aa r X i v : . [ m a t h . QA ] J un G holonomy manifolds are superconformal Lázaro O. Rodríguez Díaz ∗ . Abstract
We study the chiral de Rham complex (CDR) over a manifold M withholonomy G . We prove that the vertex algebra of global sections of theCDR associated to M contains two commuting copies of the Shatashvili-Vafa G superconformal algebra. Our proof is a tour de force , based onexplicit computations. Contents G superconformal algebra . . . . . . . . . 112.2 The Shatashvili-Vafa Spin(7) superconformal algebra . . . . . . . 12 G holonomy manifolds 165 The Chiral de Rham Complex over a G -manifold 18 G holonomy manifolds are superconformal . . . . . . . . . . . . 195.2 Proof of the Conjecture . . . . . . . . . . . . . . . . . . . . . . . 255.2.1 Linear λ -brackets . . . . . . . . . . . . . . . . . . . . . . 265.2.2 Non linear λ -brackets . . . . . . . . . . . . . . . . . . . . 295.2.3 Checking the relation . . . . . . . . . . . . . . . . . . . . 32 [Φ + λ Φ + ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 X +(2) Φ + and X +(1) Φ + . . . . . . . . . . . . . . . . . . . . . . 376.3 [Φ ± λ K ± ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3.1 (Φ +(1) K + ) + (Φ − (1) K − ) . . . . . . . . . . . . . . . . . 446.3.2 Φ ± (1) K ± . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 [ G + λ G − ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ∗ IMECC-UNICAMP, São Paulo. Supported by São Paulo State Research Council (Fapesp) grant2014/13357-7. λ . . . . . . . . . . . . . . . . . . . . . . 536.4.2 Coefficient of λ . . . . . . . . . . . . . . . . . . . . . . . 536.4.3 Coefficient of λ . . . . . . . . . . . . . . . . . . . . . . 546.5 [ G + λ Φ − ] and [ G − λ Φ + ] . . . . . . . . . . . . . . . . . . . . . . . 586.5.1 Coefficient of λ . . . . . . . . . . . . . . . . . . . . . . . 626.5.2 Coefficient of λ . . . . . . . . . . . . . . . . . . . . . . 626.6 [ G + λ Φ + ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.6.1 Coefficient of λ . . . . . . . . . . . . . . . . . . . . . . 646.6.2 Coefficient of λ . . . . . . . . . . . . . . . . . . . . . . . 656.7 [ L + λ Φ + ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Introduction
The chiral de Rham complex (CDR) was introduced in [MSV99] by Malikov,Schechtman and Vaintrob. It is a sheaf of supersymmetric vertex algebras overa smooth manifold M . Locally, over a coordinate chart, it is simply n copies of the bc − βγ system ( n is the dimension of M ), i.e., a tensor product of n copies of theClifford vertex algebra and n copies of the Weyl vertex algebra, and then extendedto M by gluing on the intersections of these coordinate charts.One of the most important facts about the CDR proved in the seminal work[MSV99] is that there exists an embedding of the N = 2 superconformal vertexalgebra into the vertex algebra of global sections of CDR when M is a Calabi-Yau manifold. This idea of looking for special vertex subalgebras of the vertexalgebra of global sections of the CDR was further investigated in [BZHS08] whereit was proved that when M is a hyperkälher manifold, the N = 4 superconformalvertex algebra appears as a subalgebra of global sections of CDR. Subsequently in[Hel09] it was shown that in fact there are two commuting copies of the N = 2 superconformal algebra ( N = 4 superconformal algebra) of half the central chargewhen M is Calabi-Yau (respectively hyperkälher).It is by now well known in the physics literature ([Oda89], [HP91], [HP93],[SV95]) that even though we can define a sigma model on an arbritary target space,in order for the theory to be supersymmetric, the target space manifold must be ofspecial holonomy. This implies the existence of covariantly constant p -forms, andthe existence of such forms on the target space manifold implies the existence ofextra elements in the chiral algebra.In [EHKZ13] a program was launched in order to understand the facts men-tioned above as a relation between special holonomy of M and the existence ofcertain subalgebras of CDR. To pursue this objective they introduced an embed-ding (Theorem 3.2) different than the one in [MSV99], of the space of differentialforms Ω ∗ ( M ) into global sections of CDR (in fact, they introduced two differentembeddings). When the manifold M has special holonomy it admits covariantlyconstant forms, so they obtain corresponding sections of CDR and the subalgebragenerated by them. In particular, they recover the result of [BZHS08, Hel09] whenM is a Calabi-Yau manifold or hyperkälher. They also constructed two commutingcopies of the Odake algebra on the space of global sections of CDR of a Calabi-Yauthreefold, and conjectured a similar result for G holonomy manifolds.More precisely they conjectured [EHKZ13, Conjecture 7.3] (Conjecture 5.1)that: if M is a manifold with G holonomy, the vertex algebra of global sectionsof CDR contains two commuting copies of the Shatashvili-Vafa G superconformalalgebra , each of these copies should be generated by the global section that comesfrom the covariantly constant -form (that defines the geometry) using the twodifferent embeddings of Ω ∗ ( M ) that they defined.The Shatashvili-Vafa G superconformal algebra appeared as the chiral algebraassociated to the sigma model with target a manifold with G holonomy in [SV95],its classical counterpart had been studied in [HP93]. It is a superconformal vertex3lgebra with six generators { L, G, Φ , K, X, M } . It is an extension of the N = 1 superconformal algebra of central charge c = 21 / (formed by the super-partners { L, G } ) by two fields Φ and K , primary of conformal weight and respectively,and their superpartners X and M (of conformal weight and respectively). TheirOPEs can be found in subsection 2.1 in the language of lambda brackets of [DK98].This algebra is a member of a two-parameter family SW ( , , of non-linear W -algebras previously studied in [Blu92], it is parametrized by ( c, ε ) ( c is thecentral charge and ε the coupling constant). The Shatashvili-Vafa G algebra is aquotient of SW ( , , with c = and ε = 0 , in other words is the only oneamong this family which has central charge c = and contains the tri-criticalIsing model as a subalgebra.The above conjecture was checked in [EHKZ13] in the case when the manifold M = R is the flat space, and when M = CY × S , where CY is a compactCalabi-Yau threefold and S is a circle, using the above mentioned result about theOdake algebra.In this paper we prove the conjecture when M is an arbitrary non-flat G man-ifold. Our approach is a tour de force , based on explicit computations, some ofthem really long, and some abstract algebraic manipulations. But the beauty ofthese after all is that we need to use many of the known identities in G geometry,all the computations are tied in a non-trivial way to the geometry of the manifold.To perform the computations we have used the Mathematica package OPEdefscreated by Kris Thielemans [Thi91] for symbolic computation of operator productexpansion, and the computer algebra system
Cadabra [Pee07] created by KasperPeeters.In section 2 we recap the basic facts about vertex superalgebras and we intro-duce the main examples that are used in the paper. In section 3 we review theconstruction of the CDR as well as the main tool (Theorem 3.2) used in the paper:the embedding of the space of differential forms Ω ∗ ( M ) into global sections ofCDR, introduced by [EHKZ13]. In section 4 we recall some background materialon G geometry as well as many of the known identities necessary for the compu-tations in the next section. In section 5 we prove the main result of this paper: Theorem.
Let M be a G holonomy manifold, then the space of global sections ofthe CDR associated to M contains two commuting copies of the Shatashvili-Vafa G superconformal algebra of central charge .The reader is referred to Theorem 5.5 for a more explicit description of thegenerators of these pairs of algebras. It is important to note that the Conjecture 5.1provides the explicit candidates (global sections of the CDR) for the generators,the challenge is to verify that they actually generate the Shatashvili-Vafa G super-conformal algebra and that the two copies commute.We develop the proof of Theorem 5.5 in four steps. First we prove in Theo-rem 5.2 that the space of global sections of CDR contains two commuting pairs of N = 1 superconformal algebras, at central charge / and / respectively, thelast one is precisely the tri-critical Ising model. Secondly in subsection 5.2.1 we4heck the linear λ -brackets that were not already computed in the way of provingTheorem 5.2. In third place we verify the non-linear λ -brackets under the assump-tion a non-linear identity (2.13) among the generators is satisfied. Finally in 5.2.3we check the non-linear identity (2.13), see Remark 5.7. Along the way of provingTheorem 5.5 we perform a lot of computations in local coordinates, for this wemake extensive use of properties of the manifold M such as: the Ricci flatness, thecontractions between the -form and its Hodge dual (see page 18), the -form isparallel, the symmetries of the Riemann curvature tensor, etc. To avoid unneces-sary calculations we take advantage of the space of global sections of the CDR isa vertex algebra, to use the axioms as well as the identities satisfied by a vertexalgebra, e.g., the Jacobi identity, the Borcherds identity, etc, to derive λ -bracketsfrom the ones already computed.It is remarkable that Borcherds identity was proved very useful computing thenon-linear λ -brackets in subsection 5.2.2. It is also engaging the appearance ofthe first Pontryagin class p ( M ) of the manifold M in the proof of the non-linearidentity (2.13) (see Remark 5.8), taking into account that the only oriented charac-teristic class of interest for a G -manifold is p ( M ) [CHNP15].A new feature of the Shatashvili-Vafa G superconformal algebra not sharedby their cousins the N = 2 superconformal algebra (the N = 4 superconformal al-gebra) in the Calabi-Yau manifold case (respectively in the hyperkälher case) is theexistence of non-linear λ -brackets among the generators, i.e., in some λ -bracketsappear the normally ordered product of generators. This is a direct consequence ofthe geometry of the manifolds, in an almost Hermitian manifold even if the metricand the complex structure satisfied some compatibility condition they are essen-tially independent, though in the G case the metric and the cross product are bothdetermined in a highly nonlinear way from the -form.It is worth to mention that our proof of the conjecture is more tortuous thanthe proofs of previous results for Calabi-Yau and hyperkälher manifolds [BZHS08,Hel09, EHKZ13], this is due to the lack of ‘good’ coordinates systems in the G -holonomy case, it would be interesting to explore if there exist some coordinatesystem in which the quantum corrections to the fields (3.4) and (3.5) get simplifiedor even vanish as happens when M is Calabi-Yau or hyperkälher.Another related question is the following. There is a proposal of how to define atopological twist of the Shatashvili-Vafa G superconformal algebra in [SV95], infact, there is a more recent approach [dBNS08]. There are known a few necessarytopological conditions a manifold with holonomy G should satisfied, but we arefar from knowing sufficient conditions. Question: can we put the ‘topological’Shatashvili-Vafa G superconformal algebra locally inside the bc − βγ systems insuch a way the obstruction to be globally well defined as a subalgebra of the CDRbe sufficient conditions for having holonomy G ? In the Calabi-Yau case this isexactly what happens, the vanishing of the first Chern class of the manifold is theobstruction to having the topological N = 2 superconformal algebra globally welldefined [MSV99, Section 4].Among the special holonomy manifolds we are missing the Spin (7) case in5imension . In view of the present work is reasonable to expect that an analogoustheorem can be proved, i.e., the space of global sections of the CDR associated toa Spin (7) -holonomy manifold contains two commuting copies of the Shatashvili-Vafa
Spin (7) superconformal algebra (subsection 2.2) of central charge (seeRemark 5.3). It follows easily from our result that for a Spin (7) -manifold of theform M × R where M is a G -holonomy manifold the above claim is true. Acknowledgements:
The author would like to thank Reimundo Heluani forgenerously sharing his insights and ideas and for the encouragement despite thelong computations; the present paper is influenced by his views about the subject.
In this section we recall some facts about vertex superalgebras, for details thereader is referred to [Kac98, DSK06].Let V be a vector superspace, i.e., a vector space with a decomposition V = V ¯0 ⊕ V ¯1 for ¯0 , ¯1 ∈ Z / Z . We call V the even space and V the odd space. If v ∈ V α , one writes p ( v ) = α and calls it the parity of v .The algebra EndV acquires a Z / Z grading by letting ( EndV ) ¯ α = { A ∈ EndV : A ( V ¯ β ) ⊂ V ¯ α + ¯ β } for ¯ α, ¯ β ∈ Z / Z .An EndV - valued field is a formal distribution of the form a ( z ) = X n ∈ Z a ( n ) z − n − , a ( n ) ∈ EndV such that for every v ∈ V , we have a ( n ) v = 0 for large enough n .We now proceed to give two different definitions of a vertex algebra that weuse freely through the text, for a proof of the equivalence of these definitions werefer to [DSK06]. Definition 2.1.
A vertex superalgebra consists of the data of a vector superspace V (the space of states) , | i ∈ V ¯0 is a vector (the vacuum vector), an even endomor-phism T , and a parity preserving linear map a → Y ( a, z ) from V to EndV -valuedfields (the state-field correspondence). This data should satisfy the following set ofaxioms: • Vacuum axioms: Y ( | i , z ) = Id, Y ( a, z ) | i | z =0 = a, T | i = 0 . • Translation invariance: [ T, Y ( a, z )] = ∂ z Y ( a, z ) . Locality: ( z − w ) n [ Y ( a, z ) , Y ( b, w )] = 0 n ≫ . Given a vertex superalgebra V and a ∈ V we expand the fields Y ( a, z ) = X n ∈ Z a ( n ) z − n − (2.1)and we call the endomorphisms a ( n ) the Fourier modes of Y ( a, z ) .In any vertex algebra we have the following identity, known as the Borcherdsidentity : X j ∈ Z + ( − j (cid:18) nj (cid:19) (cid:18) a ( m + n − j ) (cid:16) b ( k + j ) c (cid:17) − ( − n ( − p ( a ) p ( b ) b ( n + k − j ) (cid:16) a ( m + j ) c (cid:17)(cid:19) = X j ∈ Z + (cid:18) mj (cid:19) (cid:16) a ( n + j ) b (cid:17) ( m + k − j ) c, (2.2) for all a, b, c ∈ V , m, n, k ∈ Z . In fact this identity appeared in the originalBorcherds definition of vertex algebras [Bor86].Given a vertex algebra V we define two operations: [ a λ b ] = X j ≥ λ j j ! a ( j ) b, : ab := a ( − b. (2.3)The first is called the λ -bracket and the second is called the normally ordered prod-uct. These operations, the vacuum vector | i and the derivation T determine thestructure of the vertex algebra as follows from the next definition of vertex algebra.We need to introduce the definition of a Lie conformal algebra. Definition 2.2.
A Lie conformal superalgebra R = R ¯0 ⊕ R ¯1 is a Z / Z -graded C [ ∂ ] -module endowed with a parity preserving C -bilinear λ -bracket [ . λ . ] : R ⊗ R → R ⊗ C [ λ ] (2.4)which satisfies:(i) sesquilinearity: [ ∂a λ b ] = − λ [ a λ b ] , [ a λ ∂b ] = ( λ + ∂ )[ a λ b ] , (2.5)(ii) skewsymmetry: [ b λ a ] = − ( − p ( a ) p ( b ) [ a − ∂ − λ b ] , (2.6)(iii) Jacobi identity: [ a λ [ b µ c ]] = ( − p ( a ) p ( b ) [ b µ [ a λ c ]] + [[ a λ b ] λ + µ c ] , (2.7)7 efinition 2.3. A vertex algebra is a quintuple ( V, | i , ∂, [ . λ . ] , ::) which satisfiesthe following three properties:(i) ( V, ∂, [ . λ . ]) is a Lie conformal algebra,(ii) ( V, | i , ∂, ::) is a unital differential algebra satisfying • quasi-commutativity: : ab : − ( − p ( a ) p ( b ) : ba := Z − ∂ [ a λ b ] dλ (2.8)for any a, b ∈ V, • quasi-associativity: :: ab : c : − : a : bc ::=: Z ∂ dλa ! [ b λ c ] : +( − p ( a ) p ( b ) : Z ∂ dλb ! [ a λ c ] : (2.9) for any a, b, c ∈ V, The integrals in the expression above should be interpreted in the following manner.First, expand the λ -bracket. Second, put the powers of λ on the left, under thesign of integral. Finally, take the definite integral by the usual rules inside theparenthesis. The binary operation :: is called the normally ordered product of V .(iii) The λ -bracket and the normally ordered product are related by the non-commutative Wick formula: [ a λ : bc :] =: [ a λ b ] c : +( − p ( a ) p ( b ) : b [ a λ c ] : + Z λ [[ a λ b ] µ c ] dµ, (2.10)for any a, b, c ∈ V. In this paper we are concerned with a special class of vertex superalgebras,known as conformal vertex algebras. We say that a vertex algebra V is conformal ifthere is a L ∈ V such that the Fourier modes of the corresponding field Y ( L, z ) = P n ∈ Z L n z − n − satisfy:(i) the operators L n form the Virasoro algebra [ L n , L m ] = ( m − n ) L m + n + c
12 ( m − m ) δ m, − n Id V (2.11)where c ∈ C is called the central charge,(ii) L − = T ,(iii) L is a diagonalizable operator with eigenvalues bounded below.8 emark . Using (2.3) note that (2.11) is equivalent to: [ L λ L ] = ( ∂ + 2 λ ) L + cλ . (2.12)A field L ( z ) satisfying this is called a Virasoro field. Note also that the condition(ii) above implies that L ( z ) is an even field.If V is conformal vertex algebra and a ∈ V is an eigenvector of L , its eigen-value is called the conformal weight of a and is denoted by ∆( a ) = ∆ . Moreover a has conformal weight ∆ if and only if [ L λ a ] = ( ∂ + ∆ λ ) a + O ( λ ) . In the case when [ L λ a ] = ( ∂ + ∆ λ ) a the vector a and the corresponding vertexoperator Y ( a, z ) are called primary. This is equivalent to L n a = δ n, ∆ a for all n ≥ . Remark . An important feature of defining vertex algebras using Lie conformalalgebras is that: vertex algebras are to conformal Lie algebras what associativealgebras are to Lie algebras in the following sense. For any Lie conformal algebra R there exists a vertex algebra U ( R ) with an embedding of conformal algebras π : R ֒ → U ( R ) satisfying the usual universal property: for any other vertex algebra V and a map f : R ֒ → V , there exists a morphism of vertex algebras g : U ( R ) ֒ → V such that f = g ◦ π . Moreover, the algebra U ( R ) is constructed very similar asin the Lie algebra situation, any vector of U ( R ) can be obtained by products ofelements of R .This parallel with the Lie algebra case justifies the presentation of the exam-ples below. When we say that a certain set A of vectors satisfying some prescribed λ -brackets generate a vertex algebra, we first construct the corresponding Lie con-formal algebra and then we consider its universal enveloping vertex algebra, inparticular any vector of this algebra is a combination of products of elements of A and their derivatives. There are other situations (see subsections 2.1 and 2.2 )when the λ -bracket of elements in A is not linear in the elements of A but can beexpressed as combinations of products of elements of A and their derivatives. Inthis case we say that the vertex algebra is non-linearly generated [DSK05]. Remark . Note that the λ -bracket notation encodes the same information as thesingular part of the operator product expansion (OPE) of fields in a quantum fieldtheory in dimension two. In other words when we write [ A λ B ] = P N − j =0 λ j j ! A ( j ) B ,in physics notation this means that: A ( z ) B ( w ) ∼ N − X j =0 ( A ( j ) B )( w )( z − w ) j +1 , where A ( z ) , B ( w ) and ( A ( j ) B )( w ) are the fields corresponding to the vectors A , B and A ( j ) B under the state-field correspondence.9 xample 2.7 ([Kac98]) . The N = 1 (Neveu-Schwarz) superconformal vertex al-gebra.The N = 1 superconformal vertex algebra of central charge c is an extension ofthe Virasoro algebra of central charge c by an odd primary field G of conformalweight / i.e it is generated by L and G with λ -brackets: [ L λ L ] = ( ∂ + 2 λ ) L + c λ , [ L λ G ] = ( ∂ + 32 λ ) G, [ G λ G ] = 2 L + c λ . Example 2.8 ([Kac98]) . The N = 2 superconformal vertex algebra.The N = 2 superconformal vertex algebra of central charge c is generated by theVirasoro field L with λ -bracket (2.12), an even primary field J of conformal weight , and two odd primary fields G ± of conformal weight , with the λ -brackets givenby: [ L λ J ] = ( ∂ + λ ) J, [ L λ G ± ] = (cid:18) ∂ + 32 λ (cid:19) G ± , [ J λ G ± ] = ± G ± , [ J λ J ] = c λ, [ G ± λ G ± ] = 0 , [ G + λ G − ] = L + 12 ∂J + J λ + c λ . Example 2.9 ([KW04]) . The N = 4 superconformal vertex algebra.The N = 4 superconformal vertex algebra is generated by a Virasoro field L , threeprimary even fields J , J + and J − of conformal weight 1, and four primary oddfields G ± , ¯ G ± of conformal weight . The remaining non-zero λ -brackets are: [ J λ J ± ] = ± J ± , [ J λ J ] = c λ, [ J + λ J − ] = J + c λ, [ J λ G ± ] = ± G ± , [ J λ ¯ G ± ] = ± ¯ G ± , [ J + λ G − ] = G + , [ J − λ G + ] = G − , [ J + λ ¯ G − ] = − ¯ G + , [ J − λ ¯ G + ] = − ¯ G − , [ G ± λ ¯ G ± ] = ( ∂ + 2 λ ) J ± , [ G ± λ ¯ G ∓ ] = L ± ∂J ± J λ + c λ . Example 2.10.
