Probing the String Winding Sector

We probe a slice of the massive winding sector of bosonic string theory from toroidal compactifications of Double Field Theory (DFT). This string subsector corresponds to states containing one left and one right moving oscillators. We perform a generalized Kaluza Klein compactification of DFT on generic $2n$-dimensional toroidal constant backgrounds and show that, up to third order in fluctuations, the theory coincides with the corresponding effective theory of the bosonic string compactified on $n$-dimensional toroidal constant backgrounds, obtained from three-point amplitudes. The comparison between both theories is facilitated by noticing that generalized diffeomorphisms in DFT allow to fix generalized harmonic gauge conditions that help in identifying the physical degrees of freedom. These conditions manifest as conformal anomaly cancellation requirements on the string theory side. The explicit expression for the gauge invariant effective action containing the physical massless sector (gravity+antisymmetric+gauge+ scalar fields) coupled to towers of generalized Kaluza Klein massive states (corresponding to compact momentum and winding modes) is found. The action acquires a very compact form when written in terms of fields carrying $O(n,n)$ indices, and is explicitly T-duality invariant. The global algebra associated to the generalized Kaluza Klein compactification is discussed.


Introduction
Many amazing properties and symmetries of string theory can be tracked down to the extended nature of the strings. In particular, the presence of an antisymmetric tensor Bμν in the spectrum is expected because, being one dimensional, the string directly couples to it. Actually, a distinctive feature of all string theories is that, besides the metric gμν, the gravitational sector also includes the Kalb-Ramond field Bμν and a scalar dilaton φ, Double Field Theory (DFT) aims at incorporating these stringy features, and in particular information about winding, into a field theory [1]- [5]. Inspired by string compactification on tori, DFT is formulated on a doubled configuration space, with coor- Even if the original motivation is lost when choosing the section condition, DFT still provides an interesting tool for understanding underlying symmetries of string theory.
In particular, it shares the basic features of Generalized Complex Geometry [6,7] (both frameworks are based on an ordinary, undoubled, manifold) and the 2D-dimensional tangent bundle of the doubled space is an extension of the D-dimensional tangent-bundle of ordinary spacetime by its cotangent bundle, with Bμν parametrizing the structure of the fibration. Actually, some distinctive ingredients of string theory, like α ′ corrections, have been recently incorporated in these formulations [8,9].
Other solutions to the constraint equations are provided by generalized Scherk-Schwarz compactifications [10,11]. It is worth noticing that Scherk-Schwarz compactifications of DFT give rise to all the gaugings of gauged supergravity theories (not obtainable from compactifications of low energy effective supergravities) allowing for a geometric interpretation of all of them [12], albeit in a double space. In this framework, the doubled coordinates enter in a very particular way through the twist matrix, which gives rise to the constant gaugings.
While winding modes are essential for T-duality, they are not truly present and their role is not evident in these approaches. Clearly, to probe the winding sector requires to relax the section condition. Moreover, in toroidal string compactifications, winding states are massive for generic tori. Therefore, understanding the role of winding modes implies facing the massive sector of the theory and consequently dealing with an infinite number of physical states, with different spins and mass scales. However, at specific points of the compact space, some winding states become massless and an effective theory containing only massless states and enhanced gauge symmetry emerges. This scenario appears particularly suitable to identify the explicit part played by windings and a DFT description of the massless winding sector of bosonic string theory compactified on a circle was suggested in [13].
In the present work we propose a way to probe a slice of the massive winding sector of bosonic string theory in an organized fashion. Namely, we consider compactifications of DFT on generic double tori 1 . The generalized dilaton and metric fields of DFT contain bosonic string states constructed with one left and one right moving oscillators, and therefore we concentrate on this sector of the string spectrum. Even if the bosonic string is ill defined, due to the presence of tachyons, we will use it as a reference since string computations are simpler to deal with. However, for the sector we are interested in here, similar reasoning would apply for the heterotic or Type IIB string theories.
The comparison between DFT and string theory is done by expanding the generalized fields around a generic toroidal background with constant dilaton and two-form field 2 .
We then expand the DFT action up to third order in fluctuations around the constant background and contrast the result with the corresponding string theory three point amplitudes.
As a first outcome of the calculations, we find that both the DFT and string spectra containing Kaluza-Klein (KK) momenta and windings coincide as long as a "level matching" constraint (LMC) is imposed on the mode expansion of the DFT fields. Furthermore, 1 See [14] for previous work on this subject. 2 Expansions around generic backgrounds have been performed in [15] we show that the compactified DFT action (up to this order in fluctuations) is invariant under generalized gauge transformations generated by a generalized Lie derivative, if the LMC is imposed. This gauge invariance allows to choose a generalized harmonic gauge which provides a convenient "gauge fixing", as it imposes conditions on massless and massive states that can be easily identified with conformal anomaly cancellation conditions on the vertex operators creating these states in string theory. Using these conditions, we then show that cubic vertices in the DFT action can be reproduced by three point amplitudes in string theory. Actually, DFT appears to provide a straightforward way of organizing these amplitudes in an effective T-duality invariant field theory. We obtain an explicit expression for the gauge invariant effective action containing the physical massless sector (gravity+antisymmetric+gauge+scalar fields) coupled to towers of generalized Kaluza Klein (GKK) massive states (corresponding to compact momentum and winding modes).
The article is organized as follows. In Section 2 we present some basic introduction to DFT. We write the DFT action in a generalized Einstein frame and fix the gauge freedom in terms of generalized harmonic coordinates. In Section 3 we perform the expansion of the generalized fields in fluctuations around a constant generic background, we discuss the gauge fixing conditions and carry out a GKK decomposition of the fields. In Section 4 we consider the mode expansion of the fields on a double torus with constant background fields. We identify massless and massive states and examine the generalized harmonic gauge equations to distinguish physical states and Goldstone like states. The analysis of the cubic interaction terms in the effective action and the identification of unbroken symmetries is also performed. Finally the resulting gauge invariant action in d lower dimensions is presented. Section 5 is devoted to string theory amplitudes on toroidal backgrounds. The equivalence between conformal anomaly cancellation conditions on the string vertex operators and the generalized harmonic gauge conditions on the DFT fields is determined. We compute three point string scattering amplitudes of massless and massive states and show the complete agreement with the expansions in DFT. The comparison involves a huge number of terms and so it is performed with the help of a computer (cadabra program [16]). The simple example of circle compactification is worked out explicitly and the manifestly T-duality invariant effective action is also presented. A discussion on the limitations and possible extensions of this work and a brief outlook are contained in the concluding remarks in Section 6.

