)012 phenomenon associated to the fact that the string is an extended object and, therefore, it can wind around non-contractible cycles. At certain moduli points (i.e., fixed points of T-duality transformations) vector boson states, associated to definite values of windings and compact momenta become massless. These vectors, combined with massless vectors inherited from the metric and antisymmetric tensor fields, give rise to an enhanced gauge symmetry group G 1 (see for instance [25][26][27]). Further displacements on moduli space can lead to a different fixed point where, generically, other vectors associated to different values of winding and momenta will become massless leading to a different enhanced gauge group G 2 , etc. At generic points only a U(1) r+16 L × U(1) r R symmetry exists. Here, r is the number of compactified dimensions associated to the KK zero modes of the metric and antisymmetric fields and the 16 comes from Cartan generators of the ten dimensional gauge group, in the heterotic string case. The low energy effective theory, at a given moduli point, where massive states are neglected, can be described by a usual gauge field theory Lagrangian coupled to gravity with no explicit reference to any windings. By slightly moving away from this fixed moduli point, gauge symmetry gets broken. The symmetry breaking can be understood as a conventional higgsing mechanism and also, as found from a DFT approach [5,7], as associated to a dependence on moduli fields of the "will-be structure constants" fluxes.
The main aim of the present work is to write down a lower dimensional field theory able to provide a description of the enhancement phenomena occurring on toroidally compactified heterotic string. This action depends on moduli fields expectation values such that the different low energy effective field theories, associated to heterotic enhancement situations, can be reached by varying such values. Our construction is restricted to fields corresponding to low string oscillator number and includes the fields that are involved in the enhancement phenomena. Clearly a full, consistent description of the string theory would require the introduction of an infinite number of fields of all possible spins. We comment on a possible step by step completion of our construction, going beyond low energy, at the end of the article.
Very schematically, the idea is to incorporate a vector boson field A µ (x, Y) and a scalar MĪ (x, Y) into the action, in addition to the fields inherited from the usual ten dimensional metric, the dilaton and the Kalb-Ramond B 2 . All fields must depend on both d spacetime x µ coordinates as well as on internal compact toroidalY ≡ (y I , y m ,ỹ m ) coordinates. Namely, besides the y I coordinates associated to the heterotic string degrees of freedom, 2r double coordinates (y m ,ỹ m ), conjugate to momenta and windings modes (p m ,p m ), for each of the r compact dimension are considered in the spirit of DFT. A generalized mode expansion (GKK) in periodic internal coordinates would produce d dimensional fields A I (x)) with L labeling modes, depending on windings and KK momenta. As mentioned before, for certain moduli values some of these modes become massless and, when combined with KK zero modes coming from metric and B field (as well as heterotic Cartan fields) they enhance the gauge symmetry. The other modes, not participating in the enhancement process, remain very massive (with masses of the order of string mass α −1 ) and do not contribute to the low energy effective theory.

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The resulting action, in terms of the "uplifted" A µ (x, Y) and MĪ (x, Y) fields, appears to require a non-commutativity on fields introduced through a non-commutative -product in the compact space [22,23]. At the neighborhood of each specific moduli fixed point and when only the slightly massive modes that become massless at this point are kept, the usual, commutative, effective gauge theory action is recovered after integrating over the internal coordinates. The gauge symmetry gets enhanced exactly at the fixed point. Therefore, the action provides an effective interpolation among theories at different points. It is worth mentioning that enhancement can be described in DFT constructions as an enlargement of the compactification tangent space [4][5][6][7][8] at a fixed point. Here, however, the compact manifold is an r dimensional double torus and we find that this enlargement is effectively provided by Fourier modes associated to fields that "will-be massless at such point". Interestingly enough, the mentioned non-commutativity can be traced back to cocycle factors in string vertices. These factors were first mentioned in [15] but did not manifest in previous DFT constructions due to the considered level matching conditions and to the fact that calculations were performed up to third order terms in the fields.
We organize the article as follows: in section 2, we introduce the proposed action in D = d + 2r dimensions. In section 3 we perform the mode expansion and analyze the different contributions. Section 4 deals with the physical content of the action, like vector and scalar masses, Goldstone bosons, enhancement-breaking of gauge symmetries, etc. An illustrative torus compactification (r = 2) example is briefly discussed. A summary and a discussion of the limitations and possible extensions of the present work are presented in section 5. Notation and technical aspects are reserved to the appendices. A more detailed description of the -product is extended to incorporate the heterotic string gauge modes.

The effective action
In this section we present a moduli dependent field theory effective action that captures essential features of symmetry enhancement in toroidal compactification of heterotic string. The basic ingredients and notation conventions are introduced here. The reader is referred to the appendices for details.
Let us denote by Φ ≡ (g, b, A) a moduli point encoding the background metric g, the b field and Wilson line values. At a given fixed point Φ 0 on moduli space the heterotic gauge group is of the form G L × U(1) r R . The rank of G L is r L = r + 16 = 26 − d originating in the 16 Cartan generators of the ten dimensional heterotic gauge group plus the r = 10 − d vector bosons coming from Left combinations of the KK reductions of the metric and the antisymmetric tensor. Therefore, the dimension of the gauge group is dim G L = n c + r L where n c denotes the number of charged generators. These generators correspond to string vertex operators containing KK momenta and windings associated, generically, with massive fields that become massless at the fixed point. These fields will play a central role in our construction. Let us stress that n c depends on the moduli point and that, at generic points, there is no enhancement at all (n c = 0) and the generic gauge group is U(1) r L L × U(1) r R . The low energy effective action for the bosonic sector of heterotic string, at a fixed point Φ 0 with G L × U(1) r R gauge group and up to third order in the JHEP03(2019)012 fields, reads whereĪ Right indices correspond to the Abelian Right group U(1) r R and A indices label Left G L (generically non Abelian) group. We have where scalar fields M AĪ live in the (dimG L )q =0 adjoint representation of G L and carry zero vector chargeq = (q 1 , . . . ,q r ) = 0 with respect to U(1) r R Abelian Right group. H is the B field strength (with Chern-Simons interactions) defined as ϕ is the dilaton and R the scalar curvature. As mentioned above, the terms in this expression corresponding to fields originating in reductions of the 10D fields will be always present, whereas terms associated with charged fields will change when moving on moduli space. In our construction it proves convenient to separate these contributions and rewrite the above action (2.1) as Several indices are introduced: •Ī = 1, . . . r are Right indices that label the Abelian group U(1)Ī associated to Right vector bosons AĪ µ • The Left index A has been conveniently splitted as A = (Î, α) where: α = 1, . . . , n c label the Left gauge group charged generators with vector bosons A α µ . They correspond to roots of the algebra in a Cartan-Weyl basis. The field strengths introduced in (2.2) are now splitted as