The bc − βγ system.This vertex algebra is generated by four fields: b and c are odd fields, β and γ areeven fields, and the non-trivial λ -brackets between the generators are: [ β λ γ ] = 1 , [ b λ c ] = 1 . If we define G =: cβ : + : ( ∂γ ) b : , L =: ( ∂γ ) β : − : c ( ∂b ) : + : ( ∂c ) b : , { L, G } generate an N = 1 superconformal algebra of central charge (example2.7). With respect to the Virasoro L the conformal weight of the generators is asfollows: ∆ b = 12 , ∆ c = 12 , ∆ γ = 0 , ∆ β = 1 . .1 The Shatashvili-Vafa G superconformal algebra The Shatashvili-Vafa G superconformal algebra [SV95] is an extension of the N = 1 superconformal algebra { L, G } (Example 2.7) by four fields { Φ , K, X, M } such that: Φ and M are odd, K and X are even, ∆(Φ) = , ∆( K ) = 2 , ∆( X ) = 2 and ∆( M ) = . Furthermore Φ and K are primary fields. The λ -brackets aregiven by: [Φ λ Φ] = ( −
72 ) λ + 6 X, [Φ λ X ] = −
152 Φ λ − ∂ Φ , [ X λ X ] = 3524 λ − Xλ − ∂X, [ G λ Φ] = K, [ G λ X ] = − Gλ + M, [ G λ K ] = 3Φ λ + ∂ Φ , [ G λ M ] = − λ + ( L + 4 X ) λ + ∂X, [Φ λ K ] = − Gλ − (cid:18) M + 12 ∂G (cid:19) , [Φ λ M ] = 92 Kλ − (cid:18) G Φ : − ∂K (cid:19) , [ X λ K ] = − Kλ + 3 (: G Φ : − ∂K ) , [ X λ M ] = − Gλ − (cid:18) M + 94 ∂G (cid:19) λ + (cid:18) GX : − ∂M − ∂ G (cid:19) , [ K λ K ] = − λ + 6 ( X − L ) λ + 3 ∂ ( X − L ) , [ K λ M ] = −
152 Φ λ − ∂ Φ λ + 3 (: GK : − L Φ :) , [ M λ M ] = − λ + 12 (20 X − L ) λ + (cid:18) ∂X − ∂L (cid:19) λ + (cid:18) ∂ X − ∂ L − GM : +8 : LX : (cid:19) , [ L λ X ] = − λ + 2 Xλ + ∂X, [ L λ M ] = − Gλ + 52 M λ + ∂M. And the generators satisfy the following relation: GX : − K : − ∂M − ∂ G. (2.13) Remark . Unlike the previous examples the right hand side of some λ -bracketsis non-linear in the generators. This is an important feature of the algebra.This superconformal algebra appeared as the chiral algebra associated to thesigma model with target a manifold with G holonomy in [SV95] Its classicalcounterpart had been studied by Howe and Papadopoulos in [HP93]. In fact thisalgebra is a member of a two-parameter family SW ( , , previously studied in[Blu92] where the author found the family of all superconformal algebras which are11xtension of the super-Virasoro algebra, i.e., the N = 1 superconformal algebra,by two primary supercurrents of conformal weights and respectively. It is afamily parametrized by ( c, ε ) ( c is the central charge and ε the coupling constant)of non-linear W -algebras.The Shatashvili-Vafa G algebra is a quotient of SW ( , , with c = and ε = 0 , the relation (2.13) is precisely the one imposed by the quotient.In [SV95], this algebra was obtained as a free field realization in terms of sevenfree Bosons and seven free Fermions, and the relation (2.13) was trivially satis-fied. The first to note that we should impose this relation if we define the algebraabstractly (it is necessary for the Jacobi identities to be checked) was Figueroa-O’Farrill [FO97].The Shatashvili-Vafa G superconformal algebra can be obtained also as aquantum Hamiltonian reduction of osp (cid:0) | (cid:1) [HRD15]. Remark . Note that if we define ˜ X := − X and ˜Φ := i √ , then { ˜ X, ˜Φ } gen-erate an N = 1 superconformal algebra of central charge c = . This N = 1 su-perconformal algebra at this value of the central charge is known as the tri-criticalIsing model. Therefore the Shatashvili-Vafa algebra contains two N = 1 super-conformal subalgebras, the original one generated by { L, G } and the subalgebragenerated by { X, Φ } . Remark . Note that all the generators of the Shatashvili-Vafa G algebra canbe obtained as the λ -brackets of the generators in conformal weight i.e., Φ and G . The Shatashvili-Vafa G superconformal algebra has an automorphism givenby L → L , G → G , Φ → − Φ , K → − K , X → X , M → M . It is interesting tonote that the fixed vectors { L, G, X, M } generate a closed subalgebra, this algebrais a member of a one parameter family SW ( , of superconformal algebras, thisone corresponds to the parameter . Another member of this family is the nextexample. Spin(7) superconformal algebra
The Shatashvili-Vafa
Spin(7) superconformal algebra [SV95] is an extension ofthe N = 1 superconformal algebra { L, G } (Example 2.7) by an even field ¯ X ofconformal weight (not primary) and an odd field ¯ M of conformal weight (notprimary). The λ -bracket are given by: [ ¯ X λ ¯ X ] = 83 λ + 16 ¯ Xλ + 8 ∂ ¯ X, [ L λ ¯ X ] = 13 λ + ( ∂ + 2 λ ) ¯ X, [ G λ ¯ X ] = 12 Gλ + ¯ M , [ G λ ¯ M ] = 23 λ − (cid:0) L − X (cid:1) λ + ∂ ¯ X, [ ¯ X λ ¯ M ] = − Gλ − (cid:18) ∂G − M (cid:19) λ + 112 ∂ ¯ M − ∂ G − G ¯ X : , ¯ M λ ¯ M ] = − λ − (cid:18) L + 16 ¯ X (cid:19) λ − (cid:18) ∂L + 16 ∂ ¯ X (cid:19) λ − (cid:18) ∂ ¯ X + 52 ∂ L + 12 : L ¯ X : − G ¯ M : (cid:19) . Similar to the example above this algebra appeared as the chiral algebra associatedto the sigma model with target a manifold with
Spin(7) holonomy in [SV95] and itsclassical counterpart had been studied by Howe and Papadopoulos in [HP93]. Thisalgebra also belongs to a family SW ( , of superconformal algebras with oneparameter c (the central charge), the Shatashvili-Vafa Spin(7) algebra correspondsto c = 12 , see ([FOS91], [FOS92], [FO97]). In this section we review the construction of the chiral de Rham complex [MSV99]of a manifold M and remember some of its properties. We also recall a theoremfrom [EHKZ13] that provides two different embeddings of the space of differentialforms Ω ∗ ( M ) into global sections of CDR in the case when M is a smooth Rie-mannian manifold. The chiral de Rham complex can be defined in the algebraic,analytic and smooth category, in this paper we are only concerned with the smoothsetting.Let M be a C ∞ manifold, the CDR of M is a sheaf of vertex algebras, let usbegin with a local description. On a coordinate chart of M with coordinates { x i } the sections of CDR are described as follows: for each coordinate x i we have aneven field γ i , corresponding to each vector field ∂∂x i and each differential form dx i we have an odd fields b i and c i respectively, we also have an even field β i for each γ i , and { γ i , β i , b i , c i } form a bc − βγ system (see example 2.10).Now we need to take care of what happens if we change coordinates. A crucialobservation in [MSV99] is that for any change of coordinates x i → y i ( x j ) thereexists an automorphism of CDR on the intersection of the coordinate charts. Us-ing this it is possible to glue on intersections and construct a global sheaf. Moreprecisely if ˜ γ i , ˜ β i , ˜ c i and ˜ b i are the fields associated to the coordinates y i , they areexpressed in terms of the fields in the coordinates x i as follows: ˜ γ i = y i ( γ ) , ˜ c i = ∂y i ∂x j ( γ ) c j , ˜ b i = ∂x j ∂y i ( y ( γ )) b j , (3.1) ˜ β i = β j ∂x j ∂y i ( y ( γ )) + ∂ x k ∂y i ∂y l ( y ( γ )) ∂y l ∂x r c r b k . We see that the γ i transform as coordinates do, the b i transform as vector fields, the c i change as differential forms, however the β i change in a non-tensorial manner. In13he remaining of this section we collect some results about the existence of globalsections of CDR. Remark . The first thing to note is that the multiplication (normally orderedproduct) in CDR is neither associative nor commutative by the own nature of CDR:CDR is a sheaf of vertex algebras. Even though we can define a multiplication map O M × CDR → CDR , it is not associative. This implies that it is very difficult toconstruct global sections of CDR.From (3.1) follows that functions and vector fields of M give rise to global sections.However, trying to construct sections of CDR from other tensors on M is not trivialbecause of the terms on the RHS of the quasi-associativity (2.9) appearing under achange of coordinates.In [BZHS08] it was noticed that one can use the Levi-Civita connection on M tocounteract these quasi-associativity terms in order to construct sections of CDRassociated to differential two-forms. In [EHKZ13] this was generalized to higherorder forms.We can define in a local chart the following field G = ( ∂γ i ) b i + β i c i (seeexample 2.10) and ask if this field is globally defined. The answer [MSV99] isthe following, if we change coordinates x i → y i ( x j ) , and let ˜ G be the field in thecoordinates y i we obtain: ˜ G = G + ∂ ∂∂y r T r log ∂x i ∂y j ! ˜ c r . Note that if M is orientable G is globally defined. In general the field G is notglobally defined however the Fourier modes G (0) is globally defined because ˜ G differs from G by a derivative of a field. Moreover we have ( G (0) ) = ∂ . Thisendomorphism G (0) is called the supersymmetric generator, and it is an odd deriva-tion of all the n -products i.e., G (0) ( a ( n ) b ) = ( G (0) a ) ( n ) b + ( − p ( a ) a ( n ) ( G (0) b ) ,where a, b are two sections of the CDR.In [EHKZ13] were produced two different embeddings of the space of differ-ential forms Ω ∗ ( M ) into global sections of CDR in the case when M is a smoothRiemannian manifold. These embeddings depend on a choice of a metric and areexplicitly given in terms of the corresponding Levi-Civita connection. We proceedto recall their construction:Let g be a Riemannian metric on M and let { x i } be a local coordinate system.Define local sections e i ± by the equations: e i + := g ij b j + c i √ , ie i − := i (cid:0) g ij b j − c i (cid:1) √ . Note that [ e i + λ e j + ] = g ij , [ e i − λ e j − ] = − g ij , [ e i + λ e j − ] = 0 . (3.2)14efine F ± (0) := 1 , F i ...i k +( k ) := e i + . . . e i k + , F i ...i k − ( k ) := i k e i − . . . e i k − . (Here and further we adopt the following convention: when a product of fieldsappears without any colon or parenthesis, we read the normal product from rightto left, recalling that it is not associative.)Define G ± ( n,n ) = F ± ( n ) and for each ≤ s ≤ ⌊ n ⌋ we define G i ...i n ± ( n,n − s ) = Γ i k l g i k ∂γ l . . . Γ i s − k s − l s − g i s k s − ∂γ l s − F i s +1 ...i n ± ( n − s ) , where Γ ijk are the Christoffel symbols of the Levi-Civita connection associated to g , ⌊·⌋ denotes the integer part, the subscript between parenthesis denotes how many e ’s are present in the expression.Define the numbers T r,s as the coefficients of the Bessel polynomials: y r ( x ) = r X s =0 T r,s x s = r X s =0 ( r + s )!( r − s )! s !2 s x s , and let T r,s := 0 when s < or s > t . The following theorem was proved in[EHKZ13]. Theorem 3.2. [EHKZ13, Theorem 6.1]
Let ( M, g ) be a Riemannian manifold. Forany differential form w ∈ Ω ∗ ( M ) locally described by w i . . . i n dx i ∧ · · · ∧ dx i n define J ± q := 1 n ! w i ...i n E i ...i n ± ( n ) ,E ± ( n ) := [ n ] X s =0 T n − s,s G ± ( n,n − s ) . Then J ± q are well defined sections of CDR. Corollary 3.3. G (0) (cid:0) J ± q (cid:1) are also well defined sections of CDR. This follows from the discussion above about the supersymmetric generator G (0) . In fact this corollary was implicit in the original form of the Theorem 3.2in [EHKZ13] where the authors consider the CDR as a supersymmetric sheaf ofvertex superalgebras.As an exemplification of the Theorem 3.2 and because they will be used insection 5 we write the sections of CDR that correspond to , and forms:if w is a -form then J ± q = ± w ij e i ± e j ± + 12 Γ ijk g jl w il ∂γ k , (3.3)15f w is a -form then J + q = 16 w ijk e i + e j + e k + + 12 w ijk Γ imn g jm ∂γ n e k + , (3.4) J − q = ( − i )6 w ijk e i − e j − e k − + i w ijk Γ imn g jm ∂γ n e k − , if w is a -form then J + q = 124 w ijkl e i + e j + e k + e l + + 14 w ijkl Γ imn g jm ∂γ n e k + e l + + 18 w ijkl Γ im n g jm ∂γ n Γ km n g lm ∂γ n , (3.5) J − q = 124 w ijkl e i − e j − e k − e l − + ( − w ijkl Γ imn g jm ∂γ n e k − e l − + 18 w ijkl Γ im n g jm ∂γ n Γ km n g lm ∂γ n . Theorem 3.4. [EHKZ13, section 7.1 and 7.2]
Let M be a Calabi-Yau manifold, ω the Kähler form, let J ± q be the global sections given by Theorem 3.2 cor-responding to ω , then the two sets of sections of CDR { J ± q , G (0) ( J ± q ) } gen-erate two commuting copies of the N = 2 superconformal algebra of centralcharge dim M . Furthermore if we assume that M is a Calabi-Yau three-fold, let Ω and ¯Ω denote the holomorphic volume form and its complex conju-gate respectively. Let X ± and ¯ X ± be the global sections given by Theorem 3.2corresponding to Ω and ¯Ω respectively, then the two sets of sections of CDR { J ± q , G (0) ( J ± q ) , X ± , G (0) ( X ± ) , ¯ X ± , G (0) ( ¯ X ± ) } generate two commuting copiesof the Odake algebra [Oda89] . G holonomy manifolds In this section we review some properties of the manifolds with G holonomy thatwill be used extensively in the computations. The references for this section are[Joy07, Sal89].The Lie group G can be defined as the group of linear automorphism of R that preserves the cross product, where we identify R = Im ( O ) , and the crossproduct operation is induced from the octonion multiplication u × v = im (¯ v.u ) , u, v ∈ O . The cross product operation is encoded by a 3-form ϕ ∈ ∧ ( R ) ∗ defined as ϕ ( u, v, w ) = < u × v, w > . If we choose coordinates ( x , . . . , x ) on R and denote by dx ij...k the exteriorform dx i ∧ dx j ∧ · · · ∧ dx k on R , ϕ can be written as:16 = dx + dx + dx + dx − dx − dx − dx . In the other way, the identity < u, v > dvol = ( u y ϕ ) ∧ ( v y ϕ ) ∧ ϕ , (4.1)expresses the inner product of R in terms of the volume form of R and the -form ϕ . Observe that the Lie algebra g of the Lie group G can be described as: g = { (a ij ) ∈ so (7) | a ij ϕ = 0 ∀ k } . (4.2) Definition 4.1.
Let M be a smooth -dimensional manifold, and F the frame bun-dle of M . We say that M has a G -structure if there is a principal subbundle P of F with fibre G .Equivalently M has a G -structure if there exists ϕ ∈ Ω ( M ) that is pointwisemodelled in ϕ , i.e., for each x ∈ M , ϕ is the image of ϕ ( x ) under a linearoriented isomorphism T x M → R . Thus, we have a 1-1 correspondence between3-forms ϕ pointwise modelled in ϕ and G -structures P on M .Given a G -structure in M this determines in particular an orientation anda metric g in M because any G -structure is an instance of an SO (7) -structure.Furthermore, let ϕ be the 3-form that corresponds to this G -structure. Then, theidentity (4.1) determines the metric in terms of ϕ up to a conformal factor. Let Hol ( g ) denote the holonomy group of M , then Hol ( g ) ⊆ G if and only if ϕ isparallel with respect to the Levi-Civita connection of g , i.e., ∇ ϕ = 0 . Lemma 4.2. [Sal89]
The holonomy group of the Riemannian metric g induced by ϕ is contained in G if and only if dϕ = 0 and d ∗ ϕ = 0 , where ∗ denotes theHodge star for g . We say that a manifold with a G -structure is a G -manifold if Hol ( g ) ⊆ G .If we write ϕ and ∗ ϕ in local coordinates: ϕ = ϕ ijk dx ijk , ∗ ϕ = ψ ijkl dx ijkl then dϕ = 0 and d ∗ ϕ = 0 are equivalent to: ∂∂x m ϕ ijk − Γ smi ϕ sjk − Γ smj ϕ isk − Γ smk ϕ ijs , (4.3) ∂∂x m ψ ijkl − Γ smi ψ sjkl − Γ smj ψ iskl − Γ smk ψ ijsl − Γ sml ψ ijks , (4.4)for all m = 1 , . . . , . Lemma 4.3. [Sal89]
Let ( M, g ) be a Riemannian 7-manifold. If Hol ( g ) ⊆ G ,then g is Ricci-flat. M be a -dimensional manifold with G –structure given by the -form ϕ ,denote by g and ψ = ∗ ϕ the associated metric and -form respectively. Then thefollowing identities are satisfied, they are collected from [Kar09]:Contractions of ϕ and ψϕ ijk ϕ abc g ia g jb g kc =42 , (4.5) ϕ ijk ϕ abc g jb g kc =6 g ia , (4.6) ϕ ijk ϕ abc g kc = g ia g jb − g ib g ja − ψ ijab , (4.7) ϕ ijk ψ abcd g ib g jc g kd =0 , (4.8) ϕ ijk ψ abcd g jc g kd = − ϕ iab , (4.9) ϕ ijk ψ abcd g kd = g ia ϕ jbc + g ib ϕ ajc + g ic ϕ abj − g aj ϕ ibc − g bj ϕ aic − g cj ϕ abi , (4.10) ψ ijkl ψ abcd g ld = − ϕ ajk ϕ ibc − ϕ iak ϕ jbc − ϕ ija ϕ kbc + g ia g jb g kc + g ib g jc g ka + g ic g ja g kb − g ia g jc g kb − g ib g ja g kc − g ic g jb g ka − g ia ψ jkbc − g ja ψ kibc − g ka ψ ijbc + g ab ψ ijbc − g ac ψ ijkb . (4.11)Another useful identity is: ϕ ∧ ( ω ϕ ) =( − (cid:16) ψ ∧ w ♭ (cid:17) , or in coordinates ϕ [ ijk ϕ mn ] l =( − g l [ m ϕ nijk ] (4.12)where w is a vector field in M and w ♭ be the –form dual to w ; [ ] denotes theanti–symmetrization of the indices. Lemma 4.4. (Lemma 4.9 in [Kar09] ) Let M be a manifold with a G -structure andlet R ijkl be the Riemann curvature tensor of M . Then we have R ijkl ψ ijkm = 0 . The next lemma follows easily from the fact that if
Hol ( g ) ⊆ G then R ijkl ∈ Sym (g ) , and the above definition (4.2) of g Lemma 4.5.
