On effective field theory of F-theory beyond leading order
aa r X i v : . [ h e p - t h ] D ec On effective field theory of F-theory beyond leadingorder
Hamid R. Bakhtiarizadeh Department of Physics, Sirjan University of Technology, Sirjan, Iran
Abstract
We construct a proposal for effective bosonic field theory at order α ′ in twelve di-mensions, whose compactification on a circle and on a torus respectively yields eleven-dimensional and type IIB supergravity theories at eight-derivative level. The couplings( ∂G ) R , ( ∂F ) R , ( ∂F ) , ( ∂G ) and ( ∂G ) ( ∂F ) in twelve-dimensional supergrav-ity are determined with this requirement that an ansatz of these couplings should admitsa consistent truncation to the eleven-dimensional and type IIB supergravity theories.The self-duality condition of the five-form field strength in twelve dimensions is alsounderstood by considering the RR five-form field strength of type IIB theory at linearorder. [email protected] Introduction
Type IIA and most of supergravity theories in lower dimensions can be obtained directlyfrom eleven-dimensional supergravity [1], which is a low-energy description of the M-theory [2], by applying the Kaluza-Klein (KK) reduction [3] on a compact circle. Unliketype IIA supergravity, the type IIB theory cannot be obtained by reduction of eleven-dimensional supergravity on a manifold.The SL (2 , Z ) symmetry of type IIB superstring theory, which is the modular group ofa torus, provide strong evidence that the type IIB superstring theory in ten dimensionsmay have its origin in a twelve-dimensional theory known as F-theory [4]. It is shownthat type IIB superstring theory is obtained by nonperturbative compactification of F-theory on a torus for which the complex structure identified by axio-dilaton. In the otherwords, if one starts with twelve dimensions and compactifies on T , then SL (2 , Z ) getsinterpreted as the symmetry of the torus in ten dimensions. The dilaton field is notconstant in these compactifications, and since the value of the dilaton determines thestring coupling constant, thus these solutions cannot be studied perturbatively except inorientifolds [4]. The low-energy description of this theory is believed to be the twelve-dimensional supergravity [5–12]. Compactification of F-theory on various manifolds whichleads to the theories in lower dimensions has also been studied in [13–22].Till now, there is no comprehensive theory of supergravity in twelve dimensions. Someproposals have been introduced, but it seems none of them is complete. Serious difficultiesin writing the action such as the 12-dimensional minimal fermion for which should bea superpartner components with spin higher than 2 in four dimensions [23] as well asextra field degrees of freedom than elven-dimensional supergravity [10, 11], make a bigobstruction in writing the low-energy effective action of F-theory.With the present understanding, it is believed that the 12-dimensional theory shouldcontain type IIB field contents in D = 10 along with those of M-theory in D = 11. In theother words, there might exist a twelve-dimensional theory that could be compactified toten dimensions on a torus, in such a way that type IIB supergravity could be extractedas a consistent truncation. The consistency of truncation is crucial, since the solutionsof the type IIB supergravity will also be solutions of the equations of motion of thetwelve-dimensional theory. The proposed twelve-dimensional action should also be ableto consistently reproduce D = 11 supergravity action after compactification on a circle [9].The supergravity action in twelve dimensions also consists of the lowest-order actionplus an infinite number of higher-derivative terms beyond the leading order [24, 26]. Thehigher-order terms of supergraity actions have also a significant importance in the studyof particle interactions. The role of α ′ corrections to the supergravity actions is of crucialimportance in the study of various physical phenomena. Imprints of string/M/F theoryarise from corrections that are at higher order in α ′ . Upon reduction to four dimensions,such higher-order terms are of particular phenomenological interest [13,25,26]. Our mainfocus in this paper is α ′ corrections to the effective action of F-theory. In [19, 26–29],there have been some efforts in deriving α ′ corrections to low-energy effective action ofF-theory.In this paper, in fact we use the idea of Vafa’s paper [4] and generalize it to theeight-derivative level. We use the KK procedure to find these corrections. We start withmaking an ansatz for various possible couplings in twelve dimensions and putting themunder a consistent truncation to capture the possible ones in elven and ten dimensions.Then, we compare them with their counterparts in eleven-dimensional and type IIB2upergravity to derive the eight-derivative couplings in twelve dimensions. To this end,we follow the approach introduced in [9] for a consistent truncation of the bosonic fieldsof lowest-order supergravity action in D = 12 to D = 11 and D = 10. We have alreadytested the correctness of our method in [37] where we have calculated the known gaugefield corrections to eleven-dimensional supergravity [36] by a circular compactification toten dimensions.Let us now briefly review the bosonic sector of 12-dimensional supergravity action atleading order given in Ref. [9]. The twelve-dimensional Lagrangian density at leadingorder is given by L = eR − e ( ∂ψ ) − e aψ F − ee bψ G + λB ∧ dA ∧ dA , (1.1)where e = √− g , R is the Ricci scalar, ψ is the dilaton, a = i/ √ b = − i/ √ F = dA and G = dB , respectively. The coefficient λ can be obtained by comparison of thedimensionally-reduced Lagrangian in D = 11 with the bosonic sector of 11-dimensionalsupergravity, and it is √ /
4. Thus, the bosonic field content of supergavity in twelvedimensions now includes the metric g MN , the dilaton ψ , and the 3-form and 4-formpotentials A and B .As we know, the theories in D <
11 need real dilaton couplings, the theory in D = 11itself needs zero dilaton coupling, and the theories in D >
11 need imaginary dilatoncouplings. The imaginary couplings, regardless of being undesirable, are needed to makea consistent truncation to the fields of type IIB supergravity possible. In this paper, wedrop the imaginary couplings, i.e., those contain the dilaton field in twelve dimensions.It has been shown in [9] that the truncation of the twelve-dimensional Lagrangian (1.1)to the type IIB theory in D = 10 is consistent only up to linear order when the 5-formfield strength is involved. Since, as we will see in Sec. 3, the 12-dimensional 5-form fieldstrength G in D = 10 is given simply by F = dB without any Chern-Simons correctionwhereas the corresponding one in type IIB supergravity has a Chern-Simons correction as F = dB + λǫ ij dA ( i )2 ∧ dA ( j )2 . Although, the equation of motion and Bianchi identity for G makes we cannot in general, consistently impose the self-duality condition G = ⋆G , butthe self-dual 5-form ( G + ⋆G ) satisfies precisely the same equation of motion and Bianchiidentity as in type IIB supergravity. This is exactly the same condition we already haveincluded manually in calculating the couplings containing the RR 5-form field strength F in ten dimensions [33–35, 37]. As a result, one can expect that automatically obtainthe self-dual couplings of 5-form field strength G in twelve dimensions by comparing thetruncated couplings with the corresponding ones containing the RR self-dual 5-form fieldstrength in ten dimensions.The structure of the paper is arranged as follows. First, we prepare an ansatz with un-known coefficients made of all possible contractions of tensors for various eight-derivativecouplings in twelve dimensions. These bases are given in appendix. In Sec. 2, we providea consistent truncation of these bases on a circle to obtain the couplings in D = 11. In Sec.3, we make a toroidal compactification of the bases to find the couplings in D = 10. Inthe next sections we will find the 12-dimensional couplings ( ∂G ) R , ( ∂F ) R , ( ∂F ) ,( ∂G ) and ( ∂G ) ( ∂F ) , with this requirement that the couplings arising from reduc-tion of the bases to D = 10 and D = 11, should be consistent with the correspondingknown ones in type IIB and eleven-dimensional supergravity, respectively. Finally, Sec.5 is devoted to discussion. 3n this paper our notations on coordinate indices are: a,b,c,... for indices in twelvedimensions, a,b,c,... for indices in eleven dimensions, and a,b,c,... for indices in tendimensions. Let us first review the standard Kaluza-Klein procedure introduced in Ref. [9] to reducethe 12-dimensional theory, first to D = 11 and then, in the next section, to D = 10. In anobvious notation, F reduces to F , F ( i )3 , F ( ij )2 ,... after compactification on internal circleslabeled by i, j and G similarly reduces to G , G ( i )4 , G ( ij )3 ,.... The dimensional reductionof the Riemann curvature also gives rise to R , R ( i )3 , R ( ij )2 ,....Since the dimensionally-reduced theory in eleven dimensions contains more fields thanones in D = 11 supergravity, it is obvious that some of them must be set to zero. Thecrucial point is that this truncation must be consistent, i.e., setting the fields to zero mustbe consistent with their equations of motion. It has been shown that we may consistentlyset G = F (1)3 = 0 . (2.1)Setting both field strengths in (2.1) to zero simultaneously follows from their equationsof motion derived from dimensionally-reduced action at leading order.To obtain D = 11 supergravity, one should be able to reduce the remaining systemof fields further, so that in particular we have only a single independent 4-form fieldstrength, rather than two. In doing so, we take F and G (1)4 to be proportional, againto ensure that this truncation of the theory is consistent with the equations of motion.Thus one may define G (1)4 = r F , F = 1 √ F , (2.2)where F = dA is the ordinary 4-form field strength of 11-dimensional supergravity. Byapplying the definitions (2.2), one arrives at a consistent truncation of the dimensionally-reduced theory in eleven dimensions at leading order. In the following, we are going toextend the above idea to make a consistent truncation of the eight-derivative couplingsin 12-dimensional supergravity to D = 11.Our starting-points are the 12-dimensional bases given in appendix. First, we makethe ansatz (A.1) for ( ∂G ) R terms in the effective Lagrangian of F-theory at order α ′ .Then, we consistently truncate it on a circle upon the above compactification rules tocapture the coupling ( ∂F ) R in eleven dimensions. Consequently, it takes the followingform ( F , i F , i R abcd R abcd + F , h F , i R