Heterotic $\alpha$'-corrections in Double Field Theory

We extend the generalized flux formulation of Double Field Theory to include all the first order bosonic contributions to the $\alpha '$ expansion of the heterotic string low energy effective theory. The generalized tangent space and duality group are enhanced by $\alpha'$ corrections, and the gauge symmetries are generated by the usual (gauged) generalized Lie derivative in the extended space. The generalized frame receives derivative corrections through the spin connection with torsion, which is incorporated as a new degree of freedom in the extended bein. We compute the generalized fluxes and find the Riemann curvature tensor with torsion as one of their components. All the four-derivative terms of the action, Bianchi identities and equations of motion are reproduced. Using this formalism, we obtain the first order $\alpha'$ corrections to the heterotic Buscher rules. The relation of our results to alternative formulations in the literature is discussed and future research directions are outlined.


Introduction
The construction of duality invariant formulations of the supergravity limits of string theory has been an active field of research in recent years. A paradigmatic case is Double Field Theory (DFT), where T-duality is incorporated as a manifest symmetry of the universal supergravity sector [1,2]. The framework allows to incorporate heterotic vector fields [1,3], the Ramond-Ramond fields of type II theories [4,5] and the fermions that complete the supersymmetry multiplets [1,5,6]. This program led to the full covariantization of supergravities to lowest order in perturbation theory with respect to the T-duality symmetry of string theory. In the process, interesting novel geometric structures emerged, such as the generalized metric [7] and frame [1,8] including the supergravity fields as components, and a generalized Lie derivative [1,9,10] that unifies diffeomorphisms and two-form gauge transformations. In this framework, duality invariance is achieved by formally defining the theory on a double space, and the physical space on which supergravity is realized can be recovered upon enforcing the so-called strong constraint. The result is an elegant and powerful reformulation of supergravity in terms of generalized geometric quantities that make T-duality manifest. Interestingly, the duality structure of these theories is manifest even before compactification. For more details and references see [11].
A natural question is how to incorporate α ′ corrections in this context. Recently, this question was nicely addressed in [12], where a duality invariant CFT that incorporates α ′ corrections was presented. Here we consider the heterotic string, and our goal is to rewrite the massless bosonic sector of the effective low energy theory, including all first order contributions of the α ′ expansion, in the language of DFT. This comprises the action, equations of motion, Bianchi identities and duality transformations. Although conceptually our approach looks different from that in [12], we illustrate how both constructions could be connected.
The first order α ′ contributions to the heterotic string effective field theory have an interesting structure. The action includes gauge and gravitational Chern-Simons terms in the two-form field strength, in addition to quadratic terms of the Yang-Mills field strength and of the Riemann curvature tensor with torsion. These contributions were originally obtained from tree level scattering amplitudes of the massless heterotic string states [13].
An alternative method to construct the gravitational part of this action was developed in [14], making use of a symmetry that exists between the Yang-Mills and supergravity fields in ten dimensions. Since this symmetry is an essential ingredient of our construction, we briefly recall the main idea.
In d dimensional gravity, the spin-connection plays the role of an SO(1, d − 1) gauge field, that gauges the local Lorentz transformations which are part of the gauge symmetries of supergravity. Although this seems to imply that a Riemann curvature squared action can be constructed from the Yang-Mills field strength squared action, simply replacing everywhere the gauge connection by the Lorentz spin connection, these connections do not have the same behavior under supersymmetry transformations. However, the replacement of gauge by spin connection works well in the formulation of d = 10 supergravity as an SO (1,9) Yang-Mills multiplet if the spin connection has torsion and the torsion is proportional to the two-form field strength. This symmetry between the Yang-Mills gauge connection and the Lorentz torsionful spin connection will be crucial in our formalism, so we will keep it manifest all along the analysis.
Let us start by reviewing the heterotic string low energy effective action to order α ′ .
The massless bosonic degrees of freedom are a d = 10 dimensional bein e µā , a two-form B µν , n g = 496 gauge fields A µ α and a dilaton φ, where µ, ν, · · · = 1, . . . , d are spacetime indices, whileā,b, · · · = 1, . . . , d are flat Lorentz indices and α, β, · · · = 1, . . . , n g are indices in the adjoint representation of the heterotic gauge group. The action can be written as [13]- [18] is the two-form field strength. As emphasized above, the α ′ corrections include a Chern-Simons contribution from the gauge fields A µ α and a Chern-Simons contribution from the spin connection with torsion ω (−) µ Λ . These terms depend on the gauge (Lorentz) Killing metric and structure constants, which are proportional to κ αβ (κ ΛΓ ) and f αβ γ (f ΛΓ Σ ) respectively. The indices Λ, Γ, · · · = 1, . . . , n l where n l = d(d − 1)/2, are adjoint Lorentz indices. We refer to the Appendix for details on our conventions. The torsionful spin connection is where ω µāb is the usual torsionless spin connection and the two-form field strength plays the role of torsion. Note that since ω (−) µ Λ always appears in the action in terms with an α ′ factor, to O(α ′ ) the Chern Simons terms contained in the torsion in (1.3) play no role.
The second line in (1.1) contains the field strengths of the connections the latter being the Riemann tensor defined in terms of the torsionful spin connection.
Written in this form, the symmetry between the connections is manifest in the action This symmetry extends all along the Bianchi identities (BI). Indeed, the BI for the twoform, gauge and gravitational field strengths read At the level of the equations of motion (EOM), the symmetry is more subtle. The reason is that, while the gauge fields A µ α are independent degrees of freedom, the torsionful spin connection ω (−) µ Λ is not. The latter depends on the bein, the two-form and the gauge connection, and then a priori there seems to be no reason to consider its EOM. Let us then begin by writing the well known EOM to O(α ′ ) for the dilaton, bein, two-form and gauge fields (see for example [15]- [20] and references therein)   to the lemma proved in [15], to O(α ′ ) the result of varying the action with respect to the fundamental fields by first varying the explicit dependence and then adding the variation through the torsionful spin connection which implicitly also depends on them, coincides with the result of simply considering the explicit variation. We discuss this further in the Appendix. This suggests that one can still consider a formulation in which ω treated as an independent degree of freedom. This point of view is then useful in order to extend the symmetry between the gauge and torsionful spin connections to all levels, including the EOM.
In this paper, we encode all these results in the manifestly T-duality invariant DFT.
Already for the gauge sector this was done in [1,3], where the gauge fields were incorporated in an extended tangent space, enhancing the O(d, d) duality group to O(d, d + n g ).
Here, we further extend this construction to incorporate the gravitational sector to order α ′ , exploiting the above mentioned symmetry between the gauge and torsionful spin connection. Related constructions can be found in [21]- [25].
We work in the generalized flux formulation of DFT [1,8,26], which is more convenient built from the generalized metric that reduces to the square of the Riemann tensor [28].