Let M be a G -manifold and let R ijkl be the Riemann curvaturetensor of M . Then we have R ijkl ϕ ijm = 0 . G -manifold In this section we prove the main results of this paper. We begin remembering theconjecture that is the leitmotiv of this paper.18et ( M, g ) be a G -manifold, let ϕ be the corresponding defining 3-form. Let Φ ± be the global sections of CDR that correspond to ϕ by Theorem 3.2, and let K ± := G (0) (Φ ± ) . In [EHKZ13] the following conjecture was stated: Conjecture 5.1. [EHKZ13, Conjecture 7.3]
The sections pairs { Φ + , K + } and { Φ − , K − } generate two commuting copies of the Shatashvili-Vafa G supercon-formal algebra given in subsection 2.1 (see Remark 2.13). G holonomy manifolds are superconformal Before stating our main result we define some sections: X ± := 16 (cid:16) Φ ± (0) Φ ± (cid:17) , G ± := ( −
13 ) (cid:16) Φ ± (1) K ± (cid:17) , L ± := 12 (cid:16) G ± (0) G ± (cid:17) . Note that these are well defined global sections because they are defined in termsof Φ ± and K ± . Theorem 5.2.
The pairs of sections { G ± , L ± } and { Φ ± , X ± } generate two N = 1 superconformal algebras of central charge and respectively. Furthermore theplus signed sections commute with the minus signed sections.Proof. The proof is based on explicit computations and some abstract manipula-tions. To make the proof clear, every time that an explicit computation should bemade we will indicate the appropriate subsection of the section 6 where the com-putation is performed. Note that to compute a λ -bracket we can choose a localcoordinate chart and perform the computation in this chart because we are workingwith well defined global sections.We begin proving that { Φ ± , X ± } generate two N = 1 superconformal algebras ofthe desired central charges respectively. We work the plus case (the minus case issimilar).We have by (3.4) Φ + = ϕ ijk e i + e j + e k + + ϕ ijk Γ imn g jm ∂γ n e k + . (5.1)In 6.1 it is shown that [Φ + λ Φ + ] = ( − ) λ + 6 X + , (5.2)where X + is given by: X + = − ψ ijkl e i + e j + e k + e l + − ψ ijkl Γ imn g jm ∂γ n e k + e l + − ψ ijkl Γ im n g jm ∂γ n Γ km n g lm ∂γ n − g ij ∂ ( e i + ) e j + − g ij Γ jkl ∂γ k e l + e i + − Γ ijk Γ kil ∂γ j ∂γ l . (5.3)In 6.2 we see that: 19 +(2) Φ + = 0 , X +(1) Φ + = − Φ + . (5.4)Then, using skewsymmetry (2.6), we get Φ +(2) X + = 0 , Φ +(1) X + = − Φ + . Tocompute Φ +(0) X + we observe that: [Φ + λ [Φ + µ Φ + ]] = 6[Φ + λ X + ] (5.5)by (5.2). Using the Jacobi identity (2.7) on the left side of (5.5) we have: [Φ + λ [Φ + µ Φ + ]] = ( − + µ [Φ + λ Φ + ]] + [[Φ + λ Φ + ] λ + µ Φ + ]= ( − + µ X + ] + [6 X + λ + µ Φ + ]= ( − (cid:16) Φ +(0) X + + ( − )Φ + µ (cid:17) + ( − + − λ − µ − ∂ X + ]= ( − (cid:16) Φ +(0) X + + ( − )Φ + µ (cid:17) (5.6) +( − (cid:16) Φ +(0) X + + ( λ + µ + ∂ )Φ + (cid:17) . Equating (5.5) and (5.6) we obtain a polynomial equality in λ and µ . Takingthe coefficient of λ µ we get Φ +(0) X + = − ∂ Φ + . Then [Φ + λ X + ] = − Φ + λ − ∂ Φ + . (5.7)To compute [ X + λ X + ] observe that, using (5.2) and the Jacobi identity, wehave: X + λ X + ] = [ X + λ [Φ + µ Φ + ]] = [Φ + µ [ X + λ Φ + ]] + [[ X + λ Φ + ] λ + µ Φ + ] . Then X + λ X + ] = [Φ + µ − ∂ Φ + − Φ + λ ] + [[ X + λ Φ + ] λ + µ Φ + ] . (5.8)This is a polynomial identity in λ and µ . Putting µ = 0 we get: X + λ X + ] = − ∂X + − X + λ + [[ X + λ Φ + ] λ Φ + ]= − ∂X + − X + λ + [ ∂ Φ + − Φ + λ λ Φ + ]= − ∂X + − X + λ + [ − ∂ Φ + λ Φ + ] + [ − Φ + λ Φ + ] λ, Using sesquilinearity (2.5) it follows that: [ X + λ X + ] = λ − X + λ − ∂X + . (5.9)20e have proved that { Φ ± , X ± } satisfy the λ -brackets (5.2), (5.7) and (5.9) andtherefore, by Remark 2.12 we conclude that { Φ + , X + } and { Φ − , X − } generatetwo N = 1 superconformal algebras of central charge respectively. Further-more, as the e i + ’s commute with the e i − ’s (3.2), we have that Φ + commutes with Φ − and, consequently, the algebra generated by { Φ + , X + } commutes with thealgebra generated by { Φ − , X − } .Now we are going to prove that { G ± , L ± } generate two N = 1 superconformalalgebras of central charge . We compute explicitly G ± := ( −
13 ) (cid:16) Φ ± (1) K ± (cid:17) (see 6.3) and find that: G + = c i β i + ∂γ i b i + g ij b i β j + g ij Γ lik c k b j b l + g ij Γ kij ∂b k + g ij ∂γ i c j + g ij Γ kil Γ ljm ∂γ m b k ,G − = c i β i + ∂γ i b i + ( − ) g ij b i β j + ( − ) g ij Γ lik c k b j b l + ( − ) g ij Γ kij ∂b k +( − ) g ij ∂γ i c j + ( − g ij Γ kil Γ ljm ∂γ m b k . Note that if we define G := G + + G − , we have G = c i β i + ∂γ i b i , and { G, L := 12 G (0) G } generate an N = 1 superconformal algebra of central charge c = 21 , and with respect to L the conformal weight of the generators is as follows(see example 2.10 and the definition of G (0) in section 3): ∆ b i = 12 , ∆ c i = 12 , ∆ γ i = 0 , ∆ β i = 1 , i ∈ { , . . . , } . (5.10)This implies that the conformal weights of the local sections in the coordinate chartare always positive. We will use this observation below.In 6.4, we prove that [ G + λ G − ] = 0 . By definition, L + = 12 G +(0) G + and L − = 12 G − (0) G − . Then we have that { G + , L + } commute with { G − , L − } , andthat L = L + + L − . We will show that { G + , L + } is an N = 1 superconformalalgebra (the same proof works for { G − , L − } ). First we take care of [ L + λ L + ] . As ∆ L + = 2 , the observation above about conformal weights implies: [ L + λ L + ] = [ L λ L + ] = ∂L + + 2 L + λ + Aλ + Bλ , where A and B are two fields with ∆ A = 1 and ∆ B = 0 .Consequently: [ L λ A ] = ∂A + Aλ + Cλ , [ L λ B ] = ∂B, where C is a field with ∆ C = 0 , i.e., [ L λ C ] = ∂C . Using the Jacobi identity (2.7),we obtain [ L + λ [ L µ L ]] = [ L + µ [ L + λ L + ]] + [[ L + λ L + ] λ + µ L + ] . (5.11)21lso: [ L + λ [ L µ L ]] = [ L + λ ∂L + + 2 L + µ ] . (5.12)Working the right side of (5.12) : [ L + λ ∂L + + 2 L + µ ] = ( ∂ + λ ) (cid:16) ∂L + + 2 L + λ + Aλ + Bλ (cid:17) (5.13) + 2 µ (cid:16) ∂L + + 2 L + λ + Aλ + Bλ (cid:17) . Working separately the summands on the right side of (5.11) and putting µ = 0 : [ L + µ [ L + λ L + ]] | µ =0 = [ L µ ∂L + + 2 L + λ + Aλ + Bλ ] | µ =0 = ∂ L + + 2 ∂L + λ + ∂Aλ + ∂Bλ , [[ L + λ L + ] λ L + ] = [ ∂L + + 2 L + λ + Aλ + Bλ λ L ]= ( − λ ) (cid:16) ∂L + + 2 L + λ + Aλ + Bλ (cid:17) +2 λ (cid:16) ∂L + + 2 L + λ + Aλ + Bλ (cid:17) + Aλ +( − C ) λ + ( − ∂Cλ + ( − ∂ Cλ + ( − ∂Bλ . Equating the right sides of (5.11) and (5.12), putting µ = 0 , and looking at thecoefficient of λ and λ we get that C = 0 and ∂B = A respectively.Working separately the summands on the right side of (5.11) and putting λ = 0 : [ L + µ [ L + λ L + ]] | λ =0 = [ L + µ ∂L + ] = ( ∂ + µ ) (cid:16) ∂L + + 2 L + µ + Aµ + Bµ (cid:17) , [[ L + λ L + ] λ + µ L + ] | λ =0 = [ ∂L + µ L + ] = ( − µ (cid:16) ∂L + + 2 L + µ + Aµ + Bµ (cid:17) . Equating the right sides of (5.11) and (5.12), putting λ = 0 , and looking at thecoefficient of µ we get that ∂B = 0 . But this implies that A = 0 . Also, as ∆ B = 0 , we know that B is a function, therefore ∂B = 0 implies that B is aconstant.Hence [ L + λ L + ] = ( ∂ + 2 λ ) L + + c + λ , where c + is a constant.Now we compute [ L + λ G + ] . Proceeding as above we get that ∆ G + = and,hence: [ L + λ G + ] = [ L λ G + ] = ∂G + + 32 G + λ + Cλ , where C is a field with ∆ C = 12 , and [ L λ C ] = ∂C + 12 Cλ.
Using the Jacobi identity (2.7), we obtain [ G + λ [ L + µ L + ]] = [ L µ [ G + λ L + ]] + [[ G + λ L + ] λ + µ L ] . (5.14)22e also have: [ G + λ [ L + µ L + ]] = [ G + λ ( ∂ + 2 µ ) L + ] . (5.15)Working the right side of (5.15): [ G + λ ( ∂ + 2 µ ) L + ] = ( − λ + ∂ ) (cid:18) ∂G + + 32 ( − λ − ∂ ) G + + ( λ + ∂ ) C (cid:19) +2 µ ( − (cid:18) ∂G + + 32 ( − λ − ∂ ) G + + ( λ + ∂ ) C (cid:19) . Working separately the summands on the right side of (5.14) and putting µ = 0 : [ L µ [ G + λ L + ]] | µ =0 = [ L µ ∂G + + 32 G + λ + ( − C ) λ ] | µ =0 = 12 ( ∂ G + ) + 32 ∂G + λ + ( − ∂Cλ , [[ G + λ L + ] λ + µ L ] | µ =0 = [ 12 ∂G + + 32 G + λ + ( − C ) λ λ L ]= ( −
12 ) λ (cid:18) ∂G + + 32 G + λ + ( − C ) λ (cid:19) + 32 λ (cid:18) ∂G + + 32 G + λ + ( − C ) λ (cid:19) + ( −
12 ) Cλ . Equating the right sides of (5.14) and (5.15) putting µ = 0 , and looking at thecoefficient of λ we get that C = 0 which implies [ L + λ G + ] = ∂G + + 32 G + λ. Now we compute [ G + λ G + ] . By definition of L + and conformal weights positivitywe have: [ G + λ G + ] = [ G λ G + ] = 2 L + + Dλ + Eλ , with D and E two fields of conformal weights and respectively.Using the Jacobi identity (2.7) we obtain [ L + λ [ G µ G ]] = [ G + µ [ L + λ G + ]] + [[ L + λ G + ] λ + µ G + ] . (5.16)We also have: [ L + λ [ G µ G ]] = [ L + λ L ] = [ L + λ L + ] = 2( ∂ + 2 λ ) L + + 2 c + λ . (5.17)Working separately the summands on the right side of (5.16): [ G + µ [ L + λ G + ]] =[ G + µ ∂G + + 32 G + λ ]=( ∂ + µ ) (cid:16) L + + Dµ + Eµ (cid:17) + 32 λ (cid:16) L + + Dµ + Eµ (cid:17) , [ L + λ G + ] λ + µ G + ] =[ ∂G + + 32 G + λ λ + µ G + ]=( − λ + µ ) (cid:16) L + + D ( λ + µ ) + E ( λ + µ ) (cid:17) + 32 λ (cid:16) L + + D ( λ + µ ) + E ( λ + µ ) (cid:17) . Equating both the right sides of (5.16) and (5.17) and looking at the coefficient of λ with µ = 0 we get that D = 0 while looking at the coefficient of λ with µ = 0 we get that E = c + . Therefore [ G + λ G + ] = 2 L + + c + λ .We have proved that { G + , L + } and { G − , L − } are two commuting N = 1 super-conformal algebras of central charge c + and c − respectively, such that c + + c − =21 . We can calculate the central charge by computing explicitly the lambda bracket [ G + λ G + ] = [ G λ G + ] and looking at the coefficient of λ .Computing the coefficient of λ : ∂ ∂λ (cid:16)(cid:2) G + λ G + (cid:3)(cid:17)! = 74 + ( −
14 ) (cid:16) g lm,l (cid:17) ,m + 14 (cid:16) g lm Γ nlm (cid:17) ,n + 12 g lm Γ nla Γ amn + 74= 72 + 12 g lm Γ nla Γ amn + 12 (cid:16) g lm Γ nlm (cid:17) ,n = 72 + (cid:18) − (cid:19) g lm Γ nla Γ amn + 12 g lm (Γ nlm ) ,n = 72 + 12 g lm R lm = 72 . The last equality follows because G -manifolds are Ricci flat, Lemma 4.3. It isinteresting to note that, as in other places, the scalar curvature of the manifold, i.e., g ij R ij , appears here explicitly. Then, the central charge of the N = 1 superconfor-mal algebra { G + , L + } is c + = and consequently as c + + c − = 21 , the centralcharge of { G − , L − } is c − = .To conclude the proof of the theorem we only need to check that G + (resp. G − )commutes with Φ − (resp. Φ + ). This is accomplished by performing an explicitcomputation in 6.5. Remark . Note that X + (5.3) is not exactly the section produced by Theorem3.2 using the 4-form, because, besides the correction (3.5), we need to add otherterms. That this would happen was already observed in [EHKZ13] while workingthe flat case. For this reason, Conjecture 5.1 was formulated using only the -form.The expression (5.3) that we have obtained here works in any coordinate system. Remark . It should be noticed that the complexity of the explicit computationsperformed is greater than in the other holonomy cases. This is due to the lack ofspecial coordinate systems simplifying the corrections to the fields. This in turnis a reflection of our lack of knowledge about the geometry of manifolds with G holonomy. 24 .2 Proof of the Conjecture In this subsection, we prove Conjecture 5.1.
Theorem 5.5.
