Higher-derivative Heterotic Double Field Theory and Classical Double Copy
aa r X i v : . [ h e p - t h ] J a n Higher-derivative Heterotic Double FieldTheory and Classical Double Copy
Eric Lescano † and Jes´us A. Rodr´ıguez ∗ † Instituto de Astronom´ıa y F´ısica del Espacio (IAFE-CONICET-UBA)
Ciudad Universitaria, Pabell´on IAFE, 1428 Buenos Aires, Argentina ∗ Departamento de F´ısica, FCEyN, Universidad de Buenos Aires (UBA)
Ciudad Universitaria, Pabell´on 1, 1428 Buenos Aires, Argentina [email protected], [email protected]
Abstract
The Generalized Kerr-Schild Ansatz (GKSA) is a powerful tool for constructing exactsolutions in Double Field Theory (DFT). In this paper we focus in the heterotic formu-lation of DFT, considering up to four-derivative terms in the action principle, while thefield content is perturbed by the GKSA. We study the inclusion of the generalized ver-sion of the Green-Schwarz mechanism to this setup, in order to reproduce the low energyeffective heterotic supergravity upon parametrization. This formalism reproduces higher-derivative heterotic background solutions perturbed by a pair of null vectors. Then westudy higher-derivative contributions to the classical double copy structure. We use anorthogonality condition between the null vectors to impose linear perturbations on themetric and the b-field. We reproduce higher-derivative solutions for the gravity equationsin terms of a pair of U(1) gauge vectors in agreement with the KLT relation for heteroticstring theory. ontents Introduction
Double Field Theory (DFT) [1, 2] is a duality symmetric formalism that can be understoodas a generalization of D -dimensional Riemannian geometry, manifestly invariant under theaction of O ( D, D ). This group is closely related with an exact symmetry of String Theory[3]. One of the most exciting features of this formalism is that the space-time must bedoubled to accomplish O ( D, D ) as a global symmetry of the theory [4]. The generalizedcoordinates of the double space X M = ( x µ , ˜ x µ ) are in the fundamental representation of O ( D, D ), where ˜ x µ is the extra set of coordinates and M = 0 , . . . , D −
1. However, thetheory is constrained by the section condition (or strong constraint), which effectivelyremoves the dependence on ˜ x µ .The fundamental bosonic fields consist of a symmetric generalized tensor H MN ( X )and a generalized scalar d o ( X ). While the former is a multiplet of the duality group,the latter is an invariant, and they are usually referred as the generalized metric and thegeneralized dilaton. On the other hand, DFT can easily describe the low energy limit ofHeterotic String Theory [5]. This formalism requieres a generalized frame [6] instead of ageneralized metric to embed the gravitational degrees of freedom of the theory. A similarapproach is the flux formulation of DFT [7].One important issue that can be studied with DFT is the incorporation of higherderivative terms in supergravity frameworks. In [8] a biparametric family of higher-derivative duality invariant theories was presented. For particular values of the param-eters, the higher-derivative contributions reproduce the α ′ -corrections of the heteroticsupergravity, also studied in [9].In this work we are interested in considering H MN and d o as arbitrary backgroundfields that can be perturbed. We focus in the Generalized Kerr-Schild Ansatz (GKSA)introduced in [10], which is an exact and linear perturbation of the generalized backgroundmetric, H MN = H MN + κ ¯ K M K N + κK M ¯ K N , (1.1)where κ is an arbitrary parameter that allows to quantify the order of the perturbations2nd ¯ K M = ¯ P M N ¯ K N and K M = P M N K N are a pair of generalized null vectors η MN ¯ K M ¯ K N = η MN K M K N = η MN ¯ K M K N = 0 , (1.2)where η MN is an O ( D, D ) invariant metric and ¯ P MN = ( η MN + H MN ) and P MN = ( η MN − H MN ) are used to project the O ( D, D ) indices. This ansatz (plus a generalizeddilaton perturbation) is the duality invariant analogous of the ordinary Kerr-Schild ansatz[11] [12], and the inclusion of a pair of generalized null vectors is closely related withthe chiral structure of DFT. This duality invariant ansatz was recently used in differentcontext as exceptional field theory [13], supersymmetry [14], among others [15].In [16] a consistent way to impose the GKSA in a two-derivative heterotic DFT frame-work was described. In that work the authors considered a generalized metric approachto recover the leading order contributions. Here we extend the formalism by consideringhigher-derivative terms in the flux formulation of DFT. We use a systematic method forobtaining these corrections, closely related with [17]. Higher-derivative contributions re-quiere agreement between the ansatz and the generalized version of the Green-Schwarzmechanism as we show.As an application we study higher-derivative contributions to the classical double copy[18] in a heterotic supergravity background. We consider the case l. ¯ l = 0 in order tohave linear perturbations on the metric and b-field, g µν = g oµν − κl ( µ ¯ l ν ) , (1.3) b µν = b oµν − κl [ µ ¯ l ν ] . (1.4)We focus in the R ( − ) µν ab R ( − ) µν ab contributions with H µνρ = H oµνρ = 0 and we obtainthe leading four-derivative corrections to the single copy and zeroth copy. These are κ corrections to the Maxwell-like equations that described the gravity sector after identifyingthe null vectors with a pair of U (1) gauge fields, in agreement with the KLT relation forheterotic string theory. The particular case where l µ = ¯ l µ and therefore A µ = ¯ A µ is alsopresented.This work is organized as follows: In Section 2 we introduce the field content, thesymmetries and the action principle of the low energy effective heterotic supergravity, See also [19] for recent approaches to this topic. O ( D, D + K ), and we perform a suitable breaking to O ( D, D ) by identifyingthe extra gauge field with a particular flux component. With this method we obtainthe higher-derivative extension to DFT. Here we choose the free parameters of the con-struction to match with the heterotic DFT formulation. In Section 5 we parametrize thetheory in terms of the field content of heterotic supergravity. We discuss about the ten-sion between the perturbation of the gravitational sector in terms of a pair of null vectorwith respect to the absence of a perturbation for the gauge sector. Then we apply ourformalism in Section 6 to explore higher-derivative corrections to the heterotic ClassicalDouble Copy. Finally, in Section 7, we present the conclusions of the work.