An interesting application of our formalism is to determine the α ′ corrections to the Buscher rules of the heterotic massless fields. These rules play a significant role in the search of solutions to the string equations of motion, allowing to generate new solutions from old ones. Buscher derived the zero slope limit of the duality transformations of the fields from the sigma model worldsheet action [29] when there is an isometry (see also [30]). An elegant way to recover these rules is by performing a canonical transformation [31], which shows that the dual models are classically equivalent. The explicit form of the quantum corrections has been pursued using different methods and some partial results are available [32,33]. Here, we obtain the O(α ′ ) corrections to the transformation rules of the massless heterotic fields in a manifestly duality covariant way. After constructing the generalized metric and transforming it under the factorized T-duality elements of the duality group G, we get the explicit results for the α ′ corrected duality transformations of generic background fields. We show how this works for the full O(d, d, R) duality group.
The paper is organized as follows. In Section 2 we briefly review the generalized flux formulation of DFT and its gauging. We then present the heterotic setup in Section 3.
We is not strictly necessary and here instead we will consider a bigger group that contains The fields are generalized tensorial densities T M ... N ... of weight w(T ) that transform under generalized diffeomorphisms as A generalized frame EĀ M is a basis of generalized vectors of vanishing weight, and can be taken to be parameterized by some of the supergravity field degrees of freedom, namely the metric, two-form, one-form gauge fields, etc. Under generalized diffeomorphisms it transforms as The particular parameterization of the generalized frame in terms of the supergravity degrees of freedom depends on the H-gauge choice, which we do not need to specify right now. After the action of generalized diffeomorphisms, the gauge choice must be restored.
Since it parameterizes the coset G/H, the frame satisfies and so its inverse is given by EĀ M = ηĀBη M N EB N . The dilaton, instead, is contained in a density field e −2d , of weight w(e −2d ) = 1, which transforms as a measure The group of generalized diffeomorphisms closes provided a tower of closure constraints is satisfied. In particular, the transformation of a tensorial density must be itself a tensorial density where L ξ acts on a covariant object, while δ ξ faithfully transforms the object. Clearly, on tensorial densities, one has ∆ ξ T = 0. Since (2.5) is not covariant, one should impose the additional constraints that all its gauge transformations vanish as well. The result is a tower of closure constraints that restricts the space of gauge parameters and tensorial densities for which DFT is consistently defined. A stronger constraint, known as strong constraint or section condition, can be imposed where ⋄ represents any combination of fields and gauge parameters. This constraint is sufficient to satisfy the closure constraints (and hence to achieve gauge consistency), but it is not necessary [34,26]. Let us emphasize however that in this paper, for the sake of concreteness and in order to make direct contact with the heterotic supergravity theory in d = 10-dimensions, we will impose the strong constraint.
The generalized diffeomorphisms allow to define generalized fluxes which by construction transform as scalars under generalized diffeomorphisms, up to the closure constraints. When evaluated on generalized frames, the latter become Moreover, when the strong constraint is enforced, these closure constraints then simply become Bianchi identities.
Since the generalized fluxes are not H-covariant, by demanding H-invariance the action is fixed to be The action (2.10) is fully invariant under all the global and gauge symmetries, up to the closure constraints (2.5).
Varying the action with respect to the generalized dilaton and frame yields the equa- This concludes our brief summary of the gauge symmetries, action, BI and EOM of the generalized flux formulation of DFT. For more details we refer to the original papers or the reviews [11].

