A genus-4 topological recursion relation for Gromov-Witten invariants
aa r X i v : . [ m a t h . AG ] A ug A genus-4 topological recursion relation forGromov-Witten invariants
Xin Wang
Abstract
In this paper, we give a new genus-4 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds via Pixton’s relations on themoduli space of curves. As an application, we prove Pixton’s relations imply aknown topological recursion relation on M g, for genus g ≤ Let M g,n be the moduli space of genus- g , stable curves with n marked points. It iswell known that the relations among strata only involving ψ classes and boundary classesin the tautological ring of M g,n give universal equations for the Gromov-Witten invariantsof compact symplectic manifolds. Examples of genus g ≤ M g,n ,called Pixton’s relations, was proposed in [P1] and proved in [PPZ]. They are a kind ofcombinatorial formulas among strata involving κ classes, ψ classes and boundary classes.Pixton wrote a Sage program in [P2] to compute these finitely many relations for fixed g and n . We will use it to eliminate all the κ classes appearing in Pixton’s formula andobtain the kind of relations we want. In this paper, we give a new genus-4 universalequation, i.e. a genus-4 topological recursion relation on M , .To describe universal equations, we need some conventions. Let M be any compactsymplectic manifold. Define the big phase space for Gromov-Witten invariants of M to bethe product of infinitely many copies of H ∗ ( M ; C ). We will fix a basis { γ α | α = 1 , ..., N } of H ∗ ( M ; C ). An operator T on the space of vector fields on the big phase space wasintroduced in [L1] to simplify topological recursion relations. The operator T is veryuseful when we translate relations in the tautological ring of M g,n into universal equationsfor Gromov-Witten invariants. We will write the universal equations of Gromov-Witteninvariants as equations among correlators hhW · · · W k ii g which are by definition the k -th covariant derivatives of the generating functions of genus- g Gromov-Witten invariantswith respect to the trivial connection on the big phase space. we will briefly review thesedefinitions in Section 1 for completeness.In this paper, we will prove the following genus-4 universal equation
Theorem 0.1
For Gromov-Witten invariants of any compact symplectic manifold, thefollowing topological recursion relation holds for any vector field W on the big phase space: hh T ( W ) ii = A ( W ) (1)1 here A ( W ) is given by a very complicated formula only involving invariants of genus notbigger than 3. The explicit expression of A ( W ) can be found in Appendix and we will notput it here. Equivalently, this theorem corresponds to a relation in the tautological ring of M , ,representing ψ as a linear combination of boundary strata decorated with ψ classes. Sincethe relation is very long, we will not show its dual graph representations in this paper. Theinterested readers can write it carefully by themselves. Actually, our main idea behind theproof of Theorem 0.1 is: first obtain the relation representing ψ as a linear combinationof strata containing ψ classes and boundary classes on M , via Pixton’s relation, thentranslate it to equation (1) by the splitting principle of Gromov-Witten invariants.We should mention that in [L2] it was conjectured that the following type of topologicalrecursion relations hold for all genera g , hh T g ( W ) ii g = A g − ( W ) . (2)Equivalently, equation (2) gives a topological recursion relation on RH g ( M g, ), repre-senting ψ g as a linear combination of boundary strata with ψ classes. It is well known that(2) holds for g ≤ g becomes more and more complicated. However, veryinteresting topological recursion relations on RH g + r ) ( M g, ) (for any r ≥
0) were foundand proved in [LP] for all genus g ≥ ψ g + r = X g + g = g,g i > X a + b =2 g − r ( − a g g ρ ∗ (cid:16) ψ a ∗ ψ b ∗ ∩ [∆ , ∅ ( g , g )] (cid:17) . (3)Given that Pixton’s relations are conjectured to generate all relations in the tautologicalring, one would expect they imply the relations (3). We prove this in the case of genus g ≤ Theorem 0.2
In the case of genus g ≤ , Pixton’s relations imply the topological recur-sion relation (3) . As for genus g ≥
5, we conjecture that our method in the proof also works as long as weget the topological recursion relation on RH g ( M g, ).This paper is organized as follows. In Section 1, we introduce notations and reviewbasic theories needed in this paper. We prove Theorem 0.1 in Section 2 and prove Theorem0.2 in Section 3. In the Appendix, we give the precise formula of the topological recursionrelation (1). Acknowledgements.