In the first part of this section we review the D = 10 heterotic supergravity considering α and β contributions, according to [20] . Then we impose the supergravity version ofthe GKSA to perturbe the background fields. The low energy effective action that describes D = 10 heterotic string theory to first orderin α ′ is Z M e − φ o (cid:16) R o − ∂ µ φ o ∂ µ φ o −
112 ˆ H oµνρ ˆ H µνρo −
14 ( F oµνi F µνio + R ( − ) oµν ab R ( − ) µνo ab ) (cid:17) , (2.1)where µ = 0 , . . . , a = 0 , . . . , i = 1 , . . . , n with n the dimension of the Yang-Millsgroup, typically n = 496. The 10-dimensional background fields consist of a backgroundmetric g oµν = e oµa η ab e oνb , the background Kalb-Ramond field b oµν , the background gaugefield A oµi and the background dilaton φ o . The action (2.1) is written in terms of the We take α = 1 and β = 1 to simplify notation. Conventions for the field content follows [21] i.e. ,ˆ H oµνρ = 3 (cid:20) ∂ [ µ b oνρ ] − (cid:18) A io [ µ ∂ ν A oρ ] i − f ijk A ioµ A joν A koρ (cid:19) − (cid:18) ∂ [ µ Ω ( − ) cdoν Ω ( − ) oρ ] cd + 23 Ω ( − ) aboµ Ω ( − ) oνbc Ω ( − ) oρ ca (cid:19)(cid:21) ,F ioµν = 2 ∂ [ µ A ioν ] − f ijk A joµ A koν ,R ( − ) oµνab = (cid:0) − ∂ [ µ w oν ] ab + 2 w o [ µ | ac w o | ν ] cb (cid:1) , (2.2)where the spin and Hull connections are defined as w oµab = − e νo a ∂ µ e oνb + Γ σo µν e oσb e νo a , (2.3)Ω ( − ) oµab = w oµab −
12 ˆ H oµνρ e νo a e ρob , (2.4)and the Christoffel connection isΓ σo µν = 12 g σρo ( ∂ µ g oνρ + ∂ ν g oµρ − ∂ ρ g oµν ) . (2.5)The action principle (2.1) is invariant under Lorentz transformations. These transfor-mations acting on a generic vector V a reads δ Λ V a = V b Λ ba , (2.6)where Λ ab = − Λ ba is the Lorentz parameter. Contraction of indices in (2.6) make use ofthe Lorentz metric η ab .When higher-derivative terms are considered, the invariance of (2.1) requires that theKalb-Ramond field transforms with a non-covariant transformation, namely, δ Λ b oµν = − Ω ( − ) o [ µ ab ∂ ν ] Λ ab . (2.7)The previous transformation is known as the Green-Schwarz mechanism and it is a higher-derivative transformation that cannot be removed with field redefinitions.5he equations of motion for this setup are given by∆ φ o = R o + 4 g µνo ( ∇ µ ∇ ν φ o − ∂ µ φ o ∂ ν φ o ) − g µσo g ντo g ρξo ˆ H oµνρ ˆ H oστξ (2.8) − g µρo g νσo F oµν i F oρσi − g µρo g νσo R ( − ) oµνab R ( − ) oρσab = 0 , ∆ g oµν = R oµν + 2 ∇ µ ∇ ν φ o − g στo g λξo ˆ H oσλµ ˆ H oτξν − g στo F oσµi F oτνi − g στo R ( − ) oσµab R ( − ) oτνab = 0 , (2.9)∆ b oµν = g ρσo ∇ ρ (cid:16) e − φ o ˆ H oµνσ (cid:17) = 0 , (2.10)∆ A oν i = g ρµo ∇ (+ ,A ) ρ (cid:0) e − φ o F oµν i (cid:1) = 0 , (2.11)where we have defined ∇ (+ ,A ) ρ F oµνi = ∂ ρ F oµν i − Γ (+) oρµσ F oσν i − Γ (+) oρν σ F oµσi − f jki A oρj F oµνk , (2.12)Γ (+) ρoµν = Γ ρoµν + 12 ˆ H oµνσ g σρo , (2.13)which covariantizes the derivative with respect to ten dimensional diffeomorphisms andgauge transformations using the Christoffel and gauge connection respectively. Here wemention that the equations (2.8)-(2.11) are not strictly the ones obtained from variationsof the action with respect to the fundamental fields, but combinations of them. Now we are interested in imposing the supergravity version of the GKSA on the previousformulation. For simplicity we study solutions with around a Minkowski background g µν = η µν and we do not consider perturbations of the gauge field, i.e. , A µi = A oµi . Theinverse of the 10-dimensional background metric is perturbed as g µν = η µν + κl ( µ ¯ l ν ) , (2.14)where l µ and ¯ l µ are null vectors with respect to g µν and η µν , i.e. , l µ l ν g µν = l µ η µν l ν = 0 , (2.15)¯ l µ ¯ l ν g µν = ¯ l µ η µν ¯ l ν = 0 . (2.16)6he previous objects also satisfy the following relations¯ l ν ∂ ν l µ = 0 , l ν ∂ ν ¯ l µ = 0 , (2.17)which reduce to the standard goedesic condition for flat backgrounds when l and ¯ l areidentified [10]. The perturbation of the Kalb-Ramond field is given by b µν = b oµν − κ κl. ¯ l l [ µ ¯ l ν ] , (2.18)while the perturbation of the ten-dimensinal vielbein is e µa = e oµa − κ κ ( l. ¯ l ) l ( µ ¯ l ν ) e νao + 12 (cid:16) κ κ ( l. ¯ l ) (cid:17) ( l. ¯ l ) l µ l ν e νao , (2.19)and the dilaton is perturbed as showed in [14].Inspecting the Lorentz transformation rules for the exact fields we find δ Λ e µa = e µb Λ ba ,δ Λ b µν = − Ω ( − )[ µ ab ∂ ν ] Λ ab , (2.20)while the dilaton and the gauge field are Lorentz invariant. Since l a = e µa l µ and ¯ l a = e µa ¯ l µ transform as Lorentz vectors, the background b oµν receives κ -corrections to its Green-Schwarz transformation given by the κ -corrections induce in the Hull connection, δ Λ b oµν = − (Ω ( − )[ µ ab − Ω ( − ) o [ µ ab ) ∂ ν ] Λ ab . (2.21) In this section we review DFT and the GKSA following the conventions of [14]. TheGKSA was formulated in [10] as an exact and linear perturbation of the generalized back-ground metric H MN ( M, N = 0 , . . . , D −
