Formality morphism as the mechanism of \star -product associativity: how it works
aa r X i v : . [ m a t h . QA ] J u l FORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY: HOW IT WORKS RICARDO BURING AND ARTHEMY V. KISELEV ‘Symmetries & integrability of equations of mathematical physics’, (22–24 December , IM NASU Kiev, Ukraine )A bstract . The formality morphism F = {F n , n > } in Kontsevich’s deformation quantiza-tion is a collection of maps from tensor powers of the di ff erential graded Lie algebra (dgLa) ofmultivector fields to the dgLa of polydi ff erential operators on finite-dimensional a ffi ne mani-folds. Not a Lie algebra morphism by its term F alone, the entire set F is an L ∞ -morphisminstead. It induces a map of the Maurer–Cartan elements, taking Poisson bi-vectors to defor-mations µ A ⋆ A [[ ~ ]] of the usual multiplication of functions into associative noncommutative ⋆ -products of power series in ~ . The associativity of ⋆ -products is then realized, in terms ofthe Kontsevich graphs which encode polydi ff erential operators, by di ff erential consequencesof the Jacobi identity. The aim of this paper is to illustrate the work of this algebraic mech-anism for the Kontsevich ⋆ -products (in particular, with harmonic propagators). We inspecthow the Kontsevich weights are correlated for the orgraphs which occur in the associator for ⋆ and in its expansion using Leibniz graphs with the Jacobi identity at a vertex. Introduction.
The Kontsevich formality morphism F relates two di ff erential graded Lie al-gebras (dgLa). Its domain of definition is the shifted-graded vector space T ↓ [1]poly ( M r ) of mul-tivectors on an a ffi ne real finite-dimensional manifold M r ; the graded Lie algebra structureis the Schouten bracket [[ , ]] and the di ff erential is set to (the bracket with) zero by defini-tion. On the other hand, the target space of the formality morphism F is the graded vectorspace D ↓ [1]poly ( M r ) of polydi ff erential operators on M r ; the graded Lie algebra structure is theGerstenhaber bracket [ , ] G and the di ff erential d H = [ µ A , · ] is induced by using the multiplica-tion µ A in the algebra A : = C ∞ ( M r ) of functions on M r . It is readily seen that w.r.t. the abovenotation, Poisson bi-vectors P satisfying the Jacobi identity [[ P , P ]] = M r are the Mau-rer–Cartan elements (indeed, ( d ≡ P ) + [[ P , P ]] = ⋆ = µ A [[ ~ ]] + h tail = : B i , which deforms the usual multiplication µ = µ A [[ ~ ]] in A [[ ~ ]] = C ∞ ( M r ) ⊗ R R [[ ~ ]] by a tail B w.r.t. a formal parameter ~ , the requirement that ⋆ beassociative again is the Maurer–Cartan equation,[ µ, B ] G + [ B , B ] G = ⇐⇒ [ µ + B , µ + B ] G = . Here, the leading order equality [ µ, µ ] G = µ itself. Date : 1 July 2019.2010
Mathematics Subject Classification. Address : Institut f¨ur Mathematik, Johannes Gutenberg–Universit¨at, Staudingerweg 9, D-55128 Mainz,Germany.
E-mail (corresponding author): [email protected] . Address : Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University ofGroningen, P.O.Box 407, 9700 AK Groningen, The Netherlands.
E-mail : [email protected] . The Kontsevich formality mapping F = {F n : T ⊗ n poly → D poly , n > } in [14, 15] is an L ∞ -morphism which induces a map that takes Maurer–Cartan elements P , i.e. formal Poissonbi-vectors ˜ P = ~ P + ¯ o ( ~ ) on M r , to Maurer–Cartan elements , i.e. the tails B in solutions ⋆ of the associativity equation on A [[ ~ ]].The theory required to build the Kontsevich map F is standard, well reflected in the lit-erature (see [14, 15], as well as [9, 11] and references therein); a proper choice of signs isanalysed in [2, 18]. The framework of homotopy Lie algebras and L ∞ -morphisms, introducedby Schlessinger–Stashe ff [17], is available from [16], cf. [10] in the context of present paper.So, the general fact of (existence of) factorization,Assoc( ⋆ )( P )( f , g , h ) = ^ (cid:0) P , [[ P , P ]] (cid:1) ( f , g , h ) , f , g , h ∈ A [[ ~ ]] , (1)is known to the expert community. Indeed, this factorization is immediate from the construc-tion of L ∞ -morphism in [15, §6.4]. We shall inspect how this mechanism works in practice,i.e. how precisely the ⋆ -product is made associative in its perturbative expansion wheneverthe bi-vector P is Poisson, thus satisfying the Jacobi identity Jac( P ) : = [[ P , P ]] =