Double Field Theory basics
In this section we briefly review some of the basic features of DFT that are needed in our discussion.
The generalized tensors transform under generalized diffeomorphisms as The natural SO(D, D) metric is invariant under the above generalized transformations. It can be decomposed into a positivedefinite and a negative-definite metric, η| C ± , acting on each of the two D-dimensional orthogonal subspaces of the doubled space E = C + ⊕ C − , that are generated by the coordinates X M ± = xμ ±xμ. Making use of η| C ± , a positive-definite metric can be defined on E with Under O(D, D) transformations h M P , X → hX and the fields change as Upper and lower indices are lowered and raised with η MN and its inverse η MN , respectively.
It is sometimes useful to express the metric H MN in terms of a vielbein where gμν = e aμ s ab e bν , and s ab is the D dimensional Minkowski metric. A, B, · · · indices are lowered and raised with the flat SO(D, D) metric defined as and its inverse, respectively, which numerically coincide with (2.2).
From the transformation law (2.1), the generalized metric transforms as In terms of H MN , and keeping up to two derivatives, the action of DFT in the 2Ddimensional space E can be expressed as 3 where the generalized Ricci scalar is given by [4] In the original frame formulation of DFT by Siegel [1,2] the action includes extra terms that are not contained in (2.11). Up to total derivatives those can be recast as [11] ∆S = 1 In the Appendix we show that these terms vanish once the level matching condition described below, is imposed, and therefore we do not consider them in this work.
is an O(D, D) scalar and G DF T will be defined below 4 .
Comparison with string theory, as we re-discuss in more detail below, requires the level matching condition (LMC) 14) where N andN are the left and right oscillator numbers of the string and the dots stand for fields or gauge parameters. Given that g, B and φ correspond to N =N = 1, we would expect N −N = 0. However, in a compact space this difference could be a non-vanishing integer.
Even though this is a key ingredient of symmetry enhancing at certain compactification radii (see [13]), we will only consider states satisfying N −N = 0 in the main body of this article.
Introducing the 2D-dimensional momentum vector P M = (pμ, pμ), generated by the partial derivatives −i(∂μ, ∂μ) acting on the corresponding field, the constraint reads for H MN and d. In general, this constraint is not sufficient to ensure consistency. For instance, the product of fields generically does not satisfy it and the generalized transformations (2.1) fail to close.
This failure can be expected from string theory. Namely, many other terms (actually infinite) are expected to complete the effective action, containing higher derivatives but also higher spin fields. Hopefully, in the full action variations could compensate among different terms and the algebra would close. But in the truncated theory involving only massless fields with N =N = 1 in the non compact case, consistency constraints are necessary. One solution of these constraints is the so-called section condition where the dots stand for products of fields or gauge parameters. It implies that half of the coordinates drop from the theory. These coordinates can be chosen to be the dual coordinates xμ. This choice is named gravity frame since in this case the action (2.11) simply reduces to eq. (1.1) when H MN is parametrized as in (2.3) and G DF T ≡ 2κ 2 d Dx .
The section condition is sufficient to satisfy the closure constraints, but there are more general solutions [18,11] when there is a compact sector. It is important to stress that (2.11) describes more physical degrees of freedom than the standard D-dimensional action (1.1) for g, B and φ. Indeed, by introducing coordinatesxμ, and their corresponding partial derivatives 4 The overall constant G DF T was introduced in [17].
∂μ , fields can carry momentum along these directions and the backgrounds can also depend on these coordinates. Such dependence is not an artifact of DFT: backgrounds with non-trivial dependence on the coordinatesxμ cannot be described in terms of D-dimensional gravity, but are however expected to be consistent solutions of string theory. In particular, such backgrounds lead upon compactification to fully consistent effective gauged gravities with momenta along the internal coordinates associated to winding excitations. DFT contains more degrees of freedom than D-dimensional gravity, and in particular, it allows to compute observables and describe settings that cannot be accounted for in standard D-dimensional theories.
In what follows we will compactify DFT on generic tori with constant background fields and fluctuations around them. The constraints to be used will be extracted from comparison with string theory results. In our computations, the section condition must be imposed in the spacetime sector but only the LMC constraint is required in the toroidal compact space. This appears to be consistent if fluctuations are considered only up to third order. When going to higher orders, the failure of the gauge algebra to close should be interpreted as an indication that new degrees of freedom must be included. A brief discussion on this issue is offered in the concluding remarks. under which a tensor with conformal weight ∆ Φ transforms as