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for Cartan fields whereas are the field strengths for charged vectors. 2 Similarly, for scalar fields we have where a sum over repeated root indices is implicit. We are using a Cartan-Weyl basis such that f αβ γ = f αβ(−γ) = 1 (with γ = α + β) and f βαÎ = f −ααÎ = f α−αÎ = αÎ (no sum on α here) etc. Also, charged indices are contracted with the corresponding Cartan-Killing form whereas Cartan indices contract with a delta function. Finally, it proves useful to perform a further rewriting of the above action by collecting Left and Right "Cartan" indices into a unique generalized I = (Î,Ī) index spanning the vector representation of O(r l , r) duality group. The I indices are contracted with an O(r L , r) invariant metric that we will generically express in the L-R basis (also called C−basis) as In order to have a covariantly looking form in this basis we introduce the generalized vector A I µ = (AÎ µ , AĪ µ ) that incorporates the Left and Right Cartan fields respectively and define the scalars M αJ = (0, M αJ ) where Left components are projected out. We discuss this projection in (2.25) below. Also, inspired by DFT constructions [7] we introduce a generalized O(r l , r) metric H IJ and we expand on fluctuations around a flat background as where matrix elements vanish unless In terms of this metric and keeping terms up to second order in fluctuations (assuming vector fields are first order) the action 3 can be re-expressed as Iµν .
2 Here we use the convention 2A [µ B ν] = AµBν − Bν Aµ. 3 Notice that there is no scalar potential. Scalar interactions appear at fourth order in the fields when only massless states are considered.

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Field Modes L 2 N Vertex Operators The different contributions to the action above are read out from 3-point amplitudes of massless heterotic string vertices. Vertex operators for vector bosons and scalars considered here are collected in table 1 and explained below.
String vertex operators generically contain an internal factor (see appendix A for notation) where L (P) (Φ) = (l In all the cases considered hereN = 0 and N = 0, 1. Vertex operators with N = 1 correspond to KK reductions of the metric, B-field, dilaton field and heterotic vector fields in 10 dimensions. For instance, the Cartan vectors AÎ µ do originate in string vertex operators coming from KK reductions of the metric and antisymmetric field of the form

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where K µ is the space time momentum. Due to the presence of oscillators ∂ z YÎ , N = 1 and therefore LMC (2.16) reads This requirement is trivially satisfied by massless states that correspond to L ≡ l L = l R = 0 (with null windings and KK momenta), as it is indeed the case for Cartan vectors AÎ µ ≡ A (0) µÎ . On the other hand, the left handed charged vector bosons arise from vertices since N = 0. The other cases in the table 1 are understood in a similar way. Let us stress that ghost factors as well as cocycle factors must be included. At a fixed point Φ 0 and for specific values of windings and momenta (i.e., for specific values ofP) the states become massless (see (A.6)) and l (P) L (Φ 0 ) become the roots α (P) of the enhanced gauge group algebra charged generators. 5 Generically, at a different fixed point, other set ofP s will ensure (2.21), leading to a different enhanced gauge group. We will denote this set of n c GKK modes, satisfying (2.20), by Namely,Ǧ(Φ 0 ) nc encodes the n c "will-be massless charged fields at fixed point Φ 0 ". At Φ 0 , and forP ∈Ǧ(Φ 0 ) nc the A (P) µ (K) modes give rise to charged vector field A α µ (x) in the action above (similarly with charged scalars).
As stated in the Introduction the main aim of our work is to provide a unified field theory description such that at given fixed points the different effective gauge theories are reproduced. Following the suggestions in [5] we propose to consider a sort of generalized Kaluza-Klein expansion on generalized momenta L of the different fields coming into play in the enhancement process. The GKK modes in this expansion are identified with a corresponding polarization of a vertex operator. For instance, in order to describe charged vector bosons we introduce the expansion µ (x) correspond to polarization modes in (2.19). The prime in the sum indicates that LMC (2.20) must be imposed (with an abuse of notation we indicate the sum on mode JHEP03(2019)012 indexP by L). Recall that generically the sum contains an infinite number of terms even though the LMC is a severe constraint.
Generically, if the mass of the GKK components A (L) µ (x) were given by the string mass formula (A.6), as we will show to be the case, these modes would be massive. However, when moving continuously along the moduli space, for specific valuesP ∈Ǧ nc (Φ 0 ), n c vector fields A µ (x) would become massless and would lead to the enhanced G L gauge group. 6 In a similar way we introduce the GKK expansion for scalar fields by associating the fields M αĪ (x), coming from string vertex operators modes M Before addressing the other mode expansions let us note that the Right fields MĪ can be embedded into constrained fields M I = (MÎ , MĪ ) with O(r L , r) indices. Namely, they are defined as where H is the generalized metric satisfying HηH = η and P,P are projectors [18][19][20][21]  where . . . indicate higher order terms in fluctuations. Therefore MÎ = −MÎJ MJ + . . . . We see that MÎ degrees of freedom are not independent and contribute at order two or higher in fluctuations. As we show below, these will give rise to terms of order four in the action. For this reason we can set M J = (0, MJ ). Expansions of fields originating in the D = 10 metric, B field and the Cartan generators of the heterotic gauge group, namely, G µν (x, Y), B µν (x, Y), A I µ (x, Y), MÎJ (x, Y) must also be considered. Now, since the corresponding modes (first four rows in table 1) must satisfy LMC (2.18) we restrict the sum to modes obeying this constraint. For instance Recall that these modes correspond to N = 1 and, therefore, only the zero mode A