Gauged Double Field Theory
DFT can be deformed through a gauging procedure [3], parameterized by an embedding tensor that satisfies a linear and a quadratic constraint provided (any combination of) the fields and gauge parameters are further restricted to satisfy the constraints 14) The embedding tensor dictates how the gauge group is embedded in the global duality Under such a deformation, the generalized diffeomorphisms become gauged and so do the gauge transformations of the generalized frame and dilaton which in turn induce gauged contributions to the generalized fluxes After the gauging procedure, the action, equations of motion, closure constraints, etc.
take exactly the same form as in the previous section, but with hatted fluxes. In this paper we will work with a gauged DFT (GDFT), but in order to lighten the notation we will drop the hats. Let us finally comment that this gauging procedure was shown in [34] to be equivalent to a generalized Scherk-Schwarz reduction [35].

The Heterotic setup
To accommodate the O(α ′ ) corrections of the heterotic string effective theory, we take the of the quotient is then which allows to build in a symmetric d-dimensional metric g µν , a two-form B µν , n g one- The indices take values µ, ν · · · = 1, . . . , d; α, β, · · · = 1, . . . , n g and Λ, Σ, · · · = 1, . . . , n l . To make contact with the heterotic string, one has to assume that d = 10 is the dimension of the physical space-time, More concretely, if one simply replacesω µ Λ by ω , after a T-duality it will transform to a different quantity that will depend, for instance, on dual derivatives. Then, in order to construct a second order formulation that is well behaved under T-dualities, one must proceed with caution and consider a quantity that transforms consistently under O(d, d, R) and reduces to ω (−)Λ µ when the strong constraint is solved in the standard space-time coordinates. On the other hand, the DFT formalism enforces an equation of motion forω µ Λ , which must then be trivially satisfied. We will address these issues in due time, and for now just proceed by treatingω µ Λ as an independent quantity.

and the invariant metric in
G is taken to be Here, κ αβ and κ ΛΓ are proportional to the (inverse) Killing metrics in the adjoint representations of the gauge and Lorentz groups (see the Appendix), with signatures (0, n g ) and where sāb = diag(−, +, . . . , +). Here κᾱβ = e αᾱ κ αβ e ββ is numerically equivalent to κ αβ , which allows to define elements e αᾱ that preserve the Killing metric of the gauge group, and κΛΓ = e ΛΛ κ ΛΓ e ΓΓ is numerically equivalent to κ ΛΓ , which allows to define elements e ΛΛ that preserve the Killing metric of the Lorentz group.

Generalized frame and gauge transformations
Consider a generalized G-valued frame EĀ M satisfying EĀ M η M N EB N = ηĀB with a fixed H-gauge choice, and such that it has the following d-dimensional dynamical degrees of freedom: a bein e µā , a two-form B µν , n g one-forms A µ α and n l one-formsω µ Λ . Including also the elements e αᾱ and e ΛΛ introduced above, the frame can be written as where The fact that such a generalized frame exists globally means that the extended space is generalized paralellizable [36]. On the other hand, the dilaton φ is combined with the determinant of the metric g in the shifted dilaton field We now explore the action of generalized diffeomorphisms on the generalized frame and dilaton, and for simplicity we impose the section condition and pick the frame in which ∂ M = (0, 0, 0, ∂ µ ). We will assume this for the sake of concreteness in all the rest of the paper, and we will also explicitly incorporate the α ′ parameter. The generalized Lie derivative acts as where the non-vanishing fluxes f P Q M have only pure gauge or pure Lorentz indices, thus satisfying the constraint (2.14). Taking the gauge parameter in components we find

8)
L ξ e µā = ξ ρ ∂ ρ e µā + ∂ µ ξ ρ e ρā , (3.9) L ξ e αᾱ = ξ ρ ∂ ρ e αᾱ − f αγ β ξ γ e βᾱ , (3.10) where we have definedξ The last three terms in (3.14) include the gauge and Lorentz transformation of the twoform that implement the Green-Schwarz mechanism [37]. Such a transformation guarantees that the field strength of the two-form, which includes the Chern-Simons terms, is gauge and Lorentz invariant. For h sufficiently close to the identity hĀB = δĀB + ΛĀB, the above conditions impose Since H is a symmetry of the theory, one can equivalently define the gauge transformations as where the last term is introduced to restore the gauge fixing. It is easy to see that the particular gauge choice e αᾱ = const. and e ΛΛ = const. is restored through with which enforces

Generalized fluxes
Given the generalized frame and generalized Lie derivative defined in (3.4) and (3.7), respectively, we are now ready to compute the generalized fluxes (3.23) Using the above parameterization and imposing the strong constraint, one is left with the following non-vanishing components where