The sections pairs { Φ + , K + } and { Φ − , K − } generate two commut-ing copies of the Shatashvili-Vafa G superconformal algebra. In the above subsection 5.1, besides the global sections Φ ± and K ± = G (0) Φ ± ,we introduced the global sections: X ± := 16 (cid:16) Φ ± (0) Φ ± (cid:17) , G ± := ( −
13 ) (cid:16) Φ ± (1) K ± (cid:17) , L ± := 12 (cid:16) G ± (0) G ± (cid:17) . Now we introduce M ± := G (0) X ± = G ± (0) X ± . Note that, again, these sec-tions are globally well defined because they are defined in terms of X ± using thesupersymmetric generator G (0) . Remark . The reader should note that Theorem 5.5 not only proves that we havetwo pair of commuting copies of the N = 1 superconformal algebra. Namely itis implicit in the proof that { Φ ± , X ± } satisfy the commutation rules of the samenamed fields of the Shatashvili-Vafa G superconformal algebra (subsection 2.1),i.e.,: [Φ ± λ Φ ± ] = ( −
72 ) λ + 6 X ± , [Φ ± λ X ± ] = −
152 Φ ± λ − ∂ Φ ± , [ X ± λ X ± ] = 3524 λ − X ± λ − ∂X ± . It follows from the way we define the global sections (see also Remark 2.13) that { L + , X + , K + , M + } (resp. { L − , X − , K − , M − } ) can be expressed in terms of Φ + and G + (resp. Φ − and G − ). Remember also that in Theorem 5.5 we proved that { Φ + , G + } commute with { Φ − , G − } . Therefore { L + , G + , Φ + , X + , K + , M + } commute with { L − , G − Φ − , X − , K − , M − } .By the remark above, to prove the conjecture we only need to check the λ –brackets between the fields { L ± , G ± , Φ ± , X ± , K ± , M ± } satisfy the same λ -brackets of the Shatashvili-Vafa G algebra (ruling out the ones have already beenchecked in the way of proving Theorem 5.5). We work the plus case, the minuscase is similar.We prove first the linear λ -brackets (subsection 5.2.1); and then the non-linear λ -brackets (5.2.2) under the assumption that we have proved the relation (2.13): G ± X ± : − ± K ± : − ∂M ± − ∂ G ± . (5.18)Finally (subsection 5.2.3) we prove the above relation. Remark . We need the assumption (5.18) only to prove the non-linear λ -brackets, the linear ones follow without the assumption. In fact, the relation (5.18)is used only to prove the λ -bracket [Φ + λ M + ] , the others non-linear λ -brackets arededuced from this one. Similarly we could have computed first another non-linear25 -bracket using the relation, and then deduce the others ones, that is, there is noth-ing special in [Φ + λ M + ] . Even more, if we are able to check a non-linear λ -bracketwithout the assumption, then we can prove the relation (5.18) using the Jacobi iden-tity (2.7), remenber that the space of global sections of the CDR is a vertex algebra.The moral here is that we need to verify at least one non-linear identity among thefields. We opted here to check the relation (5.18).To simplify the notation we denote the sypersymmetric generator G (0) by D .Remember that D is an odd derivation of all the n -products; and that [ D, ∂ ] = 0 because D = ∂ . λ -brackets [Φ + λ K + ] By definition of G + we have that Φ +(1) K + = ( − G + . We are left with Φ +(2) K + and Φ +(0) K + , because Φ +( n ) K + = 0 for n ≥ by the positivityof the conformal weight.Computing Φ +(2) K + :By skewsymmetry (2.6), we have [Φ + λ K + ] = ( − K + − ∂ − λ Φ + ] . Then, Φ +(2) K + λ + Φ +(1) K + λ + Φ +(0) K + = − K +(2) Φ + ( − ∂ − λ ) − K +(1) Φ + ( − ∂ − λ ) − K +(0) Φ + . Expanding the right side of the last equality and looking at the coefficient of λ we get that Φ +(2) K + = − K +(2) Φ + . On the other hand, as D is an oddderivation of the n -products, we have D (Φ +(2) Φ + ) = D Φ +(2) Φ + + ( − +(2) D Φ + = K +(2) Φ + − Φ +(2) K + . This implies Φ +(2) K + = 0 .Computing Φ +(0) K + :Applying Borcherds identity (2.2) with a = Φ + , b = K + , c = | i , n = − , m = 2 , k = − , we obtain (cid:18) (cid:19) (cid:16) Φ +(0) K + (cid:17) ( − | i + (cid:18) (cid:19) (cid:16) Φ +(1) K + (cid:17) ( − | i = (cid:18) − (cid:19) Φ +(1) (cid:16) K +( − | i (cid:17) + ( − (cid:18) − (cid:19) Φ +(0) (cid:16) K +( − | i (cid:17) . Then Φ +(0) K + = 3 ∂G + + Φ +(1) ∂K + . (5.19)As we have already computed [Φ + λ Φ + ] , using sesquilinearity (2.5), we havethat Φ +(1) ∂ Φ + = 6 X + . Using that D is an odd derivation we get M + = D (Φ +(1) ∂ Φ + ) = K +(1) ∂ Φ + − Φ +(1) ∂K + . Then, Φ +(1) ∂K + = K +(1) ∂ Φ + − M + . (5.20)26e also have ∂ (cid:16) Φ +(1) K + (cid:17) = ∂ Φ +(1) K + + Φ +(1) ∂K + . Which implies ( − ∂G = ∂ Φ +(1) K + + Φ +(1) ∂K + . (5.21)As Φ +( n ) K + = 0 for n ≥ , by skewsymmetry (2.6) we have [ ∂ Φ + λ K + ] = ( − K + − λ − ∂ ∂ Φ + ]= ( − (cid:16) K +(2) ∂ Φ + ( − λ − ∂ ) + K +(1) ∂ Φ + ( − λ − ∂ ) + K +(0) ∂ Φ + (cid:17) . Then ∂ Φ +(1) K + = K +(1) ∂ Φ + − ∂ (cid:16) K +(2) ∂ Φ + (cid:17) . (5.22)Applying Borcherds identity (2.2) with a = Φ + , b = K + , c = | i , n =1 , m = − , k = 1 we obtain (cid:18) − (cid:19) (cid:16) Φ +(1) K + (cid:17) ( − | i = K +(2) (cid:16) Φ +( − | i (cid:17) + ( − K +(1) (cid:16) Φ +( − | i (cid:17) Φ +(1) K + = K +(2) ∂ Φ + + ( − K +(1) Φ + . Then using skewsymmetry (2.6) we get: K +(2) ∂ Φ + = 2Φ +(1) K + = ( − G + . (5.23)Finally substituting the equations (5.23), (5.22), (5.21), (5.20) and (5.19), weget: Φ +(0) K + = ( − M + + ( − ) ∂G + . Then we have proved that [Φ + λ K + ] = − G + λ − (cid:18) M + + 12 ∂G + (cid:19) . [ K + λ K + ] As D is an odd derivation of the n -products we have D (cid:16) Φ +( n ) K + (cid:17) = K +( n ) K + − Φ +( n ) ∂ Φ + for all n ∈ Z . Then K +( n ) K + = D (cid:16) Φ +( n ) K + (cid:17) + Φ +( n ) ∂ Φ + . (5.24)We already know [Φ + λ K + ] and [Φ + λ Φ + ] , specialising equation (5.24) for n = 3 , , , we conclude that: [ K + λ K + ] = − λ + 6 ( X + − L + ) λ + 3 ∂ ( X + − L + ) . [ G + λ Φ + ] By positivity of the conformal weight and the definition of K + := G (0) Φ + = G +(0) Φ + we have that [ G + λ Φ + ] = A λ Bλ + K + , A and B are two fields. To find A and B we perform an explicitcomputation 6.6, and found that A = B = 0 . Then we conclude: [ G + λ Φ + ] = K + . [ L + λ Φ + ] As ∆Φ + = we have [ L + λ Φ + ] = [ L λ Φ + ] = Aλ + ( ∂ + 32 λ )Φ + , where A is a field. To find A we perform and explicit computation 6.7 andfound that A =0. Then we conclude: [ L + λ Φ + ] = ( ∂ + 32 λ )Φ + . [ L + λ X + ] Applying the Jacobi identity (2.7) we have [ L + λ X + ] = [ L + λ [Φ + µ Φ + ]] = 16 (cid:16) [Φ + µ [ L + λ Φ + ]] + [[ L + λ Φ + ] λ + µ Φ + ] (cid:17) . From where [ L + λ X + ] = − λ + 2 X + λ + ∂X + . [ L + λ K + ] Applying the Jacobi identity (2.7) we have [ L + λ K + ] = [ L + λ [ G + µ Φ + ]] = [ G + µ [ L + λ Φ + ]] + [[ L + λ G + ] λ + µ Φ + ] . We conclude that: [ L + λ K + ] = ( ∂ + 2 λ ) K + . [ G + λ X + ] Computing in much the same way as [ L + λ X + ] , we obtain: [ G + λ X + ] = − G + λ + M + . [ G + λ K + ] In the same way as [ L + λ K + ] , we obtain: [ G + λ K + ] = 3Φ + λ + ∂ Φ + . [ L + λ M + ] [ L + λ [ G + µ X + ]] =[ L + λ ( − ) G + µ + M + ] (5.25) =( − )[ L + λ G + ] µ + [ L + λ M + ] . (5.26)On the other hand using the Jacobi identity (2.7) we have: [ L + λ [ G + µ X + ]] = [ G + µ [ L + λ X + ]] + [[ L + λ G + ] λ + µ X + ] . (5.27)Equating the right sides of the equations (5.25) and (5.27), expanding usingthe already known λ -brackets and then looking at the coefficients of the λ ’swe get: [ L + λ M + ] = − G + λ + 52 M + λ + ∂M + . [ G + λ M + ] Computing in much the same way as [ L + λ M + ] , we obtain: [ G + λ M + ] = − λ + ( L + + 4 X + ) λ + ∂X + , λ -brackets [Φ + λ M + ] Computing Φ +(3) M + : Φ +(3) M + = Φ +(3) (cid:16) G +(0) X + (cid:17) . Using the Borcherds identity (2.2) with m = 3 , k = 0 , n = 0 , a = G + , b = X + , and c = Φ + , we obtain: (cid:18) (cid:19) (cid:16) G +(0) X + (cid:17) (3) Φ + + (cid:18) (cid:19) (cid:16) G +(1) X + (cid:17) (2) Φ + = G +(3) (cid:16) X +(0) Φ + (cid:17) + ( − X +(0) (cid:16) G +(3) Φ + (cid:17) . implying that M +(3) Φ + = 0 and, by skewsymmetry (2.6), that Φ +(3) M + =0 .Computing Φ +(2) M + : Φ +(2) M + = Φ +(2) (cid:16) G +(0) X + (cid:17) . Using the Borcherds identity (2.2) with m = 2 , k = 0 , n = 0 , a = G + , b = X + , and c = Φ + , we obtain: (cid:18) (cid:19) (cid:16) G +(0) X + (cid:17) (2) Φ + + (cid:18) (cid:19) (cid:16) G +(1) X + (cid:17) (1) Φ + = G +(2) (cid:16) X +(0) Φ + (cid:17) + ( − X +(0) (cid:16) G +(2) Φ + (cid:17) . M +(2) Φ + = 0 and by skewsymmetry (2.6) Φ +(2) M + = 0 .Computing Φ +(1) M + : Φ +(1) M + = Φ +(1) (cid:16) G +(0) X + (cid:17) . Using the Borcherds identity (2.2) with m = 1 , k = 0 , n = 0 , a = G + , b = X + , and c = Φ + , we obtain: (cid:16) G +(0) X + (cid:17) (1) Φ + + ( − ) K + = G +(1) (cid:16) ( − ) ∂ Φ + (cid:17) . Then M +(1) Φ + = − K + and by skewsymmetry (2.6) Φ +(1) M + = K + .Computing Φ +(0) M + :By the assumption (5.18) we have G + X + : − + K + : − ∂M + − ∂ G + . Using quasi-commutativity (2.8) we have : G + X + : − : X + G + := Z − ∂ [ G + λ X + ] dλ = ∂ G + + ∂M + , and : Φ + K + : − : K + Φ + := Z − ∂ [Φ + λ K + ] dλ = ( − ∂M + . Thus, X + G + : − K + Φ + : +6 ∂M + . It follows that [Φ + λ X + G + : − K + Φ + : +6 ∂M + ] = 0 . The non-commutative Wick formula (2.10) implies: [Φ + λ X + G + :] =4 : [Φ + λ X + ] G + : +4 : X + [Φ + λ G + ] :+ 4 Z λ (cid:2) [Φ + λ X + ] µ G + (cid:3) dµ =( − K + λ + ( −
30) : Φ + G + : λ + ( −
10) : ∂ Φ + G + :+ 4 : X + K + : . We already know that [Φ + λ M + ] = K + λ + Φ +(0) M + . The non-commutative Wick formula (2.10) then gives: [Φ + λ ( −
2) : K + Φ + :] =( − (cid:0) : [Φ + λ K + ]Φ + : + : K + [Φ + λ Φ + ] :+ Z λ (cid:2) [Φ + λ K + ] µ Φ + (cid:3) dµ ! =( − K + λ + (cid:16) +(0) M + + 6 : G + Φ + : +( − ∂K + (cid:17) λ + 6 : M + Φ + : +3 : ∂G + Φ + : +( −
12) : K + X + : .
30y sesquilinearity (2.5), we have [Φ + λ ∂M + ] = 27 K + λ + (cid:16) +(0) M + + 27 ∂K + (cid:17) λ + 6 ∂ (cid:16) Φ +(0) M + (cid:17) . Looking at the coefficient of λ in [Φ + λ X + G + : − K + Φ + :+6 ∂M + ] = 0 and using that : Φ + G + : + : G + Φ + := ∂K + (by quasi-commutativity (2.8)), we obtain Φ +(0) M + = ( −
3) : G + Φ + : + ∂K + , from where we conclude that [Φ + λ M + ] = K + λ − (cid:16) G + Φ + : − ∂K + (cid:17) .2) [ K + λ M + ] Using that D is an odd derivation we get: D (cid:16) Φ +( n ) M + (cid:17) = K +( n ) M + − Φ +( n ) ∂X + . In others words, K +( n ) M + = D (cid:16) Φ +( n ) M + (cid:17) + Φ +( n ) ∂X + and, we cancompute [ K + λ M + ] in terms of already known data. We conclude that: [ K + λ M + ] = −
152 Φ + λ − ∂ Φ + λ + 3 (: G + K + : − L + Φ + :) . [ X + λ K + ] Using that D is an odd derivation we have: D (cid:16) K +( n ) M + (cid:17) = ∂ Φ +( n ) M + + K +( n ) ∂X + . Then K +( n ) ∂X + = D (cid:16) K +( n ) M + (cid:17) − ∂ Φ +( n ) M + , that is, we can compute [ K + λ X + ] in terms of already known data. We conclude that: [ X + λ K + ] = − K + λ + 3 (: G + Φ + : − ∂K + ) . [ X + λ M + ] We have [ M + λ X + ] = [ M + λ [Φ + µ Φ + ]] . Using the Jacobi identity (2.7) wecan express [ M + λ [Φ + µ Φ + ]] = ( − + µ [ M + λ Φ + ]] + [[ M + λ Φ + ] λ + µ Φ + ] . Putting µ = 0 in the above equation we can compute [ M + λ X + ] in terms ofalready known data. We obtain: [ X + λ M + ] = − G + λ − (cid:18) M + + 94 ∂G + (cid:19) λ + (cid:18) G + X + : − ∂M + − ∂ G + (cid:19) . [ M + λ M + ] Using that D is an odd derivation we have: D (cid:16) X +( n ) M + (cid:17) = M +( n ) M + + X +( n ) ∂X + . Then M +( n ) M + = D (cid:16) X +( n ) M + (cid:17) − X +( n ) ∂X + , that is, we can compute [ M + λ M + ] in terms of already known data. We conclude that: [ M + λ M + ] = − λ + 12 (20 X + − L + ) λ + (cid:18) ∂X + − ∂L + (cid:19) λ + (cid:18) ∂ X + − ∂ L + − G + M + : +8 : L + X + : (cid:19) . To conlude the proof of the Conjecture 5.1 we need to prove the relation (5.18) G + X + : − + K + : − ∂M + − ∂ G + . used in the computation of the non-linear λ -brackets [Φ + λ M + ] . We check thisrelation among the fields performing again an explicit computation. This is along calculation, even longer than the ones already performed in the proof ofTheorem 5.5. Besides the Mathematica package OPEdefs [Thi91], we haveused the computer algebra system
Cadabra [Pee07], this last software wasproved very useful for simplifying tensorial expressions with many terms. Despiteits length this is a straightforward computation.For these reasons we do not present here the computations, they are show uponline at the URL:
Remark . As in Theorem 5.5 we make extensive use of: the Ricci flatness, thecontractions on page 18, the fact that ϕ , ψ and g are parallel and the symmetriesof the Riemann curvature tensor. Unlike the computations in Theorem 5.5, weshould point out that we need to use the identity (4.12), in fact this one appearsmany times and is a key identity. It is also interesting to note the appearance ofthe first Pontryagin class p ( M ) = π Tr ( R ∧ R ) of the manifold, it appears asone of the coefficients of the term ∂γ i ∂γ j ∂γ k c l , i.e, π R ijmn R mnkl ∂γ i ∂γ j ∂γ k c l ,nevertheless this expression is identically zero because R is antisymmetric in i and j while ∂γ i commutes with ∂γ j . To perform all the computations below we are assuming that we are working in alocal coordinate chart where the volume form is constant, then we can assume that: Γ iij = ∂ log p | g | ∂x j = 0 , (6.1)32here | g | denotes the absolute value of the determinant of the metric tensor g .For convenience of the reader we recall the expression of the Riemann curvaturetensor R lijk and the Ricci tensor R ij in terms of the Christoffel symbols. We alsorecall some of its symmetries. R lijk = (Γ lik ) ,j − (Γ lij ) ,k + Γ ljs Γ sik − Γ lks Γ sij , (6.2) R iklm = R lmik , R iklm = − R kilm = − R ikml , (6.3) R iklm + R imkl + R ilmk = 0 (first Bianchi identity). (6.4) R ij = R lilk = g lm R iljm = (Γ lij ) ,l − (Γ lil ) ,j + Γ lij Γ mlm − Γ mil Γ ljm . (6.5)The Ricci tensor is symmetric. [Φ + λ Φ + ] We have Φ + = ϕ ijk e i + e j + e k + + ϕ ijk Γ imn g jm ∂γ n e k + . To simplify the notation we drop the plus subscript.a) Computing [ ϕ ijk e i e j e kλ ϕ mnl e m e n e l ] For this we first compute [ e m e n e lλ e i e j e k ] and we can assume anti–symmetrization in the indices { i, j, k } and { m, n, l } due to the future con-traction with the three form ϕ . [ e m e n e lλ e i e j e k ] =( − g im g jn g kl λ + (cid:16) g im g jn e k e l − ∂ ( g im g jn g kl ) (cid:17) λ + 9 g im e j e k e n e l − g im g jn ∂ ( e l ) e k + 36 ∂ ( g im ) g jn e k e l − ∂ ( g im ) ∂ ( g jn ) g kl . Then using the non-commutative Wick formula (2.10): [ 16 ϕ ijk e i e j e kλ ϕ mnl e m e n e l ] = 136 : ϕ mnl [ e m e n e l − λ − ∂ ϕ ijk e i e j e k ] := 136 : ϕ mnl (cid:16) : ϕ ijk [ e m e n e lµ e i e j e k ] : (cid:17) | − λ − ∂ : . Collecting terms first by the order of λ and then by the number of factors e ’s: λ
136 ( − ϕ mnl ϕ ijk g im g jn g kl = 136 ( − − . λ (cid:16) − ϕ mnl ∂ ( ϕ ijk g im g jn g kl ) − ϕ ijk ϕ mnl g im g jn e k e l + 6 ϕ mnl ϕ ijk ∂ ( g im g jn g kl ) (cid:17) = − ϕ mnl ∂ ( ϕ ijk ) g im g jn g kl − ϕ ijk ϕ mnl g im g jn e k e l = − ϕ ijk ∂ ( ϕ ijk ) − g kl e k e l = 0 . Here was used the identity (4.6) and that g ij e i e j = − g ij e j e i + g ij ∂ ( g ij ) byquasi-commutativity (2.8), then g ij e i e j = 12 g ij ∂ ( g ij ) = 12 g ij (cid:16) − Γ iab g aj − Γ jab g ia (cid:17) ∂γ b = − Γ iib ∂γ b . (6.6)We also have by (4.3) and (4.6) that ϕ ijk ∂ ( ϕ ijk ) = ϕ ijk (cid:16) Γ aib ϕ ajk + Γ ajb ϕ iak + Γ akb ϕ ija (cid:17) ∂γ b = 18Γ iib ∂γ b . (6.7)without λe i e j e k e l ϕ ijk ϕ mnl g im e j e k e n e l = 14 (cid:0) − ψ jknl + g jn g kl − g jl g kn (cid:1) e j e k e n e l = − ψ ijkl e i e j e k e l + 18 g ij g kl ∂ ( g ik ) ∂ ( g jl ) − g ij g kl ∂ ( g ij ) ∂ ( g kl )= − ψ ijkl e i e j e k e l + 14 Γ cib Γ icd ∂γ b ∂γ d + 14 g jm g ic Γ mib Γ jcd ∂γ b ∂γ d − g ij g kl ∂ ( g ij ) ∂ ( g kl ) . Here was used the identity (4.7), quasi-associativity (2.9), the equality g ij e i e j = g ij ∂ ( g ij ) proved above and that metric g is covariantly constant. ∂ ( e i ) e j (cid:16) − ϕ mnl ϕ ijk g im g jn ∂ ( e k ) e l − ϕ mnl ϕ ijk g im g jn e k ∂ ( e l ) − ϕ mnl ϕ ijk g im g jn ∂ ( e l ) e k (cid:17) = − g lk ∂ ( e l ) e k − g lk ∂ g lk . e i ∂ ( e j ) = − ∂ ( e j ) e i + Z − ∂ [ e iλ ∂e j ] dλ = − ∂ ( e j ) e i + 12 ∂ ( g ij ) .e i e j − ϕ mnl ∂ ( ϕ ijk g im g jn ) e k e l + ϕ mnl ϕ ijk ∂ ( g im ) g jn e k e l = − ϕ mnl ∂ ( ϕ ijk ) g im g jn e k e l = − ϕ mnl ϕ ajk Γ aib g im g jn ∂γ b e k e l − ϕ mnl ϕ iak Γ ajb g im g jn ∂γ b e k e l − ϕ mnl ϕ ija Γ akb g im g jn ∂γ b e k e l = − ϕ mnl ϕ ajk Γ aib g im g jn ∂γ b e k e l − ϕ mnl ϕ ija Γ akb g im g jn ∂γ b e k e l = ( −
1) ( g am g lk − g mk g la − ψ mlak ) Γ aib g im ∂γ b e k e l + ( −
12 )6 g la Γ akb ∂γ b e k e l = ψ mlak Γ aib g im ∂γ b e k e l + ( − g lk Γ iib ∂γ b e k e l + ( − g la Γ akb ∂γ b e k e l , = ψ mlak Γ aib g im ∂γ b e k e l + ( − g la Γ akb ∂γ b e k e l + ( −
12 ) g lk ∂ ( g lk )Γ iib ∂γ b , = ψ mlak Γ aib g im ∂γ b e k e l + ( − g la Γ akb ∂γ b e k e l + 14 g ij ∂ ( g ij ) g lk ∂ ( g lk ) . Here was used that dϕ = 0 , i.e., (4.3), the identities (4.7) and (4.6), and(6.6).without e ’s − ϕ mnl ∂ (cid:16) ϕ ijk g im g jn g kl (cid:17) − ϕ mnl ϕ ijk ∂ ( g im ) ∂ ( g jn ) g kl + 16 ϕ mnl ∂ (cid:18) ϕ ijk ∂ (cid:16) g im g jn g kl (cid:17)(cid:19) = − ϕ mnl ∂ ( ϕ ijk ) g im g jn g kl + 32 g lk ∂ ( g lk )= − ϕ ijk ∂ ( ϕ ijk ) + 32 g lk ∂ ( g lk )= 912 ∂ (cid:16) g ij ∂ ( g ij ) (cid:17) + 112 ∂ ( ϕ ijk ) ∂ ( ϕ ijk ) + 32 g lk ∂ ( g lk )= 34 ∂ ( g ij ) ∂ ( g ij ) + 112 ∂ ( ϕ ijk ) ∂ ( ϕ ijk ) + 94 g lk ∂ ( g lk )= −
32 Γ cbi Γ icd ∂γ b ∂γ d − g aj g ic Γ abi Γ jcd ∂γ b ∂γ d − Γ cbi Γ icd ∂γ b ∂γ d −
12 Γ ibi Γ jjd ∂γ b ∂γ d − ψ amjn Γ abi g mi Γ icd g nc ∂γ b ∂γ d + 94 g lk ∂ ( g lk )= −
52 Γ cbi Γ icd ∂γ b ∂γ d − g aj g ic Γ abi Γ jcd ∂γ b ∂γ d − ψ amjn Γ abi g mi Γ jcd g nc ∂γ b ∂γ d = − g ij ∂ ( g ij ) g lk ∂ ( g lk ) + 94 g lk ∂ g lk .