1) and an exact perturbation of the generalizedbackground dilaton d o . We work with arbitrary D until parametrization.7ince the generalized metric is an O ( D, D ) element, its perturbation has the followingform H MN = H MN + κ ( ¯ K M K N + K M ¯ K N ) , (3.1)where ¯ K M = ¯ P M N ¯ K N and K M = P M N K N are a pair of generalized null vectors η MN ¯ K M ¯ K N = η MN K M K N = η MN ¯ K M K N = 0 . (3.2)According to (3.1), the DFT projectors are P MN = P MN − κ ( ¯ K M K N + K M ¯ K N )¯ P MN = ¯ P MN + 12 κ ( ¯ K M K N + K M ¯ K N ) . (3.3)Each O ( D, D ) multiplet can be written as a sum over its projections, V M = P M N V N + ¯ P M N V N = V M + V M , (3.4)where V M is a generic double vector. When we use the underline and overline notation,we consider the background projectors P MN and ¯ P MN .The generalized background dilaton can be perturbed with a generic κ expansion, d = d o + κf , f = ∞ X n =0 κ n f n , (3.5)with n ≥ K M , ¯ K M and f obey some conditions in order to produce finite deformations in the DFT action andEOM’s. If we consider a generic double vector V N , the covariant derivative can be definedas ∇ M V N = ∂ M V N − Γ MN P V P , (3.6)where Γ MNP is the generalized affine connection. Demanding ∇ M H NP = 0 , ∇ M η NP = 0 , (3.7)and a vanishing generalized torsion Γ [ MNP ] = 0 , (3.8)8he following projections of Γ MNP = Γ
MP N are well-defined and can be perturbed,Γ
MNQ = − ¯ P QR P M S ∂ S P RN , Γ MNQ = ¯ P N R ¯ P M S ∂ S P RQ , Γ MNQ = 2 ¯ P [ N R ¯ P Q ] S ∂ S P RM , Γ MNQ = 2 ¯ P M R P [ N S ∂ S P Q ] R . (3.9)Similarly to Riemannian geometry, the generalized Ricci scalar and the generalizedRicci tensor can be constructed from different (determined) projections of the generalizedaffine connection. Following the original construction of the GKSA we impose,¯ K P ∂ P K M + K P ∂ M ¯ K P − K P ∂ P ¯ K M = 0 ,K P ∂ P ¯ K M + ¯ K P ∂ M K P − ¯ K P ∂ P K M = 0 , (3.10)and K M ∂ M f = ¯ K M ∂ M f = 0 . (3.11)Using (3.8), we can change ∂ → ∇ in (3.10) obtaining,¯ K P ∇ P K M + K P ∇ M ¯ K P − K P ∇ P ¯ K M = 0 ,K P ∇ P ¯ K M + ¯ K P ∇ M K P − ¯ K P ∇ P K M = 0 . (3.12)In the next part we explore how the previous conditions appear in the flux formalism ofDFT [7]. Then we explicitly compute the EOM’s of the field content of DFT when weimpose the GKSA in this formalism. The generalized flux formulation of DFT is closely related with the generalized frameformulation introduced in [6]. The latter is compatible with the GKSA if we considerperturbations of the form, E M A = E M A + 12 κE M B K B ¯ K A , E M A = E M A − κE M B ¯ K B K A , (3.13)where K A = E M A K M = E M A K M and ¯ K A = E M A ¯ K M = E M A ¯ K M and E MA is an O ( D, D ) /O ( D − , L × O (1 , D − R frame. In this formulation A = 0 , . . . , D − A = 0 , . . . , D − O ( D − , L and O (1 , D − R indices, respectively.9e can define flat invariant projectors as follows, P AB = E MA E M B = P AB , ¯ P AB = E MA E M B = ¯ P AB , (3.14)where P AB = E MA E M B and ¯ P AB = E MA E M B are the standard DFT flat projectors.Using these projectors we can construct two invariant metrics, η AB = E MA η MN E NB = E MA η MN E NB , (3.15) H AB = E MA H MN E NB = E MA H MN E NB . (3.16)The flat covariant derivative acting on a generic vector V B is D A V B = E A V B + W ABC V C , (3.17)where E A = √ E M A ∂ M and W ABC is the generalized spin connection that satisfies W ABC = −W ACB and W ABC = W ABC = 0 . (3.18)With the help of the generalized frames we can construct the generalized fluxes, whichare defined as F ABC = 3 E [ A ( E M B ) E MC ] , F A = √ e d ∂ M (cid:0) E M A e − d (cid:1) . (3.19)In the flux formulation of DFT, conditions (3.10) and (3.11) become K A E A ¯ K C + K A ¯ K B F ABC = 0 , ¯ K A E A K C + ¯ K A K B F ABC = 0 , (3.20)and K A E A f = ¯ K A E A f = 0 . (3.21)It is straightforward to check that the previous conditions are double Lorentz invariantusing δ Γ E MA = E M B Γ BA , δ Γ E MA = E M B Γ BA (3.22)10here Γ AB = − Γ BA is the double Lorentz parameter.Only the totally antisymmetric and trace parts of W ABC can be determined in termsof E M A and d , W [ ABC ] = − F ABC , (3.23) W BAB = −F A , (3.24)the latter arising from partial integration with the dilaton density. Using these identifi-cations, conditions (3.20) and (3.21) can be written as K A D A ¯ K B = ¯ K A D A K B = 0 ,K A D A f = K A D A f = 0 , (3.25)where D A is the background covariant derivative. As we mentioned before, the gener-alized Ricci scalar and the generalized Ricci tensor are completely determined in termsof the degrees of freedom of DFT and, particularly, can be written in terms of differentprojections of the fluxes, R = 2 E A F A + F A F A − F ABC F ABC − F ABC F ABC , (3.26) R AB = E A F B − E C F ABC + F CDA F DBC − F C F ABC . (3.27)The previous projections of the fluxes can be computed using (3.13) and imposing(3.20) and (3.21), F ABC = F ABC − κK