29)
D µ e αᾱ = ∂ µ e αᾱ + f αβ γ A µ β e γᾱ , (3.30) We then readily identify all the covariant building blocks of the theory, namely the fieldstrengths of the dilaton (3.25), the two-form (3.26), the bein (3.27) (which is the antisymmetrized spin connection), the gauge fields (3.28), the extra one-forms (3.29) (which is nothing but the Riemann tensor whenω µ Γ is identified with the spin connection) and the covariant derivatives of the gauge and Lorentz beins (3.30) and (3.31). Of course, the last two quantities are just pure gauge as we showed above, so we expect them not to appear in the action. Moreover, since the action is quadratic in fluxes, one can already anticipate the presence of the Riemann squared term induced by α ′ -corrections. Although somehow expected, the fact that the Riemann tensor appears as one of the components of a generalized flux is very interesting. As discussed in [28], the Riemann tensor is not a component of the generalized Riemann tensor introduced in [38], nor can it be generated from a combination of derivatives of the generalized metric. Here, the extension of the tangent space permits to accommodate a spin connection, whose field strength is the Riemann tensor, which then appears as a generalized flux component.
For the sake of completion, let us now compute the checked fluxes (2.11) Note that the fluxesFāᾱβ andFāΛΓ vanish, signaling the fact that no kinetic term of the gauge and Lorentz beins will appear in the action. Also, note that the checked fluxes carry the information of the couplings in the action.

Generalized Bianchi identities
We have shown that the closure of the algebra of generalized Lie derivatives leads to a set of closure constraints (2.9), that become BI when the strong constraint is enforced.
In terms of fluxes, they read Let us then compute their components to show how they match the BI of the heterotic string. The non-vanishing components are

The action
Having computed the components of the fluxes (3.24) and their checked projections (3.34), it is now straightforward to compute the action In components this reads and after an integration by parts we are left with where g µν = eā µ sābeb ν and R = g µν R µν .
This confirms our expectations related to the appearance of the Riemann squared term, and the absence of kinetic terms for the gauge and Lorentz beins. Modulo the identification ofω µ Λ with ω (−) µ Λ , the action precisely matches (1.1), the low energy effective action of the heterotic string to order α ′ .

Equations of motion
As a final step, we now compute the EOM of the theory. As discussed above, all the EOM are condensed in (2.12), the generalized EOM that depend on the generalized fluxes The non-vanishing components of these equations are where we use the convention Gāb = e ρā eb ν g µρ ∆g µν , (3.61) We have defined in terms of a torsionful connection A well known lemma discussed in [15] proves that this is indeed the case. In fact, replacing coordinates. This will allow us to promote this formulation to a consistent second order formalism.

Generalized metric formulation
An alternative formulation of GDFT can be performed in terms of the generalized metric.
The inverse generalized metric is given by and it is straightforward to compute its components The action of GDFT was given in terms of the generalized metric in [3] and it has the following form One can check that this action is equivalent (up to strong constraint violating terms) to (2.10), and one can equally compute the BI and EOM in terms of the generalized metric.
Since the results agree with those obtained in previous sections through the generalized flux formulation, we do not pursue this analysis here. However, the generalized metric is more convenient than the generalized frame formulation to discuss duality symmetries.
This is because the generalized metric is H-invariant, and therefore, the action of the duality group G must not be compensated by gauge-fixing H-transformations. We make use of this advantage in the following subsection to compute the α ′ -corrections to the heterotic Buscher rules induced by factorized T-dualities.

T-duality, α ′ corrected Buscher rules and O(d, d, R)
We are now in a good position to compute the α ′ corrections to the Buscher rules, and more generally to discuss the role of the O(d, d, R) symmetry. In the absence of α ′ corrections, the Buscher rules were derived by Buscher [29] from the sigma model formulation of string theory, and they determine how the metric and two-form degrees of freedom mix g ′ (g, B) and B ′ (g, B) under factorized T-dualities. Other derivations can be found in [31,30] and α ′ corrections were explored in [32,33], and references therein.
Here we apply a different, more direct, strategy. We have seen that the generalized where

