M. Kontsevich's graph complexes and universal structures on graded symplectic manifolds I
MM. Kontsevich’s graph complexes and universal structureson graded symplectic manifolds I.
Kevin MorandDepartment of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107, South Korea [email protected]
Abstract
In the formulation of his celebrated
Formality conjecture , M. Kontsevich introduced a universal version of thedeformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformationcomplex takes the form of a differential graded Lie algebra of graphs, denoted fGC , together with an injectivemorphism towards the Chevalley–Eilenberg complex associated with the Schouten algebra. The latter morphismis given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebraas the algebra of functions on a graded symplectic manifold of degree . The ambition of the present series of worksis to generalise this construction to graded symplectic manifolds of arbitrary degree n ≥ . The correspondinggraph model is given by the full Kontsevich graph complex fGC d where d = n + 1 stands for the dimension ofthe associated AKSZ type σ -model. This generalisation is instrumental to classify universal structures on gradedsymplectic manifolds. In particular, the zeroth cohomology of the full graph complex fGC d is shown to act via Lie ∞ -automorphisms on the algebra of functions on graded symplectic manifolds of degree n . This generalisesthe known action of the Grothendieck–Teichmüller algebra grt (cid:39) H ( fGC ) on the space of polyvector fields.This extended action can in turn be used to generate new universal deformations of Hamiltonian functions,generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees.As an application of the general formalism, new universal flows on the space of Courant algebroids are presented. Contents
Gra d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 The full graph complex fGC d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Cohomology of the full graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 The directed graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 n -algebroids 29 n = 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Courant algebroids ( n = 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 a r X i v : . [ m a t h . QA ] A ug Introduction
In a seminal 97’ preprint [79], M. Kontsevich proved his celebrated formality theorem by constructing an explicit
Lie ∞ quasi-isomorphism U Φ : T poly ∼ −→ D poly (1.1)between T poly , the graded Lie algebra of polyvector fields on the affine space R m , and D poly , the Hochschilddifferential graded Lie algebra (dg Lie algebra) of multidifferential operators on R m , and such that the firstTaylor coefficient coincides with the Hochschild–Kostant–Rosenberg (HKR) quasi-isomorphism of complexes .An important corollary of the formality theorem is that it provides an explicit bijective map : ˆ U Φ : FPoiss ∼ −→ Star (1.2)between the set
FPoiss of (equivalence classes of) formal Poisson structures on R m and the set Star of (equivalenceclasses of) formal associative deformations of the algebra of functions on R m (also called star products ).The bijection (1.2) straightforwardly induces a quantization map Poiss → Star assigning to any Poisson bivector π ∈ Poiss on R m an equivalence class of star products [ ∗ ] ∈ Star quantizing π .An important characteristic of Kontsevich’s formality morphism is that it is given by universal formulas i.e. formulas applying without distinction to all affine spaces of all finite dimensions and which are defined “graphi-cally” via grafting of existing structures on T poly without resorting to additional data. Such formality morphismswere called stable in [32]. Informally, these are Lie ∞ quasi-isomorphisms whose Taylor coefficients can be writ-ten as a sum over Kontsevich’s admissible graphs [79] where the coefficient in front of each graph is given bya weight function, cf. e.g. [114]. The master equation ensuring that the Taylor maps assemble to a Lie ∞ -morphism thus boils down to a series of identities on the weights. Although these equations are algebraic, theonly known explicit solutions make use of transcendental methods involving integrals over (compactificationsof) configuration spaces of points.Kontsevich’s formality theorem indisputably constitutes the most remarkable result in the field of deformationquantization, providing a complete solution to the quantization problem formulated in [14, 13]. However, thetranscendental methods involved in the construction are generically difficult to handle thus calling for morealgebraic tools allowing to address issues in formality theory and deformation quantization. Such algebraicmethods have in fact been introduced by M. Kontsevich prior to [79] in the formulation of his Formality conjecture around 93’-94’ [78] ( cf. also [112]). More precisely, M. Kontsevich defined in [78] a universal version of thedeformation theory for formality morphisms. Recall that, on very general grounds, any dg Lie algebra g is quasi-isomorphic (as a Lie ∞ -algebra) to its cohomology H ( g ) endowed with a certain Lie ∞ -structure obtained from thedg Lie algebra structure on g via the homotopy transfer theorem. This allows in particular to address formalityquestions by studying the space of Lie ∞ -structures on H ( g ) . Going back to the case at hand, the relevantdeformation theory is therefore controlled by the Chevalley–Eilenberg dg Lie algebra CE ( T poly ) associated withthe Schouten algebra of polyvector fields. In [78], M. Kontsevich introduced a universal version of CE ( T poly ) inthe guise of a dg Lie algebra of graphs, denoted fGC , together with an injective morphism fGC (cid:44) → CE ( T poly ) (1.3)given by local formulas. The morphism (1.3) allows to reformulate questions regarding formality morphisms onaffine manifolds (in the stable setting) into purely algebraic questions on the cohomology of the graph complex The subscript Φ in (1.1) will be hereafter interpreted as denoting a Drinfel’d associator. The proof that a
Lie ∞ quasi-isomorphism between two dg Lie algebras induces a bijection between the associated Delignegroupoids [49] can be found in [79, 26] for the nilpotent case and in [118] for the pro-nilpotent case. Via composition of the bijective map (1.2) with the canonical “formalisation map”
Poiss → FPoiss : π (cid:55)→ (cid:15) π where (cid:15) is a formalparameter. See [33, 38] for a recursive construction of formality morphisms over rationals. GC . In particular, obstructions to the existence of a stable formality morphism live in H ( fGC ) while suchmorphisms can be shown to be classified by H ( fGC ) . More precisely, it was shown by V. A. Dolgushev in[32] that the exponentiation of the (pro-nilpotent) graded Lie algebra H ( fGC ) acts regularly on the space SQI of (homotopy classes of) stable
Lie ∞ quasi-isomorphisms of the form (1.1) so that SQI is a torsor (or principalhomogeneous space) for the pro-unipotent group exp (cid:0) H ( fGC ) (cid:1) . Furthermore, T. Willwacher constructedin [113] an explicit isomorphism of Lie algebras H ( GC ) (cid:39) grt where grt stands for the Grothendieck–Teichmüller Lie algebra. Combining these two results leads to a full characterisation of the set SQI of stableformality maps as a
GRT -torsor where GRT stands for the (pro-unipotent ) Grothendieck–Teichmüller group GRT = exp( grt ) . The Grothendieck–Teichmüller group was introduced by V. Drinfel’d in [40] in virtueof its relation to the absolute Galois group Gal ( ¯ Q / Q ) and the theory of quasi-Hopf algebras. Since then,the Grothendieck–Teichmüller group (together with the GRT -torsor of Drinfel’d associators) have found anumber of applications in various areas of mathematics including the Kashiwara–Vergne conjecture in Lietheory [2, 4, 1, 103], quantization of Lie bialgebras [42, 110], the study of multiple zeta values [82, 18, 46],rational homotopy automorphisms of the E -operad [44, 113], etc. The action of the Grothendieck–Teichmüller group on formality morphisms can be traced back to an action of
GRT (cid:39) exp (cid:0) H ( fGC ) (cid:1) on T poly via Lie ∞ -automorphisms . Explicitly, to any graph cocycle γ ∈ H ( fGC ) oneassociates a Lie ∞ -automorphism U Γ : T poly ∼ −→ T poly where Γ := exp( γ ) ∈ exp (cid:0) H ( fGC ) (cid:1) . Composition with(1.1) leads to a new formality morphism U Φ · Γ := U Φ ◦ U Γ : T poly ∼ −→ D poly . Furthermore, the bijection betweenDeligne groupoids derived from U Γ ( cf. footnote 2) induces a universal deformation map ˆ U Γ : FPoiss ∼ −→ FPoiss .In particular, the latter can be used to map Poisson bivectors π ( cf. footnote 3) towards universal formal Poissonstructures deforming π . At first order, such universal deformations can be interpreted as maps from cocycles in H ( GC ) to universal flows on the space of Poisson bivectors. The first example of such flows is the so-called tetrahedral flow introduced by M. Kontsevich in [78, Section 5.3], cf. Section 6.1.Remarkably, Kontsevich’s solution to the quantization problem of [14, 13] is inspired by ideas coming fromstring theory. Explicitly, Kontsevich’s quantization formula can be interpreted [79, 24] as the Feynman diagramexpansion of a 2-dimensional topological field theory – the
Poisson σ -model – introduced in [64, 61, 102]. Asnoted in [25], the quantization of the Poisson σ -model can be best interpreted within the AKSZ formalism[6]. The latter deals with theories living on the space of maps between a source manifold of dimension d anda target manifold classically endowed with a structure of differential graded symplectic manifold of degree n and such that d = n + 1 . The first and simplest example of such construction is provided by the Poisson σ -model where the source is of dimension d = 2 and the target is the (shifted) cotangent bundle of a (finitedimensional) Poisson manifold. More generally, we will refer to the geometrical structure necessary to define aAKSZ σ -model in dimension d as a symplectic Lie n -algebroid , with d = n +1 . While symplectic Lie -algebroidsidentify with Poisson manifolds, symplectic Lie -algebroids correspond to Courant algebroids . The latter firstappeared implicitly in the study of integrable Dirac structures [39, 28] before their precise geometric structure Or equivalently non-trivial universal deformations of the Schouten graded Lie algebra as a
Lie ∞ -algebra. This fact was conjectured by M. Kontsevich in [80] based on the relations between the transcendental formulas involved in hisformality morphism and the theory of mixed Tate motives. The Grothendieck–Teichmüller group and Drinfel’d associators alsoappear in D. Tamarkin’s approach to formality [109, 57] via either the use of the Etingof–Kazdhan quantization of Lie bialgebrasor the formality of little disks operad, cf.
Section 6.1 for details. There are different versions of the Grothendieck–Teichmüller group, the most important ones being a profinite version (cid:99) GT , apro- l version GT l and a pro-unipotent version GT . The latter is isomorphic to a graded version of the group, denoted GRT . Wewill only be concerned with the exponentiation
GRT = exp( grt ) related to GRT via
GRT = K × (cid:110) GRT where the action of themultiplicative group is via rescaling, cf. [116] for details. Inspired by A. Grothendieck’s suggestion in his
Esquisse d’un Programme [53] of studying the combinatorial properties ofGal (¯ Q / Q ) via its natural action on the tour of Teichmüller groupoids. We refer to [113] (see also [88]) for the affine space case, [66] for a globalisation to any smooth manifolds and [37] for ageneralisation to the sheaf of polyvector fields on any smooth algebraic variety. Various examples of flows on the space of Poisson bivectors have recently been systematically investigated in a series of worksby A. V. Kiselev and collaborators, see [15, 16, 20, 21, 22, 19, 71]. Also referred to as a
NPQ -manifold of degree n , cf. Section 3 below.