D K [ A F BC ] D , (3.28) F ABC = F ABC + κ (cid:18) K [ C D B ] K A + K A E [ B K C ] − K D K A F DBC (cid:19) , (3.29) F ABC = F ABC − κ (cid:18) K [ C D B ] K A + K A E [ B K C ] − K D K A F DBC (cid:19) , (3.30) F A = F A − κ (cid:16) K A D B K B + F BAC K B K C + 4 D A f (cid:17) . (3.31)Replacing the previous expressions in (3.26) the generalized Ricci scalar can be writtenas R = R + κ h − K A K B E B F A − D A (cid:16) K A D B K B + F BAC K B K C (cid:17) + F ABC K C D B K A + F ABC K A E B K C − D A D A f i + κ (cid:2) E A f E A f (cid:3) , (3.32)11nd therefore in the case f = const . , the generalized Ricci scalar can be linearized.With a similar procedure the generalized Ricci tensor can be written as, R AB = R AB + κ R ( κ ) AB + κ R ( κ ) AB , (3.33)where R ( κ ) AB = − D A (cid:16) K B D C K C (cid:17) + 12 E C (cid:0) K C D B K A (cid:1) − E C (cid:0) K B D C K A (cid:1) + 12 E C (cid:0) K A E B K C (cid:1) − E C (cid:0) K A E C K B (cid:1) − K D K C E A F DBC − E A K D K C F DBC − K D E A K C F DBC − E C K D K A F DBC − K D E C K A F DBC − K D K A E C F DBC + 12 K D K C E D F ABC − K C D B K E F CEA + 12 K B D C K E F CEA − K E E B K C F CEA + 12 K E E C K B F CEA + 12 K A D D K C F BCD − K D D A K C F BCD + 12 K C E D K A F BCD − K C E A K D F BCD + 12 K C D B K A F C + 12 K C K A E C F B − K B D C K A F C + 12 K A E B K C F C − K A E C K B F C − K E K C F EDA F BCD + 12 K D K E F DBC F CEA − K D K A F DBC F C − K E K D F EDC F ABC − D A D B f , (3.34)and R ( κ ) AB = 12 K D K C (cid:0) E D K A (cid:1) ( E C K B ) + 14 K D K C K A ( D D E C K B ) − K C K A K B (cid:16) E C D D K D (cid:17) − K C K A K E E C (cid:16) K D F DBE (cid:17) − K C K A ( E C D B f ) + (cid:0) K B D C K A (cid:1) (cid:0) D C f (cid:1) − (cid:0) K A E B K C (cid:1) (cid:0) D C f (cid:1) + (cid:0) K A E C K B (cid:1) (cid:0) D C f (cid:1) + (cid:0) K D K A F DBC (cid:1) (cid:0) D C f (cid:1) . (3.35)As can be appreciated, the EOM of the generalized metric contains quadratic termseven if f = 0, and unlike general relativity there no exist α and α such that the quadraticterms can be written as R ( κ ) AB = α κ ¯ K A ¯ K C R ( κ ) CB + α κK B K C R ( κ ) AC . The previous equation shows that the equation of motion of the generalized metriccannot be linearized when the GKSA is considered. Nevertheless, upon breaking the globalO(D,D) invariance and using the equation of motion of g µν and b µν , it is straighforwardto probe that the quadratic contributions vanish when f = 0, as showed in [10].12 Higher-derivative Double Field Theory
Higher-derivative extensions in DFT were analyzed in several works [8] [9]. An iterativeprocedure to find an infinite tower of this kind of terms was recently given in [17]. In thatwork the authors consider an O ( D, D + K ) multiplet b H MN , which is a generalized metricconstrained to be an element of O ( D, D + K ) with invariant metric b η MN b H MP b η PQ b H N Q = b η MN , (4.1) M , N = 0 , . . . , d − K . In this formulation, K is the dimension of a gauge group K and therefore ˆ H is parametrized by a generalized metric which is an O ( D, D ) elementand by a generalized constrained O ( D, D ) vector field.The generalized frame b E MA relates the generalized metric b H MN with the flat gener-alized metric b H AB , and the O ( D, D + K ) invariant metric b η MN with its flat version b η AB (which we assume to be constant) as follows b H MN = b E MA b H AB b E N B (4.2) b η MN = b E MA b η AB b E N B . (4.3)Since the idea of this formalism is to cast the O ( D, D + K ) formulation in terms of O ( D, D ) frame multiplets, the extended Lorentz subgroup O (1 , D − K ) → O (1 , D − × O ( K ) is broken such that the flat indices now split as A = ( A , A ) = ( A , A , α ), andtransform respectively under O ( D − , × O (1 , D − × O ( K ) .Under this splitting we have b H MN = − b E M A b E N A + b E M A b E N A + b E M α b E N α , (4.4) b η MN = b E M A b E N A + b E M A b E N A + b E M α b E N α , (4.5)where we use the convention that P AB , ¯ P AB and κ αβ raise and lower indices once weparametrize the O ( D − , × O (1 , D − K ) projectors as b P AB = P AB , b ¯ P AB = P AB
00 0 κ αβ . (4.6)13e introduce the following O ( D, D ) multiplets H MN , C M α , d , (4.7)and the O ( D, D ) /O ( D − , × O (1 , D −