T-duality covariant Lorentz connection
We would like to show that there is a field-dependent object that transforms under fac- and reduces to ω The generalized frame transforms as follows where T m n is a global element of O(d, d, R), i.e.
T m p η pq T n q = η mn , (4.20) and Λmn is a local double Lorentz transformation that satisfies with sāb the Minkowski metric.
We now explore how the components of the generalized frame transform under O(d, d, R).
The elements of this group factorize in GL(d) transformations, B-shifts, and factorized T-dualities. The first two preserve the triangular form of the generalized frame, but the latter do not. Then, in order to restore the gauge one has to compensate with a local double Lorentz transformation. A factorized T-duality in the z direction has the form The transformations for the bein are only defined up to the diagonal part of the double Lorentz group. One can check that they reproduce the transformations (4.7)-(4.11) for the metric and two form to lowest order in α ′ .
On the other hand, one can define the O(d, d) generalized fluxes (also called generalized coefficients of anholonomy) in terms of the generalized frame The components of these fluxes are detailed in [26]. Under a global T-duality these objects are manifestly invariant, but after the compensating double Lorentz transformation they transform in a non-trivial way [26] F ′mnp = 3Λ [m|q ∂qΛ |nr Λp ]r + ΛmqΛnrΛpsFqrs . (4.28) To understand the impact of these transformations, it is convenient to perform an SO(2, 2d − 2) rotation on flat indices through the element This rotation connects the frame formalism in [1] and [8] with that in [26]. Under this rotation, the quantities defined above becomê Then, if we defineω we find that under a factorized T-duality it transforms as (4.38) Moreover, one can check [8] that under the definition (4.37),ω µbc exactly reduces to ω (−) µbc defined in (1.3), when the strong constraint is solved in the supergravity frame.

Comparison with Double α ′ -Geometry
Having computed the generalized metric, it is instructive to compare our approach with that of the double α ′ -geometry presented in [12]. There, it was realized that α ′ corrections can be obtained from a duality covariant CFT construction. In that approach, the generalized Lie derivative receives an α ′ correction, T-dualities are not corrected, and the tangent space is the usual double tangent space. In contrast, here we preserve the form of the generalized Lie derivative and extend the duality group by enhancing the generalized tangent space. It is then natural to ask if these two seemingly different approaches can be reconciled.
In [12], both the inner product and the generalized Lie derivative receive higher derivative corrections (we introduce α ′ explicitly to make the comparison with our results clearer) Here, we use the same convention of the previous subsection, namely m, n, · · · = 1, . . . , 2d are O(d, d) indices, which are raised and lowered with the O(d, d) invariant metric η mn .
With this convention, the strong constraint reads η mn ∂ m ∂ n ⋄ = 0.

Now consider an extended tangent space with generalized vectors
can be used to define an invariant metric in the extended space The key observation is that the usual inner product and gauged generalized Lie derivative in the extended space after implementing the strong constraint. In particular, when the constraint is solved in the frame in which everything depends only on the supergravity coordinates, one finds Then, we see that the α ′ corrections to the O(d, d) inner product and generalized Lie derivative of [12] can be encoded in an extended space in which the inner product and generalized Lie derivative take the usual expressions. This parallelism, while promising in order to reconcile both approaches, deserves a better understanding.

