Supersymmetry, T-duality and Heterotic $\alpha'$-corrections

Higher-derivative interactions and transformation rules of the fields in the effective field theories of the massless string states are strongly constrained by space-time symmetries and dualities. Here we use an exact formulation of ten dimensional ${\cal N}=1$ supergravity coupled to Yang-Mills with manifest T-duality symmetry to construct the first order $\alpha'$-corrections of the heterotic string effective action. The theory contains a supersymmetric and T-duality covariant generalization of the Green-Schwarz mechanism that determines the modifications to the leading order supersymmetry transformation rules of the fields. We compute the resulting field-dependent deformations of the coefficients in the supersymmetry algebra and construct the invariant action, with up to and including four-derivative terms of all the massless bosonic and fermionic fields of the heterotic string spectrum.


Introduction
At low energy, or small curvature, heterotic string theory reduces to ten dimensional N = 1 supergravity coupled to super Yang-Mills [1]. Successive terms in the α ′ -expansion may be expressed as higher-derivative interactions that are strongly constrained by the symmetries of string theory. There are several reasons to study the higher-order terms in the effective field theories of the massless string modes. They are needed to evaluate the stringy effects on solutions to the supergravity equations of motion [2,3], they play a central role in the tests of duality conjectures [4], in the microstate counting of black hole entropy [5] and in moduli stabilization [6].The swampland program [7] has revealed that the effective field theories of low energy physics and cosmology are limited by their couplings to quantum gravity [8], and together with the string lamppost principle [9], reinforces the interest in the restrictions imposed by string theory on the higher-derivative corrections to General Relativity.
The first few orders of the heterotic string α ′ -expansion are known explicitly. The interactions of the bosonic fields up to O(α ′3 ) were originally determined from the computation of scattering amplitudes of the massless string states at tree [1,10] and one loop [11] levels in the string coupling and from conformal anomaly cancellations [12]. The contributions of the fermionic fields have been computed using supersymmetry and superspace methods [13]- [20]. Supersymmetry completely fixes the leading order terms [13] and it often provides an elegant underlying explanation of the higher-derivative corrections.
But it holds iteratively in powers of α ′ and the transformation rules of the fields demand order by order modifications that are further restricted by other string symmetries and dualities.
In particular, the effective field theories for the massless string fields exhibit a global O(n, n; R) symmetry when the fields are independent of n spatial coordinates. This continuous T-duality symmetry holds to all orders in α ′ [21] (see also [22]- [25]) and it has been explicitly displayed recently for the quadratic and some of the quartic interactions of the bosonic fields in [26,27]. This feature motivated the construction of field theories with T-duality covariant structures, such as double field theory (DFT) [28,29] and generalized geometry [30], which provide reformulations of the string (super)gravities in which the global duality invariance is made manifest.
In the duality covariant frameworks, the standard local symmetries are generalized to larger groups: diffeomorphism invariance is extended to also include the gauge transformations of the two-form and the tangent space is enhanced with an extended Lorentz symmetry. Interestingly, the duality covariant gauge transformations completely determine the lowest order field interactions in string (super)gravities even before dimensional reduction (for reviews see [31] and references therein). Moreover, extensions of the duality group [32,33] as well as enhancings of the gauge structure of DFT [34,35] allowed to reproduce the four-derivative interactions of the massless bosonic heterotic string fields.
Supersymmetry can be naturally incorporated in the duality covariant formulations [36]- [41]. A supersymmetric and manifestly O(10, 10 + n g ) covariant DFT reformulation of ten dimensional N = 1 supergravity coupled to n g abelian vector multiplets was introduced in [37,38]. Although it is formally constructed on a 20 + n g dimensional space-time, the apparent inconsistency of supergravity beyond eleven dimensions is avoided through a strong constraint that admits solutions removing the field dependence With the motivation to further understand the structure of the heterotic string α ′expansion, in this paper we perform a perturbative expansion of the formal exact construction of [41] and obtain the first order corrections to N = 1 supersymmetric DFT. Further parameterizing the duality multiplets in terms of supergravity and super Yang-Mills mul-tiplets, we show that the supersymmetric duality covariant generalized Green-Schwarz transformation completely fixes the first order deformations of the transformation rules of the fields. We also construct the invariant action with up to and including four-derivative terms of all the massless bosonic and fermionic fields of the heterotic string and up to bilinear terms in fermions.
The paper is organized as follows. In section 2 we review the basic features of the N = 1 supersymmetric DFT introduced in [38] and we trivially extend it to incorporate non-abelian gauge vectors. In section 3, after briefly recalling the relevant aspects of the duality covariant mechanism proposed in [41], we extract the first order corrections to the transformation rules of the O(10, 10+n g ) generalized fields from those of the O(10, 10+k) multiplets, and obtain the manifestly duality covariant and gauge invariant N = 1 supersymmetric DFT action to O(α ′ ). We then parameterize the O(10, 10 + n g ) fields in terms of supergravity and super Yang-Mills multiplets in section 4 and find the relations between the duality and the local gauge covariant structures. We discuss the deformations induced from the generalized Green-Schwarz transformation on the transformation rules of the supergravity fields and compare with previous results in the literature. Finally, in section 5 we present the first order α ′ -corrections of the heterotic string effective action including up to bilinear terms in fermions. Conclusions are the subject of section 6. The conventions used throughout the paper and some useful gamma function identities are included in appendix A. Details of the proof of closure of the symmetry algebra on the duality multiplets are contained in appendix B. Finally, in appendix C we compute the deformed supersymmetry algebra on the supergravity multiplets and prove the supersymmetric invariance of the first order corrections in the heterotic string effective action.