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G µν (x) is the d dimensional metric field whereas non-zero modes would describe massive gravitons, etc. In most of the considerations below only these zero modes will be needed.
Notice that a generic, moduli dependent, field φ(x, Y) ≡ A µ (x, Y), G µν (x, Y), . . . could be interpreted as an uplifting of d dimensional fields to d + r + r l dimensions with r + r l periodic. 7 A Lagrangian L(x, Y) in terms of these fields, when integrated over the d+r will lead to an action in d space-time dimensions after periodic coordinates are integrated out, where the physical fields will be the φ(x) (L) GKK modes. Our expectation is that such action includes the effective low energy heterotic effective action (2.13) for different fixed moduli point Φ 0 .
A crucial point is how to generate a non-Abelian structure out of these fields in order to give rise to enhancements at fixed points. We will see that the job is accomplished by a new so called "star product" [22,23], which we denote by , accounting for non-commutativity.
In the following we present the action and subsequently we discuss its particular features. Let us assume that we are able to write down a full field theory action S het (Φ) by computing all possible heterotic string theory amplitudes. This action should include an infinite number of fields, let us call them Φ µ 1 µ 2 ...;N ,N (x, Y), of all possible spins and oscillator numbersN , N that must be mode expanded with the corresponding level matching condition . Among all these contributions we isolate the action piece, that we call S enh (Φ), containing up to third order terms (and some fourth order as we discuss below) and involving fields coming form 10D KK reductions G µν , B µν , AJ µ , ϕ, MÎ ,J , AĴ µ and the extra fields A µ , MĪ . These fields are associated to oscillator numbers N = 0, 1, respectively, andN = 0. Their corresponding modes are collected in table 1 as well as their associated string vertex operators.
Therefore, we split the full action into where the term S encodes all other (infinite) contributions that we are not explicitly considering here. These include higher spin fields, fields associated to oscillator numbers N > 1, higher order terms in fluctuations, etc. S enh (Φ) is the action we are going to deal with, given by As we have already emphasized the different terms in the action are expressed in terms of the fields introduced above. These fields depend on the compact coordinate Y and can 7 Recall that Heterotic coordinates can be thought of as coordinates on a 16 dimensional torus with a chiral projection.

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therefore be mode expanded. Integration over Y will produce an effective action in d spacetime dimensions. In the next section we perform the mode expansions and integrate over internal coordinates in order to obtain a d-dimensional space time action. Before presenting these computations let us first discuss the general structure and the kind of information we expect this action to contain. Fields originating in D = 10 KK reductions, i.e. G µν , B µν , ϕ, MÎ ,J , A J ≡ (AĴ µ , AJ µ ), require a mode expansion with the constraint L 2 = 0 whereas fields A µ , M I , associated to enhancements, require 1 2 L 2 = 1. It appears somewhat unnatural to indicate what kind of constrained mode expansion must be performed in each case. However, these LMC constraints might be implemented in the Lagrangian through, for instance, Lagrange multipliers. Thus, if we indicate by φ N (x, Y) a field such that its mode expansion must be restricted to 1 2 L 2 = 1 − N , in a DFT language we would require In the cases considered here N = 0, 1 label the number of Left indices. Clearly, the so called strong constraint of DFT (see for instance [15,16]) cannot be satisfied if enhancement phenomena are included. The term 1 2 (η J I − H J I ) acts as a covariant O(r l , r) projector. If just the zero modes are kept we notice that the first two rows in (2.13) are formally reproduced with g µν = G , etc. However, a non trivial action of theproduct arises whenever non zero modes come into play as it happens, for instance, in products of fields associated to enhancements (and thus requiring expansions with δ( 1 2 L 2 , 1) constraint). We provide a more detailed discussion of this situation in the next section. Also, the different terms in the action are now defined as (2.31) ) by generalizing (2.7). Finally, the three form H is defined as Whenever a product of two fields appears a -product must be used. For instance, the generalized metric must be expressed in terms of fluctuations as in (2.11) but with a replacing the ordinary product.
All the fields that we are considering contain modes that are massless at some specific values of moduli Φ. This is always the case for the zero modes G I (x) become massless at a point Φ 0 for momenta inǦ(Φ 0 ) nc (see (2.22)). These are the modes JHEP03(2019)012 that participate in the enhancement phenomena. When approaching a point Φ 0 in moduli space the light spectrum will contain the zero mode massless fields plus the n c slightly massive modes inǦ(Φ 0 ) nc , all other fields having masses of the order of the string mass. When moving to some other fixed point Φ 1 other set of modes (intersections can occur) iň G(Φ 1 ) nc will become light. 8 Therefore, the action (2.29) can be splitted as The first (second) splitting is convenient when Φ is close to Φ 0 (Φ 1 ). In this case, at low energies, the second term in the action (and also S above), containing heavy states (of order α −1 ) and light states in interaction with them, does not contribute. We will be left with the effective S eff (Φ ≈ Φ 0 ) low energy action and similarly for Φ ≈ Φ 1 etc. At Φ = Φ 0 all fields in S eff (Φ 0 ) become massless and the effective action should reproduce (2.5) with gauge group G L × U(1) r R . The -product plays a crucial role in reproducing the non-Abelian group structure. When slightly moving away from Φ 0 , the gauge symmetry should break, generically, to U(1) r+16 should contain massless and massive physical states correctly transforming under the Abelian groups.
Besides these features addressed in the next section when mode expansions are performed, we stress that S enh (Φ) appears to encode some relevant information about very massive states as discussed in an explicit example in 4.5.