The author is partially supported by the fundamental ResearchFunds of Shandong University. We would like to thank professor Xiaobo Liu for helpfuldiscussion. We also thank Felix Janda for lots of helpful comments on the paper.2
Preliminaries
Let M g,n be the moduli space of stable curves of genus- g with n -marked points. Theboundary strata of the moduli M g,n of fixed topological type correspond to stable graphs.Formally, a stable graph is the structureΓ = ( V, E, L, g )satisfying the following properties: • V is the vertex set with a genus function g : V → Z , • E is the edge set, • L is the set of legs (corresponding to the set of markings), • the pair ( V, E ) defines a connected graph, • for each vertex v , the stability condition holds:2 g ( v ) − n ( v ) > , where n ( v ) is the degree of the vertex v in the graph Γ.The genus of a stable graph is defined by g (Γ) := X v ∈ V g ( v ) + h (Γ) . To each stable graph Γ, we associate the moduli space M Γ := Y v ∈ V M g ( v ) ,n ( v ) . Then there is a canonical morphism ξ Γ : M Γ → M g,n Recall that the strata algebra S g,n on M g,n is defined to be a finite dimensional algebra,with additive basis the isomorphism classes [Γ , γ ], where Γ is a stable graph of genus g with n legs and γ is a class on M Γ which is a product of monomials in κ classes oneach vertex of Γ and ψ classes on each half edge. Push forward along ξ Γ defines a ringhomomorphism from strata algebra to cohomology ring q : S g,n → H ∗ ( M g,n )[Γ , γ ] → ( ξ Γ ) ∗ ( γ ) . The tautological ring RH ∗ ( M g,n ) is by definition the image of q . An element in the kernelof q is call a tautological relation on M g,n .We define tautological classes R ( g, n, r ; σ, a , ..., a n ) associated to the data:3 g, n ∈ Z ≥ in the stable range 2 g − n > • a , ..., a n are nonnegative integers not 2 mod 3, • σ : a partition with no parts 2 mod 3, • r ∈ Z ≥ satisfying | σ | + X i a i ≤ r − g − , | σ | + X i a i ≡ r − g − σ = 1 n n n n · ·· , | σ | := P i =2 mod in i .The elements R ( g, n, r ; σ, a , ..., a n ) are expressed as sums over stable graphs of genus g with n legs. R ( g, n, r ; σ, a , ..., a n ) := X Γ ∈ G g,n | Aut (Γ) | ( ξ Γ ) ∗ ( R Γ ( g, n, r ; σ, a , ..., a n )) (4)where G g,n denotes the finite set of stable graphs of genus g with n legs (up to isomor-phism). Pixton’s relations then takes R ( g, n, r ; σ, a , ..., a n ) = 0 ∈ H r ( M g,n , Q ) . (5)Before writing the formula for R Γ ( g, n, r ; σ, a , ..., a n ), a few definitions are required.We first introduce the following two power series: A ( T ) = X n ≥ (6 n )!(3 n )!(2 n )! T n ,B ( T ) = X n ≥ n + 16 n − n )!(3 n )!(2 n )! T n . These two series play a central role in Pixton’s relations. Let b C i ( T, ζ ) = T i A ( ζ T ) b C i +1 ( T, ζ ) = ζ T i B ( ζ T )where ζ is a formal parameter satisfying ζ = 1.For any polynomial in variable T , F ( T ) = X n a n T n + ζ X n b n T n , { F ( T ) } := X n a n T n K n, + X n b n T n K n, . For each vertex v ∈ V of a stable graph, we introduce an auxiliary variable ζ v and imposethe conditions ζ v ζ v ′ = ζ v ′ ζ v , ζ v = 1 . The variables ζ v will be responsible for keeping track of a local parity condition at eachvertex.For each stable graph Γ ∈ G g,n , define b κ Γ ( K e ,a · · · K e l ,a l ) := X τ ∈ S l Y c cycle in τ ( X v ∈ V (Γ) κ ( v ) P i ∈ c e i ζ P i ∈ c a i v ) (6)We denote by R Γ ( g, n, r ; σ, a , ..., a n ) ∈ H r ( M g,n ; Q ) the class R Γ ( g, n, r ; σ, a , ..., a n )= 12 h (Γ) hb κ Γ (cid:0) e { − b C } { b C σ } · · · { b C σ l } (cid:1) n Y i =1 b C a i ( ψ i T, ζ v i ) Y e ∈ E ∆ e i T r −| E | Q v ζ g ( v )+1 v where h (Γ) = | E (Γ) | − | V (Γ) | + 1 is the number of loops in Γ, marking i corresponds tohalf-edge h i on vertex v i , and g ( v ) is the genus of vertex v . The subscript T r −| E | Q v ζ g ( v )+1 v indicates the coefficient of the monomial T r −| E | Q v ζ g ( v )+1 v after the product inside thebrackets is expanded. For each edge e ∈ E ,∆ e = A ( ζ ψ T ) ζ B ( ζ ψ T ) + ζ B ( ζ ψ T ) A ( ζ ψ T ) + ζ + ζ ( ψ + ψ ) T , where ζ , ζ are the ζ variables assigned to the vertices adjacent to the edge e and ψ , ψ are the ψ classes corresponding to the half edges. The numerator of ∆ e is divisible by thedenominator due to the identity A ( T ) B ( − T ) + A ( − T ) B ( T ) = − . Let M be a compact symplectic manifold. The small phase space is by definition H ∗ ( M ; C )and the big phase space is defined to be P := Q ∞ n =0 H ∗ ( M, C ). Let { γ , ..., γ N } be afixed basis of H ∗ ( M, C ), where γ is the identity element of the cohomology ring of M .The corresponding basis for the n -th copy of H ∗ ( M, C ) in this product is denoted by { τ n ( γ α ) | α = 1 , . . . , N } for n ≥