The leading order theory
In this section we review the basic features of the DFT reformulation of N = 1 supergravity coupled to n g vector multiplets in ten dimensions that was introduced in [38], mainly to establish the notation. The frame formalism used in [42] is most useful to achieve a manifestly O(10, 10 + n g ) covariant rewriting of heterotic supergravity truncated to the Cartan subalgebra of SO (32) or E 8 × E 8 for n g = 16. Employing gauged DFT [43], we further include the full set of non-abelian gauge fields and recover the leading order terms of heterotic supergravity. The group invariant symmetric and invertible O(10, 10 + n g ) metric is H MN is also an element of O(10, 10 + n g ), constrained as It is convenient to define the projectors satisfying the usual properties and related with the generalized vielbein in the following way We use the convention that P AB , P AB and their inverse lower and raise projected indices.
The generalized Lie derivative acts as where the partial derivatives ∂ M belong to the fundamental representation of O(10, 10+n g ) and the so-called fluxes or gaugings f MNP are a set of constants [42] verifying linear and quadratic constraints Consistency of the construction requires constraints which restrict the coordinate dependence of fields and gauge parameters. The strong constraint where · · · refers to products of fields, will be assumed throughout. This constraint locally removes the field dependence on 10 + n g coordinates, so that fermions can be effectively defined in a 10-dimensional tangent space 1 .
The local O(9, 1) L × O(1, 9 + n g ) R double Lorentz symmetry is parameterized by an infinitesimal parameter Γ AB satisfying 9) in order to preserve the invariance of η AB and H AB . The two projections of a generic where the Γ A B and Γ A B components generate the O(9, 1) L and O(1, 9 + n g ) R transformations leaving P AB and P AB invariant, respectively, and δ Λ H AB = 0 implies Γ AB = 0.
The fields transform under double Lorentz variations as where the O(9, 1) L gamma matrices can be chosen to be conventional gamma matrices in ten dimensions, satisfying Some useful identities for the product of gamma matrices are listed in Appendix A.1.
The Lorentz and space-time covariant derivatives act on generic vectors as Only the totally antisymmetric and trace parts of ω ABC can be determined in terms of E M A and d, namely the latter arising from partial integration with the dilaton density for arbitrary V and V A . Only the combinations with the same projection on the last two indices are non-vanishing.
The covariant derivatives of the (adjoint) gravitino and dilatino are The supersymmetry transformation rules are parameterized by an infinitesimal Majorana fermion ǫ transforming as a spinor of O (1,9) Putting all together, the generalized fields obey the transformation rules In Appendix B.1 we review the algebra of these transformations, and show that it closes up to terms with two fermions, with the following parameters where the C f -bracket is defined as The transformation rules (2.19) leave the following action invariant, up to bilinear terms in fermions, where L B is the generalized Ricci scalar, which can be written as up to terms that vanish under the strong constraint, and the fermionic Lagrangian is Using the Bianchi identity it is useful to rewrite The supersymmetry variation of the bosonic piece of the action gives where we have used and The supersymmetry transformation rules define the following Lichnerowicz principle and then, the supersymmetric variation of the fermionic piece of the action

Parameterization and choice of section
To make contact with ten dimensional N = 1 supergravity coupled to n g vector multiplets, we split the G and H indices as M = ( µ , µ , i) and A = (A, A), respectively with A = a, A = (a, i), µ , µ , a, a = 0, . . . , 9, i, i = 1, . . . , n g , and parameterize the generalized fields as follows: -Generalized frame where e a and e a are two vielbein for the same ten dimensional metric. To guarantee that the number of DFT and supergravity degrees of freedom agree, we gauge fix e µ a = e µ a , e µa = e µa , and identify e µ a , e µa with the supergravity vielbein e µ a , e µa , a, b = 0, . . . , 9, respectively, i.e. g µν = e µ a g ab e ν b , with g ab the Minkowski metric. C µν = b µν + 1 2 A i µ A νi , with A i µ being the gauge connection. For consistency, we also need to impose with e i i the (inverse) vielbein for the Killing metric of the SO(32) or E 8 × E 8 gauge group, η ij = e i i η ij e j j , as required for modular invariance of the heterotic string.