0. Tothe same extent as our paper [6] justifies a similar factorization, [[ P , Q ( P )]] = ^ (cid:0) P , [[ P , P ]] (cid:1) ,of the Poisson cocycle condition for universal deformations ˙ P = Q ( P ) of Poisson struc-tures , we presently motivate the findings in [5] for ⋆ mod ¯ o ( ~ ), proceeding to the next order ⋆ mod ¯ o ( ~ ) from [7] (and higher orders, recently available from [3]). Let us emphasizethat the theoretical constructions and algorithms (contained in the computer-assisted proofscheme under study and in the tools for graph weight calculation) would still work at arbi-trarily high orders of expansion ⋆ mod ¯ o ( ~ k ) as k → ∞ . Explicit factorization (1) up to ¯ o ( ~ k )helps us build the star-product ⋆ mod ¯ o ( ~ k ) by using a self-starting iterative process, becausethe Jacobi identity for P is the only obstruction to the associativity of ⋆ . Specifically, theKontsevich weights of graphs on fewer vertices (yet with a number of edges such that theydo not show up in the perturbative expansion of ⋆ ) dictate the coe ffi cients of Leibniz or-graphs in operator ^ at higher orders in ~ . These weights in the r.-h.s. of (1) constrain thehigher-order weights of the Kontsevich orgraphs in the expansion of ⋆ -product itself. This isimportant also in the context of a number-theoretic open problem about the (ir)rational value(const ∈ Q \ { } ) · ζ (3) /π + (const ∈ Q ) of a graph weight at ~ in ⋆ (see [12] and [3]).Our paper is structured as follows. First, we fix notation and recall some basic facts fromrelevant theory. Secondly, we provide three examples which illustrate the work of formalitymorphism in solving Eq. (1). Specifically, we read the operators ^ k = ^ mod ¯ o ( ~ k ) satisfyingAssoc( ⋆ )( P )( f , g , h ) mod ¯ o ( ~ k ) = ^ k (cid:0) P , [[ P , P ]] (cid:1) ( f , g , h ) (1 ′ )at k =
2, 3, and 4. This corresponds to the expansions ⋆ mod ¯ o ( ~ k ) in [15], [5], and [7],respectively. One can then continue with k = ,
6; these expansions are in [3]. Independently,one can probe such factorizations using other stable formality morphisms: for instance, theones which correspond to a di ff erent star-product, the weights in which are determined by alogarithmic propagator instead of the harmonic one (see [1]). In fact, the morphism F is a quasi-isomorphism (see [15, Th. 6.3]), inducing a bijection between the sets ofgauge-equivalence classes of Maurer–Cartan elements. Universal w.r.t. all Poisson brackets on all finite-dimensional a ffi ne manifolds, such infinitesimal deforma-tions were pioneered in [14]; explicit examples of these flows ˙ P = Q ( P ) are given in [4, 8, 6]. Note that both the approaches – to noncommutative associative ⋆ -products and deformations of Poissonstructures – rely on the same calculus of oriented graphs by Kontsevich [13, 14, 15]. ORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY 3
1. T wo differential graded L ie algebra structures Let M r be an r -dimensional a ffi ne real manifold (we set k = R for simplicity). In the algebra A : = C ∞ ( M r ) of smooth functions, denote by µ A (or equivalently, by the dot · ) the usual com-mutative, associative, bi-linear multiplication. The space of formal power series in ~ over A will be A [[ ~ ]] and the ~ -linear multiplication in it is µ (instead of µ A [[ ~ ]] ). Consider two dif-ferential graded Lie algebra stuctures. First, we have that the shifted-graded space T ↓ [1]poly ( M r )of multivector fields on M r is equipped with the shifted-graded skew-symmetric Schoutenbracket [[ , ]] (itself bi-linear by construction and satisfying the shifted-graded Jacobi iden-tity); the di ff erential is set to zero. Secondly, the vector space D ↓ [1]poly ( M r ) of polydi ff erentialoperators (linear in each argument but not necessarily skew over the set of arguments or aderivation in any of them) is graded by using the number of arguments m : by definition,let deg( θ ( m arguments)) : = m −
1. For instance, deg( µ A ) =
1. The Lie algebra structureon D ↓ [1]poly ( M r ) is the Gerstenhaber bracket [ , ] G ; for two homogeneous operators Φ and Φ it equals [ Φ , Φ ] G = Φ ~ ◦ Φ − ( − ) deg Φ · deg Φ Φ ~ ◦ Φ , where the directed, non-associativeinsertion product is, by definition( Φ ~ ◦ Φ )( a , . . . , a k + k ) = k X i = ( − ) ik Φ (cid:0) a ⊗ . . . ⊗ a i − ⊗ Φ ( a i ⊗ . . . ⊗ a i + k ) ⊗ a i + k + ⊗ . . . ⊗ a k + k (cid:1) . In the above, Φ i : A ⊗ ( k i + → A so that a j ∈ A . Like [[ · , · ]], the Gerstenhaber bracket satisfiesthe shifted-graded Jacobi identity. The Hochshild di ff erential on D ↓ [1]poly ( M r ) is d H = [ µ A , · ] G ;indeed, its square vanishes, d H =
0, due to the Jacobi identity for [ , ] G into which one plugsthe equality [ µ A , µ A ] G = Example 1.
The associativity of the product µ A in the algebra of functions A = C ∞ ( M r ) isthe statement that µ (1) A ( µ (2) A ( a , a ) , a ) + ( − ( i = · (deg µ A = µ (1) A ( a , µ (2) A ( a , a )) − ( − ) (deg µ (1) A = · (deg µ (2) A = (cid:8) µ (1) A ( µ (1) A ( a , a ) , a ) − µ (2) A ( a , µ (1) A ( a , a )) (cid:9) = (cid:8) ( a · a ) · a − a · ( a · a ) (cid:9) = . So, the associator Assoc( µ A )( a , a , a ) = [ µ A , µ A ] G ( a , a , a ) = a j ∈ A .2. T he M aurer –C artan elements In every di ff erential graded Lie algebra with a Lie bracket [ , ], the Maurer–Cartan (MC)elements are solutions of degree 1 for the Maurer–Cartan equationd α + [ α, α ] = , (2)where d is the di ff erential (equal, we recall, to zero identically on T ↓ [1]poly ( M r ) and d H = [ µ A , · ] G on D ↓ [1]poly ( M r ). Likewise, the Lie algebra structure[ · , · ] is the Schouten bracket [[ · , · ]] and Ger-stenhaber bracket [ · , · ] G , respectively.)Now tensor the degree-one parts of both dgLa structures with ~ · k [[ ~ ]], i.e. with formalpower series starting at ~ , and, preserving the notation (that is, extending the brackets andthe di ff erentials by ~ -linearity), consider the same Maurer–Cartan equation (2). Let us studyits formal power series solutions α = ~ α + · · · . R. BURING AND A. V. KISELEV
So far, in the Poisson world we have that the Maurer–Cartan bi-vectors are formal Poissonstructures 0 + ~ P + ¯ o ( ~ ) satisfying (2), which is [[ ~ P + ¯ o ( ~ ) , ~ P + ¯ o ( ~ )]] = ff erential. In the world of associative structures, the Maurer–Cartan elements are the tails B in expansions ⋆ = µ + B , so that the associativity equation [ ⋆, ⋆ ] G = µ, µ ] G = µ, B ] G + [ B , B ] G = , which is again (2). 3. T he L ∞ - morphisms Our goal is to have (and use) a morphism T ↓ [1]poly ( M r ) → D ↓ [1]poly ( M r ) which would induce a mapthat takes Maurer–Cartan elements in the Poisson world to Maurer–Cartan elements in theassociative world.The leading term F , i.e. the first approximation to the morphism which we consider, isthe Hochschild–Kostant–Rosenberg (HKR) map (obviously, extended by linearity), F : ξ ∧ . . . ∧ ξ m m ! X σ ∈ S m ( − ) σ ξ σ (1) ⊗ . . . ⊗ ξ σ ( m ) , which takes a split multi-vector to a polydi ff erential operator (in fact, an m -vector). Moreexplicitly, we have that F : ( ξ ∧ . . . ∧ ξ m ) (cid:18) a ⊗ . . . ⊗ a m m ! X σ ∈ S m ( − ) σ Y mi = ξ σ ( i ) ( a i ) (cid:19) , (3)here a j ∈ A : = C ∞ ( M r ). For zero-vectors h ∈ A , one has F : h (1 h ). Claim 1 ([15, §4.6.2]) . The leading term, map F , is not a Lie algebra morphism (which,if it were, would take the Schouten bracket of multivectors to the Gerstenhaber bracket ofpolydi ff erential operators).Proof (by counterexample). Take two bi-vectors; their Schouten bracket is a tri-vector, butthe Gerstenhaber bracket of two bi-vectors is a di ff erential operator which has homogeneouscomponents of di ff erential orders (2,1,1) and (1,1,2). And in general, those components donot vanish. (cid:3) The construction of not a single map F but of an entire collection F = {F n , n > } ofmaps does nevertheless yield a well-defined mapping of the Maurer–Cartan elements fromthe two di ff erential graded Lie algebras. Theorem 2 ([15, Main Theorem]) . There exists a collection of linear maps F = {F n : T ↓ [1]poly ( M r ) ⊗ n → D ↓ [1]poly ( M r ) , n > } such that F is the HKR map (3) and F is an L ∞ -morphism of the two dif-ferential graded Lie algebras : (cid:0) T ↓ [1]poly ( M r ) , [[ · , · ]] , d = (cid:1) → (cid:0) D ↓ [1]poly ( M r ) , [ · , · ] G , d H = [ µ A , · ] G (cid:1) .Namely, (1) each component F n is homogeneous of own grading − n, (2) each morphism F n is graded skew-symmetric, i.e. F n ( . . . , ξ, η, . . . ) = − ( − ) deg( ξ ) · deg( η ) F n ( . . . , η, ξ, . . . ) for ξ, η homogeneous, The name ‘Formality’ for the collection F of maps is motivated by Theorem 4.10 in [15] and by the maintheorem in loc. cit. ORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY 5 (3) for each n > and (homogeneous) multivectors ξ , . . . , ξ n ∈ T ↓ [1]poly ( M r ) , we have that(cf. [11, §3.6] ) d H ( F n ( ξ , . . . , ξ n )) − ( − ) n − n X i = ( − ) u F n ( ξ , . . . , d ξ i , . . . , ξ n ) + X p + q = np , q > X σ ∈ S p , q ( − ) pn + t (cid:2) F p ( ξ σ (1) , . . . , ξ σ ( p ) ) , F q ( ξ σ ( p + , . . . , ξ σ ( n ) ) (cid:3) G = ( − ) n X i < j ( − ) s F n − (cid:0) [ ξ i , ξ j ] , ξ , . . . , b ξ i , . . . , b ξ j , . . . , ξ n (cid:1) . (4) In the above formula, σ runs through the set of ( p , q ) -shu ffl es, i.e. all permutations σ ∈ S n such that σ (1) < . . . < σ ( p ) and independently σ ( p + < . . . < σ ( n ); theexponents t and s are the numbers of transpositions of odd elements which we countwhen passing ( t ) from ( F p , F q , ξ , . . . , ξ n ) to ( F p , ξ σ (1) , . . . , ξ σ ( p ) , F q , ξ σ ( p + , . . . , ξ σ ( n ) ) ,and ( s ) from ( ξ , . . . , ξ n ) to ( ξ i , ξ j , ξ , . . . , b ξ , . . . , b ξ j , . . . , ξ n ) . Remark . Let n : =
1, then equality (4) in Theorem 2 isd H ◦ F − ( − ) − · ( − ) u = d ,ξ ) ( d ,ξ ) F ◦ d = ⇐⇒ d H ◦ F = F ◦ d , whence F is a morphism of complexes. • Let n : =
2, then for any homogeneous multivectors ξ and ξ , F (cid:0) [[ ξ , ξ ]] (cid:1) − (cid:2) F ( ξ ) , F ( ξ ) (cid:3) G = d H (cid:0) F ( ξ , ξ ) (cid:1) + F (cid:0) ( d = ξ ) , ξ (cid:1) + ( − ) deg ξ F (cid:0) ξ , ( d = ξ ) (cid:1) , so that in our case F is “almost” a Lie algebra morphism but for the discrepancy whichis controlled by the di ff erential of the (value of the) succeeding map F in the sequence F = {F n , n > } . Big formula (4) shows in precisely which sense this is also the case forhigher homotopies F n , n > L ∞ -morphism F . Indeed, an L ∞ -morphism is a mapbetween dgLas which, in every term, almost preserves the bracket up to a homotopy d H ◦ { . . . } provided by the next term.Even though neither F nor the entire collection F = {F n , n > } is a dgLa morphism,their defining property (4) guarantees that F gives us a well defined mapping of the Maurer–Cartan elements (which, we recall, are formal Poisson bi-vectors and tails B of associative(non)commutative multiplcations ⋆ = µ + B on A [[ ~ ]], respectively). Corollary 3.