35n this chain of equalities was used that ∂ (cid:16) ϕ ijk ∂ ( ϕ ijk ) (cid:17) = ∂ ( ϕ ijk ) ∂ ( ϕ ijk ) + ϕ ijk ∂ ( ϕ ijk ) and ϕ ijk ∂ ( ϕ ijk ) = 9 g ij ∂ ( g ij ) ; this last identity is proved combining (6.6)and (6.7). We also used that dϕ = 0 i.e., (4.3), that g is covariantly constantand the identity (4.7).b) Computing [ ϕ ijk e i e j e kλ ϕ mnl Γ mab g na ∂γ b e l ] + [ ϕ ijk Γ iab g ja ∂γ b e kλ ϕ mnl e m e n e l ] . [ 16 ϕ ijk e i e j e kλ ϕ mnl Γ mab g na ∂γ b e l ] = 14 ϕ ijk ϕ mnl g il g na Γ mab ∂γ b e j e k . Using skew-symmetry (2.6) and arranging the indices we have: [ 12 ϕ ijk Γ iab g ja ∂γ b e kλ ϕ mnl e m e n e l ] = [ 16 ϕ ijk e i e j e kλ ϕ mnl Γ mab g na ∂γ b e l ] , then [ 16 ϕ ijk e i e j e kλ ϕ mnl Γ mab g na ∂γ b e l ] + [ 12 ϕ ijk Γ iab g ja ∂γ b e kλ ϕ mnl e m e n e l ]= 12 ϕ ijk ϕ mnl g il g na Γ mab ∂γ b e j e k = 12 (cid:0) g jm g kn − g jn g km − ψ jkmn (cid:1) g na Γ mab ∂γ b e j e k = 12 g jm Γ mkb ∂γ b e j e k + ( −
12 ) g km Γ mjb ∂γ b e j e k + ( −
12 ) ψ jkmn g na Γ mab ∂γ b e j e k =( −
12 ) ψ jkmn g na Γ mab ∂γ b e j e k + ( − g km Γ mjb ∂γ b e j e k + 12 g jm ∂ ( g jk )Γ mkb ∂γ b =( −
12 ) ψ jkmn g na Γ mab ∂γ b e j e k + ( − g km Γ mjb ∂γ b e j e k + ( −
12 ) g jm g ck Γ jcd Γ mkd ∂γ b ∂γ d + ( −
12 )Γ kcd Γ ckb ∂γ b ∂γ d . Here were used the identity (4.7), (6.6) and that g is covariantly constant.c) Computing [ ϕ ijk Γ irs g jr ∂γ s e kλ ϕ mnl Γ mab g na ∂γ b e l ] . [ 12 ϕ ijk Γ irs g jr ∂γ s e kλ ϕ mnl Γ mab g na ∂γ b e l ]= 14 ϕ mnl ϕ ijk g lk Γ mab g na ∂γ b Γ irs g jr ∂γ s = 14 (cid:0) g mi g nj − g mj g ni − ψ mnij (cid:1) Γ mab g na ∂γ b Γ irs g jr ∂γ s = − ψ mnij Γ mab g na ∂γ b Γ irs g jr ∂γ s + 14 g mi g na Γ mab Γ ins ∂γ b ∂γ s −
14 Γ mib Γ ims ∂γ b ∂γ s . a ) , b ) and c ) to obtain [Φ + λ Φ + ] = ( − ) λ + 6 X + , (6.8)where X + is: X + = − ψ ijkl e i + e j + e k + e l + − ψ ijkl Γ imn g jm ∂γ n e k + e l + − ψ ijkl Γ im n g jm ∂γ n Γ km n g lm ∂γ n − g ij ∂ ( e i + ) e j + − g ij Γ jkl ∂γ k e l + e i + − Γ ijk Γ kil ∂γ j ∂γ l ,X − = − ψ ijkl e i − e j − e k − e l − + ψ ijkl Γ imn g jm ∂γ n e k − e l − − ψ ijkl Γ im n g jm ∂γ n Γ km n g lm ∂γ n + g ij ∂ ( e i − ) e j − + g ij Γ jkl ∂γ k e l − e i − − Γ ijk Γ kil ∂γ j ∂γ l . To get this precise form of X + we only need to manipulate a little more the termswithout e ’s that come from a ) , b ) and c ) : − ψ amjn Γ abi g mi Γ jcd g nc ∂γ b ∂γ d − cbi Γ icd ∂γ b ∂γ d − g aj g ic Γ abi Γ jcd ∂γ b ∂γ d + g lk ∂ g lk = − ψ amjn Γ abi g mi Γ jcd g nc ∂γ b ∂γ d −
32 Γ cbi Γ icd ∂γ b ∂γ d + 34 ∂ (cid:16) g ij ∂ ( g ij ) (cid:17) = − ψ amjn Γ abi g mi Γ jcd g nc ∂γ b ∂γ d −
32 Γ cbi Γ icd ∂γ b ∂γ d . The first equality is proved using: ∂ ( g ij ∂ ( g ij )) = ∂ ( g ij ) ∂ ( g ij ) + g ij ∂ ( g ij )=( − cbi Γ icd ∂γ b ∂γ d + ( − g aj g ic Γ abi Γ jcd ∂γ b ∂γ d + g ij ∂ ( g ij ) , which itself is proved using that g is covariantly constant. The second equalityfollows from (6.6) and (6.1). X +(2) Φ + and X +(1) Φ + To simplify the notation we drop the “+” subscript.Below we only compute the λ -bracket between the sumands of X and Φ contribut-ing to X (2) Φ and X (1) Φ . 37) Computing [ − ψ abcd e a e b e c e dλ ϕ ijk e i e j e k ] . [ − ψ abcd e a e b e c e dλ ϕ ijk e i e j e k ]= (cid:18) − ψ abcd ϕ ijk g ai g bj g ck e d (cid:19) λ + (cid:18) − ψ abcd ϕ ijk g bi g dj e a e c e k − ψ abcd ϕ ijk g bi g cj ∂ ( g dk ) e a + 16 ϕ ijk ∂ ( ψ abcd g bi g cj g dk e a ) (cid:19) λ + terms without λ = (cid:18) − ϕ ack e a e c e k + 16 ϕ ijk ∂ ( ψ abcd ) g bi g cj g dk e a (cid:19) λ + terms without λ = (cid:16) − ϕ ijk e i e j e k + ( − ϕ ijk Γ ilm g lj ∂γ m e k (cid:17) λ + terms without λ. Here were used the identities (4.8) and (4.9), and that d ∗ ϕ = 0 , i.e., (4.4).b) Computing [ − ψ abcd Γ amn g bm ∂γ n e c e dλ ϕ ijk e i e j e k ] . [ − ψ abcd Γ amn g bm ∂γ n e c e dλ ϕ ijk e i e j e k ]= (cid:18) ϕ ijk ψ abcd Γ amn g bm ∂γ n g ci g dj e k (cid:19) λ + terms without λ = (cid:16) − ϕ abk Γ amn g bm ∂γ n e k (cid:17) + terms without λ. Here was used the identity (4.9).c) Computing [ − g lb ∂ ( e b ) e lλ ϕ ijk e i e j e k ] . [ − g lb ∂ ( e b ) e lλ ϕ ijk e i e j e k ]= (cid:18) ( −
12 ) ϕ ijk g ij e k (cid:19) λ + (cid:18) ( −
14 ) ϕ ijk e i e j e k + ( −
12 ) ϕ ijk g lb ∂ ( g bi g kl e j )+ 12 ϕ ijk ∂ ( g ik e j ) (cid:19) λ + terms without λ = (cid:18) ( −
14 ) ϕ ijk e i e j e k (cid:19) λ + terms without λ. d) Computing [ − g lb ∂ ( e b ) e lλ ϕ ijk Γ imn g jm ∂γ n e k ] . [ − g lb ∂ ( e b ) e lλ ϕ ijk Γ imn g jm ∂γ n e k ]= (cid:18) − ϕ ijk Γ imn g jm ∂γ n e k (cid:19) λ + terms without λ.
38) Computing [ − g am Γ anb ∂γ b e n e mλ ϕ ijk e i e j e k ] . [ − g am Γ anb ∂γ b e n e mλ ϕ ijk e i e j e k ]= (cid:18) − ϕ ijk Γ inb g nj ∂γ b e k (cid:19) λ + terms without λ. Combining a ) , b ) , c ) , d ) and e ) we get that X +(2) Φ + = 0 and X +(1) Φ + = − Φ + . [Φ ± λ K ± ] To perform this computation we express Φ ± explicitly in terms of the bc − βγ system. Φ + = √ ϕ ijk c i c j c k + √ ϕ ijk g il c j c k b l + √ ϕ ijk g il g jm c k b l b m + √ ϕ ijk g il g jm g kn b l b m b n + √ ϕ ijk Γ imn g jm ∂γ n g kl b l + √ ϕ ijk Γ imn g jm ∂γ n c k , Φ − = i √ ϕ ijk c i c j c k − i √ ϕ ijk g il c j c k b l + i √ ϕ ijk g il g jm c k b l b m − i √ ϕ ijk g il g jm g kn b l b m b n + i √ ϕ ijk Γ imn g jm ∂γ n g kl b l − i √ ϕ ijk Γ imn g jm ∂γ n c k . As by definition K ± := G (0) (Φ ± ) we need to apply G (0) to each of the sum-mands of Φ ± . To simplify the notation we denote G (0) simply by D . D ( ϕ ijk c i c j c k ) = ( ϕ ijk,l c l ) c i c j c k + 3 ϕ ijk ∂γ i c j c k ,D ( ϕ ijk g il c j c k b l ) = D ( ϕ ljk c j c k b l )=( ϕ ljk,m c m ) c j c k b l + 2 ϕ ljk ∂γ j c k b l + ϕ ljk c j c k β l ,D ( ϕ ijk g il g jm c k b l b m ) = D ( ϕ lmk c k b l b m )=( ϕ lmk,n c n ) c k b l b m + ϕ lmk ∂γ k b l b m + 2 ϕ lmk c k b l β m ,D ( ϕ ijk g il g jm g kn b l b m b n ) = D ( ϕ lmn b l b m b n )=( ϕ lmn,a c a ) b l b m b n + 3 ϕ lmn b l b m β n , ( ϕ ijk Γ imn g jm ∂γ n g kl b l ) = D ( ϕ ijk Γ imn g jm g kl ∂γ n b l )=( F ln,a c a ) ∂γ n b l + F ln ∂c n b l + F ln ∂γ n β l , where F ln := ϕ ijk Γ imn g jm g kl , D ( ϕ ijk Γ imn g jm ∂γ n c k ) = ( F kn,a c a ) ∂γ n c k + F kn ∂c n c k + F kn ∂γ n ∂γ k , where F kn := ϕ ijk Γ imn g jm .Now we collect the non-zero λ -bracket between the summands of Φ ± and thesummands of K ± that contains terms with λ . To compute this λ -brackets we usedthe Mathematica package [Thi91]: [ ϕ ijk c i c j c kλ D ( ϕ ijk g il g jm c k b l b m )]= (cid:16) ϕ ijk ϕ jkn,s c i c n c s + 6 ϕ ijk ϕ kjn ∂γ n c i +6 ϕ kmn ϕ ijk,m c i c j c n (cid:17) λ + terms without λ, [ ϕ ijk c i c j c kλ D ( ϕ ijk g il g jm g kn b l b m b n )]= (cid:16) ϕ ijk ϕ ijk,s c s + 18 ϕ jkn ϕ ijk,n c i (cid:17) λ (cid:16) ( − ϕ ijk ϕ ljk,s c i c s b l + ( − ϕ ijk ∂ϕ sjk,s c i +6 ∂ϕ ijk ϕ ijk,s c s + ( − ϕ ijk ϕ jkn c i β n +18 ϕ lkn ϕ ijk,n c i c j b l + 18 ϕ jkn ϕ ijk,n ∂c i (cid:17) λ + terms without λ, [ ϕ ijk g il c j c k b lλ D ( ϕ ijk g il c j c k b l )]= (cid:16) ϕ lmi ϕ ijk,l c j c k c m + 4 ϕ lmi ϕ ijl,s c j c m c s + ( − ϕ ijk ϕ kin ∂γ j c n (cid:17) λ + terms without λ, [ ϕ ijk g il c j c k b lλ D ( ϕ ijk g il g jm c k b l b m )]= (cid:16) ϕ lji ϕ ijk,l c k + ( − ϕ lji ϕ ijl,s c s + ( − ϕ ijk ϕ kmi,j c m (cid:17) λ (cid:16) ( − ϕ lmj ϕ ijk,l c k c m b i + ( − ϕ kmj ϕ ijk,s c m c s b i +( − ϕ kmj ∂ ( ϕ ijk,i ) c m + ( − ϕ lji ϕ ijk,s c k c s b l +2 ∂ ( ϕ lji ) ϕ ijk,l c k + ( − ∂ ( ϕ kji ) ϕ ijk,s c s + 2 ϕ ijk ϕ lji ∂γ k b l +4 ϕ ijk ϕ kmi c m β j + ( − ϕ ijk ϕ lmi,j c k c m b l +( − ϕ ijk ϕ kmn,j c m c n b i + ( − ϕ ijk ϕ kmi,j ∂c m (cid:17) λ + terms without λ, ϕ ijk g il c j c k b lλ D ( ϕ ijk g il g jm g kn b l b m b n )]= (cid:16) ϕ lmk ϕ ijk,l c m b i b j + 6 ϕ lkj ϕ ijk,s c s b i b l + 6 ϕ lkj ∂ ( ϕ ijk,i ) b l +6 ∂ ( ϕ lkj ) ϕ ijk,l b i + ( − ϕ ijk ϕ lij b l β k +12 ϕ ijk ϕ lmj,k c m b i b l + 6 ϕ ijk ϕ lij,k ∂b l (cid:17) λ + terms without λ, [ ϕ ijk g il c j c k b lλ D ( ϕ ijk Γ imn g jm ∂γ n g kl b l )]= (cid:16) ( − F ij ϕ jin c n (cid:17) λ (cid:16) ( − ϕ kin F ij,k ∂γ j c n + F ij ϕ jmn c m c n b i + ( − F ij ϕ jin ∂c n + ( − F ij ∂ϕ jin c n (cid:17) λ + terms without λ, [ ϕ ijk g il c j c k b lλ D ( ϕ ijk Γ imn g jm ∂γ n c k )]= (cid:16) F ji ϕ imn c j c m c n (cid:17) λ + terms without λ, [ ϕ ijk g il g jm c k b l b mλ D ( ϕ ijk c i c j c k )]= (cid:16) − ϕ ijk ϕ lmj,i c k c l c m − ϕ ijk ϕ lji,s c k c l c s + 6 ϕ ijk ϕ lji ∂γ l c k (cid:17) λ + terms without λ, [ ϕ ijk g il g jm c k b l b mλ D ( ϕ ijk g il c j c k b l )]= (cid:16) ( − ϕ ijk ϕ kmj,i c m + 2 ϕ ijk ϕ kij,s c s + 2 ϕ lij ϕ ijk,l c k (cid:17) λ (cid:16) ϕ ijk ϕ lmj,i c k c m b l + 2 ϕ ijk ϕ lji,s c k c s b l +( − ϕ ijk ϕ kmn,j c m c n b i + 4 ϕ ijk ϕ kmj,s c m c s b i +2 ϕ ijk ∂ ( ϕ lji,l ) c k + 2 ∂ ( ϕ ijk ) ϕ kij,s c s − ∂ ( ϕ ijk ) ϕ kmj,i c m +2 ∂ ( ϕ ijk ) ϕ kjn,i c n + ( − ϕ ijk ϕ kmj ∂γ m b i +2 ϕ ijk ϕ lji c k β l + ( − ϕ lmj ϕ ijk,l c k c m b i + 2 ϕ lij ϕ ijk,l ∂c k (cid:17) λ + terms without λ, [ ϕ ijk g il g jm c k b l b mλ D ( ϕ ijk g il g jm c k b l b m )]= (cid:16) ϕ kmn ϕ ijk,m c n b i b j + 8 ϕ ijk ϕ lkn,j c n b i b l − ϕ ijk ϕ lkj,s c s b i b l + 4 ϕ ijk ϕ kmi b j β m +4 ϕ ijk ∂ ( ϕ lkj,l ) b i + 4 ∂ ( ϕ ijk ) ϕ lkj,i b l +4 ϕ kmj ϕ ijk,m ∂b i (cid:17) λ + terms without λ, ϕ ijk g il g jm c k b l b mλ D ( ϕ ijk g il g jm g kn b l b m b n )]= (cid:16) ϕ ijk ϕ lmk,i b j b l b m (cid:17) λ + terms without λ, [ ϕ ijk g il g jm c k b l b mλ D ( ϕ ijk Γ imn g jm ∂γ n g kl b l )]= (cid:16) ϕ ijk F ki b j (cid:17) λ (cid:16) ϕ ijk F km,i ∂γ m b j +2 ϕ ijk F lj c k b i b l + 2 ϕ ijk F ki ∂b j + 2 ∂ ( ϕ ijk ) F ki b j (cid:17) λ + terms without λ, [ ϕ ijk g il g jm c k b l b mλ D ( ϕ ijk Γ imn g jm ∂γ n c k )]= (cid:16) ϕ ijk F ij c k (cid:17) λ (cid:16) ϕ ijk F il,j ∂γ l c k + 2 ϕ ijk F mi c k c m b j +2 ϕ ijk F ij ∂c k + 2 ∂ ( ϕ ijk ) F ij c k (cid:17) λ + terms without λ, [ ϕ ijk g il g jm g kn b l b m b nλ D ( ϕ ijk c i c j c k )]= (cid:16) ϕ ijk ϕ lkj,i c l + 6 ϕ ijk ϕ ijk,s c s (cid:17) λ (cid:16) ϕ ijk ϕ lmj,k c l c m b i + 18 ϕ ijk ϕ ljk,s c l c s b i +18 ∂ ( ϕ ijk ) ϕ lkj,i c l + 6 ∂ ( ϕ ijk ) ϕ ijk,s c s +18 ϕ ijk ϕ lkj ∂γ l b i (cid:17) λ + terms without λ, [ ϕ ijk g il g jm g kn b l b m b nλ D ( ϕ ijk g il c j c k b l )]= (cid:16) ϕ ijk ϕ lmj,k c m b i b l + 6 ϕ ijk ϕ ljk,s c s b i b l +6 ϕ ijk ∂ ( ϕ lkj,l ) b i + 6 ∂ ( ϕ ijk ) ϕ lkj,i b l + 6 ϕ ijk ϕ lkj b i β l +6 ϕ lmk ϕ ijk,l c m b i b j + 6 ϕ ljk ϕ ijk,l ∂b i (cid:17) λ + terms without λ, [ ϕ ijk g il g jm g kn b l b m b nλ D ( ϕ ijk g il g jm c k b l b m )]= (cid:16) ϕ ijk ϕ lmi,j b k b l b m (cid:17) λ + terms without λ, [ ϕ ijk g il g jm g kn b l b m b nλ D ( ϕ ijk Γ imn g jm ∂γ n g kl b l )]= (cid:16) ϕ ijk F li b j b k b l (cid:17) λ + terms without λ, ϕ ijk g il g jm g kn b l b m b nλ D ( ϕ ijk Γ imn g jm ∂γ n c k )]= (cid:16) ϕ ijk F ij b k (cid:17) λ (cid:16) ϕ ijk F il,j ∂γ l b k +3 ϕ ijk F mi c m b j b k + 6 ϕ ijk F ij ∂b k + 6 ∂ ( ϕ ijk ) F ij b k (cid:17) λ + terms without λ, [ F lm ∂γ m b lλ D ( ϕ ijk g il c j c k b l )]= (cid:16) F ji ϕ ijk c k (cid:17) λ (cid:16) F li ϕ ijk c j c k b l +2 ϕ ijk F km,i ∂γ m c j + 2 ∂ ( F ji ) ϕ ijk c k (cid:17) λ + terms without λ, [ F lm ∂γ m b lλ D ( ϕ ijk g il g jm c k b l b m )]= (cid:16) F kj ϕ ijk b i (cid:17) λ (cid:16) F lj ϕ ijk c k b i b l + ( − ϕ ijk F km,j ∂γ m b i +2 ∂ ( F kj ) ϕ ijk b i (cid:17) λ + terms without λ, [ F lm ∂γ m b lλ D ( ϕ ijk g il g jm g kn b l b m b n )] = (cid:16) F lk ϕ ijk b i b j b l (cid:17) λ + terms without λ, [ F lm ∂γ m b lλ D ( ϕ ijk Γ imn g jm ∂γ n g kl b l )] = (cid:16) F jm F ij ∂γ m b