Einstein frame and harmonic coordinates
We list the conformal weights of the tensors introduced in the previous section in table 1.
Taking Ω = d and integrating by parts, we can express (2.11) in the Einstein frame as This action (2.20) behaves similarly to the more familiar Einstein-Hilbert action in many aspects. In particular, the equations of motion are greatly simplified by taking a harmonic coordinate condition to fix the gauge freedom under generalized diffeomorphisms. This can be achieved by requiring the coordinates X R to be solutions of the Laplacian equation which amounts to the gauge fixing condition 5 Making use of these conditions and integrating by parts, the action (2.11) can be expressed in harmonic coordinates in a particularly compact form

25)
5 It can be shown that, in terms of the generalized connection of [1,2,19], this is equivalent to requiring or, in Einstein-like frame, It is also interesting to express the gauge fixing condition in terms of g, B and φ. For standard D-dimensional gravity backgrounds withpμ = 0, one may easily check that its components reduce to gμν∂ν Bλμ = 0 .
In particular, for vanishing dilaton φ = 0, the first equation is the usual harmonic gauge fixing condition of General Relativity. More generally, for generic DFT backgrounds, the gauge fixing conditions for B, g and φ read

Perturbative DFT
The physical content of a quantum field theory can be recast in terms of its S-matrix elements, that are usually computed perturbatively. In the particular case of General Relativity, perturbative computations are however specially complex due to the huge number of vertices, rendering most of the brute force computations of scattering amplitudes infeasible. Fortunately, the field theory limit of Kawai-Lewellen-Tye (KLT) relations [20] allows to express gravity amplitudes in terms of two copies of gluon amplitudes, which are much simpler to compute. In particular, starting from gluon amplitudes and using KLT relations, it has been possible to construct a Lagrangian for gravity [21]. The resulting Lagrangian is particularly simple and is related to the usual Einstein-Hilbert action by non-linear redefinitions and gauge fixing similar to those used in [22].
For simplicity, we consider H MN and d to be constant. Due to the presence of two metrics, namely H MN and η MN , (3.1) can be inverted in two different ways: by making use of η MN or by using the geometric series for matrices. We thus obtain for the inverse where we have introduced the short-hand notation AṀ ≡ H MN A N , AṀ ≡ H MN A N , and it is useful to note that, up to first order, The single fieldĥ MN ≡ η MP η MQĥ PQ therefore encodes an infinite set of operators when expressed in terms of the background metric H MN .
Note also that by construction may easily check the following relation For comparison with string theory results, it proves convenient to look at fluctuations in the so-called modified Einstein frame, namely the Einstein frame discussed above with the vacuum value of the generalized dilaton e −2d extracted out 6 . Thus, the generalized metric is, up to first

Expansion of DFT in fluctuations
Following the discussion above, by using (3.5) we expand the DFT harmonic gauge fixed action (2.26) into background and quantum fluctuations. We get, up to third order in fluctuations, (3.6) 6 In what follows Einstein frame means modified Einstein frame.
Recall that in terms of fields, the fluctuationsĥ MN =ĥ (1)MN +ĥ (2)MN + . . . contain contributions from higher orders. In particular, terms quadratic inĥ MN could give third order interaction terms. However, this is not the case. Actually, integrating by parts the term (2) ∂ Nĥ(1)KL , one gets the equations of motion (see (4.8) below), and so this cubic term vanishes on shell. The same conclusion holds for the second term. Therefore, the third order terms in the action involve only the first order fluctuations of the generalized fields, and we finally have the Lagrangian (3.6) withĥ KL =ĥ (1)KL .
Before compactification, in a flat background and to first order in fluctuations, gμν = ημν + hμν , Bμν = bμν, we havê Then from the second order terms in fluctuations and imposing the strong constraint in the gravity frame (namely, dropping the dependence on thexμ coordinates), we recover the quadratic terms in the action (1.1) in the de Donder gauge [23,25]. Actually, in the string frame we get Transforming this action into momentum space, we obtain the propagators for h, b andφ The first lesson to be drawn from this calculation is that the strong constraint must be imposed on the space-time coordinates in order to recover ordinary gravity theories, as expected.

Generalized Kaluza-Klein compactification
The generalized Kaluza-Klein (GKK) decomposition of the generalized metric reads where now the M, N indices split into spacetime µ, ν, · · · indices taking the values 0, · · · , d − 1, and internal doubled indices M, N, · · · = 1, · · · , 2(D − d). We have introduced the combination A N µ denote the vectors and M M N is the scalar matrix defined below. In terms of components, the constant generalized background metric reads now where m, n, · · · = 1, · · · , D − d. The fluctuations up to first order arê The matrix h M N encoding the scalar field content reads where h nl , b nl are the scalar fields derived from the higher dimensional graviton and antisymmetric fields, respectively.
From the definition of the generalized dilaton (2.13) and recalling that d = d +d, we have In Einstein frame, the only fluctuations that are modified arễ The harmonic gauge conditions in the Einstein frame (∂ MH MN = 0) become, in terms of fluctuations, where we have used the strong constraint in the spacetime sector. Therefore, when specifying values for the index N , we have We will discuss the link between this set of equations and the vanishing of conformal anomalies in string theory in section (5) below.