The natural ~ -linear extension of F , now acting on the space of formal powerseries in ~ with coe ffi cients in T ↓ [1]poly ( M r ) and with zero free term by the rule ξ X n > n ! F n ( ξ, . . . , ξ ) , takes the Maurer–Cartan elements ˜ P = ~ P + ¯ o ( ~ ) to the Maurer–Cartan elements B = P n > n ! F n ( ˜ P , . . . , ˜ P ) = ~ ˜ P + ¯ o ( ~ ) . (Note that the HKR map F , extended by ~ -linearity, stillis an identity mapping on multivectors, now viewed as special polydi ff erential operators.) In plain terms, for a bivector P itself Poisson, formal Poisson structures ˜ P = ~ P + ¯ o ( ~ )satisfying [[ ˜ P , ˜ P ]] = F to the tails B = ~ P + ¯ o ( ~ ) such that ⋆ = µ + B isassociative and its leading order deformation term is a given Poisson structure P . The exponent u is not essential for us now because the di ff erential d on T ↓ [1]poly ( M r ) is set equal to zeroidentically, so that the entire term with u does not contribute (recall F n is linear). R. BURING AND A. V. KISELEV
Proof (of Corollary 3).
Let us presently consider the restricted case when ˜ P = ~ P , withoutany higher order tail ¯ o ( ~ ). The Maurer–Cartan equation in D ↓ [1]poly ( M r ) ⊗ ~k [[ ~ ]] is [ µ, B ] G + [ B , B ] G =
0, where B = P n > n ! F n ( ˜ P , . . . , ˜ P ) and we let ˜ P = ~ P , so that B = P n > ~ n n ! F n ( P , . . . , P ). Let us plug this formal power series in the l.-h.s. of the above equation. Equating thecoe ffi cients at powers ~ n and multiplying by n !, we obtain the expression[ µ, F n ( P , . . . , P )] G + X p + q = np , q > n ! p ! q ! (cid:2) F p ( P , . . . , P ) , F q ( P , . . . , P ) (cid:3) G . It is readily seen that now the sum P σ ∈ S p , q in (4) over the set of ( p , q )-shu ffl es of n = p + q identical copies of an object P just counts the number of ways to pick p copies going first in anordered string of length n . To balance the signs, we note at once that by item 2 in Theorem 2,see above, F p ( . . . , P ( α ) , P ( α + , . . . ) = + F p ( . . . , P ( α + , P ( α ) , . . . ) because bi-vector’s shifteddegree is +
1, so that no ( p , q )-shu ffl es of ( P , . . . , P ) contribute with any sign factor. The onlysign contribution that remains stems from the symbol F q of grading 1 − q transported along p copies of odd-degree bi-vector P ; this yields t = (1 − p ) · q and ( − ) pn + t = ( − ) p · ( p + q ) · ( − ) (1 − q ) · p = ( − ) p · ( p + = + .The left-hand side of the Maurer–Cartan equation (2) is, by the above, expressed by theleft-hand side of (4) which the L ∞ -morphism F satisfies. In the right-hand side of (4), wenow obtain (with, actually, whatever sign factors) the values of linear mappings F n − at twicethe Jacobiator [[ ˜ P , ˜ P ]] as one of the arguments. All these values are therefore zero, whichimplies that the right-hand side of the Maurer–Cartan equation (2) vanishes, so that the tail B indeed is a Maurer–Cartan element in the Hochschild cochain complex (in other words, thestar-product ⋆ = µ + B is associative).This completes the proof in the restricted case when ˜ P = ~ P . Formal power series bi-vectors ˜ P = ~ P + ¯ o ( ~ ) refer to the same count of signs as above, yet the calculation ofmultiplicities at ~ n (for all possible lexicographically ordered p - and q -tuples of n arguments)is an extensive exercise in combinatorics. (cid:3) Corollary 4.
Because the right-hand side of (2) in the above reasoning is determined bythe right-hand side of (4) , we read o ff an explicit formula of the operator ^ that solves thefactorization problem Assoc( ⋆ )( P )( f , g , h ) = ^ (cid:0) P , [[ P , P ]] (cid:1) ( f , g , h ) , f , g , h ∈ A [[ ~ ]] . (1) Indeed, the operator is ^ = · X n > ~ n n ! · c n · F n − (cid:0) [[ P , P ]] , P , . . . , P (cid:1) . (5)But what are the coe ffi cients c n ∈ R equal to? Let us find it out.4. E xplicit construction of the formality morphism F The first explicit formula for the formality morphism F which we study in this paper was dis-covered by Kontsevich in [15, §6.4], providing an expansion of every term F n using weighteddecorated graphs: F = n F n = X m > X Γ ∈ G n , m W Γ · U Γ o . Here Γ belongs to the set G n , m of oriented graphs on n internal vertices (i.e. arrowtails), m sinks (from which no arrows start), and 2 n + m − > ORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY 7 vertex there is an ordering of outgoing edges. By decorating each edge with a summationindex that runs from 1 to r , by viewing each edge as a derivation ∂/∂ x α of the arrowheadvertex content, by placing n multivectors from an ordered tuple of arguments of F n intothe respective vertices, now taking the sum over all indices of the resulting products of thecontent of vertices, and skew-symmetrizing over the n -tuple of (shifted-)graded multivectors,we realize each graph at hand as a polydi ff erential operator T ↓ [1]poly ( M r ) ⊗ n → D ↓ [1]poly ( M r ) whosearguments are multivectors. Note that the value F n ( ξ , . . . , ξ n ) itself is, by construction, adi ff erential operator w.r.t. the contents of sinks of the graph Γ . All of this is discussed indetail in [13, 14, 15] or [4, 5, 7].The formula for the harmonic weights W Γ ∈ R is given in [15, §6.2]; it is W Γ = n Y k = k )! ! · π ) n + m − Z ¯ C + n , m ^ e ∈ E Γ d φ e , where k ) is the number of edges star ting from vertex k , d ϕ e is the “harmonic angle”di ff erential 1-form associated to the edge e , and the integration domain ¯ C + n , m is the connectedcomponent of ¯ C n , m which is the closure of configurations where points q j , 1 j m on R are placed in increasing order: q < · · · < q m . For convenience, let us also define w Γ = (cid:18) n Y k = k )! (cid:19) · W Γ . The convenience is that by summing over labelled graphs Γ , we actually sum over the equiv-alence classes [ Γ ] (i.e. over unlabeled graphs) with multiplicities ( w Γ / W Γ ) · n ! / Γ ). Thedivision by the volume Γ ) of the symmetry group eliminates the repetitions of graphswhich di ff er only by a labeling of vertices but, modulo such, do not di ff er by the labeling ofordered edge tuples (issued from the vertices which are matched by a symmetry).Let us remember that the integrand in the formula of W Γ is defined in terms of the harmonicpropagator; other propagators (e.g. logarithmic, or other members of the family interpolatingbetween harmonic and logarithmic [1]) would give other formality morphisms. A path inte-gral realization of the ⋆ -product itself and of the components F n in the formality morphismis proposed in [10].To calculate the graph weights W Γ in practice, we employ methods which were outlinedin [7], as well as [12, App. E] (about the cyclic weight relations), and [3] that puts those realvalues in the context of Riemann multiple zeta functions and polylogarithms. Examples ofsuch decorated oriented graphs Γ and their weights W Γ will be given in the next section.4.1. Sum over equivalence classes.