1) frames which satisfy η MN = H MP η P Q H NQ H MN = −E M A E NA + E M A E NA ,η MN = E M A E NA + E M A E NA (4.8)and we demand ¯ P M N C N α = 0 , P M N C N α = C M α . (4.9)It is straighforward to find the relation between the O ( D, D ) multiplets and the com-ponents of the O ( D, D + K ) multiplets. Given the following parameterization [22] b H MN = e H MN e C M β ( e C T ) αN e N αβ , b η MN = η MN κ αβ , (4.10)one obtains a non-polynomial relation given by e H MN = H MN + 2 C Mα (cid:0) κ + C T η − C (cid:1) − αβ ( C T ) βN , e C Mα = 2 C Mβ (cid:0) κ + C T η − C (cid:1) − βγ κ γα , (4.11) e N αβ = − κ αβ + 2 κ αγ (cid:0) κ + C T η − C (cid:1) − γδ κ δβ . The tilded fields are also O ( D, D ) multiplets, which are constrained by the requirement(4.1), that reads e H M P e H NP + e C M α e C Nα = η MN , (4.12) e H M P e C P α + e C M β e N βα = 0 , (4.13) e C P α e C P β + e N γα e N γβ = κ αβ . (4.14)On the other hand introducing the following definitions∆ αβ = κ αβ + C Mα C Mβ (4.15)Ξ M N = η M N + C Mα C Nα (4.16)14e can parametrize b E M A = E M A b E M A = (Ξ − ) M P E P A b E M α = C M γ (∆ − ) γβ e βα b E αA = 0 b E αA = −E P A (Ξ − ) P Q C Qα b E αα = (∆ − ) αβ e β α , (4.17)which verify (4.4) and (4.5). Here we have introduced a constant e αβ that identifies thegauge indices α with α and satisfies e αα κ αβ e ββ = κ αβ . In this work, we explicitly implement the following identification for the gauge group K , K = O (1 , D − ⊂ O (1 , D − K ) , (4.18)which is enough to include 4-derivative terms in the ordinary DFT action, as was discussedin appendix A of [17]. The idea is to identify the gauge degrees of freedom C M α with(derivatives of) the generalized frame E M A .Let us first begin by introducing the generators ( t α ) AB that relate objects with gaugeand ¯ P -projected adjoint Lorentz indices A α = ( t α ) BA A AB , A AB = A α ( t α ) AB . (4.19)This implies that ( t α ) AB ( t α ) CD = − δ A [ C δ BD ] , ( t α ) AB ( t β ) BA = κ αβ , (4.20)and [ t α , t β ] = f αβγ t γ . (4.21)We define C ABC = √ E M A C M α ( t α ) BC (4.22)15hich we identify with F ABC , as its index structure suggests. It is important to remarkthat this method is valid only to include four-derivative terms in the action principle, asdiscussed in [17].Mimicking the previous procedure, but starting with an O ( D + K, D ) invariant theoryresults in an equivalent O ( D, D ) formalism up to a Z transformation. Therefore, themost general higher-order action principle in terms of O ( D, D ) fields is a biparametricaction with the following form, S = Z d d Xe − d (cid:16) R + a R ( − ) + b R (+) (cid:17) (4.23)where R (+) is R (+) = − h ( E A E B F BCD ) F ACD + ( E A E B F ACD ) F BCD + 2( E A F BCD ) F ACD F B +( E A F ACD )( E B F BCD ) + ( E A F BCD )( E A F BCD ) + 2( E A F B ) F BCD F ACD +( E A F BCD ) F C CD F ABC − ( E A F BCD ) F C CD F ABC + 2( E A F ACD ) F BCD F B − E A F BCD ) F ACE F BED + 43 F EAC F BED F C CD F ABC + F BCD F ACD F B F A + F ACE F BED F ACG F BGD − F
BCE F AED F ACG F BGD − F
ABD F DCD F CCD F ABC i , in agreement with [8], and was determined through the corrections to the extended gen-eralized fluxes using (4.17),ˆ F ABC = F ABC + 12 (cid:16) E A F CD [ B + F ECD F AE [ B (cid:17) F C ] CD , ˆ F ABC = F ABC − F DEF F EF [ A F DBC ] , ˆ F ABC = F ABC + 32 (cid:18) E [ A F CDB − F D [ AB F DCD − F C E [ A F BED (cid:19) F C ] CD , ˆ F A = F A − h F BCD F ACD F B + E B (cid:16) F BCD F ACD (cid:17)i . (4.24) R ( − ) coincides with R (+) modulo a Z transformation. Here a and b are undeterminedconstants and there exists an infinite amount of first-order duality invariant theories, someof them not related to String Theory [23]. In this work we focus in the case a = 0 and b = 1in order to match with the higher-derivative heterotic supergravity after parametrization.16 .2 Generalized Green-Schwarz transformations The biparametric higher-derivative DFT action (4.23) is invariant under generalized Lorentztransformations only if the generalized frame receives a higher-derivative correction to itsLorentz transformation, δ (1) E MA = −√ E M B F BCD E N A ∂ N Γ CD ,δ (1) E MA = √ F ACD ¯ P MN ∂ N Γ CD , (4.25)where Γ AB was defined in (3.22). Equations (4.25) mimic a Green-Schwarz mechanism,but in a DFT scenario. Using the previous expressions it is straightforward to obtain thefollowing transformations δ (1) F ABC = √ δ (1) E NA ( ∂ N E M B ) E MC + 2 E [ B E N C ] δ (1) E NA +2 √ δ (1) E N [ B ∂ N E M C ] E MA − E [ C δ (1) E NB ] ) E N A ,δ (1) F ABC = √ δ (1) E NA ( ∂ N E M B ) E MC + 2 E [ B E N C ] δ (1) E NA +2 √ δ (1) E N [ B ∂ N E M C ] E MA − E [ C δ (1) E NB ] ) E N A . (4.26)Imposing the GKSA and demanding δ (1) K A = 0 , δ (1) ¯ K A = 0 , (4.27)it is straightforward to obtain the higher-derivative corrections to the generalized Lorentztransformations acting on the backgrounds, δ (1) E MA = − E M B F BBC E A Γ BC − κE M B (cid:16) F BBC K C ¯ K A E C Γ BC − F BBC K C ¯ K C E A Γ BC − F BCD K B ¯ K B E A Γ CD + K B E B ¯ K C E A Γ BC + ¯ K C E B K B E A Γ BC (cid:17) − κ E M B (cid:16) − F BBC K D K C ¯ K A ¯ K C E D Γ BC − F BCD K B K C ¯ K A ¯ K B E C Γ CD + 12 K B K C ¯ K A E B ¯ K C E C Γ BC + 12 K C ¯ K A ¯ K C E B K B E C Γ BC (cid:17) (4.28)17nd δ (1) E MA = E M B F ACD E B Γ CD + κE M B (cid:16) F ACD K C ¯ K B E C Γ CD − F CBC K C ¯ K D E A Γ CD − F ECD K A ¯ K E E B Γ CD + K A E C ¯ K D E B Γ CD + ¯ K D E C K A E B Γ CD (cid:17) + κ E M B (cid:16) − F CAC K D K C ¯ K B ¯ K D E D Γ CD − F ECD K A K C ¯ K B ¯ K E E C Γ CD + 12 K