Toroidal compactification
We consider the mode expansion of fields on an internal 2n-dimensional double torus with constant background (metric, dilaton and antisymmetric fields) turned on. It corresponds to a compactification on 2(D − d) = 2n circles, which are generically non-orthogonal since the background metric is in general non-diagonal 7 .
The internal coordinates Y M = (ỹ m , y m ) have periodicitỹ where R (m) andR (m) = α ′ R (m) −1 denote the radii of the m-th cycle and its dual, respectively.
The internal momenta are encoded in the O(n, n) vector P M of components

2)
n m and w m being the integer momentum and winding numbers.
On the torus background, the non-trivial identifications (4.1) are only preserved by O(n, n) transformations with integer-valued matrix entries. Thus, the O(n, n, R) symmetry is broken to the discrete O(n, n, Z) group.
The mode expansion of the generalized metric would be where the dependence on the dual space time coordinatesx µ has been dropped. The expansion of the component fields is and similarly ford(x, Y), gauge parameters, etc.
The sum over P involves, in principle, all integer values of momenta and windings (n m , w m ).
Possible constraints are indicated with a prime on the sum. Also, since all the fields we are dealing with are real, we require Remember that we have dropped the field dependence on dual space-time coordinates, or in DFT words, we have imposed the strong constraint in order to stay in the gravity frame. This means that there will be a 1 2κ 2 overall factor in the action, where κ is now the gravitational constant in d + 2n dimensions. In terms of the DFT coupling above it would formally read Due to the contributions from both, a circle and its dual, the usual R dependent volume factor of dimensional reduction is not present here, and instead an α ′ factor is left, namely Furthermore, we use that We will see below that the dependence on radii shows up when vector fields are redefined in order to have integer U (1) charges. Also a scaling factor appears through the expectation value of the generalized dilaton e −2d containing both the determinant of the background metricḠ and dilatonφ fields.

Quadratic terms and masses
We first concentrate on the quadratic terms in the action. Inserting the GKK expansion in the first line of the Lagrangian (3.6), we obtain where we have redefinedd → (D − 2) 1/2d , and by using (3.15), The equations of motion read Interestingly enough, these expressions not only reproduce the propagators for the gravity multiplet 8 but they also contain the propagators for GKK states. In particular, we can identify the mass squared of the GKK (P) modes 9 as This is exactly the mass squared of string states on generic toroidal backgrounds for N + N − 2 = 0. We expect this condition is satisfied since we started with N =N = 1. However, the string states also satisfy the LMC, namely Therefore, it appears that in order to recover the string theory results, we must consider the following constrained GKK expansion and similarly ford(x, Y).
Let us look at the transformation of the compactified action under the generalized diffemorphisms (2.1). From the discussion above, we know that this variation should be proportional to terms that vanish if the strong constraint ∂ P ⊗ ∂ P = 0 is imposed. Moreover, since the space-time part already satisfies it, the transformation must be proportional to ∂ P ⊗ ∂ P = 0, where now P labels the internal compact coordinates. Since the variation is proportional to the gauge parameter, it can be written as ∂ P ξ M J P M , with J P M a product of generalized metric and dilaton fields with a ∂ P derivative acting on one of them. By mode expanding the generalized fields, these derivatives lead to a Q i P Q jP factor times a δ 2n ( i Q i ) requiring total momentum conservation. If up to third order terms in fluctuations are kept in the action, momentum conservation and level matching Q i 2 = 0 for each field (including ξ M ) leads to Q i · Q j = 0 and we conclude that the action, up to this order, is invariant under generalized diffeomorphisms.

Physical degrees of freedom
The mass formula (4.11) is generic and does not allow us to isolate physical states. For instance, h (P)µ M (x) seems to denote 2(D − d) massive vector states. However, we know that some of these vectors must be absorbed by the gravitational and two-form fields to become massive. Actually, the harmonic gauge condition allows to identify the physical degrees of freedom. In order to see this, first recall the expected physical fields in lower dimensions.
With n compact dimensions, we can write Starting with the metric in D dimensions, decomposing the indices into D − n spacetime and n internal indices, for massless states (corresponding to zero modes in the KK expansion) we f. for g µν , n vectors g µm leading to n(D − n − 2) d.o.f. and 1 2 n(n + 1) scalars g mn , consistent with the first equation. On the other hand, if the states are massive, we must decompose them as in the second equation, corresponding to a massive symmetric two-tensor, n − 1 massive vectors and 1 2 n(n − 1) scalars. We can understand this combination by interpreting that a scalar is eaten by a massless vector to become massive, leaving 1 2 n(n + 1) − n = 1 2 n(n − 1) scalars and n massive vectors with (D − n − 1) degrees of freedom. However, one of these vectors is eaten by the massless graviton to become massive, leaving a massive two-tensor with 1 2 f., and n − 1 massive vectors.
A similar computation can be done for the antisymmetric tensor. Namely, a massless two- The first equation leads to the familiar KK decomposition in terms of a massless two-tensor b µν , n massless vectors b µm and 1 2 n(n − 1) massless scalars b mn . For the massive case, a massless antisymmetric tensor eats a massless vector, leaving a massive antisymmetric tensor with 1 2 f. The n − 1 massless vectors left eat n − 1 scalars to become n − 1 massive vectors, leaving 1 2 n(n − 1) − (n − 1) = 1 2 (n − 2)(n − 1) massive scalars.
On the whole, a massive GKK level is characterized by the generalized momentum P, with P 2 = 0, and it contains a spin two symmetric tensor (which can be decomposed into a traceless tensor + trace), an antisymmetric tensor, 2(n − 1) vectors and n(n − 1) scalars, all mass degenerate with mass M 2 = PMP. Note that a non-equivalent level P ′ = hP will have the same mass if h is an O(n, n) transformation, namely h is a duality transformation. Recall that, in the n = 1 double circle case no extra massive vectors or scalars are present.
In (both spacetime and internal) momentum space, the generalized harmonic gauge conditions (3.21) for the modesĥ where we have used that −k 2 = M 2 is the (squared) mass of the states as given in (4.11).
This is an indication that there is a physical massive field satisfying ∂ µĥ ′(P) µN (x) = 0. The field combinations (Pĥ (P) ) N play the role of eaten Goldstone fields to provide the physical degrees of freedom. Let us analyze them in terms of component fields. Using (3.13), (4.15) can be decomposed into graviton, antisymmetric tensor and vector field polarization tensors as