Courant σ -model was constructed in [63, 100]. Higher examples of symplectic Lie n -algebroids can be found e.g. in [65, 83, 54].An interesting open problem that arises from what precedes concerns the possibility of generalising the interplaybetween deformation quantization results (on the algebraic side) and quantization of AKSZ-type of models (onthe field theoretic side). Motivated by this problem, the ambition of the present paper is to generalise some ofthe algebraic methods introduced by M. Kontsevich in [78] for Poisson manifolds to the case of higher symplecticLie n -algebroids. Our main tool in this endeavour is given by a universal version of the Chevalley–Eilenbergdg Lie algebra associated with the deformation complex of symplectic Lie n -algebroids for arbitrary values of n ≥ . Explicitly, this graph model takes the form of an injective morphism of dg Lie algebras: fGC d (cid:44) → CE ( T ( n ) poly ) (1.4)thus generalising (1.3) to any d ≥ . Here, fGC d stands for the generalisation of Kontsevich’s graph complexto arbitrary dimension d ( cf. e.g. [113]) and the dg Lie algebra T ( n ) poly – referred to hereafter as the n -Schoutenalgebra – controls the deformation theory of symplectic Lie n -algebroids, for d = n + 1 . The morphism (1.4)will allow us to take advantage of the available results regarding the cohomology of fGC d in order to providea classification of the universal structures on graded symplectic manifolds of arbitrary (positive) degree. Inparticular, we propose a classification of Lie ∞ -algebra structures deforming the n -Schouten algebra in a non-trivial way as well as of Lie ∞ -automorphisms of the n -Schouten algebra T ( n ) poly . The latter yield in particularnew universal flows on the space of symplectic Lie n -algebroids.The present paper will focus on universal structures in the stable setting i.e. we consider cochains of theChevalley–Eilenberg algebra obtained from graphs belonging to the Kontsevich graph complex of undirectedgraphs fGC d (or its directed analogue dfGC d ). A direct consequence of this choice is that the only incarnation ofthe Grothendieck–Teichmüller Lie algebra as a universal structure occurs in dimension d = 2 where we recoverthe above mentioned action of GRT on T poly via Lie ∞ -automorphisms. In higher dimensions, the universalstructures are insensitive to grt and are in fact classified by loop cocycles. In order to obtain universalstructures induced from the Grothendieck–Teichmüller Lie algebra in dimensions d > , we will need to moveaway from the stable setting to enter the (multi)-oriented regime. This will be the subject of the companionpaper [93] which will generalise the present discussion to universal structures induced by (multi)-oriented graphs[115, 119, 120, 89, 90] allowing in particular to provide incarnations of the Grothendieck–Teichmüller algebrainto the deformation theory of higher symplectic Lie n -algebroids. Summary and main results
After displaying our conventions and notations in Section 2, we dedicate Sections 3 and 4 to a review – aimedat non-experts – of the principal tools and notions involved in the rest of the paper. In Section 3, we recall thebasic concepts of graded geometry, detail the hierarchy of structures endowing graded manifolds (namely N , NP and NPQ -manifolds) and discuss their associated (non-graded) geometric counterparts. In Section 4, we departfrom the geometric to the algebraic realm and review the construction of the Kontsevich’s full graph complex fGC d generalising fGC to arbitrary dimension d . The differential graded Lie algebra structure on fGC d is bestintroduced as a convolution Lie algebra from the graph operad Gra d whose construction we review. We concludethe section by recalling some known facts regarding the (even and odd) cohomology of fGC d . Building on thelast two sections, we introduce our main results in Section 5. We start by displaying a tower of representations Gra d (cid:44) → End C ∞ ( V ) for all d > where V stands for an arbitrary NP -manifold of degree n , such that d = n + 1 .This tower of morphism of operads will in turn induce a tower of injective morphisms of dg Lie algebras fGC d (cid:44) → CE ( T ( n ) poly ) thus providing a universal version of the Chevalley–Eilenberg complex for the n -Schouten See Definition 4.2 below. T ( n ) poly in the stable setting. Using this universal model, we show in particular ( cf. Corollary 5.6) that thepro-unipotent group exp (cid:0) H ( fGC d ) (cid:1) acts via Lie ∞ -automorphisms on T ( n ) poly . More generally, universal structureson graded symplectic manifolds in the stable setting are classified in Proposition 5.8. We conclude the sectionby discussing Hamiltonian deformations and Hamiltonian flows. In particular, we present a canonical mapfrom the zeroth cohomology of fGC d to universal flows on the space of Hamiltonian functions – see Proposition5.12 – thus generalising Kontsevich’s construction from Poisson bivectors to higher symplectic Lie n -algebroids.Furthermore, we show that this map is not surjective in general by exhibiting a new class of Hamiltonian flows– dubbed conformal hereafter – induced by elements in H − d ( fGC d ) , cf. Proposition 5.13. Finally, Section 6is devoted to illustrate some of the machinery developed in Section 5 to the case of
NPQ -manifolds of degrees1 and 2, respectively. After reviewing some known applications in the case n = 1 (corresponding to Poissonmanifolds), we turn to the case n = 2 and present new results concerning deformations of Courant algebroids. Inparticular, we obtain an explicit expression for the unique deformation map for Courant algebroids and displaya large class of conformal Hamiltonian flows given by trivalent graphs (modulo IHX relations). We concludeby a discussion regarding the implications of our results to the deformation quantization problem for Courantalgebroids. Acknowledgements
We are indebted to Thomas Basile for numerous stimulating discussions regarding various aspects of the presentwork as well as for valuable feedback on a preliminary version of the manuscript. We are also grateful toVasily A. Dolgushev, Noriaki Ikeda and Hsuan–Yi Liao for useful exchanges and to Clément Berthiere forprecious help in dealing with L A TEX issues. This work was supported by the Korean Research Fellowship GrantN ◦ Suspension
We will work over a ground field K of characteristic zero. Let V = (cid:76) k ∈ Z V k be a graded vector space over K .The suspended graded vector space V [ k ] is defined as V [ k ] n = V k + n so that the suspension map s : V [ k ] → V is of degree k . Invariants and coinvariants
Let G be a group and denote K (cid:104) G (cid:105) the associated group ring over K . A (right) representation of G is a (right)module M over the group ring K (cid:104) G (cid:105) . Letting M be a right K (cid:104) G (cid:105) -module, we define the two following spaces: • Invariants : M G := { m ∈ M | m · g = m for all g ∈ G }• Coinvariants : M G := M/ { m · g − m | g ∈ G and m ∈ M } Note that while the space of invariants is a subspace of M , the space of coinvariants (or space of orbits) isdefined as a quotient of M by the group action. In other words, there are natural maps M G i (cid:44) −→ M p − (cid:16) M G where i is injective and p surjective. If M is a right K (cid:104) G (cid:105) -module and N a left K (cid:104) G (cid:105) -module, then M ⊗ N isa right K (cid:104) G (cid:105) -module under the diagonal right action ( M ⊗ N ) × G → M ⊗ N : ( a, b ) × g (cid:55)→ ( a · g, g − · b ) .The associated space of coinvariants is then denoted M ⊗ G N . Letting M, N be two right K (cid:104) G (cid:105) -modules, alinear map f : M → N will be said G - equivariant if it is a morphism in the category of K (cid:104) G (cid:105) -modules i.e. if f ( x · g ) = f ( x ) · g for all x ∈ M and g ∈ G . The space of G -equivariant maps will be denoted Hom G ( M, N ) . Or equivalently for the graded Poisson algebra of functions C ∞ ( V ) , being isomorphic to T ( n ) poly through degree suspension. ymmetric group S N The symmetric group S N is defined as the group of automorphisms of the set { , , . . . , N } . An element σ ∈ S N is called a permutation and is defined by its image { σ (1) , σ (2) , . . . , σ ( N ) } .The composition σ · τ of twopermutations σ, τ ∈ S N is given by { , , . . . , N } τ (cid:55)→ { τ (1) , τ (2) , . . . , τ ( N ) } σ (cid:55)→ (cid:8) σ (cid:0) τ (1) (cid:1) , σ (cid:0) τ (2) (cid:1) , . . . , σ (cid:0) τ ( N ) (cid:1)(cid:9) .In the following, we will often represent a permutation σ by the × N matrix σ = (cid:18) · · · Nσ (1) σ (2) · · · σ ( N ) (cid:19) .We define two actions of the symmetric group S N on V ⊗ N : • Right action: V ⊗ N × S N → V ⊗ N as ( v , . . . , v N ) · σ = ( v σ (1) , . . . , v σ ( N ) ) • Left action: S N × V ⊗ N → V ⊗ N as σ · ( v , . . . , v N ) = ( v σ − (1) , . . . , v σ − ( N ) ) . Example 2.1.
Let σ, τ ∈ S defined as σ := (cid:18) (cid:19) and τ := (cid:18) (cid:19) and admitting the following inverses: σ − = (cid:18) (cid:19) and τ − = (cid:18) (cid:19) . We compute the following compositions: σ · τ = (cid:18) (cid:19) , τ · σ = (cid:18) (cid:19) . (2.1)Now, denoting v := ( v , v , v ) , one can check that: ( v · σ ) · τ = ( v , v , v ) · τ = ( v , v , v ) = v · ( σ · τ ) σ · ( τ · v ) = σ · ( v , v , v ) = ( v , v , v ) = ( σ · τ ) · v. The previous actions on V ⊗ N induce dual actions of the symmetric group S N on Hom ( V ⊗ N , V ) : • Right action: Hom ( V ⊗ N , V ) × S N → Hom ( V ⊗ N , V ) as ( f · σ )( v , . . . , v N ) = f ( v σ − (1) , . . . , v σ − ( N ) ) • Left action: S N × Hom ( V ⊗ N , V ) → Hom ( V ⊗ N , V ) as ( σ · f )( v , . . . , v N ) = f ( v σ (1) , . . . , v σ ( N ) ) Example 2.2.
