aa r X i v : . [ m a t h . AG ] A ug GETZLER RELATION AND VIRASORO CONJECTURE FORGENUS ONE
YIJIE LIN
Abstract.
We derive explicit universal equations for primary Gromov-Witteninvariants by applying Getzler’s genus one relation to quantum powers of Eulervector field. As an application, we provide some evidences for the genus-1Virasoro conjecture. Introduction
The Virasoro conjecture of Eguchi-Hori-Xiong [5] predicts some mysterious re-lations between Gromov-Witten invariants in all genera, that is, a sequence ofoperators are conjectured to annihilate the generating functions of Gromov-Witteninvariants of smooth projective varieties. It is equivalent to Witten conjecture [19]proved by Kontsevich [11] when the underlying mainfold is one point. For mani-folds with semisimple quantum product, this conjecture was proved in lower genera[3, 4, 6, 12, 13, 16, 17], and completely solved by Teleman [18]. The genus-0 partof the Virasoro conjecture has been proved firstly in [12] without assumption ofsemisimplicity, and later by other authors [3, 6]. It is still open for the general caseof the genus-1 Virasoro conjecture.In [13], it is proved that the genus-1 Virasoro conjecture can be reduced to somesimple equation derived by restricting the genus-1 L -constraint on the small phasespace. This equation is motivated by using Getzler’s genus one relation [7] as follows G ( E k , E k , E k , E k ) = 0 , (1)where E k is k th quantum power of Euler vector field, and the definition of G ispresented in section 2. Furtherly, the author in [15] shows that to prove the genus-1 Virasoro conjecture, it is enough to prove the property that the derivative ofthat simple equation along the direction of any vector field vanishes. Besides theidentity of the ordinary cohomology ring satisfying this property, he also find onemore vector field called quantum volume element by the computation of X α G ( E, E, γ α , γ α ) = 0 , (2)where γ α and γ α belong to the space of cohomology classes.In this paper, we will consider the following form G ( E k , γ α , γ β , γ σ ) = 0 . (3)The explicit formula of equation (3) is computed in section 3 (cf. Theorem 3.8),which implies the explicit expressions (cf. Theorems 3.13 and 3.16) for the followingtwo equations G ( E k , E k , γ α , γ β ) = 0 , (4) G ( E k , E k , E k , γ α ) = 0 . (5)The explicit formula (5) generalizes equation (1), and its equivalent result, i.e.,Theorem 4.1 implies some generalized version (cf. Corollaries 4.3 and 4.4) of theVirasoro type relation for { Φ k } in [13]. And formula (4) implies equation (2), andprovides more evidences (cf. Theorem 4.6) for the genus-1 Virasoro conjecture bycomputing X µ G ( E k , E k , γ µ , γ µ ◦ γ α ) = 0 . These evidences give an alternative proof of the genus-1 Virasoro conjecture for anymanifold with semisimple quantum cohomology (cf. Corollary 4.7). We also derivesome new relation (cf. Theorem 4.8) from the general equation (3).An outline of this paper is as follows. In section 2, we recall some basic defini-tions, present some known facts for the genus-1 Virasoro conjecture and recollectuniversal equations for primary Gromov-Witten invariants which will be used later.In section 3, we firstly obtain an explicit formula for the first derivative of Φ k , andthen derive explicit universal equations from Getzler’s genus one relation involvingsome quantum powers of Euler vector field. In section 4, we consider applicationsto the genus-1 Virasoro conjecture. Acknowledgements.
The author would like to thank Professor Xiaobo Liu formany helpful suggestions, and Professor Jian Zhou for his encouragement.2.
Preliminaries
In this section, we recall Gromov-Witten invariants, quantum product, and someknown facts for the genus-1 Virasoro conjecture. We will also recollect some uni-versal equations for primary Gromov-Witten invariants, and fix some notation.2.1.
Gromov-Witten invariants, quantum product and Virasoro conjec-ture.
Let X be a smooth projective variety of dimension d , and denote by N thedimension of the space of cohomology classes H ∗ ( X, C ). We assume H odd ( X ; C ) = 0for simplicity, and fix a basis { γ , · · · , γ N } of H ∗ ( X, C ). Let γ be the identity ofthe cohomology ring of X and γ α ∈ H p α ,q α ( X, C ) for each α . Let η αβ = R X γ α ∪ γ β be the intersection form on H ∗ ( X, C ), and C = ( C βα ) be the matrix satisfying c ( X ) ∪ γ α = X β C βα γ β . The symmetric matrices η = ( η αβ ) and η − = ( η αβ ) are used to lower and raiseindices. For example, γ α = P µ γ µ η µα . Let b α = p α − ( d − η αβ = 0 or η αβ = 0, then b α = 1 − b β . For Λ ∈ H ( X, Z ), let M g,k ( X, Λ) be the moduli space of stable map with [ M g,k ( X, Λ)] vir as its virtualfundamental class, and L i the tautological line bundle over M g,k ( X, Λ). Let e γ , · ·· , e γ k ∈ H ∗ ( X, C ), the genus- g descendant Gromov-Witten invariants are defined by h τ n ( e γ ) · · · τ n k ( e γ k ) i g = X Λ ∈ H ( X, Z ) q Λ Z [ M g,k ( X, Λ)] vir k Y i =1 c ( L ) n i ∪ ev ∗ i ( e γ i ) , (6)where q Λ is in Novikov ring with the product defined by q Λ q Λ = q Λ +Λ , and ev i : M g,k ( X, Λ) −→ X is the i -th evaluation map ( C ; x , · · · , x k ; f ) f ( x i ). One can refer [2] for more details. As in [13], for any τ n ( γ α ), one associates a parameter t αn , and call the space of all T = { t αn : n ∈ Z ≥ , α = 1 , ··· , N } the big phase space andits subspace spanned by { T | t αn = 0 if n > } the small phase space. For simplicity,we identify τ n ( γ α ) with tangent vector field ∂∂t αn on the big phase space, and on thesmall phase space, we write t α as t α and identify γ α with ∂∂t α . If we restrict theinvariant (6) on the small phase space, i.e., setting n i = 0 for all 1 ≤ i ≤ k , theresulting invariants are called primary Gromov-Witten invariants.The generating function of genus- g Gromov-Witten invariants is defined as F g ( T ) := X k ≥ k ! X n ,α , ··· ,n k ,α k t α n · · · t α k n k h τ n ( γ α ) · · · τ n k ( γ α k ) i g . And the generating function for Gromov-Witten invariants involving all genera isdefined to be Z ( T ; λ ) := exp X g ≥ λ g − F g ( T ) . As in [17], we define k -point (correlation) function hhW W · · · W k ii g := X n ,α , ··· ,n k ,α k f n ,α · · · f kn k ,α k ∂ k ∂t α n ∂t α n · · · ∂t α k n k F g for vector fields W i = P n,α f in,α ∂∂t αn . Let ∇ be the covariant derivative defined by ∇ W V = X n,α ( W f n,α ) τ n ( γ α ) (7)for any vector fields W and V = P n,α f n,α τ n ( γ α ). It is simple to show that[ V , W ] = ∇ V W − ∇ W V (8)and WhhV · · · V k ii g = hhWV · · · V k ii g + k X i =1 hhV · · · ( ∇ W V i ) · · · V k ii g (9)for any vector fields V , W and V i .Next, as in [17], we define the quantum product of any two vector fields V and W by V ◦ W = X α hhVW γ α ii γ α . Obviously, this product is commutative by definition and associative by the follow-ing well known generalized WDVV equation hh{V ◦ V }V V ii = hh{V ◦ V }V V ii . (10)It follows easily from formula (9) that ∇ U ( V ◦ W ) = ( ∇ U V ) ◦ W + V ◦ ( ∇ U W ) + X α hhUVW γ α ii γ α (11)for any vector fields U , V , and W . Define G ( V ) := X n,α ( n + b α ) f n,α τ n ( γ α ) (12) YIJIE LIN for any vector field V = P n,α f n,α τ n ( γ α ). We have the following easy observations X µ hh{V ◦ γ µ }{V ◦ V ◦ γ µ }V · · · V k ii g = X µ hh{V ◦ γ µ }{V ◦ V ◦ γ µ }V · · · V k ii g = X µ hh{V ◦ V ◦ γ µ }{V ◦ γ µ }V · · · V k ii g (13)and X µ hh G ( V ◦ V ◦ γ µ ) {V ◦ V ◦ γ µ }V · · · V k ii g = X µ hh G ( V ◦ V ◦ γ µ ) {V ◦ V ◦ γ µ }V · · · V k ii g = X µ hh G ( V ◦ V ◦ V ◦ γ µ ) {V ◦ γ µ }V · · · V k ii g = X µ hh G ( V ◦ γ µ ) {V ◦ V ◦ V ◦ γ µ }V · · · V k ii g (14)for any vector fields V i (1 ≤ i ≤ k ).We recollect the following Virasoro operators defined in [12] L − : = X m,α ˜ t αm ∂∂t αm − + 12 λ X α,β η αβ t α t β ,L : = X m,α ( m + b α )˜ t αm ∂∂t αm + X m,α,β C βα ˜ t αm ∂∂t βm − + 12 λ X α,β C αβ t α t β + 124 (cid:18) − d χ ( X ) − Z X c ( X ) ∪ c d − ( X ) (cid:19) , and for n ≥ L n : = X m,α,β m + n X j =0 A ( j ) α ( m, n )( C j ) βα ˜ t αm ∂∂t βm + n − j + λ X α,β,γ n − X j =0 n − j − X k =0 B ( j ) α ( k, n )( C j ) βα η αγ ∂∂t γk ∂∂t βn − k − − j + 12 λ X α,β ( C n +1 ) αβ t α t β , where A ( j ) α ( m, n ) := Γ( b α + m + n + 1)Γ( b α + m ) X m ≤ l 1. The Virasoro conjecture is Conjecture 2.1. L n Z ≡ (called the L n -constraint) for all n ≥ − . It is well known that the L − -constraint and the L -constraint hold for all mani-folds. In fact, the L − -constraint is equivalent to the string equation, and L Z = 0is true by Hori [9]. Assume that L n Z ( T ; λ ) = (cid:26) X g ≥ Ω g,n λ g − (cid:27) Z ( T ; λ ) . Then the L n -constraint is equivalent to Ω g,n = 0 for all g ≥ 0. As in [12], equationΩ g,n = 0 is called the genus- g L n -constraint, and the so called genus- g Virasoroconjecture predicts that the genus- g L n -constraint is true for all n ≥ − hh· · ·ii g is used again to denotesome correlation function on the small phase space which is written as hh· · ·ii g,s in[13]. We start with Euler vector field on the small phase space, which is defined by E := c ( X ) + X α ( b + 1 − b α ) t α γ α . It is showed in [12] that the following quasi-homogeneity equation holds hh E ii g = (3 − d )(1 − g ) F g + 12 δ g, X