The action for GKK modes
In this section we perform the expansion of the fields in the above action in terms of GKK modes, compute the -products for these modes and finally integrate over the internal coordinates Y in order to obtain the moduli dependent d dimensional effective action. In particular, we will show that after integrating out the massive modes, the massless GKK modes at a self-dual point, eq. (2.29) give rise to the gauge enhanced action (2.13).
The particular -product we consider here is a generalization of the one proposed in [22,23] to the case of the heterotic string. It is described in appendix B. For two mode expanded fields it reads where a phase l 1 ·l 2 = p 1mp 2m + p 1Ip I 2 dependent on the KK momenta l 1 of the first field and the windingsl 2 of second mode is generated (see (B.4)). The first term corresponds to a sum over the internal compactification lattice indices. The sum over heterotic directions 8 There will always be modes that remain very massive, as for instance G

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is constrained by a chiral projection that eliminates Right heterotic momenta. It can be expressed as in terms of Spin(32), 9 weights and E IJ = G IJ + B IJ (see appendix B). The prime in the sum indicates that the constraint 1 2 L 2 i = 1 − N i (i = 1, 2) must be imposed for the field with subindex N i . By using that by recalling that the exponents are just integer multiples of π and by using LMC we can rewrite the above product as where L = L 1 + L 2 . By comparing to (3.1) we see that the -product is non commutative unless the phase e iπ( 1 For the cases we are considering here we notice that fields with similar LMC commute whereas for N 1 = 0, N 2 = 1 (or viceversa) they anticommute. Let us proceed to consider the product of three fields. In this case, by using associativity (see (B.3)), we have If we integrate over internal coordinates, due to momentum conservation L = L 1 +L 2 +L 3 = 0, the second phase becomes e −iπ 1 2 L 2 3 and thereforẽ In this case,f where we have used (3.3) above with L 1 + L 2 = −L 3 . A similar phase is obtained if 2 ↔ 3. 9 We will mainly refer to Spin(32) but results are valid for E8 × E8 as well.

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We conclude that the product of three fields with N i = 0, as it is the case for charged fields participating in the enhancements, the phasesf L 1 L 2 L 3 are completely antisymmetric under index permutation. This result is valid both for massless and massive states. On the contrary, for modes originating in 10D fields, N i = 1, and the phase becomes irrelevant so the -product reduces to just the ordinary product.
In the following subsections we analyze the different contributions in the action (2.29) in terms of their mode expansions.

The vectors kinetic term
Let us analyze first d d xdY F µν F µν . In order to do it let us consider the Fourier 10 component (see (2.31)) F (L) µν as follows where: Here, L 3 = L − L 2 , and we have definedf L−LI as µν depends on moduli point. Let us look at the contributions, at low energy, at self-dual points. On the one hand, from the last sum we keep the zero mode contribution A giving rise to Cartan vector fields. On the other hand the light modes on the first sum correspond toP ∈Ǧ nc (Φ 0 ). Thus, when sliding to Φ = Φ 0 (3.11) will reduce to (2.7) as long as we identify where L i is in a one to one correspondence with the positive roots α i ≡ α (L i ) , i = 1, . . . , nc 2 , (and −L i with −α i ≡ α (−L i ) ) of the enhanced group (the underlying reason for this iden- (3.14) as the algebra structure constants for charged generators. Therefore, (3.14) becomes the field strength for the JHEP03(2019)012 charged fields of the corresponding gauge theory. Then, up to third order in fluctuations we can write We have thus matched the first term of the third row of action (2.13). The second term of the same row is reproduced by H IJ F I µν F J µν in (2.29) since, when focusing only on massless GKK modes,

Scalars kinetic term
Following similar steps as above we can write The massless scalars are provided by the zero modes M (0) IJ . In this case, the last two terms drop out and (2.8) is reproduced.

Scalar potential and other couplings
The scalar potential is such that it vanishes (up to third order in fluctuations) for massless states, it is O(r L , r) invariant and it reproduces the scalar potential away from the enhancing point for scalars that would be massless at such point, as computed in [7]. It appears that the most general form is It is worth noticing that the first term above, when opened up in terms of fluctuations, contains a fourth-order term (at the enhancement point) of the following schematic form: and is part of the fourth order scalar potential [8]. To complete the full fourth-order terms in the potential more terms are needed. For instance we would need an extra term of the form ∂ I φĪ ∂ I φĪ φJ φJ among others. We leave the analysis of these extra terms for future work.
Finally our action (2.29) contains a last term, M I F µν F I µν , which gives rise to the last term of (2.13) at the enhancement point (actually this term is always present and away from the self-dual point it gives rise to the adequate coupling between a massive scalar, a massive vector and a massless U(1) R vector).

Breaking and enhancement of gauge symmetry along moduli space
We have shown the explicit mode expansions for some of the terms appearing in the action. Computation of other terms proceed by following similar steps.
Several interesting results like vector and scalar masses, presence of would-be Goldstone bosons, etc. can be straightforwardly read out from these expansions. We discuss some of these issues below.

Vector masses
Vector boson masses can be extracted from the fourth term in the covariant derivative-like term (3.17) where we have used the string theory result m 2 A = 2l 2 R = L · H − η · L for the masses of the charged vector fields. We also observe that there are no mass terms for Cartan vectors A I µ so, generically, the gauge group is U(1) r L × U(1) r . At Φ = Φ 0 and forP ∈Ǧ nc (Φ 0 ) vector bosons become massless leading to gauge symmetry enhancement.
Finally we check the normalizations. Since the kinetic term of the vectors reads: when adding (3.17) we find which is the Proca Lagrangian with signature (− + + + . . .) (with a global different normalization) of a vector with mass m A .