0. Let t αn be the coordinates on P with respect to thestandard basis { τ n ( γ α ) | α = 1 , . . . , N, n ≥ } . We will identify τ n ( γ α ) with ∂∂t αn as vectorfields on the big phase space. If n < τ n ( γ α ) is understood to be the zero vector field.5e will also write τ ( γ α ) simply as γ α . We use τ + and τ − to denote the operators whichshift the level of descendants, i.e. τ ± (cid:16) X n,α f n,α τ n ( γ α ) (cid:17) = X n,α f n,α τ n ± ( γ α )where f n,α are functions on the big phase space.Define η = ( η αβ ) to be the matrix of intersection pairing on H ∗ ( M ; C ) in this basis { γ , ..., γ N } . We will use η = ( η αβ ) and η − = ( η αβ ) to lower and raise indices, forexample γ α := η αβ γ β for any α . Here we use the summation convention that repeatedindices should be summed over their entire ranges.Let h τ n ( γ α ) · · · τ n k ( γ α k ) i g,β := Z [ M g,n ( M,β )] vir k Y i =1 (Ψ i ∪ ev ∗ i ( γ α i ))be the genus- g , degree- β , descendant Gromov-Witten invariant associated to γ α , ..., γ α k and nonnegative integers n , ..., n k (cf. [RT], [LT]). Here M g,n ( M, β ) is the moduli spaceof stable maps from genus- g , k -marked curves to M of degree β ∈ H ( M ; Z ). Ψ i is the firstChern class of the tautological line bundle over M g,n ( M, β ) whose geometric fiber is thecotangent space of the domain curve at the i -th marked point and ev i : M g,n ( M, β ) → M is the i -th evaluation map for all i = 1 , ..., k and [ M g,n ( M, β )] vir is the virtual fundamentalclass. The genus- g generating function is defined to be F g := X k ≥ X α ,...,α k X n ,...,n k k ! t α n · · · t α k n k X β q β h τ n ( γ α ) · · · τ n k ( γ α k ) i g,β where q β belongs to the Novikov ring. This function is understood as a formal powerseries in variables { t αn } with coefficients in the Novikov ring.Define a k -tensor hh · · · ii g by hh W W · · · W k ii g := X m ,α ,...,m k ,α k f m ,α · · · f km k ,α k ∂ k ∂t α m · · · ∂t α k m k F g (7)for vector fields W i = P m,α f im,α ∂∂t αm where f im,α are functions on the big phase space.This tensor is called k - point ( correlation ) f unction .For any vector fields W and W on the big phase space, the quantum product of W and W is defined by W ◦ W := hh W W γ α ii γ α . Define the vector field T ( W ) := τ + ( W ) − hhW γ α ii γ α for any vector field W . The operator T was introduced in [L1] in order to simplifytopological recursion relations of Gromov-Witten invariants. Let ∇ be the trivial flat6onnection on the big phase space with respect to the coordinates { t αn } . Then the covariantderivative of the quantum product is given by ∇ W ( W ◦ W ) = ( ∇ W W ) ◦ W + W ◦ ( ∇ W W ) + hhW W W γ α ii γ α (8)and the covariant derivative of the operator T satisfies ∇ W T ( W ) = T ( ∇ W W ) − W ◦ W (9)for any vector fields W , W and W . We need these formulas to compute derivatives ofuniversal equations. Obviously, Pixton’s relations give lots of tautological relations on M g,n for any g and n . What we need is to eliminate the kappa classes in Pixton’s relations to get explicittopological recursion relations on M g,n .The proof of Theorem 0.1 mainly contains five steps. Here we should mention that,by using his sage code, Pixton has checked the presence of Getzler’s relation [Ge1] in RH ( M , ) and Belorousski-Pandharipande relation [BP] in RH ( M , ) (cf. [P1]).Actually, the equation (1) is also obtained with the aid of Pixton’s sage code. Inprinciple, we can use this method for all g and n to get topological recursion relations.Here we focus on relations on M , .Firstly, find all the stable graphs up to isomorphism. In fact, we can define a degenerateoperation on any vertex of a stable graph: simply splitting one vertex with genus g ( v )into two vertexes with genus g ( v ) and g ( v ) respectively, such that g ( v ) = g ( v ) + g ( v ).We notice that any stable graph Γ of genus- g = P v ∈ V g ( v ) + h (Γ) with n -legs can beobtained from a graph with only one vertex attached with genus- P v ∈ V (Γ) g ( v ), h (Γ) loopsand n -legs via finite (equal to the number of ordinary edges of Γ) steps of degeneration.Then we identify graphs which are isomorphic to each other.Secondly, we decorate each stable graph with some κ classes on the vertex and ψ classeson the half edge and legs, and identify the isomorphic ones. So we get the canonical linearbasis of strata algebra S g,n . Then we reorder the basis so that strata with κ classes arein the front.Thirdly, let R g,n be the set of all elements in the strata algebra S g,n produced as follows:choose a dual graph Γ for a boundary stratum of M g,n , pick one of the components S g ′ ,n ′ in S Γ = S g ,n × ... × S g m ,n m , take the product of a relation R ( g ′ , n ′ , d ; σ, a , ..., a n ) = 0 onthe chosen component together with arbitrary classes on the other components, and pushforward along the gluing map S Γ → S g,n . So the set R g,n give us linear equations betweenthe generators of the strata algebra. For our purpose, we should rewrite all these linearequations with respect to our list of strata and get the corresponding coefficients matrix R . Fourthly, by technics in linear algebra, we can transform the matrix R into the reducedrow echelon form. Then we can get many linear independent tautological relations amongstrata only involving ψ classes and boundary classes.7astly, we translate the tautological relations obtained above into differential equationsfor generating functions of Gromov-Witten invariants, here each ψ class corresponds toan insertion of the operator T . Actually we can get many linear independent differentialequations, from which we can choose the equation (1). Remark 2.1
Actually, the first 3 steps in the above algorithm is essentially the same asthe method in Pixton’s sage program except the reordering of the strata generators. Usingthis method, we have also checked for the presence of all known topological recursionrelations (cf. [Ge2], [KL1] and [KL2]). To our knowledge, Lin and Zhou have done somesimilar work (cf. [Li]). On the other hand, we get many new relations for genus-2 andgenus-3 and their relation with higher genus Virasoro conjecture will be studied in theforthcoming papers.