The sum in Kontsevich’s formula is over labeled graphs:internal vertices are numbered from 1 to n , and the edges starting from each internal vertex k are numbered from 1 to k ). Under a re-labeling σ : Γ Γ σ of internal vertices andedges it is seen from the definitions that the operator U Γ and the weight W Γ enjoy the sameskew-symmetry property (as remarked in [15, §6.5]), whence W Γ · U Γ = W Γ σ · U Γ σ . It followsthat the sum over labeled graphs can be replaced by a sum over equivalence classes [ Γ ] ofgraphs, modulo labeling of internal vertices and edges. For this it remains to count the sizeof an equivalence class: the edges can be labeled in Q nk = k )! ways, while the n internalvertices can be labeled in n ! / Γ ) ways. It is the values w Γ instead of W Γ which are calculated by software [3]. R. BURING AND A. V. KISELEV
Example 2.
The double wedge on two ground vertices has only one possible labeling ofvertices, due to the automorphism that interchanges the wedges.We denote by M Γ = (cid:0) Q nk = k )! (cid:1) · n ! / Γ ) the multiplicity of the graph Γ , and let¯ G n , m be the set of equivalence classes [ Γ ] modulo labeling of Γ ∈ G n , m . The formula for theformality morphism can then be rewritten as F = n F n = X m > X [ Γ ] ∈ ¯ G n , m M Γ · W Γ · U Γ o ;here the Γ in M Γ · W Γ · U Γ is any representative of [ Γ ]. Any ambiguity in signs (due to thechoice of representative) in the latter two factors is cancelled in their product. Note that thefactor (cid:0) Q nk = k )! (cid:1) in M Γ kills the corresponding factor in W Γ , as remarked in [15, §6.5].4.2. The coe ffi cient of a graph in the ⋆ -product. The ⋆ -product associated to a Poissonstructure P is given by Corollary 3: ⋆ = µ + X n > ~ n n ! F n ( P , . . . , P ) = µ + X n > ~ n n ! X [ Γ ] ∈ ¯ G n , M Γ · W Γ · U Γ ( P , . . . , P ) . For a graph Γ ∈ G n , such that each internal vertex has two outgoing edges (these are theonly graphs that contribute, because we insert bi-vectors) we have M Γ = n · n ! / Γ ).In total, the coe ffi cient of U Γ ( P , . . . , P ) at ~ n is 2 n / Γ ) · W Γ = w Γ / Γ ). The skew-symmetrization without prefactor of bi-vector coe ffi cients in U Γ ( P , . . . , P ) provides an extrafactor 2 n . Example 3 (at ~ ) . The coe ffi cient of the wedge graph is 1 / P , hencewe recover P .4.3. The coe ffi cient of a Leibniz graph in the associator. The factorizing operator ^ forAssoc( ⋆ ) is given by Corollary 4: ^ = · X n > ~ n n ! · c n · F n − (cid:0) [[ P , P ]] , P , . . . , P (cid:1) = · X n > ~ n n ! · c n · X [ Γ ] ∈ ¯ G n − , M Γ · W Γ · U Γ (cid:0) [[ P , P ]] , P , . . . , P (cid:1) . For a graph Γ ∈ G n − , where one internal vertex has three outgoing edges and the rest havetwo, we have M Γ = · n − · ( n − / Γ ). In total, the coe ffi cient of U Γ ([[ P , P ]] , P , . . . , P )at ~ n is (cid:20) · n ! · c n · · n − · ( n − (cid:21) · W Γ Γ ) = (cid:20) · c n n (cid:21) · w Γ Γ )The skew-symmetrization without prefactor of bi- and tri-vector coe ffi cients in the operator U Γ ([[ P , P ]] , P , . . . , P ) provides an extra factor 3! · n − . Example 4 (at ~ ) . The coe ffi cient of the tripod graph is c · and the operator is 3! · [[ P , P ]],hence we recover c [[ P , P ]] = Jac( P ). (The right-hand side is known from the associator,e.g. from [5].) This yields c = /
3. In addition, we see that the HKR map F acts here bythe identity on [[ P , P ]]. ORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY 9 In the next section, we shall find that at ~ n , the coe ffi cients of our Leibniz graphs (withJac( P ) inserted instead of [[ P , P ]]) are[[ P , P ]]Jac( P ) · (cid:20) · n − (cid:21) · (cid:20) · c n n (cid:21) · w Γ Γ ) = n · w Γ Γ ) , so 3! · n · c n n = n . We deduce that c n = n / = n / Conjecture.