abcd R abcd + F , i F , i R abce R abcd + F , i F , h R abce R abcd − F , i F , e R abce R abcd + F , e F , d R abce R abcd + F , i F , i R abef R abcd + F , i F , h R abef R abcd − F , i F , f R abef R abcd + F , e F , f R abef R abcd + F , i F , i R abef R abcd + F , i F , h R abef R abcd F , i F , f R abef R abcd + F , d F , f R abef R abcd + F , e F , d R abef R abcd + F , i F , i R acbd R abcd + F , h F , i R acbd R abcd + F , i F , i R acbe R abcd + F , i F , h R acbe R abcd − F , i F , e R acbe R abcd + F , e F , d R acbe R abcd + F , i F , i R acef R abcd + F , i F , h R acef R abcd − F , i F , f R acef R abcd + F , e F , f R acef R abcd + F , i F , i R acef R abcd − F , d F , i R acef R abcd + F , i F , h R acef R abcd − F , i F , f R acef R abcd + F , d F , f R acef R abcd − F , i F , d R acef R abcd + F , e F , d R acef R abcd + F , b F , d R acef R abcd + F , i F , i R aecf R abcd + F , i F , h R aecf R abcd + F , i F , i R aecf R abcd + F , i F , h R aecf R abcd − F , i F , e R aecf R abcd + F , f F , e R aecf R abcd − F , i F , f R aecf R abcd + F , e F , f R aecf R abcd + F , i F , i R aecf R abcd + F , i F , h R aecf R abcd + F , f F , d R aecf R abcd − F , i F , f R aecf R abcd + F , d F , f R aecf R abcd + F , e F , d R aecf R abcd + F , d F , e R aecf R abcd + F , i F , i R aefg R abcd + a F , i F , h R aefg R abcd − F , e F , i R aefg R abcd − F , i F , e R aefg R abcd + F , g F , e R aefg R abcd + F , e F , g R aefg R abcd − F , i F , d R aefg R abcd + F , g F , d R aefg R abcd + F , e F , d R aefg R abcd + F , i F , i R aefg R abcd + a F , i F , h R aefg R abcd − F , i F , g R aefg R abcd + F , f F , g R aefg R abcd + F , c F , g R aefg R abcd + F , i F , i R aefg R abcd − F , e F , i R aefg R abcd − F , c F , i R aefg R abcd + a F , i F , h R aefg R abcd − F , i F , g R aefg R abcd + F , e F , g R aefg R abcd + F , c F , g R aefg R abcd − F , i F , e R aefg R abcd + F , f F , e R aefg R abcd + F , c F , e R aefg R abcd + F , b F , e R aefg R abcd + F , i F , i R aefg R abcd + a F , i F , h R aefg R abcd − F , i F , g R aefg R abcd + F , d F , g R aefg R abcd − F , i F , d R aefg R abcd + F , f F , d R aefg R abcd + F , e F , b R aefg R abcd − F , i F , d R aefg R abcd + F , e F , d R aefg R abcd + F , c F , d R aefg R abcd − F , i F , e R aefg R abcd + F , d F , e R aefg R abcd + F , b F , e R aefg R abcd + a F , i F , i R abcd R efgh − a F , i F , h R abcd R efgh + F , g F , h R abcd R efgh + F , f F , h R abcd R efgh + F , g F , f R abcd R efgh + F , g F , d R abcd R efgh a F , i F , i R abcd R efgh − a F , i F , h R abcd R efgh + F , g F , h R abcd R efgh + a F , i F , i R abcd R efgh − a F , i F , h R abcd R efgh + F , f F , h R abcd R efgh + F , g F , f R abcd R efgh − a F , i F , d R abcd R efgh + F , g F , d R abcd R efgh + F , f F , d R abcd R efgh + a F , i F , i R abcd R efgh − a F , i F , h R abcd R efgh + F , e F , h R abcd R efgh + F , c F , h R abcd R efgh + F , b F , h R abcd R efgh − a F , i F , f R abcd R efgh + F , e F , f R abcd R efgh + F , c F , f R abcd R efgh + a F , i F , i R abcd R efgh − a F , i F , h R abcd R efgh + F , d F , h R abcd R efgh − a F , i F , d R abcd R efgh + F , e F , d R abcd R efgh ) . (2.3)In obtaining this result, we have taken the Riemann curvature in eleven dimensions as12-dimensional one. In our notations this means R = R . This is plausible at least forlinearized Riemann curvature where we have considered in this article.Now, we consider reduction of the basis (A.2) constructed of ( ∂F ) R terms in 12-dimensional supergravity action. Dimensional reduction of the ansatz to eleven dimen-sions by compactifying it on a circle leads to ( ∂F ) R terms based on the above com-pactification rules. It can easily be seen from Eq. (2.2) that ( ∂F ) R coupling in elevendimensions has the same structure as 12-dimensional one but with an overall factor 2 / ∂F ) terms in twelve dimensions is given by (A.3). Putting it under acircular reduction gives rise to ( ∂F ) coupling in eleven dimensions. According to Eqs.(2.1) and (2.2), we observe that this coupling has the same form as the ansatz (A.3)which is now multiplied by an overall factor 4 / ∂F ) which aregiven by ( − ( d + d ) F , h F , e F , j F , d − ( + + d ) F , h F , e F , j F , g + (cid:0) − ( d + ) (cid:1) F , h F , e F , h F , g + (cid:0) − ( d + d ) (cid:1) F , h F , e F , j F , j +( − + − ) F , g F , e F , j F , j +( + ) F , e F , e F , j F , j + (cid:0) − ( d + d ) (cid:1) F , h F , e F , j F , i − ( + d − ) F , g F , e F , j F , i +( + ) F , e F , e F , j F , i + ( d + d ) F , b F , e F , j F , i +( − − ) F , h F , e F , j F , h +( − + + ) F , h F , e F , f F , h +( + + ) F , h F , e F , e F , h ( d − − ) F , h F , e F , j F , g +( + − ) F , h F , e F , f F , g +( − − − ) F , g F , e F , j F , g +( + + ) F , h F , e F , e F , g +( d + + ) F , h F , e F , j F , j +( + d + ) F , h F , e F , j F , i +( d − + ) F , g F , e F , j F , j +( d − + ) F , g F , e F , j F , i +( − − + d ) F , h F , e F , j F , g +( − + ) F , h F , e F , j F , j + ( d + ) F , g F , e F , j F , j + (cid:0) − d + ( d − ) (cid:1) F , g F , e F , f F , j +( − + ) F , h F , e F , j F , i + ( d + ) F , g F , e F , j F , i +( d + + ) F , h F , e F , f F , d +( − − + d ) F , g F , e F , j F , f + ( d + d − ) F , g F , e F , j F , g + (cid:0) − d + ( d + ) (cid:1) F , g F , e F , f F , g +( − d + − ) F , g F , e F , j F , j + (cid:0) d + ( d + d ) (cid:1) F , e F , e F , j F , j +( − d − − ) F , g F , e F , d F , j + (cid:0) − d + ( d + ) (cid:1) F , g F , e F , j F , j +( − d − + ) F , f F , e F , j F , j + ( d + + ) F , e F , e F , j F , j +( − d + d + ) F , f F , e F , j F , j +( − d + − ) F , g F , e F , j F , i +( d + + ) F , e F , e F , j F , i +( − d + + ) F , g F , e F , j F , i +( − d − + ) F , f F , e F , j F , i + ( d + + ) F , e F , e F , j F , i +( + + ) F , c F , e F , j F , i +( − d + d + ) F , f F , e F , j F , i +( + d − ) F , h F , e F , j F , e +( d + ) F , h F , e F , h F , e +( − d − − ) F , g F , e F , j F , d + (cid:0) − d + ( d − ) (cid:1) F , g F , e F , j F , e + (cid:0) d + ( d + ) (cid:1) F , g F , e F , g F , e − d − − ) F , g F , e F , j F , g +( d + + ) F , h F , e F , d F , g + (cid:0) − d + ( d − ) (cid:1) F , g F , e F , j F , g + (cid:0) d + ( d + ) (cid:1) F , g F , e F , e F , g +( d +
25 d ) F , e F , e F , j F , j +( d + ) F , e F , e F , i F , j +( d + ) F , d F , e F , i F , j − ( d + + ) F , h F , e F , j F , d +( − − + ) F , h F , e F , f F , d +( + − ) F , g F , e F , j F , d +( − d + ) F , g F , e F , f F , d +( − d − d − ) F , f F , e F
4f ghi , j F , d +( − + ) F , h F , e F , j F , e +( − + + d ) F , g F , e F , f F , e +( − − d − ) F , g F , e F , j F , f +( − − d + ) F , g F , e F , j F , f +( − − d − ) F , f F , e F , j F , f +( + d + ) F , g F , e F , e F , f +( − + d + d ) F , f F , e F , j F , f +( + d + d ) F , g F , e F , d F , f +( + d + d ) F , f F , e F , f F , e ) , (2.4)in eleven dimensions. Finally, we consider the basis (A.5) for the ( ∂G ) ( ∂F ) part ofhigher-order terms in twelve dimensions. Under dimensional reduction on a circle, onecan get the ( ∂F ) terms in eleven dimensions, that are ( − ( e + e ) F , h F , e F , j F , d +( − − + − e − − ) F , h F , e F , j F , g +( − − e − + ) F , h F , e F , h F , g +( − + − + ) F , h F , e F , j F , j +( − − e − + − + ) F , g F , e F , j F , j +( + ) F , e F , e F , j F , j + (cid:0) e + − ( e + e ) (cid:1) F , h F , e F , j F , i +( − − e − + − ) F , g F , e F , j F , i +( + + ) F , e F , e F , j F , i + ( e + e ) F , b F , e F , j F , i + (cid:0) − − ( e + e + e − e ) (cid:1) F , h F , e F , j F , h +( − e + + − + + ) F , h F , e F , f F , h +( + + ) F , h F , e F , e F , h e + − − − − ) F , h F , e F , j F , g +( − e + − + + e ) F , h F , e F , f F , g +( − − + e ) F , g F , e F , j F , g + (cid:0) e + + ( e + e ) (cid:1) F , h F , e F , e F , g + (cid:0) e + + ( e + e ) (cid:1) F , h F , e F , j F , j + (cid:0) e + e + ( e + e ) (cid:1) F , h F , e F , j F , i +( − e + − + ) F , g F , e F , j F , j +( − e + − + ) F , g F , e F , j F , i +( − − + + e − + e ) F , h F , e F , j F , g +( − + + + − + e ) F , h F , e F , j F , j +( + + + e ) F , g F , e F , j F , j +( e − + − − e − e ) F , g F , e F , f F , j +( − + e + − + e ) F , h F , e F , j F , i +( + + + e ) F , g F , e F , j F , i +( e + + ) F , h F , e F , f F , d +( e − − − e − − e ) F , g F , e F , j F , f +( − − − e − + e + ) F , g F , e F , j F , g +( − e + − − e ) F , g F , e F , f F , g +( − − + + e + e ) F , g F , e F , j F , j + (cid:0) e + ( e + e ) (cid:1) F , e F , e F , j F , j +( − − + e + e ) F , g F , e F , d F , j +( + + + e ) F , g F , e F , j F , j +( e + − − e + + e ) F , f F , e F , j F , j +( + + e ) F , e F , e F , j F , j +( e + + e + ) F , f F , e F , j F , j +( e + − − + + e ) F , g F , e F , j F , i +( e + + + ) F , e F , e F , j F , i +( + + + e ) F , g F , e F , j F , i +( e + e + − − e + ) F , f F , e F , j F , i +( e + + + ) F , e F , e F , j F , i +( e + + ) F , c F , e F , j F , i +( e + + e + ) F , f F , e F , j F , i +( e + − − + + e ) F , h F , e F , j F , e +( + e ) F , h F , e F , h F , e +( − − − + e ) F , g F , e F , j F , d +( − e + − − + − e ) F , g F , e F , j F , e +( + + e ) F , g F , e F , g F , e (cid:0) e − − ( e + e ) (cid:1) F , g F , e F , j F , g +( e + + ) F , h F , e F , d F , g +( e + + − − + e ) F , g F , e F , j F , g +( + + e ) F , g F , e F , e F , g +( e + ) F , e F , e F , j F , j +( e + + ) F , e F , e F , i F , j + ( e + ) F , d F , e F , i F , j +( − − − − − e − ) F , h F , e F , j F , d +( − e − + ) F , h F , e F , f F , d +( − + + e − + + e ) F , g F , e F , j F , d +( e + ) F , g F , e F , f F , d +( e − e + e − ) F , f F , e F