35)
χ i being the standard gaugino field.
The non-abelian gauge sector is trivially incorporated through the gaugings that deform the generalized Lie derivative (2.6a) as The γ-functions γ a = γ a δ a a verify the Clifford algebra {γ a , γ b } = 2g ab . The gauge fixing e µ a = e µ a implies δe µ a = δe µ a , and (2.11) lead to where Λ ab denotes the generator of O (1,9) transformations that parameterizes Γ ab .
The additional gauge fixings δE i i = 0 and δE µ i = 0 lead respectively to where we have parameterized ξ M = (ξ µ , ξ µ , ξ i ) and Λ ai , Λ ij are introduced for convenience, as we will discuss in section 4.
Solving the strong constraint in the supergravity frame, parameterizing (2.18) and using the non-vanishing determined components of the generalized spin connection listed in Appendix A.2, we recover the leading order supersymmetry transformation rules of the coupled ten dimensional N = 1 supergravity and Yang-Mills fields, namely where w (+) µab = w µab + 1 2 H µab is the spin connection with torsion given by the field strength of the b-field µνρ the Yang-Mills Chern-Simons form The Lorentz transformations of the supergravity and super Yang-Mills multiplets obtained from (2.11) are 42) and the gauge transformations derived from (2.6) are where the second term in the gauge transformation of the b-field is the gauge sector of the Green-Schwarz transformation required for anomaly cancellation.
Parameterizing the DFT action (2.22), using the fluxes listed in Appendix A.2, we get (2.44) We use standard notation defined in Appendix A. Both the action and the transformation rules match the corresponding ones in [16], with the field redefinitions specified in 3 The first order α ′ -corrections In this section we construct the first order corrections to N = 1 supersymmetric DFT, performing a perturbative expansion of the exact formalism developed in [41].
The duality structure of the first order α ′ -corrections to heterotic supergravity was originally considered in [32,33]. Exploiting a symmetry between the gauge and torsionful spin connections that exists in ten dimensional heterotic supergravity [15,16], the duality group was extended to O(10, 10+n g +n l ), with n g (n l ) the dimension of the heterotic gauge (Lorentz) group. In this construction, the gaugings in the generalized Lie derivative (2.6a) preserve a residual O(10, 10) global symmetry. Including one-form fields in the GL (10) parameterization of the generalized vielbein, the formalism reproduces the first order corrections to the interactions of the bosonic fields in the heterotic effective field theory.
The lack of manifest duality covariance and the difficulties to incorporate higher orders of the α ′ -expansion in these formulations motivated the search of alternative frameworks.
A deformation of the gauge structure of DFT was proposed in [34], introducing a gen-

The generalized Bergshoeff-de Roo identification
The theory has a global O(10, 10+k) symmetry, where k is the dimension of the O(1, 9+k) group. This differs from the construction of the previous section, where the duality group is O(10, 10 + n g ) and n g denotes the dimension of the SO(32) or E 8 × E 8 heterotic gauge group. In the construction of [41] instead the gauge sector encodes the higher derivatives.
The vielbein E M A is an element of O(10, 10 + k), parameterized in terms of O(10, 10) The gauge freedom is used to set E αā to zero and the bijective map e α β relates the Cartan-Killing metrics of O(k), κ αβ and κ αβ , as The parameterization (3.1) preserves the constraint where η MN and η AB are the invariant metrics of O(10, 10 + k) and O(9, 1) The generalized O(10, 10 + k) gravitino splits as Ψ A = (0, Ψā, Ψ α ), where Ψ a is a generalized O(10, 10) gravitino and Ψ α is a gaugino of the O(1, 9 + k) R gauge group, that will later be identified with a function of the O(10, 10) generalized fields. The gamma matrices are γ A = (γ a , 0, 0), with γ a the O(9, 1) L gamma matrices verifying (2.12).
The transformation rules of the O(10, 10 + k) fields have the same functional form as Equivalent constraints to (2.7) and (2.8) must be imposed, i.e.
The gauge fixing δE α a = 0 implies and δe α α = 0 determines (3.14) The gauge generators (t α ) A B implement the map Parameterizing δE M a one gets In order to eliminate these extra degrees of freedom, it is convenient to define which allows to establish the generalized Bergshoeff-de Roo identification between the generalized gauge and spin connections 20) and to determine Ψ CD as the generalized gravitino curvature since both sides of (3.20) and (3.21) transform in the same way. The main steps of the demonstration can be found in [41].
We now proceed to extract the first order α ′ -corrections to the transformation rules of the O(10, 10 + n g ) generalized fields.