Goldstone bosons
From the same scalars kinetic factors above we find the terms As also discussed in [7] this coupling indicates that, for a given vector boson A (this is exactly the Goldstone combination which can be read from the vertex operators in string theory [4]). In fact, at enhancement point Φ 0 and forP ∈Ǧ nc (Φ 0 ) these terms are not present since l R = 0. However, when sliding away from Φ 0 , l R = 0, and these n c combinations provide the longitudinal components of the n c corresponding A µ (L) massive vector bosons. Namely,

Scalar masses
The masses of the scalar fields can be read from the quadratic terms in the scalar potential (3.20) as expected from string theory. They coincide with the masses of the corresponding vector boson modes. As in the vector case, it is still necessary to check for the relative coefficients, so we must compare the above terms with the scalar kinetic term

Duality and gauge invariance
Let us close this section by commenting on T-duality and gauge invariance. We notice that, even if the different terms in the action (2.13) are written in an O(r L , r) invariant way, by index contraction, the effect of the -product on T-duality is not transparent. On the one hand we expect that, whenever windings and momenta are included, the symmetry group becomes O(r L , r, Z). Consider for instance (3.4). Each term on the expansion contains a Fourier mode labeled byP, encoding momenta and winding modes (2.15), as well as an exponential term e iL.Y where the exponent is explicitly O(r L , r, Z) covariant. On the other hand, a h ∈ O(r L , r, Z) rotation leads toP →P = hP but also to e iπl 1 l 2 φ (P 1 ) (x)ψ (P 2 ) (x) → e iπl 1 l 2 φ (P 1 ) (x)ψ (P 2 ) (x), where the heterotic part is expressed in terms of 16 windings and momenta as discussed in (B.4). However, ifP satisfies LMC so doesP and, since we are summing over all modes satisfying LMC, the sum will be invariant.
Notice that, if we restricted our analysis to a set of GKK momenta inǦ(Φ 0 ) nc , the above transformations will take us out of this set. Namely, theP will not become massless at Φ 0 . However,P ∈Ǧ(hΦ 0 ) nc , and therefore their associated fields will become massless JHEP03(2019)012 at the transformed moduli point hΦ 0 (note that the mass terms are invariant when transforming both the background and momenta and windings). We will illustrate this fact in an example below. Let us stress that if any of the fields contained an O(r L , r) index, as for example M (P) I , it must appear contracted in an invariant way as it indeed happens in the action.
The action we are dealing with contains massive and massless states. At a U(1) r L × U(1) r generic points, besides the 2r + 16 Abelian vectors and the gravity sector fields, all other vector and scalar fields will be massive. Recall that a field Φ(x) (L) carries charge (L) I =f L−LI with respect to the Abelian factor A (I) µ . Therefore, under a U I (1) gauge transformation and therefore, gauge invariance should be ensured by a derivative of the form . In fact, it can be checked that this is indeed the case for the covariant derivative of scalars in eq. (3.17) as well as for the derivative of the massive vectors as given in eq. (3.11). For instance, in the latter case we have that where we have used momentum conservation L = L 2 + L 3 . Therefore F (L) µν F µν(−L) terms in the action are U(1) invariant.
On the other hand, at a given fixed point the Abelian gauge group is enhanced to some non-Abelian gauge group G, and all fields in the theory, massless and massive, should organize into G irreducible representations. We have shown that, at a fixed point Φ 0 or close to it, after very massive states are integrated out (see discussion around (2.35), a well defined low energy effective gauge theory is obtained. The light modes that define this theory are the zero modes coming from 10D fields KK reductions plus modes inǦ(Φ 0 ) nc .
However, if we were to consider the other (very massive) modes, as the ones appearing in the mode summations we are dealing with, we expect to run into trouble. Indeed, generically, these massive modes will fill gauge multiplets that will contain modes associated to higher oscillator numbers. This appears as a limitation of our construction restricted to N = 0, 1 modes.
Indeed, assume that K = (k L , k R ) with k R = 0, k 2 L = 2 encode the charged gauge vector boson modes A (K) µ of the group G and let us call the currents associated to these vectors J (K) . From a string theory analysis, if we start with some massive field Φ(x) (L) with mass m (L) and 1 2 L 2 = 1, its OPE with the current should lead to another field Φ(x) (S) with S = (s L , s R ) = L + K and the same mass, in order to be part of the same multiplet. Thus, JHEP03(2019)012 by using (A.2) and (A.7) we find that Φ(x) (S) mode should haveN s =N B +N F +Ẽ 0 = 0 and a left oscillator mode N s such that or, equivalently, 1 2 S 2 = 1 − N s . Therefore, even if we started with a field corresponding to zero oscillator number we conclude that other values must be generically included. Presumably full consistency would be attained if δ( 1 2 S 2 , 1 − N ) LMC is allowed in for all possible values of N . However, this would imply introducing higher spin fields, as expected form string theory.
Interestingly enough, it appears that consistency (at tree level) can be achieved up to first mass level, with N = 0, 1 as we are indeed considering here. We discuss this issue in the example below.
Finally recall that several consistency conditions are expected to be satisfied by physical states. For instance, physical massive vectors must satisfy ∂ µ A B µ = 0 , etc. In string theory such conditions arise from conformal invariance. Namely, physical fields must satisfy the adequate OPE with the stress energy tensor. It was shown in ref. [28], in the case of the bosonic string and for some specific fields, that these conditions can be understood from generalized diffeomorphism invariance. However, as mentioned above, when level matching conditions as 1 2 L 2 = 1 (or 1 2 L 2 = 0 for non zero modes) are considered our analysis points towards a modification of the generalized diffeomorphism algebra in order to incorporate the -product. Therefore consistency conditions expected from diffeomorphism invariance need further investigation in these cases.
In what follows we illustrate some of the issues discussed above in an explicit example for the torus case.