4f ghi , j F , d +( − + e − ) F , h F , e F , j F , e +( − − e + e + + e ) F , g F , e F , f F , e +( − e − + e − ) F , g F , e F , j F , f +( e + e + − − + e ) F , g F , e F , j F , f +( − e − − + e ) F , f F , e F , j F , f +( e + + e + ) F , g F , e F , e F , f +( e + e + e + e − − ) F , f F , e F , j F , f +( e + e + ) F , g F , e F , d F , f +( e + e + ) F , f F , e F , f F , e ) . (2.5)In the next section we shall attempt the truncation of the 12-dimensional Lagrangiansto give type IIB supergravity. Here also we exploit the approach introduced in Ref. [9] for reduction of fields on a furthercircle. One can make a consistent truncation by setting to zero those field strengths thatare not present in the field content of type IIB supergravity, i.e., F = G (1)4 = G (2)4 = G (12)3 = 0 . (3.1)As the eleven-dimensional case, it can be seen from the equations of motion for thesefields that just when the conditions (3.1) are imposed, the truncation of these fields is aconsistent one.By taking into account the Chern-Simons modifications to the various field strengths,one finds that the fields F (1)3 and F (2)3 are given by F (1)3 = dA (1)2 , F (2)3 = dA (2)2 − χdA (1)2 . (3.2)These are precisely the same structures of the NSNS and RR 3-form field strengthsrespectively, in type IIB supergravity. i.e., F (1)3 = H , F (2)3 = F . (3.3)10ote that before applying the condition (3.1) to the Lagrangian obtained from toroidalreduction, there are in total three 3-form field strengths in dimensionally-reduced theory,that are G (12)3 , F (1)3 and F (2)3 . Since G (12)3 is a singlet under the SL (2 , R ) symmetry, it isclear that it should be excluded from the 10-dimensional theory; the remaining two 3-form field strengths form the required doublet under SL (2 , R ). Furthermore, a consistenttruncation to ten dimensions requires that the 5-form field strength G is the ordinaryRR 5-form in type IIB supergravity. i.e., G = F . (3.4)This implies that the RR 5-form in type IIB theory is given at linear order, which issufficient for the purposes we follow in this paper. Due to the reasons noted in theprevious section, for the linearized Riemann curvature we can set R = R in dimensionalreduction directly from twelve to ten dimensions. At the following, we would like totruncate 12-dimensional Lagrangians to ten dimensions according to the compactificationrules (3.1), (3.3) and (3.4) for toroidal reduction.Let us first consider the ansatz (A.1) consists of ( ∂G ) R terms in twelve dimensions.Our calculations shows that compactification of the ansatz on a torus results in ( ∂F ) R coupling in ten dimensions upon the rule (3.4). The obtained coupling has the same formas ( ∂G ) R in which the 12-dimensional 5-form field strength G is replaced by the RR5-form F in ten dimensions.Similarly, toroidal reduction of the basis (A.2) containing the ( ∂F ) R terms in thesupergravity action in twelve dimensions gives rise to the ( ∂H ) R and ( ∂F ) R termsin ten dimension. The ( ∂F ) R part has the same structure as ( ∂H ) R in which theKalb-Ramond field strength H is replaced by RR 3-form field strength F . However the( ∂H ) R terms are given by − H , g H , d R bdgh R cefh − b H , d H , h R abde R cfgh − b H , d H , d R abeh R cfgh − H , g H , d R bdeh R cfgh + H , d H , d R begh R cfgh + b H , d H , h R aebf R cgdh + H , g H , d R bgeh R chdf + b H , d H , h R aebf R chdg + H , d H , d R bgeh R chfg + H , f H , d R cfgh R degh + H , f H , d R cegh R dfgh − H , g H , d R bech R dgfh + H , f H , d R cgeh R dgfh − H , g H , d R bcgh R dhef + H , f H , d R cgfh R dheg − H , d H , d R cfgh R efgh − H , e H , d R dfgh R efgh + H , g H , d R bdch R egfh − H , f H , d R bgch R egfh + H , d H , d R bgch R egfh + H , f H , d R cgdh R egfh + H , c H , d R efgh R efgh . (3.5)At the following, we are going to examine KK reduction of the ansatz (A.3), including( ∂F ) terms in 12-dimensional theory, on a torus to find the couplings in ten dimensions.A consistent truncation leads to the couplings ( ∂F ) ( ∂H ) , ( ∂H ) and ( ∂F ) in tendimensions. Among other terms, the ( ∂F ) ( ∂H ) takes the following form − F , d F , g H , h H , c − F , d F , h H , f H , c − F , d F , g H , f H , c − c F , d F , h H , h H , g − F , d F , h H , h H , g − c F , d F , g H , h H , f F , d F , h H , h H , d + c F , d F , g H , h H ,, h + F , d F , d H , h H , h + F , d F , a H , h H , h + F , d F , h H , d H , g + c F , d F , d H , h H , g − F , d F , g H , e H , g + F , d F , d H , e H , g + F , d F , h H , b H , g − F , d F , g H , b H , g − F , g F , d H , h H , g + F , d F , d H , h H , g + F , d F , a H , h H , g + F , g F , d H , d H , g − c F , d F , h H , f H , d + F , f F , d H , h H , e − F , f F , d H , h H , f + F , f F , d H , h H , h + F , g F , d H , h H , f + F , g F , d H , c H , f + F , d F , h H , e H , c − F , d F , h H , e H , h + F , g F , d H , h H , h + F , g F , d H , e H , h + F , g F , d H , e H , c − F , g F , d H , h H , g − F , f F , d H , e H , f + F , d F , h H , d H , h + c F , d F , g H , h H , h + F , f F , d H , h H , h + F , d F , d H , h H , h + F , e F , d H , h H , h − F , d F , g H , h H , c − F , d F , g H , b H , c + F , d F , g H , h H , g + c F , d F , d H , h H , g + F , f F , d H , h H , g + F , d F , a H , h H , g − F , d F , g H , c H , f − F , g F , d H , h H , f + F , d F , a H , h H , f + F , g F , d H , c H , f − F , f F , d H , c H , f + F , d F , d H , c H , f − F , f F , d H , h H , f + F , d F , d H , h H , h − c F , d F , g H , h H , c + F , d F , g H , e H , c − F , g F , d H , h H , c + F , g F , d H , e H , c + F , f F , d H , h H , c + F , d F , d H , e H , c − F , f F , d H , e H , c − F , e F , d H , h H , c + F , f F , d H , d H , e + F , d F , d H , e H , c , (3.6)in 10-dimensional theory. We also observe that the ( ∂F ) terms has the same structureas ( ∂H ) in which the B-field strength H is replaced by RR 3-form F . This result wasexpected because as remarked above, these fields transforms as a doublet under SL (2 , R )symmetry of type IIB theory. Anyway, ( ∂H ) terms are given as − H , f H , d H , h H , h + H , d H , d H , h H , h + H , d H , d H , h H , g − H , f H , d H , h H , f + H , f H , d H , h H , h + H , f H , d H , h H , h − H , f H , d H , e H , h + ( + c ) H , g H , d H , e H , c +( + c ) H , f H , d H , h H , e + H , f H , d H , h H , f − H , f H , d H , e H , f − c H , f H , d H , c H , h + H , f H , d H , h H , h + H , d H , d H , h H , h +( − c + c + ) H , e H , d H , h H , h − H , e H , d H , h H , g H , d H , d H , h H , g + ( + c ) H , b H , d H , h H , g − ( c + ) H , f H , d H , h H , c − c H , f H , d H , h H , d + c H , f H , d H , f H , d + ( − − c ) H , f H , d H , h H , f + c H , f H , d H , d H , f + ( c +
16 c ) H , d H , d H , h H , h + ( + c ) H , f H , d H , h H , c + ( − c − ) H , f H , d H , e H , c − H , e H , d H , h H , c + H , f H , d H , h H , e − c H , e H , d H , h H , e + H , f H , d H , d H , e + c H , e H , d H , h H , e + ( c + c ) H , f H , d H , c H , e . (3.7)Now, we consider a toroidal compactification of the basis (A.4). We find out that aconsistent truncation to ten dimensions only gives rise to the coupling ( ∂F ) which hasthe same form as the ansatz (A.4) in which the 5-form field strength G is replaced bythe RR 5-form F .Finally, by a consistent reduction of the 12-dimensional basis (A.5) to D = 10, wearrive at the couplings ( ∂F ) ( ∂H ) and ( ∂F ) ( ∂F ) . The former has the same formas the latter in which the Kalb-Ramond field strength H is replaced by RR 3-form fieldstrength F . However, the ( ∂F ) ( ∂H ) part will be e H , d H , g F , j F , c + e H , d H , h F , f F , c + e H , d H , g F , j F , c + e H , d H , g F , f F , c + e H , d H , h F , j F , j + e H , d H , h F , j F , i + e H , d H , h F , j F , g + e H , d H , h F , h F , g + e H , d H , h F , e F , g + e H , d H , h F , j F , j + e H , d H , h F , f F , j + e H , d H , h F , j F , i + e H , d H , h F , j F , h + e H , d H , h F , f F , h + e H , d H , h F , j F , g + e H , d H , h F , f F , g + e H , d H , g F , j F , j + e H , d H , g F , g F , j + e H , d H , g F , j F , i + e H , d H , h F , j F , g + e H , d H , h F , h F , g + e H , d H , g F , j F , f + e H , d H , g F , g F , f + H , g H , d F , j F , f + H , g H , d F , g F , f + e H , d H , h F , j F , j + e H , d H , g F , j F , j + e H , d H , d F , j F , j + e H , d H , g F , e F , j + H , g H , d F , j F , j + H , f H , d F , j F , j + H , d H , d F , j F , j + e H , d H , a F , j F , j + e H , d H , h F , j F , i + e H , d H , g F , j F , i + e H , d H , d F , j F , i + H , g H , d F , j F , i + H , f H , d F , j F , i + H , d H , d F , j F , i + e H , d H , a F , j F , i + e H , d H , h F , j F , d + e H , d H , h F , h F , d + e H , d H , h F , j F , h + e H , d H , h F , e F , h + e H , d H , h F , d F , h + e H , d H , h F , j F , g + e H , d H , h F , e F , g + e H , d H , g F , j F , g + e H , d H , d F , j F , g + e H , d H , h F , d F , g e H , d H , g F , e F , g + e H , d H , d F , e F , g + e H , d H , h F , b F , g + e H , d H , g F , b F , g + e H , d H , d F , b F , g + H , g H , d F , j F , g + H , g H , d F , e F , g + H , g H , d F , d F , g + e H , d H , h F , f F , d + H , f H , d F , j F , e + e H , d H , g F , j F , f + e H , d H , g F , e F , f + e H , d H , g F , b F , f + H , g H , d F , j F , f + H , g H , d F , e F , f + H , f H , d F , j F , f + H , d H , d F , j F , f + e H , d H , a F , j F , f + H , g H , d F , d F , f + H , f H , d F , e F , f + H , d H , d F , e F , f + e H , d H , a F , j , e F , f + e H , d H , h F , j F , j + e H , d H , h F , j F , i + H , g H , d F , j F , j + H , g H , d F , j F , i + e H , d H , h F , j F , g + e H , d H , h F , h F , g + e H , d H , h F , c F , g + H , f H , d F , j F , j + H , f H , d F , j F , i + H , g H , d F , j F , f + H , g H , d F , g F , f + H , g H , d F , c F , f + e H , d H , h F , j F , j + e H , d H , h F , e F , j + H , g H , d F , j F , j + H , g H , d F , e F , j + H , f H , d F , j F , j + H , f H , d F , e F , j + e H , d H , h F , j F , i + H , g H , d F , j F , i + H , f H , d F , j F , i + e H , d H , h F , j F , c + e H , d H , h F , h F , c + e H , d H , h F , e F , c + e H , d H , h F , j F , h + e H , d H , h F , e F , h + e H , d H , h F , c F , h + H , g H , d F , e F , c + e H , d H , h F , j F , g + e H , d H , h F , e F , g + e H , d H , h F , c F , g + H , g H , d F , j F , g + H , g H , d F , e F , g + H , g H , d F , c F , g + e H , d H , h F , f F , c + H , f H , d F , j F , e + H , f H , d F , f F , e + H , g H , d F , j F , f + H , g H , d F , e F , f + H , g H , d F , c F , f + H , f H , d F , j F , f + H , f H , d F , e F , f + e H , d H , h F , j F , j + e H , d H , g F , j F , j + e H , d H , d F , j F , j + e H , d H , g F , c F , j + H , g H , d F , j F , j + H , f H , d F , j F , j + H , d H , d F , j F , j + e H , d H , a F , j F , j + H , f H , d F , c F , j + H , f H , d F , j F , j + H , e H , d F , j F , j + H , d H , d F , j F , j + H , b H , d F , j F , j + H , e H , d F , j F , j + e H , d H , h F , j F , i + e H , d H , g F , j F , i e H , d H , d F , j F , i + H , g H , d F , j F , i + H , f H , d F , j F , i + H , d H , d F , j F , i + e H , d H , a F , j F , i + H , f H , d F , j F , i + H , e H , d F , j F , i + H , d H , d F , j F , i + H , b H , d F , j F , i + H , e H , d F , j F , i + e H , d H , h F , j F , d + e H , d H , h F , h F , d + e H , d H , h F , j F , h + e H , d H , h F , d F , h + e H , d H , g F , j F , c + e H , d H , g F , g F , c + e H , d H , g F , b F , c + H , g H , d F , j F , d + H , g H , d F , g F , d + e H , d H , h F , j F , g + e H , d H , g F , j F , g + e H , d H , d F , j F , g + e H , d H , h F , d F , g + e H , d H , h F , c F , g + e