Let σ, τ ∈ S as in Example 2.1 and f ∈ Hom ( V ⊗ , V ) . Denoting v := ( v , v , v ) , one can check that: (cid:0) ( f · σ ) · τ (cid:1) v = ( f · σ )( v , v , v ) = ( v , v , v ) = (cid:0) f · ( σ · τ ) (cid:1) v (cid:0) σ · ( τ · f ) (cid:1) v = ( τ · f )( v , v , v ) = f ( v , v , v ) = (cid:0) ( σ · τ ) · f (cid:1) v. In the following, we will denote sgn N the signature representation of S N i.e. the one-dimensional K (cid:104) S N (cid:105) -module associating to each permutation σ ∈ S N its signature | σ | ∈ {− , } .A collection of right K (cid:104) S N (cid:105) -modules M ( N ) for N ≥ will be referred to as a S - module . (Un)shuffles Let p, q ∈ N . A shuffle of type ( p, q ) is a permutation σ ∈ S p + q such that σ sends { , . . . , p + q } to { i , . . . , i p | j , . . . , j q } where i < · · · < i p and j < · · · < j q . Example 2.3 (Shuffles) . • Sh (1 ,
1) = { (1 | , (2 | }• Sh (1 ,
2) = { (1 | , (2 | , (3 | }• Sh (2 ,
1) = { (12 | , (13 | , (23 | }• Sh (1 ,
3) = { (1 | , (2 | , (3 | , (4 | } Sh (2 ,
2) = { (12 | , (13 | , (14 | , (23 | , (24 | , (34 | }• Sh (3 ,
1) = { (123 | , (124 | , (134 | , (234 | } The set of shuffles of type ( p, q ) is denoted Sh ( p, q ) . Since a shuffle σ ∈ Sh ( p, q ) is completely determined bythe set { i , . . . , i p } , there are (cid:0) p + qp (cid:1) shuffles of type ( p, q ) . A unshuffle of type ( p, q ) is a permutation σ ∈ S p + q such that the inverse permutation σ − is a shuffle of type ( p, q ) . The set of unshuffles of type ( p, q ) is denotedSh − ( p, q ) . Example 2.4 (Unshuffles) . • Sh − (1 ,
1) = { (12) , (21) }• Sh − (1 ,
2) = { (123) , (213) , (231) }• Sh − (2 ,
1) = { (123) , (132) , (312) }• Sh − (1 ,
3) = { (1234) , (2134) , (2314) , (2341) }• Sh − (2 ,
2) = { (1234) , (1324) , (1342) , (3124) , (3142) , (3412) }• Sh − (3 ,
1) = { (1234) , (1243) , (1423) , (4123) } Operads
We will consider operads in the category of (graded) vector spaces over K . Our conventions will mostly follow theones of the book [85]. We will denote Ass , Lie and
Com the operads of (graded) vector spaces encoding (graded)associative, Lie and commutative associative algebras without unit, respectively. The cooperad governingcocommutative coassociative algebras without counit will be denoted coCom . The latter is defined explicitly as coCom = for n = 0 K for all n > where K stands for the trivial representation of S n .Letting O be an operad in the category of graded vector spaces, the set of graded vector spaces: O{ d } ( N ) := O ( N )[ d (1 − N )] for d even O ( N ) ⊗ sgn N [ d (1 − N )] for d odd (2.2)assemble to a S -module. Endowing this S -module with the partial composition maps, identity and right-actionsof O defines the d - suspended operad O{ d } . Alternatively, the d -suspended operad O{ d } can be characterisedas the unique operad for which the set of algebras of the operad O on a graded vector space V are in one-to-onecorrespondence with the set of algebras of O{ d } on the suspended graded vector space V [ d ] . In particular, End V { d } = End V [ d ] where End V denotes the endomorphism operad associated with the graded vector space V . The aim of the present section is to provide a short introduction to graded manifolds as well as their (non-graded)geometric counterparts. The latter objects are defined as the geometrical data associated with graded mani-folds – understood as manifolds endowed with a grading of the corresponding structure sheaf – supplementedwith some additional graded structures. We will only deal with N - graded manifolds for which the corresponding degree assigned to each local coordinate is a non-negative integer. Cf. [87] for precise definitions. -graded manifolds Letting V be a N -graded manifold of degree n , we will denote C ∞ ( V ) the associated algebra of functions. Thesubvector space of homogeneous functions of degree k will be denoted C ∞| k ( V ) ⊂ C ∞ ( V ) so that C ∞ ( V ) = (cid:76) k ≥ C ∞| k ( V ) is a graded algebra. Moreover, C ∞ ( V ) is a filtered algebra. Letting A k denote the (graded)subalgebra of C ∞ ( V ) locally generated by functions of degree ≤ k , there is an increasing sequence: A ⊂ A ⊂ · · · ⊂ A n = C ∞ ( V ) (3.1)where we have C ∞| k ( V ) = A k / A k − so that C ∞ ( V ) = (cid:76) k ≥ C ∞| k ( V ) is the graded algebra associated withthe filtration (3.1). Corresponding to this filtration, there is a tower of fibrations M = M ← M ← · · · ← M n = V (3.2)where M is an ordinary smooth manifold – referred to as the base – and such that C ∞ ( M ) = C ∞| ( V ) = A .Furthermore, M is a vector bundle over M and for all k ≥ , M k ← M k +1 is an affine fibration, cf. [99] fordetails.The geometry of the fibration underlying graded manifolds can be enriched by introducing additional (hierar-chised) data on V : • A NP - manifold ( V , ω ) of degree n is a N -graded manifold V endowed with a symplectic 2-form ω ofintrinsic degree n . • A NPQ - manifold ( V , ω, Q ) of degree n is a N -graded manifold V endowed with a symplectic 2-form ω of intrinsic degree n and a homological vector field Q ( i.e. Q is of degree and satisfies Q = 0 ) such that L Q ω = 0 .These additional data induce some extra geometric structures on the fibration 3.2. We will refer to the (non-graded) geometrical data associated with NPQ -manifolds of degree n as symplectic Lie n -algebroids . NP -manifolds Endowing a graded manifold with a symplectic ( i.e. non-degenerate and closed) 2-form has a number ofconsequences. First of all, the existence of a symplectic -form of degree n on a N -graded manifold V constrains the degree of V to not exceed n . Secondly, it can be shown than any homogeneous symplectic 2-form of degree n ≥ is exact. These two properties can be used in order to provide a local presentation à la Darboux of NP -manifolds. We distinguish between odd and even cases as follows: • n oddWe introduce a set of homogeneous coordinates (cid:26) x µ , ψ α i i , χ α i n − i , p µn (cid:27) where i ∈ (cid:8) , . . . , ( n − (cid:9) .The symplectic 2-form of odd degree n can thus be written as: ω = dx µ ∧ dp µ + ( n − (cid:88) i =1 dψ α i ∧ dχ α i . (3.3)The associated Poisson bracket of degree − n acts as follows: (cid:8) f, g (cid:9) ω = ( − f ∂f∂x µ ∂g∂p µ + ∂f∂p µ ∂g∂x µ + ( n − (cid:88) i =1 (cid:40) ( − f ( i +1) ∂f∂ψ α i ∂g∂χ α i + ( − if ∂f∂χ α i ∂g∂ψ α i (cid:41) (3.4)on homogeneous functions f ∈ C ∞| f ( V ) and g ∈ C ∞| g ( V ) . Cf.
Lemma 2.4 in [99]. Cf.
Lemma 2.2 in [99]. The subscript denotes the corresponding degree. n evenThe corresponding set of homogeneous coordinates reads (cid:40) x µ , ψ α i i , ξ an/ , χ α i n − i , p µn (cid:41) where i ∈ (cid:8) , . . . , n − (cid:9) .The symplectic 2-form of even degree n is written as: ω = dx µ ∧ dp µ + n − (cid:88) i =1 dψ α i ∧ dχ α i + 12 κ ab dξ a ∧ dξ b (3.5)where the bilinear form κ is non-degenerate and symmetric (resp. skewsymmetric ) for n/ odd (resp.even) i.e. κ ab = − ( − n/ κ ba .The associated Poisson bracket thus takes the form: (cid:8) f, g (cid:9) ω = ∂f∂x µ ∂g∂p µ − ∂f∂p µ ∂g∂x µ + n − (cid:88) i =1 (cid:40) ( − if ∂f∂ψ α i ∂g∂χ α i − ( − i ( f +1) ∂f∂χ α i ∂g∂ψ α i (cid:41) + ( − fn/ ∂f∂ξ a κ ab ∂g∂ξ b . (3.6)It can be checked that the Poisson brackets (3.4) and (3.6) satisfy the following properties:1. (cid:8) f, g (cid:9) ω = − ( − n ( − fg (cid:8) g, f (cid:9) ω (cid:8) f, g · h (cid:9) ω = (cid:8) f, g (cid:9) ω · h + ( − g ( f − n ) g · (cid:8) f, h (cid:9) ω (cid:8)(cid:8) f, g (cid:9) ω , h (cid:9) ω + ( − f ( g + h ) (cid:8)(cid:8) g, h (cid:9) ω , f (cid:9) ω + ( − h ( f + g ) (cid:8)(cid:8) h, f (cid:9) ω , g (cid:9) ω = 0 for all homogeneous functions f ∈ C ∞| f ( V ) , g ∈ C ∞| g ( V ) and h ∈ C ∞| h ( V ) . NPQ -manifolds
We now turn to
NPQ -manifolds and start by pointing out that the latter can be equivalently described in termsof a Poisson bracket together with a Hamiltonian function i.e. as a triplet ( V , {· , ·} ω , H ) where:1. V is a N -graded manifold.2. {· , ·} ω is a non-degenerate Poisson bracket of degree − n acting on the graded algebra of functions on V .3. H is a Hamiltonian function i.e. a homogeneous function of degree n + 1 being nilpotent with respect tothe graded Poisson bracket i.e. (cid:8) H , H (cid:9) ω = 0 . The set of Hamiltonian functions will be denoted Ham .Equivalence between the homological and Hamiltonian presentations of
NPQ -manifolds of degree n is realisedby identifying ω as the symplectic -form of degree n dual to {· , ·} ω and defining the privileged vector field Q ∈ Γ ( T V ) of degree 1 on V as Q = (cid:8) H , · (cid:9) ω . The nilpotency of H ensures that Q is homological i.e. [ Q , Q ] = 0 , with [ · , · ] the graded Lie bracket on V .The importance of NPQ -manifolds (or equivalently symplectic Lie n -algebroids) stems from the fact that thesenaturally form the target space of the classical action associated with AKSZ-type σ -models [6] for which thesource manifold has dimension d = n + 1 .We conclude this brief survey by displaying examples of symplectic Lie n -algebroids in low degrees. Note that, whenever n = 4 k (for some integer k = 0 , , , . . . ) the indices of type a, b, . . . should run over an even number ofdimensions in order to ensure the existence of a skewsymmetric invertible bilinear form κ . In other words, the triplet (cid:0) C ∞ ( V ) , · , {· , ·} ω (cid:1) is a Ger n +1 -algebra, cf. footnote 31 below. Indeed, it follows from Cartan’s homotopy formula that the compatibility relation between the symplectic -form and the vectorfield ensures that the latter is Hamiltonian, cf. Lemma 2.2 in [99]. xample 3.1 (Symplectic Lie n -algebroids) . • n = 0 (Symplectic manifolds)The manifold is coordinatised by a unique set of homogeneous coordinates ξ a of degree 0, with a ∈{ , . . . , D } and D the (even) dimension of the manifold. The manifold is thus non-graded (or bosonic) i.e. V identifies with its base M . The symplectic 2-form of degree takes the usual form ω = κ ab dξ a ∧ dξ b where the bilinear