First we use operator T to translate equation (3) into a differential equation hh T g + r ( W ) ii g = X g + g = g,g i > X a + b =2 g − r ( − a g g hhW T a ( γ α ) ii g hh T b ( γ α ) ii g (10)where g ≥ r ≥ W is an arbitrary vector field on the big phase space.Via the operator T , we can reformulate the genus-0 topological recursion relation as hh T ( W ) W W ii = 0 (11)and the genus-1 topological recursion relation as hh T ( W ) ii = 124 hhW γ α γ α ii (12)for any vector fields W and W i .Taking derivatives of equation (12) and combining with equation (9), we have hh T ( W ) Vii = hh{W ◦ V}ii + 124 hhWV γ α γ α ii . To prove theorem 0.2, we only need to consider the following nontrivial cases. g = 2 Due to the dimension constraints, we only need to show that equation (10) holds for r = 0, i.e. hh T ( W ) ii = 12 hhW T ( γ α ) ii hh T ( γ α ) ii . (13)8e recall the genus-2 Mumford relation (cf. [Ge2]) as formulated in [L1]: hh T ( W ) ii = 710 hh γ α ii hh{ γ α ◦ W}ii + 110 hh γ α { γ α ◦ W}ii − hhW{ γ α ◦ γ α }ii + 13240 hhW γ α γ α γ β ii hh γ β ii + 1960 hhW γ α γ α γ β γ β ii (14)for any vector field W .By equation (14), (11) and (12) and their derivatives, both sides of equation (13) equalto the expression 11152 hh ∆∆ Wii . g = 3 Due to the dimension constraints, we only need to equation (10) holds for r = 0 and r = 1. For simplicity, we here only give the proof of the case for r = 0 and the other casecan be similarly done, i.e. hh T ( W ) ii = 13 hhW T ( γ α ) ii hh T ( γ α ) ii − hhW T ( γ α ) ii hh γ α ii . (15)To prove equation (15), we need the genus-3 topological recursion relation (cf. [KL1]): hh T ( W ) ii = − hhW T ( γ α ◦ γ α ) ii + 542 hh T ( γ α ) {W ◦ γ α }ii + 13168 hh T ( γ α ) ii hh γ α W γ β γ β ii + 4121 hh T ( γ α ) ii hh{ γ α ◦ W}ii − hh{W ◦ γ α ◦ γ α }ii + 1280 hhW γ α ii hh γ α { γ β ◦ γ β }ii − hh γ α ii hh γ α W{ γ β ◦ γ β }ii − hh γ α ii hh γ α γ β ii hh γ β W γ µ γ µ ii − hhW γ α ii hh γ α γ β γ β γ µ ii hh γ µ ii + 23504 hh γ α ii hh γ α W γ β γ β γ µ ii hh γ µ ii + 11140 hh γ α γ β ii hh γ α { γ β ◦ W}ii − hh γ α ii hh γ α γ β ii hh{ γ β ◦ W}ii + 2105 hhW γ α ii hh{ γ α ◦ γ β }ii hh γ β ii + 89210 hh γ α ii hh γ α W γ β γ µ ii hh γ β ii hh γ µ ii − hh γ α ii hh γ α γ β { γ β ◦ W}ii + 1140 hhW γ α γ β ii hh{ γ α ◦ γ β }ii + 23140 hh γ α γ β ii hh γ α γ β W γ µ ii hh γ µ ii − hh γ α γ β ii hh{ γ α ◦ γ β }Wii − hhW γ α ii hh γ α γ β γ β γ µ γ µ ii + 138064 hh γ α ii hh γ α W γ β γ β γ µ γ µ ii hhW γ α γ β ii hh γ α γ β γ µ γ µ ii + 416720 hh γ α γ β ii hh γ α γ β W γ µ γ µ ii + 153760 hhW γ α γ α γ β γ β γ µ γ µ ii − hh{W ◦ γ α }ii hh γ α γ β γ β ii − hhW γ α γ α { γ β ◦ γ β }ii − hh γ α γ α γ β ii hh γ β W γ µ γ µ ii − hh γ α γ α γ β { γ β ◦ W}ii + 13780 hhW γ α γ β γ µ ii hh γ α γ β γ µ ii + 1252 hh γ α γ β γ µ ii hhW γ α γ β γ µ ii . (16)By equations (16), (11), (12), (14) and their derivatives, both sides of