H , d H , g F , c F , g + e H , d H , d F , c F , g + H , g H , d F , j F , g + H , g H , d F , d F , g + H , f H , d F , j F , c + H , f H , d F , f F , c + H , f H , d F , j F , d + H , f H , d F , f F , d + e H , d H , g F , j F , f + e H , d H , g F , c F , f + H , g H , d F , j F , f + H , f H , d F , j F , f + H , d H , d F , j F , f + e H , d H , a F , j F , f + H , g H , d F , d F , f + H , g H , d F , c F , f + H , f H , d F , c F , f + H , d H , d F , c F , f + e H , d H , a F , c F , f + H , f H , d F , j F , f + H , f H , d F , d F , f + H , d H , d F , j F , j + H , c H , d F , j F , j + H , d H , d F , i F , j + H , c H , d F , i F , j + e H , d H , h F , j F , c + e H , d H , h F , e F , c + e H , d H , g F , j F , c + e H , d H , d F , j F , c + e H H , g F , e F , c + e H , d H , d F , e F , c + e H , d H , g F , b F , c + H , g H , d F , j F , c + H , g H , d F , e F , c + H , f H , d F , j F , c + H , f H , d F , e F , c + H , d H , d F , e F , c + e H , d H , a F , e F , c + H , f H , d F , e F , c + H , e H , d F , j F , c + H , f H , d F , b F , c + e H , d H , h F , j F , d + e H , d H , h F , e F , d + H , g H , d F , j F , d + H , g H , d F , e F , d + H , f H , d F , e F , d + H , f H , d F , j F , e + H , f H , d F , c F , e + H , f H , d F , j F , e + H , e H , d F , j F , e + H , d H , d F , j F , e + H , b H , d F , j F , e + H , f H , d F , d F , e + H , e H , d F , j F , e + H , f H , d F , c F , e + H , e H , d F , e F , c H , d H , d F , e F , c + H , b H , d F , e F , c + H , e H , d F , e F , d . (3.8)Having obtained the complete dimensionally-reduced Lagrangians at order α ′ in D =11 and D = 10 dimensions, we shall now find the unknown coefficients in 12-dimensionalbases by comparing the dimensionally-reduced Lagrangians with the corresponding knownones in type IIB and 11-dimensional supergravity theories, respectively. To find the terms containing two 5-form field strengths and two Riemann curvatures ineffective action of F-theory, we use the fact that ( ∂F ) R terms in D = 10 obtainedfrom toroidal reduction of the basis (A.1) should be match with the corresponding onesin type IIB supergravity where have already been found in [31–34]. Demanding this holdfixes the unknown coefficients to the values (cid:26) a → − − a + 2 a − a − a − a − a , a → + 643 + 2 a + a a + a − a + a + a − a − a , a → a + 2 a + 2 a + 20 a + 4 a +8 a + a + a + a − a , a → − a + a − a − a − a + a a − a − a − a + a − a − a , a → − a − a − a − a − a − a − a − a − a − a − a + a a a − a − a − a − a ,a → a + 2 a − a + 2 a + 20 a + 4 a + 2 a + a a + a + 4 a + a + a + a − a − a − a , a → − a − a − a − a − a − a − a − a − a , a → − − a a a a + a a − a + 2 a a − a − a − a , a → + 2563 + 4 a + a + 8 a + 2 a − a − a , a → − a − a − a − a − a − a + 3 a a − a − a + a
2+ 3 a a a a a a − a + 3 a − a − a − a − a − a − a − a − , a → − − a − a − a − a − a + 2 a +3 a + 3 a + a + 3 a + 3 a + 3 a − a + a a a − a − a − a , a → + 2563 + 6 a + 3 a a + 3 a − a − a + a a − a − a − a − a − a − a , a → − − a a − a − a − a − a + a + a a a a a a a a a a a − a − a − a , a → a + a a + a a − a , a → a + 4 a a + 8 a + 4 a + a + 16 a + a + 2 a + 8 a + a a − a + a − a + 2 a + 2 a − a , a → a − a − a − a − a − a + 4 a +2 a − a , a → − a + a a − a + 6 a + 3 a a + 3 a + a + a a − a − a − a − a − a − a − a − a − a − a , a → −
32 + a a a a a
16 + a a a
16 + a a − a − a − a − a − a − a o . (4.1)These conditions should also be able to ensure a consistent truncation to the couplings(2.3) in eleven dimensions. To justify this, we put them into the couplings (2.3) andrewrite the result in terms of independent variables. It can be seen that no unknowncoefficient remains unfixed. This indicates that the above conditions properly fix thecoefficients. Inserting the conditions (4.1) into the basis (A.1) yields e − L ( ∂G ) R = −
323 (6 G ceghi,j G dfghj,i R abef R abcd − G ceghi,j G dghij,f R abef R abcd +12 G beghi,j G dfghj,i R aecf R abcd − G bghij,e G dghij,f R aecf R abcd +8 G behij,c G dfhij,g R aefg R abcd − G cf hij,b G dghij,e R aefg R abcd +48 G bcehi,j G fghij,d R aefg R abcd − G bchij,e G fghij,d R aefg R abcd +12 G behij ,c G fghij,d R aefg R abcd − G abeij ,g G cdfij,h R abcd R efgh +3 G abef i,j G cdghi,j R abcd R efgh ) , (4.2)for the couplings between two 5-form field strengths and two Riemann curvatures intwelve dimensions. Note that in writing the above result, the self-duality condition isalso imposed, automatically. Because the dimensionally-reduced coupling ( ∂F ) R hasbeen compared with the corresponding self-dual one in type IIB supergravity. Now we are in a position to determine the corrections consist of two 4-form field strengthsand two Riemann tensors to the 12-dimensional supergravity action. As previously notedin Sec. 3, dimensional reduction of the ansatz (A.2) on a torus leads to the ( ∂H ) R and ( ∂F ) R couplings in 10-dimensional theory. It now should be possible to matchthe result (3.5) for the former with known expressions for the two-B-field-two-gravitonamplitudes in type IIB superstring theory, where were computed along time ago by Grossand Sloan [30]. This fixes the unknown coefficients as (cid:26) b → b − , b → − b + 2 b − b + 256 , b → b − b − ,b → − b − b + 3 b + 256 , b → − b b − b , b → b ,b → b − b b b − , b → b − b b − b b ,b → − b b − b , b → − b b , b → − b b − b , → − b − , b → b − b − , b → − b + 2 b + 8 b + 1024 ,b → b + 256 , b → b − b − b − , b → b − b − } . (4.3)Since ( ∂F ) R terms have the same structure as ( ∂H ) R in which the B-field strengthis replaced by RR 3-form field strength, it is obvious that comparing them with their10-dimensional counterparts results in the same constraints on the unknown coefficients.By inserting the above conditions into the basis (A.2) and writing the result in terms ofindependent variables, we are left with some unknown coefficients that have not yet beenfixed. This means there should be at least one extra constraint on the coefficients. Theremaining coefficients can be found with this requirement that the ( ∂F ) R couplings,arising from circular reduction of 12-dimensional theory, should be consistent with thecorresponding ones in 11-dimensional supergravity. A precise investigation shows thatthis dimensionally-reduced coupling results from two contributions. The first one comesfrom reduction of the basis (A.2), which has the same structure as ( ∂F ) R ansatz intwelve dimensions but with an overall factor 2 /
3, and the other one arises from reductionof ( ∂F ) R terms to eleven dimensions which is given by (2.3). By adding these twocontributions together and comparing the result with the corresponding expression intype IIB supergravity [36,37], one ends up with the following extra constraint on unknowncoefficients b → − b b b ∂F ) R couplings in twelve dimensions, that are e − L ( ∂F ) R =323 (3 F eghi,f F fghi,e R abcd R abcd + 8 F bfgh,i F efgh,i R abcd R eacd + 4 F fghi,b F fghi,e R abcd R eacd − F bf gh,i F degh,i R abcd R eaf c +24 F bghi,d F ef gh,i R abcd R eaf c − F eghi,b F fghi,d R abcd R eaf c +24 F dghi,b F fghi,e R abcd R eaf c + 12 F aegh,i F bfgh,i R abcd R ef cd − F eghi,a F fghi,b R abcd R ef cd − F bghi,f F cehi,a R abcd R efgd +24 F behi,a F cfhi,g R abcd R efgd − F cehi,a F fghi,b R abcd R efgd +96 F behi,a F fghi,c R abcd R efgd − F bchi,a F fghi,e R abcd R efgd − F acgi,e F bdhi,f R abcd R efgh + 12 F cdgh,i F iabe,f R abcd R efgh ) . (4.5) To find the corrections including four 4-form field strengths to effective action of F-theory,we first match the coupling ( ∂F ) ( ∂H ) presented in (3.6), which has been obtained fromtoroidal reduction of the basis (A.3), with the corresponding one in type IIB supergravityfounded in [31–34]. This fixes the unknown coefficients to be (cid:26) c → , c → −
128 + c − c , c →
649 + c , c → − c + 3 c ,c → − c − c − c c , c → − c − c + 3 c + 4 c − c , → − c c − c − c + c , c → − − c − c
18 + c ,c → −
29 + c , c → −
64 + c − c , c → − c + 16 c − c + c ,c → −
768 + c c + 18 c − c − c − c , c → − c + 9 c +2 c + c + c − c , c →
128 + 2 c − c , c → − c − c (cid:27) . (4.6)Comparison of the dimensionally-reduced coupling ( ∂F ) with its 10-dimensional coun-terpart also yields the similar constraints. Before putting the above conditions into thebasis (A.3), we have to make sure that they correctly gives the other couplings in elevenand ten dimensions. In doing so, we first substitute them into the ( ∂F ) coupling ineleven dimensions, which has the same structure as the 12-dimensional one, but with anoverall factor 4 /
9. After writing the result in terms of independent variables, we observethat they exactly produces the known ( ∂F ) couplings in eleven dimensions obtainedin [36, 37]. As a further consistency check, applying the constraints (4.6) to the couplings(3.7) also yields the correct result for ( ∂H ) couplings in ten dimensions calculatedin [30]. By inserting the above conditions into the basis (A.3), one ends up with thefollowing coupling for four 4-form field strengths in twelve dimensions: e − L ( ∂F ) =29 (576 F afgh,b F bcde,a F cdf j,i F eghi,j − F bcde,a F bfgh,a F cdf j,i F eghj,i +2304 F bcde,a F bfgh,a F df ij ,c F eghj,i − F acfg,b F bcde,a F df ij,h F eghj,i +864 F bcde,a F bcfg,a F df ij,h F eghj,i + 576 F bcde,a F cdf j,b F ehij,g F fghi,a − F aghi,j F bcde,a F cdej ,f F fghi,b − F aeij,h F bcde,a F cfgh,b F fgij,d + 32 F bcde,a F bcdf ,a F ehij,g F fhij,g − F acde,j F bcde,a F fghi,b F ghij ,f + 576 F bcde,a F bfgh,a F cdej,i F ghij,f − F bcde,a F bcfg,a F ehij,d F ghij,f − F bcde,a F bcdf ,a F ghij,e F ghij,f − F bcde,a F bcde,a F ghij,f F ghij,f ) . (4.7) As already mentioned in Sec. 2, dimensional reduction of the 12-dimensional basis (A.4)on a 2-torus only gives rise the coupling ( ∂F ) in ten dimensions. It has the sameform as ( ∂G ) in which the 5-form field strength G is replaced by ordinary RR 5-formfield strength F in ten dimensions. The corresponding self-dual coupling in type IIBsupergravity have been obtained previously in [31, 35]. Matching the results yields thefollowing values for unknown coefficients: (cid:26) d → d d − d + 2 d d − , d → − d + 18 d − d +6 d + 2 d + 1536 , d → − d − d − d + d − d − d − , d → − d d − d + d − d − − d − d − d − d , d → d + 2 d +2 d − d − d , d → − d d − d + d d + d − d − d d − d − d − d − d , d → − d
36 + d d
12 + d d − d + d d + d + d d − d − d − d − d − d − d − d − d − d − d − d − d − d , d → − d − d + 3 d − d − d − d + 3 d d d − d + d − d − d − d + 18 d + 9 d + 3 d + 3 d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − , d → − d − d − d − d − d − d − , d → − d − d − d + d + d + d − d + 8 d + 16 d +40 d + 2 d + 2 d + 10 d − , d → d − d + d
36 + d
18 + d
16 + 3 d d d
18 + d