form κ is non-degenerate and skewsymmetric i.e. κ ab = − κ ba .The associated Poisson bracket thus takes the form: (cid:8) f, g (cid:9) ω = ∂f∂ξ a κ ab ∂g∂ξ b . (3.7)The absence of degree 1 coordinates prevents the existence of a Hamiltonian function H (of would-bedegree 1) in this case.Symplectic Lie- algebroids are thus in one-to-one correspondence with (ordinary) symplectic manifolds. • n = 1 (Poisson manifolds)The set of homogeneous coordinates takes the form (cid:26) x µ , p µ (cid:27) .The symplectic 2-form of odd degree can thus be written as ω = dx µ ∧ dp µ while the associated Poissonbracket of degree − acts as follows: (cid:8) f, g (cid:9) ω = ( − f ∂f∂x µ ∂g∂p µ + ∂f∂p µ ∂g∂x µ (3.8)on homogeneous functions f ∈ C ∞| f ( V ) and g ∈ C ∞| g ( V ) . Up to degree suspension, {· , ·} ω identifies withthe Schouten bracket acting on polyvector fields.The most general function of degree reads H = π µν ( x ) p µ p ν with π a bivector, i.e. π µν = − π νµ .It can be checked that (cid:8) H , H (cid:9) ω = 0 ⇔ π ρ [ λ ∂ ρ π µν ] = 0 i.e. H is Hamiltonian if and only if π is aPoisson bivector. It follows that symplectic Lie- algebroids are in one-to-one correspondence with Poissonmanifolds. • n = 2 (Courant algebroids)The set of homogeneous coordinates can be decomposed as (cid:26) x µ , ξ a , p µ (cid:27) . The symplectic 2-form of odddegree can thus be written as ω = dx µ ∧ dp µ + κ ab dξ a ∧ dξ b where the bilinear form κ is non-degenerateand symmetric i.e. κ ab = κ ba . The associated Poisson bracket of degree − acts as follows: (cid:8) f, g (cid:9) ω = ∂f∂x µ ∂g∂p µ − ∂f∂p µ ∂g∂x µ + ( − f ∂f∂ξ a κ ab ∂g∂ξ b . (3.9)on homogeneous functions f ∈ C ∞| f ( V ) and g ∈ C ∞| g ( V ) .The most general function of degree reads H = ρ aµ ξ a p µ + T abc ξ a ξ b ξ c where T abc is totally skewsym-metric. It can be checked that the nilpotency condition (cid:8) H , H (cid:9) ω = 0 is equivalent to the three followingconstraints:1. C µν := ρ aµ κ ab ρ bν = 0 (3.10)2. C µab := ρ cµ κ cd T dab + 2 ρ [ aλ ∂ λ ρ b ] µ = 0 (3.11)3. C abcd := T e [ ab κ ef T cd ] f + ρ [ aµ ∂ µ T bcd ] = 0 . (3.12)As will be reviewed in Section 6.2, symplectic Lie- algebroids are in one-to-one correspondence withCourant algebroids. 10 Graph complexes
The aim of the present section is to review a particular family of graph complexes introduced by M. Kontsevichin [77, 76, 78]. The former is most clearly defined in terms of the convolution Lie algebra constructed from asuitable graph operad. We start by reviewing the construction of this graph operad – denoted Gra d hereafter – from a combinatorial point of view before turning to the definition of the so-called full graph complex fGC d .After reviewing results regarding the cohomology of fGC d , we conclude by presenting a variant of the full graphcomplex whose elements are directed graphs. The material covered in this section is standard and can be foundfor example in [113, 35, 37]. Gra d Our starting point towards a definition of the graph operad
Gra d will be the set of multidigraphs (or quivers) i.e. directed graphs which are allowed to contain multiple edges and loops . The set of multidigraphs with N vertices and k directed edges will be denoted gra N,k . A typical example of multidigraph is given in Figure 1.1 32 4 5 iiiiii ivvvi Figure 1: Example of graph in gra , There is a natural right-action of the semi-direct product S k (cid:110) S × k on elements of gra N,k by permutation of theordering ( S k ) and flipping of the directions of the edges ( S × k ) . We will consider the 1-dimensional signaturerepresentation sgn k (resp. sgn ⊗ k ) as a left K (cid:10) S k (cid:110) S × k (cid:11) -module with trivial action of S × k (resp. S k ).For all N ≥ and d ∈ Z , we define the collection of graded vector spaces Gra d ( N ) as: Graph complexes come in many variants. As shown in [48, 86, 27], to any cyclic operad O one can associate a class of O -graphcomplexes. In particular, O = Ass corresponds to the class of ribbon graphs computing cohomology of moduli spaces of curves[95, 96] while the graph complex for O = Lie computes cohomology of outer automorphisms of free groups [29]. We will solely beinterested in the case O = Com . Also, graph complexes come in two dual versions: a homological version in which the boundaryoperator acts via “collapsing” of edges [77, 76] and a cohomological one in which the coboundary operator acts by “blowing up”edges [78, 113]. We will hereafter focus on the cohomological version. In Section 5, we will relate the integer d (in the case when d ≥ ) to the dimension of the source of the relevant AKSZ σ -modelon which Gra d will be shown to act. In other words, we will consider d = n + 1 where n is the degree of the corresponding NPQ -manifold, cf.
Section 3. Formally, a multidigraph is defined as a four-tuple γ = ( V γ , E γ , s, t ) where: • V γ is a set whose elements are called vertices . • E γ is a set whose elements are called edges . • The map s : E γ → V γ assigns to each edge its source . • The map t : E γ → V γ assigns to each edge its target .An edge e ∈ E γ such that s ( e ) = t ( e ) is called a loop (or tadpole ) while a pair of edges e , e ∈ E γ such that s ( e ) = s ( e ) and t ( e ) = t ( e ) are called double edges . The set of edges connecting a given vertex v ∈ V γ will be denoted E γ ( v ) . We willmostly deal with labeled multidigraphs i.e. multidigraphs endowed with two bijective maps l V : V γ → [ | V γ | ] and l E : E γ → [ | E γ | ] where | V γ | (resp. | E γ | ) denotes the number of vertices (resp. edges) of γ and [ n ] := { , , . . . , n } . While depicting multidigraphspictorially, we will represent edges by arrows from source to target vertices. To avoid ambiguity, labelling will be performed usingHindu-Arabic numerals for vertices and Roman numerals for edges. Note that we do not assume any compatibility between thelabelling of vertices and edges a priori . Note that the definition of a multidigraph does not assume connectedness. d even: Gra d ( N ) := (cid:89) k ≥ (cid:0) K (cid:10) gra N,k (cid:11) ⊗ S k (cid:110)S × k sgn k (cid:1) [ k ( d − (4.1) • d odd: Gra d ( N ) := (cid:89) k ≥ (cid:0) K (cid:10) gra N,k (cid:11) ⊗ S k (cid:110)S × k sgn ⊗ k (cid:1) [ k ( d − (4.2)where the subscript stands for taking coinvariants with respect to the diagonal right action of S k (cid:110) S × k and theterm between brackets denotes degree suspension ( cf. Section 2 for conventions).Elements of
Gra d ( N ) are linear combinations of equivalence classes of graphs in gra N,k , for arbitrary k ≥ .Two graphs γ, γ (cid:48) ∈ gra N,k will be said equivalent ( i.e. γ ∼ γ (cid:48) ) if one of the two following condition holds:1. There exists an element σ ∈ S × k such that Φ dir σ ( γ ) = ( − d | σ | γ (cid:48) where Φ dir σ stands for the automorphismof gra N,k that flips the direction of the edges according to σ , e.g. i ∼ ( − d i . (4.3)2. There exists an element σ ∈ S k such that Φ order σ ( γ ) = ( − ( d +1) | σ | γ (cid:48) where Φ order σ stands for the automor-phism of gra N,k that permutes the order of the edges according to σ , e.g. i ii ∼ ( − d +1 ii i . (4.4)According to the degree suspension in (4.1)-(4.2), each edge is assigned an intrinsic degree − d , so that thedegree of an element γ ∈ gra N,k as seen in
Gra d ( N ) is given by | γ | = k (1 − d ) .It is also clear from their definition that graded vector spaces Gra d ( N ) for different d of same parity only differby their degree assignment and are thus isomorphic to each other.Following [101], we will call zero graph a graph γ ∈ gra N,k which equals minus itself in
Gra d ( N ) and thusbelongs to the zero class in Gra d ( N ) . It follows that a graph admitting an automorphism that permutes theedges ordering by an odd permutation is a zero graph whenever d is even. In particular, graphs admittingmultiple edges are zero graphs for d even . On the other hand, a graph admitting an automorphism that flipsan odd number of edges is automatically a zero graph whenever d is odd. In particular, graphs with tadpolesare zero graphs for d odd .For all N ≥ , the symmetric group S N acts naturally on the right on the graded vector space Gra d ( N ) bypermuting the label of vertices as { , , . . . , N } σ (cid:55)→ (cid:8) σ − (1) , σ − (2) , . . . , σ − ( N ) (cid:9) .We will denote Σ N : Gra d ( N ) × S N → Gra d ( N ) the corresponding right action .In other words, the set of graded vector spaces { Gra d ( N ) } N ≥ assemble to a S -module over K .The S -module { Gra d ( N ) } N ≥ can further be given the structure of an operad by endowing it with partial com-position operations. Explicitly, we define partial composition operations ◦ i : gra M,j ⊗ gra N,k → gra M + N − , j + k for all ≤ i ≤ M as: γ ◦ i γ (cid:48) = (cid:88) f ∈ Hom ( E γ ( v i ) ,V γ (cid:48) ) γ ◦ fi γ (cid:48) (4.5) Whenever d is even, the double edges graph 1 2 iii satisfies 1 2 iii ∼ − iii and is thus a zero graph in Gra d (2) . Whenever d is odd, the tadpole graph 1 satisfies 1 ∼ − Gra d (1) . For example, letting σ ∈ S be defined as σ := (cid:18) (cid:19) , the right-action of σ on the graph γ ∈ Gra d (3) defined as γ := i ii reads Σ ( γ | σ ) = i ii . v i is the i th vertex of γ ∈ gra M,j and the sum is performed over homomorphisms of sets between the set E γ ( v i ) of edges of γ connecting v i and the set V γ (cid:48) of vertices of γ (cid:48) ∈ gra N,k .The operation ◦ fi consists in first inserting the graph γ (cid:48) in place of the vertex v i ∈ γ and then reconnecting theelements in E γ ( v i ) to vertices of γ (cid:48) along the map f .As for labelling of vertices and edges, we follow the following rules: • The labels of the first i − vertices of γ are left unchanged. • The labels of the vertices of γ (cid:48) are shifted up by i − . • The last M − i vertices of γ are shifted up by N − . • All edges originating from γ are declared smaller than all edges originating from γ (cid:48) .The partial composition operations ◦ i can be checked to be equivariant with respect to the right-action of S k (cid:110)S × k on gra N,k allowing to define partial composition operations ◦ i : Gra d ( M ) ⊗ Gra d ( N ) → Gra d ( M + N − .1 23 i iiiii ◦ i = iiiiii iv + iiiiii iv + i iiiii iv + i iiiii iv Figure 2: Example of partial composition