For all n > , the coe ffi cients in (5) are c n = n / = n / (hence, the coe ffi cientsof markers Γ for equivalence classes [ Γ ] of the Leibniz graphs in (5) are n · w Γ / Γ ) ),although it still remains to be explained how exactly this follows from the L ∞ condition (4) .
5. E xamples
Let P be a Poisson bi-vector on an a ffi ne manifold M r . We inspect the asssociativity of thestar-product ⋆ = µ + P n > ~ n n ! F n ( P , . . . , P ) given by Corollary 3 by illustrating the work of thefactorization mechanism from Corollary 4. The powers of deformation parameter ~ providea natural filtration ~ · A (2) + ~ · A (3) + ~ · A (4) + ¯ o ( ~ ) so that we verify the vanishing ofAssoc( ⋆ )( P )( · , · , · ) mod ¯ o ( ~ ) for ⋆ mod ¯ o ( ~ ) order by order.At ~ there is nothing to do (indeed, the usual multiplication is associative). All contribu-tion to the associator of ⋆ at ~ cancels out because the leading deformation term ~ P in thestar-product ⋆ = µ + ~ P + ¯ o ( ~ ) is a bi-derivation. The order ~ was discussed in Example 4in §4.3. Remark . In all our reasoning at any order ~ n > , the Jacobiator in Leibniz graphs is expanded(w.r.t. the three cyclic permutations of its arguments) into the Kontsevich graphs, built ofwedges, in such a way that the internal edge, connecting two Poisson bi-vectors in Jac( P ), isproclaimed Left by construction. Specifically, the algorithm to expand each Leibniz graphsis as follows:(1) Split the trivalent vertex with ordered targets ( a , b , c ) into two wedges: the first wedgestands on a and b (in that order), and the second wedge stands on the first wedge-top and c (in that order), so that the internal edge of the Jacobiator is marked Left,preceding the Right edge towards c .(2) Re-direct the edges (if any) which had the tri-valent vertex as their target, to one of thewedge-tops; take the sum over all possible combinations (this is the iterated Leibnizrule).(3) Take the sum over cyclic permutations of the targets of the edges which (initially)have ( a , b , c ) as their targets (this is the expansion of the Jacobiator).5.1. The order ~ . To factorize the next order expansion of the associator, Assoc( ⋆ )( P )mod ¯ o ( ~ ) = ~ · A (2) + ~ · A (3) + ¯ o ( ~ ), at ~ in the operator ^ in the right-hand side of (1),we use graphs on n − = m = n − + m − = ~ , two internal vertices in the Leibniz graphs in the r.-h.s. of factorization (1) aremanifestly di ff erent: one vertex, containg the bi-vector P , is a source of two outgoing edges,and the other, with [[ P , P ]], of three. Therefore, the automorphism groups of such Leibnizgraphs (under relabellings of internal vertices of the same valency but with the sinks fixed)can only be trivial, i.e. one-element. (This will not necessarily be the case of Leibniz graphson ( n − + ~ > : compare Examples 8 vs 9 on p. 13 below, where theweight of a graph is divided further by the size of its automorphism group.) The coe ffi cient of ~ in the factorizing operator ^ ,coe ff ( ^ , ~ ) = · · c · X [ Γ ] ∈ ¯ G , M Γ · W Γ · U Γ (cid:0) [[ P , P ]] , P , . . . , P (cid:1) , expands into a sum of
24 admissible oriented graphs. Indeed, there are six essentiallydi ff erent oriented graph topologies, filtered by the number of sinks on which the tri-vector[[ P , P ]] and bi-vector P stand; the ordering of sinks in the associator then yields 3 + + × + × + =
24 oriented graphs. (None of them is a zero orgraph.) As we recallfrom [5], only thirteen of them actually occur with nonzero coe ffi cients in the term A (3) ∼ ~ inAssoc( ⋆ )( P )), the remaining eleven have zero weights. The weights of 15 relevant orientedLeibniz graphs from [5] are listed in Table 1. T able
1. Weights w Γ of oriented Leibniz graphs Γ in coe ff ( ^ , ~ ).( S f ) = [ ; ] ( S g ) = [ ; ] ( S h ) = [ ; ] − ( I f ) = [ ; ] ( I g ) = [ ; ] ( S h ) = [ ; ] − ( S f ) = [ ; ] ( I g ) = [ ; ] − ( I h ) = [ ; ] − ( I f ) = [ ; ] ( I h ) = [ ; ] − ( I g ) = [ ; ] 0( S g ) = [ ; ] 0 ( I f ) = [ ; ] ( I h ) = [ ; ] − Here we let by definition I f : = ∂ j (cid:0) Jac( P )( P i j , g , h ) (cid:1) ∂ i f = ✑✑✑ r r rr ❅❅❘(cid:0)(cid:0)✠ r ❅❅❅❘(cid:0)(cid:0)✠ ★✧ ✥✦ ❄✕ r j − ✑✑✑ r r rr ❍❍❍❥✟✟✟✙ r (cid:0)(cid:0)✠✁✁✁☛ ★✧ ✥✦ ❄✕ r R j − ✑✑✑ r r rr ❅❅❘(cid:0)(cid:0)✠ r (cid:0)(cid:0)(cid:0)✠ ❅❅❘ ★✧ ✥✦ ❄✕ r j = . Likewise, I g : = ∂ j (cid:0) Jac( P )( f , P i j , h ) (cid:1) · ∂ i g and I h : = ∂ j (cid:0) Jac( P )( f , g , P i j ) · ∂ i h , respectively. We also set S f : = P i j ∂ j Jac( P )( ∂ i f , g , h ) = i r r rr ❅❅❘(cid:0)(cid:0)✠ r ❅❅❅❘(cid:0)(cid:0)✠ ★✧ ✥✦r ❆❆❆❯❍❍❥ − i r r rr ❍❍❍❥✟✟✟✙ r (cid:0)(cid:0)✠✁✁✁☛ ★✧ ✥✦r ❆❆❆❯❍❍❥ L R − i r r rr ❅❅❘(cid:0)(cid:0)✠ r (cid:0)(cid:0)(cid:0)✠ ❅❅❘ ★✧ ✥✦r ❆❆❆❯❍❍❥ = . Similarly, we let S g : = P i j ∂ j Jac( P )( f , ∂ i g , h ) = S h : = P i j ∂ j Jac( P )( f , g , ∂ i h ) =
0. Notethat after all the Leibniz rules are reworked, each of the six graphs I f , . . . , S h – with the Jacobi-ator Jac( P ) = [[ P , P ]] at the tri-valent vertex – splits into several homogeneous components,like ( I f ) or ( S h ) ; taken alone, each of the components encodes a zero polydi ff erentialoperator of respective orders. Claim 5.