Induced transformation rules on O(10, 10) multiplets
The covariant transformation rules (3.7) induce higher derivative deformations on the transformations (2.19) of the O(10, 10 + n g ) fields. In this section, we work out the first order modifications, expanding the coefficients ( To simplify the presentation, we turn off the gauge sector of the O(10, 10 + n g ) multiplets, i.e. we take n g = 0, and obtain the induced transformation rules of the O(10, 10) fields. The gauge sector will be trivially included in the next subsection.
It is convenient to first express the components of the generalized O(10, 10 + k) fluxes we get the first order deformations where we used (1−X R )g 2 , the superscripts (2) and (3) refer to the number of derivatives, and we defined The transformation rules (3.7) take the following form:

− Vielbein
The identification E M a = E M a implies δE M a = δE M a , and from (3.7a) we get Using the gauge fixing (3.13) and the following relation which holds for any function f , one gets The second term in the r.h.s. of this expression allows to identify T ab with the Γ ab component of the Lorentz parameter (2.9). The third term contains the deformation which is the leading order of the O(10, 10) covariant generalization of the Green-Schwarz transformation [34]. And finally, the last term in (3.28) contains the first order correction to the supersymmetry transformation rule (2.18a), namely Following a similar reasoning, one can see that the other projection transforms as where we have identified where we have kept the leading order terms in the O(10, 10 + k) gaugino identification (3.21). Note that there are two corrections to the Lorentz transformations. The first term in the right hand side can be interpreted as a generalized Green-Schwarz transformation and the second one depends on the gravitino curvature, that we now define.
− Gravitino curvature To leading order in (3.21), the induced O(10, 10) gravitino curvature is, From (3.7c), we find that it obeys the transformation rule The first order corrections to the transformation rules of the generalized dilatino (2.19e) that are obtained from (3.7d) are Note that the transformation rules of the dilaton (2.19c) as well as the diffeomorphisms on all the fields are not corrected.

Including the heterotic gauge sector
It is now trivial to include the gauge sector of the O(10, 10 + n g ) formulation. We simply In Appendix B.2 we show that the algebra of these transformation rules closes, up to terms with two fermions, with the following field-dependent parameters