i (cid:17) λ + terms without λ, [ F lm ∂γ m b lλ D ( ϕ ijk Γ imn g jm ∂γ n c k )] = (cid:16) F im F ji ∂γ m c j (cid:17) λ + terms without λ, [ F ji ∂γ i c j λ D ( ϕ ijk g il c j c k b l )] = (cid:16) F ji ϕ imn c j c m c n (cid:17) λ + terms without λ, [ F ji ∂γ i c j λ D ( ϕ ijk g il g jm c k b l b m )]= (cid:16) F ji ϕ ijn c n (cid:17) λ (cid:16) F ji ϕ lin c j c n b l +2 ϕ jmn F ji,m ∂γ i c n + 2 ∂ ( F ji ) ϕ ijn c n (cid:17) λ + terms without λ, [ F ji ∂γ i c j λ D ( ϕ ijk g il g jm g kn b l b m b n )]= (cid:16) F ji ϕ ijm b m (cid:17) λ (cid:16) F ji ϕ lmi c j b l b m +6 ϕ ljn F ji,n ∂γ i b l + 6 ∂ ( F ji ) ϕ ijl b l (cid:17) λ + terms without λ, [ F ji ∂γ i c j λ D ( ϕ ijk Γ imn g jm ∂γ n g kl b l )] = (cid:16) F ji F im ∂γ m c j (cid:17) λ + terms without λ. .3.1 (Φ +(1) K + ) + (Φ − (1) K − ) Now we want to compute (Φ +(1) K + ) + (Φ − (1) K − ) , it should be notice that the λ -brackets used to compute Φ +(1) K + and Φ − (1) K − are the same modulo a sign.Then to compute (Φ +(1) K + ) + (Φ − (1) K − ) we only need to take into account the λ -brackets that have the same sign an consider each one twice.We compute (Φ +(1) K + ) + (Φ − (1) K − ) analyzing the coefficient of each type ofterm that appears. All the coefficients were obtained after a long but straightfor-ward computation using identities (4.6) and (4.7), and the fact that dϕ = 0 and ∇ g = 0 , except the coefficient of ∂γ i c j that is more involved and is detailed below.coefficient of b i b j b k : ,coefficient of c i c j b k : ,coefficient of ∂c i : ,coefficient of ∂γ i b i : ( − ∂γ i b i ,coefficient of c i β i : ( − c i β i ,coefficient of ∂γ i c j : . Computations to obtain the coefficient of ∂γ i c j :Denote by A the terms of type ∂γ i c j that appear in the computations of the c i β i coefficient due to quasi-associativity (2.9): A = − ∂ϕ ijk ϕ jkn,n c i + ( −
18 ) ∂ϕ jkn ϕ ijk,n c i + 14 ∂ϕ ijk ϕ kmi,j c m + 14 ∂ϕ kmi ϕ ijk,j c m + 18 ∂ϕ ijk ϕ lji,l c k + 18 ∂ϕ lji ϕ ijk,l c k , Collecting the other terms that contain ∂γ i c j : denote by A the sum of the termsthat does not contain derivatives of the Christoffel symbols, denote by A the sumof the terms containing derivatives of the Christoffel symbols.We have: A = 12 ϕ mli ϕ nls (Γ imn ) ,r ∂γ r c s + 12 ϕ srl ϕ mli (Γ imn ) ,s ∂γ n c r . Using the identity (4.7) we get: A = 12 g is g mn (Γ imn ) ,r ∂γ r c s + ( −
12 ) g ri g sm (Γ imn ) ,s ∂γ n c r + 12 (Γ smn ) ,s ∂γ n c m + 12 ψ s mr i (Γ imn ) ,s ∂γ n c r . Let R denote the Riemann curvature, using the identity (6.2) we can work the first44wo summands of A : g is g mn (Γ imn ) ,r ∂γ r c s = 12 g is g mn R inrm ∂γ r c s + 12 g is g mn (Γ ir n ) ,m ∂γ r c s + ( −
12 ) g is g mn Γ ir a Γ am n ∂γ r c s + 12 g is g mn Γ im a Γ ar n ∂γ r c s = 12 R sr ∂γ r c s + 12 g is g mn (Γ ir n ) ,m ∂γ r c s + ( −
12 )Γ ias Γ ijj ∂γ s c a + 12 Γ iaj Γ ijs ∂γ s c a , g ri g sm (Γ imn ) ,s ∂γ n c r =( −
12 ) g ri g sm R insm ∂γ n c r + ( −
12 ) g ri g sm (Γ is n ) ,m ∂γ n c r + 12 g ri g sm Γ is a Γ am n ∂γ n c r + ( −
12 ) g ri g sm Γ im a Γ as n ∂γ n c r =( −
12 ) g ri g sm (Γ is n ) ,m ∂γ n c r = − g is g mn (Γ ir n ) ,m ∂γ r c s , then A = 12 R sr ∂γ r c s + ( −
12 )Γ ias Γ ijj ∂γ s c a + 12 Γ iaj Γ ijs ∂γ s c a + 12 (Γ smn ) ,s ∂γ n c m + 12 ψ s mr i (Γ imn ) ,s ∂γ n c r . We also have A + A =( −
12 )Γ ia j Γ ji s ∂γ s c a + 12 Γ ias Γ ijj ∂γ s c a + ( −
12 )Γ iaj Γ ijs ∂γ s c a + 12 Γ ji k Γ il s ψ k la j ∂γ s c a , A + A + A =( −
12 )Γ ia j Γ ji s ∂γ s c a + 12 Γ ji k Γ il s ψ k la j ∂γ s c a + 12 R sr ∂γ r c s + 12 (Γ smn ) ,s ∂γ n c m + 12 ψ s mr i (Γ imn ) ,s ∂γ n c r =( 12 (Γ smn ) ,s ∂γ n c m + ( −
12 )Γ ia j Γ ji s ∂γ s c a ) + ( 12 ψ s mr i (Γ imn ) ,s ∂γ n c r + 12 ψ s mr i Γ ia s Γ am n ∂γ n c r ) + 12 R sr ∂γ r c s = 12 R mn ∂γ n c m + 12 ψ s mr i R imsn ∂γ n c r + 12 R sr ∂γ r c s = R mn ∂γ n c m + 12 R abcd ψ cdlm g am ∂γ b c l , =0 . To conclude the last equality is zero we use the Lemma 4.3 and Lemma 4.4.Finally we have proved that (Φ +(1) K + ) + (Φ − (1) K − ) = ( − ∂γ i b i + ( − c i β i . Φ ± (1) K ± Now we want to compute Φ +(1) K + (cid:16) Φ − (1) K − (cid:17) , as was noted in 6.3.1 the λ -brackets used to compute Φ +(1) K + and Φ − (1) K − are exactly the same mod-ulo a sign. Then to compute (Φ +(1) K + ) (cid:16) Φ − (1) K − (cid:17) we only need to con-sider the λ -brackets that change sign and remember to add one half of the sum (Φ +(1) K + ) + (Φ − (1) K − ) . We compute Φ +(1) K + (cid:16) Φ − (1) K − (cid:17) analyzing the co-efficient of each type of term that appears. All the coefficients were obtained aftera long but straightforward computation using identities (4.6) and (4.7), and the factthat dϕ = 0 and ∇ g = 0 , except the coefficient of ∂γ i b j that is more involved andis detailed below.coefficient of b i β j : ( − ) g ij b i β j , coefficient of ∂b i : ( − ) g ij Γ kij ∂b k ,coefficient of c i c j c k : ,coefficient of c i b j b k : ( − ) g ij Γ lik c k b j b l , coefficient of ∂γ i c j : ( − ) g ij ∂γ i c j , coefficient of ∂γ i b j : ( − g ij Γ kil Γ ljm ∂γ m b k .46 omputations to obtain the coefficient of ∂γ i b j .Denote by A the terms of type ∂γ i b j that appear in the computations of the b i β j coefficient due to quasi-associativity (2.9): A =( −
116 ) ∂ ( ϕ ijk ) ϕ lij,k b l + ( −
116 ) ∂ ( ϕ lij ) ϕ ijk,k b l + 116 ∂ ( ϕ ijk ) ϕ lkj,l b i + 116 ∂ ( ϕ lkj ) ϕ ijk,l b i + 18 ∂ ( ϕ ijk ) ϕ kmi,m b j + 18 ∂ ( ϕ kmi ) ϕ ijk,m b j . Collecting the other terms that contain ∂γ i b j : denote by A the sum of the termsthat does not contain derivatives of the Christoffel symbols, denote by A the sumof the terms containing derivatives of the Christoffel symbols.We have: A + A =( −
12 )Γ ji s Γ aij ∂γ s b a + 12 Γ ai s Γ ijj ∂γ s b a + ( −
72 )Γ ai j Γ ijs ∂γ s b a + ( −
12 ) ψ a klj Γ ji k Γ il s ∂γ s b a ,A = 12 g lk (Γ ik l ) ,s ∂γ s b i + ( −
12 ) g rm (Γ im s ) ,r ∂γ s b i + 12 g am (Γ im s ) ,i ∂γ s b a + 12 ψ ramn g in (Γ im s ) ,r ∂γ s b a . Let R denote the Riemann curvature, using the identity (6.2) we can write: g lk (Γ ik l ) ,s ∂γ s b i = 12 g kl R ilsk ∂γ s b i + 12 g kl (Γ is l ) ,k ∂γ s b i + ( −
12 ) g kl Γ is a Γ ak l ∂γ s b i + 12 g kl Γ ik a Γ as l ∂γ s b i , and using identity (6.5) we have g am (Γ im s ) ,i ∂γ s b a = 12 g am R ms ∂γ s b a + 12 g am Γ jm l Γ ls j ∂γ s b a . Then A = 12 R is ∂γ s b i + ( −
12 ) g kl Γ is a Γ ak l ∂γ s b i + + 12 g kl Γ ik a Γ as l ∂γ s b i + 12 R as ∂γ s b a + 12 g am Γ jm l Γ ls j ∂γ s b a + 12 ψ ramn g in (Γ im s ) ,r ∂γ s b a . ( −
12 ) ψ a klj Γ ji k Γ il s ∂γ s b a + 12 ψ ramn g in (Γ im s ) ,r ∂γ s b a =( −
12 ) ψ a klj Γ jk i Γ il s ∂γ s b a + ( −
12 ) ψ a klj (Γ jl s ) ,k ∂γ s b a =( −
12 ) ψ a klj R jlks ∂γ s b a = 12 R ksjl ψ jl ki g ai ∂γ s b a , then A + A + A =( − g kl Γ ik a Γ as l ∂γ s b i + R ij ∂γ j b i + 12 R ksjl ψ jl ki g ai ∂γ s b a =( − g ij Γ kil Γ ljm ∂γ m b k . To establish the last equality we use the Lemma 4.3 and Lemma 4.4.Finally we have proved that: Φ +(1) K + = − c i β i − ∂γ i b i − g ij b i β j − g ij Γ lik c k b j b l − g ij Γ kij ∂b k − g ij ∂γ i c j − g ij Γ kil Γ ljm ∂γ m b k , Φ − (1) K − = − c i β i − ∂γ i b i + ( ) g ij b i β j + ( ) g ij Γ lik c k b j b l + ( ) g ij Γ kij ∂b k + ( ) g ij ∂γ i c j + 3 g ij Γ kil Γ ljm ∂γ m b k . [ G + λ G − ] Now we compute [ G + λ G − ] , G + = c i β i + ∂γ i b i + g ij b i β j + g ij Γ lik c k b j b l + g ij Γ kij ∂b k + g ij ∂γ i c j + g ij Γ kil Γ ljm ∂γ m b k ,G − = c i β i + ∂γ i b i + ( − ) g ij b i β j + ( − ) g ij Γ lik c k b j b l + ( − ) g ij Γ kij ∂b k +( − ) g ij ∂γ i c j + ( − g ij Γ kil Γ ljm ∂γ m b k . We list the non-zero λ -brackets between the summands of G + and the summandsof G − , to compute these we used the Mathematica package [Thi91]. [ c i β iλ ∂γ j b j ] = λ + ( c i b i ) λ + ∂c i b i + ∂γ j β j , [ c i β iλ g lm b l β m ] = (cid:16) ( − g lm,l ) ,m (cid:17) λ (cid:16) g lm,l β m + ( − g lm,i ) ,m c i b l (cid:17) λ + g lm β l β m + g lm,i c i b l β m + ( − g lm,i ) ,m ∂c i b l + ∂ ( g lm,l ) β m , c i β iλ g lm Γ nla c a b m b n ]= (cid:16) ( g lm Γ nla ) ,n c a b m + ( − g lm Γ nla ) ,m c a b n (cid:17) λ + ( g lm Γ nla ) c a b m β n + ( − g lm Γ nla ) c a b n β m + ( − g lm Γ nla ) ,i c a c i b m b n + ∂ [( g lm Γ nla ) ,n ] c a b m + ( − ∂ [( g lm Γ nla ) ,m ] c a b n , [ c i β iλ g lm Γ nlm ∂b n ] = (cid:16) ( g lm Γ nlm ) ,n (cid:17) λ (cid:16) ( g lm Γ nlm ) β n + ∂ [( g lm Γ nlm ) ,n ] (cid:17) λ ( g lm Γ nlm ) ∂β n + ( g lm Γ nlm ) ,i c i ∂b n + 12 ∂ [( g lm Γ nlm ) ,n ] , [ c i β iλ g lm ∂γ l c m ] = g lm ∂c l c m + ( g lm ) ,i ∂γ l c i c m , [ c i β iλ g lm Γ nla Γ amr ∂γ r b n ]= (cid:16) g lm Γ nla Γ amn (cid:17) λ (cid:16) g lm Γ nla Γ ami c i b n + ( g lm Γ nla Γ amr ) ,n ∂γ r (cid:17) λ + g lm Γ nla Γ ami ∂c i b n + ( g lm Γ nla Γ amr ) ∂γ r β n + ( g lm Γ nla Γ amr ) ,i ∂γ r c i b n + ∂ [( g lm Γ nla Γ amr ) ,n ] ∂γ r , [ ∂γ i b iλ c j β j ] = λ + (cid:16) ( − c i b i (cid:17) λ + ( − c i ∂b i + ∂γ i β i , [ ∂γ i b iλ g lm b l β m ] = g lm ∂b m b l , [ ∂γ i b iλ g lm Γ nla c a b m b n ] = g lm Γ nli ∂γ i b m b n , [ ∂γ i b iλ g lm ∂γ l c m ] = g lm ∂γ l ∂γ m , [ g lm b l β mλ c i β i ] = (cid:16) ( − g lm,l ) ,m (cid:17) λ (cid:16) ( − g lm,l ) β m + ( g lm,i ) ,m c i b l +( − ∂ [( g lm,l ) ,m ] (cid:17) λ + g lm β l β m + g lm,i c i b l β m + ( − g lm,l ∂β m + ( g lm,i ) ,m c i ∂b l + ∂ [( g lm,i ) ,m ] c i b l + ( −
12 ) ∂ [( g lm,l ) ,m ] , [ g lm b l β mλ ∂γ i b i ] =( − g lm b m ∂b l , g lm b l β mλ g ij b i β j ]= (cid:16) ( g lm,j )( g ij,m ) b i b l (cid:17) λ + ( − g lm g ij,m b i b l β j + g lm ( g ij,m ) ,j b i ∂b l + g ij ( g lm,j ) b i b l β m + g ij ( g lm,j ) ,m b i ∂b l + g ij ∂ [( g lm,j ) ,m ] b i b l + g lm,j g ij,m b i ∂b l + g lm,j ∂ [( g ij,m )] b i b l + ∂ ( g lm )( g ij,m ) ,j b i b l , [ g lm b l β mλ g ij Γ aik c k b j b a ]= (cid:16) ( − g lm ( g ij Γ ail ) ,m b a b j (cid:17) λ + ( − g lm ( g ij Γ ail ) b a b j β m + g lm ( g ij Γ aik ) ,m c k b a b j b l + ( − g lm ∂ [( g ij Γ ail ) ,m ] b a b j + ( − ∂ ( g lm )( g ij Γ ail ) ,m b a b j , [ g lm b l β mλ g ij Γ kij ∂b k ] =( − g lm ( g ij Γ kij ) ,m ∂b k b l , [ g lm b l β mλ g ij ∂γ i c j ]= 12 λ + (cid:16) ( − c j b j + g lm ( g il ) ,m ∂γ i + g lm ∂ ( g lm ) (cid:17) λ + ( − c j ∂b j + ∂γ i β i + ( − g lm ( g ij ) ,m ∂γ i c j b l + g lm ∂ [( g il ) ,m ] ∂γ i + ( − ∂ ( g lm ) g mj c j b l + 2 ∂ ( g lm )( g il ) ,m ∂γ i + 12 g lm ∂ ( g lm ) + ∂ ( g il )( g lm ) ,m ∂γ i , [ g lm b l β mλ g ij Γ kin Γ nja ∂γ a b k ]= (cid:16) ( − g lm g ij Γ kin Γ njm b k b l (cid:17) λ + ( − g lm g ij Γ kin Γ njm b k ∂b l + ( − g lm ( g ij Γ kin Γ nja ) ,m ∂γ a b k b l + ( − ∂ ( g lm ) g ij Γ kin Γ njm b k b l , [ g ij Γ lik c k b j b lλ c m β m ]= (cid:16) ( − g ij Γ mik ) ,m c k b j + ( g im Γ lik ) ,m c k b l (cid:17) λ + ( g ij Γ mik ) c k b j β m + ( − g im Γ lik ) c k b l β m + ( − g ij Γ lik ) ,m c k c m b j b l ( − g ij Γ mik ) ,m c k ∂b j + ( g im Γ lik ) ,m c k ∂b l + ( − g ij Γ mik ) ,m ∂c k b j + ( g im Γ lik ) ,m ∂c k b l , [ g ij Γ lik c k b j b lλ ∂γ m b m ] = g ij Γ lim ∂γ m b j b l , g ij Γ lik c k b j b lλ g mn b m β n ]= (cid:16) ( − g mn ( g ij Γ lim ) ,n b j b l (cid:17) λ + ( g ij Γ lim ) g mn b j b l β n + ( − g mn ( g ij Γ lim ) ,n b j ∂b l + ( − g mn ( g ij Γ lik ) ,n c k b j b l b m + ( − g mn ( g ij Γ lim ) ,n ∂b j b l , [ g ij Γ lik c k b j b lλ g mn Γ rma c a b n b r ]= (cid:16) g ij Γ lir g mn Γ rml b j b n + ( − g ij Γ lin g mn Γ rml b j b r +( − g ij Γ lir g mn Γ rmj b l b n + g ij Γ lin g mn Γ rmj b l b r (cid:17) λ + g ij Γ lik g mn Γ kma c a b j b l