SU(3) example
We consider the 2-torus compactification case in order to provide a specific example of the issues presented above. The generalized momentum encoding KK and winding modes iš P = (P I , p 1 , p 2 ;p 1 ,p 2 ). At a generic moduli point Φ = (g, b, A) non zero momenta lead to massive states. The massless vectors arise from zero modes AÎ and lead to the generic group U(1) 2 R × U(1) 16 gauge group. Enhancements will occur at specific moduli. As an example, let us look at moduli point Φ ≡ (g, b, 0) with turned off Wilson lines. The set of momenta that would lead to massless states at this pointrecall (2.22)-isǦ(Φ) 480 = {P ≡ (α; 0, 0; 0, 0)} where α ≡ (±1, ±1, 0, . . .) are the SO(32) roots. Thus, when sliding to this moduli point these sates become massless and, together with the Cartan vectors should lead to an enhancement to U(1) 2 R × SO(32) gauge group. Actually, we see from (3.1) that f L−LI ≡ f α−αI = α I providing the right structure constants involving charged fields and Cartans. Moreover, the phases arising from the -product-see JHEP03(2019)012 for instance (3.11) for the field strength modes-now read from (3.9) and (3.2) is the SO(32) Cartan-Weyl metric for I ≥ J and zero otherwise. These values exactly correspond to structure constants involving three charged fields (see for instance [1]). We conclude that S eff (Φ ≡ (g, b, 0)) corresponds to a well defined U(1) 2 R × SO(32) gauge theory.
Moduli points Φ ≡ (g, b, 0) can lead to further enhancements for specific values of g and b on the compactification 2-torus. It proves convenient to rewrite this point as Φ = (T, U, 0) where U = U 1 + iU 2 and T = T 1 + iT 2 are the complex and the Kähler structure of the torus respectively. They are defined in terms of the metric and b field as , 0) which is obtained from the SU(3) Cartan matrix and b field whereas at Φ 1 = (i, i, 0), associated to g mn = 2 0 0 2 (4.15) and b = 0, an enhancement to (SU(2) × SU(2)) L group occurs. At the SU(3) point (some basic information and notation is presented in appendix C), Left and Right momenta (A.2) become (4.17) whereas at SU(2) × SU(2) we have where we have set α = 1.
Having discussed the low energy effective action arising from action (2.29) at different moduli points, we propose to explore the contributions from massive states. Let us concentrate in the Φ 0 point.
For instance, the massive state l L = − 2 3 , − 4 3 , not contributing to the low energy theory, must now be considered. Interestingly enough, this state corresponds to the lowest weight of the symmetric 6 representation (whereas 2 3 , 4 3 corresponds to the highest weight of6). Indeed, it is easy to see that when shifting withP 0 vectors we obtain the mode vectors 3 ) ). We notice that all states have l 2 R = 2 3 but l 2 L = 8 3 for s = 1, 3, 6 whereas l 2 L = 2 3 for s = 2, 4, 5. Indeed, they satisfy with N s = 0 for s = 1, 3, 6 and N s = 1 for s = 2, 4, 5. Moreover, for these values of N s , all states have the same mass α m 2 = 4 3 , as it must be if they all belong to the same multiplet. Thus we conclude that states with oscillator numbers N = 1, 0 must be mixed in order to build up the 6 massive symmetric representation. These results are collected in table 3.
These states, even though they are very massive, are indeed present in our construction. As an illustration let us consider the massive scalar fields with mass α m 2 = 4 3 . The N = 0 states correspond to the modes MJ (x) (Λs,q R ) with r = 1, 3, 6 in the GKK expansion of MJ (x, Y) whereas states with N = 1, MJ m (x) (Λs,q R ) , with s = 2, 4, 5, are contained in MJ m (x, Y) expansion. It is worth noticing that in the N = 1 case there are two states

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Before presenting some explicit examples recall that a well defined covariant derivative on fields Φ s in this multiplet must read where T α l and T I are the matrices corresponding to SU(3) charged and Cartan generators, respectively, in the 6 representation. They are collected (in Cartan-Weyl basis) in appendix C. Let us consider the derivative (4.23) for the state M where we have used that Λ 1 = 4 3 α 1 + 2 3 α 2 = √ 2, 2 3 . Also we denotef Λsα j Λr =f sα j r . By using thatf 1α 1 2 =f 1α 3 4 = 1 and by defining this derivative can be recast in the form (4.25) with (T α 1 ) 12 = (T α 3 ) 14 = √ 2 and (T 1 ) 11 = √ 2, (T 2 ) 11 = 2 3 in exact correspondence with (C.5). Nevertheless, we should check that these definitions are consistent for the six states. Consider, for instance, the derivative of M (Λ 2 ) iJ field. We have noticed above that this field appears contracted with root α 1 in order to define the field Φ 2J with the correct transformation properties. This indicates that we must actually compute the derivative of Φ 2J . Thus, by projecting in (3.19) we find Interestingly enough we see that (T 1 ) 22 = 0, (T 2 ) 22 = 2 3 as expected from (C.5). Also, sincef 2(−α 1 )1 = −f 2α 3 5 =f 2α 2 4 =f 2α 1 3 = 1, consistency requires the extra definitions in order to have (T −α 1 ) 21 = √ 2 and (T α 2 ) 24 = (T α 3 ) 25 = 1 (see (C.5)). However, we have already defined Φ 4J in (4.27), so the only way to obtain a consistent description is to have The α l appearing in the contraction with M The physical degrees of freedom (DOF) conditions above can be interpreted from different perspectives. From a string theory point of view this requirement arises from conformal invariance. Namely, by looking at the OPE of the stress energy tensor with the vertex operators associated to different N = 1 modes above (r = 2, 4, 5) an anomalous term For massless modes, corresponding to L = 0, we recover the expected gauge conditions for the g µν , the B field contained in H µν (x) (0) and the gauge vectors A mµ ≡ H µn (x). However, for massive modes, a consistency requirements ∂ µ H µn (x) (L) = 0 is needed for a Proca field to have the right number of degrees of freedom. Similarly ∂ µ H µν (x) (L) = 0 for massive symmetric and anti-symmetric tensors. We conclude that which correspond to physical state conditions (4.31) and (4.33) respectively.

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Finally let us provide a third way of looking at physical DOF conditions. Given the two N = 1 fields, m = 1, 2 we have seen that we can combine them into two (orthogonal) linear independent combinations. For instance, in the case of M and correspond to weights ±Λ s with s = 2, 4, 5 and U(1) 2 R charges q R = ± − 2 3 , − 1 3 as the vector and scalars states discussed above.
With this observation in mind, the decoupling can be understood (see [28] for a related discussion in a DFT context) as follows: the scalars Λ m s M (Λs) are "eaten" by the corresponding vector boson to become massive. At the same time a massive graviton mode g (Λs) µν "eats" the vector boson to become a massive graviton with the correct degrees of freedom. Indeed, this can be explicitly shown by following the construction in ref. [28]. Namely, we can write write the massive vector meson as such that the Λ s · A (Λs) µ projection is "eaten" by the massive graviton (4.38) and the remaining physical states A s ∝ α l .A (Λs) mµ , with α l · Λ s = 0 satisfy ∂ µ A µs = 0. Moreover, by noticing that the weights in the fundamental representation of SU(3) correspond to modes Λ s ≡ (0, 1), (1, −1), (0, −1), we see that gravitons organize into multiplets of (3) (− 2 3 ,− 1 3 ) (and3 2 3 , 1 3 ) of the gauge group U(1) 2 R × SU(3). Let us close this section by stressing that the action (2.29) appears to contain very non trivial information even for states with masses of the order of the string mass. This is what the analysis of the covariant derivative of the massive symmetric representation in the above example indicates. Again, going to higher massive states would require the introduction of N > 1 and call for further investigation.