A K C ¯ K B E C ¯ K D E C Γ CD + 12 K C ¯ K B ¯ K D E C K A E C Γ CD (cid:17) . (4.29)At this point is important to observe that the generalized Green-Schwarz transformationmust respect the constraints of the GKSA. Inspecting the null condition we need, δ (1) ( ¯ K M ) ¯ K M = 0 , δ (1) ( K M ) K M = 0 . (4.30)Since the generalized Green-Schwarz transformation acts in the following way δ (1) ¯ K M = δ (1) ( E oM A ) ¯ K A and δ (1) ¯ K M = δ (1) ( E oM ¯ A ) ¯ K ¯ A , the conditions (4.30) are trivially satisfied.Finally we observe that the generalized geodesic equations (3.20) forces, √ K A δ (1) E MA ∂ M ¯ K C + K A ¯ K B δ (1) F ABC = 0 , √ K A δ (1) E MA ∂ M K C + ¯ K A K B δ (1) F ABC = 0 . (4.31)In the next part of the work we break the duality group in order to obtain the low energyeffective heterotic supergravity with higher derivative terms. We start by taking D = 10 and promoting the duality group to H = O (9 , L × O (1 , n ) R with n = 496 in order to describe the 10-dimensional heterotic supergravity. We split theindices as M = ( µ , µ , i ) and A = ( a, a, i ). The generalized frame is parametrized in thefollowing way, E M A = E µa E µa E ia E µa E µa E ia E µi E µi E ii = 1 √ − ˜ e oµa − C oρµ ˜ e ρoa ˜ e µo a − A oρi ˜ e ρoa , ˜ e oµa − C oρµ ˜ e ρoa ˜ e µo a − A oρi ˜ e ρoa √ A oµi e ii √ e ii , (5.1)18here C oµν = b oµν + A oµi A oνi . The invariant projectors of DFT are parametrized in thefollowing way P ab = − η ab δ aa δ bb , P ab = η ab δ aa δ bb . (5.2)According to the previous parametrization, the generalized metric takes the followingform, H MN = ˜ g µνo − ˜ g µρo C oρν − ˜ g µρo A oρi − ˜ g νρo C oρµ ˜ g oµν + C oρµ C oσν ˜ g ρσo + A oµi κ ij A oν j C oρµ ˜ g ρσo A oσi + A oµj κ ji − ˜ g νρo A oρi C oρν ˜ g ρσo A oσi + A oνj κ ij κ ij + A oρi ˜ g ρσo A oσj , (5.3)On the other hand K M and ¯ K M can be parametrized as K M = 1 √ l µ − l µ − C oρµ l ρ − A oiρ l ρ , ¯ K M = 1 √ ¯ l µ ¯ l µ − C oρµ ¯ l ρ − A oiρ ¯ l ρ . (5.4)We impose the standard gauge fixing for the double Lorentz group,˜ e oµa η ab ˜ e oνb = ˜ e oµa η ab ˜ e oνb = ˜ g µν , (5.5)with η ab the ten dimensional flat metric, a, b = 0 , . . . , e − d = p ˜ ge − φ . (5.6)Using (5.1), the parametrization of the generalized fluxes is F abc = − (cid:18) ˜ w abc + 12 ˜ H abc (cid:19) δ aa δ bb δ cc , (5.7) F abc = (cid:18) ˜ w abc −
12 ˜ H abc (cid:19) δ aa δ bb δ cc , (5.8) F abc = 3 (cid:18) ˜ w [ abc ] −
16 ˜ H abc (cid:19) δ aa δ bb δ cc , (5.9) F abc = − (cid:18) ˜ w [ abc ] + 16 ˜ H abc (cid:19) δ aa δ bb δ cc (5.10) F a = = (cid:16) ∂ µ ˜ e µa + ˜ e µa ˜ e νb ∂ µ ˜ e bν − e µa ∂ µ ˜ φ (cid:17) δ aa . (5.11)19he previous parametrization reproduce the low energy heterotic supergravity withhigher-derivative terms. While ˜ g µν and b µν are consistently perturbed by a pair of nullvectors l and ¯ l as in (2.14) and (2.18), the perturbations of the gauge field A oµi aresupressed by the O (10 ,
10 + n ) invariance. One of the most interesting aspects of higher-derivative DFT is the need of field redef-initions to match with standard heterotic supergravity using (5.1). From (4.28) it isstraightforward to show that ˜ g µν transforms under Lorentz and gauge transformations as, δ Λ ˜ g µν = − Ω ( − )( µ ab ∂ ν ) Λ ab . (5.12)Therefore it is mandatory to consider a metric and dilaton redefinition of the form,˜ g µν = g µν −
12 Ω ( − ) µab Ω ( − ) ν ab , (5.13) p − ˜ ge − φ = √− ge − φ , (5.14)and hence δ Λ g µν = 0 as in (2.20). An interesting observation here is that g µν and ˜ g µν must respect the null condition as well as (2.17). The anomalous transformation of thebackground b oµν can be easily obtained parametrizing (4.28) or (4.29) and imposing (5.13), δ (1)Λ b oµν = − Ω ( − )[ µ ab ∂ ν ] Λ ab , (5.15)in agreement with (2.21). Similarly, it is straightforward to check that the rest of thefundamental fields do not receive higher-derivative Lorentz contributions. The field redefinition (5.13) can be interpreted as a higher-derivative ansatz of the form, g µν = η µν + κl a ¯ l b ( e ( µo a −
14 Ω ( − )( µcd Ω ( − ) a cd )( e ν ) o b −
14 Ω ( − ) ν ) ef Ω ( − ) b ef ) , (5.16)where l µ l ν ( −
14 Ω ( − ) µcd Ω ( − ) νcd ) = l µ l ν ( −
14 Ω ( − ) µo cd Ω ( − ) νcdo ) (5.17)20nd ¯ l µ ¯ l ν ( −
14 Ω ( − ) µcd Ω ( − ) νcd ) = ¯ l µ ¯ l ν ( −
14 Ω ( − ) µo cd Ω ( − ) νcdo ) . (5.18)The previous conditions show that the field redefinition (5.13) has to respect the nullconditions of the l and ¯ l vectors when higher-derivative terms are incorporated. A similarargument can be applied for the parametrization of the generalized geodesic equation, i.e. , ( −
14 Ω ( − ) µcd Ω ( − ) νcd ) l µ ∂ ν ¯ l ρ = ( −
14 Ω ( − ) µo cd Ω ( − ) νcdo ) l µ ∂ ν ¯ l ρ (5.19)and ( −
14 Ω ( − ) µcd Ω ( − ) νcd )¯ l µ ∂ ν l ρ = ( −