equation (15) canbe reduced to the following expression75760 hh{W ◦ ∆ ◦ ∆ }ii + 112903040 hh{W ∆∆∆ }ii + 19967680 hh{W ◦ ∆ } ∆ γ α γ α ii + 1120960 hhW{ ∆ ◦ ∆ } γ α γ α ii + 160480 hh{W ◦ γ α } γ α ∆∆ ii + 111520 hh{W ◦ ∆ ◦ γ α } γ α γ β γ β ii . g = 4 Due to the dimension constraints, we only need to equation (10) holds for r = 0, r = 1and r = 2. For simplicity, we only prove the case for r = 0 and the other case can besimilarly done, i.e. hh T ( W ) ii = − hhW T ( γ α ) ii hh T ( γ α ) ii + 14 hhW T ( γ α ) ii hh T ( γ α ) ii − hhW T ( γ α ) ii hh γ α ii . (17)By the equations (17), (11), (12), (16) and their derivatives, we can reduce both sidesof equation (17) to an expression only involving genus-0 and genus-1 invariants. As theexpression itself is long and not so important, we omit it here. AppendixA Expression of A ( W ) In this appendix, we give the explicit expression of A ( W ) in equation (1). A ( W )= − hh γ α W γ β ii hh T ( γ α ) T ( γ β ) ii − hh γ α γ α γ β ii hh T ( W ) γ β ii + 1372 hh γ α γ α γ β ii hhW T ( γ β ) ii
10 4772 hh γ α γ α γ β ii hhW γ β γ σ ii hh T ( γ σ ) ii − hh γ α γ α W γ β ii hh T ( γ β ) ii + 12714 hh T ( γ α ) ii hhW γ α γ β ii hh T ( γ β ) ii + 8521 hh γ α ii hhW γ α γ β ii hh γ σ ii hh γ β T ( γ σ ) ii + 10342 hh γ α ii hh γ β ii hh γ α γ β γ σ ii hh T ( W ) γ σ ii − hh γ α ii hh γ β ii hh γ α γ β γ σ ii hhW T ( γ σ ) ii + 16330 hh γ α ii hh γ β ii hhW γ α γ β γ σ ii hh T ( γ σ ) ii − hh γ α ii hh γ α γ β ii hhW γ β γ σ ii hh T ( γ σ ) ii − hh γ α ii hhW γ β ii hh γ α γ β γ σ ii hh T ( γ σ ) ii − hh γ α ii hhW γ α γ β ii hh γ σ ii hh γ µ ii hh γ β γ σ γ µ ii − hh γ α ii hhW γ α γ β ii hh γ σ ii hh γ σ γ µ ii hh γ β γ µ ii − hh γ α ii hh γ β ii hh γ α γ β γ σ ii hhW γ σ γ µ ii hh γ µ ii + 3235 hh γ α ii hh γ β ii hh γ α γ β γ σ ii hh γ µ ii hhW γ σ γ µ ii + 970 hh γ α ii hh γ β ii hh γ α γ β γ σ ii hh γ σ γ µ ii hhW γ µ ii − hh γ α ii hh γ β ii hhW γ α γ β γ σ ii hh γ µ ii hh γ σ γ µ ii − hh γ α ii hh γ β ii hh γ σ ii hh γ α γ β γ σ γ µ ii hhW γ µ ii + 463840 hh γ α ii hh γ β ii hh γ σ ii hhW γ α γ β γ σ γ µ ii hh γ µ ii − hh γ α ii hh γ β ii hh γ β γ σ ii hh γ α γ σ γ µ ii hhW γ µ ii + 11735 hh γ α ii hh γ β ii hh γ β γ σ ii hhW γ σ γ µ ii hh γ α γ µ ii + 13984 hh γ α W γ β ii hh γ β γ σ ii hh γ α T ( γ σ ) ii + 107168 hh γ α γ α γ β ii hh γ β γ σ ii hh T ( W ) γ σ ii − hh γ α γ α γ β ii hh γ β γ σ ii hhW T ( γ σ ) ii − hh γ α γ α γ β ii hhW γ σ ii hh γ β T ( γ σ ) ii + 767672 hh γ α γ α γ β ii hhW γ β γ σ ii hh T ( γ σ ) ii + 229504 hh γ α γ α W γ β ii hh γ σ ii hh T ( γ β ) γ σ ii + 767672 hh γ α γ α W γ β ii hh γ β γ σ ii hh T ( γ σ ) ii + 71210 hh γ β ii hhW γ β γ σ ii hh γ α T ( γ α ) γ σ ii + 124 hh γ β ii hhW γ β γ σ ii hh γ α γ α T ( γ σ ) ii − hh γ β ii hh γ α γ β γ σ ii hh T ( γ α ) W γ σ ii + 4342 hh γ β ii hh γ α γ β γ σ ii hh γ α T ( W ) γ σ ii + 144917560 hh γ β ii hh γ α W γ β γ σ ii