Gra d (3) ◦ Gra d (2) → Gra d (4) The partial composition operations ◦ i on Gra d preserve the number of edges and thus have zero intrinsic degree.Further, they can be checked to satisfy the following properties for all γ m ∈ Gra d ( m ) : • Sequential composition : ( γ m ◦ j γ n ) ◦ i γ p = γ m ◦ j ( γ n ◦ i − j +1 γ p ) for all j (cid:54) i (cid:54) j + n − . (4.6) • Parallel composition : ( γ m ◦ j γ n ) ◦ i γ p = ( − | γ n || γ p | ( γ m ◦ i − n +1 γ p ) ◦ j γ n for all i (cid:62) j + n. (4.7)Finally, the partial composition operations are equivariant with respect to the right-action Σ N of S N on Gra d ( N ) .The previous properties ensure that the S -module { Gra d ( N ) } N ≥ is naturally endowed with a structure ofoperad [113]: Proposition 4.1 (Operad
Gra d ) . For all d ∈ Z , one can define an operad in the category of graded vectorspaces as the quadruplet (cid:0) Gra d , Σ , ◦ i , id (cid:1) where: • The set of graded vector spaces { Gra d ( N ) } N ≥ endowed with the set of natural right-actions Σ N : Gra d ( N ) × S N → Gra d ( N ) is a S -module. • ◦ i : Gra d ( M ) ⊗ Gra d ( N ) → Gra d ( M + N − is the set of equivariant partial composition operations definedin eq. (4.5) . • The identity element id ∈ Gra d (1) is defined as the graph id := of degree . As usual, representations of the graded operad
Gra d (or Gra d -algebras) are ordered pairs ( V, ρ ) where V is agraded vector space and ρ : Gra d → End V is a morphism of operads, with End V the endomorphism operad on V , see [85] for details. 13 niversal structures The notion of universal (or stable ) structures was first introduced in [78] to characterise a subclass of cochains inthe Chevalley–Eilenberg algebra of polyvector fields CE ( T poly ) . The terminology referred to the fact that suchcochains are defined “graphically” via grafting of existing structures on T poly without resorting to additionaldata and thus independently of the dimension of the underlying manifold. Such universal cochains were thenargued to constitute natural candidate recipients for the possible obstructions to the existence of a formalitymorphism. The corresponding class of formality morphisms was then precisely defined in [32] in terms of theoperads OC and KGra ( cf. definitions therein). Informally, these are Lie ∞ quasi-isomorphisms whose Taylorcoefficients can be written as a sum over Kontsevich admissible graphs [79], independently of the dimension .The definition we adopt here is adapted from [7]: Definition 4.2 (Universal structure) . Let P be an operad in the category of graded vector spaces and V agraded vector space. A P -algebra structure on V will be said universal if the action of the operad P on V factors through a graph operad G as P −→ G −→ End V .Whenever the graph operad G is given by the Kontsevich’s graph operad Gra d (or its directed avatar dGra d ) forsome d ∈ Z , the corresponding structure will be said universal in the stable setting or stable for short .An important example of universal structures in the stable setting is given by Ger d -algebras : Proposition 4.3 (T. Willwacher [113]) . For all d ∈ Z , there is a natural embedding of operads i d : Ger d (cid:44) −→ Gra d . Explicitly, the embedding of operads i d is defined by the following action on generators a ∧ a , (cid:8) a , a (cid:9) ∈ Ger d (2) : • i d ( a ∧ a ) = Γ with ∧ the graded commutative associative product of degree • i d ( (cid:8) a , a (cid:9) ) = Γ with {· , ·} the graded Lie bracket of degree − d where Γ and Γ ∈ Gra d (2) are respectively defined as Γ := , Γ := i . (4.8) Two such morphisms thus only differ by their weight function, the latter depending on the choice of a Drinfel’d associator. While all the universal structures described in the present work will be stable in the above sense, we will encounter somenon-stable universal structures in the companion paper [93] ( cf. also [108] for an example of non-stable universal structure onpolyvector fields). A Ger d -algebra is a triplet (cid:0) g , ∧ , [ · , · ] (cid:1) such that:1. (cid:0) g , ∧ (cid:1) is a Com -algebra.2. (cid:0) g [ d − , [ · , · ] (cid:1) is a Lie -algebra.3. The bracket [ · , · ] is a bi-derivation with respect to the product ∧ .We will denote Ger d the operad whose associated representations are Ger d -algebras. Note that, in order to explicitly state the thirdcompatibility relation, one needs first to pullback one of the defining maps along the suspension map s : g [ d − → g of degree d − so that both products act on the same space. Explicitly, one can define the pushforward {· , ·} of the graded Lie bracket [ · , · ] on g as {· , ·} := s ◦ [ · , · ] ◦ ( s − ⊗ s − ) so that (cid:8) a, b (cid:9) = ( − ( d − a s ◦ (cid:2) s − ( a ) , s − ( b ) (cid:3) for all a, b ∈ g . The pushforward bracket {· , ·} isof degree − d and satisfies the following properties: • graded-(skew)symmetric i.e. (cid:8) a, b (cid:9) = ( − d ( − ab (cid:8) b, a (cid:9) • graded-Jacobi identity i.e. (cid:8)(cid:8) a, b (cid:9) , c (cid:9) + ( − a ( b + c ) (cid:8)(cid:8) b, c (cid:9) , a (cid:9) + ( − c ( a + b ) (cid:8)(cid:8) c, a (cid:9) , b (cid:9) = 0 .The graded Poisson identity on g thus reads (cid:8) a, b ∧ c (cid:9) = (cid:8) a, b (cid:9) ∧ c + ( − b ( a +1 − d ) b ∧ (cid:8) a, c (cid:9) .The notion of Ger -algebra identifies with the one of Poisson algebra for which both binary operations are of zero degree on g . Thecase d = 2 was first introduced by M. Gerstenhaber in [47] in order to characterise the natural structure living on the Hochschildcohomology of an associative algebra. For this reason, Ger -algebras are usually referred to as a Gerstenhaber algebras . Notethat the definition of
Ger d -algebras coincides with the one of e d -algebras ( cf. e.g. Section 13.3.16 of [85]) for d ≥ while e -algebrasare conventionally chosen to be associative algebras.
14t follows from Proposition 4.3 that any
Gra d -algebra is endowed with a universal structure of Ger d -algebra.In particular, the previous embedding provides a canonical morphism of operads Lie { − d } → Gra d so that any Gra d -algebra is naturally endowed with a Lie bracket of degree − d .We conclude by pointing out that in the case d = 1 , there is a natural embedding of operads Ass (cid:44) −→ Gra mapping the generator m ∈ Ass (2) ( i.e. the associative binary product) of the associative operad Ass to theelement 1 ... 2 ∈ Gra (2) . The latter is explicitly defined as the infinite sum of graphs [70]:1 ... 2 := (cid:88) j ≥ j ! j edges . (4.9)As a result, Gra -algebras are naturally endowed with a universal associative product. fGC d We now turn to the definition of the full graph complex, denoted fGC d hereafter. The differential on fGC d stemsfrom a richer structure – namely a pre-Lie structure – defined in terms of the graph operad Gra d using one ofthe following equivalent constructions:1. The pre-Lie algebra associated with the suspended operad Gra d { d } .2. The convolution pre-Lie algebra Hom S ( coCom , Gra d { d } ) .3. The deformation complex of the trivial operad morphism Lie { − d } → Gra d .We pass on the explicit unfolding of these definitions and merely present the final result: Proposition 4.4 (Pre-Lie structure on fGC d ) . For all d ∈ Z , the couple (cid:0) fGC d , ◦ (cid:1) where: • The graded vector space fGC d is defined as : – d even: fGC d := (cid:89) N ≥ (cid:0) Gra d ( N )[ d (1 − N )] (cid:1) S N – d odd: fGC d := (cid:89) N ≥ (cid:0) Gra d ( N ) ⊗ sgn N [ d (1 − N )] (cid:1) S N where the superscript stands for taking invariants with respect to the right action of S N with sgn N the1-dimensional signature representation of S N . The terms between brackets denote degree suspension . • The binary operation ◦ : fGC d ⊗ fGC d → fGC d is of degree and defined via the formula γ ◦ γ (cid:48) = (cid:88) σ ∈ Sh − ( N (cid:48) ,N − ( − d | σ | Σ N + N (cid:48) − (cid:0) γ ◦ γ (cid:48) (cid:12)(cid:12) σ ) (4.10) where Σ N : Gra d ( N ) × S N → Gra d ( N ) denotes the right action defined previously while N, N (cid:48) stand for thenumber of vertices in the homogeneous graphs γ, γ (cid:48) , respectively. The sum is performed over the unshufflesof type ( N (cid:48) , N − and | σ | denotes the signature of the permutation σ ∈ S N + N (cid:48) − .is a graded pre-Lie algebra i.e. ( γ ◦ γ ) ◦ γ − γ ◦ ( γ ◦ γ ) = ( − | γ || γ | (cid:0) ( γ ◦ γ ) ◦ γ − γ ◦ ( γ ◦ γ ) (cid:1) for all γ m ∈ fGC d . (4.11) See eq.(5.14) below for an example. We refer to [85, 91] for generic constructions and to [113, 35] for applications to the case at hand. The sign conventions used relatively to the action of the various symmetry groups are summed up in Table 1. According to the suspension, the degree of an element γ ∈ fGC d with N vertices and k edges is given by | γ | = d ( N − k (1 − d ) . × k S k S N d even + − + d odd − + − Table 1: Symmetries of graphs in fGC d Proposition 4.4 can be reformulated as the existence of a morphism of operads preLie → End fGC d . Composingwith the morphism of operads Lie → preLie allows to endow fGC d with a structure of graded Lie algebra throughthe commutator (graded) Lie bracket [ · , · ] defined as: [ γ , γ ] = γ ◦ γ − ( − | γ || γ | γ ◦ γ . (4.12)For all d ∈ Z , it can be checked that the element Γ ∈ fGC d (cid:0) cf. (4.8) (cid:1) is a Maurer–Cartan element for thegraded Lie algebra ( fGC d , [ · , · ]) i.e. [Γ , Γ ] = 0 . This property allows to define the differential operator δ := [Γ , · ] acting through the adjoint action associated with the Maurer–Cartan element. The latter can beshown to square to zero as well as to preserve the graded Lie bracket.We sum up the previous discussion by the following proposition: Proposition 4.5.