14 Ω ( − ) µo cd Ω ( − ) νcdo )¯ l µ ∂ ν l ρ . (5.20)These conditions are 2-derivative constraints over the background solutions in order topreserve the covariance under the Buscher rules (see also [10]). Higher-derivative terms can be easily incorporated in the classical double copy prescriptionof the low energy limit of heterotic string theory. We assume that the geometry admitsone Killing vector ξ µ such that the Lie Derivative L ξ acting on an exact field vanishes.Moreover we choose a coordinate system where ξ µ is covariantly constant, i.e. , ∇ oµ ξ ν = ∇ o [ µ ξ ν ] = 0 . (6.1)We normalize the null vectors to satisfy, ξ µ η µν l µ = ξ µ η µν ¯ l µ = 1 . (6.2)In order to obtain the leading order terms with up to four derivatives in the single andzeroth copy, we perturbe the gravity contributions to (2.9) by considering A oµi = const.,21 = φ o = const. and ˆ H µνρ = ˆ H oµνρ = 0. Then we contract it with ξ µ and ξ µ ξ ν , ξ µ ∆ (1) g µν = − ξ µ g στ R σµab R τν ab , (6.3) ξ µ ξ ν ∆ (1) g µν = − ξ µ g στ R σµab R τνab , (6.4)to find the single copy and the zeroth copy, respectively. We perturb around a Minkowskibackground, and we impose l. ¯ l = 0 to work with a linear perturbation on the metric and b -field, g µν = η µν − κl ( µ ¯ l ν ) , (6.5) b µν = b oµν − κl [ µ ¯ l ν ] , (6.6)and we assume that l and ¯ l satisfy geodesic equations, l µ ∂ µ l ν = 0 , ¯ l µ ∂ µ ¯ l ν = 0 . (6.7)in addition to equations (2.17), for simplicity. After imposing the previous ansatz thespin connection takes the following form w µbc = w oµbc − κe νo [ b e λo c ] ∂ ν (cid:0) l ( µ ¯ l λ ) (cid:1) + κ e νo [ b e λo c ] (cid:0) l ν ¯ l λ ¯ l ρ ∂ µ l ρ + l µ ¯ l λ ¯ l ρ ∂ ν l ρ − l λ ¯ l µ ¯ l ρ ∂ ν l ρ (cid:1) , (6.8)while the Riemann tensor is R µν ab = ∂ [ µ | ρ ¯ l | ν ] e oρ [ a l b ] − ∂ [ ν | l σ ∂ [ a ¯ l | µ ] e oσb ] + ∂ [ µ | ρ l σ e oρ [ a e oσb ] ¯ l | ν ] + ∂ [ µ ¯ l ν ] ∂ [ a l σ e oσb ] + ∂ [ µ | ρ l | ν ] e oρ [ a ¯ l b ] − ∂ [ µ | ¯ l ρ e oρ [ a ∂ b ] l | ν ] + ∂ [ µ | ρ ¯ l σ e oρ [ a e oσb ] l | ν ] + ∂ [ µ l ν ] ∂ [ a ¯ l σ e oσb ] + O ( κ ) . In these backgrounds, the Killing vector is just a constant vector, ∂ µ ξ ν = 0. The leadingorder contributions to (6.4) were studied in [16] and match with a Maxwell-like equationafter identifing l µ = A µ and ¯ l µ = ¯ A µ , where A µ and ¯ A µ are a pair of U (1) gauge vectors.The next order are κ terms that come from the linear perturbation of the Riemanntensor. The higher-derivative dynamic of these solutions contain contributions from both A µ and ¯ A while the leading order Maxwell-like equations depends only on one of them[16]. We present our computations in appendix A.22 .2 Kerr-Schild ansatz Let us focus in the case l µ = ¯ l µ . Interestingly enough, it is not possible to perturbatethe generalized metric of DFT with a single generalized vector. Instead, we are able toidentified the vectors in the supergravity approach in order to recover the Kerr-Schildansatz. In this case the spin connection is given by, w µab = w oµab − κ ( ∂ [ a l µ l b ] + ∂ [ a l ρ e oρb ] l µ ) , (6.9)and the Riemann tensor takes the following form, R µν ab = κ (2 ∂ [ µ | ρ l | ν ] e oρ [ a l b ] − ∂ [ µ | l ρ e oρ [ a ∂ b ] l | ν ] + 2 ∂ [ µ | ρ l σ e oρ [ a e oσb ] l | ν ] +2 ∂ [ µ l ν ] ∂ [ a l σ e oσb ] ) + κ ( − ∂ γ l [ µ ∂ [ a l γ l b ] l | ν ] + ∂ σ l [ µ ∂ σ l γ e oγ [ a l b ] l | ν ] ) . (6.10)The leading contributions to the single copy, in this case, are given by − ξ µ g στ R σµab R τνab = 14 κ ( ∂ µ A ρ ∂ σ A ρ ∂ µσ A ν − ∂ µ A ν ∂ ρ A σ ∂ µρ A σ − ∂ µρ A σ ∂ µρ A σ A ν − ∂ µ A ρ ∂ ρ A σ ∂ µσ A ν + ∂ µ A ν ∂ ρ A σ ∂ ρσ A µ + ∂ µρ A σ ∂ µσ A ρ A ν + ∂ µ A ν ∂ ρ A σ ∂ µσ A ρ − ∂ µ A ρ ∂ σ A ρ ∂ ν σ A µ + ∂ ν A µ ∂ ρ A σ ∂ ρσ A µ − ∂ µ A ρ ∂ ρ A σ ∂ ν µ A σ + ∂ ν A µ ∂ ρ A σ ∂ µρ A σ + 2 ∂ µ A ρ ∂ ρ A σ ∂ νσ A µ − ∂ ν A µ ∂ ρ A σ ∂ µσ A ρ ) (6.11)where ∂ µν ⋆ = ∂ µ ( ∂ ν ⋆ ) and we omit terms depending on ξ µ . Similarly, the zeroth copytakes the form, − ξ µ ξ ν g στ R σµab R τν ab = − / κ ( ∂ µν A ρ ∂ µν A ρ − ∂ µν A ρ ∂ µρ A ν ) . (6.12)In this case condition (5.17) is trivially satisfied, while ∂ ρ A µ ∂ a A ρ ∂ σ e oνa A ν A σ = 0 (6.13)is requiered to satisfy (5.19). We study the heterotic formulation of DFT when higher-derivative terms are included, andthe field content is perturbed with the GKSA. We start by adapting the GKSA to the flux23ormulation of DFT. Then we compute a higher-derivative extension for DFT consideringmultiplets of O ( D, D + K ) and we choose the free parameters of the formalism to studythe heterotic case. The double Lorentz symmetry receives a generalized Green-Schwarztransformation that must respect the constraints of the GKSA.Upon parametrization, we reproduce the low energy heterotic supergravity with higher-derivative terms. Higher-derivative field redefinitions are requiered to match with thestandard transformation rules. The gravitational field content, g µν and b µν , is consis-tently perturbed by a pair of null vectors l and ¯ l . Interestingly enough, the perturbationsof the gauge field A µi are supressed by the O (10 , n ) invariance, using the flux formula-tion of DFT. This last point indicates a tension between the generalized metric formalismand the generalized frame formalism upon parametrization.As an application, we study the higher-derivative contributions to the classical dou-ble copy. We focus in the single and zeroth copy, starting from a flat background andconsidering linear perturbations thanks to a orthogonality condition between l and ¯ l . Inthis scenario we obtain four-derivative corrections to the κ Maxwell-like equations pre-viously discussed in [16]. The particular case where l µ = ¯ l µ and therefore A µ = ¯ A µ is alsopresented. Acknowledgements