First order corrections to N = 1 supersymmetric DFT
The invariant action under the transformation rules (3.7) is clearly of the same functional form as (2.22) but it depends on the O(10, 10 + k) multiplets, namely Hence it contains higher derivatives of the O(10, 10 + n g ) multiplets.
The transformation rules (3.7) define the following Lichnerowicz principle, and then the O(10, 10 + k) generalized Ricci scalar determines the corrections to the generalized Dirac operator.
In terms of the O(10, 10 + n g ) generalized fluxes, the O(10, 10 + k) generalized Ricci scalar is, up to first order, where R was defined in (2.25). Replacing the expressions (3.23) with the overlined indices extended to include the gauge sector (i.e. c, d, ... → C, D, ...), R (1) may be written as Note that it depends on the generalized gravitino through F * aBC . Similarly, we may define where L F was introduced in (2.23) and the first order corrections are given by F , up to bilinear terms in fermions. We have explicitly verified that the action To find the relations between both sets of fields, it is convenient to first work out the parameterizations of the generalized fluxes and curvatures and their transformation rules. From the first order terms in the action (3.47), we see that only the leading order expressions are necessary. We denote the parameterization of F * aCD aŝ where the hats distinguish objects that contain fermions and the collective indices of the tangent space C = (c, i) include the gauge indices. In terms of supergravity and super Yang-Mills fields, the components arê with w The generalized gravitino curvature Ψ AB is parameterized as is the parameterization of the generalized flux component F ABi .
The transformation rule (4.9) contains, other than the standard Lorentz transformations, the supersymmetry variation of the torsionful spin connection [15,16] δ ǫŵ the supersymmetry and gauge transformations of the Yang-Mills field strength, and δ ξFµci = f ijk ξ jF Similarly, from the transformation rule of the generalized gravitino curvature cdAB γ cd ǫ (4.12) we obtain where we have definedR which has componentŝ In particular, (4.13) contains the supersymmetry transformation rule of the supergravity gravitino curvature µνab is the two-form curvature computed from the torsionful spin connectionŵ [15,16]. Now we turn to the parameterization of the elementary fields. We start from the deformed transformation rules of the components E M a and E M a given in (3.37a) and (3.37b). Of course, different definitions lead to supergravity multiplets that obey different transformation rules. An interesting one is the following whereT ab =F aciFb ci andT =F i acF ac i . The quadratic terms in spin and gauge connections are known to be necessary in order to remove the non-standard Lorentz transformations of the supergravity vielbein e µ a and dilaton φ fields [34,35]. Together with the gauge covariantT terms, these parameterizations determine e µ a and φ fields that obey the leading order supersymmetry and Lorentz transformation rules (2.39a) and (2.42). To get this result, the gauge fixings e µ a = e µ a ≡ e µ a , δE i i = 0 and δE µ i = 0 are used to absorb several terms into the Lorentz parameters. As a consequence, the following parameterization is needed for the duality covariant gravitino Interestingly, these parameterizations induce a deformation of the gravitino supersymmetry variation (2.39c) that can be absorbed into the torsion of the spin connection through the following modification of the two-form curvature The Yang-Mills Chern-Simons form C (g) µνρ was defined in (2.41), the coefficient The gaugino bilinear terms in (4.22) may be absorbed into the first order deformation of the Yang-Mills Chern-Simons form replacing A i µ →Â µ jk , but this is not convenient for reasons that will become clear shortly.
The modified three-form H µνρ (4.22) may be rewritten as the compact expression Likewise, a parameterization of the dilatino analogous to (4.21) also induces the replacement of the lowest order H µνρ by H µνρ in the supersymmetry transformation rule (2.39c), so that the combination ρ = 2 λ + γ a ψ a and its supersymmetry transformation rule are not deformed, i.e. ρ = ρ and δ ǫ ρ = δ (0) ǫ ρ. From δE µ i and δΨ i in (3.37), one can see that the gauge and gaugino transformation rules are not deformed and hence it is not necessary to redefine these fields.
Finally, from the transformation rules of the components E µā or E µa , and using the parameterizations defined above, we get This compact expression contains information about the gauge, Lorentz and supersymmetry transformations of the b−field, which we now analyze separately.
Expanding the first term in (4.27) one gets (4.28) The first term in the r.h.s. is the Lorentz sector of the Green-Schwarz transformation [44], which requires the Lorentz Chern-Simons form (4.24) in H µνρ . It cannot be eliminated through redefinitions of the b-field [34]. The bilinear fermionic terms inŵ in order to compare with standard results. With this redefinition (4.22) becomes (4.31) Finally the third term in (4.28) together with the second term in (4.27) contain the first order deformations of the supersymmetry transformation of b µν , i.e. (4.32) The first term in (4.32) was originally introduced in [14] to restore manifest Lorentz covariance to the supersymmetry variation of the b-field curvature. It was later reobtained in [15] as a consequence of the assumption that the Yang-Mills and torsionful spin connections should appear symmetrically in ten dimensional N = 1 supergravity coupled to super Yang-Mills. The second term in (4.32) reflects the ̺ deformation of the Killing metric (4.23) in the zeroth order supersymmetry transformation (2.39b). These two terms are the obvious analogs of the Lorentz and Yang-Mills Green-Schwarz transformations as already noticed in [14]. Here, these transformations follow directly from the manifestly duality covariant formulation of the theory.
Interestingly, the second term in (4.27) can be obtained from the leading order transformation of the 2-form in (2.39b) with the identifications A i µ ↔Ω µ CD , χ i ↔ Ψ CD , i.e. a generalization of the symmetry A i µ ↔ŵ (−)cd µ , χ i ↔ ψ cd that was used in [15,16]   Before turning to the construction of the invariant action under the modified transformations, we analyze the deformations that were proposed in references [15,16]. In particular, we wonder if there is a parameterization of the duality covariant vielbein in terms of a gauge covariant one that transforms as proposed in [15] or [16], i.e.
respectively, written here in our conventions. Note that we only examine the gauge dependent terms since the gravitational sectors coincide up to the order we are considering.
Specifically, we search for a quantity E µ a such that e µ a = e µ a + E µ a and δ (1) e µ a = δ (0) E µ a . (4.38) The most general expressions that can reproduce either one of (4.37) can be schematically written as or as where the terms between parenthesis refer to all possible contractions of indices and numbers of γ-matrices, numerated by the supraindex m, while ψ . and ψ .. denote the gravitino and gravitino curvature, respectively. We found that neither of (4.37) can be reproduced. It is a straightforward though heavy exercise to parameterize the action (3.47). Interestingly, using Bianchi identities and integrations by parts, the action of the theory to O(α ′ ) may be written in the following compact form: where we have taken b = α ′ and defined / H = γ µνρ H µνρ and As expected, the bosonic fields reproduce the expression obtained from the scattering amplitudes of the heterotic string massless fields up to first order in α ′ and field redefinitions [10], i.e.  The supersymmetric invariance of the action (5.1) is shown in appendix C. It simply results from the observation that both the action and the transformation rules of the fields have the same structure as the corresponding ones in [16], albeit with collective indices, except for the terms contained in the parameter Λ ci = 1 2 √ 2ǭ γ c χ i , which cancel in the variation of the action.

Outlook and final remarks
In this paper we have obtained the first order corrections to N = 1 supersymmetric DFT performing a perturbative expansion of the exact supersymmetric and duality covariant framework introduced in [41]. The action has the same functional form as the leading order one constructed in [38], but it is expressed in terms of O(10, 10 + k) multiplets, where k is the dimension of the O(1, 9 + k) group. Decomposing the O(10, 10 + k) duality group in terms of O(10, 10 + n g ) multiplets, the theory contains higher derivative terms to all orders. We kept all the terms with up to and including four derivatives of the fields and bilinears in fermions.
The transformation rules of the O(10, 10 + k) multiplets obey a closed algebra and induce higher-derivative deformations on those of the O(10, 10 + n g ) fields. In particular, they produce a supersymmetric generalization of the duality covariant Green-Schwarz transformation that was found in [34]. We showed that the algebra of deformations closes up to first order and constructed the invariant action with up to and including four derivatives of the O(10, 10 + n g ) multiplets and bilinears in fermions.
To make contact with the heterotic string low energy effective field theory, we parameterized the duality covariant multiplets in terms of supergravity and super Yang-Mills fields. The inclusion of higher-derivative terms requires unconventional non-covariant field redefinitions in the parameterizations of the duality covariant structures. The definitions that reproduce the four-derivative interactions of the bosonic fields of the heterotic string effective action were found in [34,35]. Here, we worked with a set of fields related to the latter through gauge covariant redefinitions. Except for the two-form, the fields defined in section 4 obey the leading order transformation rules with a modification of the two-form curvature in the supersymmetry variations. The Lorentz and non-abelian gauge transformations of the two-form are deformed by the standard Green-Schwarz mechanism, as expected, and its supersymmetry transformations are deformed by Green-Schwarz-like terms plus some extra Yang-Mills dependent higher-derivative terms.
The deformed transformations obey a closed algebra, which guarantees the existence of an invariant action. We constructed such action in section 5, by parameterizing the manifestly duality covariant expression (3.47) in terms of the fields that obey supersymmetry transformation rules with the minimal set of deformations. As expected, the interactions of the bosonic fields agree with the results obtained from the heterotic string scattering amplitudes [10], up to terms proportional to the leading order equations of motion. To our knowledge, the three-derivative low energy interactions involving fermions have not been constructed directly from string theory. The action and transformation rules that we have obtained follow from an exact supersymmetric and duality covariant formalism. Hence the theory avoids an iterative procedure which only guarantees consistency up to a given order. Moreover, supersymmetry is manifest to all orders and dimensional reductions will preserve the expected T-duality symmetry of the theory.
Supersymmetric extensions of the Yang-Mills and Lorentz Chern-Simons forms have been constructed using the Noether method. In particular, a supersymmetric L(R) + L(R 2 ) invariant was obtained in [15,16] from the leading order action (2.44), using the symmetry between the gauge and torsionful spin connections. The three-derivative terms that are independent of the Yang-Mills fields in the action (5.1) coincide with those results.
But not surprisingly, the Yang-Mills field-dependent terms disagree with the corresponding expressions of the L(RF 2 ) + L(F 4 ) invariants proposed in those references, since the deformations of the transformation rules differ by Yang-Mills field-dependent terms.
The supersymmetric and T-duality covariant generalized Green-Schwarz transformation strongly restricts the modifications to the leading order supersymmetry transformation rules, and in particular, it does not allow the proposals of [15,16]. As argued in section 4 this does not imply that the latter are in conflict with string theory. In order to establish if they are compatible with the required T-duality symmetry, the corresponding invariant action should be dimensionally reduced.
The effort employed in the construction of the higher-derivative fermionic sector of the heterotic string effective field theory is justified for various reasons. First of all, an intriguing consequence of the duality covariant formalism is the natural appearance of the generalized collective tangent space indices C, D, ..., which allows to include the higher-derivative Yang-Mills field-dependent terms into gravitational structures such aŝ R µνCD ,Ω µCD or Ψ CD . In particular, it leads to relatively mild modifications of the leading order supersymmetry transformation rules of the fields, which permits the use of the leading order Killing spinor equations to obtain classical solutions containing higher-derivative corrections [2]. These features not only simplify the construction of new supersymmetric solutions but also allow to easily extend the known solutions for the gravitational sector to the Yang-Mills sector.
The fermionic contributions to the action are also relevant for applications to fourdimensional physics. Both the superpotential and D-terms can be more easily computed from the fermionic couplings [6] and the higher derivative corrections to these terms as well as to the Yukawa couplings could also have interesting consequences for string phenomenology and moduli fixing.
An obvious natural extension of our work would be to determine further interactions beyond the first order. The quartic interactions of the Yang-Mills fields that we have reproduced are mirrored by corresponding quartic Riemann curvature terms [10]. Consequently, we expect that the higher orders of perturbation will reproduce these higherderivative corrections. It would be interesting to see if the generalized structures with capital indices persist to higher orders. If they do, the formulation would contain information about higher than four-point functions in the string scattering amplitudes.
Nevertheless, there is another quartic Riemann curvature structure that has no analog in the Yang-Mills sector [10]. At tree level, these terms are proportional to the transcendental coefficient ζ(3). The analysis of the higher-derivative terms is technically more challenging but also more interesting, since further duality covariant structures, or even a more drastic change of scheme, seem to be necessary as advocated in [45].
Performing a generalized Scherk-Schwarz compactification of the sub-leading corrections to N = 1 supersymmetric DFT would be another promising line of research, as this would produce higher-derivative corrections to lower dimensional gauged supergravities [46,35]. We hope to return to these and related questions in the future.