Summary and outlook
A striking and distinctive feature of string compactifications is that, at certain values of the compactification background -namely a point in moduli space-compact momenta and winding modes can combine to generate new (let us say n c ) massless vector bosons leading to an enhancement of the gauge symmetry group. Different enhancement can occur for other values of moduli and, generically, for other values of winding and momenta. In the notation presented above (2.22) we would say that, for a given number of compact JHEP03(2019)012 dimensions r, several setsǦ(Φ i ) n i c of generalized momentaP could exist. These lead to enhancement at moduli point Φ i where n i c vector bosons and scalars become massless. The structure gets richer for lower space-time dimensions.
We have shown that the heterotic low energy effective theory at each Φ i is obtained by considering fields associated toP ∈Ǧ(Φ i ) n i c modes and the zero modes arising from fields in the gravity sector whereas all other, very massive modes, are integrated out. Slight displacements from Φ i can be interpreted as a Higgs mechanism. Actually, when moving along moduli space some (or all) of these fields become massive whereas other fields become lighter at a different point. Therefore, a moduli dependent description able to account for these different enhancements implies handling an infinite number of fields. In this work we were able to identify some guiding lines towards this description, which is encoded in a moduli dependent effective action where a non-commutative -product plays a central role.
The proposed action, written in d space time dimensions, contains a generically infinite number of fields labeled by allowed momenta and winding modes. In principle, this action could have been obtained by carefully looking at string 3-point amplitudes of vertex operators associated with these modes. We have shown that these infinite fields in d dimensions can be understood as modes of a GKK expansion in the internal double torus and heterotic coordinates Y ≡ (y I , y m L , y m R ), providing an uplifting to higher dimensions. In this sense the action can be seen as a Kaluza-Klein inspired rewriting of a double field theory (see for instance ref. [32]), where coordinates are split into space-time coordinates (that could be formally doubled) and internal double coordinates. However, once compact coordinates come into play we noticed that a -product that introduces a non-commutativity in the target compact space is called for. Indeed, it is this non-commutativity that leads to the adequate factors to reproduce the structure constants. As we have shown in an example, this non-commutativity also appears to have the right features to reproduce the generator matrix elements in higher order representations where massive states live, as required by gauge invariance. It would be interesting to trace the origin of this product for the heterotic string case [33] back. In the context of bosonic string it was shown in [22,23] to be associated to non-commutativity of string coordinate zero modes.
An interesting result of the construction is that, close to a given enhancement point Φ 0 , by keeping just the n c slightly massive fields, the Higgs mechanism can be cast in terms off moduli dependent "structure like constants" that become the enhanced group structure constants at Φ 0 . This description provides a field theory stringy version of the gauge symmetry breaking-enhancement mechanism. This fact was already addressed in the context of DFT in [5,7] where it was shown that constantsf (Φ) can be interpreted as DFT Scherk -Schwarz [16,[29][30][31] compactifications generalized fluxes. These fluxes can be read from the DFT generalized diffeomorphism algebra. Actually, it is worth noticing that these fluxes were explicitly constructed from a generalized frame only in the circle case where a SU(2) enhancing at the self dual radio R =R = √ α occurs [4][5][6][7]. Interestingly enough, the SU(2) case is the only situation where the -product is not needed (essentially due to the absence of a b field). Difficulties in going beyond this case were mentioned in [5,6]. In refs. [6,8] a connection among these difficulties and vertex operators cocycle factors was suggested. The non-commutative product could provide a solution for this problem since JHEP03(2019)012 the appears as a manifestation of the cocycle factors in the DFT context. Let us stress that the -product is not needed, at third order in fluctuations, if fields satisfy L 2 = 0 level matching condition. This is why it did not manifest in original DFT constructions but would be required in a DFT formulation including four (or higher) order terms in the fields, where cocycle factors would be required, as it stressed in [15].
Actually, the problem already arises at third order when the Lie algebra of three charged fields with 1 2 L 2 = 1 LMC is considered, which is just the situation where theproduct phase is relevant. Moreover, since -product is providing cocycle factors, four order terms (or higher) could be consistently considered in DFT. In fact, we have shown that this appears to be the case in a partial computation of the fourth order scalar potential.
As mentioned above, a modified version of generalized diffeomorphisms is called for to handle these cases. The detailed construction is left for future investigation.
In our construction we started by proposing mode expansions restricted by the level matching constraint 1 2 L 2 = 1 (corresponding to N = 0 oscillators) necessary to contain massless vectors at the enhancement point. Even if it effectively interpolates among different enhancement points, we stressed that new ingredients must be incorporated. In particular, at first mass level, we noticed that for massive states to organize into multiplets of the enhanced group G, N = 1 oscillator number is also required. Since we had already included the N = 1 case, to tackle the gravity sector, we showed in an example that indeed massive vector and scalar states nicely fill G multiplets for first massive level. This happens to be the case also for gravity sector massive modes. However, for higher masses, other oscillator numbers are expected (this was was noticed in [28]). Namely, if we consider next to first massive level, in order to complete a G multiplet, a level matching condition with N > 1 is required. 13 We see that a simple gauge symmetry consistency check points towards the necessity of including massive higher spin fields and higher derivative terms in the action (see [34,35] for a discussion from another perspective), as is in fact expected from string theory. Let us stress that gauge invariance underscores the limitations of the construction but at the same time it is a guide for consistent extensions. Indeed, gauge invariance provides a tool to systematically include higher spin modes and α corrections by looking for consistency all the way from the very first massive levels up to the highest ones.
Throughout our construction we have made intensive use of DFT tools. In particular, before mode expanding, all fields are expressed in terms of higher dimensional coordinates. However, a fully higher dimensional version is still lacking in the sense that fields are written here in terms of space time d dimensional indices. Formally it appears rather straightforward. On the one hand, the new fields we are introducing here associated to N = 0, can be cast in terms of a D dimensional "charged vector" field Y)) and, on the other hand, the sectors originating in the generalized metric in 10-dimensions were already addressed in [28] (up to a third order expansion) in terms of a generalized metric. However, it appears that the latter must be modified by the presence of the new fields, as required by gauge invariance. Moreover, the form of generalized diffeomorphism and -product should be understood.