36 + d d d − d + 2 d + 4 d + 10 d + d d
2+ 5 d − d − d − d − d − d − d − d − d − , d → d d
9+ 8 d d d + 6 d + d d d d d d d − d +4 d + 4 d d d d d d d d + 4 d d d − d − d + 2 d d d − d − d − d − d − d − d − d − d − d − d − d − d , d → − d d d − d − d + 2 d d + 4 d d + 4 d d − d + 4 d d − d − d +16 d + 8 d + 2 d d d d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − , d → − d
18 + d − d + d − d + d
6+ 3 d d − d + 2 d + d d − d − d − d − d − d − d + d − d − d − d − d − d − d − d − d − d − d , d → d d + 4 d + d d d d d d d − d + 3 d d − d + 18 d + 6 d + 3 d + 3 d d − d + 3 d + 9 d +9 d − d − d + 3 d − d − d − d − d − d − d − d − d − d − d − d − d − , d → − d d − d + d
16 + d d d
16 + d − d − d − d − d − d − d − d − d − d ,d → − d − d , − d
16 + d d d
32 + d
16 + d d
32 + d − d − d − d − d − d − d − d − d − d − d − d − d − d , d → − d + 9 d + 3 d + 9 d − d + 27 d + d d + d + 4 d − d + 3 d d − d − d − d + 3 d − d + d + 4 d + 8 d − d − d − d + 36 d + 18 d − d + 6 d + 6 d + 2 d + 6 d − d − d − d − d , d → − d + 6 d + d + 6 d − d − d + 24 d − d + 4 d − d − d + 8 d − d − d + 2 d − d − d − d − d − d + 48 d + 24 d − d + 8 d + 8 d + 4 d + 2048 , d → − d d d − d − d + d + d + 2 d + 4 d − d + 2 d d − d + 8 d + 4 d + 4 d d − d − d − d − d − d − d − − d − d − d − d , d → − d d − d − d + 2 d d + 2 d + 8 d − d +2 d − d + 16 d − d − d − d − d − d − d + d − d − d − d − d − d , d → − d
72 + d − d + d
24 + d
24 + d d
16 + d
16 + d
16 + d d d d d + d − d − d − d + d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d − d , d → d − d − d − d − d + d d d d d d + 3 d + 5 d + d d d − d − d − d − d − d − d − (cid:27) . (4.8)Before applying these conditions to the ansatz (A.4), we have to make sure that theseconditions provide a consistent truncation to ( ∂F ) couplings in 11-dimensional super-gravity. To this end, we first substitute them into the dimensionally-reduced coupling(2.4) and then write the result in terms of independent variables. We find out that nounfixed coefficient remains and as a result the above conditions precisely produce the( ∂F ) couplings in 11-dimensional supergravity. This indicates that we are safe to applythe conditions (4.8) to obtain the ( ∂G ) couplings in twelve dimensions. In doing so,we put them into the basis (A.4). This leads to the following Lagrangian for ( ∂G ) intwelve dimensions: e − L ( ∂G ) = 21
329 (216 G abghi,j G abcde,f G cdgjk,l G efhik,l + 36 G abf gh,i G abcde,f G cdijk,l G eghjk,l +108 G abf gh,i G abcde,f G cdjkl,i G eghjk,l + 192 G abcgh,i G abcde,f G dfgjk,l G ehijl,k − G abghi,j G abcde,f G cfgkl,d G ehjkl,i + 48 G abcgh,i G abcde,f G dgjkl,f G ehjkl,i − G abcgh,i G abcde,f G dejkl,i G ghjkl,f + 8 G abcgh,i G abcde,f G dejkl,f G ghjkl,i +48 G abcdg,h G abcde,f G ehijk,l G gijkl,f + 9 G abcdg,h G abcde,f G eijkl,h G gijkl,f − G abcgh,i G abcde,f G dejkl,f G gijkl,h + 128 G abcgh,i G abcde,f G df jkl,e G gijkl,h − G abcdg,h G abcde,f G ef ijk,l G gijkl,h − G abcdg,h G abcde,f G eijkl,f G gijkl,h − G abcdf,e G abcde,f G ghijl,k G ghijk,l − G abf gh,i G abcde,f G cdgjk,l G hijkl,e − G abcgh,i G abcde,f G dfgjk,l G hijkl,e + 96 G abcgh,i G abcde,f G df jkl,g G hijkl,e − G abcf g,h G abcde,f G dgijk,l G hijkl,e − G abcdg,h G abcde,f G f ijkl,g G hijkl,e +16 G abcdf ,g G abcde,f G ghijk,l G hijkl,e ) . (4.9) Finally, we are going to find the couplings including two 5-form field strengths and two 4-form field strengths in the low-energy effective action of F-theory at eight-derivative level.As discussed in Sec. 3, a consistent truncation of the 12-dimensional ansatz (A.5) to tendimensions leads to the couplings ( ∂F ) ( ∂H ) given by Eq. (3.8) and ( ∂F ) ( ∂F ) . Thelatter has the same form as the former in which the B-field strength H is replaced by theRR 3-form F . By comparing the couplings (3.8) or ( ∂F ) ( ∂F ) with the correspondingself-dual ones in type IIB supergravity, which have already been obtained in Refs. [31–35],we easily find the following relations between unknown coefficients { e → e + e + e − e − e − e − , e → e + e + e − e − e − e , e → e + e + e − e − e , e → − e − e − e − e − e − e − e − e − e − e − e − e , e → − e − e + e + e − e + e − e + 8 e + 8 e +32 e + 2 e + 2 e + 8 e , e → e e + e + 4 e e e +2 e + 6 e + 2 e e e + 2 e + 2 e + 2 e + 4 e − e − , e → − e e e e e − e − e − e − e − e − e − e , e → − e e e e
12 + e e + e e e + e − e + 2 e + e + e e + e − e − e − e − e − e − e − e − e − e − e , e → − e − e − e + 2 e − e − e + 3 e − e + 3 e − e + 3 e − e +18 e + 18 e + 72 e + 9 e e e + 3 e − e + 120 e +30 e + 24 e + 6 e + e + 2 e + 3 e − e − e − e − e − , → − e − e − e − e − e − e − e − e + 329 ,e → − e
12 + e
12 + e e e − e + e − e + 2 e e e e + 2 e + 8 e + e e e + e − e + e e e + 10 e + 8 e + 2 e + e e + e e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e , e → e
12 + e e
12 + e
12 + e
12 + e
12 + e e e e
12 + e + e e
12 + e + e − e − e − e − e + e e + e e − e + e − e − e − e − e − e − e − e − e − e − e − e − e − − e , e → e
10 + e
20 + e
20 + e
10+ 3 e
10 + 3 e
10 + e
10 + 3 e e e
20 + 3 e − e − e + e − e − e − e − e − e − e − e − e − e − e − e − e − e − e
10 + 3215 − e − e − e − e , e → − e − e − e − e − e − e − e − e − e − e − e − e − e − , e → − e − e + e + e − e + 6 e + 2 e + 2 e + 3 e + 3 e − e − e − e + 6 e + e + e − e + e e − e − e ,e → − e − e − e + e + e − e − e , e → e − e + 6 e − e − e − e − e − e − e + e e + 3 e + 6 e − e + 3 e e + 3 e − e + 6 e + 18 e + 3 e + 18 e − e + 6 e +8 e − e − e + 9 e − e − e + e − e + 36 e − e + 36 e +12 e + 36 e + 4 e + 8 e + 6 e − e − e − e − e − e +2 e − e − e − e − e − e + 6 e + 12 e + 6 e + 12 e +4 e + 3 e − e + 36 e − e − e − e − e + 3 e + 6 e + 3 e − e − e + e e − e − , e → − e − e − e − e + 3 e − e − e − e + 18 e − e + 2 e − e + 6 e ,e → − e − e − e − e − e − e − e − e − e − e − e − e , e → − e + 2 e + 4 e + e + e + e + 3 e + 3 e − e +3 e − e − e + 6 e + 2 e + 3 e + e + 2 e + 6 e − e − e − e − e + 2 e − e + 4 e + 12 e − e − e − e − e − e + 3 e + 3 e + 2 e + 6 e − e + 3 e − e − e − e + 3 e e , e → e + e − e − e − e − e + 2 e + e + 4 e − e + e − e − e − , e → e − e − e + 3 e − e + 2 e − e + 4 e + 12 e − e − e − e − e − , e → e + 3 e +2 e − e − e + 2 e + 2 e + e + 4 e − e − e − ,e → e − e − e + 2 e − e + 3 e − e − e + 6 e + 18 e +4 e + 3 e + 9 e + 2 e − e + 18 e − e − e − e − e − e + 2 e − e − e − e + 6 e − e + e − , e → − e + 2 e +4 e − e − e − e − e − e + 3 e e e − e − e +3 e − e + 8 e − e − e − e − e − e + 9 e + 4 e +4 e − e + 9 e − e + 9 e − e − e + 12 e + 9 e + 72 e +72 e + 288 e − e + 4 e + 3 e + 18 e + 18 e + 72 e + 2 e + 6 e +8 e + 24 e − e + 12 e − e + 2 e + 6 e − e − e + 12 e − e − e + 2 e + 2 e − e − , e → − e − e − e − e − e − e − e − e − e − e − e − e + 64 , e → e − e + 8 e − e − e + e − e − e + e − e + 6 e + 3 e − e + 3 e +18 e + 3 e + 18 e − e + 8 e + 8 e − e − e + 3 e − e − e − e − e − e + 36 e − e + 36 e + 12 e + 36 e + 4 e + 8 e +6 e − e − e − e − e − e + 2 e − e − e − e − e − e + e e + 3 e + 6 e + 12 e + 6 e + 12 e + 4 e +3 e − e + 36 e − e − e − e − e + 6 e + e + 3 e − e + 2 e + 6 e + e + 2 e − e − e − , e → − e + 2 e − e − e + 3 e − e + 4 e − e − e − e − e − e +9 e − e + 9 e − e + 9 e − e − e + 12 e + 9 e + 72 e +72 e + 288 e − e + 4 e + 3 e + 18 e + 18 e + 72 e + 4 e + 12 e − e + 12 e − e − e − e + 12 e − e − e , e → − e + e − e − e + 3 e − e + 3 e − e + 3 e − e + 12 e + 12 e +48 e + 3 e + 3 e + 12 e + 2 e − e + 2 e + e − e − e − e ,e → − e e e e e + e − e + 2 e + 3 e + 6 e − e + 2 e − e + 12 e − e + 12 e − e − e − e − e − e − e + e − e + 12 e − e − e − e o . (4.10)By substituting these conditions into the ansatz (A.5) and writing it in terms ofindependent variables, it can be seen that the conditions above do not fix all unknowncoefficients. To find the other ones, we use the fact that the ansatz (A.5) should also beable to produce the couplings ( ∂F ) in eleven dimensions. A detailed calculation showsthat the ( ∂F ) terms in eleven dimensions arise from three distinct contributions. Thefirst one is obtained from circular reduction of 12-dimensional ( ∂F ) couplings to eleven24imensions. The second one comes from compactification of ( ∂G ) on a circle, which isgiven by (2.4) and the last contribution results from reduction of the 12-dimensional basis(A.5) to eleven dimensions, i.e., the couplings (2.5). Adding these three contributionstogether, along with this constraint that the third part must satisfy the conditions (4.10),truly gives the coupling ( ∂F ) in eleven dimensions. The counterpart of this couplingin D = 11 has already been calculated in [36, 37]. Comparing them yields the followingconstraints on those coefficients which have not been fixed in conditions (4.10), that are { e → e − e − e + 2 e − e − e − e − e + 2 e − e − e + 3 e − e + 2 e − e − e − e − e − e + 4 e − e +2 e − e + e + 2 e + 704 , e → − e − e + 2 e + 8 e + 2 e +2 e + 2 e + e + e − e − e + 4 e + 2 e − e + 2 e − e − e + 4 e + e + e − , e → e − e + 3 e + 2 e + e + 3 e e + e − e + e − e − e + 3 e e − e − e − e +9 e + 9 e + 2 e − e + 18 e + 18 e − e + 9 e + 9 e − e − e + e − e + e + 3 e − e − e − e − , e → e + e − e + e − e − e − e − e − e − e − e + 9 e + 18 e + e + 2 e + 4 e − e − e + 2 e − e − e − e + 144 e − e + 9 e + 9 e − e + 18 e − e + 144 e + 2 e − e + 9 e +18 e − e + 2 e − e + 18 e − e + 2 e + 6 e + 4 e + 12 e − e + 6 e + 3 e − e − e − e − e − e + 2 e − e − e − e − e − e − e − e + 2 e − , e → e − e + e e e − e − e − e − e − e − e − e , e → − e e − e − e − e − e − e + 3 e e − e − e + 3 e − e − e + 72 e + 9 e e e + 144 e − e + 9 e − e − e + 36 e + 9 e +9 e − e − e + 3 e + 3 e − e − e − e − e + 3 e − e − e − e − e − e − e − e − e − e − e − ,e → e − e − e + 2 e + e − e − e − e − e + 2 e − e + e − , e → − e − e + e + 4 e − e + 2 e − e − e − e + e − e − e + e − e − e + 2 e − e − e + 96 e +6 e + 6 e + 96 e + 384 e − e − e − e − e + 96 e + 24 e +12 e − e − e + 4 e + 2 e − e − e − e + 3 e + 2 e − e − e + e − e + 3 e − e − e − e − e + 2 e + e − e − e − e , e → − e − e − e + 2 e + 3 e + 3 e − e − e + 2 e − e + 16 e + e + 2 e + 3 e + 2 e + 18 e + 36 e +9 e + 8 e + 3 e − e − e − e − e − e + 12 e + 3 e − e e + 3 e + 36 e − e + 4 e − e − e − e − e − e − e − e − e − e − e − e + 18 e + 18 e − e + 36 e − e − e − e + 18 e + 12 e − e − e + 6 e + 6 e + 6 e +6 e + 6 e − e + 9 e − e + 18 e + e − e + 2 e + 2 e + 6 e +6 e + 36 e + 3 e − e + 3 e + 3 e − e + 9 e + 9 e − e − ,e → − e e + 2 e + 2 e + e + 2 e + 2 e + e e + 3 e + 2 e +6 e + e + 3 e + e + e − e − , e → e + 3 e e − e + e − e − e + 144 e + 36 e + 9 e + 3 e e − e − e − e − e − e − e − e − , e → − e − e − e − e + 32 e − e − e + 8 e + 2 e − , e → e e e + e − e − e + 4 e − e − e − e − e − e − e + 4 e − e − e + 2 e − e + e − e + e + 4 e + e − e − e − e , e → − e