Conclusions
In shell. An important outcome of this enhancement is that the Riemann curvature tensor with torsion appears as one of the components of the generalized fluxes. In this way, being quadratic in fluxes, the generalized action successfully reproduces the curvature squared term of the heterotic effective theory. Hence, the construction allows to circumvent the issue raised in [28] about the absence of a T-duality invariant four-derivative object, built from the generalized metric, that reduces to the square of the Riemann tensor.
The gauging preserves a remnant O(d, d) global symmetry that allowed us to compute the explicit α ′ corrections to the Buscher rules. Indeed, acting on the extended generalized metric with factorized T-duality transformations, we have found the first order α ′ corrections to the transformation rules of the massless bosonic heterotic fields.
These transformations serve as a solution generating mechanism, as new solutions of the heterotic EOM can be found by applying these rules to known solutions.
Several subsequent directions to extend these results suggest themselves. One obvious course of future action is the construction of higher derivative terms. The ultimate goal is to incorporate all order α ′ corrections in a duality invariant formulation. This is clearly a difficult problem and a more modest target would be to understand these corrections order by order. Using duality symmetries to determine higher derivative corrections to supergravity has been a prolific area of research in recent years (for example, see [40] and references therein). It is possible that higher order corrections require further enhancements of the duality group and additional extensions of the tangent space, in order to allow for more degrees of freedom into a yet larger generalized bein.
The supersymmetric extension is another direction of interest. Supersymmetric DFT was constructed in [1,5,6] and more recently in gauged DFT in [41]. As explained in [15], the symmetry between the gauge and gravitational connections extends to the fermionic sector as well (more specifically the symmetry interchanges the gauginos with the curva-ture of the gravitinos), and this can be useful in the construction of the supersymmetric extension of our work.
It would also be interesting to explore α ′ corrections in the bosonic string and Type II superstring theories and see if they can be cast in a duality invariant form, similar to the one considered here. One can already make contact with Type II theories by letting the heterotic gauge group be embedded in the holonomy group. In this case, due to the symmetry between gauge and gravitational connections, the order α ′ terms in the action cancel each other, in concordance with the fact that Type II theories only receive corrections of order α ′3 and higher. The bosonic case will be discussed in a separate work [42].
From a more phenomenological perspective, compactifying this theory to lower dimensions would allow to study the quantum corrections to the low energy effective couplings and scalar potential. Compactifications in manifolds with SU(3) structure were performed in [43], and it is also of interest to study supersymmetry preserving generalized Scherk-Schwarz compactifications along the lines of [35] in this context. The deformations of the moduli space induced by α ′ corrections may have important consequences in the search of vacua and the construction of sensible cosmological models in string theory. Moreover, it would allow to explore the relation between α ′ corrections and non-geometry, particularly the duality orbits of non-geometric fluxes discussed in [44], where the non-geometric effects are expected to be of order α ′ .
Note 1. At early stages of this work we received a preliminary version of [25], which contains some of the building blocks of our paper. This includes the extended tangent space, inner product and generalized Lie derivative. We would like to emphasize that a similar discussion on the relation between this formalism and the one in [12] was first posed in [25].
Note 2. Soon after our work was posted, the papers [45] appeared, which aim to describe bosonic and heterotic α ′ corrections following the approach in [12].
Note the different conventions used for the Killing metrics in the gauge (A.1) and Lorentz (A.5) sectors.

A.2 Comments on the equations of motion
In this Appendix we outline the procedure to obtain the equations of motion to O(α ′ ), i.e.
(1.10)-(1.13). For further details we refer to [15]- [20] and references therein. Specifically we only focus on the EOM for the two-form and the bein, since their derivation is subtle due the fact that they are both implicitly contained in the torsionful spin connection.
We consider first the equation of motion for the two-form. Explicitly, the Lagrangian does not depend on B µν , but on its derivatives. Implicitly, it depends on first and second derivatives of B µν through the torsionful spin connection and its derivatives. Therefore, the full variation of the action is which can be rewritten, after integrating by parts, as It is straightforward to compute the expression multiplying δ B ω (−) in (A.9), that we denote δ ω (−) L, and the result is Here, the first term is what we denoted ∆B in (1.12) and the second one is proportional to both ∆B and ∆g defined in (1.11) (see for example (3.74)). This is a very useful and well known result obtained in [15]. However, because of the complete antisymmetry in the indices λ, ρ, ν, the dependence on ∆g (which is symmetric) vanishes. One ends with where the linear differential operator is defined bŷ (A.14) We then see from (A.13) that the EOM for the B-field takes the following schematic form We then see that the EOM for the bein and two-form, properly computed from the full variation of the action, are equivalent (to O(α ′ )) to the EOM that one would obtain by only varying the action with respect to the explicit dependence of the fields, and treating the torsionful spin connection as an independent field.