The triplet ( fGC d , δ, [ · , · ]) is a dg Lie algebra . Forgetting the Lie bracket, we refer to the couple ( fGC d , δ ) as the full graph complex .We conclude by displaying distinguished examples of graphs in fGC d :1 2 3 = 13 (cid:16) i + i + i (cid:17) Figure 3: Example of graph in fGC d Example 4.6. • The graph 2 3 is a cocycle in the even and odd graph complexes. • The tadpole graph 1 is a cocycle in the even graph complex and a zero graph in the odd graphcomplex. • The multi-arrows graph 2 3 – sometimes referred to as the “ Θ -graph” – is a cocycle in the odd graphcomplex and a zero graph in the even graph complex. In retrospect, it can be checked that the choices made in Table 1 are the only ones ensuring that δ ≡ , cf. [117]. Note that the dg Lie algebra ( fGC d , δ, [ · , · ]) can be defined from the onset as the deformation complex of the (non-trivial) operadmorphism Lie { − d } → Gra d defined in Section 4.1, cf. [113]. As is customary, we will represent a given element of fGC d as a linear combination of undirected graphs with black verticessince taking invariants with respect to S N makes the vertices undistinguishable. In order to obtain an explicit element of fGC d from such a graph, one needs to go through the following steps ( cf. Figure 3 for an example):1. Choose an ordering of the edges.2. Choose an orientation of the edges.3. Sum over all possible ways of assigning labels to the vertices.4. Divide by the order of the symmetry of the given graph.Note that the overall sign is left ambiguous. The Θ -graph cocycle can be promoted to a Maurer–Cartan element in ( fGC , δ, [ · , · ]) as the sum ofmulti-arrows graph [70]: 1 ... 2 := (cid:88) k ≥ k + 1)! k + 1 edges . (4.13) We now collect some known results regarding the cohomology of the full graph complex fGC d . In the following,we will let fGC con d denote the sub-dg Lie algebra of fGC d spanned by connected graphs. Furthermore, we define GC d as the subcomplex of fGC con d spanned by graphs without tadpoles for which all vertices have valence atleast 3. The latter subcomplex was introduced in [78] and is sometimes referred to as the Kontsevich graphcomplex . As noted in [113], the full graph complex can be described in terms of its connected component as fGC d = (cid:98) S ( fGC con d [ − d ])[ d ] . In other words, computing the cohomology of fGC d reduces to computing thecohomology of its connected component fGC con d . The latter admits the following decomposition: Theorem 4.7 (Kontsevich [77, 76], Willwacher [113]) . The connected part of the full graph complex satisfies: H • ( fGC con d ) = H • ( GC d ) ⊕ (cid:77) k =2 d +1 mod k ≥ K [ d − k ] (4.14) where the class corresponding to K [ d − k ] is represented by a loop L k with k edges, cf. Figure 4 . For symmetry reasons, the only non-zero loop classes are represented by: • d even Loops L k with k = 4 j + 1 edges, j ≥ • d odd Loops L k with k = 4 j + 3 edges, j ≥ .1 2 3 1 23 1 23 4 321 5 4Figure 4: Loop graphs L k for k ∈ { , . . . , } It follows from Proposition 4.7 that the cohomology of fGC con d is located in GC d , up to some known (loop)classes. We now focus on the cohomology of GC d , for d = 2 , (see e.g. [70, 45] for summary and [11, 70] forcomputer generated tables). Cohomology of GC One of the major results of [113] is the following theorem:
Theorem 4.8 (Willwacher [113]) . The cohomology of the Kontsevich graph complex GC satisfies:1. Lower bound : H ≤− ( GC ) = The obstruction to the prolongation of the Θ -graph to a full Maurer–Cartan element lies in H ( fGC con ) = K (cid:104) L (cid:105) , cf. Section4.3. Since the obstruction to the prolongation of the Θ -graph at order k ≥ has Betti number k + 2 , it never hits the loop graph L of Betti number . The prolongation of the Θ -graph to a Maurer–Cartan element in fGC is thus unobstructed at all orders. In the case d = 2 . For any graded vector space V , we will let (cid:98) S ( V ) denote the (completed) symmetric product space of the graded vector space V defined as (cid:98) S ( V ) := (cid:89) j ≥ ( V ⊗ j ) S j . . Dominant degree : H ( GC ) (cid:39) grt as Lie algebras where grt stands for the Grothendieck–TeichmüllerLie algebra.3. Upper bound : H ≥ b − ( GC ) = 0 where b stands for the first Betti number defined as b = k − N + 1 for a connected graph in gra N,k . Combining Theorems 4.7 and 4.8 leads to a complete characterisation of the connected part of the full graphcomplex for d = 2 in low degrees:1. H < − ( fGC con ) = H − ( fGC con ) = K (cid:104) L (cid:105) H ( fGC con ) = H ( GC ) (cid:39) grt as Lie algebras.Explicit representatives of classes in the dominant degree H ( GC ) can be constructed according to: Theorem 4.9 (Willwacher [113]) . For every integer j ≥ , there exists a non-trivial cocycle γ j +1 ∈ H ( GC ) admitting a non-zero coefficient in front of the wheel with j + 1 spokes, cf. Figure 5 . The Grothendieck–Teichmüller Lie algebra grt is known to contain a series of non-trivial elements σ , σ , . . . indexed by an odd integer . In fact, the Drinfel’d–Deligne–Ihara conjecture states that there is an isomorphismof Lie algebras between grt and the (degree completion of) the free Lie algebra generated by the odd elements { σ j +1 } j ≥ . Part of the conjecture has been proved by F. Brown in [18] who showed that these elements generatea free Lie subalgebra of grt . In order to fully prove the conjecture, it remains to be shown that this free Liesubalgebra identifies with grt .In [113], T. Willwacher provides an explicit isomorphism of Lie algebras H ( GC ) (cid:39) grt under which the seriesof odd elements σ j +1 ∈ grt is mapped to the series of graphs γ j +1 in H ( GC ) . An explicit transcendentalformula for the cocycles γ j +1 is given in [97] as a sum over gra j +2 , j +2 where the coefficients are given by explicitconverging integrals over the configuration space of n points in C \ { , } . However, a purely combinatorialconstruction of the γ j +1 ’s is still missing.21 34 321 5 46 3217 56 48Figure 5: Wheel graphs for j ∈ { , , } Regarding higher degrees, it is a difficult open conjecture (Drinfel’d, Kontsevich) that H ( GC ) = 0 whilecomputer experiments have exhibited sporadic classes in H ≥ ( GC ) . Cohomology of GC The cohomology of the odd graph complex can be characterised in low degrees in a similar way as in the evencase as (see e.g. [10, 70]): Note that the first Betti number endows the dg Lie algebra fGC d with an additional grading. It is more generally defined as b = k − N + c where c denotes the number of connected components. Relatively to the bigrading given by both | γ | and b , thegraded Lie bracket is of bidegree | while the differential is of bidegree | . The odd elements { σ j +1 } j ≥ are the homogeneous components of odd degrees of the element ψ ∈ grt defined such that g = exp( ψ ) is the unique element of GRT sending the Knizhnik–Zamolodchikov associator Φ KZ to the anti-Knizhnik–Zamolodchikovassociator Φ KZ , cf. e.g. [97]. Upper bound : H ≥− ( GC ) = 0 Dominant degree : The dominant level of the odd graph complex GC is located in degree − .The corresponding cohomology space H − ( GC ) can be shown to be spanned by trivalent graphs ( cf. Figure 6 for examples ) modulo the so-called IHX relation reading (see e.g. [10]): = + . (4.15)The cohomology space H − ( GC ) is furthermore endowed with a structure of unital Com -algebra wherethe rôle of the unit is played by the Θ -graph . In fact, there is a morphism of commutative algebras: K (cid:104) t, ω , ω , . . . (cid:105) / (cid:0) ω p ω q − ω ω p + q , P (cid:1) → H − ( GC ) (4.16)for a certain (explicitly known) polynomial P , cf. [73, 74, 75, 111]. The map (4.16) is conjectured to bean isomorphism, up to a 1-dimensional class represented by the Θ -graph. A BC D E F
Figure 6: Non-trivial connected trivalent graphs in GC for N = 4 ( A, B ) and N = 6 ( C, D, E, F ).3.
Lower bound : H ≤− b − ( GC ) = 0 where b = k − N + 1 is the first Betti number.Regarding higher degrees, computer experiments have shown that there exist sporadic classes in H − ( GC ) . We conclude this review of graph complexes by presenting an important variant of the full graph complex knownas the full directed graph complex dfGC d . Following similar steps as for fGC d , we start by defining, for all N ≥ , the graded vector space dGra d ( N ) as: • d even: dGra d ( N ) := (cid:89) k ≥ (cid:0) K (cid:10) gra N,k (cid:11) ⊗ S k sgn k (cid:1) [ k ( d − • d odd: dGra d ( N ) := (cid:89) k ≥ (cid:0) K (cid:10) gra N,k (cid:11) S k (cid:1) [ k ( d − Trivalent graphs in the odd graph complex are usually depicted as chord diagrams where each intersection of three lines standsfor a vertex. Note that modding by the IHX relation ensures that the trivalent graphs in Figure 6 satisfy the equivalence relations A ∼ B and C ∼ D ∼ E ∼ F . The tetrahedron graph B is sometimes denoted t in the literature. The commutative product is defined as follows. Let γ, γ (cid:48) be two trivalent graphs. Remove one vertex of γ so that the resultinggraph has now three dangling edges. Remove one vertex of γ (cid:48) and insert the graph obtained from γ by connecting the danglingedges. The obtained graph is trivalent and modding by the IHX relation ensures that the procedure is independent of the choiceof vertices and that the resulting product is commutative. For example, one can check that A · B = F , cf. Figure 6. S k and the termbetween brackets denotes degree suspension.In other words, the definition of dGra d ( N ) differs from the one of Gra d ( N ) by relaxing the modding out by S ⊗ k .As a result, we deal with directed graphs i.e. whose edge orientation is fixed. Similarly to the undirected case,the set of graded vector spaces { dGra d ( N ) } N ≥ assemble to an operad dGra d .There is an injective morphism of operads O (cid:126) r : Gra d (cid:44) → dGra d (4.17)called the orientation morphism and defined by sending each undirected graph into a sum of directed graphsobtained by interpreting each undirected edge as a sum of directed edges in both directions, cf. Figure 7.The operad dGra d yields a dg Lie algebra, denoted dfGC d where the differential is induced by the Maurer–Cartanelement 2 3 := + ( − d dfGC con d the sub-dg Lie algebra spanned byconnected graphs. The morphism of operads (4.17) induces a morphism of dg Lie algebras s ∗ O (cid:126) r : fGC con d (cid:44) → dfGC con d . (4.18) s ∗ O (cid:126) r (cid:0)
21 34 (cid:1) = 24
21 34 + 8
21 34 + 24
21 34 + 8
21 34Figure 7: Orientation morphismThe following result was shown by T. Willwacher in [113], cf. also [36].
Theorem 4.10.