Gravitons
The first equation in (4.16) can be recast as where we have used the third equation in (4.16). Thus, we have an effective symmetric tensor with polarizationh is constructed from the original graviton polarization tensor, one vector field (P·A) ν and a scalar field P · M ·h · M · P, as expected from the above counting of degrees of freedom.

Antisymmetric tensor
We can proceed similarly with the antisymmetric field. Namely, the second equation in (4.16) can be rewritten as and using the third equation in (4.16), the last term reads However, this term vanishes at first order 11 , and then we are left with an effective antisymmetric 11 In fact, this can be easily seen by rewriting the condition P 2 = 0. Namely = P 2 + 2P · M ·h · P + O h 2 and, therefore P · M ·h · P = 0 where the original polarization b (P) µν "eats" a vector (P·M·A (P) ) ν , in agreement with the discussion above.

Vectors
The third equation (4.16) directly tells us that there are massive vector polarizations Indeed, this appears to be the case. If P · A (P) selects some combination, then P · M · A (P) selects an independent one. Actually, M acts effectively by changing lower to upper indices (see (3.3)).
The physical states found above should be interpreted from the generalized gauge trans- In terms of GKK modes, the gauge transformed fields will bẽ M . (4.27) In order to fix the gauge parameters, we first impose the conditions ν . By also requiring that since this is the scalar absorbed by the graviton, this coefficient is fixed. Finally, the gauge transformations required to gauge away the non-physical fields read (4.31) By noticing that M · P · Λ (P)N = 0 and P · Λ (P)N = − 1 2M 2 P · M ·h (P) · M · P, and using the LMC (P 2 = 0), it is easy to check that Interestingly enough, since in the harmonic gauge then the physical vectors satisfy Moreover, it can also be checked that the physical fields M N are invariant under generic linearized diffeomorphisms δ ξ (P) = (δ ξ µ(P) , δξ(P) µ , δ Λ M (P) ) as given in (4.27). This is due to the fact that these combinations correspond to physical fields. The situation is analogous in electromagnetism where the electric and magnetic fields are a gauge invariant combination.
M N would be the physical combinations for the internal symmetries (symmetries associated to δ ξ (0) must still be fixed).
Finally let us discuss the splitting of the symmetric tensor into a traceless part and a trace contribution. Of course the splitting can be performed just by adding and subtracting the trace. Let us consider a trace contribution of the formh µ ), where we have used the freedom of including a diffeomorphism parameter χ ν and f d is a numerical factor (different for massive and massless states). The parameters χ ν are chosen such that k µ ǫ φ µν (k) = 0. For the massless modes, this leads to the requirement k µ χ (0)µ = −1, whereas for massive modes with M 2 = P·M·P, we find χ (P) µ = 1 2M 2 k µ . Therefore, the polarization tensor for the traceless symmetric graviton is with f d = 1 d−2 for the massless modes and f d = 1 d−1 for the massive ones. However, we still have the freedom to fix the traceh ′λ λ . A convenient choice is T r(h ′ ) = h ′λ λ = 4φ, where φ is the dilaton field, which amounts to settingd = 0 (see 3.15).
We notice that the choiced = 0 eliminates the lastd dependent term from the Lagrangian (3.6). However the dilaton part is now included in the previous terms due to the splitting Finally, the cubic order Lagrangian to be considered is where only the physical fields identified above must be considered.
Recall that, even if diffeomorphisms have been used in order to fix the physical degrees of freedom, the expression of the action in the harmonic gauge can still be used since these transformations, up to first order in the fields and on shell, do preserve the gauge. More Since the gauge parameters ξ M are already first order in the fields, we obtain (using LMC) The second term vanishes due to the e.o.m (see (4.8)) and it can be easily checked that ∂ M ξ M andH MN ∂ N ξ M are identically zero for the ξ M parameters found above. Thus, the harmonic gauge does not completely fix the gauge freedom, and we can still use the remaining symmetries to gauge away the Goldstone bosons.