3+ 4 e e e − e − (cid:27) . (4.11)The above two sets of conditions are direct consequence of consistent truncation of the 12-dimensional theory to ten and eleven dimensions, respectively. It is possible to substitutethem into the basis (A.5) to find the couplings between two 5-form and two 4-form fieldstrengths. By doing so, one arrives at: e − L ( ∂G ) ( ∂F ) =445 (495 F abcd,e F fghi,e G abfgj,k G cdhij,k − F abcd,e F fghi,j G abef k,g G cdhik,j +7920 F afgh,i F abcd,e G bcfij,k G degjk,h − F afgh,i F abcd,e G bcf jk,i G degjk,h − F abcd,e F fghi,j G abfgk,c G dehjk,i + 3960 F afgh,i F abcd,e G bcfgj,k G dehjk,i − F afgh,i F abcd,e G bcf jk,g G dehjk,i − F afgh,i F abcd,e G bcf jk,g G deijk,h − F aefg,h F abcd,e G bchij,k G dfgik,j − F abcd,e F efgh,i G abijk,c G dfgjk,h +4320 F aefg,h F abcd,e G bchij,k G dfijk,g − F aefg,h F abcd,e G bcijk,h G dfijk,g − F abfg,h F abcd,e G cehij,k G dfijk,g − F abfg,h F abcd,e G ceijk,h G dfijk,g − F abcd,e F fghi,e G abcf j,k G dghij,k + 960 F abcd,e F efgh,i G abcjk,f G dghij,k − F abcd,e F efgh,i G abf jk,c G dghij,k − F afgh,i F abcd,e G bcef j,k G dghij,k − F aefg,h F abcd,e G bcf ij,k G dghij,k + 720 F afgh,e F abcd,e G bcf ij,k G dghij,k − F abcd,e F efgh,a G bcf ij,k G dghij,k − F aefg,h F abcd,e G bcijk,f G dghij,k − F abcd,e F fghi,j G abcf k,j G dghik,e + 720 F abcd,e F fghi,j G abcek,f G dghik,j − F abcd,e F fghi,j G abcf k,e G dghik,j − F aefg,b F abcd,e G cdhij,k G fghik,j + 120 F abfg,h F abcd,e G cdijk,e G fgijk,h − F abfg,h F abcd,e G ceijk,d G fhijk,g +40 F abcf,g F abcd,e G dhijk,e G fhijk,g + 240 F aefg,b F abcd,e G chijk,f G ghijk,d − F abef,g F abcd,e G f hijk,c G ghijk,d + 40 F abcf,g F abcd,e G dhijk,f G ghijk,e + 40 F abcf,g F abcd,e G dhijk,e G ghijk,f − F abcf,g F abcd,e G ehijk,d G ghijk,f F abef ,c F abcd,e G ghijk,f G ghijk,d − F abcd,f F abcd,e G ghijk,f G ghijk,e ) . (4.12)This coupling obviously is self-dual, because the result is a consequence of comparisonwith the self-dual coupling in type IIB supergravity. In this paper, we have shown that there exists a possible candidate field theory for F-theory at order α ′ , whose dimensional reduction to D = 11 and D = 10 admits aconsistent truncation to the bosonic sector of eleven-dimensional and type IIB super-gravity theories. Let us conclude by discussing some issues and possible applications ofour results.As already discussed in [9] for 12-dimensional supergravity at leading order, we haveobtained a twelve-dimensional theory at eight-derivative level that contains more bosonicdegrees of freedom than are in its dimensionally-reduced M-theory or type IIB theory.Alternatively, one could choose an another method, and carry out precisely these fieldtruncations already in twelve dimensions. But, they cannot be performed in a twelve-dimensionally covariant manner, and thus this formulation of field theory in twelve dimen-sions would have only eleven-dimensional or ten-dimensional covariance. In summary, wehave two ways in formulation of the theory, a covariant twelve-dimensional theory withextra degrees of freedom, or non-covariant one with the correct degrees of freedom. Butin this paper we have employed the former, since the truncation required to obtain eleven-dimensional supergravity is different from the one required to obtain type IIB theory, itwould sound that the two would only be unified in twelve dimensions if the covarianttwelve-dimensional theory is taken as the starting point [9].Finally, let us comment on the applications of our results and future directions. Italso would be interesting to obtain the couplings containing the dilaton field with theapproach introduced in this paper. Upon compactification on elliptically fibered Calabi-Yau fourfolds, one can obtain the non-trivial vacuum for the axio-dilaton which leads toa new, N = 1, α ′ correction to the four-dimensional effective action, as noted in [24],which makes it phenomenologically attractive. One also can explore the implications offlux compactifications for the moduli-space problem [38]. Brane solutions [9, 39–41] andapplications in cosmology [42] will become interesting future directions.Furthermore, it is possible to employ the algorithm introduced in [33] for reducing thenumber of terms of the couplings obtained in this paper and write them in its minimal-term form. This imposes all symmetries including mono-term (antisymmetry propertyof field strengths and symmetries of Riemann tensors) as well as multi-term symmetries(the Bianchi identities for field strengths and Riemann tensors). A The bases for higher-order terms
Here we collect the bases for higher-order terms that have been used in this paper, but dueto their size are unpleasant to include in the main body of the paper. Note that the basesin sections A.1, A.4 and A.5 have been calculated by the “AllContractions” command in“xTras” package of Mathematica that does not take the Bianchi identities into accountand accordingly does not necessarily return an irreducible basis of contractions, but itdoes always gives a complete basis. 27 .1 The ( ∂G ) R basis There are 120 possible ( ∂G ) R terms in the action, which satisfy the on-shell conditions,and those are: a G efghi,j G efghi,j R abcd R abcd + a G efghj,i G efghi,j R abcd R abcd + a G dfghi,j G efghi,j R abce R abcd + a G dfghi,j G efghj,i R abce R abcd + a G dfghi,j G fghij,e R abce R abcd + a G fghij,e G fghij ,d R abce R abcd + a G ceghi,j G dfghi,j R abef R abcd + a G ceghi,j G dfghj,i R abef R abcd + a G ceghi,j G dghij,f R abef R abcd + a G cghij,e G dghij,f R abef R abcd + a G cdghi,j G efghi,j R abef R abcd + a G cdghi,j G efghj,i R abef R abcd + a G cdghi,j G eghij,f R abef R abcd + a G cghij ,d G eghij,f R abef R abcd + a G cghij,e G fghij,d R abef R abcd + a G efghi,j G efghi,j R acbd R abcd + a G efghj,i G efghi,j R acbd R abcd + a G dfghi,j G efghi,j R acbe R abcd + a G dfghi,j G efghj,i R acbe R abcd + a G dfghi,j G fghij,e R acbe R abcd + a G fghij,e G fghij,d R acbe R abcd + a G beghi,j G dfghi,j R acef R abcd + a G beghi,j G dfghj,i R acef R abcd + a G beghi,j G dghij,f R acef R abcd + a G bghij,e G dghij,f R acef R abcd + a G bdghi,j G efghi,j R acef R abcd + a G bghij,d G efghi,j R acef R abcd + a G bdghi,j G efghj,i R acef R abcd + a G bdghi,j G eghij,f R acef R abcd + a G bghij,d G eghij,f R acef R abcd + a G beghi,j G fghij,d R acef R abcd + a G bghij,e G fghij,d R acef R abcd + a G eghij,b G fghij,d R acef R abcd + a G bf ghi,j G deghi,j R aecf R abcd + a G bf ghi,j G deghj,i R aecf R abcd + a G beghi,j G dfghi,j R aecf R abcd + a G beghi,j G dfghj,i R aecf R abcd + a G bf ghi,j G dghij,e R aecf R abcd + a G bghij,f G dghij,e R aecf R abcd + a G beghi,j G dghij,f R aecf R abcd + a G bghij,e G dghij,f R aecf R abcd + a G bdghi,j G efghi,j R aecf R abcd + a G bdghi,j G efghj,i R aecf R abcd + a G bghij ,f G eghij,d R aecf R abcd + a G bdghi,j G eghij,f R aecf R abcd + a G bghij ,d G eghij,f R aecf R abcd + a G bghij,e G fghij,d R aecf R abcd + a G bghij,d G fghij,e R aecf R abcd + a G bfghi,j G cdehi,j R aefg R abcd + a G bfghi,j G cdehj,i R aefg R abcd + a G bf hij,e G cdghi,j R aefg R abcd + a G bfghi,j G cdhij,e R aefg R abcd + a G bf hij,g G cdhij,e R aefg R abcd + a G bf hij ,e G cdhij,g R aefg R abcd + a G bfghi,j G cehij,d R aefg R abcd + a G bf hij ,g G cehij,d R aefg R abcd + a G bf hij,e G cghij,d R aefg R abcd + a G bcf hi,j G deghi,j R aefg R abcd + a G bcf hi,j G deghj,i R aefg R abcd + a G bcf hi,j G dehij,g R aefg R abcd + a G bchij,f G dehij,g R aefg R abcd + a G bf hij ,c G dehij,g R aefg R abcd + a G bcehi,j G dfghi,j R aefg R abcd + a G bchij,e G dfghi,j R aefg R abcd + a G behij ,c G dfghi,j R aefg R abcd + a G bcehi,j G dfghj,i R aefg R abcd + a G bcehi,j G dfhij,g R aefg R abcd + a G bchij,e G dfhij,g R aefg R abcd + a G behij ,c G dfhij,g R aefg R abcd + a G bcf hi,j G dghij,e R aefg R abcd + a G bchij,f G dghij,e R aefg R abcd + a G bf hij ,c G dghij,e R aefg R abcd + a G cf hij,b G dghij,e R aefg R abcd + a G bcdhi,j G efghi,j R aefg R abcd a G bcdhi,j G efghj,i R aefg R abcd + a G bcdhi,j G efhij,g R aefg R abcd + a G bchij,d G efhij,g R aefg R abcd + a G bcf hi,j G eghij,d R aefg R abcd + a G bchij,f G eghij,d R aefg R abcd + a G cdhij,e G fghij,b R aefg R abcd + a G bcehi,j G fghij,d R aefg R abcd + a G bchij,e G fghij,d R aefg R abcd + a G behij ,c G fghij,d R aefg R abcd + a G bcdhi,j G fghij,e R aefg R abcd + a G bchij,d G fghij,e R aefg R abcd + a G cdhij,b G fghij,e R aefg R abcd + a G acegi,j G bdfhi,j R abcd R efgh + a G acegi,j G bdfhj,i R abcd R efgh + a G acegi,j G bdfij,h R abcd R efgh + a G aceij ,g G bdfij,h R abcd R efgh + a G aceij,f G bdgij,h R abcd R efgh + a G aceij,g G bdhij,f R abcd R efgh + a G aceij,g G bfhij,d R abcd R efgh + a G abegi,j G cdfhi,j R abcd R efgh + a G abegi,j G cdfhj,i R abcd R efgh + a G abegi,j G cdfij,h R abcd R efgh + a G abeij ,g G cdfij,h R abcd R efgh + a G abef i,j G cdghi,j R abcd R efgh + a G abef i,j G cdghj,i R abcd R efgh + a G abef i,j G cdgij,h R abcd R efgh + a G abeij,f G cdgij,h R abcd R efgh + a G abeij,g G cdhij,f R abcd R efgh + a G abegi,j G cfhij,d R abcd R efgh + a G abeij,g G cfhij,d R abcd R efgh + a G abeij,f G cghij,d R abcd R efgh + a G abcei,j G dfghi,j R abcd R efgh + a G abcei,j G dfghj,i R abcd R efgh + a G abcei,j G dfgij,h R abcd R efgh + a G abcij ,e G dfgij,h R abcd R efgh + a G abeij,c G dfgij,h R abcd R efgh + a G aceij ,b G dfgij,h R abcd R efgh + a G abcei,j G dghij,f R abcd R efgh + a G abcij ,e G dghij,f R abcd R efgh + a G abeij,c G dghij,f R abcd R efgh + a G abcdi,j G efghi,j R abcd R efgh + a G abcdi,j G efghj,i R abcd R efgh + a G abcdi,j G efgij,h R abcd R efgh + a G abcij,d G efgij,h R abcd R efgh + a G abcei,j G fghij,d R abcd R efgh + a G abcij ,e G fghij,d R abcd R efgh , (A.1)where comma on the indices of field strengths refers to a partial derivative with respectto the index afterwards. A.2 The ( ∂F ) R basis By imposing the linearised lowest-order equations of motion [36], one obtains 24 possibleterms of the form ( ∂F ) R in the action b F aghi,e F bdfi,c R abcd R efgh + b F acgi,e F bdfi,h R abcd R efgh + b F acgi,e F bdhi,f R abcd R efgh + b F cdgh,i F iabe,f R abcd R efgh + b F bchi,a F fghi,e R abcd R efgd + b F behi,a F fghi,c R abcd R efgd + b F behi,a F cfhi,g R abcd R efgd + b F cehi,a F fghi,b R abcd R efgd + b F bghi,f F cehi,a R abcd R efgd + b F abfh,i F cegh,i R abcd R efgd + b F abf i,h F cegh,i R abcd R efgd + b F bahi,g F ef hi,c R abcd R efgd + b F bf gh,i F degh,i R abcd R eaf c + b F bdgh,i F efgh,i R abcd R eaf c + b F dghi,e F f ghi,b R abcd R eaf c + b F dghi,b F fghi,e R abcd R eaf c + b F eghi,b F fghi,d R abcd R eaf c + b F bghi,d F ef gh,i R abcd R eaf c + b F aegh,i F bfgh,i R abcd R ef cd + b F eghi,a F fghi,b R abcd R ef cd + b F bghi,f F eghi,a R abcd R ef cd + b F fghi,b F