b n + ( − g ij Γ lik g mk Γ rma c a b j b l b r + g ij Γ lik g mn Γ rml c k b j b n b r + ( − g ij Γ lik g mn Γ rmj c k b l b n b r + g ij Γ lik g mn Γ kml ∂b j b n + ( − g ij Γ lik g mk Γ rml ∂b j b r + ( − g ij Γ lik g mn Γ kmj ∂b l b n + g ij Γ lik g mk Γ rmj ∂b l b r + ∂ ( g ij Γ lik ) g mn Γ kml b j b n + ( − ∂ ( g ij Γ lik ) g mk Γ rml b j b r + ( − ∂ ( g ij Γ lik ) g mn Γ kmj b l b n + ∂ ( g ij Γ lik ) g mk Γ rmj b l b r , [ g ij Γ lik c k b j b lλ g mn Γ amn ∂b a ] = (cid:16) g ij Γ lik g mn Γ kmn b j b l (cid:17) λ + g ij Γ lik g mn Γ kmn b j ∂b l + g ij Γ lik g mn Γ kmn ∂b j b l + ∂ ( g ij Γ lik ) g mn Γ kmn b j b l , [ g ij Γ lik c k b j b lλ g mn ∂γ m c n ] = g ij Γ lik g lm ∂γ m c k b j + ( − lik ∂γ i c k b l , [ g ij Γ lik c k b j b lλ g mn Γ ams Γ snr ∂γ r b a ] = g ij Γ lik g mn Γ kms Γ snr ∂γ r b j b l , [ g ij Γ kij ∂b kλ c m β m ] = (cid:16) ( g ij Γ kij ) ,k (cid:17) λ (cid:16) ( − g ij Γ kij ) β k (cid:17) λ + ( g ij Γ kij ) ,m c m ∂b k + ( − ∂ ( g ij Γ kij ) β k , [ g ij Γ kij ∂b kλ g lm b l β m ] =( − g lm ( g ij Γ kij ) ,m ∂b k b l , [ g ij Γ kij ∂b kλ g lm Γ nla c a b m b n ]= (cid:16) ( − g ij Γ kij g lm Γ nlk b m b n (cid:17) λ + ( − ∂ ( g ij Γ kij ) g lm Γ nlk b m b n , [ g ij Γ kij ∂b kλ g lm ∂γ l c m ] = (cid:16) ( − g ij Γ kij g lk ∂γ l (cid:17) λ + ( − ∂ ( g ij Γ kij ) g lk ∂γ l , g mn ∂γ m c nλ c i β i ] =( − g mn c m ∂c n + ( g mn ) ,i ∂γ m c i c n , [ g mn ∂γ m c nλ ∂γ i b i ] = g mn ∂γ m ∂γ n , [ g mn ∂γ m c nλ g ij b i β j ]= λ (cid:16) c i b i + ( − g ij ( g mi ) ,j ∂γ m + g ij ∂ ( g ij ) (cid:17) λ + ∂c i b i + ∂γ j β j + ( − g ij ( g mn ) ,j ∂γ m c n b i + ( − g ij ( g mi ) ,j ∂ γ m + ∂ ( g mn ) g im c n b i + 12 ∂ ( g ij ) g ij + ∂ ( g ij )( g mi ) ,j ∂γ m + ∂ ( g mi )( g ij ) ,j ∂γ m , [ g mn ∂γ m c nλ g ij Γ lik c k b j b l ] = g lm g ij Γ lik ∂γ m c k b j + ( − lik ∂γ i c k b l , [ g mn ∂γ m c nλ g ij Γ kij ∂b k ]= (cid:16) g mk g ij Γ kij ∂γ m (cid:17) λ + g mk g ij Γ kij ∂ γ m + ∂ ( g mk ) g ij Γ kij ∂γ m , [ g mn ∂γ m c nλ g ij Γ kil Γ lja ∂γ a b k ] = g mn g ij Γ nil Γ lja ∂γ a ∂γ m , [ g ij Γ kil Γ lja ∂γ a b kλ c m β m ]= (cid:16) g ij Γ kil Γ ljk (cid:17) λ (cid:16) ( − g ij Γ kil Γ ljm c m b k +( − g ij Γ kil Γ lja ) ,k ∂γ a + ∂ ( g ij Γ kil Γ ljk ) (cid:17) λ + ( − g ij Γ kil Γ ljm c m ∂b k + ( g ij Γ kil Γ lja ) ∂γ a β k + ( g ij Γ kil Γ lja ) ,m ∂γ a c m b k + ( − g ij Γ kil Γ lja ) ,k ∂ γ a + ( − ∂ ( g ij Γ kil Γ ljm ) c m b k + ∂ ( g ij Γ kil Γ ljk ) , [ g ij Γ kil Γ lja ∂γ a b kλ g mn b m β n ]= (cid:16) g ij Γ kil Γ lja g ma b k b m (cid:17) λ + g ij Γ kil Γ lja g ma ∂b k b m ( − g ij Γ kil Γ lja ) ,n g mn ∂γ a b k b m + ∂ ( g ij Γ kil Γ lja ) g ma b k b m , [ g ij Γ kil Γ lja ∂γ a b kλ g mn Γ rms c s b n b r ] = g ij Γ kil Γ lja g mn Γ rmk ∂γ a b n b r , [ g ij Γ kil Γ lja ∂γ a b kλ g mn ∂γ m c n ] = g ij Γ kil Γ lja g mk ∂γ a ∂γ m . .4.1 Coefficient of λ We prove that it is zero after an easy computation using only that the metric iscovariantly constant. λ We compute the coefficient of each type of term that appears: • coefficient c i b j : 0 , ( − g lm Γ nla ) ,n c a b m + ( − g lm Γ nla Γ ami c i b n = g sm Γ lsn Γ nla c a b m + ( − g lm (Γ nla ) ,n c a b m = g sm (Γ nsa ) ,n c a b m + ( − g sm R sa c a b m + ( − g lm (Γ nla ) ,n c a b m =( − g sm R sa c a b m =0 , by Lemma (4.3). • coefficient ∂γ i : 0 , ( − g ij Γ kil Γ lja ) ,k ∂γ a + 12 ∂ ( g ij Γ kil Γ ljk ) . Expanding each summand separately: ( − g ij Γ kil Γ lja ) ,k ∂γ a =( − g ij,k Γ kil Γ lja ∂γ a + ( − g ij (Γ kil ) ,k Γ lja ∂γ a + ( − g ij Γ kil (Γ lja ) ,k ∂γ a =( g sj Γ isk + g is Γ jsk )Γ kil Γ lja ∂γ a + ( − g ij (Γ mik Γ klm )Γ lja ∂γ a + ( − g ij R il Γ lja ∂γ a + ( − g ij Γ kil h R ljka + (Γ ljk ) ,a − Γ lks Γ sja + Γ las Γ sjk i ∂γ a =( − g ij R il Γ lja ∂γ a + ( − g ij Γ kil (Γ ljk ) ,a ∂γ a + g ij Γ kil Γ lks Γ sja ∂γ a . Above, to cancel out the summand involving the curvature tensor we usedthat the Riemann’s curvature tensor is antisymmetric in the first two indices(6.3). ∂ ( g ij Γ kil Γ ljk )= 12 g ij,a Γ kil Γ ljk ∂γ a + 12 g ij (Γ kil ) ,a Γ ljk ∂γ a + 12 g ij Γ kil (Γ ljk ) ,a ∂γ a = 12 (cid:16) − g sj Γ isa − g is Γ jsa (cid:17) Γ kil Γ ljk ∂γ a + g ij Γ kil (Γ ljk ) ,a ∂γ a =( − g sj Γ isa Γ kil Γ ljk ∂γ a + g ij Γ kil (Γ ljk ) ,a ∂γ a . Combining the two expansions and using Lemma 4.3 we get the desiredresult. 53 .4.3 Coefficient of λ We compute the coefficient of each type of term that appears: • coefficient ∂c i b j : 0 ,
14 ( g lm,i ) ,m ∂c i b l + ( −
12 ) g lm Γ nla Γ ami ∂c i b n + ( −
14 )( g ij Γ mik ) ,m ∂c k b j + 14 ( g im Γ lik ) ,m ∂c k b l =( −
12 ) g lm Γ nla Γ ami ∂c i b n + ( −
12 )( g ij Γ mik ) ,m ∂c k b j = 12 g aj Γ ima Γ mik ∂c k b j + ( −
12 ) g ij (Γ mik ) ,m ∂c k b j =( −
12 ) g sj R sk ∂c k b j =0 , by Lemma 4.3. • coefficient ∂γ i β j : 0 . • coefficient c i b j β k : 0 . • coefficient c i ∂b j : 0 , ( −
12 ) g lm Γ nla Γ ami c i ∂b n + ( −
12 )( g ij Γ mik ) ,m c k ∂b j = 12 g aj Γ ima Γ mik c k ∂b j + ( −
12 )( g ij Γ mik ) ,m c k ∂b j =( −
12 ) g sj R sk c k ∂b j =0 , by Lemma 4.3. • coefficient c i c j b m b n : 0 . • coefficient ∂γ i c j b k : 0 ,using that the metric is covariantly constant we get: ( −
12 ) ∂ h ( g lk Γ nlm ) ,n i c m b k + ( −
12 ) ∂ ( g ij Γ kil Γ ljm ) c m b k =( −
12 ) ∂ h ( g lk Γ nlm ) ,n + g ij Γ kil Γ ljm i c m b k =( −
12 ) ∂ [ g sm R sa ] c a b m =0 ,
54y Lemma 4.3. • coefficient ∂β m : 0 . • coefficient ∂c i c j : 0 . • coefficient ∂γ i c j c k : 0 . • coefficient b i b j β k : 0 ,using that the metric is covariantly constant we get: ( −
12 ) (cid:16) g lm g ia Γ jam (cid:17) b i b l β j = 0 , because the factor between parentheses is symmetric in i and l .Due to the quasi-associativity (2.9) we obtain some terms of type ∂γ i b j b k : Q =( −
12 ) ∂ ( g lm ) (cid:16) g aj Γ iam (cid:17) ,j b i b l + ( −
12 ) g lm,j ∂ (cid:16) g aj Γ iam (cid:17) b i b l + ( −
12 ) ∂ ( g lm ) (cid:16) g ia Γ jam (cid:17) ,j b i b l + ( −
12 ) g lm,j ∂ (cid:16) g ia Γ jam (cid:17) b i b l + 12 ∂ ( g lm ) (cid:16) g ij Γ ail (cid:17) ,m b a b j + 12 g lm,m ∂ (cid:16) g ij Γ ail (cid:17) b a b j . (6.9) • coefficient γ i γ j : 0 ,using that the metric is covariantly constant we get: ( −
12 ) ∂ [( g lm Γ nla Γ amr ) ,n ] ∂γ r + 14 ∂ ( g ij Γ kil Γ ljk ) + ( −
18 ) g ij ∂ ( g ij )+ ( −
18 ) ∂ ( g ij ) g ij + 12 g rm g al Γ lrs Γ aim ∂γ s ∂γ i + 12 Γ mrs Γ rmi ∂γ s ∂γ i =0 . Here we used the following three identities −
18 ) ∂ ( g ij g ij ) = ( −
18 ) ∂ g ij g ij + ( −
14 ) ∂g ij ∂g ij + ( −
18 ) g ij ∂ g ij , ∂g ij ∂g ij = ( −
12 )Γ lis Γ ilr ∂γ s ∂γ r + ( −
12 ) g aj g il Γ ais Γ jlr ∂γ s ∂γ r , −
12 ) ∂ [( g lm Γ nla Γ amr ) ,n ] ∂γ r +( −
12 )( g lm Γ nla Γ amr ) ,n ∂ γ r + 14 ∂ ( g ij Γ kil Γ ljk ) . The last identity follows taking the derivative of the equality: − g ij Γ kil Γ lja ) ,k ∂γ a + 12 ∂ ( g ij Γ kil Γ ljk ) (this is exactly the coefficientof ∂γ i in the terms with λ in the subsection 6.4.2 above).55ue to the quasi-associativity (2.9) we obtain some terms of type ∂ γ i : Q = 12 ( g ij Γ kil Γ lja ) ,k ∂ γ a . (6.10) • coefficient ∂ γ i : 0 ,just using that the metric is covariantly constant and taking into account theterm (6.10) that comes from the coefficient of γ i γ j . • coefficient c i b j b k b l : 0 , ( −
12 ) g lm g ij (Γ aik ) ,m c k b a b j b l + ( −
12 ) g ij g mn Γ lik Γ kma c a b j b l b n + 12 g lm g sj Γ ism Γ aik c k b a b j b l = 12 g lm g ij R aikm c k b a b j b l + ( −
12 ) g lm g ij (Γ aim ) ,k c k b a b j b l + ( −
12 ) g lm g ij Γ akx Γ xim c k b a b j b l + 12 g lm g ij Γ amx Γ xik c k b a b j b l + ( −
12 ) g ij g mn Γ lik Γ kma c a b j b l b n + 12 g lm g sj Γ ism Γ aik c k b a b j b l = − R ajlk c k b a b j b l = − R [ ajl ] k c k b a b j b l =0 . Here [ ] denotes the anti-symmetrization of the indices, the last equalityfollows from the Bianchi’s first identity (6.4) which implies that R [ ajl ] k = 0 . • coefficient ∂b i b j : 0 ,denote by A terms containing derivatives of the Christoffel symbols are: A = 12 g lm g ia (Γ jam ) ,j b i ∂b l + ( −
12 ) g lm g ij (Γ kij ) ,m b l ∂b k + 12 g ij g am (Γ laj ) ,m b i ∂b l , denote by A the sum of terms that doesn’t contain derivatives of theChristoffel symbols: A = ( −
12 )Γ jii Γ laj b a ∂b l + ( −
12 )Γ jli Γ aij b a ∂b l + ( 12 )Γ jai Γ ilj b a ∂b l . Replacing the derivatives of the Christoffel symbol in A using (6.2) and566.5) we get: A = 12 g lm g ia R am b i ∂b l + 12 g lm g ia Γ xay Γ ymx b i ∂b l + 12 g lm g ij R kijm b l ∂b k + ( −
12 ) g lm g ij (Γ kim ) ,j b l ∂b k + ( −
12 ) g lm g ij Γ kjs Γ sim b l ∂b k + 12 g lm g ij Γ kms Γ sij b l ∂b k + 12 g ij g am R ljam b i ∂b l + 12 g ij g am (Γ ljm ) ,a b i ∂b l + 12 g ij g am Γ las Γ sjm b i ∂b l + ( −
12 ) g ij g am Γ lms Γ sja b i ∂b l = 12 g lm g ia R am b i ∂b l + 12 g lm g ia Γ xay Γ ymx b i ∂b l + 12 g lm g ij R kijm b l ∂b k + 12 g lm g ij Γ kms Γ sij b l ∂b k + 12 g ij g am R ljam b i ∂b l + ( −
12 ) g ij g am Γ lms Γ sja b i ∂b l , then A + A = 12 g lm g ia R am b i ∂b l + 12 g lm g ij R kijm b l ∂b k + 12 g ij g am R ljam b i ∂b l = 12 g ij g am R ljam b i ∂b l =0 , using the symmetries (6.3). • coefficient ∂γ i b j b k : 0 ,due to the quasi-associativity in the computations of the b i b j β k coefficientwe need to take into account (6.9).Collecting the terms that contain ∂γ i b j b k : denote by A the sum of theterms that does not contain derivative of the Christoffel symbols, denote by A the sum of the terms containing derivative of the Christoffel symbols. Wehave: A = 12 g lm,j g aj,s Γ iam ∂γ s b i b l + g lm g ij,m Γ kin Γ nja ∂γ a b k b l + ( − g ij g mn Γ lik Γ kms Γ snr ∂γ r b j b l = 12 g lx g ak Γ mxj Γ jks Γ iam ∂γ s b i b l + ( −
12 ) g lm g xj Γ ixm Γ kin Γ nja ∂γ a b k b l + ( − g lm g ix Γ jxm Γ kin Γ nja ∂γ a b k b l + ( − g ij g mn Γ lik Γ kms Γ snr ∂γ r b j b l , = g rm g ia Γ lrj (Γ jam ) ,s ∂γ s b i b l + ( −
12 ) g rm g ij Γ lrs (Γ ail ) ,m ∂γ s b a b j + ( −
12 ) g lr g ij Γ mrs (Γ ail ) ,m ∂γ s b a b j + g lm g ij Γ kin (Γ nja ) ,m ∂γ a b k b l + g lm g ij Γ nja (Γ kin ) ,m ∂γ a b k b l . Substituting the derivative of the Christoffel symbols in the second, third andfourth summand of A by the Riemann curvature tensor (6.2) we get: A = g rm g ia Γ lrj (Γ jam ) ,s ∂γ s b i b l + 12 g rm g ij Γ lrs R alim ∂γ s b a b j + ( −
12 ) g rm g ij Γ lrs (Γ alm ) ,i ∂γ s b a b j + ( −
12 ) g rm g ij Γ lrs Γ aix Γ xlm ∂γ s b a b j + 12 g rm g ij Γ lrs Γ amx Γ xli ∂γ s b a b j + 12 g lr g ij Γ mrs R alim ∂γ s b a b j + ( −
12 ) g lr g ij Γ mrs (Γ alm ) ,i ∂γ s b a b j + ( −
12 ) g lr g ij Γ mrs Γ aix Γ xlm ∂γ s b a b j + 12 g lr g ij Γ mrs Γ amx Γ xli ∂γ s b a b j + ( − g lm g ij Γ kin R njam ∂γ a b k b l + g lm g ij Γ kin (Γ njm ) ,a ∂γ a b k b l + g lm g ij Γ kin Γ nax Γ xjm ∂γ a b k b l + ( − g lm g ij Γ kin Γ nmx Γ xja ∂γ a b k b l + g lm g ij Γ nja (Γ kin ) ,m ∂γ a b k b l =( − g rm g ij Γ lrs Γ aix Γ xlm ∂γ s b a b j + ( −
12 ) g lm g ij Γ kin Γ nmx Γ xja ∂γ a b k b l + 12 g lr g ij Γ mrs Γ amx Γ xli ∂γ s b a b j + g lm g ij Γ kin Γ nax Γ xjm ∂γ a b k b l . Finally we check that A + A = 0 . We conclude that [ G + λ G − ] = 0 . [ G + λ Φ − ] and [ G − λ Φ + ] Now we compute [ G + λ Φ − ] , [ G − λ Φ + ] is computed similarly. G + = 12 c i β i + 12 ∂γ i b i + 12 g ij b i β j + 12 g ij Γ lik c k b j b l + 12 g ij Γ kij ∂b k + 12 g ij ∂γ i c j + g ij Γ kil Γ ljm ∂γ m b k , Φ − = i √ ϕ ijk c i c j c k + − i √ ϕ ijk g il c j c k b l + i √ ϕ ijk g il g jm c k b l b m + − i √ ϕ ijk g il g jm g kn b l b m b n + i √ ϕ ijk Γ imn g jm ∂γ n g kl b l − i √ ϕ ijk Γ imn g jm ∂γ n c k .