α,β C αβ t α t β − δ g, Z X c ( X ) ∪ c d − ( X ) . And its derivatives are hh Eυ · · · υ k ii g = k X i =1 hh υ · · · G ( υ i ) · · · υ k ii g − (2 g + k − b + 1) hh υ · · · υ k ii g + δ g, ∇ kυ , ··· ,υ k (cid:18) X α,β C αβ t α t β (cid:19) (16)for any vector fields υ i (1 ≤ i ≤ k ) on the small phase space. In particular, we have X α hh Eυ υ γ α ii γ α = G ( υ ) ◦ υ + υ ◦ G ( υ ) − G ( υ ◦ υ ) − b υ ◦ υ . (17)It is shown in [15] that ∇ υ E = − G ( υ ) + ( b + 1) υ. (18)More generally, we have Lemma 2.2. For all k ≥ and any vector field υ on the small phase space, let E k be the k th quantum power of E , we have ∇ υ E k = k X i =1 G ( E i − ) ◦ υ ◦ E k − i − k X i =1 G ( υ ◦ E k − i ) ◦ E i − + kυ ◦ E k − . (19) YIJIE LIN Proof. We will prove this lemma by induction on k . By the definition (7) and E = γ , it holds for k = 0 and also for k = 1 due to formula (18). Assumeequation (19) holds for k ≤ n . For k = n + 1, by equations (11) and (17), we have ∇ υ E n +1 = ∇ υ ( E ◦ E n )= ( ∇ υ E ) ◦ E n + E ◦ ( ∇ υ E n ) + X α hh EE n υγ α ii γ α = − G ( υ ) ◦ E n + ( b + 1) υ ◦ E n + n X i =1 G ( E i − ) ◦ υ ◦ E n − i +1 − n X i =1 G ( υ ◦ E n − i ) ◦ E i + nυ ◦ E n + G ( E n ) ◦ υ + G ( υ ) ◦ E n − G ( E n ◦ υ ) − b E n ◦ υ = n +1 X i =1 G ( E i − ) ◦ υ ◦ E n +1 − i − n +1 X i =1 G ( υ ◦ E n +1 − i ) ◦ E i − + ( n + 1) υ ◦ E n . The prooof is completed. (cid:3) Then by equations (8) and (19), we have Corollary 2.3 ([3, 8, 13]) . For k, m ≥ , [ E k , E m ] = ( m − k ) E m + k − . Secondly, functions Φ k are defined in terms of genus-0 data in [13]. They areΦ = 0 , Φ = − Z X c ( X ) ∪ c d − ( X ) , Φ k = − k − X m =0 X α,β,σ b α hh γ E m γ α ii hh γ α E k − − m γ β ii hh γ β γ σ γ σ ii − k − X m =0 X α,β b α b β hh γ α E m γ β ii hh γ β E k − − m γ α ii + k X σ hh γ σ E k − γ σ ii , (20)for k ≥ 2. It follows from the definitions of quantum product and G that24Φ k = − k − X i =0 hh G ( E i )∆ E k − i − ii − k X µ hh E k − γ µ γ µ ii +6 k − X i =0 X µ hh G ( E i ◦ γ µ ) G ( γ µ ) E k − i − ii , (21)where ∆ = P α γ α ◦ γ α and k ≥ Remark 2.4. It is easy to check that the expression (20) holds for k = 0 . It alsoholds for k = 1 by the string equation (22) and the following equality [1] derived by Borisov X α b α (1 − b α ) − b + 16 χ ( X ) = − Z X c ( X ) ∪ c d − ( X ) . The same argument is applied in the proof of γ Φ = 2Φ in [13] . Therefore, weadopt equation (20) or (21) as the definition of Φ k for all k ≥ . And notice thatit is shown in [13] that hh E k ii = Φ k for k = 0 , . One important result is Theorem 2.5 ([13]) . For any manifold X , the genus-1 Virasoro conjecture holdsif and only if hh E ii = Φ . It is shown in [14] that for any given k ≥ 2, the genus-1 Virasoro conjecture holdsif and only if hh E k ii = Φ k . Due to the equation E ( hh E ii − Φ ) = hh E ii − Φ proved in [13], this conjecture is reduced to prove that υ ( hh E ii − Φ ) = 0 forany υ ∈ H ∗ ( X, C ). It is verified in [13] that γ ( hh E ii − Φ ) = 0. And one moreevidence was discovered in the following Theorem 2.6 ([15]) . For all smooth projective varieties, we have ∆( hh E ii − Φ ) = 0 . We will find more evidences in Section 4. By the fact γ ( hh E k ii − Φ k ) = k ( hh E k − ii − Φ k − ) proved in Lemma 6.3 of [13], it is easy to show that for anygiven k ≥ 2, the genus-1 Virasoro conjecture holds if and only if υ ( hh E k ii − Φ k ) = 0for any υ ∈ H ∗ ( X, C ). Remark 2.7. By the above argument, to prove the genus-1 Virasoro conjecture, itis also reduced to verify that for any given k, l ≥ , υ l υ l − · · · υ ( hh E k ii − Φ k ) = 0 holds for any υ , · · · , υ l ∈ H ∗ ( X, C ) . Notice that the computation in Section 3 onlyyields the first derivative of {hh E k ii − Φ k } , hence to derive its l- th derivative, onemay take derivatives of our results (cf. Remark 3.9) in section 3 or by applying thepull-back of Getzler’s genus one relation to quantum powers of Euler vector field. Universal equations for Gromov-Witten invariants. In this subsection,we breifly recall some universal equations for primary Gromov-Witten invariants.Firstly, the string equation on the small phase space shows that Lemma 2.8 ([13]) . hh γ γ α γ β ii = η αβ , (22) hh γ γ α · · · γ α k ii = 0 if k ≥ , (23) hh γ γ α · · · γ α k ii g = 0 if g ≥ and k ≥ . (24)Next, the important universal equation for genus-0 primary Gromov-Witten in-variants is the WDVV equation (10), which is implicitly used for computationthroughout the paper. Its first derivative has the form hh{ υ ◦ υ } υ υ υ ii = hh{ υ ◦ υ } υ υ υ ii + hh{ υ ◦ υ } υ υ υ ii − hh{ υ ◦ υ } υ υ υ ii . (25)It is well known that equation (25) is the topological recursion relation derived fromKeel relation [10] on the moduli space of stable curves M , . And it implies YIJIE LIN Lemma 2.9 ([13]) . For any vector field υ on the small phase space, let υ k be the k th quantum power of υ . Then for any α, β and µ , we have hh υ k γ α γ β γ µ ii = − k − X i =1 hh υ k − i ( γ α ◦ γ β ◦ υ i − ) υγ µ ii + k X i =1 hh ( υ k − i ◦ γ α )( γ β ◦ υ i − ) υγ µ ii . It follows easily from Keel relation on M , that hh{ υ ◦ υ } υ υ υ υ ii = X ρ hh υ υ υ γ ρ ii hh γ ρ υ υ υ ii + X ρ hh υ υ υ γ ρ ii hh γ ρ υ υ υ ii + hh{ υ ◦ υ } υ υ υ υ ii + hh{ υ ◦ υ } υ υ υ υ ii − X ρ hh υ υ υ γ ρ ii hh γ ρ υ υ υ ii − X ρ hh υ υ υ γ ρ ii hh γ ρ υ υ υ ii −hh{ υ ◦ υ } υ υ υ υ ii , (26)which is also derived from the second derivative of the WDVV equation (10). Wewill use the following lemma for the computation in Section 3. Lemma 2.10. For any vector field υ on the small phase space, let υ k be the k th quantum power of υ . Then for any α, β , µ , and σ , we have hh υ k γ α γ β γ µ γ σ ii = − k − X i =1 X ρ hh υ k − i γ σ υγ ρ ii hh γ ρ γ α { γ β ◦ υ i − } γ µ ii − k − X i =1 hh υ k − i { γ α ◦ γ β ◦ υ i − } υγ µ γ σ ii − k − X i =1 X ρ hh υ k − i γ µ υγ ρ ii hh γ ρ γ α { γ β ◦ υ i − } γ σ ii + k − X i =1 X ρ hh υ k − i γ α γ σ γ ρ ii hh γ ρ { γ β ◦ υ i − } υγ µ ii + k − X i =1 X ρ hh υ k − i γ α γ µ γ ρ ii hh γ ρ { γ β ◦ υ i − } υγ σ ii + k X i =1 hh{ υ k − i ◦ γ α }{ γ β ◦ υ i − } υγ µ γ σ ii Proof. As in the proof of Lemma 4.2 in [13], the proof follows by choosing υ = υ k − , υ = υ , υ = γ α , υ = γ β , υ = γ µ , and υ = γ σ in equation (26) and repeatedlyusing the resulting formula. (cid:3) The essential universal equation for genus-1 primary Gromov-Witten invariantsis derived from Getzler’s genus one relation [7]. Adopting the presentation in [13], it has the following form (we call it Getzler’s genus one relation again) G ( υ , υ , υ , υ ) = G ( υ , υ , υ , υ ) + G ( υ , υ , υ , υ ) = 0 , (27)where G ( υ , υ , υ , υ )= X h ∈ S X α,β (cid:26) hh υ h (1) υ h (2) υ h (3) γ α ii hh γ α υ h (4) γ β γ β ii + 124 hh υ h (1) υ h (2) υ h (3) υ h (4) γ α ii hh γ α γ β γ β ii − hh υ h (1) υ h (2) γ α γ β ii hh γ α γ β υ h (3) υ h (4) ii (cid:27) , and G ( υ , υ , υ , υ )= 3 X h ∈ S hh{ υ h (1) ◦ υ h (2) }{ υ h (3) ◦ υ h (4) }ii − X h ∈ S hh{ υ h (1) ◦ υ h (2) ◦ υ h (3) } υ h (4) ii − X h ∈ S X α hh{ υ h (1) ◦ υ h (2) } υ h (3) υ h (4) γ α ii hh γ α ii +2 X h ∈ S X α hh υ h (1) υ h (2) υ h (3) γ α ii hh{ γ α ◦ υ h (4) }ii , for any vector fields υ , υ , υ , υ on the small phase space. Remark 2.11. By string equations (23) and (24) , we have G ( γ , υ , υ , υ ) ≡ and G ( γ , υ , υ , υ ) ≡ for any vector fields υ , υ , υ on the small phase space. For simplicity, we will use the following notational conventions: Repeated indicesare summed over their entire meaningful range unless otherwise indicated.3. Universal equations from Getzler’s genus one relation In this section, we will derive an explicit formula for the first derivative of Φ k , andthen obtain an explicit universal equation, i.e., Theorem 3.8, by computing Getzler’sgenus one relation (27) when one of vector fields is substituted by quantum powerof Euler vector field. With this theorem, we can derive other explicit universalequations for the cases involving more quantum powers of Euler vector field. Theseexplicit universal equations from Getzler’s genus one relation will be called Getzlerequations. To see how much information of Getzler equations can be applied in thegenus-1 Virasoro conjecture, we always reduce higher k -point correlation functionsinvolving some quantum power of Euler vector field to lower l -point correlationfunctions, and use the first derivative of Φ k for possible simplification.3.1. An explicit formula for the first derivative of Φ k . The following lemmais useful for reducing 4-point functions to 3-point functions. Lemma 3.1. For all m ≥ , hh E m γ α γ β γ µ ii 00 YIJIE LIN = − m X i =1 hh G ( E m − i ) { E i − ◦ γ α ◦ γ β } γ µ ii − m X i =1 hh G ( E i − ◦ γ α ◦ γ β ) E m − i γ µ ii + m X i =1 hh G ( E m − i ◦ γ α ) { E i − ◦ γ β } γ µ ii + m X i =1 hh G ( E i − ◦ γ β ) { E m − i ◦ γ α } γ µ ii . Proof. Using Lemma 2.9, equations (16) and (10), we have hh E m γ α γ β γ µ ii = − m − X i =1 hh G ( E m − i ) { E i − ◦ γ α ◦ γ β } γ µ ii − m − X i =1 hh G ( E i − ◦ γ α ◦ γ β ) E m − i γ µ ii + m X i =1 hh G ( E m − i ◦ γ α ) { E i − ◦ γ β } γ µ ii + m X i =1 hh G ( E i − ◦ γ β ) { E m − i ◦ γ α } γ µ ii − m − X i =1 hh E m − { γ α ◦ γ β } G ( γ µ ) ii + m X i =1 hh E m − { γ α ◦ γ β } G ( γ µ ) ii +( b + 1) (cid:26) m − X i =1 hh E m − { γ α ◦ γ β } γ µ ii − m X i =1 hh E m − { γ α ◦ γ β } γ µ ii (cid:27) . The proof follows by using the following equation (29). (cid:3) In particular, it follows from Lemma 3.1 that hh E m γ α γ β γ µ ii γ µ = − m X i =1 G ( E m − i ) ◦ E i − ◦ γ α ◦ γ β − m X i =1 G ( E i − ◦ γ α ◦ γ β ) ◦ E m − i + m X i =1 G ( E m − i ◦ γ α ) ◦ E i − ◦ γ β + m X i =1 G ( E i − ◦ γ β ) ◦ E m − i ◦ γ α . (28)The following lemma is useful for simplification. Lemma 3.2. For any vector field υ i ( ≤ i ≤ ) on the small phase space, we have hh G ( υ ◦ υ ) υ υ ii = hh υ υ { υ ◦ υ }ii − hh υ υ G ( υ ◦ υ ) ii , (29) G ( υ ◦ γ µ ) ◦ γ µ = G ( υ ◦ γ µ ) ◦ γ µ = 12 ∆ ◦ υ , (30) hh G ( υ ◦ υ ◦ υ ◦ γ µ ) G ( γ µ ) υ ii = hh{ G ( υ ◦ γ µ ) ◦ G ( γ µ ) ◦ υ } υ υ ii , (31) hh{ G ( υ ◦ γ µ ) ◦ G ( υ ◦ γ µ ) } υ υ ii = hh{ G ( υ ◦ υ ◦ γ µ ) ◦ G ( γ µ ) } υ υ ii . (32) Proof. As η αβ = 0 requires that b α = 1 − b β , hh G ( υ ◦ υ ) υ υ ii = b α hh υ υ γ α iihh γ α υ υ ii = b α η αβ hh υ υ γ α ii hh γ β υ υ ii = (1 − b β ) η αβ hh υ υ γ α ii hh γ β υ υ ii = (1 − b β ) hh υ υ γ β ii hh γ β υ υ ii . The equation (29) follows. By the same argument, the equation (30) also holds.The equation (31) follows from equations (29) and (30). The last equation can beproved as follows hh{ G ( υ ◦ γ µ ) ◦ G ( υ ◦ γ µ ) } υ υ ii = b α b β hh υ γ µ γ α ii hh υ γ µ γ β ii hh{ γ α ◦ γ β } υ υ ii = b α b β hh υ { υ ◦ γ α } γ β ii hh{ γ α ◦ γ β } υ υ ii = hh{ G ( υ ◦ υ ◦ γ µ ) ◦ G ( γ µ ) } υ υ ii . (cid:3) Now, an explicit formula for the first derivative of Φ k is presented below. Theorem 3.3. For all k ≥ , and any α , γ α Φ k = − k − X i =1 hh G ( E i ) { E k − i − ◦ γ α } γ µ γ µ ii − k − X i =0 ( k − i − hh{ G ( E i ) ◦ E k − i − } ∆ γ α ii + k − X i =0 ( − k + 2 i + 2) hh G (∆ ◦ E k − i − ) E i γ α ii − k − X i =1 i X j =1 hh G ( E k − i − ◦ ∆) G ( E i − j ◦ γ α ) E j − ii − k − X i =1 i X j =1 hh{ G ( E k − i − ) ◦ ∆ } G ( E i − j ◦ γ α ) E j − ii +12 k − X i =1 i hh{ G ( E k − i − ◦ γ µ ) ◦ G ( γ µ ) } E i − γ α ii − k ( k − hh ∆ E k − γ α ii . Proof. It is trivial for k = 0 , 1. It remains to prove this theorem when k ≥ γ α Φ k = − k − X i =0 hh G ( ∇ γ α E i ) E k − i − ∆ ii − k − X i =0 hh E k − i − G ( E i )∆ γ α ii − k − X i =0 hh G ( E i ) {∇ γ α E k − i − } ∆ ii − k − X i =0 hh{ G ( E i ) ◦ E k − i − } γ α γ σ γ σ ii − k − X i =0 b µ b β hh E i γ α γ µ γ β ii hh γ β E k − i − γ µ ii − k − X i =0 b µ b β hh{∇ γ α E i } γ µ γ β ii hh γ β E k − i − γ µ ii +2 k hh E k − γ σ γ σ γ α ii + 2 k hh{∇ γ α E k − } γ σ γ σ ii . (33)By equation (25), we have hh{ G ( E i ) ◦ E k − i − } γ α γ σ γ σ ii = hh{ E k − i − ◦ γ α } γ σ γ σ G ( E i ) ii + hh E k − i − γ σ γ α { G ( E i ) ◦ γ σ }ii 02 YIJIE LIN −hh E k − i − { γ α ◦ γ σ } γ σ G ( E i ) ii . (34)Notice that b µ b β hh E i γ α γ µ γ β ii hh γ β E k − i − γ µ ii = b µ (1 − b β ) hh E i γ α γ µ γ β ii hh γ β E k − i − γ µ ii = hh E i G ( γ µ ) { E k − i − ◦ γ µ } γ α ii − hh E i G ( γ µ ) G ( E k − i − ◦ γ µ ) γ α ii (35)and b µ b β hh{∇ γ α E i } γ µ γ β ii hh γ β E k − i − γ µ ii = [1 − (1 − b µ )] hh G (( ∇ γ α E i ) ◦ γ µ ) E k − i − γ µ ii = hh G (( ∇ γ α E i ) ◦ γ µ ) E k − i − γ µ ii − hh G (( ∇ γ α E i ) ◦ γ µ ) E k − i − G ( γ µ ) ii . (36)The proof follows by firstly substituting equations (34), (35) and (36) into theequality (33), and secondly applying Lemma 3.1 and formula (19) to the resultingequality, and then using Lemma 3.2 for simplification. (cid:3) Therefore, we have an alternative direct proof of Corollary 3.4 ([13]) . For any smooth projective variety X , we have E k Φ m − E m Φ k = ( m − k )Φ k + m − . Proof. We only consider the cases for k + m ≥ 2, since other cases are trivial. UsingTheorem 3.3, Lemma 3.1 and Lemma 3.2, we have24 E m Φ k = k X i =1 m X j =1 hh G ( E i − ) G ( E j − ) { ∆ ◦ E k + m − i − j }ii − b k + m − X i =1 hh G ( E i − )∆ E k + m − i − ii + k − X i =1 k + m − i − X j =1 hh G ( E i ) G (∆ ◦ E j − ) E k + m − i − j − ii − k + m − X i = m k + m − i − X j =1 hh G ( E i ) G (∆ ◦ E j − ) E k + m − i − j − ii − k − X i =1 (2 k + m − i − hh G ( E i )∆ E k + m − i − ii + k + m − X i = m (2 m + k − i − hh G ( E i )∆ E k + m − i − ii +6 k − X i =1 ( k − i ) hh G ( E i − ◦ γ µ ) G ( γ µ ) E k + m − i − ii − k + m − X i = m +1 ( m − i ) hh G ( E i − ◦ γ µ ) G ( γ µ ) E k + m − i − ii − ( k − k + b · δ k ≥ ) hh E m + k − γ µ γ µ ii . where if k ≥ 2, then δ k ≥ = 1, otherwise δ k ≥ = 0. Then it follows that24 E m Φ k − E k Φ m = (cid:26) − k + m − X i =0 hh G ( E i )∆ E k + m − i − ii − ( k + m − hh E m + k − γ µ γ µ ii +6 k + m − X i =1 hh G ( E i − ◦ γ µ ) G ( γ µ ) E k + m − i − ii (cid:27) × ( k − m ) . The proof is completed by equation (21). (cid:3) Getzler equation G ( E k , γ α , γ β , γ σ ) = 0 . We start with the computation ofthe function G . Lemma 3.5. For any α, β, σ , G ( E k , γ α , γ β , γ σ )= − { γ α ◦ γ β ◦ γ σ }hh E k ii + 24 k hh E k − ◦ γ α ◦ γ β ◦ γ σ ii +12 X g ∈ S hh{ E k ◦ γ ς g (1) }{ γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S hh{ E k ◦ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) ii +12 X g ∈ S k X i =1 hh G ( E i − ◦ γ ς g (1) ) ◦ E k − i ◦ γ ς g (2) ◦ γ ς g (3) ii − X g ∈ S k X i =1 hh G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) ◦ E k − i ◦ γ ς g (3) ii where k ≥ and { ς , ς , ς } = { α, β, σ } .Proof. By the definition of G , we have G ( E k , γ α , γ β , γ σ )= 12 X g ∈ S hh{ E k ◦ γ ς g (1) }{ γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S hh{ E k ◦ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) }ii − hh{ γ α ◦ γ β ◦ γ σ } E k ii − X g ∈ S hh{ E k ◦ γ ς g (1) } γ ς g (2) γ ς g (3) γ µ ii hh γ µ ii − X g ∈ S hh E k { γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ µ ii hh γ µ ii +6 X g ∈ S hh E k γ ς g (1) γ ς g (2) γ µ ii hh γ µ ◦ γ ς g (3) ii +12 hh γ α γ β γ σ γ µ ii hh γ µ ◦ E k ii . (37)Firstly, it follows from equations (9) and (19) that hh{ γ α ◦ γ β ◦ γ σ } E k ii = { γ α ◦ γ β ◦ γ σ }hh E k ii − k − X i =0 hh G ( E i ) ◦ E k − i − ◦ γ α ◦ γ β ◦ γ σ ii + k X i =1 hh G ( E k − i ◦ γ α ◦ γ β ◦ γ σ ) ◦ E i − ii − k hh E k − ◦ γ α ◦ γ β ◦ γ σ ii . (38) Using equation (25), we have hh γ α γ β γ σ γ µ ii hh γ µ ◦ E k ii = hh{ E k ◦ γ µ } γ α γ β γ σ ii hh γ µ ii = hh E k γ ς g (1) γ ς g (2) { γ µ ◦ γ ς g (3) }ii hh γ µ ii + hh{ E k ◦ γ ς g (1) } γ ς g (2) γ ς g (3) γ µ ii hh γ µ ii −hh{ γ ς g (1) ◦ γ ς g (3) } E k γ ς g (2) γ µ ii hh γ µ ii for any g ∈ S . Hence it is easy to show that12 hh γ α γ β γ σ γ µ ii hh γ µ ◦ E k ii − X g ∈ S hh{ E k ◦ γ ς g (1) } γ ς g (2) γ ς g (3) γ µ ii hh γ µ ii = 2 X g ∈ S hh E k γ ς g (1) γ ς g (2) γ µ ii hh γ µ ◦ γ ς g (3) ii − X g ∈ S hh E k { γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ µ ii hh γ µ ii . (39)By substituting equations (38) and (39) into equation (37), the proof is completedby using Lemma 3.1. (cid:3) Remark 3.6. Although it is simple to show that there are other equivalent expres-sions for G ( E k , γ α , γ β , γ σ ) by equations (9) and (19) , for example, G ( E k , γ α , γ β , γ σ )= − { γ α ◦ γ β ◦ γ σ }hh E k ii + 24 k hh E k − ◦ γ α ◦ γ β ◦ γ σ ii +12 X g ∈ S { γ ς g (1) ◦ γ ς g (2) }hh E k ◦ γ ς g (3) ii − X g ∈ S γ ς g (1) hh E k ◦ γ ς g (2) ◦ γ ς g (3) ii +72 hh γ α γ β γ σ γ µ ii hh γ µ ◦ E k ii , (40) we adopt the expression of G in Lemma 3.5 which is useful for later applica-tions in Section 3 and 4. Notice that equation (40) can not be used to derive G ( E k , γ α , γ β , E m ) by substituting E m into the position of γ σ directly. By the definition of G , we have for any α, β, σ , G ( E k , γ α , γ β , γ σ )= hh E k γ α γ β γ σ ∆ ii + 12 X g ∈ S hh E k γ ς g (1) γ ς g (2) γ µ ii hh γ µ γ ς g (3) γ ρ γ ρ ii − X g ∈ S hh E k γ ς g (1) γ µ γ ρ ii hh γ µ γ ρ γ ς g (2) γ ς g (3) ii + hh E k γ ρ γ ρ γ µ ii hh γ µ γ α γ β γ σ ii , (41)where k ≥ { ς , ς , ς } = { α, β, σ } . Since the computation is quite involved,we put some intermediate results in Appendix A. With these preparation, we havethe following explicit formula. Lemma 3.7. For any α, β, σ , G ( E k , γ α , γ β , γ σ )= 24 { γ α ◦ γ β ◦ γ σ } Φ k − k hh{ ∆ ◦ E k − } γ α γ β γ σ ii + 12 X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ) γ µ γ µ { E i − ◦ γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ◦ γ ς g (2) ) γ µ γ µ { E i − ◦ γ ς g (3) }ii +2 k X g ∈ S hh G ( E k − ◦ γ ς g (1) ◦ γ µ ) γ ς g (2) γ ς g (3) γ µ ii − k X g ∈ S hh{ E k − ◦ γ ς g (1) } γ µ γ µ { γ ς g (2) ◦ γ ς g (3) }ii +2 X g ∈ S k − X i =1 ( k − i ) hh{ G ( E k − i − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E i − { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k − X i =1 i hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii − X g ∈ S k − X i =1 ( k − i ) hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − k k − X i =1 hh G (∆ ◦ E k − i − ) E i − { γ α ◦ γ β ◦ γ σ }ii +5 k ( k − hh ∆ E k − { γ α ◦ γ β ◦ γ σ }ii , where k ≥ and { ς , ς , ς } = { α, β, σ } .Proof. Firstly, by substituting Lemmas A.5, A.6, A.7, A.8, A.9 into Lemma A.1,and substituting Lemma A.11 into Lemma A.2, and combining with Lemmas A.3and A.4, we obtain an expression for the function G by equation (41). Thensubsituting Lemma A.10 to the resulting expression, and using Theorem 3.3 andLemma 3.2 for simplification, a tedious computation shows that G ( E k , γ α , γ β , γ σ )= 24 { γ α ◦ γ β ◦ γ σ } Φ k − ( b + k ) hh{ ∆ ◦ E k − } γ α γ β γ σ ii − k X i =1 hh{ G ( E k − i ) ◦ E i − } γ µ γ µ { γ α ◦ γ β ◦ γ σ }ii − X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ◦ γ ς g (2) ) γ µ γ µ { E i − ◦ γ ς g (3) }ii + X g ∈ S k X i =1 hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (2) } γ µ γ µ { E i − ◦ γ ς g (3) }ii − X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ E i − ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ µ ii + 12 X g ∈ S k X i =1 hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii 06 YIJIE LIN + 13 X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ E i − ◦ γ ς g (1) ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii − X g ∈ S k − X i =1 hh{ ∆ ◦ E i − } γ ς g (1) γ ς g (2) { G ( E k − i ) ◦ γ ς g (3) }ii − X g ∈ S k − X i =1 hh{ ∆ ◦ E i − }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) G ( E k − i ) ii + 12 X g ∈ S k X i =1 hh{ ∆ ◦ E i − } γ ς g (1) γ ς g (2) G ( E k − i ◦ γ ς g (3) ) ii + k X g ∈ S hh G ( E k − ◦ γ ς g (1) ◦ γ µ ) γ ς g (2) γ ς g (3) γ µ ii − X g ∈ S k X i =1 hh{ G ( E k − i ◦ γ ς g (1) ) ◦ E i − ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii − k X g ∈ S hh{ G ( E k − ◦ γ µ ) ◦ γ ς g (1) } γ ς g (2) γ ς g (3) γ µ ii − k k − X i =1 hh{ G ( E k − i − ◦ γ µ ) ◦ G ( γ µ ) } E i − { γ α ◦ γ β ◦ γ σ }ii +3 X g ∈ S k − X i =1 ( k − i ) hh{ G ( E k − i − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E i − { γ ς g (2) ◦ γ ς g (3) }ii − k X i =1 i − X j =1 hh{ G (∆ ◦ E k − i ) ◦ G ( E j − ) } E i − j − { γ α ◦ γ β ◦ γ σ }ii +2 k X i =1 i − X j =1 hh G ( E k − i ) E i − j − G (∆ ◦ E j − ◦ γ α ◦ γ β ◦ γ σ ) ii + 12 X g ∈ S k X i =1 i − X j =1 hh{ G (∆ ◦ E k − i ) ◦ E j − } G ( E i − j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k − X i =1 i hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii − X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E k − i } G ( E j − ◦ γ ς g (1) ◦ γ ς g (2) ) G ( E i − j − ◦ γ ς g (3) ) ii + X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E k − i ◦ γ ς g (1) } G ( E j − ◦ γ ς g (2) ) G ( E i − j − ◦ γ ς g (3) ) ii − k k − X i =0 hh G (∆ ◦ E k − i − ) E i { γ α ◦ γ β ◦ γ σ }ii +2 k ( k − hh ∆ E k − { γ α ◦ γ β ◦ γ σ }ii . The proof follows from some tedious manipulation again, that is, substituting Lem-mas A.12, A.13, A.15 into the above expression, and using Lemma A.14 to cancelthe redundant terms in the resulting expression, and then using Lemma 3.2 forpossible simplification. (cid:3) The main result of this subsection is the following Theorem 3.8. For any α, β, σ , { γ α ◦ γ β ◦ γ σ } ( hh E k ii − Φ k )= 12 X g ∈ S hh{ E k ◦ γ ς g (1) }{ γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S hh{ E k ◦ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) }ii +12 X g ∈ S k X i =1 hh G ( E i − ◦ γ ς g (1) ) ◦ E k − i ◦ γ ς g (2) ◦ γ ς g (3) ii − X g ∈ S k X i =1 hh G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) ◦ E k − i ◦ γ ς g (3) ii +24 k hh E k − ◦ γ α ◦ γ β ◦ γ σ ii − k hh{ ∆ ◦ E k − } γ α γ β γ σ ii + 12 X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ) γ µ γ µ { E i − ◦ γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ◦ γ ς g (2) ) γ µ γ µ { E i − ◦ γ ς g (3) }ii +2 k X g ∈ S hh G ( E k − ◦ γ ς g (1) ◦ γ µ ) γ ς g (2) γ ς g (3) γ µ ii − k X g ∈ S hh{ E k − ◦ γ ς g (1) } γ µ γ µ { γ ς g (2) ◦ γ ς g (3) }ii +2 X g ∈ S k − X i =1 ( k − i ) hh{ G ( E k − i − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E i − { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k − X i =1 i hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii − X g ∈ S k − X i =1 ( k − i ) hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − k k − X i =1 hh G (∆ ◦ E k − i − ) E i − { γ α ◦ γ β ◦ γ σ }ii +5 k ( k − hh ∆ E k − { γ α ◦ γ β ◦ γ σ }ii , where k ≥ and { ς , ς , ς } = { α, β, σ } .Proof. It follows easily from Lemma 3.5, Lemma 3.7 and equation (27). (cid:3) Remark 3.9. In Lemma 3.5, Lemma 3.7, and Theorem 3.8, one can replace γ α , γ β ,and γ σ by υ , υ , and υ respectively for any vector fields υ , υ , and υ on the small phase space. The same holds for the following Lemmas 3.10, 3.12, 3.14, 3.15, andTheorems 3.13, 3.16. Getzler equation G ( E k , E k , γ α , γ β ) = 0 . In this subsection, the followingexplicit formulas for functions G , G and G are derived from the above subsection. Lemma 3.10. For k , k ≥ , and any α, β , G ( E k , E k , γ α , γ β )= 24 { γ α ◦ γ β }hh E b K ii − X m =1 { E b K − k m ◦ γ α ◦ γ β }hh E k m ii +24 X h ∈ S hh{ E k h (1) ◦ γ α }{ E k h (2) ◦ γ β }ii − X g ∈ S hh{ E b K ◦ γ ς g (1) } γ ς g (2) ii +24 X g ∈ S X m =1 k m X i =1 hh G ( E i − ◦ γ ς g (1) ) ◦ E b K − i ◦ γ ς g (2) ii − X g ∈ S b K X i =1 hh G ( E i − ◦ γ ς g (1) ) ◦ E b K − i ◦ γ ς g (2) ii , where b K = k + k and { ς , ς } = { α, β } .Proof. The proof follows easily from Lemma 3.5 and equations (9) and (19). (cid:3) Remark 3.11. As in Remark 3.6, we also have the following equivalent form G ( E k , E k , γ α , γ β )= 24 { γ α ◦ γ β }hh E b K ii − X m =1 { E b K − k m ◦ γ α ◦ γ β }hh E k m ii +12 X g ∈ S X h ∈ S { E k h (1) ◦ γ ς g (1) }hh E k h (2) ◦ γ ς g (2) ii − X g ∈ S γ ς g (1) hh E b K ◦ γ ς g (2) ii +12 X g ∈ S X m =1 k m X i =1 hh G ( E i − ◦ γ ς g (1) ) ◦ E b K − i ◦ γ ς g (2) ii − X m =1 k m X i =1 hh G ( E i − ◦ γ α ◦ γ β ) ◦ E b K − i ii + 24 b K hh E b K − ◦ γ α ◦ γ β ii , by equations (9) and (19) . Lemma 3.12. For k , k ≥ , and any α, β , G ( E k , E k , γ α , γ β )= − { γ α ◦ γ β } Φ b K + 24 X m =1 { E b K − k m ◦ γ α ◦ γ β } Φ k m − X g ∈ S b K X i =1 hh G ( E i − ◦ γ ς g (1) ) γ µ γ µ { E b K − i ◦ γ ς g (2) }ii + X g ∈ S X m =1 k m X i =1 hh G ( E i − ◦ γ ς g (1) ) γ µ γ µ { E b K − i ◦ γ ς g (2) }ii − X g ∈ S k X i =1 k X j =1 hh{ ∆ ◦ E i + j − } G ( E b K − i − j ◦ γ ς g (1) ) γ ς g (2) ii +6 X g ∈ S k X i =1 k X j =1 hh{ G ( E i + j − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E b K − i − j γ ς g (2) ii − k k hh ∆ E b K − { γ α ◦ γ β }ii , where b K = k + k and { ς , ς } = { α, β } .Proof. By Lemma 3.7, we get the expression of G ( E k , E k , γ α , γ β ) which containsthree terms: hh G ( E b K − ◦ γ µ ) γ α γ β γ µ ii , hh{ E k − ◦ γ α } γ µ γ µ { γ β ◦ E k }ii , and hh{ E k − ◦ γ β } γ µ γ µ { γ α ◦ E k }ii . They are computed as follows. hh G ( E b K − ◦ γ µ ) γ α γ β γ µ ii = hh G ( γ µ ) γ α γ β { E b K − ◦ γ µ }ii = hh E b K − { γ β ◦ γ µ } γ α G ( γ µ ) ii + hh{ E b K − ◦ γ α } γ µ G ( γ µ ) γ β ii −hh E b K − { γ α ◦ γ β } γ µ G ( γ µ ) ii , hh{ E k − ◦ γ α } γ µ γ µ { γ β ◦ E k }ii = hh E k { γ β ◦ γ µ } γ µ { E k − ◦ γ α }ii + hh{ E b K − ◦ γ α } γ µ γ µ γ β ii −hh E k γ β γ µ { E k − ◦ γ α ◦ γ µ }ii , hh{ E k − ◦ γ β } γ µ γ µ { γ α ◦ E k }ii = hh E k − { γ β ◦ γ µ } γ µ { E k ◦ γ α }ii + hh{ E b K − ◦ γ α } γ µ γ µ γ β ii −hh E k − γ β γ µ { E k ◦ γ α ◦ γ µ }ii . The proof is completed by Lemmas 3.1, 3.2 and Theorem 3.3. (cid:3) Theorem 3.13. For k , k ≥ , and any α, β , { γ α ◦ γ β } ( hh E b K ii − Φ b K ) − X m =1 { E b K − k m ◦ γ α ◦ γ β } ( hh E k m ii − Φ k m )= − X h ∈ S hh{ E k h (1) ◦ γ α }{ E k h (2) ◦ γ β }ii + 24 X g ∈ S hh{ E b K ◦ γ ς g (1) } γ ς g (2) ii − X g ∈ S X m =1 k m X i =1 hh G ( E i − ◦ γ ς g (1) ) ◦ E b K − i ◦ γ ς g (2) ii +24 X g ∈ S b K X i =1 hh G ( E i − ◦ γ ς g (1) ) ◦ E b K − i ◦ γ ς g (2) ii + X g ∈ S b K X i =1 hh G ( E i − ◦ γ ς g (1) ) γ µ γ µ { E b K − i ◦ γ ς g (2) }ii − X g ∈ S X m =1 k m X i =1 hh G ( E i − ◦ γ ς g (1) ) γ µ γ µ { E b K − i ◦ γ ς g (2) }ii 00 YIJIE LIN + X g ∈ S k X i =1 k X j =1 hh{ ∆ ◦ E i + j − } G ( E b K − i − j ◦ γ ς g (1) ) γ ς g (2) ii − X g ∈ S k X i =1 k X j =1 hh{ G ( E i + j − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E b K − i − j γ ς g (2) ii +2 k k hh ∆ E b K − { γ α ◦ γ β }ii , where b K = k + k and { ς , ς } = { α, β } .Proof. It follows easily from Lemma 3.10, Lemma 3.12 and equation (27). (cid:3) Getzler equation G ( E k , E k , E k , γ α ) = 0 . In this subsection, we will de-rive the following results from subsection 3.3. Lemma 3.14. For k , k , k ≥ , and any α , G ( E k , E k , E k , γ α )= − γ α hh E e K ii + 24 X i =1 { E k i ◦ γ α }hh E e K − k i ii − X i =1 { E e K − k i ◦ γ α }hh E k i ii , where e K = k + k + k .Proof. It follows easily from Lemma 3.10 and equations (9) and (19). (cid:3) Lemma 3.15. For k , k , k ≥ , and any α , G ( E k , E k , E k , γ α )= 24 γ α Φ e K − X i =1 { E k i ◦ γ α } Φ e K − k i + 24 X i =1 { E e K − k i ◦ γ α } Φ k i , where e K = k + k + k .Proof. It follows from Lemmas 3.12, 3.1, 3.2 and Theorem 3.3. (cid:3) Theorem 3.16. For k , k , k ≥ , and any α , γ α ( hh E e K ii − Φ e K ) = X i =1 { E k i ◦ γ α } ( hh E e K − k i ii − Φ e K − k i ) − X i =1 { E e K − k i ◦ γ α } ( hh E k i ii − Φ k i ) , where e K = k + k + k .Proof. It follows easily from Lemma 3.14, Lemma 3.15 and equation (27). (cid:3) Getzler equation G ( E k , E k , E k , E k ) = 0 . The case in this subsectionhas been studied in [13]. We present alternative symmetric expressions as follows. Lemma 3.17. For k , k , k , k ≥ , G ( E k , E k , E k , E k )= 36 K hh E K − ii − X i =1 E k i hh E K − k i ii + 3 X g ∈ S E k g (1) + k g (2) hh E k g (3) + k g (4) ii , where K = k + k + k + k .Proof. It follows from Lemma 3.14 and Corollary 2.3. (cid:3) Remark 