hh T ( γ α ) γ σ ii + 65168 hh γ β ii hh γ α γ α γ β γ σ ii hh T ( W ) γ σ ii − hh γ β ii hh γ α γ α γ β γ σ ii hhW T ( γ σ ) ii + 810730240 hh γ β ii hh γ α γ α W γ β γ σ ii hh T ( γ σ ) ii + 36377560 hhW γ β ii hh γ α γ β γ σ ii hh T ( γ α ) γ σ ii − hhW γ β ii hh γ α γ α γ β γ σ ii hh T ( γ σ ) ii − hh γ α γ β ii hh γ α γ β γ σ ii hh T ( W ) γ σ ii + 3595115120 hh γ α γ β ii hh γ α γ β γ σ ii hhW T ( γ σ ) ii − hh γ α γ β ii hh γ α W γ β γ σ ii hh T ( γ σ ) ii − hh γ α W γ β ii hh γ α γ β γ σ ii hh T ( γ σ ) ii − hh γ α γ α γ β ii hhW γ β γ σ ii hh T ( γ σ ) ii − hh γ α W γ β ii hh γ σ ii hh γ µ ii hh γ α γ β γ σ γ µ ii − hh γ α W γ β ii hh γ σ ii hh γ σ γ µ ii hh γ α γ β γ µ ii + 79140 hh γ α W γ β ii hh γ σ ii hh γ β γ µ ii hh γ α γ σ γ µ ii hh γ α W γ β ii hh γ β γ σ ii hh γ σ γ µ ii hh γ α γ µ ii − hh γ α γ α γ β ii hhW γ β γ σ ii hh γ µ ii hh γ σ γ µ ii + 1742 hh γ α γ α γ β ii hh γ σ ii hhW γ σ γ µ ii hh γ β γ µ ii + 15331100800 hh γ α γ α γ β ii hh γ σ ii hh γ µ ii hhW γ β γ σ γ µ ii − hh γ α γ α γ β ii hh γ σ ii hh γ σ γ µ ii hhW γ β γ µ ii − hh γ α γ α γ β ii hh γ σ ii hh γ β γ µ ii hhW γ σ γ µ ii + 1788150400 hh γ α γ α γ β ii hh γ σ ii hh γ β γ σ γ µ ii hhW γ µ ii − hh γ α γ α γ β ii hh γ β γ σ ii hhW γ σ γ µ ii hh γ µ ii + 6571400 hh γ α γ α γ β ii hh γ β γ σ ii hh γ σ γ µ ii hhW γ µ ii + 201733600 hh γ α γ α W γ β ii hh γ σ ii hh γ µ ii hh γ β γ σ γ µ ii − hh γ α γ α W γ β ii hh γ σ ii hh γ σ γ µ ii hh γ β γ µ ii − hh γ β ii hhW γ β γ σ ii hh γ α γ σ γ µ ii hh γ α γ µ ii − hh γ β ii hhW γ β γ σ ii hh γ µ ii hh γ α γ α γ σ γ µ ii − hh γ β ii hhW γ β γ σ ii hh γ σ γ µ ii hh γ α γ α γ µ ii − hh γ β ii hhW γ β γ σ ii hh γ α γ µ ii hh γ α γ σ γ µ ii + 39773024 hh γ β ii hh γ α γ β γ σ ii hh γ α γ σ γ µ ii hhW γ µ ii − hh γ β ii hh γ α γ β γ σ ii hh γ α W γ σ γ µ ii hh γ µ ii − hh γ β ii hh γ α γ β γ σ ii hh γ µ ii hh γ α W γ σ γ µ ii − hh γ β ii hh γ α γ β γ σ ii hh γ σ γ µ ii hh γ α W γ µ ii + 20696300 hh γ β ii hh γ α γ β γ σ ii hhW γ µ ii hh γ α γ σ γ µ ii − hh γ β ii hh γ α W γ β γ σ ii hh γ α γ σ γ µ ii hh γ µ ii − hh γ β ii hh γ α W γ β γ σ ii hh γ µ ii hh γ α γ σ γ µ ii − hh γ β ii hh γ α W γ β γ σ ii hh γ σ γ µ ii hh γ α γ µ ii − hh γ β ii hh γ α γ α γ β γ σ ii hhW γ σ γ µ ii hh γ µ ii + 13097200 hh γ β ii hh γ α γ α γ β γ σ ii hh γ µ ii hhW γ σ γ µ ii + 1078150400 hh γ β ii hh γ α γ α γ β γ σ ii hh γ σ γ µ ii hhW γ µ ii − hh γ β ii hh γ α γ α W γ β γ σ ii hh γ µ ii hh γ σ γ µ ii − hh γ β ii hh γ σ ii hh γ β γ σ γ µ ii hh γ α γ α W γ µ ii − hh γ β ii hh γ σ ii hhW γ β γ σ γ µ ii hh γ α γ α γ µ ii − hh γ β ii hh γ σ ii hh γ α γ β γ σ γ µ ii hh γ α W γ µ ii + 350910800 hh γ β ii hh γ σ ii hh γ α W γ β γ σ γ µ ii hh γ α γ µ ii − hh γ β ii hh γ σ ii hh γ α γ α γ β γ σ γ µ ii hhW γ µ ii + 23113302400 hh γ β ii hh γ σ ii hh γ α γ α W γ β γ σ γ µ ii hh γ µ ii + 109210 hh γ β ii hh γ β γ σ ii hhW γ σ γ µ ii hh γ α γ α γ µ ii + 