fghi,e R abcd R eacd b F bfgh,i F efgh,i R abcd R eacd + b F eghi,f F fghi,e R abcd R abcd . (A.2) A.3 The ( ∂F ) basis The basis for the ( ∂F ) terms (at linearised on-shell level) [36] is also given by 24 terms,that are c F aehj,i F bcde,a F cfgh,b F dgij ,f + c F afgh,b F bcde,a F cdf j,i F eghi,j + c F bcde,a F bfgh,a F cdf j,i F eghj,i + c F adij ,f F bcde,a F cfgh,b F eghj,i + c F bcde,a F bfgh,a F df ij,c F eghj,i + c F acfg,b F bcde,a F df ij,h F eghj,i + c F bcde,a F bcfg,a F df ij,h F eghj,i + c F bcde,a F cdf j ,b F ehij,g F fghi,a + c F aghi,j F bcde,a F cdej ,f F fghi,b + c F aehi,j F bcde,a F cdgj ,f F fghi,b + c F acdj ,f F bcde,a F ehij,g F fghi,b + c F bcde,a F bfgh,a F cdej,i F fghi,j + c F bcde,a F cdej,i F fghi,a F fghj ,b + c F acfg,b F bcde,a F deij,h F fghj,i + c F bcde,a F bfgh,a F ehij,d F fgij ,c + c F aeij,h F bcde,a F cfgh,b F fgij,d + c F acfg,b F bcde,a F ehij,d F fgij,h + c F bcde,a F bcfg,a F ehij,g F f hij,d + c F bcde,a F bcdf ,a F ehij,g F fhij,g + c F acde,j F bcde,a F fghi,b F ghij,f + c F bcde,a F bfgh,a F cdej,i F ghij,f + c F bcde,a F bcfg,a F ehij ,d F ghij,f + c F bcde,a F bcdf ,a F ghij,e F ghij,f + c F bcde,a F bcde,a F ghij,f F ghij,f . (A.3) A.4 The ( ∂G ) basis The basis for the ( ∂G ) terms consists, at least at on-shell level, of 109 elements: d G abf gh,i G abcde,f G cgijk,l G dhjkl,e + d G abghi,j G abcde,f G cfgkl,h G dijkl,e + d G abghi,j G abcde,f G cdgj k,l G efhik,l + d G abghi,j G abcde,f G cdgj k,l G efhil,k + d G abghi,j G abcde,f G cdgj k,l G efhkl,i + d G abghi,j G abcde,f G cdghk,l G efijk,l + d G abghi,j G abcde,f G cdghk,l G efijl,k + d G abf gh,i G abcde,f G cdijk,l G eghjk,l + d G abf gh,i G abcde,f G cdjkl,i G eghjk,l + d G abf gh,i G abcde,f G cdijk,l G eghjl,k + d G abcgh,i G abcde,f G dfijk,l G egjkl,h + d G abcgh,i G abcde,f G df jkl,i G egjkl,h + d G abf gh,i G abcde,f G cdgjk,l G ehijk,l + d G abghi,f G abcde,f G cdgjk,l G ehijk,l + d G abf gh,i G abcde,f G cdjkl,g G ehijk,l + d G abcgh,i G abcde,f G dfgjk,l G ehijk,l + d G abcf g,h G abcde,f G dgijk,l G ehijk,l + d G abcgh,f G abcde,f G dgijk,l G ehijk,l + d G abf gh,i G abcde,f G cdgjk,l G ehijl,k + d G abghi,f G abcde,f G cdgjk,l G ehijl,k + d G abcgh,i G abcde,f G dfgjk,l G ehijl,k + d G abcf g,h G abcde,f G dgijk,l G ehijl,k + d G abcgh,f G abcde,f G dgijk,l G ehijl,k + d G abf gh,c G abcde,f G dgijk,l G ehijl,k + d G abghi,j G abcde,f G cdgjk,l G ehikl,f + d G abghi,j G abcde,f G cdgkl,j G ehikl,f + d G abf gh,i G abcde,f G cdgjk,l G ehjkl,i + d G abghi,j G abcde,f G cfgkl,d G ehjkl,i + d G abcgh,i G abcde,f G dfgjk,l G ehjkl,i + d G abcgh,i G abcde,f G df jkl,g G ehjkl,i + d G abcgh,i G abcde,f G dgjkl,f G ehjkl,i + d G abf gh,i G abcde,f G cdgjk,l G eijkl,h + d G abcgh,i G abcde,f G dfgjk,l G eijkl,h + d G abcgh,i G abcde,f G df jkl,g G eijkl,h + d G abcf g,h G abcde,f G dgijk,l G eijkl,h + d G abcgh,i G abcde,f G dgjkl,f G eijkl,h + d G abcgh,i G abcde,f G deijk,l G fghjk,l + d G abcgh,i G abcde,f G deijk,l G fghjl,k d G abcdg,h G abcde,f G ehijk,l G fgijk,l + d G abcdg,h G abcde,f G ehijk,l G fgijl,k + d G abcgh,i G abcde,f G deijk,l G fgjkl,h + d G abcgh,i G abcde,f G degjk,l G fhijk,l + d G abcgh,i G abcde,f G dejkl,g G fhijk,l + d G abcdg,h G abcde,f G egijk,l G fhijk,l + d G abcdg,h G abcde,f G eijkl,g G fhijk,l + d G abcgh,i G abcde,f G degjk,l G fhijl,k + d G abcdg,h G abcde,f G egijkl G fhijl,k + d G abghi,j G abcde,f G cdgj k,l G fhikl,e + d G abghi,j G abcde,f G cdgkl,j G fhikl,e + d G abghi,j G abcde,f G cdj kl,g G fhikl,e + d G abcgh,i G abcde,f G dijkl,g G fhjkl,e + d G abcgh,i G abcde,f G degjk,l G fhjkl,i + d G abcgh,i G abcde,f G dejkl,g G fhjkl,i + d G abcdg,h G abcde,f G ehijk,l G fijkl,g + d G abcgh,i G abcde,f G degjk,l G fijkl,h + d G abcgh,i G abcde,f G dejkl,g G fijkl,h + d G abcdg,h G abcde,f G egijk,l G fijkl,h + d G abcdg,h G abcde,f G eijkl,g G fijkl,h + d G abf gh,i G abcde,f G cdjkl,e G ghijk,l + d G abcgh,i G abcde,f G def jk,l G ghijk,l + d G abcf g,h G abcde,f G deijk,l G ghijk,l + d G abcgh,f G abcde,f G deijk,l G ghijk,l + d G abcf g,h G abcde,f G dijkl,e G ghijk,l + d G abcdg,h G abcde,f G ef ijk,l G ghijk,l + d G abcdf ,g G abcde,f G ehijk,l G ghijk,l + d G abcdg,f G abcde,f G ehijk,l G ghijk,l + d G abcde,g G abcde,f G f hijk,l G ghijk,l + d G abcgh,i G abcde,f G def jk,l G ghijl,k + d G abcf g,h G abcde,f G deijk,l G ghijl,k + d G abcgh,f G abcde,f G deijk,l G ghijl,k + d G abf gh,c G abcde,f G deijk,l G ghijl,k + d G abcdg,h G abcde,f G ef ijk,l G ghijl,k + d G abcdf ,g G abcde,f G ehijk,l G ghijl,k + d G abcdg,f G abcde,f G ehijk,l G ghijl,k + d G abcf g,d G abcde,f G ehijk,l G ghijl,k + d G abcde,g G abcde,f G f hijk,l G ghijl,k + d G abf gh,i G abcde,f G cdijk,l G ghjkl,e + d G abcgh,i G abcde,f G deijk,l G ghjkl,f + d G abcgh,i G abcde,f G dejkl,i G ghjkl,f + d G abcgh,i G abcde,f G def jk,l G ghjkl,i + d G abcgh,i G abcde,f G dejkl,f G ghjkl,i + d G abcf g,h G abcde,f G dhijk,l G gijkl,e + d G abcdg,h G abcde,f G ehijk,l G gijkl,f + d G abcdg,h G abcde,f G eijkl,h G gijkl,f + d G abcgh,i G abcde,f G def jk,l G gijkl,h + d G abcf g,h G abcde,f G deijk,l G gijkl,h + d G abcgh,i G abcde,f G dejkl,f G gijkl,h + d G abcgh,i G abcde,f G df jkl,e G gijkl,h + d G abcdg,h G abcde,f G ef ijk,l G gijkl,h + d G abcdg,h G abcde,f G eijkl,f G gijkl,h + d G abcde,f G abcde,f G ghijk,l G ghijk,l + d G abcde,f G abcde,f G ghijl,k G ghijk,l + d G abcdf,e G abcde,f G ghijl,k G ghijk,l + d G abf gh,i G abcde,f G cdgjk,l G hijkl,e + d G abcgh,i G abcde,f G dfgjk,l G hijkl,e + d G abcgh,i G abcde,f G df jkl,g G hijkl,e + d G abcf g,h G abcde,f G dgijk,l G hijkl,e + d G abcdg,h G abcde,f G f ijkl,g G hijkl,e + d G abcdf ,g G abcde,f G ghijk,l G hijkl,e + d G abcgh,i G abcde,f G degjk,l G hijkl,f + d G abcgh,i G abcde,f G dejkl,g G hijkl,f + d G abcdg,h G abcde,f G eijkl,g G hijkl,f + d G abcf g,h G abcde,f G deijk,l G hijkl,g + d G abcdg,h G abcde,f G ef ijk,l G hijkl,g + d G abcdf ,g G abcde,f G ehijk,l G hijkl,g + d G abcdg,h G abcde,f G eijkl,f G hijkl,g + d G abcde,g G abcde,f G f hijk,l G hijkl,g + d G abcdg,h G abcde,f G f ijkl,e G hijkl,g + d G abcde,g G abcde,f G hijkl,g G hijkl,f . (A.4) A.5 The ( ∂G ) ( ∂F ) basis There are 352 independent on-shell terms in ( ∂G ) ( ∂F ) basis which can be written as e F abcd,e F fghi,j G abefj ,k G cdghk,i + e F abcd,e F fghi,j G abef k,j G cdghk,i e F abcd,e F fghi,j G abej k,f G cdghk,i + e F abcd,e F efgh,i G abfij,k G cdgjk,h + e F abcd,e F fghi,j G abefg,k G cdhij,k + e F abcd,e F efgh,i G abfgj,k G cdhij,k + e F abcd,e F fghi,e G abfgj,k G cdhij,k + e F abcd,e F fghi,j G abefg,k G cdhik,j + e F abcd,e F fghi,j G abef k,g G cdhik,j + e F abcd,e F efgh,i G abfgj,k G cdhik,j + e F abcd,e F fghi,e G abfgj,k G cdhik,j + e F abcd,e F fghi,j G abfgk,e G cdhik,j + e F abcd,e F fghi,j G abefg,k G cdhjk,i + e F abcd,e F fghi,j G abef k,g G cdhjk,i + e F abcd,e F efgh,i G abfgj,k G cdhjk,i + e F abcd,e F fghi,e G abfgj,k G cdhjk,i + e F abcd,e F fghi,j G abfgk,e G cdhjk,i + e F abcd,e F efgh,i G abf jk,g G cdhjk,i + e F abcd,e F fghi,e G abf jk,g G cdhjk,i + e F abcd,e F efgh,i G abfgj,k G cdijk,h + e F abcd,e F efgh,i G abf jk,g G cdijk,h + e F abcd,e F fghi,j G abfj k,g G cehik,d + e F abcd,e F efgh,i G abfij,k G cghjk,d + e F abcd,e F efgh,i G abf jk,i G cghjk,d + e F abcd,e F efgh,i G abijk,f G cghjk,d + e F aefg,h F abcd,e G bfhij,k G cgijk,d + e F abcd,e F fghi,j G abefg,k G chijk,d + e F abcd,e F fghi,j G abef k,g G chijk,d + e F abcd,e F efgh,i G abfgj,k G chijk,d + e F abcd,e F efgh,i G abf jk,g G chijk,d + e F abcd,e F fghi,e G abf jk,g G chijk,d + e F abcd,e F efgh,i G afgjk,b G chijk,d + e F afgh,i F abcd,e G bef jk,g G chijk,d + e F aefg,h F abcd,e G bfgij,k G chijk,d + e F aefg,h F abcd,e G bf ijk,g G chijk,d + e F abcd,e F fghi,j G abcfj ,k G deghi,k + e F afgh,i F abcd,e G bcfij,k G deghj,k + e F abcd,e F fghi,j G abcfj ,k G deghk,i + e F abcd,e F fghi,j G abcf k,j G deghk,i + e F abcd,e F fghi,j G abcj k,f G deghk,i + e F abcd,e F fghi,j G abfj k,c G deghk,i + e F afgh,i F abcd,e G bcfij,k G deghk,j + e F afgh,i F abcd,e G bcfij,k G degjk,h + e F afgh,i F abcd,e G bcf jk,i G degjk,h + e F afgh,i F abcd,e G bcijk,f G degjk,h + e F abcd,e F fghi,j G abcfg,k G dehij,k + e F abcd,e F fghi,j G abcf k,g G dehij,k + e F afgh,i F abcd,e G bcfgj,k G dehij,k + e F afgh,i F abcd,e G bcf jk,g G dehij,k + e F abcd,e F fghi,j G abcfg,k G dehik,j + e F abcd,e F fghi,j G abcf k,g G dehik,j + e F abcd,e F fghi,j G abfgk,c G dehik,j + e F afgh,i F abcd,e G bcfgj,k G dehik,j + e F abcd,e F fghi,j G abcfg,k G dehjk,i + e F abcd,e F fghi,j G abcf k,g G dehjk,i + e F abcd,e F fghi,j G abfgk,c G dehjk,i + e F afgh,i F abcd,e G bcfgj,k G dehjk,i + e F afgh,i F abcd,e G bcf jk,g G dehjk,i + e F afgh,i F abcd,e G bcfgj,k G deijk,h + e F afgh,i F abcd,e G bcf jk,g G deijk,h + e F abcd,e F efgh,i G abcij,k G dfghj,k + e F abcd,e F efgh,i G abcjk,i G dfghj,k + e F abcd,e F efgh,i G abijk,c G dfghj,k + e F abcd,e F fghi,j G abcej,k G dfghk,i + e F abcd,e F fghi,j G abcek,j G dfghk,i + e F abcd,e F efgh,i G abcij,k G dfghk,j + e F aefg,h F abcd,e G bchij,k G dfgij,k + e F aefg,h F abcd,e G bcijk,h G dfgij,k + e F aefg,h F abcd,e G bchij,k G dfgik,j + e F abcd,e F efgh,i G abcij,k G dfgjk,h + e F abcd,e F efgh,i G abcjk,i G dfgjk,h + e F abcd,e F efgh,i G abijk,c G dfgjk,h + e F afgh,i F abcd,e G bceij,k G dfgjk,h + e F afgh,i F abcd,e G bcejk,i G dfgjk,h + e F aefg,h F abcd,e G bchij,k G dfijk,g + e F aefg,h F abcd,e G bcijk,h G dfijk,g + e F abfg,h F abcd,e G cehij,k G dfijk,g + e F abfg,h F abcd,e G ceijk,h G dfijk,g + e F abcd,e F fghi,j G abcef ,k G dghij,k + e F abcd,e F efgh,i G abcf j,k G dghij,k + e F abcd,e F fghi,e G abcf j,k G dghij,k + e F abcd,e F efgh,i G abcjk,f G dghij,k e F abcd,e F efgh,i G abf jk,c G dghij,k + e F afgh,i F abcd,e G bcef j,k G dghij,k + e F aefg,h F abcd,e G bcf ij,k G dghij,k + e F afgh,e F abcd,e G bcf ij,k G dghij,k + e F abcd,e F efgh,a G bcf ij,k G dghij,k + e F aefg,h F abcd,e G bcijk,f G dghij,k + e F abfg,h F abcd,e G cef ij,k G dghij,k + e F abef,g F abcd,e G cf hij,k G dghij,k + e F abfg,e F abcd,e G cf hij,k G dghij,k + e F aefg,b F abcd,e G cf hij,k G dghij,k + e F abcd,e F fghi,j G abcfj ,k G dghik,e + e F abcd,e F fghi,j G abcf k,j G dghik,e + e F abcd,e F fghi,j G abcef ,k G dghik,j + e F abcd,e F fghi,j G abcek,f G dghik,j + e F abcd,e F efgh,i G abcf j,k G dghik,j + e F abcd,e F fghi,e G abcf j,k G dghik,j + e F abcd,e F fghi,j G abcf k,e G dghik,j + e F afgh,i F abcd,e G bcef j,k G dghik,j + e F aefg,h F abcd,e G bcf ij,k G dghik,j + e F afgh,e F abcd,e G bcf ij,k G dghik,j + e F abcd,e F efgh,a G bcf ij,k G dghik,j + e F abfg,h F abcd,e G cef ij,k G dghik,j + e F abef,g F abcd,e G cf hij,k G dghik,j + e F abfg,e F abcd,e G cf