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Finally let us mention that even if we have restricted our analysis to the bosonic sector of the heterotic string, the inclusion of fermions could also be addressed by invoking supersymmetry, generalizing the discussion in [7] (see also [36][37][38]) where "will-be massless fermions at a fixed point", specifically for modes inǦ nc , were considered. From a duality invariant field theory point of view, an uplift including fermions would require an analysis from an Extended Field Theory (EFT) [39,40] in order to include magnetic modes. The recent work in in ref. [41] might be helpful in this direction.

A Some heterotic string basics
We summarize here some string theory ingredients (that can be found in string books) needed in the body of the article. We mainly concentrate in the SO (32)  where g mn , B mn are internal metric and antisymmetric tensor components, A m are Wilson lines and p n andp n are integers corresponding to KK momenta and windings, respectively. P I are Spin(32) weight components. More schematically, by defining the vectorP = (P I , p n ,p n ) and L = (L I L , l m L , l m R ) we can write Notice that R(Φ) encodes the dependence on moduli. The mass formulas for string states are (we mainly use the notation in [3]) where N = N B ,N =N B +N F +Ē 0 where N B ,N B are the bosonic L and R-oscillator numbers,N F is the R fermion oscillator number andĒ 0 = − 1 2 (0) for NS (R) sector. The level matching condition is 1 2 m 2 L − 1 2 m 2 R = 0 or, in terms of above notation 1 2 L 2 =p.p + 1 2 P 2 = (1 − N +N ). (A.7) In our discussion we restrict toN B = 0, N F = 1 2 , namelyN = 0. The "charged vectors" sector, corresponds to N = 0, i.e. L 2 = 2. Massless vectors are a particular case with 1 2 l 2 L = 1, l R = 0. 14 As is well known, there are 10 − d + 16 Left gauge bosons corresponding to 16 Cartan generators ∂ z Y Iψµ of the original gauge algebra as well as 10 − d KK Left gauge bosons coming from a Left combination of the metric and antisymmetric field ∂ z Y mψµ . The 10 − d Right combinations ∂ z X µψm with m = 1, . . . 10 − d generate the Right Abelian group. These states have l R = 0 and l L = 0, with vanishing winding and KK momenta.

B The -product
A -product, was proposed in [22,23] in order to incorporate, in a "Double Field theory" description, information about bosonic string vertex cocycle factors. IfP ≡ (p m ,p m ) is an O(n, n) vector encoding information about winding numbersp m and Kaluza-Klein (KK) compact momenta p m , then for two fields depending on the compact double coordinatě Y ≡ (y m ,ỹ m ) their proposed -product reads (φ 1 φ 2 )(x, Y) = is the Fourier mode of the star product. It is straightforward to show that the -product is indeed associative. Namely 3 (x)e i(L+P 3 ).Y = P 1 ,P 2 ,P 3 e iπ(p 1 +p 2 )·p 3 e iπp 1 ·p 2 φ (P 1 ) 3 (x)e i(P 1 +P 2 +P 3 ).Y = P 1 ,P 2 ,P 3 e iπp 1 ·(p 2 +p 3 ) e iπp 2 ·p 3 φ Interestingly enough, the appearance of the phases can be traced back as a noncommutativity of the string compact coordinates zero modes (see [24]).
If the sum over φ (P i ) i modes is constrained by LMC's, namely to modes satisfying δ( 1 2 P 2 i , 1 − N i ) then the same proof goes through if we define Here we just extend this product to account for heterotic string degrees of freedom in a O(r L , r) context. Actually, since it is possible to interpret the heterotic string momenta P I as originating in a 16 dimensional torus [27] with some winding and momenta (p I , p I ) (with I = 1, . . . 16) we can generalize above expression by including a phase that contains not only the compactified winding and momenta but also the gauge ones. More concretely, P I L , P I R can be computed using similar expressions as (A.2) above (no Wilson lines) but by imposing P I R = 0. Then, P I ≡ P I L root vectors are obtained with, G IJ the Cartan Weyl metric of Spin(32) and B IJ = G IJ = −B JI for I > J. It is possible to check then that for two vectors P 1 , P 2 we havep I 1 p 2I = 1 2 P I 1 E IJ P 2J where E IJ = G IJ + B IJ . Therefore, for the heterotic string we would have (see (A.1) above) L = (l L , l R ) ≡ (L I L , l m L , l m R ) and Y = (y l , y R ) ≡ (y I , y m L , y m R ) and using that we recover the expression in (3.4). Notice that the phase (P 1 , P 2 ) = e iπ 1 2 P 1 EP 2 introduces a notion of ordering for Spin(32) roots. For two adjacent roots in the corresponding Dynkin diagram E IJ = −1 for I > J and vanishes otherwise. This provides an adequate representation of structure constants for Spin(32) charged operator algebra. Namely [E P 1 , E P 2 ] = (P 1 , P 2 ) E P 3 (see e.g. the construction in [1]).

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The same reasoning holds for the full enhanced group. At the enhancement point Φ 0 withP ∈Ǧ nc (Φ 0 ), p r = 0, lÎ become the roots of the gauge group and from equations (A.2) above we can express windings and momenta in terms of the l L such that l 1 ·l 2 = p 1mp