Unbroken symmetries
The fact that physical fields can be defined by absorbing "Goldstone like fields" is associated to the spontaneous symmetry breaking by the background. This issue has been extensively discussed in the literature about KK compactification (see for instance [26,27,28]).
Actually, most of the generalized diffeomorphisms are spontaneously broken by the choice of vacuum, namely g µν = η µν , In fact, only the zero modes ξ where H and B M are spacetime three-form and two-form, respectively. We will see that fields in the Lagrangian do group into these combinations.

Cubic terms and effective action
Once the physical states have been identified, we proceed to consider the third order action 13 (4.37). By splitting the indices of the fluctuations into spacetime and internal components, the Lagrangian containing only physical fields, reads 12 The algebra of diffeomorphisms is discussed in the Appendix. 13 An alternative proposal of this action can be found in [14].
where we have included cubic interactions plus some higher order terms required by spacetime diffeomorphism and gauge invariances.
Here (4.42) and the derivatives are The Lagrangian (4.41) has a rather compact expression due to the explicit O(n, n) invariant setting. The fields here depend on both space time and internal coordinates and must still be mode expanded in generalized momenta, according to (4.13). Modes correspond to physical fields, in terms of which the contributions acquire a more familiar shape. Recall that, when acting on the field mode (P), −i∂ M → P M is just the charge operator.
The action contains both kinetic and cubic interaction terms of massless and massive fields.
Covariant derivatives and Chern-Simons terms in the antisymmetric tensor field strength appear as usual. For instance, the derivative D µ in (4.43) leads, when mode expanded and acting on a generic field Φ (P) (x), to the covariant derivative where P M is the electric charge with respect to the U (1) M gauge field A We know from (4.2) that fields charged under (A ) and therefore, in order to have integer charge, field redefinitions  provides a generalization of previous constructions (see for instance [29,30]) where KK compactifications of gravity were considered, in diverse phenomenological proposals.
For comparison with string theory amplitudes we will be interested in the on shell action.

String theory amplitudes
In this section we consider string theory with constant toroidal backgrounds G pn and B mp for the metric and antisymmetric tensor, respectively. We analyze the vertex operators creating On the other hand, consistency requirements on the full vertex operator, once the harmonic gauge was chosen, fix the physical polarizations and it is with these operators, corresponding to physical degrees of freedom, that all scattering amplitudes are computed.

Conformal anomalies and DFT harmonic condition
It is known that the cancellation of conformal anomalies at the string world sheet level manifests as gauge symmetry requirements on the target space fields. This is indeed the case here. The different vertex operators corresponding to two-tensor, vector and scalar fields will generically have anomalous OPEs (Operator Product Expansions) with the world sheet stress energy tensor.
For massless fields, the cancellation of anomalous terms leads to the familiar gauge conditions k µ ǫ G µν (k) = 0, k µ ǫ M µ (k) = 0, etc. for the polarization tensors of gravitons, vectors, etc. These correspond to equations (4.16) for zero generalized momentum.
For massive fields, a combination of the different vertex operators must be considered, such that the sum of the different anomalous contributions cancel. This is, indeed, a world sheet manifestation of the Higgs mechanism. The conditions for cancellation of the anomalous terms can be written in an O(n, n) language and can be shown to coincide with the harmonic gauge conditions found in DFT.
The vertex operators we are interested in are, up to normalizations, The label G generically denotes a symmetric traceless, antisymmetric or trace polarization, The vertex operator momenta are k aL = e a m p mL , k aR = e a m p mR , The stress energy tensor is and similarly for the right moving one. The OPEs are [: k a L ǫ a Lρ ∂X ρ e ik·X+ik L ·Y +ik R ·Ȳ :] + . . . , [: k ρ ǫ a Rρ ∂Ȳ a e ik·X+k L Y +k RȲ :] + . . . , [: k a L φ ab ∂Ȳ b e ik·X+ik L ·Y +ik R ·Ȳ :] + . . . .
Since k 2 L = −k 2 , the vertex operators have the correct conformal weight h = 1 (and similarlȳ h = 1), however, there are cubic anomalies which suggest that the physical fields should be created from combinations of these operators. Consider then the vertex associated with the massive graviton with constant α, β, γ, δ. From the OPE with T andT , the anomaly cancellation conditions are Choosing 2α = γ = β, the sum of the first two equations leads to where we have defined and we have made the identifications Therefore, (5.5) is nothing but the first harmonic gauge condition in (3.21) in momentum space.
On the other hand, by subtracting the first two equations in (5.4), we obtain which can be written as as found in (4.29).
The other two equations involving the scalars lead to which can be shown to coincide with the third equation of the harmonic gauge conditions in (3.21) when choosing δ = 1 2 γ and establishing the identification with DFT scalar fields (3.14) Thus, the physical vertex operator for the massive graviton is The effective symmetric polarization tensor can be shown to coincide with (4.18).
Similar steps can be followed for the Kalb-Ramond field and the second equation in (3.21) is obtained.
In the next section we introduce the physical vertex operators used in the computation of scattering amplitudes. The anomaly free conditions on the polarizations coincide with those of the physical fields redefined through the use of the harmonic gauge condition.