68593150 hh γ β ii hh γ β γ σ ii hh γ α γ σ γ µ ii hh γ α W γ µ ii + 1935112600 hh γ β ii hh γ β γ σ ii hh γ α W γ σ γ µ ii hh γ α γ µ ii + 391710080 hh γ β ii hh γ β γ σ ii hh γ α γ α γ σ γ µ ii hhW γ µ ii − hh γ β ii hhW γ σ ii hh γ β γ σ γ µ ii hh γ α γ α γ µ ii − hh γ β ii hhW γ σ ii hh γ α γ σ γ µ ii hh γ α γ β γ µ ii + 1837137800 hh γ β ii hhW γ σ ii hh γ α γ β γ σ γ µ ii hh γ α γ µ ii + 11231800 hh γ β ii hhW γ β γ σ ii hh γ α γ σ γ µ ii hh γ α γ µ ii − hhW γ β ii hh γ α γ β γ σ ii hh γ α γ σ γ µ ii hh γ µ ii + 4330715120 hhW γ β ii hh γ α γ β γ σ ii hh γ σ γ µ ii hh γ α γ µ ii − hhW γ β ii hh γ β γ σ ii hh γ α γ σ γ µ ii hh γ α γ µ ii
12 154017560 hh γ α γ β ii hh γ α γ β γ σ ii hhW γ σ γ µ ii hh γ µ ii + 171680 hh γ β W γ σ ii hh γ α γ α T ( γ β ) γ σ ii − hh γ β γ β γ σ ii hh γ α γ α W T ( γ σ ) ii + 371811814400 hh γ β γ β W γ σ ii hh γ α γ α T ( γ σ ) ii − hh γ α γ β γ σ ii hh T ( γ α ) γ β W γ σ ii + 70320160 hh γ α γ β γ σ ii hh γ α γ β T ( W ) γ σ ii + 4537151200 hh γ α γ β W γ σ ii hh T ( γ α ) γ β γ σ ii − hh γ α γ β γ β γ σ ii hh T ( γ α ) W γ σ ii + 160 hh γ α γ β γ β γ σ ii hh γ α T ( W ) γ σ ii − hh γ α γ β γ β W γ σ ii hh T ( γ α ) γ σ ii + 1120 hh γ α γ α γ β γ β γ σ ii hh T ( W ) γ σ ii − hh γ α γ α γ β γ β γ σ ii hhW T ( γ σ ) ii + 1319604800 hh γ α γ α γ β γ β W γ σ ii hh T ( γ σ ) ii − hh γ β W γ σ ii hh γ α γ σ γ µ ii hh γ α γ β γ µ ii + 1915040 hh γ β W γ σ ii hh γ α γ α γ µ ii hh γ β γ σ γ µ ii − hh γ β W γ σ ii hh γ α γ α γ σ γ µ ii hh γ β γ µ ii − hh γ β W γ σ ii hh γ µ ii hh γ α γ α γ β γ σ γ µ ii + 48750400 hh γ β W γ σ ii hh γ σ γ µ ii hh γ α γ α γ β γ µ ii − hh γ β W γ σ ii hh γ β γ σ γ µ ii hh γ α γ α γ µ ii − hh γ β W γ σ ii hh γ α γ µ ii hh γ α γ β γ σ γ µ ii + 13210 hh γ β W γ σ ii hh γ α γ σ γ µ ii hh γ α γ β γ µ ii − hh γ β γ β γ σ ii hhW γ σ γ µ ii hh γ α γ α γ µ ii − hh γ β γ β γ σ ii hh γ α γ α γ µ ii hhW γ σ γ µ ii + 4259201600 hh γ β γ β γ σ ii hh γ α γ α W γ µ ii hh γ σ γ µ ii + 21659918144000 hh γ β γ β γ σ ii hh γ µ ii hh γ α γ α W γ σ γ µ ii − hh γ β γ β γ σ ii hh γ σ γ µ ii hh γ α γ α W γ µ ii + 423472592000 hh γ β γ β γ σ ii hhW γ µ ii hh γ α γ α γ σ γ µ ii − hh γ β γ β γ σ ii hhW γ σ γ µ ii hh γ α γ α γ µ ii + 1048271296000 hh γ β γ β γ σ ii hh γ α γ µ ii hh γ α W γ σ γ µ ii − hh γ β γ β γ σ ii hh γ α γ σ γ µ ii hh γ α W γ µ ii + 10643918144000 hh γ β γ β W γ σ ii hh γ µ ii hh γ α γ α γ σ γ µ ii − hh γ β γ β W γ σ ii hh γ σ γ µ ii hh γ α γ α γ µ ii + 683299072000 hh γ β γ β W γ σ ii hh γ α γ µ ii hh γ α γ σ γ µ ii + 971620 hh γ α γ β γ σ ii hh γ β γ σ γ µ ii hh γ α W γ µ ii − hh γ α γ β γ σ ii hh γ β W γ σ γ µ ii hh γ α γ µ ii + 500271209600 hh γ α γ β γ σ ii hh γ α γ β γ σ γ µ ii hhW γ µ ii − hh γ α γ β γ σ ii hh γ α γ β W γ σ γ µ ii hh γ µ ii − hh γ α γ β γ σ ii hh γ µ ii hh γ α γ β W γ σ γ µ ii − hh γ α γ β γ σ ii hh γ σ γ µ ii hh γ α γ β W γ µ ii + 1601216000 hh γ α γ β γ σ ii hhW γ µ ii hh γ α γ β γ σ γ µ ii + 187375600 hh γ α γ β γ σ ii hhW γ σ γ µ ii hh γ α γ β γ µ ii − hh γ α γ β W γ σ ii hh γ α γ β γ σ γ µ ii hh γ µ ii − hh γ α γ β W γ σ ii hh γ µ ii hh γ α γ β γ σ γ µ ii + 38657151200 hh γ α γ β W γ σ ii hh γ σ γ µ ii hh γ α γ β γ µ ii