A Conventions and definitions
In this appendix we introduce the conventions and definitions used throughout the paper. Space-time and tangent space Lorentz indices are denoted µ, ν, . . . and a, b, . . . , respectively.
The covariant derivative acting on a gauge tensor G µ ci and on a spinor ǫ is, respectively, and the torsionful spin connection The commutator of covariant derivatives acting on gauge tensors and spinors is where the Riemann tensor is defined as and the Yang-Mills field strength is The Ricci tensor and scalar are (A.11)

A.2 Leading order components of the generalized fluxes
Using the parameterizations introduced in section 2 and solving the strong constraint in the supergravity frame, the non-vanishing determined components of the generalized spin connection are, to leading order, and f ijk are the structure constants of the SO(32) or E 8 × E 8 gauge groups.

A.3 The leading order action and equations of motion
Here we rewrite the zeroth order action (2.44) in terms of the dilatino λ of the supergravity multiplet and compare with the corresponding expression in [16]. We also list the leading order equations of motion of all the massless fields derived from it.
Rewriting the generalized dilatino ρ = 2λ + γ µ ψ µ in terms of λ and ψ and integrating by parts, the action (2.44) takes the form It matches the corresponding expression in [16] with the following field redefinitions: The leading order equations of motion of all the massless fields, written in terms of ρ, B Algebra of transformations of O(10, 10 + n g ) fields In this appendix we show that the algebra of transformation rules closes, up to terms with two fermions. We first review the algebra of zeroth order transformations (2.19) and in B.2 we include the first order corrections. We define [δ 1 , δ 2 ] = −δ 12 .

B.1 Leading order algebra
We focus on the algebra determined by the leading order transformations (2.19) and show that it closes with the parameters (2.20). We split the algebra of transformations on the generalized fields into the following commutators: − Supersymmetry transformations of the dilaton where we have usedǭ 1 γ a ǫ 2 = −ǭ 2 γ a ǫ 1 and ǫ 1 γ abc ǫ 2 = ǫ 2 γ abc ǫ 1 , and defined − Diffeomorphisms on the dilaton − Mixed supersymmetry and double Lorentz transformations on the dilaton − Mixed diffeomorphisms and supersymmetry variations on the dilaton − Supersymmetry variations of the frame Projecting with E M C , we get where we have used (2.14) and ξ ′M 12 is the generalization of (B.2), i.e.
Projecting with E M c we find Following similar steps, we get (B.14) − Diffeomorphisms and double Lorentz variations of the frame Note that ξ ′′M 12 in (B.4) does not contain the second and third terms in the r.h.s. of this expression, due to the strong constraint.
− Mixed diffeomorphisms and supersymmetry variations of the frame − Mixed double Lorentz and supersymmetry variations of the frame − Mixed diffeomorphisms and supersymmetry transformations of the gravitino − Mixed supersymmetry and double Lorentz transformations of the gravitino − Diffeomorphisms and double Lorentz transformations of the gravitino − Mixed supersymmetry and double Lorentz transformations of the dilatino Summarizing we have found, up to bi-linear terms in fermions, The commutator of supersymmetry variations on the gravitino and dilatino as well as the missing terms δ ξ ′ 12 ρ and δ ξ ′ 12 Ψ A are not included as they are of higher order in fermions.