Multiplied by a common factor (cid:0) [[ P , P ]] / Jac( P ) (cid:1) · k − = · = , the Leibnizgraph weights from Table 1 at ~ fully reproduce the factorization which was found in the Yet, these seemingly ‘unnecessary’ graphs can contribute to the cyclic weight relations (see [12, App. E]):zero values of some of such graph weights can simplify the system of linear relations between nonzero weights. To get the values, one uses the software [3] by Banks–Panzer–Pym or, independently, exact symbolic orapproximate numeric methods from [7], also taking into account the cyclic weight relations from [12, App. E]. In [5], the indices i and j were interchanged in the definitions of both I g and I h (compare the expressionof I f ); that typo is now corrected in the above formulae. ORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY 11 main Claim in [5] , namely : A (3)221 = ( S f ) , A (3)122 = ( S g ) , A (3)212 = − ( S h ) , A (3)111 = ( I f − I h ) , A (3)112 = (cid:0) I f + I g − S h (cid:1) , A (3)121 = ( I f − I h ) , A (3)211 = (cid:0) S f − I g − I h (cid:1) . Otherwise speaking, the sum of these Leibniz oriented graphs with these weights (times 2 · = ~ -proportional term in the associator Assoc( ⋆ )( P )( f , g , h ). Proof scheme.
The encodings of weighted Kontsevich-graph expansions of the homogeneouscomponents of the weighted Leibniz graphs I f , . . . , S h , which show up in the associatorat ~ and which are processed according to the algorithm in Remark 2, are listed in Appen-dix A. Reducing that collection modulo skew symmetry at internal vertices, we reproduce,as desired, the entire term A (3) in the expansion ~ · A (2) + ~ · A (3) + ¯ o ( ~ ) of the associatorAssoc( ⋆ )( P ) mod ¯ o ( ~ ). (cid:3) Three examples, corresponding to the leftmost column of equalities in Claim 5, illustratethis scheme at order ~ . The three cases di ff er in that for A (3)221 in Example 5, there is just oneLeibniz graph without any arrows acting on the Jacobiator vertex. In the other Example 6for A (3)121 , there are two Leibniz graphs still without Leibniz-rule actions on the Jacobiatorsin them, so that we aim to show how similar terms are collected. Finally, in Example 7about A (3)111 there are two Leibniz graphs with one Leibniz rule action per either graph: anarrow targets the two internal vertices in the Jacobiator. Example 5.
Take the Leibniz graph ( S f ) = [ ; ]. Its weight is 1 /
12. Multiplyingthe Leibniz graph by 8 times its weight and expanding the Jacobiator (there are no Leibnizrules to expand) yields the sum of three Kontsevich graphs: (cid:0) [ ; ; ] + [ ; ; ] + [ ; ; ] (cid:1) . This is identically equal to the di ff erential order (2 , ,
1) homogeneous part A (3)221 of Assoc( ⋆ )( P ) at ~ . For instance, these terms are listed in [7, App. D]. Example 6.
Take the Leibniz graphs ( I f ) = [ ; ] and ( I h ) = [ ; ]. Theirweights are 1 /
24 and − /
24, respectively; multiply them by 8. Expanding the Jacobiatorin the linear combination ( I f − I h ) yields the sum of Kontsevich graphs (cid:0) [ ; ; ] + [ ; ; ] + [ ; ; ] − [ ; ; ] − [ ; ; ] − [ ; ; ] (cid:1) . The two Leibniz graphshave a Kontsevich graph in common: [ ; ; ] = [ ; ; ] (recall that internal vertexlabels can be permuted at no cost and the swap L ⇄ R at a wedge costs a minus sign). Thisgives one cancellation; the remaining four terms equal A (3)121 as listed in [7, App. D]. Example 7.