Physical Vertex Operators on the Torus
In the same way that we found the anomaly free combinations of vertex operators (or equivalently, the harmonic gauge conditions), we can impose that each one of the vertex operators (5.1) be anomaly free. This would give the conditions to be satisfied by the physical polarizations, that now we distinguish with a prime. Note that this procedure will give identically zero polarizations for massive vectors and scalars in the case of only one compact dimension, thus confirming that there are no such degrees of freedom on a circle compactification.
The anomaly cancellation conditions for vectors are (5.11) The first two equations can be combined as which are equivalent to Namely, the conditions found in (4.29) after gauge fixing. In the same way, for scalars we find which can be expressed in terms ofh ′ mn and b ′ mn as the following two conditions Finally, the dilaton vertex can be written as as found in (4.36) by identifyingk ν ≡ χ Thus, we have obtained the requirements that physical polarizations must satisfy.
Notice that the two approaches to deal with vertex operators provide different information on the theory: The first one displays a built in Higgs mechanism exhibiting the Goldstone bosons. The second one deals with the physical degrees of freedom once the gauge was chosen.
Of course, one can obtain the latter using the former, as was shown in the previous section. We will use physical polarization tensors to compute scattering amplitudes.

Three-point interaction terms
In this section we consider three point functions of the massless and massive string states created by the vertex operators described above. The resulting amplitudes are then compared with the DFT action (4.37), evaluated on shell. We sketch the computation here and provide some details for the circle case in the Appendix.
For the sake of clarity we first concentrate on the circle compactification. This case is particularly simple since neither physical massive vectors nor massive scalars are present. The string S-matrix three-point amplitudes are presented in (A.2). When mode expanding (4.37) and by using the identifications (5.7) and (5.9) between string polarization tensors and DFT fields polarizations, complete agreement is achieved if we further identify where g c is the closed string coupling.
The effective U (1) × U (1) gauge invariant action, containing massless as well as massive states with these S-matrix elements, can be written down. By including terms required from gauge invariance and diffeomorphism invariance, this action reads where Φ denotes the massless scalar; h We have introduced the following definitions Here indices are raised with the inverse of the metric tensor g µν ,q is the charge operator, complex conjugation is denoted with * and, under charge conjugation, the momentum or winding The kinetic terms of the symmetric massive states produce the known Fierz Pauli Lagrangian [24,25], and the T-duality symmetry R ↔R, n ↔ w is manifest. This action coincides with (4.41) when specified for the circle case.

Strings vs DFT on generic tori
Generalizing the results obtained for the circle to generic tori is formally straightforward.
However, the number of terms involved is much bigger. The massless sector contains, besides the graviton, dilaton, antisymmetric and scalar fields, the 2n gauge fields associated to U (1) n × U (1) n . The massive sector includes now, generically, massive vectors and scalars. The comparison of DFT cubic interactions contained in the mode expansion of (4.37) with three point scattering amplitudes computed using the vertex operators (5.1) is now performed with the help of the symbolic algebra computer program XCadabra [16]. Our algorithm compares three point scattering amplitudes of string states and DFT cubic interaction terms by systematic use of momentum conservation and on shell conditions 14 .
As an example of the calculated quantities, we present the result of the scattering amplitude between one antisymmetric tensor b µν (with momentum k 1 µ , and charges p 1 m and w 1 m ), one vector A m Lµ (with momentum k 2 µ , and charges p 2 m and w 2 m ) and one antisymmetric scalar b mn (with momentum k 3 µ , and charges p 3 m and w 3 m ).
In the DFT action there is only one place where the interaction vertex can be found, namely Splitting the double internal indices, in order to exhibit the explicit contributions of b mn and h mn scalars, one can collect the required interactions and compute the three point amplitude.
The result is As is well known, physical states in string theory are selected by ensuring cancellation of conformal anomalies in the world sheet. We found that the DFT manifestation of these require-ments is the invariance under generalized diffeomorphisms. By using such invariance, we have shown that a generalized harmonic gauge condition can be chosen, and established a correspondence with conditions derived from string theory. Interestingly enough, this gauge choice allows to identify the different Goldstone modes that are absorbed to generate physical fields. Besides the gravity multiplet and massless vectors associated to the compactified gravitational and antisymmetric fields, physical massive fields correspond to massive symmetric and antisymmetric tensors, vectors and scalars charged under the G n gauge group. The charges, corresponding to momentum and winding numbers, are simply encoded in the generalized DFT momenta P.
Generalizing known results in KK compactifications, we found the infinite global symmetry algebra associated to infinite local generalized parameters. In particular, it contains a finite Poincaré ×SO(1, 2) n × SO(1, 2) n subalgebra and massive states should organize in its (infinite dimensional) representations.
Of course the effective action reproducing the three-point amplitudes of these physical massless and massive string states is not a low energy effective action since all possible massive levels are involved. The action provides an organized truncation of string theory. However this truncation is incomplete since it contains states with masses of the order or higher than those of string states with N and/orN = 1 that were not included here. Indeed, we know from string theory that new fields involving higher spins (associated with N and/orN = 1) appear in the spectrum and play a crucial role in higher point-amplitudes. In DFT language, higher order O(n, n) generalized tensors, incorporating these missing string degrees of freedom, are expected.
We also know that a gauge symmetry enhancing, associated to the presence of windings, occurs in string theory at self dual points. This enhancing involves states with N −N = 0 (e.g. N,N = 0, ±1) and for this reason it cannot be seen in our construction. In [13], a DFT description of gauge enhancing in circle compactification at self dual radius R 0 was provided.
There, it is shown that enhancing from U (1) × U (1) to SU (2) × SU (2)  Massive particles with spin larger than 2 would also be needed if higher powers of momentum were considered. Actually, the three-point functions presented in the Appendix contain higher powers of momentum that we have not included since they go beyond the aim of this paper.
However, these higher order terms lead to higher derivative contributions to the effective action which would of course be necessary if quantum corrections were considered. In particular, the inclusion of higher order terms in curvature invariants is known to demand the addition of massive tensors in order to fix the short-distance violations of causality [31], and the Regge behavior required for the resolution of the causality problem [32] also calls for higher order tensors in DFT.
Certainly, the effective theory we have constructed does not work as a fundamental theory.
Nevertheless, despite the absence of essential ingredients for full consistency, it might be appealing by itself. It encodes an effective gauge invariant theory with a massless sector containing gravity, antisymmetric tensor plus gauge bosons and scalars coupled to towers of GKK massive modes. It is interesting to notice that, even if a given field has a zero mode, it spreads out into towers of momenta and windings. The simplest case of a non-zero graviton mass is an interesting theoretical possibility since it was not until recently that a consistent non-linear theory of massive gravity could be constructed [33].
Even in this simple toroidal scenario it could be interesting to look at possible phenomenological consequences and to explore them in more detail. This aspect is beyond the scope of the present work but let us signal some new features that could be worth exploring. Many scenarios including KK excitations have been proposed in the literature for different physical models. These proposals deserve being reconsidered in this GKK scenario including windings as well as other fields. On the one hand new fields, associated to antisymmetric tensor and dilaton, can be present. Also a new energy scale is built in. In fact, even at the circle level two different energy scales λ KK = 1/R and λ windings = 1/R appear now which can lead to relevant physical consequences.
For instance, the type of models proposed in [29] in the large extra dimensions scenario of [34] appear to be drastically modified. There, toroidal bulk KK gravity modes were coupled to Standard Model fields with radii λ KK M string ∼ T eV . However now, besides the fact that other fields are present, the λ windings energy scale will also be present. Leaving aside stringy gauge symmetry enhancing, R =R self dual point situations, where both windings and KK modes contribute on the same footing, are also possible.
KK universal scenarios for dark matter [35] have been extensively discussed. The consistent incorporation of massive antisymmetric tensors coupled to Einstein gravity plus other massless and massive fields could be also appealing in this context (see for example [36]). More complex situations, that would require generalizations of this simpler toroidal case, provide attractive candidates for dark matter [37]. Phenomenology of massive KK gravitons at the LHC was recently discussed in [38], composite Higgs models associated to bulk KK modes have been considered in [39], etc.
The ideas developed here could in principle be extended to GKK reductions in which the starting theory has non Abelian gauge fields already in higher dimensions (e.g. the heterotic string). These are just plausible roads of research that call for careful study.