B.2 First order algebra
We now work out the algebra of first order transformations (3.37) and show that it closes with the parameters (3.38), up to terms with two fermions. Here we denote δ ≡ δ (0) + δ (1) and [δ 1 , δ 2 ] = δ 12 . We split the algebra as in the previous section.
− Double Lorentz transformations on the generalized frame Repeating the procedure for E M a , we find defined in (B.31) and − Mixed supersymmetry and double Lorentz transformations on the generalized frame Using we get the first order contribution to the mixed transformation rules of The first two terms are a Lorentz transformation with parameter From the second line, only one term survives after commuting the gamma matrices, which corresponds to a first order supersymmetric variation with zeroth order parameter cd . In the same way, from the remaining terms we find a first-order supersymmetry pa- The first line is a zeroth order Lorentz transformation with parameter Commuting the gamma matrices in the first term of the second line, the second contribution in the fourth line is canceled and we get again a supersymmetry transformation with zeroth order parameter ǫ ′′ 12 = − 1 2 ǫ [1 γ cd Λ 2]cd . Finally, commuting the gamma matrices in the second term of the third line, various cancellations leave a supersymmetry transformation with first order parameter (B.37).

− Supersymmetry variations on the generalized frame
The first and last terms of the r.h.s. combine into a Lorentz transformation with while the other terms form a diffeomorphism with first order parameter The same result holds for E M C δ ǫ 1 , δ ǫ 2 E Ma , while − Mixed diffeomorphism and Lorentz variations of the generalized frame Recalling that diffeomorphisms are not deformed, we get to first order which is a first-order Lorentz transformation with a zeroth order parameter. We use the This case is similar to the previous one. We start with which is a first order supersymmetry transformation with a zeroth order parameter. It is straightforward to see that the same result holds for E Ma .
− Double Lorentz variations on the generalized gravitino After some straightforward manipulations, we finally obtain Lorentz transformations with the following parameters CD − Mixed Lorentz and supersymmetry transformations on the generalized gravitino Acd γ cd ǫ 1 + δ Commuting the gamma matrices in the second term of the r.h.s, and combining it with the corresponding term in the (1 ↔ 2) operation, we recognize a supersymmetry transformation with zeroth order parameter ǫ ′ 12 = − 1 2ǭ [1 γ ab Λ 2]ab . The first term in the second line together with the corresponding term in the (1 ↔ 2) operation, gives a zeroth order supersymmetry transformation with first order parameter The remaining terms cancel and then we get In the second line (adding the (1 ↔ 2) operation) we recognize a Lorentz transformation with first and zeroth order parameters In equations (3.38) of the main text we collect the parameters that appear in this algebra of first order transformation rules.

C Supersymmetry of heterotic string effective action
In the first part of this appendix we prove that the higher-derivative deformations of the transformation rules of the supergravity fields satisfy a closed algebra up to O(α ′ ) and up to terms with two fermions. In the second part, we show that the action (5.1) is invariant under these supersymmetry transformations.

C.1 Supersymmetry algebra
It is well known that the algebra of leading order transformations of supergravity and super Yang-Mills fields closes. Moreover, the replacement H µνρ → H µνρ in the supersymmetry transformations of the gravitino and dilatino does not affect the leading order closure on any field except for the b-field. Hence we focus on the algebra of first order transformation rules on b µν .
It is convenient to first look at the brackets acting on b µν = b µν + b 8 A k [µ χ i γ ν] χ j f ijk . Up to first order and bilinear terms in fermions, we need the following transformation rules: µab γ ab ǫ , (C.1a) We exclude the diffeomorphisms since it is trivial to see that all the transformation rules of b µν (i.e. Lorentz, supersymmetry, abelian and non-abelian gauge transformations) transform as tensors under diffeomorphisms and hence their commutators are trivial.
Therefore, we compute the brackets The first term in the r.h.s. gives Adding both contributions, we get To see the algebra of transformations on b µν , note that and it is easy to see that the second term in the r.h.s. vanishes. Rewriting (C.5) in terms of supergravity and super Yang-Mills fields, the brackets that mix supersymmetry with Lorentz and abelian gauge transformations vanish, while the supersymmetry algebra gives (C.10)