The morphism s ∗ O (cid:126) r : fGC con d (cid:44) → dfGC con d is a quasi-isomorphism of dg Lie algebras. Theorem 4.10 implies that the study of the cohomology of the directed graph complex boils down to the one ofthe full graph complex, so that essentially nothing new appears when going from undirected to directed graphs.However, the directed graph complex constitutes a useful intermediary when considering representations ofthe Kontsevich’s graph complex, cf. [19]. Furthermore, the directed graph complex possesses two interestingsubcomplexes spanned by oriented and sourced graphs, respectively, which have recently been shown to providesome incarnations of the Grothendieck–Teichmüller Lie algebra grt in higher dimensions , see [115, 119, 120,89, 90] for details and [93] for an application to representations of grt on higher symplectic Lie n -algebroids. In the formulation of his “
Formality conjecture ” [78], M. Kontsevich introduced a universal version of thedeformation complex of the Schouten algebra of polyvector fields, in the guise of an injective morphism fGC (cid:44) → CE ( T poly ) . As shown in [113, 66], this morphism of dg Lie algebras can be best understood as originating from amorphism of operads Gra (cid:44) → End C ∞ ( V ) where V := T ∗ [1] M is a degree 1 NP -manifold whose associated gradedPoisson algebra of functions is isomorphic to T poly endowed with the Schouten bracket (up to suspension).The aim of the present section is to generalise Kontsevich’s construction from d = 2 to arbitrary d > .In other words, we will introduce a tower of representations Gra d (cid:44) → End C ∞ ( V ) with V an arbitrary NP -manifoldof degree n , such that d = n + 1 . This tower of morphism of operads will in turn induce a tower of injectivemorphisms of dg Lie algebras fGC d (cid:44) → CE ( T ( n ) poly ) . Cochains in the image of this map will be called universal That is, for values of d > . fGC d (as recalled in Section 4.3) will allow to classify universal structureson NP -manifolds.As for notation, we will let ( V , ω ) be a NP -manifold of arbitrary degree n ∈ N with d = n + 1 and denote {· , ·} ω the associated Poisson bracket of degree − n . We will make use of the local presentation of NP -manifoldsprovided in Section 3. By analogy with the n = 1 case, we will denote T ( n ) poly := C ∞ ( V ) [ n ] the n -suspension ofthe graded algebra of functions on V along the suspension map s : T ( n ) poly → C ∞ ( V ) of degree | s | = n . We will alsodenote CE ( T ( n ) poly ) the graded Chevalley–Eilenberg cochain space associated with T ( n ) poly and [ · , · ] S the pullback ofthe Poisson bracket {· , ·} ω by the suspension map s : T ( n ) poly → C ∞ ( V ) i.e. [ · , · ] S = s − ◦ {· , ·} ω ◦ ( s ⊗ s ) . It can bechecked that [ · , · ] S is a graded Lie bracket of degree 0, thus endowing T ( n ) poly with a (universal) structure of gradedLie algebra. Pursuing with the previous analogy, we will refer to (cid:0) T ( n ) poly , [ · , · ] S (cid:1) as the n - Schouten algebra. Wewill further denote δ S := (cid:2) [ · , · ] S , · (cid:3) NR – where [ · , · ] NR is the Nijenhuis–Richardson bracket, cf. footnote 47 – theChevalley–Eilenberg differential associated with the Schouten bracket. Proposition 5.1.
The graded algebra of functions on V is endowed with a structure of a dGra d -algebra. The corresponding morphism of operads of graded vector spaces will de denoted dRep ( d ) : dGra d (cid:44) → End C ∞ ( V ) and defined explicitly as the sequence (cid:110) dRep ( d ) N (cid:111) N ≥ of maps dRep ( d ) N : dGra d ( N ) ⊗ C ∞ ( V ) ⊗ N → C ∞ ( V ) reading,for all γ ∈ dGra d ( N ) : dRep ( d ) N ( γ )( f ⊗ · · · ⊗ f N ) = µ N (cid:0)(cid:89) ( i,j ) ∈ E γ ¯∆ ij ( f ⊗ · · · ⊗ f N ) (cid:1) (5.3)where • The f i ’s are functions on V . • The symbol µ N denotes the multiplication map on N elements: µ N : C ∞ ( V ) ⊗ N → C ∞ ( V ): f ⊗ f ⊗ · · · ⊗ f N (cid:55)→ f · f · · · f N (5.4) • The product is performed over the set of edges E γ . For each edge ( i, j ) ∈ E γ connecting vertices labeledby integers i and j , the derivative operator ¯∆ ij is defined as: – d even: ¯∆ ij = ∂∂x µ ( i ) ∂∂p ( j ) µ + ∂∂ψ α k ( i ) ∂∂χ ( j ) α k (5.5) – d odd: ¯∆ ij = ∂∂x µ ( i ) ∂∂p ( j ) µ + ∂∂ψ α k ( i ) ∂∂χ ( j ) α k + 12 ∂∂ξ a ( i ) κ ab ∂∂ξ b ( j ) (5.6)where the sub(super)scripts ( i ) or ( j ) indicate that the derivative acts on the i -th or j -th factor in thetensor product. Proof.
The maps dRep ( d ) N can be checked to satisfy the three following properties:1. dRep ( d )1 (cid:0) (cid:1) = id C ∞ ( V ) Letting g be a graded vector space, the graded Chevalley–Eilenberg cochain space (in the adjoint representation) is defined as CE ( g ) := (cid:77) k ∈ Z CE k ( g ) where CE k ( g ) := (cid:77) i + j = k Hom i ( g ∧ j +1 , g ) . (5.1)The latter is endowed with a pre-Lie algebra structure through the Nijenhuis–Richardson product defined as f ◦ NR g = (cid:88) σ ∈ Sh − ( q,p − ( − | σ | ( − ( p − | g | Σ p + q − (cid:0) f ◦ g (cid:12)(cid:12) σ ) (5.2)for all f ∈ Hom | f | ( g ∧ p , g ) and g ∈ Hom | g | ( g ∧ q , g ) .The commutator [ f, g ] NR := f ◦ NR g − ( − | f || g | g ◦ NR f is a graded Lie bracket referred to as the Nijenhuis–Richardson bracket . dRep ( d ) M + N − ( γ ◦ dGra i γ (cid:48) ) = dRep ( d ) M ( γ ) ◦ End i dRep ( d ) N ( γ (cid:48) ) for all γ ∈ dGra d ( M ) and γ (cid:48) ∈ dGra d ( N ) where thepartial composition maps of the endomorphism operad take the form: θ ◦ End i θ (cid:48) = θ ◦ (cid:0) ⊗ i − ⊗ θ (cid:48) ⊗ ⊗ M − i (cid:1) (5.7)for all θ ∈ Hom (cid:0) C ∞ ( V ) ⊗ M , C ∞ ( V ) (cid:1) and θ (cid:48) ∈ Hom (cid:0) C ∞ ( V ) ⊗ N , C ∞ ( V ) (cid:1) .3. dRep ( d ) N (cid:0) Σ dGra N ( γ | σ ) (cid:1) = Σ End N (cid:0) dRep ( d ) N ( γ ) | σ (cid:1) where the endomorphism operad right action reads Σ End N ( θ | σ )( f , . . . , f N ) := θ ( f σ − , . . . , f σ − N ) ) (5.8)for all f i ∈ C ∞ ( V ) , θ ∈ Hom (cid:0) C ∞ ( V ) ⊗ N , C ∞ ( V ) (cid:1) and σ ∈ S N .The three above properties ensure that the maps (cid:110) dRep ( d ) N (cid:111) N ≥ assemble to form a morphism of operads.Composing the representation morphism dRep ( d ) : dGra d (cid:44) → End C ∞ ( V ) with the orientation morphism O (cid:126) r : Gra d (cid:44) → dGra d (cid:0) see eq.(4.17) (cid:1) endows the algebra of functions C ∞ ( V ) with a structure of Gra d -algebra through the morphism Rep ( d ) : Gra d O (cid:126) r (cid:44) −→ dGra d dRep ( d ) −→ End C ∞ ( V ) .The maps (cid:110) Rep ( d ) N (cid:111) N ≥ can be defined explicitly in a form similar to eq.(5.3) as Rep ( d ) N ( γ )( f ⊗ · · · ⊗ f N ) = µ N (cid:0)(cid:89) ( i,j ) ∈ E γ ∆ ij ( f ⊗ · · · ⊗ f N ) (cid:1) (5.9)where one traded the ¯∆ operators with: • d even: ∆ ij = ∂∂x µ ( i ) ∂∂p ( j ) µ + ∂∂p ( i ) µ ∂∂x µ ( j ) + ∂∂ψ α k ( i ) ∂∂χ ( j ) α k + ∂∂χ ( i ) α k ∂∂ψ α k ( j ) (5.10) • d odd: ∆ ij = ∂∂x µ ( i ) ∂∂p ( j ) µ − ∂∂p ( i ) µ ∂∂x µ ( j ) + ∂∂ψ α k ( i ) ∂∂χ ( j ) α k − ( − k ∂∂χ ( i ) α k ∂∂ψ α k ( j ) + ∂∂ξ a ( i ) κ ab ∂∂ξ b ( j ) . (5.11)The differential operator ∆ ij enjoys the following properties :1. | ∆ ij | = 1 − d consistently with the grading of an edge in Gra d .2. ∆ ij ∆ kl = − ( − d ∆ kl ∆ ij consistently with the fact that permuting two edges in graphs in Gra d brings asign only for even d .3. ∆ ij = ( − d ∆ ji consistently with the fact that flipping the orientation of an edge in graphs in Gra d bringsa sign only for odd d .The tower of morphisms Rep ( d ) : Gra d (cid:44) → End C ∞ ( V ) generalises to all d the Kontsevich morphism for d = 2 , cf. [78, 113, 66] and more recently [21, 19, 101, 71]. Universal structures on C ∞ ( V ) The morphism
Rep ( d ) allows to define universal structures (cid:0) in the sense of Definition 4.2 (cid:1) on the algebra offunctions C ∞ ( V ) .In particular, it was noted earlier ( cf. Section 3) that the algebra of functions on V was naturally endowed witha structure of Ger d -algebra. This statement can be refined as follows: It will be shown in [93] how the morphism
Rep ( d ) factors through the operad of multi-directed graphs. In contrast, the differential operator ¯∆ ij only satisfies properties 1-2 consistently with the definition of dGra d which omits tomod out by S ⊗ k , cf. Section 4.4. roposition 5.2. The
Ger d -algebra structure on C ∞ ( V ) is universal i.e. the action of the operad Ger d factorsthrough: Ger d i d (cid:44) −→ Gra d Rep ( d ) −→ End C ∞ ( V ) (5.12) where i d : Ger d (cid:44) −→ Gra d is the natural embedding of operads defined in Proposition 4.3 .Proof.