hij,k G dghik,j + e F aefg,b F abcd,e G cf hij,k G dghik,j + e F afgh,i F abcd,e G bcfij,k G dghjk,e + e F afgh,i F abcd,e G bcf jk,i G dghjk,e + e F abcd,e F fghi,j G abcef ,k G dghjk,i + e F abcd,e F fghi,j G abcek,f G dghjk,i + e F abcd,e F efgh,i G abcf j,k G dghjk,i + e F abcd,e F fghi,e G abcf j,k G dghjk,i + e F abcd,e F fghi,j G abcf k,e G dghjk,i + e F abcd,e F efgh,i G abcjk,f G dghjk,i + e F abcd,e F fghi,e G abcjk,f G dghjk,i + e F abcd,e F fghi,j G abef k,c G dghjk,i + e F abcd,e F efgh,i G abf jk,c G dghjk,i + e F abcd,e F fghi,e G abf jk,c G dghjk,i + e F afgh,i F abcd,e G bcef j,k G dghjk,i + e F afgh,i F abcd,e G bcejk,f G dghjk,i + e F afgh,i F abcd,e G bcf jk,e G dghjk,i + e F abcd,e F efgh,i G abcf j,k G dgijk,h + e F abcd,e F efgh,i G abcjk,f G dgijk,h + e F abcd,e F efgh,i G abf jk,c G dgijk,h + e F afgh,i F abcd,e G bcef j,k G dgijk,h + e F afgh,i F abcd,e G bcejk,f G dgijk,h + e F aefg,h F abcd,e G bcf ij,k G dgijk,h + e F afgh,e F abcd,e G bcf ij,k G dgijk,h + e F abcd,e F efgh,a G bcf ij,k G dgijk,h + e F afgh,i F abcd,e G bcf jk,e G dgijk,h + e F aefg,h F abcd,e G bcijk,f G dgijk,h + e F afgh,e F abcd,e G bcijk,f G dgijk,h + e F abcd,e F efgh,a G bcijk,f G dgijk,h + e F afgh,i F abcd,e G bef jk,c G dgijk,h + e F aefg,h F abcd,e G bf ijk,c G dgijk,h + e F afgh,e F abcd,e G bf ijk,c G dgijk,h + e F abcd,e F efgh,a G bf ijk,c G dgijk,h + e F abfg,h F abcd,e G cef ij,k G dgijk,h + e F abfg,h F abcd,e G ceijk,f G dgijk,h + e F abfg,h F abcd,e G cf ijk,e G dgijk,h + e F abcd,e F fghi,j G abcfg,k G dhijk,e + e F abcd,e F fghi,j G abcf k,g G dhijk,e + e F afgh,i F abcd,e G bcf jk,g G dhijk,e + e F abef,g F abcd,e G cghij,k G dhijk,f + e F aefg,h F abcd,e G bcf ij,k G dhijk,g + e F aefg,h F abcd,e G bcijk,f G dhijk,g + e F aefg,h F abcd,e G bf ijk,c G dhijk,g + e F abfg,h F abcd,e G cef ij,k G dhijk,g + e F abfg,h F abcd,e G ceijk,f G dhijk,g + e F abef,g F abcd,e G cf hij,k G dhijk,g + e F abfg,e F abcd,e G cf hij,k G dhijk,g + e F aefg,b F abcd,e G cf hij,k G dhijk,g + e F abfg,h F abcd,e G cf ijk,e G dhijk,g + e F abef,g F abcd,e G chijk,f G dhijk,g + e F abfg,e F abcd,e G chijk,f G dhijk,g + e F aefg,b F abcd,e G chijk,f G dhijk,g + e F abcd,e F fghi,j G abcdj ,k G efghi,k + e F afgh,i F abcd,e G bcdij,k G efghj,k + e F abcd,e F fghi,j G abcdj ,k G efghk,i + e F abcd,e F fghi,j G abcdk,j G efghk,i + e F abcd,e F fghi,j G abcj k,d G efghk,i e F afgh,i F abcd,e G bcdij,k G efghk,j + e F abfg,h F abcd,e G cdhij,k G efgij,k + e F abfg,h F abcd,e G cdhij,k G efgik,j + e F afgh,i F abcd,e G bcdij,k G efgjk,h + e F afgh,i F abcd,e G bcdjk,i G efgjk,h + e F afgh,i F abcd,e G bcijk,d G efgjk,h + e F abcf,g F abcd,e G dghij,k G efhij,k + e F abcf,g F abcd,e G dghij,k G efhik,j + e F abfg,h F abcd,e G cdhij,k G efijk,g + e F abfg,h F abcd,e G cdijk,h G efijk,g + e F abfg,h F abcd,e G chijk,d G efijk,g + e F abcd,e F fghi,j G abcdf ,k G eghij,k + e F abcd,e F fghi,j G abcdk,f G eghij,k + e F afgh,i F abcd,e G bcdf j,k G eghij,k + e F afgh,i F abcd,e G bcdjk,f G eghij,k + e F abfg,h F abcd,e G cdf ij,k G eghij,k + e F abfg,h F abcd,e G cdijk,f G eghij,k + e F abcf,g F abcd,e G df hij,k G eghij,k + e F abcf,g F abcd,e G dhijk,f G eghij,k + e F abcd,e F fghi,j G abcfj ,k G eghik,d + e F abcd,e F fghi,j G abcf k,j G eghik,d + e F abcd,e F fghi,j G abcjk,f G eghik,d + e F abcd,e F fghi,j G abcdf ,k G eghik,j + e F abcd,e F fghi,j G abcdk,f G eghik,j + e F abcd,e F fghi,j G abcf k,d G eghik,j + e F afgh,i F abcd,e G bcdf j,k G eghik,j + e F abfg,h F abcd,e G cdf ij,k G eghik,j + e F abcf,g F abcd,e G df hij,k G eghik,j + e F afgh,i F abcd,e G bcfij,k G eghjk,d + e F afgh,i F abcd,e G bcf jk,i G eghjk,d + e F afgh,i F abcd,e G bcijk,f G eghjk,d + e F abcd,e F fghi,j G abcdf ,k G eghjk,i + e F abcd,e F fghi,j G abcdk,f G eghjk,i + e F abcd,e F fghi,j G abcf k,d G eghjk,i + e F afgh,i F abcd,e G bcdf j,k G eghjk,i + e F afgh,i F abcd,e G bcdjk,f G eghjk,i + e F afgh,i F abcd,e G bcf jk,d G eghjk,i + e F abfg,h F abcd,e G chijk,f G egijk,d + e F afgh,i F abcd,e G bcdf j,k G egijk,h + e F afgh,i F abcd,e G bcdjk,f G egijk,h + e F afgh,i F abcd,e G bcf jk,d G egijk,h + e F abfg,h F abcd,e G cdf ij,k G egijk,h + e F abfg,h F abcd,e G cdijk,f G egijk,h + e F abfg,h F abcd,e G cf ijk,d G egijk,h + e F abcd,e F fghi,j G abcfg,k G ehijk,d + e F abcd,e F fghi,j G abcf k,g G ehijk,d + e F afgh,i F abcd,e G bcf jk,g G ehijk,d + e F abcf,g F abcd,e G dghij,k G ehijk,f + e F abcf,g F abcd,e G dhijk,g G ehijk,f + e F abfg,h F abcd,e G cdf ij,k G ehijk,g + e F abfg,h F abcd,e G cdijk,f G ehijk,g + e F abfg,h F abcd,e G cf ijk,d G ehijk,g + e F abcf,g F abcd,e G df hij,k G ehijk,g + e F abcf,g F abcd,e G dhijk,f G ehijk,g + e F abcd,e F fghi,j G abcde,k G fghij,k + e F abcd,e F efgh,i G abcdj,k G fghij,k + e F abcd,e F fghi,e G abcdj,k G fghij,k + e F abcd,e F efgh,i G abcjk,d G fghij,k + e F afgh,i F abcd,e G bcdej,k G fghij,k + e F aefg,h F abcd,e G bcdij,k G fghij,k + e F afgh,e F abcd,e G bcdij,k G fghij,k + e F abcd,e F efgh,a G bcdij,k G fghij,k + e F aefg,h F abcd,e G bcijk,d G fghij,k + e F abfg,h F abcd,e G cdeij,k G fghij,k + e F abef,g F abcd,e G cdhij,k G fghij,k + e F abfg,e F abcd,e G cdhij,k G fghij,k + e F aefg,b F abcd,e G cdhij,k G fghij,k + e F abef,g F abcd,e G chijk,d G fghij,k + e F abcf,g F abcd,e G dehij,k G fghij,k + e F abce,f F abcd,e G dghij,k G fghij,k + e F abcf ,e F abcd,e G dghij,k G fghij,k + e F abef ,c F abcd,e G dghij,k G fghij,k + e F abcd,f F abcd,e G eghij,k G fghij,k + e F abcd,e F fghi,j G abcdj ,k G fghik,e + e F abcd,e F fghi,j G abcdk,j G fghik,e + e F abcd,e F fghi,j G abcde,k G fghik,j + e F abcd,e F efgh,i G abcdj,k G fghik,j + e F abcd,e F fghi,e G abcdj,k G fghik,j + e F abcd,e F fghi,j G abcdk,e G fghik,j + e F afgh,i F abcd,e G bcdej,k G fghik,j e F aefg,h F abcd,e G bcdij,k G fghik,j + e F afgh,e F abcd,e G bcdij,k G fghik,j + e F abcd,e F efgh,a G bcdij,k G fghik,j + e F abfg,h F abcd,e G cdeij,k G fghik,j + e F abef,g F abcd,e G cdhij,k G fghik,j + e F abfg,e F abcd,e G cdhij,k G fghik,j + e F aefg,b F abcd,e G cdhij,k G fghik,j + e F abcf,g F abcd,e G dehij,k G fghik,j + e F abce,f F abcd,e G dghij,k G fghik,j + e F abcf ,e F abcd,e G dghij,k G fghik,j + e F abef ,c F abcd,e G dghij,k G fghik,j + e F abcd,f F abcd,e G eghij,k G fghik,j + e F abcd,e F efgh,i G abcij,k G fghjk,d + e F abcd,e F efgh,i G abcjk,i G fghjk,d + e F abcd,e F efgh,i G abijk,c G fghjk,d + e F afgh,i F abcd,e G bcdij,k G fghjk,e + e F afgh,i F abcd,e G bcdjk,i G fghjk,e + e F abcd,e F fghi,j G abcde,k G fghjk,i + e F abcd,e F efgh,i G abcdj,k G fghjk,i + e F abcd,e F fghi,e G abcdj,k G fghjk,i + e F abcd,e F fghi,j G abcdk,e G fghjk,i + e F abcd,e F fghi,j G abcek,d G fghjk,i + e F abcd,e F efgh,i G abcjk,d G fghjk,i + e F abcd,e F fghi,e G abcjk,d G fghjk,i + e F afgh,i F abcd,e G bcdej,k G fghjk,i + e F afgh,i F abcd,e G bcdjk,e G fghjk,i + e F aefg,h F abcd,e G bchij,k G fgijk,d + e F aefg,h F abcd,e G bcijk,h G fgijk,d + e F aefg,h F abcd,e G bhijk,c G fgijk,d + e F abfg,h F abcd,e G cdhij,k G fgijk,e + e F abfg,h F abcd,e G cdijk,h G fgijk,e + e F abcd,e F efgh,i G abcdj,k G fgijk,h + e F abcd,e F efgh,i G abcjk,d G fgijk,h + e F afgh,i F abcd,e G bcdej,k G fgijk,h + e F aefg,h F abcd,e G bcdij,k G fgijk,h + e F afgh,e F abcd,e G bcdij,k G fgijk,h + e F abcd,e F efgh,a G bcdij,k G fgijk,h + e F afgh,i F abcd,e G bcdjk,e G fgijk,h + e F afgh,i F abcd,e G bcejk,d G fgijk,h + e F aefg,h F abcd,e G bcijk,d G fgijk,h + e F afgh,e F abcd,e G bcijk,d G fgijk,h + e F abcd,e F efgh,a G bcijk,d G fgijk,h + e F abfg,h F abcd,e G cdeij,k G fgijk,h + e F abfg,h F abcd,e G cdijk,e G fgijk,h + e F abef,g F abcd,e G cghij,k G fhijk,d + e F abef,g F abcd,e G chijk,g G fhijk,d + e F abcf,g F abcd,e G dghij,k G fhijk,e + e F abcf,g F abcd,e G dhijk,g G fhijk,e + e F aefg,h F abcd,e G bcdij,k G fhijk,g + e F aefg,h F abcd,e G bcijk,d G fhijk,g + e F abfg,h F abcd,e G cdeij,k G fhijk,g + e F abef,g F abcd,e G cdhij,k G fhijk,g + e F abfg,e F abcd,e G cdhij,k G fhijk,g + e F aefg,b F abcd,e G cdhij,k G fhijk,g + e F abfg,h F abcd,e G cdijk,e G fhijk,g + e F abfg,h F abcd,e G ceijk,d G fhijk,g + e F abef,g F abcd,e G chijk,d G fhijk,g + e F abfg,e F abcd,e G chijk,d G fhijk,g + e F aefg,b F abcd,e G chijk,d G fhijk,g + e F abcf,g F abcd,e G dehij,k G fhijk,g + e F abcf,g F abcd,e G dhijk,e G fhijk,g + e F abcd,e F abcd,e G fghij,k G fghij,k + e F abce,d F abcd,e G fghij,k G fghij,k + e F abcd,e F abcd,e G fghik,j G fghij,k + e F abce,d F abcd,e G fghik,j G fghij,k + e F abcd,e F fghi,j G abcef ,k G ghijk,d + e F abcd,e F fghi,j G abcek,f G ghijk,d + e F abcd,e F efgh,i G abcf j,k G ghijk,d + e F abcd,e F fghi,e G abcf j,k G ghijk,d + e F abcd,e F efgh,i G abcjk,f G ghijk,d + e F abcd,e F fghi,e G abcjk,f G ghijk,d + e F abcd,e F efgh,i G abf jk,c G ghijk,d + e F afgh,i F abcd,e G bcef j,k G ghijk,d + e F afgh,i F abcd,e G bcejk,f G ghijk,d + e F aefg,h F abcd,e G bcf ij,k G ghijk,d + e F afgh,e F abcd,e G bcf ij,k G ghijk,d + e F abcd,e F efgh,a G bcf ij,k G ghijk,d + e F aefg,h F abcd,e G bcijk,f G ghijk,d + e F afgh,e F abcd,e G bcijk,f G ghijk,d + e F abcd,e F efgh,a G bcijk,f G ghijk,d e F aefg,h F abcd,e G bf ijk,c G ghijk,d + e F abfg,h F abcd,e G cef ij,k G ghijk,d + e F abfg,h F abcd,e G ceijk,f G ghijk,d + e F abef,g F abcd,e G cf hij,k G ghijk,d + e F abef,g F abcd,e G chijk,f G ghijk,d + e F abfg,e F abcd,e G chijk,f G ghijk,d + e F aefg,b F abcd,e G chijk,f G ghijk,d + e F abcf,g F abcd,e G ehijk,f G ghijk,d + e F abce,f F abcd,e G f ghij,k G ghijk,d + e F abef,g F abcd,e G f hijk,c G ghijk,d + e F abcd,e F fghi,j G abcdf ,k G ghijk,e + e F abcd,e F fghi,j G abcdk,f G ghijk,e + e F afgh,i F abcd,e G bcdf j,k G ghijk,e + e F afgh,i F abcd,e G bcdjk,f G ghijk,e + e F abfg,h F abcd,e G cdf ij,k G ghijk,e + e F abfg,h F abcd,e G cdijk,f G ghijk,e + e F abcf,g F abcd,e G dhijk,f G ghijk,e + e F abef,g F abcd,e G cdhij,k G ghijk,f + e F abef,g F abcd,e G chijk,d G ghijk,f + e F abcf,g F abcd,e G dehij,k G ghijk,f + e F abce,f F abcd,e G dghij,k G ghijk,f + e F abcf ,e F abcd,e G dghij,k G ghijk,f + e F abef ,c F abcd,e G dghij,k G ghijk,f + e F abcf,g F abcd,e G dhijk,e G ghijk,f + e F abcd,f F abcd,e G eghij,k G ghijk,f + e F abcf,g F abcd,e G ehijk,d G ghijk,f + e F abce,f F abcd,e G ghijk,f G ghijk,d + e F abcf ,e F abcd,e G ghijk,f G ghijk,d + e F abef ,c F abcd,e G ghijk,f G ghijk,d + e F abcd,f F abcd,e G ghijk,f G ghijk,e . (A.5) Acknowledgement
We would like to thank the authors of Ref. [43] for developing the excellent Mathematicapackage “xTras” which we have used extensively for symbolic calculations. This work hasbeen financially supported by the research deputy of Sirjan University of Technology.
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