A.1 Extra terms in the DFT action
In the original frame formulation of DFT by Siegel [1,2]. the action contains extra terms that are not contained in (2.11). Up to total derivatives those can be recast as [11] ∆S = d 2D X e −2d 1 2 Here we show that these terms vanish once the level matching condition (2.15) is imposed.
To show the vanishing of the term proportional to SĀB we consider the following integral A little of algebra, making use of the property H P Q η P Q = 0, shows that Similarly, for the term in (A.1) proportional to ηĀB we consider the integral And therefore, transforming into momentum space and imposing the level-matching condition (2.15), we get ∆S = 0.

A.2 String computations
The results of three-point scattering amplitudes in bosonic string theory are presented here for the case of one compact dimension on a circle of radius R. They are computed with the vertex operators defined in (5.1). We first collect the amplitudes involving only massless states and then the ones containing at least one massive state. We use a shorthand notation with h, b, φ, A,Ã denoting graviton, antisymmetric tensor, scalar and vector fields. Recall that no massive vectors or scalars appear in the circle compactification and the massive fields are only h and b. Dots indicate contractions with Minkowski space-time metric η µν .

A.3 Algebra of diffeomorphisms
Following the discussion in [26,27], we can associate a global infinite parameter algebra to the infinite modes ξ P(M) (x) of the GKK expansion of the parameters of local transformations, in much the same way as a global Poincaré algebra is associated to general coordinate transformations. It is easy to check that these operators generate an algebra that corresponds to the direct generalization of the algebra found in [26]. Namely, 2 close a Poincaré⊗SO(1, 2) algebra. In the same way, exchanging 1 ↔ 2, namely, windings with momenta, another SO(1, 2) algebra is obtained. Thus, finally the original Poincaré algebra is enlarged to Poincaré ⊗SO(1, 2) 2 . It was shown in [28] that, in the circle case in field theory, the massive KK states organize into an infinite dimensional (non-unitary) R representation of SO (1,2). In DFT on the circle, windings and momenta are decoupled, so massive KK momenta states will fill up the infinite dimensional representation of the first algebra whereas windings will organize in a similar representation of the second one, namely (R, 1) + (1, R).
In the generic case we can proceed in the same way by choosing the GKK momenta at the position L, M L = 0, ±1 with all other components vanishing. In this case we would have Poincaré ⊗SO(1, 2) 2n . Since massive states with M 2 = P · M · P mix windings and momenta the analysis of representations is more involved and we will not perform it in the present work.
Even if the above algebra is a symmetry of the original Lagrangian, it is broken to Poincaré