Take the Leibniz graphs ( I f ) = [ ; ] and ( I h ) = [ ; ]. Theirweights are 1 /
48 and − /
48, respectively; multiply them by 8. Expanding the Jacobiator and To collect and compare the Kontsevich orgraphs (built of wedges, i.e. ordered edge pairs issued frominternal vertices), we can bring every such graph to its normal form, that is, represent it using the minimal base-( + ≺ ≺ . . . ≺ m − .) the Leibniz rule in the linear combination ( I f − I h ) yields the sum of Kontsevich graphs: (cid:0) [ ; ; ] + [ ; ; ] + [ ; ; ] + [ ; ; ] + [ ; ; ] + [ ; ; ] − [ ; ; ] − [ ; ; ] − [ ; ; ] − [ ; ; ] − [ ; ; ] − [ ; ; ] (cid:1) . Two pairs of graphs cancel; namely [ ; ; ] = [ ; ; ] and [ ; ; ] = [ ; ; ].The remaining eight terms equal A (3)111 as listed in [7, App. D].5.2. The order ~ . Let us proceed with the term A (4) at ~ in the associator Assoc( ⋆ )( P )( · , · , · )mod ¯ o ( ~ ). The numbers of Kontsevich oriented graphs in the star-product expansion growas fast as ⋆ = ~ · ( = + ~ · ( = + ~ · ( = + ~ · ( = + ~ · ( = ++ ~ · ( = + ~ · ( = + ¯ o ( ~ );here we report the count of all nonzero-weight Kontsevich oriented graphs. Counting themmodulo automorphisms (which may also swap the sinks), Banks, Panzer, and Pym obtain thenumbers ( ~ : 1, ~ : 1, ~ : 3, ~ : 8, ~ : 133, ~ : 1209, ~ : 33268). This shows thatat orders ~ k > , the use of graph-processing software is indispensible in the task of verifyingfactorization (1) using weighted graph expansion (5) of the operator ^ .Specifically, the number of Kontsevich oriented graphs at ~ k in the left-hand side of thefactorization problem Assoc( ⋆ )( P )( · , · , · ) = ^ (cid:0) P , [[ P , P ]] (cid:1) ( · , · , · ), and the number of Leibnizgraphs which assemble with nonzero coe ffi cients to a solution ^ in the right-hand side ispresented in Table 2. At ~ , the expansion of Assoc( ⋆ )( P ) mod ¯ o ( ~ ) requires 241 nonzeroT able
2. Number of graphs in either side of the factorization. k ff , | {z } Reference §4.3, [15] §5.1, [5] §5.2, [7] [3]coe ffi cients of Leibniz graphs on 3 sinks, 2 = n − P and oneinternal vertex for the tri-vector [[ P , P ]], and therefore, 2( n − + = n + − = Remark . Again, this set of Leibniz graphs is well structured. Indeed, it is a disjoint unionof homogeneous di ff erential operators arranged according to their di ff erential orders w.r.t. thesinks, e.g., (1 , , , , , , , , , , Example 8.
The Leibniz graph L : = [ ; ; ] of di ff erential orders (3 , ,
1) has theweight 1 /
24 according to [3]. Multiplied by a universal (for all graphs at ~ ) factor 2 = / ( L )) = / ⇄ (cid:0) [ ; ; ; ] + [ ; ; ; ] + [ ; ; ; ] (cid:1) by the definition of Jacobi’s identity.This sum of three weighted Kontsevich orgraphs reproduces exactly A (4)331 , which is knownfrom [7, Table 8 in App. D]. ORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY 13 Example 9.
The Leibniz graph L : = [ ; ; ] of di ff erential orders (3 , ,
2) has theweight 1 /
24 according to [3]. Multiplied now by a universal (for all graphs at ~ ) factor2 =
16 and the factor 1 / ( L )) =
1, it expands to (cid:0) [ ; ; ; ] + [ ; ; ; ] + [ ; ; ; ] (cid:1) . This sum reproduces A (4)322 (again, see [7, Table 8 in App. D]). Example 10.
Consider at the di ff erential order (1 , ,
2) at ~ the three Leibniz graphs L (1)132 : = [ ; ; ], L (2)132 : = [ ; ; ], and L (3)132 : = [ ; ; ]. They have no symmetries, i.e.their automorphism groups are one-element, and their weights are W ( L (1)132 ) = / W ( L (2)132 ) = /
48, and W ( L (3)132 ) = /
48, respectively. Pre-multiplied by their weights and universal factor2 =
16, these Leibniz graphs expand to (cid:0) [ ; ; ; ] + [ ; ; ; ] + [ ; ; ; ] (cid:1) + (cid:0) [ ; ; ; ] + [ ; ; ; ] + [ ; ; ; ] (cid:1) + (cid:0) [ ; ; ; ] + [ ; ; ; ] + [ ; ; ; ] (cid:1) . There is one cancellation, since [ ; ; ; ] = − [ ; ; ; ]. The remaining seventerms reproduce exactly A (4)132 ; that component is known from [7, Table 8 in App. D].Actually, there was another Leibniz graph at this homogeneity order, L (4)132 : = [ ; ; ],but its weight is zero and hence it does not contribute. (Indeed, we get an independent veri-fication of this by having already balanced the entire homogeneous component at di ff erentialorders (1 , ,
2) in the associator.)
Intermediate conclusion.
We have experimentally found the constants c k in Corollary 4which balance the Kontsevich graph expansion of the ~ k -term A ( k ) in the associator against anexpansion of the respective term at ~ k in the r.-h.s. of (1) using the weighted Leibniz graphs.Namely, we conjecture c k = k / i < j in the L ∞ condition (4) (perhaps, in combination with di ff erentnormalizations of the objects which we consider) still remains to be explained, similar tothe reasoning in [2, 18] where the signs are fixed. Note that both in the associator, which isquadratic w.r.t. the weights of Kontsevich graphs in ⋆ , and in the operator ^ , which is linearin the Kontsevich weights of Leibniz graphs, the weight values are provided simultaneously,by using identical techniques (for instance, from [3]). Indeed, the weights are provided bythe integral formula which is universal with respect to all the graphs under study [15].A ppendix A. E ncodings of weighted K ontsevich - graph expansions for ( p , q , r )- homogeneous components ( I f , . . . , S h ) pqr ORMALITY MORPHISM AS THE MECHANISM OF ⋆ -PRODUCT ASSOCIATIVITY 15 Acknowledgements.
The first author thanks the Organisers of international workshop ‘Sym-metries & integrability of equations of Mathematical Physics’ (22–24 December 2018, IMNASU Kiev, Ukraine) for helpful discussions and warm atmosphere during the meeting.A part of this research was done while RB was visiting at RUG and AVK was visiting atJGU Mainz (supported by IM JGU via project 5020 and JBI RUG project 106552). Theresearch of AVK is supported by the IH ´ES (partially, by the Nokia Fund).R eferences [1]
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