3.18. By Corollary 2.3, Lemma 3.17 is equivalent to Corollary 3.2 of [13] . Lemma 3.19. For k , k , k , k ≥ , G ( E k , E k , E k , E k )= − K Φ K − + 24 X i =1 E k i Φ K − k i − X g ∈ S E k g (1) + k g (2) Φ k g (3) + k g (4) , where K = k + k + k + k .Proof. It follows from Lemma 3.15 and Corollary 3.4. (cid:3) Remark 3.20. It is easy to show that Lemma 3.19 implies the definition of Φ k ( k ≥ ) given in [13] . Theorem 3.21. For k , k , k , k ≥ , K ( hh E K − ii − Φ K − )= 8 X i =1 E k i ( hh E K − k i ii − Φ K − k i ) − X g ∈ S E k g (1) + k g (2) ( hh E k g (3) + k g (4) ii − Φ k g (3) + k g (4) ) , where K = k + k + k + k .Proof. It follows from Lemma 3.17, Lemma 3.19 and equation (27). (cid:3) Application to the genus-1 Virasoro Conjecture In this section, we apply Getzler equations derived in Section 3 to study thegenus-1 Virasoro conjecture. Since Getzler equation G ( E k , E k , E k , E k ) = 0has been analyzed in great detail in [13], we begin with the application of Get-zler equation G ( E k , E k , E k , γ α ) = 0. It provides the following relations among {hh E k ii − Φ k } k ≥ . Theorem 4.1. For k ≥ , and any α , γ α ( hh E k ii − Φ k ) = 12 k ( k − { E k − ◦ γ α } ( hh E ii − Φ ) . Proof. It is trivial for k = 0 , , . For k ≥ 3, let k = k − k = k = 1 in Theorem3.16, we have γ α (cid:0) hh E k ii − Φ k (cid:1) = 2 { E ◦ γ α } (cid:0) hh E k − ii − Φ k − (cid:1) − { E ◦ γ α } (cid:0) hh E k − ii − Φ k − (cid:1) + { E k − ◦ γ α } (cid:0) hh E ii − Φ (cid:1) . (42)Suppose that the theorem holds for k ≤ n ( n ≥ γ α (cid:0) hh E n +1 ii − Φ n +1 (cid:1) = 2 { E ◦ γ α } ( hh E n ii − Φ n ) − { E ◦ γ α } (cid:0) hh E n − ii − Φ n − (cid:1) + { E n − ◦ γ α } (cid:0) hh E ii − Φ (cid:1) = n ( n − { E n − ◦ γ α } (cid:0) hh E ii − Φ (cid:1) + { E n − ◦ γ α } (cid:0) hh E ii − Φ (cid:1) − 12 ( n − n − { E n − ◦ γ α } (cid:0) hh E ii − Φ (cid:1) = 12 n ( n + 1) { E n − ◦ γ α } (cid:0) hh E ii − Φ (cid:1) . The proof is completed by induction on k . (cid:3) Remark 4.2. By Theorem 4.1, we have { γ α ◦ γ β } ( hh E b K ii − Φ b K ) − X m =1 { E b K − k m ◦ γ α ◦ γ β } ( hh E k m ii − Φ k m )= 2 k k b K ( b K − { γ α ◦ γ β } ( hh E b K ii − Φ b K ) which gives an equivalent equation in Theorem 3.13. It follows easily from Theorem 4.1 that Corollary 4.3. For any α , m − km + k γ α ( hh E m + k ii − Φ m + k )= { E k ◦ γ α } ( hh E m ii − Φ m ) − { E m ◦ γ α } ( hh E k ii − Φ k ) . where k + m > . Corollary 4.4. For any k ≥ and m > satisfying m + k ≥ , and any α , { E k ◦ γ α } hh E m ii − Φ m m = m − m + k )( m + k − γ α ( hh E m + k ii − Φ m + k ) . It is easy to show that Theorem 4.1 is equivalent to Theorem 3.16. Due to thefact γ ( hh E k ii − Φ k ) = k ( hh E k − ii − Φ k − ), Corollary 4.3 implies Virasoro typerelation for { Φ k } in Theorem 6.1 of [13] by setting γ α = γ and Corollary 2.3, whileCorollary 4.4 implies Lemma 6.3 in [13] if γ α = γ .Next, we deal with more general case, i.e., Getzler equation G ( E k , E k , γ α , γ β ) =0. It involves more complicated genus-1 data and genus-0 4-point functions. Butthanks to formulas (13) and (14), we may consider the following universal equation G ( E k , E k , γ µ , γ µ ◦ γ α ) = 0 . Actually, we have Theorem 4.5. For k , k ≥ , and any α , { ∆ ◦ γ α } ( hh E k + k ii − Φ k + k )= { ∆ ◦ E k ◦ γ α } ( hh E k ii − Φ k ) + { ∆ ◦ E k ◦ γ α } ( hh E k ii − Φ k ) . Proof. By Lemmas 3.10, 3.12 and equations (13), (14), (30), it is easy to show that G ( E k , E k , γ µ , γ µ ◦ γ α )= 24 { ∆ ◦ γ α }hh E k + k ii − { ∆ ◦ E k ◦ γ α }hh E k ii − { ∆ ◦ E k ◦ γ α }hh E k ii and G ( E k , E k , γ µ , γ µ ◦ γ α )= − { ∆ ◦ γ α } Φ k + k + 24 { ∆ ◦ E k ◦ γ α } Φ k + 24 { ∆ ◦ E k ◦ γ α } Φ k . The proof is completed by equation (27). (cid:3) It is equivalent to the following evidences for the genus-1 Virasoro conjecture. Theorem 4.6. For any smooth projective variety X , we have (∆ ◦ γ α )( hh E k ii − Φ k ) = 0 , for all k ≥ and α ∈ { , , . . . , N } .Proof. Let k = k = 1 in Theorem 4.5, we have(∆ ◦ γ α )( hh E ii − Φ ) = 0 . Set k = k ( k ≥ 2) and k = 1 in Theorem 4.5, we have(∆ ◦ γ α )( hh E k +1 ii − Φ k +1 ) = (∆ ◦ E ◦ γ α )( hh E k ii − Φ k ) . Since α is arbitrary, the proof is completed. (cid:3) By setting k = 2 and γ α = γ , Theorem 4.6 implies Theorem 1.1 of [15]. Hence itprovides more evidences for the genus-1 Virasoro conjecture. Since by the WDVVequation (10), we have∆ ◦ γ α = X µ γ µ ◦ γ µ ◦ γ α = X σ,µ,β hh γ α γ β γ σ ii hh γ σ γ µ γ µ ii γ β = X σ,µ,β hh γ α γ σ γ µ ii hh γ σ γ µ γ β ii γ β . Hence by Theorem 4.6, we have for any k ≥ A A . . . A N A A . . . A N ... ... ... A N A N . . . A NN γ ( hh E k ii − Φ k ) γ ( hh E k ii − Φ k )... γ N ( hh E k ii − Φ k ) = where A α,β = X σ,µ hh γ α γ σ γ µ ii hh γ σ γ µ γ β ii . It is obviously that A α,β = A β,α . If the following symmetric matrix A A . . . A N A A . . . A N ... ... ... A N A N . . . A NN = X σ,µ hh γ γ σ γ µ ii hh γ γ σ γ µ ii ... hh γ N γ σ γ µ ii hh γ σ γ µ γ ii hh γ σ γ µ γ ii ... hh γ σ γ µ γ N ii T is invertible, then γ α ( hh E k ii − Φ k ) = 0 for any α ∈ { , , ··· , N } , which is equivalentto γ α ( hh E k ii − Φ k ) = 0 for any α ∈ { , , · · · , N } , and then the genus-1 Virasoroconjecture holds.The above argument gives an alternative proof of Corollary 4.7 ([4, 13, 16, 18]) . For any compact symplectic manifold with semisim-ple quantum cohomology, the genus-1 Virasoro conjecture holds. Proof. Follow the notation in [16] and restrict everything on the small phase space.Let {E i } be the idempotents which span the space of primary vector fields on thesmall phase space. Recall the following formulas derived in [16]: E i ◦ E j = δ ij E i ; γ α = N X i =1 ψ iα E i √ g i ; ∆ = N X i =1 g i E i , where g α = kE α k and ( ψ αβ ) is a invertible matrix (see [16] for more detail). Hence,we have ∆ ◦ γ α = N X i =1 g − i ψ iα E i . By Theorem 4.6, we have for any k ≥ g − ψ g − ψ . . . g − N ψ N g − ψ g − ψ . . . g − N ψ N ... ... ... g − ψ N g − ψ N . . . g − N ψ NN E ( hh E k ii − Φ k ) E ( hh E k ii − Φ k )... E N ( hh E k ii − Φ k ) = The proof is completed due to the fact that the matrix ( ψ αβ ) is invertible. (cid:3) Finally, we obtain one new relation from Getzler equation G ( E k , γ α , γ β , γ σ ) = 0.The equation is G ( E k , γ α , γ α ◦ γ β ◦ γ µ , γ β ) = 0 since in this case, using equations(13), (14) and (30), we have G ( E k , γ α , γ α ◦ γ β ◦ γ µ , γ β ) = − { ∆ ◦ γ µ }hh E k ii + 24 k hh ∆ ◦ E k − ◦ γ µ ii (43)by Lemma 3.5. Actually, we have Theorem 4.8. For all k ≥ and any µ , hh ∆ ◦ E k − ◦ γ µ ii = 5 hh{ ∆ ◦ E k − } γ α γ α { ∆ ◦ γ µ }ii +2 hh{ ∆ ◦ E k − } γ α { γ α ◦ γ µ } ∆ ii − hh{ ∆ ◦ E k − } G ( γ α ◦ γ β ◦ γ µ ) γ α γ β ii − hh{ ∆ ◦ E k − ◦ γ µ } G ( γ α ◦ γ β ) γ α γ β ii − k − X i =1 hh{ G (∆ ◦ E i − ◦ γ α ) ◦ G ( γ α ) } E k − i − { ∆ ◦ γ µ }ii +4 k − X i =1 hh G (∆ ◦ E i − ) E k − i − { ∆ ◦ γ µ }ii − k − X i =1 hh G (∆ ◦ E i − ◦ γ µ ) E k − i − ∆ ii +( k − hh ∆ E k − γ µ ii . Proof. By Lemma 3.7, and using formulas (13), (14), (25) and Lemmas 3.1, 3.2, itcan be verified that G ( E k , γ α , γ α ◦ γ β ◦ γ µ , γ β ) = 24 { ∆ ◦ γ µ } Φ k − k hh{ ∆ ◦ E k − } γ α γ α { ∆ ◦ γ µ }ii − k hh{ ∆ ◦ E k − } γ α { γ α ◦ γ µ } ∆ ii +6 k hh{ ∆ ◦ E k − } G ( γ α ◦ γ β ◦ γ µ ) γ α γ β ii +6 k hh{ ∆ ◦ E k − ◦ γ µ } G ( γ α ◦ γ β ) γ α γ β ii +6 k k − X i =1 hh{ G (∆ ◦ E i − ◦ γ α ) ◦ G ( γ α ) } E k − i − { ∆ ◦ γ µ }ii − k k − X i =1 hh G (∆ ◦ E i − ) E k − i − { ∆ ◦ γ µ }ii +3 k k − X i =1 hh G (∆ ◦ E i − ◦ γ µ ) E k − i − ∆ ii − k ( k − hh ∆ E k − γ µ ii . (44)By Theorem 4.6, we have { ∆ ◦ γ µ } ( hh E k ii − Φ k ) = 0 . Together with equations (43), (44) and (27), the proof is completed. (cid:3) In particular, we have Corollary 4.9. hh ∆ ii = 724 hh ∆ γ µ γ µ ∆ ii − hh ∆ G ( γ α ◦ γ β ) γ α γ β ii . Remark 4.10. Theorem 4.8 can be also obtained from G ( E k , γ α , γ α , ∆ ◦ γ µ ) + G ( E k , γ α , γ α ◦ γ µ , ∆) = 0 . Appendix A.In this appendix, we present the following results which are used in Lemma 3.7.Notice that { α, β, σ } = { ς , ς , ς } is implicit below. We start with the computationof each term on the right-hand side of equation (41) as follows. Lemma A.1. For any α, β, σ , hh E k γ α γ β γ σ ∆ ii = − X g ∈ S k − X i =1 hh{ G ( E k − i ) ◦ ∆ } γ ς g (1) γ ς g (2) { E i − ◦ γ ς g (3) }ii + 16 X g ∈ S k X i =1 hh{ G (∆ ◦ E k − i ) } γ ς g (1) γ ς g (2) { E i − ◦ γ ς g (3) }ii − b X g ∈ S hh ∆ γ ς g (1) γ ς g (2) { E k − ◦ γ ς g (3) }ii − X g ∈ S k X i =1 hh{ ∆ ◦ E k − i } γ ς g (1) γ ς g (2) { E i − ◦ γ ς g (3) }ii 06 YIJIE LIN − X g ∈ S k − X i =1 hh G ( E k − i )∆ γ ς g (1) { E i − ◦ γ ς g (2) ◦ γ ς g (3) }ii + 12 X g ∈ S k − X i =1 hh{ ∆ ◦ E i − } γ ς g (1) γ ς g (2) G ( E k − i ◦ γ ς g (3) ) ii + 13 X g ∈ S hh{ ∆ ◦ E k − } γ ς g (1) γ ς g (2) G ( γ ς g (3) ) ii + 16 X g ∈ S hh ∆ γ ς g (1) { E k − ◦ γ ς g (2) } G ( γ ς g (3) ) ii − X g ∈ S k − X i =1 hh{ ∆ ◦ E i − } γ ς g (1) γ ς g (2) { G ( E k − i ) ◦ γ ς g (3) }ii + 16 X g ∈ S k X i =1 i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E j − } G ( E i − j − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii + 16 b X g ∈ S k − X i =1 hh{ ∆ ◦ E k − i − } G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii + 12 X g ∈ S k X i =1 i − X j =1 hh E k − i G ( E j − ◦ γ ς g (1) ◦ γ ς g (2) ) G (∆ ◦ E i − j − ◦ γ ς g (3) ) ii − X g ∈ S k − X i =1 hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (1) ◦ γ ς g (2) ) G ( γ ς g (3) ) ii − X g ∈ S k X i =1 i − X j =1 hh{ G (∆ ◦ E k − i ) ◦ E j − } G ( E i − j − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii − k − X i =1 k − i X j =1 hh{ G ( E k − i − j ) ◦ E i + j − } ∆ { γ α ◦ γ β ◦ γ σ }ii − 16 ( b + 2) X g ∈ S k − X i =1 k − i X j =1 hh{ ∆ ◦ E k − j − } G ( E j − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii + 16 ( b + 4) X g ∈ S k − X i =1 k − i X j =1 hh{ ∆ ◦ E k − j − } G ( E j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E j − } G ( E i − j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − b X g ∈ S k − X i =1 hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 i − X j =1 hh{ E k − i ◦ γ ς g (1) } G (∆ ◦ E j − ◦ γ ς g (2) ) G ( E i − j − ◦ γ ς g (3) ) ii + 16 X g ∈ S k − X i =1 hh{ E i − ◦ γ ς g (1) } G (∆ ◦ E k − i − ◦ γ ς g (2) ) G ( γ ς g (3) ) ii + 23 X g ∈ S k X i =1 i − X j =1 hh{ G (∆ ◦ E k − i ) ◦ E j − } G ( E i − j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k − X i =1 hh{ G (∆ ◦ E i − ) ◦ E k − i − }{ γ ς g (1) ◦ γ ς g (2) } G ( γ ς g (3) ) ii + 16 X g ∈ S k − X i =1 hh{ ∆ ◦ E k − i − ◦ γ ς g (1) } G ( E i − ◦ γ ς g (2) ) G ( γ ς g (3) ) ii − b X g ∈ S k − X i =1 k − i X j =1 hh E k − i − j G (∆ ◦ E i + j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii + 13 X g ∈ S k X i =1 i − X j =1 hh{ G ( E k − i ) ◦ E j − } G (∆ ◦ E i − j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − b X g ∈ S k − X i =1 hh E k − i − G (∆ ◦ E i − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − k − X i =1 i − X j =1 hh{ G (∆ ◦ E i − j − ) ◦ G ( E k − i ) } E j − { γ α ◦ γ β ◦ γ σ }ii + 16 b X g ∈ S k − X i =1 k − i X j =1 hh E k − i − j G (∆ ◦ γ ς g (1) ◦ γ ς g (2) ◦ E i + j − ) γ ς g (3) ii , where k ≥ and { ς , ς , ς } = { α, β, σ } .Proof. By Lemma 2.10, we have hh E k γ α γ β γ σ ∆ ii = − k − X i =1 hh EE k − i ∆ γ ρ ii hh γ ρ γ α { γ β ◦ E i − } γ σ ii − k − X i =1 hh EE k − i { γ α ◦ γ β ◦ E i − } γ σ ∆ ii − k − X i =1 hh EE k − i γ σ γ ρ ii hh γ ρ γ α { γ β ◦ E i − } ∆ ii + k − X i =1 hh E k − i γ α ∆ γ ρ ii hh γ ρ { γ β ◦ E i − } Eγ σ ii + k − X i =1 hh E k − i γ α γ σ γ ρ ii hh γ ρ { γ β ◦ E i − } E ∆ ii + k X i =1 hh{ E k − i ◦ γ α }{ γ β ◦ E i − } Eγ σ ∆ ii . We compute each term of right-hand side of the above equation as follows. Byequation (17), we have hh EE k − i ∆ γ ρ ii hh γ ρ γ α { γ β ◦ E i − } γ σ ii = hh{ G ( E k − i ) ◦ ∆ } γ α γ σ { E i − ◦ γ β }ii + hh{ G (∆) ◦ E k − i } γ α γ σ { E i − ◦ γ β }ii −hh G (∆ ◦ E k − i ) γ α γ σ { E i − ◦ γ β }ii − b hh{ ∆ ◦ E k − i } γ α γ σ { E i − ◦ γ β }ii , hh EE k − i γ σ γ ρ ii hh γ ρ γ α { γ β ◦ E i − } ∆ ii = hh{ G ( E k − i ) ◦ γ σ } γ α { E i − ◦ γ β } ∆ ii + hh{ E k − i ◦ G ( γ σ ) } γ α { E i − ◦ γ β } ∆ ii −hh G ( E k − i ◦ γ σ ) γ α { E i − ◦ γ β } ∆ ii − b hh{ E k − i ◦ γ σ } γ α { E i − ◦ γ β } ∆ ii , hh E k − i γ α ∆ γ ρ ii hh γ ρ { γ β ◦ E i − } Eγ σ ii = hh E k − i ∆ γ α { G ( E i − ◦ γ β ) ◦ γ σ }ii + hh E k − i ∆ γ α { G ( γ σ ) ◦ E i − ◦ γ β }ii −hh E k − i ∆ γ α G ( E i − ◦ γ β ◦ γ σ ) ii − b hh E k − i ∆ γ α { E i − ◦ γ β ◦ γ σ }ii , hh E k − i γ α γ σ γ ρ ii hh γ ρ { γ β ◦ E i − } E ∆ ii = hh E k − i γ α γ σ { G ( E i − ◦ γ β ) ◦ ∆ }ii + hh E k − i γ α γ σ { G (∆) ◦ E i − ◦ γ β }ii −hh E k − i γ α γ σ G ( E i − ◦ γ β ◦ ∆) ii − b hh E k − i γ α γ σ { E i − ◦ γ β ◦ ∆ }ii . By equation (16), we have hh EE k − i { γ α ◦ γ β ◦ E i − } γ σ ∆ ii = hh G ( E k − i )∆ γ σ { E i − ◦ γ α ◦ γ β }ii + hh E k − i ∆ γ σ G ( E i − ◦ γ α ◦ γ β ) ii + hh E k − i ∆ { E i − ◦ γ α ◦ γ β } G ( γ σ ) ii + hh E k − i γ σ { E i − ◦ γ α ◦ γ β } G (∆) ii − b + 1) hh E k − i γ σ { E i − ◦ γ α ◦ γ β } ∆ ii and hh{ E k − i ◦ γ α }{ γ β ◦ E i − } Eγ σ ∆ ii = hh G ( E k − i ◦ γ α ) { γ β ◦ E i − } γ σ ∆ ii + hh{ E k − i ◦ γ α } G ( γ β ◦ E i − ) γ σ ∆ ii + hh{ E k − i ◦ γ α }{ γ β ◦ E i − } G ( γ σ )∆ ii + hh{ E k − i ◦ γ α }{ γ β ◦ E i − } γ σ G (∆) ii − b + 1) hh{ E k − i ◦ γ α }{ γ β ◦ E i − } γ σ ∆ ii . Next, using equation (25) to change the type of the following 4-point functions hh{ G ( E k − i ) ◦ γ σ } γ α { E i − ◦ γ β } ∆ ii = hh{ ∆ ◦ E i − } γ α γ β { G ( E k − i ) ◦ γ σ }ii + hh E i − ∆ { γ α ◦ γ β }{ G ( E k − i ) ◦ γ σ }ii −hh E i − γ β { ∆ ◦ γ α }{ G ( E k − i ) ◦ γ σ }ii , hh{ G ( γ σ ) ◦ E k − i } γ α { E i − ◦ γ β } ∆ ii = hh{ ∆ ◦ E k − i }{ E i − ◦ γ β } γ α G ( γ σ ) ii + hh E k − i ∆ { E i − ◦ γ β }{ γ α ◦ G ( γ σ ) }ii −hh E k − i { E i − ◦ γ β }{ ∆ ◦ γ α } G ( γ σ ) ii , hh G ( E k − i ◦ γ σ ) γ α { E i − ◦ γ β } ∆ ii = hh{ ∆ ◦ E i − } γ α γ β G ( E k − i ◦ γ σ ) ii + hh E i − ∆ { γ α ◦ γ β } G ( E k − i ◦ γ σ ) ii −hh E i − γ β { ∆ ◦ γ α } G ( E k − i ◦ γ σ ) ii , hh{ E k − i ◦ γ σ } γ α { E i − ◦ γ β } ∆ ii = hh{ ∆ ◦ E k − i }{ E i − ◦ γ β } γ α γ σ ii + hh E k − i ∆ { E i − ◦ γ β }{ γ α ◦ γ σ }ii −hh E k − i { E i − ◦ γ β }{ ∆ ◦ γ α } γ σ ii , hh G ( E k − i ◦ γ α ) { γ β ◦ E i − } γ σ ∆ ii = hh{ ∆ ◦ E i − } γ β γ σ G ( E k − i ◦ γ α ) ii + hh E i − ∆ { γ β ◦ γ σ } G ( E k − i ◦ γ α ) ii −hh E i − { ∆ ◦ γ σ } γ β G ( E k − i ◦ γ α ) ii , hh{ E k − i ◦ γ α } G ( γ β ◦ E i − ) γ σ ∆ ii = hh{ ∆ ◦ E k − i } γ α γ σ G ( E i − ◦ γ β ) ii + hh E k − i ∆ { γ α ◦ γ σ } G ( E i − ◦ γ β ) ii −hh E k − i { ∆ ◦ γ σ } γ α G ( E k − i ◦ γ β ) ii , hh{ E k − i ◦ γ α }{ γ β ◦ E i − } G ( γ σ )∆ ii = hh{ ∆ ◦ E k − i } γ α { γ β ◦ E i − } G ( γ σ ) ii + hh E k − i ∆ { γ α ◦ γ β ◦ E i − } G ( γ σ ) ii −hh E k − i γ α { ∆ ◦ γ β ◦ E i − } G ( γ σ ) ii , hh{ E k − i ◦ γ α }{ γ β ◦ E i − } γ σ G (∆) ii = hh{ G (∆) ◦ E k − i } γ α { γ β ◦ E i − } γ σ ii + hh E k − i γ σ { γ α ◦ γ β ◦ E i − } G (∆) ii −hh E k − i γ α γ σ { G (∆) ◦ γ β ◦ E i − }ii , hh{ E k − i ◦ γ α }{ γ β ◦ E i − } γ σ ∆ ii = hh{ ∆ ◦ E k − i } γ α { γ β ◦ E i − } γ σ ii + hh E k − i γ σ { γ α ◦ γ β ◦ E i − } ∆ ii −hh E k − i γ α γ σ { ∆ ◦ γ β ◦ E i − }ii . The proof is completed by a tedious computation, i.e., by firstly collecting all theabove results and using Lemma 3.1 to reduce 4-point functions to 3-point func-tions, and secondly using Lemma 3.2 to simplify the resulting expression, and thensymmetrizing the result. (cid:3) Lemma A.2. For any α, β, σ , X g ∈ S hh E k γ ς g (1) γ ς g (2) γ µ ii hh γ µ γ ς g (3) γ ρ γ ρ ii = − X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (2) } γ µ γ µ { E i − ◦ γ ς g (3) }ii − X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ◦ γ ς g (2) ) γ µ γ µ { E i − ◦ γ ς g (3) }ii + X g ∈ S k X i =1 hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (2) } γ µ γ µ { E i − ◦ γ ς g (3) }ii − X g ∈ S k X i =1 i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E j − } G ( E i − j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 i − X j =1 hh{ G ( E k − i ) ◦ E j − } G (∆ ◦ E i − j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii 00 YIJIE LIN +3 k X i =1 ( i − hh{ G ( E k − i ) ◦ E i − } ∆ { γ α ◦ γ β ◦ γ σ }ii − X g ∈ S k X i =1 k − i X j =1 hh{ ∆ ◦ E j − } G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) G ( E k − i − j ◦ γ ς g (3) ) ii − X g ∈ S k X i =1 k − i X j =1 hh E k − i − j G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) G (∆ ◦ E j − ◦ γ ς g (3) ) ii + 12 X g ∈ S k X i =1 ( k − i ) hh{ ∆ ◦ E k − i − } G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii + X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E j − ◦ γ ς g (1) } G ( E k − i ◦ γ ς g (2) ) G ( E i − j − ◦ γ ς g (3) ) ii + X g ∈ S k X i =1 i − X j =1 hh{ E i − j − ◦ γ ς g (1) } G ( E k − i ◦ γ ς g (2) ) G (∆ ◦ E j − ◦ γ ς g (3) ) ii − X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E i − } G ( E k − i ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii , where k ≥ and { ς , ς , ς } = { α, β, σ } .Proof. It follows from equation (28) that hh E k γ ς g (1) γ ς g (2) γ µ ii hh γ µ γ ς g (3) γ ρ γ ρ ii = − k X i =1 hh{ G ( E k − i ) ◦ E i − ◦ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ ρ γ ρ ii − k X i =1 hh{ G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) ◦ E k − i } γ ς g (3) γ ρ γ ρ ii + k X i =1 hh{ G ( E k − i ◦ γ ς g (1) ) ◦ E i − ◦ γ ς g (2) } γ ς g (3) γ ρ γ ρ ii + k X i =1 hh{ G ( E i − ◦ γ ς g (2) ) ◦ E k − i ◦ γ ς g (1) } γ ς g (3) γ ρ γ ρ ii . Using equation (25), we have hh{ G ( E k − i ) ◦ E i − ◦ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ ρ γ ρ ii = hh{ G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (2) } γ ρ γ ρ { E i − ◦ γ ς g (3) }ii + hh E i − γ ς g (3) γ ρ { G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (2) ◦ γ ρ }ii −hh E i − γ ρ { G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (2) }{ γ ς g (3) ◦ γ ρ }ii , hh{ G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) ◦ E k − i } γ ς g (3) γ ρ γ ρ ii = hh G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ρ γ ρ { E k − i ◦ γ ς g (3) ii + hh E k − i γ ς g (3) γ ρ { G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) ◦ γ ρ }ii −hh E k − i γ ρ { γ ς g (3) ◦ γ ρ } G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) ii , hh{ G ( E k − i ◦ γ ς g (1) ) ◦ E i − ◦ γ ς g (2) } γ ς g (3) γ ρ γ ρ ii = hh E i − γ ς g (3) γ ρ { G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (2) ◦ γ ρ }ii + hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (2) } γ ρ γ ρ { E i − ◦ γ ς g (3) }ii −hh E i − γ ρ { γ ρ ◦ γ ς g (3) }{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (2) }ii , and hh{ G ( E i − ◦ γ ς g (2) ) ◦ E k − i ◦ γ ς g (1) } γ ς g (3) γ ρ γ ρ ii = hh E k − i γ ς g (3) γ ρ { G ( E i − ◦ γ ς g (2) ) ◦ γ ς g (1) ◦ γ ρ }ii + hh{ G ( E i − ◦ γ ς g (2) ) ◦ γ ς g (1) } γ ρ γ ρ { E k − i ◦ γ ς g (3) }ii −hh E k − i γ ρ { γ ρ ◦ γ ς g (3) }{ G ( E i − ◦ γ ς g (2) ) ◦ γ ς g (1) }ii . The remainder of the argument is analogous to that in Lemma A.1. (cid:3) Lemma A.3. For any α, β, σ , X g ∈ S hh E k γ ς g (1) γ µ γ ρ ii hh γ µ γ ρ γ ς g (2) γ ς g (3) ii = − X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ γ ς g (1) ◦ γ µ } γ ς g (2) γ ς g (3) { E i − ◦ γ µ }ii − X g ∈ S k X i =1 hh G ( E i − ◦ γ ς g (1) ◦ γ µ ) γ ς g (2) γ ς g (3) { E k − i ◦ γ µ }ii + X g ∈ S k X i =1 hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ µ } γ ς g (2) γ ς g (3) { E i − ◦ γ µ }ii + X g ∈ S k X i =1 hh{ G ( E i − ◦ γ µ ) ◦ γ ς g (1) } γ ς g (2) γ ς g (3) { E k − i ◦ γ µ }ii − k X i =1 k − i X j =1 hh G ( E i + j − ◦ γ α ◦ γ β ◦ γ σ ◦ γ µ ) G ( γ µ ) E k − i − j ii +3 X g ∈ S k X i =1 k − i X j =1 hh{ G ( E k − j − ◦ γ ς g (1) ◦ γ ς g (2) ◦ γ µ ) ◦ G ( γ µ ) } E j − γ ς g (3) ii − X g ∈ S k X i =1 k − i X j =1 hh{ G ( E k − j − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E j − { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 k − i X j =1 hh{ G ( E i + j − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E k − i − j { γ ς g (2) ◦ γ ς g (3) }ii +6 k X i =1 k − i X j =1 hh{ G ( E k − j − ◦ γ µ ) ◦ G ( γ µ ) } E j − { γ α ◦ γ β ◦ γ σ }ii , where k ≥ and { ς , ς , ς } = { α, β, σ } . Proof. The proof is completed by the same argument in that of Lemma A.2. (cid:3) Lemma A.4. For k ≥ , and any α, β, σ , hh E k γ ρ γ ρ γ µ ii hh γ µ γ α γ β γ σ ii = − k X i =1 hh{ G ( E k − i ) ◦ ∆ ◦ E i − } γ α γ β γ σ ii − k X i =1 hh{ G (∆ ◦ E i − ) ◦ E k − i } γ α γ β γ σ ii + k X i =1 hh{ ∆ ◦ E k − } γ α γ β γ σ ii . Proof. It follows from equations (28) and (30). (cid:3) By equation (41) and Lemmas A.1, A.2, A.3, A.4, one can obtain an expressionfor G ( E k , γ α , γ β , γ σ ). To simplify this expression, we have to reduce the numberof 4-point functions. In order to cancel redundant 4-point functions, the strategyis to repeatedly use equation (25) to obtain some common 4-point functions, andthen use Lemma 3.1 and Lemma 3.2 for possible reduction. The following resultsare used to maximally reduce the number of 4-point functions. Lemma A.5. hh{ G ( E k − i ) ◦ ∆ } γ ς g (1) γ ς g (2) { E i − ◦ γ ς g (3) }ii = hh{ G ( E k − i ) ◦ ∆ ◦ E i − } γ α γ β γ σ ii − i − X j =1 hh{ G ( E k − i ) ◦ E i − j − } ∆ G ( E j − ◦ γ α ◦ γ β ◦ γ σ ) ii + i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E j − } G ( E i − j − ◦ γ ς g (2) ◦ γ ς g (3) ) γ ς g (1) ii + i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E i − j − } G ( E j − ◦ γ ς g (1) ◦ γ ς g (3) ) γ ς g (2) ii − i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E j − } G ( E i − j − ◦ γ ς g (3) ) { γ ς g (1) ◦ γ ς g (2) }ii . Proof. By equation (25), we have hh{ G ( E k − i ) ◦ ∆ } γ ς g (1) γ ς g (2) { E i − ◦ γ ς g (3) }ii = hh{ G ( E k − i ) ◦ ∆ ◦ E i − } γ α γ β γ σ ii + hh E i − { γ ς g (2) ◦ γ ς g (3) } γ ς g (1) { G ( E k − i ) ◦ ∆ }ii −hh E i − γ ς g (1) γ ς g (3) { G ( E k − i ) ◦ ∆ ◦ γ ς g (2) }ii . The proof is completed by using Lemma 3.1. (cid:3) By the same argument as in Lemma A.5, we have the following results, i.e.,Lemmas A.6, A.7, A.8, A.9. Lemma A.6. hh G (∆ ◦ E k − i ) γ ς g (1) γ ς g (2) { E i − ◦ γ ς g (3) }ii = hh{ G (∆ ◦ E k − i ) ◦ E i − } γ α γ β γ σ ii − i − X j =1 hh G (∆ ◦ E k − i ) E i − j − G ( E j − ◦ γ α ◦ γ β ◦ γ σ ) ii + i − X j =1 hh{ G (∆ ◦ E k − i ) ◦ E j − } G ( E i − j − ◦ γ ς g (2) ◦ γ ς g (3) ) γ ς g (1) ii + i − X j =1 hh{ G (∆ ◦ E k − i ) ◦ E i − j − } G ( E j − ◦ γ ς g (1) ◦ γ ς g (3) ) γ ς g (2) ii − i − X j =1 hh{ G (∆ ◦ E k − i ) ◦ E j − } G ( E i − j − ◦ γ ς g (3) ) { γ ς g (1) ◦ γ ς g (2) }ii . Lemma A.7. hh{ ∆ ◦ E k − i } γ ς g (1) γ ς g (2) { E i − ◦ γ ς g (3) }ii = hh{ ∆ ◦ E k − } γ α γ β γ σ ii − i − X j =1 hh ∆ E k − j − G ( E j − ◦ γ α ◦ γ β ◦ γ σ ) ii + i − X j =1 hh{ ∆ ◦ E k − i + j − } G ( E i − j − ◦ γ ς g (2) ◦ γ ς g (3) ) γ ς g (1) ii + i − X j =1 hh{ ∆ ◦ E k − j − } G ( E j − ◦ γ ς g (1) ◦ γ ς g (3) ) γ ς g (2) ii − i − X j =1 hh{ ∆ ◦ E k − i + j − } G ( E i − j − ◦ γ ς g (3) ) { γ ς g (1) ◦ γ ς g (2) }ii . Lemma A.8. hh G ( E k − i )∆ γ ς g (1) { E i − ◦ γ ς g (2) ◦ γ ς g (3) }ii = hh{ ∆ ◦ E i − }{ γ ς g (2) ◦ γ ς g (3) } γ ς g (1) G ( E k − i ) ii + i − X j =1 hh{ G (∆ ◦ E i − j − ) ◦ G ( E k − i ) } E j − { γ α ◦ γ β ◦ γ σ }ii + i − X j =1 hh{ G ( E k − i ) ◦ E i − j − } ∆ G ( E j − ◦ γ α ◦ γ β ◦ γ σ ) ii − i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E j − } G ( E i − j − ◦ γ ς g (2) ◦ γ ς g (3) ) γ ς g (1) ii − i − X j =1 hh{ G ( E k − i ) ◦ E i − j − } G (∆ ◦ E j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii . Lemma A.9. hh{ E k − ◦ γ ς g (2) } ∆ γ ς g (1) G ( γ ς g (3) ) ii = hh{ ∆ ◦ E k − } γ ς g (1) γ ς g (2) G ( γ ς g (3) ) ii 04 YIJIE LIN + k − X i =1 hh{ G (∆ ◦ E k − i − ) ◦ E i − }{ γ ς g (1) ◦ γ ς g (2) } G ( γ ς g (3) ) ii + k − X i =1 hh{ ∆ ◦ E k − i − } G ( E i − ◦ γ ς g (1) ◦ γ ς g (2) ) G ( γ ς g (3) ) ii − k − X i =1 hh{ ∆ ◦ E i − } G ( E k − i − ◦ γ ς g (2) ) { γ ς g (1) ◦ G ( γ ς g (3) ) }ii − k − X i =1 hh E k − i − G (∆ ◦ E i − ◦ γ ς g (1) ) { γ ς g (2) ◦ G ( γ ς g (3) ) }ii . Next, the following two lemmas produce the 4-point function appearing in { γ α ◦ γ β ◦ γ σ } Φ k which is used for simplification. Lemma A.10. k X i =1 hh{ G ( E k − i ) ◦ E i − ◦ ∆ } γ α γ β γ σ ii = − k − X i =1 hh{ E i − ◦ γ α ◦ γ β ◦ γ σ } γ µ γ µ G ( E k − i ) ii + 13 X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ E i − ◦ γ ς g (1) ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii + k X i =1 hh{ G ( E k − i ) ◦ E i − } γ µ γ µ { γ α ◦ γ β ◦ γ σ }ii − k X i =1 i − X j =1 hh G ( E k − i ) E i − j − G (∆ ◦ E j − ◦ γ α ◦ γ β ◦ γ σ ) ii − X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) { E i − ◦ γ µ }ii − k X i =1 i − X j =1 hh{ G ( E k − i ) ◦ E i − j − } ∆ G ( E j − ◦ γ α ◦ γ β ◦ γ σ ) ii + k X i =1 ( i − hh{ G ( E k − i ) ◦ E i − } ∆ { γ α ◦ γ β ◦ γ σ }ii . Proof. By equation (25), we have hh{ G ( E k − i ) ◦ E i − ◦ ∆ } γ α γ β γ σ ii = hh{ G ( E k − i ) ◦ E i − ◦ γ µ ◦ γ µ } γ α γ β γ σ ii = hh{ G ( E k − i ) ◦ E i − ◦ γ µ } γ α γ β { γ µ ◦ γ σ }ii + hh{ G ( E k − i ) ◦ E i − ◦ γ α ◦ γ µ } γ β γ σ γ µ ii −hh{ G ( E k − i ) ◦ E i − ◦ γ µ } γ β { γ α ◦ γ σ } γ µ ii = hh{ G ( E k − i ) ◦ E i − ◦ γ σ ◦ γ µ } γ α γ β γ µ ii + hh{ G ( E k − i ) ◦ E i − ◦ γ α ◦ γ µ } γ β γ σ γ µ ii −hh{ G ( E k − i ) ◦ E i − ◦ γ µ } γ β { γ α ◦ γ σ } γ µ ii . Using equation (25) again, we have hh{ G ( E k − i ) ◦ E i − ◦ γ µ } γ β { γ α ◦ γ σ } γ µ ii = hh{ G ( E k − i ) ◦ E i − }{ γ β ◦ γ µ }{ γ α ◦ γ σ } γ µ ii + hh{ G ( E k − i ) ◦ E i − ◦ γ α ◦ γ σ } γ µ γ µ γ β ii −hh{ G ( E k − i ) ◦ E i − } γ µ γ µ { γ α ◦ γ β ◦ γ σ }ii , hh{ G ( E k − i ) ◦ E i − }{ γ β ◦ γ µ }{ γ α ◦ γ σ } γ µ ii = hh{ E i − ◦ γ µ }{ γ β ◦ γ µ }{ γ α ◦ γ σ } G ( E k − i ) ii + hh E i − γ µ { γ β ◦ γ µ }{ G ( E k − i ) ◦ γ α ◦ γ σ }ii −hh E i − { γ β ◦ γ µ }{ γ α ◦ γ σ ◦ γ µ } G ( E k − i ) ii , hh{ G ( E k − i ) ◦ E i − ◦ γ α ◦ γ σ } γ µ γ µ γ β ii = hh{ E i − ◦ γ α ◦ γ σ } γ β γ µ { G ( E k − i ) ◦ γ µ }ii + hh{ E i − ◦ γ α ◦ γ β ◦ γ σ } γ µ γ µ G ( E k − i ) ii −hh{ E i − ◦ γ α ◦ γ σ } G ( E k − i ) γ µ { γ β ◦ γ µ }ii , hh{ E i − ◦ γ α ◦ γ σ } γ β γ µ { G ( E k − i ) ◦ γ µ }ii = hh{ E i − ◦ γ µ }{ γ α ◦ γ σ } γ β { G ( E k − i ) ◦ γ µ }ii + hh E i − γ β γ µ { G ( E k − i ) ◦ γ α ◦ γ σ ◦ γ µ }ii −hh E i − { γ α ◦ γ σ } γ β { G ( E k − i ) ◦ ∆ }ii , and hh{ E i − ◦ γ α ◦ γ σ } G ( E k − i ) γ µ { γ β ◦ γ µ }ii = hh{ E i − ◦ γ µ }{ γ α ◦ γ σ }{ γ β ◦ γ µ } G ( E k − i ) ii + hh E i − γ µ { γ α ◦ γ β ◦ γ σ ◦ γ µ } G ( E k − i ) ii −hh E i − { γ α ◦ γ σ }{ γ β ◦ ∆ } G ( E k − i ) ii . Collecting all the above equations and using Lemma 3.1 and equation (30), andthen symmetrizing the resulting expression, the proof follows. (cid:3) Lemma A.11. hh{ G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (2) } γ µ γ µ { E i − ◦ γ ς g (3) }ii = hh{ E i − ◦ γ α ◦ γ β ◦ γ σ } γ µ γ µ G ( E k − i ) ii + hh{ G ( E k − i ) ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) { E i − ◦ γ µ }ii −hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ E i − ◦ γ µ }{ γ ς g (3) ◦ γ µ }ii + i − X j =1 hh{ G ( E k − i ) ◦ E i − j − } ∆ G ( E j − ◦ γ α ◦ γ β ◦ γ σ ) ii − i − X j =1 hh{ G ( E k − i ) ◦ ∆ ◦ E j − } G ( E i − j − ◦ γ ς g (3) ) { γ ς g (1) ◦ γ ς g (2) }ii . Proof. By equation (25), we have hh{ G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (2) } γ µ γ µ { E i − ◦ γ ς g (3) }ii = hh{ E i − ◦ γ α ◦ γ β ◦ γ σ } γ µ γ µ G ( E k − i ) ii + hh{ G ( E k − i ) ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) }{ E i − ◦ γ ς g (3) } γ µ ii −hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) } γ µ { E i − ◦ γ ς g (3) ◦ γ µ }ii , and hh{ G ( E k − i ) ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) }{ E i − ◦ γ ς g (3) } γ µ ii = hh{ G ( E k − i ) ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) { E i − ◦ γ µ }ii + hh E i − γ µ { G ( E k − i ) ◦ γ µ }{ γ α ◦ γ β ◦ γ σ }ii −hh E i − γ ς g (3) { γ ς g (1) ◦ γ ς g (2) ◦ γ µ }{ G ( E k − i ) ◦ γ µ }ii . Notice that hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) } γ µ { E i − ◦ γ ς g (3) ◦ γ µ }ii = hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ E i − ◦ γ µ }{ γ ς g (3) ◦ γ µ }ii . The proof is completed by using Lemma 3.1 and equation (30). (cid:3) The following results are useful for further simplification. Lemma A.12. − X g ∈ S k X i =1 hh{ G ( E k − i ◦ γ ς g (1) ) ◦ E i − ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii − X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ E i − ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ µ ii + 13 X g ∈ S k X i =1 hh{ G ( E k − i ) ◦ E i − ◦ γ ς g (1) ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii = − X g ∈ S k X i =1 hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (2) } γ µ γ µ { E i − ◦ γ ς g (3) }ii − X g ∈ S k X i =1 hh{ ∆ ◦ E i − } G ( E k − i ◦ γ ς g (1) ) γ ς g (2) γ ς g (3) ii + X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ) γ ς g (2) { γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii − X g ∈ S k X i =1 hh{ ∆ ◦ E i − }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) G ( E k − i ) ii + 13 X g ∈ S k X i =1 hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) ◦ γ µ } γ ς g (3) { E i − ◦ γ µ }ii + 16 X g ∈ S k X i =1 hh{ ∆ ◦ E i − } γ ς g (1) γ ς g (2) { G ( E k − i ) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii − X g ∈ S k X i =1 i − X j =1 hh{ G (∆ ◦ E i − j − ) ◦ E j − } G ( E k − i ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E i − j − ◦ γ ς g (1) } G ( E k − i ◦ γ ς g (2) ) G ( E j − ◦ γ ς g (3) ) ii + X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E i − } G ( E k − i ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii . Proof. Using equation (25), we have hh{ G ( E k − i ◦ γ ς g (1) ) ◦ E i − ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii = hh{ ∆ ◦ E i − } G ( E k − i ◦ γ ς g (1) ) γ ς g (2) γ ς g (3) ii + hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (3) }{ E i − ◦ γ µ } γ µ γ ς g (2) ii −hh{ E i − ◦ γ µ } G ( E k − i ◦ γ ς g (1) ) γ ς g (2) { γ ς g (3) ◦ γ µ }ii , hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (3) }{ E i − ◦ γ µ } γ µ γ ς g (2) ii = hh E i − ∆ γ ς g (2) { G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (3) }ii + hh{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (3) } γ µ γ µ { E i − ◦ γ ς g (2) }ii −hh E i − γ µ { γ ς g (3) ◦ γ µ }{ G ( E k − i ◦ γ ς g (1) ) ◦ γ ς g (3) }ii , and hh{ G ( E k − i ) ◦ E i − ◦ γ µ }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ µ ii = hh{ G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) γ µ { E i − ◦ γ µ }ii + hh{ ∆ ◦ E i − } G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) } γ ς g (3) ii −hh{ E i − ◦ γ µ } G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) ◦ γ µ } γ ς g (3) ii . Since by equation (25) again, we have hh{ G ( E k − i ) ◦ E i − ◦ γ ς g (1) ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii = hh{ E i − ◦ γ µ } γ ς g (2) γ µ { G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (3) }ii + hh{ ∆ ◦ E i − }{ G ( E k − i ) ◦ γ ς g (1) } γ ς g (2) γ ς g (3) ii −hh{ G ( E k − i ) ◦ γ ς g (1) }{ E i − ◦ γ µ } γ ς g (2) { γ ς g (3) ◦ γ µ }ii , and hh{ G ( E k − i ) ◦ γ ς g (1) }{ E i − ◦ γ µ } γ ς g (2) { γ ς g (3) ◦ γ µ }ii = hh{ G ( E k − i ) ◦ E i − ◦ γ µ } γ ς g (1) γ ς g (2) { γ ς g (3) ◦ γ µ }ii + hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii 08 YIJIE LIN −hh G ( E k − i ) γ ς g (1) { γ ς g (3) ◦ γ µ }{ E i − ◦ γ ς g (2) ◦ γ µ }ii = hh{ G ( E k − i ) ◦ γ ς g (3) ◦ E i − ◦ γ µ } γ ς g (1) γ ς g (2) γ µ ii + hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii −hh G ( E k − i ) γ ς g (1) { γ ς g (2) ◦ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii . Hence we have hh{ G ( E k − i ) ◦ γ ς g (1) ◦ E i − ◦ γ µ } γ ς g (2) γ ς g (3) γ µ ii + hh{ G ( E k − i ) ◦ γ ς g (3) ◦ E i − ◦ γ µ } γ ς g (1) γ ς g (2) γ µ ii = hh{ G ( E k − i ) ◦ γ ς g (1) ◦ γ ς g (3) } γ ς g (2) γ µ { E i − ◦ γ µ }ii + hh{ ∆ ◦ E i − } γ ς g (2) γ ς g (3) { G ( E k − i ) ◦ γ ς g (1) }ii −hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii + hh G ( E k − i ) γ ς g (1) { γ ς g (2) ◦ γ ς g (2) ◦ γ µ }{ E i − ◦ γ µ }ii . The proof follows by using Lemma 3.1 and equation (30). (cid:3) Lemma A.13. − k X i =1 hh{ G ( E k − i ) ◦ E i − } γ µ γ µ { γ α ◦ γ β ◦ γ σ }ii = 2 k X i =1 i − X j =1 hh{ G (∆ ◦ E i − j − ) ◦ G ( E k − i ) } E j − { γ α ◦ γ β ◦ γ σ }ii − k X i =1 i − X j =1 hh G ( E k − i ) E i − j − G (∆ ◦ E j − ◦ γ α ◦ γ β ◦ γ σ ) ii − X g ∈ S k X i =1 hh G ( E k − i ) γ ς g (1) { γ ς g (2) ◦ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii − X g ∈ S k X i =1 hh G ( E k − i ) { γ ς g (1) ◦ γ ς g (2) }{ γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii + 13 X g ∈ S k X i =1 hh G ( E k − i ) { ∆ ◦ E i − }{ γ ς g (1) ◦ γ ς g (2) } γ ς g (3) ii . Proof. By equation (25), we have hh{ G ( E k − i ) ◦ E i − } γ µ γ µ { γ α ◦ γ β ◦ γ σ }ii = hh{ G ( E k − i ) ◦ E i − } γ µ γ α { γ β ◦ γ σ ◦ γ µ }ii + hh{ G ( E k − i ) ◦ E i − } γ µ { γ α ◦ γ µ }{ γ β ◦ γ σ }ii −hh{ G ( E k − i ) ◦ E i − } ∆ γ α { γ β ◦ γ σ }ii , hh{ G ( E k − i ) ◦ E i − } γ µ γ α { γ β ◦ γ σ ◦ γ µ }ii = hh G ( E k − i ) γ α { γ β ◦ γ σ ◦ γ µ }{ E i − ◦ γ µ }ii + hh E i − { G ( E k − i ) ◦ γ α } γ µ { γ β ◦ γ σ ◦ γ µ }ii −hh E i − { γ α ◦ γ µ }{ γ β ◦ γ σ ◦ γ µ } G ( E k − i ) ii , hh{ G ( E k − i ) ◦ E i − } γ µ { γ α ◦ γ µ }{ γ β ◦ γ σ }ii = hh G ( E k − i ) { γ β ◦ γ σ }{ γ α ◦ γ µ }{ E i − ◦ γ µ }ii + hh E i − { γ β ◦ γ σ } γ µ { G ( E k − i ) ◦ γ α ◦ γ µ }ii −hh E i − { γ β ◦ γ σ }{ γ α ◦ ∆ } G ( E k − i ) ii , and hh{ G ( E k − i ) ◦ E i − } ∆ γ α { γ β ◦ γ σ }ii = hh G ( E k − i ) { ∆ ◦ E i − }{ γ β ◦ γ σ } γ α ii + hh E i − ∆ { γ β ◦ γ σ }{ G ( E k − i ) ◦ γ α }ii −hh E i − { ∆ ◦ γ α }{ γ β ◦ γ σ } G ( E k − i ) ii . The proof is completed by using Lemma 3.1 and equation (30). (cid:3) Lemma A.14. X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ) γ ς g (2) { γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii = 12 X g ∈ S k X i =1 hh{ ∆ ◦ E i − } G ( E k − i ◦ γ ς g (1) ) γ ς g (2) γ ς g (3) ii + 12 X g ∈ S k X i =1 hh G ( E k − i ◦ γ ς g (1) ) γ µ γ µ { E i − ◦ γ ς g (2) ◦ γ ς g (3) }ii + 12 X g ∈ S k X i =1 i − X j =1 hh{ G (∆ ◦ E i − j − ) ◦ E j − } G ( E k − i ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii + 12 X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E i − j − } G ( E k − i ◦ γ ς g (1) ) G ( E j − ◦ γ ς g (2) ◦ γ ς g (3) ) ii − X g ∈ S k X i =1 i − X j =1 hh{ ∆ ◦ E i − } G ( E k − i ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii . Proof. By equation (25), we have hh G ( E k − i ◦ γ ς g (1) ) γ ς g (2) { γ ς g (3) ◦ γ µ }{ E i − ◦ γ µ }ii + hh G ( E k − i ◦ γ ς g (1) ) γ ς g (3) { γ ς g (2) ◦ γ µ }{ E i − ◦ γ µ }ii = hh{ ∆ ◦ E i − } G ( E k − i ◦ γ ς g (1) ) γ ς g (2) γ ς g (3) ii + hh G ( E k − i ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) } γ µ { E i − ◦ γ µ }ii , and hh G ( E k − i ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) } γ µ { E i − ◦ γ µ }ii = hh G ( E k − i ◦ γ ς g (1) ) γ µ γ µ { E i − ◦ γ ς g (2) ◦ γ ς g (3) }ii + hh E i − ∆ { γ ς g (2) ◦ γ ς g (3) } G ( E k − i ◦ γ ς g (1) ) ii −hh E i − γ µ { γ ς g (2) ◦ γ ς g (3) ◦ γ µ } G ( E k − i ◦ γ ς g (1) ) ii . The proof is completed by using Lemma 3.1 and equation (30). (cid:3) Lemma A.15. − X g ∈ S hh{ G ( E k − ◦ γ µ ) ◦ γ ς g (1) } γ ς g (2) γ ς g (3) γ µ ii = − X g ∈ S hh{ E k − ◦ γ ς g (1) } γ µ γ µ { γ ς g (2) ◦ γ ς g (3) }ii + 12 X g ∈ S k − X j =1 hh E k − j − G (∆ ◦ E j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − k − X j =1 hh G (∆ ◦ E k − j − ) E j − { γ α ◦ γ β ◦ γ σ }ii − X g ∈ S k − X j =1 hh{ ∆ ◦ E k − j − } G ( E j − ◦ γ ς g (1) ) { γ ς g (2) ◦ γ ς g (3) }ii − X g ∈ S k − X j =1 hh{ G ( E k − j − ◦ γ ς g (1) ◦ γ µ ) ◦ G ( γ µ ) } E j − { γ ς g (2) ◦ γ ς g (3) }ii +6 k − X j =1 hh{ G ( E k − j − ◦ γ µ ) ◦ G ( γ µ ) } E j − { γ α ◦ γ β ◦ γ σ }ii + 12 X g ∈ S k − X j =1 hh{ ∆ ◦ E k − j − } G ( E j − ◦ γ ς g (1) ◦ γ ς g (2) ) γ ς g (3) ii − hh{ ∆ ◦ E k − } γ α γ β γ σ ii + X g ∈ S hh G ( E k − ◦ γ ς g (1) ◦ γ µ ) γ ς g (2) γ ς g (3) γ µ ii . Proof. Using equation (25), we have hh{ G ( E k − ◦ γ µ ) ◦ γ ς g (1) } γ ς g (2) γ ς g (3) γ µ ii = hh G ( E k − ◦ γ µ ) { γ ς g (1) ◦ γ ς g (3) } γ ς g (2) γ µ ii + 12 hh{ ∆ ◦ E k − } γ α γ β γ σ ii −hh G ( E k − ◦ γ µ ) γ ς g (1) γ ς g (2) { γ µ ◦ γ ς g (3) }ii = hh G ( γ µ ) { γ ς g (1) ◦ γ ς g (3) } γ ς g (2) { E k − ◦ γ µ }ii + 12 hh{ ∆ ◦ E k − } γ α γ β γ σ ii −hh{ G ( E k − ◦ γ ς g (3) ◦ γ µ ) } γ ς g (1) γ ς g (2) γ µ ii , and hh G ( γ µ ) { γ ς g (1) ◦ γ ς g (3) } γ ς g (2) { E k − ◦ γ µ }ii = 12 hh E k − ∆ γ ς g (2) { γ ς g (1) ◦ γ ς g (3) }ii + hh{ E k − ◦ γ ς g (2) } γ µ G ( γ µ ) { γ ς g (1) ◦ γ ς g (3) }ii −hh E k − γ µ { γ ς g (1) ◦ γ ς g (3) }{ γ ς g (2) ◦ G ( γ µ ) }ii = 12 hh{ E k − ◦ γ ς g (2) } γ µ γ µ { γ ς g (1) ◦ γ ς g (3) }ii + 12 hh E k − ∆ γ ς g (2) { γ ς g (1) ◦ γ ς g (3) }ii −hh E k − γ µ { γ ς g (1) ◦ γ ς g (3) }{ γ ς g (2) ◦ G ( γ µ ) }ii . The proof is completed by using Lemma 3.1. (cid:3) References [1] L. 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