13 3331302400 hh γ α γ β γ β γ σ ii hh γ α γ σ γ µ ii hhW γ µ ii − hh γ α γ β γ β γ σ ii hh γ α W γ σ γ µ ii hh γ µ ii + 495839072000 hh γ α γ β γ β γ σ ii hh γ µ ii hh γ α W γ σ γ µ ii − hh γ α γ β γ β γ σ ii hh γ σ γ µ ii hh γ α W γ µ ii + 761339072000 hh γ α γ β γ β γ σ ii hhW γ µ ii hh γ α γ σ γ µ ii − hh γ α γ β γ β W γ σ ii hh γ α γ σ γ µ ii hh γ µ ii + 814439072000 hh γ α γ β γ β W γ σ ii hh γ µ ii hh γ α γ σ γ µ ii + 1611440 hh γ α γ β γ β W γ σ ii hh γ σ γ µ ii hh γ α γ µ ii − hh γ α γ α γ β γ β γ σ ii hhW γ σ γ µ ii hh γ µ ii + 4958318144000 hh γ α γ α γ β γ β γ σ ii hh γ µ ii hhW γ σ γ µ ii + 7613318144000 hh γ α γ α γ β γ β γ σ ii hh γ σ γ µ ii hhW γ µ ii + 7082318144000 hh γ α γ α γ β γ β W γ σ ii hh γ µ ii hh γ σ γ µ ii − hh γ σ ii hhW γ σ γ µ ii hh γ α γ α γ β γ β γ µ ii − hh γ σ ii hh γ β γ σ γ µ ii hh γ α γ α γ β W γ µ ii − hh γ σ ii hh γ β W γ σ γ µ ii hh γ α γ α γ β γ µ ii − hh γ σ ii hh γ β γ β γ σ γ µ ii hh γ α γ α W γ µ ii − hh γ σ ii hh γ β γ β W γ σ γ µ ii hh γ α γ α γ µ ii + 32527756000 hh γ σ ii hh γ α γ β γ σ γ µ ii hh γ α γ β W γ µ ii + 13579252000 hh γ σ ii hh γ α γ β W γ σ γ µ ii hh γ α γ β γ µ ii − hh γ σ ii hh γ α γ β γ β γ σ γ µ ii hh γ α W γ µ ii + 228784000 hh γ σ ii hh γ α γ β γ β W γ σ γ µ ii hh γ α γ µ ii − hh γ σ ii hh γ α γ α γ β γ β γ σ γ µ ii hhW γ µ ii + 334919072000 hh γ σ ii hh γ α γ α γ β γ β W γ σ γ µ ii hh γ µ ii − hhW γ σ ii hh γ β γ σ γ µ ii hh γ α γ α γ β γ µ ii + 20812918144000 hhW γ σ ii hh γ β γ β γ σ γ µ ii hh γ α γ α γ µ ii − hhW γ σ ii hh γ α γ β γ σ γ µ ii hh γ α γ β γ µ ii − hhW γ σ ii hh γ α γ β γ β γ σ γ µ ii hh γ α γ µ ii + 77571512000 hh γ β γ σ ii hh γ β γ σ γ µ ii hh γ α γ α W γ µ ii + 731271512000 hh γ β γ σ ii hh γ β W γ σ γ µ ii hh γ α γ α γ µ ii + 171728000 hh γ β γ σ ii hh γ α γ β γ σ γ µ ii hh γ α W γ µ ii − hh γ β γ σ ii hh γ α γ β W γ σ γ µ ii hh γ α γ µ ii + 14029189000 hh γ β W γ σ ii hh γ β γ σ γ µ ii hh γ α γ α γ µ ii + 33716800 hh γ β γ β γ σ ii hhW γ σ γ µ ii hh γ α γ α γ µ ii − hh γ σ W γ µ ii hh γ α γ α γ β γ β γ σ γ µ ii + 840136288000 hh γ σ γ σ γ µ ii hh γ α γ α γ β γ β W γ µ ii + 75136288000 hh γ σ γ σ W γ µ ii hh γ α γ α γ β γ β γ µ ii − hh γ β γ σ γ µ ii hh γ α γ α γ β γ σ W γ µ ii − hh γ β γ σ W γ µ ii hh γ α γ α γ β γ σ γ µ ii − hh γ β γ σ γ σ γ µ ii hh γ α γ α γ β W γ µ ii + 22693024000 hh γ β γ σ γ σ W γ µ ii hh γ α γ α γ β γ µ ii − hh γ β γ β γ σ γ σ γ µ ii hh γ α γ α W γ µ ii + 158312096000 hh γ β γ β γ σ γ σ W γ µ ii hh γ α γ α γ µ ii + 48431701000 hh γ α γ β γ σ γ µ ii hh γ α γ β γ σ W γ µ ii − hh γ α γ β γ σ W γ µ ii hh γ α γ β γ σ γ µ ii
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