We sincerely thank K. Lee and K. Cho for exceedingly interesting remarks and comments.Support by CONICET is also gratefully acknowledged.24
Results of Section 6.1
The full expression for the Riemann tensor (6.9) is R µν ab = κR (1) µν ab + κ R (2) µν ab + κ R (3) µν ab + κ R (4) µν ab , (A.1)where R (1) µν ab = ∂ [ µ | ρ ¯ l | ν ] e oρ [ a l b ] − ∂ [ ν | l σ ∂ [ a ¯ l | µ ] e oσb ] + ∂ [ µ | ρ l σ e oρ [ a e oσb ] ¯ l | ν ] + ∂ [ µ ¯ l ν ] ∂ [ a l σ e oσb ] + ∂ [ µ | ρ l | ν ] e oρ [ a ¯ l b ] − ∂ [ µ | ¯ l ρ e oρ [ a ∂ b ] l | ν ] + ∂ [ µ | ρ ¯ l σ e oρ [ a e oσb ] l | ν ] + ∂ [ µ l ν ] ∂ [ a ¯ l σ e oσb ] ,R (2) µν ab = 14 (cid:16) ∂ [ µ | ¯ l ρ ∂ | ν ] l σ e oρ [ a l b ] ¯ l σ + ∂ [ µ ¯ l σ ∂ ν ] l σ ¯ l [ a l b ] − ∂ [ µ | l ρ ∂ | ν ] l σ e oρ [ a ¯ l b ] ¯ l σ + ∂ [ µ | ρ l σ e oρ [ a l b ] ¯ l | ν ] ¯ l σ + ∂ [ µ ¯ l ν ] ∂ [ a l σ l b ] ¯ l σ + ∂ [ µ ¯ l γ ∂ [ a l γ l b ] ¯ l ν ] − ∂ [ µ | l ρ e oρ [ a ∂ b ] l γ ¯ l | ν ] ¯ l γ − ∂ [ µ | ρ l σ e oρ [ a ¯ l b ] ¯ l σ l | ν ] + ∂ [ µ | ¯ l ρ e oρ [ a ∂ b ] l γ ¯ l γ l | ν ] − ∂ [ µ ¯ l γ ∂ [ a l γ ¯ l b ] l ν ] − ∂ [ µ l ν ] ∂ [ a l σ ¯ l b ] ¯ l σ + ∂ [ a ¯ l [ µ ∂ b ] l γ ¯ l γ l ν ] − ∂ γ ¯ l [ µ | ∂ [ a l γ l b ] ¯ l | ν ] − ∂ γ l [ µ | ∂ [ a l γ ¯ l b ] ¯ l | ν ] − ∂ ǫ ¯ l σ e oσ [ a ∂ b ] l ǫ ¯ l [ µ l ν ] − ∂ [ a l [ µ | ∂ b ] l γ ¯ l | ν ] ¯ l γ + ∂ [ a ¯ l ǫ ∂ b ] l ǫ ¯ l [ µ l ν ] − ∂ γ ¯ l [ µ | ∂ [ a ¯ l γ l b ] l | ν ] − ∂ γ l ǫ ∂ [ a ¯ l γ e oǫb ] ¯ l [ µ l ν ] − ∂ γ l [ µ ∂ [ a ¯ l γ ¯ l b ] l ν ] + ∂ σ ¯ l [ µ | ∂ σ l γ e oγ [ a l b ] ¯ l | ν ] + ∂ σ ¯ l [ µ | ∂ σ l | ν ] ¯ l [ a l b ] + ∂ σ ¯ l [ µ | ∂ σ ¯ l γ e oγ [ a l b ] l | ν ] + ∂ σ l [ µ | ∂ σ l γ e oγ [ a ¯ l b ] ¯ l | ν ] + ∂ γ ¯ l σ ∂ γ l ǫ e oσ [ a e oǫb ] ¯ l [ µ l ν ] + ∂ γ ¯ l σ ∂ γ l [ µ e oσ [ a ¯ l b ] l ν ] (cid:17) ,R (3) µν ab = 116 (cid:16) − ∂ [ µ | l ρ ∂ [ a l γ l b ] ¯ l | ν ] ¯ l ρ ¯ l γ − ∂ [ µ | l ρ ∂ [ a l γ ¯ l b ] ¯ l ρ ¯ l γ l | ν ] − ∂ γ l ǫ ∂ [ a | l γ ¯ l b ] ¯ l [ µ ¯ l ǫ l ν ] − ∂ γ l ǫ ∂ [ a ¯ l γ l b ] ¯ l [ µ ¯ l ǫ l ν ] − ∂ σ ¯ l [ µ | ∂ σ l γ ¯ l [ a l b ] ¯ l γ l | ν ] + ∂ γ l σ ∂ γ l ǫ e oσ [ a ¯ l b ] ¯ l [ µ ¯ l ǫ l ν ] − ∂ σ l [ µ | ∂ σ l γ ¯ l [ a l b ] ¯ l | ν ] ¯ l γ + ∂ γ ¯ l σ ∂ γ l ǫ e oσ [ a l b ] ¯ l [ µ ¯ l ǫ l ν ] (cid:17) ,R (4) µν ab = − ∂ γ l σ ∂ γ l ǫ ¯ l [ a l b ] ¯ l [ µ | ¯ l σ ¯ l ǫ l | ν ] . ingle copy Using (6.9) it is straighforward to compute the higher-derivative κ corrections to thesingle copy in this background, − ξ µ g στ R ( − ) σµab R ( − ) τν ab = 116 κ (cid:16) ∂ µ A ρ ∂ σ A ρ ∂ µσ ¯ A ν − ∂ µ ¯ A ν ∂ ρ A σ ∂ µρ A σ − ∂ µρ A σ ∂ µρ A σ ¯ A ν − ∂ µρ A ν ∂ µρ A σ ¯ A σ − ∂ µ ¯ A ρ ∂ σ A ν ∂ µσ A ρ − ∂ µρ ¯ A σ ∂ µρ A σ A ν − ∂ µ A ρ ∂ ρ A σ ∂ µσ ¯ A ν + ∂ µ ¯ A ν ∂ ρ A σ ∂ ρσ A µ + ∂ µρ A σ ∂ µσ A ρ ¯ A ν + ∂ µ ¯ A ν ∂ ρ A σ ∂ µσ A ρ − ∂ µ ¯ A ρ ∂ ρ A σ ∂ µσ A ν + ∂ µ ¯ A ρ ∂ σ A ν ∂ µρ A σ + ∂ µρ ¯ A σ ∂ µσ A ρ A ν + ∂ µ ¯ A ρ ∂ σ A ν ∂ ρσ A µ − ∂ µ A ρ ∂ σ A ρ ∂ ν σ ¯ A µ + ∂ µ ¯ A ρ ∂ ν A σ ∂ µρ A σ − ∂ µ ¯ A ρ ∂ ρ A σ ∂ ν µ A σ + ∂ ν ¯ A µ ∂ ρ A σ ∂ µρ A σ + ∂ νµ A ρ ∂ µρ A σ ¯ A σ + ∂ ν ¯ A µ ∂ ρ A σ ∂ ρσ A µ − ∂ µ A ρ ∂ ρ A σ ∂ ν µ ¯ A σ + ∂ µ ¯ A ρ ∂ ν A σ ∂ µσ A ρ + 2 ∂ µ A ρ ∂ ρ A σ ∂ νσ ¯ A µ − ∂ µ ¯ A ρ ∂ ν A σ ∂ ρσ A µ + 2 ∂ µ ¯ A ρ ∂ ρ A σ ∂ νσ A µ − ∂ ν ¯ A µ ∂ ρ A σ ∂ µσ A ρ + ∂ µρ ¯ A ν ∂ µρ A σ ¯ A σ +2 ∂ µ ¯ A ρ ∂ σ A ρ ∂ µσ ¯ A ν − ∂ µ ¯ A ν ∂ ρ A σ ∂ µρ ¯ A σ − ∂ µρ ¯ A σ ∂ µρ A σ ¯ A ν + ∂ µ ¯ A ρ ∂ σ ¯ A ρ ∂ µσ A ν − ∂ µ ¯ A ρ ∂ σ A ν ∂ µσ ¯ A ρ − ∂ µρ ¯ A σ ∂ µρ ¯ A σ A ν − ∂ µ ¯ A ρ ∂ σ A µ ∂ ρσ ¯ A ν + ∂ µ ¯ A ν ∂ ρ A σ ∂ ρσ ¯ A µ + ∂ µρ ¯ A σ ∂ µσ A ρ ¯ A ν + ∂ µ ¯ A ν ∂ ρ A σ ∂ µσ ¯ A ρ − ∂ µ ¯ A ρ ∂ ρ ¯ A σ ∂ µσ A ν + ∂ µ ¯ A ρ ∂ σ A ν ∂ µρ ¯ A σ + ∂ µρ ¯ A σ ∂ µσ ¯ A ρ A ν + ∂ µ ¯ A ρ ∂ σ A ν ∂ ρσ ¯ A µ − ∂ νµ ¯ A ρ ∂ µρ A σ ¯ A σ − ∂ µ ¯ A ρ ∂ σ A ρ ∂ ν σ ¯ A µ − ∂ µ ¯ A ρ ∂ σ A ρ ∂ ν µ ¯ A σ + ∂ µ ¯ A ρ ∂ ν A σ ∂ µρ ¯ A σ − ∂ µ ¯ A ρ ∂ ρ ¯ A σ ∂ νµ A σ + ∂ ν ¯ A µ ∂ ρ A σ ∂ µρ ¯ A σ − ∂ µ ¯ A ρ ∂ σ ¯ A ρ ∂ ν σ A µ + ∂ ν ¯ A µ ∂ ρ A σ ∂ ρσ ¯ A µ − ∂ µ ¯ A ρ ∂ σ A µ ∂ ν σ ¯ A ρ + ∂ µ ¯ A ρ ∂ ν A σ ∂ µσ ¯ A ρ +2 ∂ µ ¯ A ρ ∂ σ A µ ∂ νρ ¯ A σ − ∂ µ ¯ A ρ ∂ ν A σ ∂ ρσ ¯ A µ + 2 ∂ µ ¯ A ρ ∂ ρ ¯ A σ ∂ νσ A µ − ∂ ν ¯ A µ ∂ ρ A σ ∂ µσ ¯ A ρ (cid:17) . where we focus in the terms that do not depend on ξ µ . Zeroth copy
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