58e list the non-zero λ -brackets between the summands of G + and the summandsof Φ − , to compute these we used the Mathematica package [Thi91]. [ c i β iλ ϕ lmn c l c m c n ] = ϕ lmn,i c i c l c m c n = (cid:0) ϕ amn Γ ail + ϕ lan Γ aim + ϕ lma Γ ain (cid:1) c i c l c m c n =0 , [ c i β iλ ϕ lmn c m c n b l ] = (cid:16) ϕ lmn,l c m c n (cid:17) λ + ϕ lmn c m c n β l + ϕ lmn,i c i c m c n b l + ∂ (cid:16) ϕ lmn,l (cid:17) c m c n , [ c i β iλ ϕ lmn c n b l b m ] =2 (cid:16) ϕ lmn,m c n b l (cid:17) λ + 2 ϕ lmn c n b l β m + ϕ lmn,i c i c n b l b m + 2 ∂ (cid:16) ϕ lmn,m (cid:17) c n b l , [ c i β iλ ϕ lmn b l b m b n ] =3 (cid:16) ϕ lmn,n b l b m (cid:17) λ + 3 ϕ lmn b l b m β n + ϕ lmn,i c i b l b m b n + 3 ∂ (cid:16) ϕ lmn,n (cid:17) b l b m , [ c i β iλ ϕ mnl Γ lms ∂γ s b n ] = (cid:18) ϕ mnl Γ lms c s b n + (cid:16) ϕ mnl Γ lms (cid:17) ,n ∂γ s (cid:19) λ + ϕ mnl Γ lms ∂c s b n + (cid:16) ϕ mnl Γ lms (cid:17) ∂γ s β n + (cid:16) ϕ mnl Γ lms (cid:17) ,i ∂γ s c i b n + ∂ [ (cid:16) ϕ mnl Γ lms (cid:17) ,n ] ∂γ s , [ c i β iλ ϕ ml n Γ lms ∂γ s c n ] = (cid:16) ϕ ml n Γ lms c s c n (cid:17) λ + ϕ ml n Γ lms ∂c s c n + (cid:16) ϕ ml n Γ lms (cid:17) ,i ∂γ s c i c n , [ ∂γ i b iλ ϕ lmn c l c m c n ] =3 ϕ lmn ∂γ n c l c m , [ ∂γ i b iλ ϕ lmn c m c n b l ] =2 ϕ lmn ∂γ m c n b l , [ ∂γ i b iλ ϕ lmn c n b l b m ] = ϕ lmn ∂γ n b l b m , [ ∂γ i b iλ ϕ ml n Γ lms ∂γ s c n ] = ϕ ml n Γ lms ∂γ n ∂γ s , g ij b i β j λ ϕ lmn c l c m c n ] =3 (cid:16) g ni ϕ lmn,i c l c m (cid:17) λ + 3 (cid:16) g ni ϕ lmn (cid:17) c l c m β i + ( − g ij ϕ lmn,j c l c m c n b i + 3 g ni ∂ [ ϕ lmn,i ] c l c m + 6 ∂ ( g ni ) ϕ lmn,i c l c m + 3 ∂ (cid:0) ϕ lmn (cid:1) g ni,i c l c m , [ g ij b i β j λ ϕ lmn c m c n b l ] =( − (cid:16) g ni ϕ lmn,i c m b l (cid:17) λ + 2 (cid:16) g mi ϕ lmn (cid:17) c n b l β i + g ij ϕ lmn,j c m c n b i b l + 2 g mi ∂ [ ϕ lmn,i ] c n b l + 4 ∂ ( g mi ) ϕ lmn,i c n b l + 2 ∂ (cid:16) ϕ lmn (cid:17) g mi,i c n b l , [ g ij b i β j λ ϕ lmn c n b l b m ] = (cid:16) g ni ϕ lmn,i b l b m (cid:17) λ + (cid:16) g ni ϕ lmn (cid:17) b l b m β i + ( − g ij ϕ lmn,j c n b i b l b m + g ni ∂ (cid:16) ϕ lmn,i (cid:17) b l b m + 2 ∂ (cid:16) g ni (cid:17) ϕ lmn,i b l b m + ∂ (cid:16) ϕ lmn (cid:17) g ni,i b l b m , [ g ij b i β j λ ϕ lmn b l b m b n ] = g ij ϕ lmn,j b i b l b m b n , [ g ij b i β j λ ϕ mnl Γ lms ∂γ s b n ] = (cid:16) g ij ϕ mnl Γ lmj b i b n (cid:17) λ + g ij ϕ mnl Γ lmj ∂b i b n + g ij (cid:16) ϕ mnl Γ lms (cid:17) ,j ∂γ s b i b n + ∂ ( g ij ) ϕ mnl Γ lmj b i b n , [ g ij b i β j λ ϕ ml n Γ lms ∂γ s c n ]= (cid:16) ( − g ij ϕ ml n Γ lmj c n b i + g nj ( ϕ ml n Γ lms ) ,j ∂γ s + ∂ ( g ns ) ϕ ml n Γ lms (cid:17) λ + ( − g is ϕ ml n Γ lms c n ∂b i + (cid:16) g nj ϕ ml n Γ lms (cid:17) ∂γ s β j + ( − g ij (cid:16) ϕ ml n Γ lms (cid:17) ,j ∂γ s c n b i + g nj ∂ [ (cid:16) ϕ ml n Γ lms (cid:17) ,j ] ∂γ s + ( − ∂ ( g is ) ϕ ml n Γ lms c n b i + 2 ∂ ( g nj ) (cid:16) ϕ ml n Γ lms (cid:17) ,j ∂γ s + 12 ∂ ( g ns ) ϕ ml n Γ lms + g nj,j ∂ (cid:16) ϕ ml n Γ lms (cid:17) ∂γ s , [ g ij Γ rik c k b j b rλ ϕ lmn c l c m c n ] = (cid:16) g in Γ mik ϕ lmn c k c l (cid:17) λ + 3 g ij Γ nik ϕ lmn c k c l c m b j + 3 g im Γ rik ϕ lmn c k c l c n b r + 6 g in Γ mik ϕ lmn ∂c k c l + 6 ∂ (cid:16) g in Γ mik (cid:17) ϕ lmn c k c l , g ij Γ rik c k b j b rλ ϕ lmn c m c n b l ]= (cid:16) g in Γ mik ϕ lmn c k b l + 2 g ij Γ nil ϕ lmn c m b j + 2 g im Γ ril ϕ lmn c n b r + 2 ∂ (cid:16) g in Γ mil (cid:17) ϕ lmn (cid:19) λ + 2 g ij Γ nik ϕ lmn c k c m b j b l + 2 g in Γ rik ϕ lmn c k c m b l b r + g ij Γ ril ϕ lmn c m c n b j b r + 2 g ij Γ nil ϕ lmn c m ∂b j + 2 g im Γ ril ϕ lmn c n ∂b r + 2 g in Γ mik ϕ lmn ∂c k b l + 2 ∂ (cid:16) g in Γ mik (cid:17) ϕ lmn c k b l + 2 ∂ (cid:16) g ij Γ nil (cid:17) ϕ lmn c m b j + 2 ∂ (cid:16) g im Γ ril (cid:17) ϕ lmn c n b r + ∂ (cid:16) g in Γ mil (cid:17) ϕ lmn , [ g ij Γ rik c k b j b rλ ϕ lmn c n b l b m ]= (cid:16) g ij Γ nim ϕ lmn b j b l + 2 g in Γ rim ϕ lmn b l b r (cid:17) λ + 2 g in Γ rim ϕ lmn b l ∂b r + g ij Γ nik ϕ lmn c k b j b l b m + ( − g in Γ rik ϕ lmn c k b l b m b r + 2 g ij Γ ril ϕ lmn c n b j b m b r + 2 g ij Γ nim ϕ lmn ∂b j b l + 2 ∂ (cid:16) g ij Γ nim (cid:17) ϕ lmn b j b l + 2 ∂ (cid:16) g in Γ rim (cid:17) ϕ lmn b l b r , [ g ij Γ rik c k b j b rλ ϕ lmn b l b m b n ] =3 g ij Γ rin ϕ lmn b j b l b m b r , [ g ij Γ rik c k b j b rλ ϕ mnl Γ lms ∂γ s b n ] = g ij Γ rin Γ lms ϕ mnl ∂γ s b j b r , [ g ij Γ rik c k b j b rλ ϕ ml n Γ lms ∂γ s c n ]= g ij Γ nik Γ lms ϕ ml n ∂γ s c k b j + ( − g in Γ rik Γ lms ϕ ml n ∂γ s c k b r , [ g ij Γ kij ∂b kλ ϕ lmn c l c m c n ] = (cid:16) g ij Γ mij ϕ lmn c l c n (cid:17) λ + 3 ∂ (cid:16) g ij Γ mij (cid:17) ϕ lmn c l c n , [ g ij Γ kij ∂b kλ ϕ lmn c m c n b l ] = (cid:16) g ij Γ nij ϕ lmn c m b l (cid:17) λ + 2 ∂ (cid:16) g ij Γ nij (cid:17) ϕ lmn c m b l , [ g ij Γ kij ∂b kλ ϕ lmn c n b l b m ]= (cid:16) ( − g ij Γ nij ϕ lmn b l b m (cid:17) λ + ( − ∂ ( g ij Γ nij ) ϕ lmn b l b m , [ g ij Γ kij ∂b kλ ϕ ml n Γ lms ∂γ s c n ]= (cid:16) ( − g ij Γ nij ϕ ml n Γ lms ∂γ s (cid:17) λ + ( − ∂ (cid:16) g ij Γ nij (cid:17) ϕ ml n Γ lms ∂γ s , g ij ∂γ i c j λ ϕ lmn c m c n b l ] = g il ϕ lmn ∂γ i c m c n , [ g ij ∂γ i c jλ ϕ lmn c n b l b m ] =2 g im ϕ lmn ∂γ i c n b l , [ g ij ∂γ i c j λ ϕ lmn b l b m b n ] =3 g in ϕ lmn ∂γ i b l b m , [ g ij ∂γ i c j λ ϕ mnl Γ lms ∂γ s b n ] = g in ϕ mnl Γ lms ∂γ i ∂γ s , [ g ij Γ kia Γ ajr ∂γ r b kλ ϕ lmn c l c m c n ] =3 g ij Γ nia Γ ajr ϕ lmn ∂γ r c l c m , [ g ij Γ kia Γ ajr ∂γ r b kλ ϕ lmn c m c n b l ] =( − g ij Γ nia Γ ajr ϕ lmn ∂γ r c m b l , [ g ij Γ kia Γ ajr ∂γ r b kλ ϕ lmn c n b l b m ] = g ij Γ nia Γ ajr ϕ lmn ∂γ r b l b m , [ g ij Γ kia Γ ajr ∂γ r b kλ ϕ ml n Γ lms ∂γ s c n ] = g ij Γ nia Γ ajr ϕ ml n Γ lms ∂γ r ∂γ s . λ We prove that it is zero after an easy computation using only that the metric iscovariantly constant and that dϕ = 0 , (4.3). λ We compute the coefficient of each type of term that appears: • Coefficient of c i c j β k : . • Coefficient of c i c j c k b l : , we use that dϕ = 0 . • Coefficient of ∂γ s c i c j :After some simplifications using that dϕ = 0 and ∇ g = 0 we arrived at thefollowing expression: ( − i √ (cid:16) g ij Γ klm Γ ljs ϕ kin ∂γ s c m c n + ( − g ij Γ ksl Γ ljm ϕ kin ∂γ s c m c n + g ij (cid:16) Γ kjs (cid:17) ,m ϕ kin ∂γ s c m c n + ( − g ij (cid:16) Γ kjm (cid:17) ,s ϕ kin ∂γ s c m c n (cid:19) =( − i √ g ij ϕ kin R kjms ∂γ s c m c n =0 , the last equality is zero by Lemma 4.5.62 Coefficient of c i b j β k : . • Coefficient of c i c j b l b m : , we use that dϕ = 0 . • Coefficient of ∂γ s c i b j :After some simplifications using that dϕ = 0 and ∇ g = 0 we arrived at thefollowing expression: i √ (cid:18) ϕ jki (cid:16) Γ ijs (cid:17) ,l ∂γ s c l b k + ( − ϕ jki (cid:16) Γ ijl (cid:17) ,s ∂γ s c l b k + ϕ jki Γ ila Γ ajs ∂γ s c l b k + ( − ϕ jki Γ isa Γ ajl ∂γ s c l b k (cid:17) + i √ (cid:18) g ij ϕ ml n (cid:16) Γ lms (cid:17) ,j ∂γ s c n b i + ( − g ij ϕ ml n (cid:16) Γ lmj (cid:17) ,s ∂γ s c n b i + g ij ϕ ml n Γ lja Γ ams ∂γ s c n b i + ( − g ij ϕ ml n Γ lsa Γ amj ∂γ s c n b i (cid:17) = i √ ϕ jki R ijls ∂γ s c l b k + i √ g ij ϕ ml n R lmjs ∂γ s c n b i =0 , the last equality is zero by Lemma 4.5. • Coefficient of b i b j β k : . • Coefficient of c l b i b j b k : , we use that dϕ = 0 . • Coefficient of ∂γ s b i b j :After some simplifications using that dϕ = 0 and ∇ g = 0 we arrived at thefollowing expression: i √ (cid:18) g ij ϕ mnl (cid:16) Γ lms (cid:17) ,j ∂γ s b i b n + ( − g ij ϕ mnl (cid:16) Γ lmj (cid:17) ,s ∂γ s b i b n + g ij ϕ mnl Γ lja Γ ams ∂γ s b i b n + ( − g ij ϕ mnl Γ lsa Γ amj ∂γ s b i b n (cid:17) = i √ g ij ϕ mnl R lmjs ∂γ s b i b n =0 , the last equality is zero by Lemma 4.5. • Coefficient of c i ∂b j : . • Coefficient of ∂c i b j : . 63 Coefficient of ∂c i c j : . • Coefficient of b i b j b k b l : , we use that dϕ = 0 . • Coefficient of ∂b i b j : . • Coefficient of ∂γ s β i : . • Coefficient of ∂γ s ∂γ r :After some simplifications using that dϕ = 0 and ∇ g = 0 we arrived at thefollowing expression: ( − i )2 √ (cid:18) g ij Γ kir (cid:16) Γ lkm (cid:17) ,s ϕ mjl ∂γ s ∂γ r + ( − g ij Γ kir (cid:16) Γ lms (cid:17) ,k ϕ mjl ∂γ s ∂γ r + g ij Γ kir (cid:16) Γ lsa Γ amk (cid:17) ϕ mjl ∂γ s ∂γ r + ( − g ij Γ kir (cid:16) Γ lka Γ ams (cid:17) ϕ mjl ∂γ s ∂γ r (cid:19) + ( − i )2 √ (cid:16) g ij Γ ljs (Γ min ) ,r ϕ nlm ∂γ r ∂γ s + ( − g ij Γ ljs (Γ mnr ) ,i ϕ nlm ∂γ r ∂γ s + g ij Γ ljs (Γ mra Γ ani ) ϕ nlm ∂γ r ∂γ s + ( − g ij Γ ljs (Γ mia Γ anr ) ϕ nlm ∂γ r ∂γ s (cid:17) = ( − i )2 √ g ij Γ kir ϕ mjl R lmsk ∂γ s ∂γ r + ( − i )2 √ g ij Γ ljs ϕ nlm R mnri ∂γ r ∂γ s =0 , the last equality is zero by Lemma 4.5.Then we conclude that [ G + λ Φ − ] = 0 , similarly we obtain [ G − λ Φ + ] = 0 . [ G + λ Φ + ] G = c i β i + ∂γ i b i , Φ + = 112 √ ϕ ijk c i c j c k + 14 √ ϕ ijk g il c j c k b l + 14 √ ϕ ijk g il g jm c k b l b m + 112 √ ϕ ijk g il g jm g kn b l b m b n + 12 √ ϕ ijk Γ imn g jm ∂γ n g kl b l + 12 √ ϕ ijk Γ imn g jm ∂γ n c k . We can use the λ -brackets performed in section 6.5. λ We realize inmediatly that the coefficient of λ is zero because there are not λ terms. 64 .6.2 Coefficient of λ • Coefficient of c i c j : , we use that ∇ ϕ = 0 . • Coefficient of c i b j : , we use that ∇ ϕ = 0 . • Coefficient of b i b j : √ ϕ lmn,n b l b m = 0 .As ∇ ϕ = 0 (4.3) we have, ϕ lmn,n = − ϕ amn Γ lan − ϕ lan Γ man − ϕ lma Γ nan = 0 , to conclude that the last equality is zero we used the symmetries of ϕ and(6.1). • Coefficient of γ s : √ (cid:16) ϕ mnl Γ lms (cid:17) ,n ∂γ s = 0 . As ∇ ϕ = 0 (4.3) we have (cid:16) ϕ mnl Γ lms (cid:17) ,n = ϕ mnl ,n Γ lms + ϕ mnl (cid:16) Γ lms (cid:17) ,n = (cid:0) ϕ mna Γ aln − ϕ anl Γ man − ϕ mal Γ nan (cid:1) Γ lms + ϕ mnl (cid:16) Γ lms (cid:17) ,n = ϕ mna Γ aln Γ lms + ϕ mnl (cid:16) Γ lms (cid:17) ,n = ϕ mnl (cid:16) Γ lms (cid:17) ,n + ( − ϕ mnl (cid:16) Γ lmn (cid:17) ,s + ϕ mnl Γ lna Γ ams + ( − ϕ mnl Γ lsa Γ amn = ϕ mnl R lmns =0 . That the last equality is zero follows by Lemma 4.5.Then [ G + λ φ + ] = K + and similarly [ G − λ φ − ] = K − . [ L + λ Φ + ] L = ∂γ i β i − c i ∂b i + 12 ∂c i b i , + = 112 √ ϕ ijk c i c j c k + 14 √ ϕ ijk g il c j c k b l + 14 √ ϕ ijk g il g jm c k b l b m + 112 √ ϕ ijk g il g jm g kn b l b m b n + 12 √ ϕ ijk Γ imn g jm ∂γ n g kl b l + 12 √ ϕ ijk Γ imn g jm ∂γ n c k . 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