The statement follows straightforwardly from:
Rep ( d )2 (Γ )( f ⊗ g ) = f · g. , Rep ( d )2 (Γ )( f ⊗ g ) = (cid:8) f, g (cid:9) ω . (5.13)In the case d = 2 , Proposition 5.2 can be completed by stating that the graded algebra of functions C ∞ ( V ) isnaturally endowed with a universal structure of Batalin–Vilkovisky algebra , cf. [92].The BV Laplacian ∆ is then defined as the image of the tadpole L = i.e. ∆ := Rep (2)1 (cid:0) L (cid:1) .In the case d = 1 , the classification recalled in Section 3 ensures that a NP -manifold of degree 0 is in fact a(non-graded) symplectic manifold ( M , κ ) . In that case, the chain of morphisms of operads Ass i (cid:44) −→ Gra Rep (1) −→ End C ∞ ( M ) endows the algebra of functions on the symplectic manifold M with a universal associative structure f ∗ GM g = Rep (1)2 ( )( f ⊗ g ) , where the graph 1 ... 2 ∈ Gra (2) is defined in (4.9), cf. [70].The induced associative product is the Groenewold–Moyal product [52, 94] reading explicitly as: ( f ∗ GM g )( ξ ) := exp (cid:16) (cid:15) κ ab ∂∂ζ a ∂∂η b (cid:17) f ( ζ ) g ( η ) (cid:12)(cid:12)(cid:12) ζ = η = ξ (5.14)where (cid:15) is a formal parameter. Universal cochains of the Chevalley–Eilenberg algebra
The following proposition allows to define a universal graph model for (cid:0) CE ( T ( n ) poly ) : Proposition 5.3.
The morphism of operads
Rep ( d ) : Gra d (cid:44) → End C ∞ ( V ) induces a morphism of dg Lie algebras s ∗ Rep ( d ) : (cid:0) fGC d , δ, [ · , · ] (cid:1) (cid:44) → (cid:0) CE ( T ( n ) poly ) , δ S , [ · , · ] NR (cid:1) . (5.15) Proof.
The proof follows straightforwardly from the equivariance (5.8) of the morphism
Rep ( d ) and from theequality s ∗ Rep ( d ) (Γ ) = [ · , · ] S .Pursuing with the terminology introduced in Definition 4.1, Chevalley–Eilenberg cochains in the image of(5.15) will be referred to as universal . In other words, the dg Lie algebra of graphs (cid:0) fGC d , δ, [ · , · ] (cid:1) provides auniversal version of the Chevalley–Eilenberg dg Lie algebra (cid:0) CE ( T ( n ) poly ) , δ S , [ · , · ] NR (cid:1) . The former thus controlsthe deformation theory – in the universal setting – of the n -Schouten Lie algebra as a Lie ∞ -algebra. The twofollowing corollaries make this fact explicit: A Batalin–Vilkovisky algebra is a
Ger -algebra (cid:0) g , ∧ , {· , ·} (cid:1) such that there exists a unary operator ∆ of degree − satisfying:1. ∆ = 0 ∆( a ∧ b ∧ c ) − ∆( a ∧ b ) ∧ c + ∆ a ∧ b ∧ c − ( − a a ∧ ∆( b ∧ c ) − ( − b ( a − b ∧ ∆( a ∧ c ) + ( − a a ∧ ∆ b ∧ c + ( − a + b a ∧ b ∧ ∆ c = 0 ∆( a ∧ b ) − ∆ a ∧ b − ( − a a ∧ ∆ b = (cid:8) a, b (cid:9) . Note that not all conceivable deformations of T ( n ) poly are universal. For example, letting H ∈ Ω ( M ) be a closed 3-form, onecan define a non-universal deformation of T (1) poly = T poly by defining a higher bracket of arity as l ∈ Hom − (cid:0) T poly ∧ , T poly (cid:1) ∈ CE ( T poly ) as l ( X , X , X ) = H ( X , X , X ) , where the X i ’s are polyvector fields. Denoting l the usual Schouten bracket,the triplet ( T poly ( M ) , l , l ) forms a L ∞ -algebra. Associated Maurer–Cartan elements are so-called twisted Poisson structures [72, 105] i.e. bivectors π ∈ γ ( ∧ T M ) satisfying [ π, π ] S = 13 H ( π, π, π ) . The latter can be interpreted as Dirac structures for thestandard Courant algebroid twisted by H , cf. [104]. orollary 5.4. Maurer–Cartan elements for the dg Lie algebra (cid:0) fGC d , δ, [ · , · ] (cid:1) are mapped via s ∗ Rep ( d ) touniversal deformations of the graded Lie algebra ( T ( n ) poly , [ · , · ] S ) as a Lie ∞ -algebra. Example 5.5 (Groenewold–Moyal commutator) . Let ( M , κ ) be a symplectic manifold.The Maurer–Cartan element (4.13) prolongating the Θ -graph is mapped via s ∗ Rep (1) to the (essentially unique)universal deformation of ( C ∞ ( M ) , {· , ·} κ ) as a Lie algebra where the Poisson bracket is deformed into the Groenewold–Moyal commutator [ f, g ] GM := f ∗ GM g − g ∗ GM f on C ∞ ( M ) constructed from (5.14), cf. [70].We refer to Example 5.9 for an example of a deformation of the -Schouten algebra as a genuine Lie ∞ -algebra.The Lie algebra H ( fGC d ) being pro-nilpotent , one defines the pro-unipotent group exp (cid:0) H ( fGC d ) (cid:1) as [116]: • Group elements are elements of H ( fGC d ) , viewed as a set. • The unit is ∈ H ( fGC d ) . • The inverse map sends γ to − γ . • The group operation is defined as γ · γ = BCH ( γ , γ ) where BCH stands for the Baker–Campbell–Hausdorff formula. Corollary 5.6.
The pro-unipotent group exp (cid:0) H ( fGC d ) (cid:1) acts via Lie ∞ -automorphisms on the n -Schouten al-gebra.Proof. The proof is identical to the one of the case n = 1 ( cf. e.g. Theorem 1. in [66], based on [113]) thatwe review for completeness. Let γ ∈ H ( fGC d ) be a non-trivial cocycle in fGC d . The morphism s ∗ Rep ( d ) ofdg Lie algebras introduced in Proposition 5.3 maps γ to a zero degree Chevalley–Eilenberg cocycle for the n -Schouten algebra denoted s ∗ Rep ( d ) ( γ ) . In other words, s ∗ Rep ( d ) ( γ ) ∈ H (cid:0) CE ( T ( n ) poly ) (cid:1) is a Lie ∞ -derivationof (cid:0) T ( n ) poly , [ · , · ] S (cid:1) . This ensures that exp (cid:0) s ∗ Rep ( d ) ( γ ) (cid:1) is a Lie ∞ -automorphism of (cid:0) T ( n ) poly , [ · , · ] S (cid:1) . Furthermore,since exp (cid:0) H ( fGC d ) (cid:1) is pro-unipotent, to any element Γ ∈ exp (cid:0) H ( fGC d ) (cid:1) one can associate a unique element γ ∈ H ( fGC d ) so that Γ = exp( γ ) . We can thus define a Lie ∞ -action via its Taylor coefficients: U N : exp (cid:0) H ( fGC d ) (cid:1) × T ( n ) poly ∧ N → T ( n ) poly : (cid:0) Γ , X , . . . , X N (cid:1) (cid:55)→ exp (cid:0) s ∗ Rep ( d ) ( γ ) (cid:1) ( X , . . . , X N ) for all N ≥ . Classification of universal structures
We now make use of the results regarding cohomology of the full graph complex as reviewed in Section 4.3 inorder to provide a classification of universal structures on the n -Schouten algebra for all n ∈ N – where theterm universal structures will refer to :1. Universal Lie ∞ -automorphisms of the n -Schouten algebra2. Universal deformations of the n -Schouten algebra as a Lie ∞ -algebra.Since the case n = 1 has already been addressed in the literature, we treat it separately: Note that n = 0 so that the ordinary ( i.e. non-graded) Lie algebra ( C ∞ ( M ) , {· , ·} κ ) identifies with the -Schouten algebra ( T (0) poly , [ · , · ] S ) . In this case, the deformation complex (cid:0) CE ( T ( n ) poly ) , δ S , [ · , · ] NR (cid:1) controls the deformation theory of ( C ∞ ( M ) , {· , ·} κ ) as an ordinary Lie algebra. See [32] for a proof of the d = 2 case. The proof for all d is identical. We will focus on universal structures obtained from connected graphs.
24. Recall from Theorem 4.8 that there exists an isomorphism of Lie algebras H ( fGC con ) (cid:39) grt . By Corollary5.6, it follows that the Grothendieck–Teichmüller group GRT := exp( grt ) acts via Lie ∞ -automorphismson the Schouten algebra T poly , see [113, 66].2. As noted earlier, it is a difficult open conjecture (Drinfel’d, Kontsevich) that H ( fGC con ) = . If theconjecture holds, then there are no universal deformations of the Schouten algebra as a
Lie ∞ -algebra i.e. ( T poly , [ · , · ] S ) is rigid as a universal Lie ∞ -algebra.We now turn to the case n (cid:54) = 1 . The following classification of the cohomology of the connected part of the fullgraph complex in low degrees is obtained from the various bounds collected in Section 4.3: Lemma 5.7.
The cohomology of the (connected part of ) the full graph complex fGC con d in low degrees for all d (cid:54) = 2 is given by: • Degree : H ( fGC con j +3 ) = K (cid:104) L j +3 (cid:105) for all j ≥ and trivial otherwise. • Degree : H ( fGC con ) = K (cid:104) Θ (cid:105) , H ( fGC con j +4 ) = K (cid:104) L j +5 (cid:105) for all j ≥ − and trivial otherwise. • Degree : H ( fGC con j +1 ) = K (cid:104) L j +3 (cid:105) for all j ≥ and trivial otherwise.Note that the only non-loop cocycle in this classification is given by the Θ -graph. The universal Lie ∞ -structureinduced by the Maurer–Cartan element (4.13) prolongating the latter is given by the Groenewold–Moyal bracketon symplectic manifolds – cf. Example 5.5 – which constitutes the unique universal structure in dimension d = 1 .It follows that the only universal structures in dimension d > are induced by loop classes : Proposition 5.8 (Loop induced universal structures) . Let us denote L k the loop graph with k edges.1. Let ( V , ω ) be a NP -manifold of (even) degree n = 4 j + 2 , j ≥ . There is a unique universal Lie ∞ -automorphism of the n -Schouten algebra (cid:0) T ( n ) poly , [ · , · ] S (cid:1) . The latter is induced by the loop cocycle L j +3 .2. Let ( V , ω ) be a NP -manifold of (odd) degree n = 4 j + 3 , j ≥ . There is a unique universal deformation ofthe n -Schouten algebra (cid:0) T ( n ) poly , [ · , · ] S (cid:1) as a Lie ∞ -algebra. The latter is induced by the loop cocycle L j +5 .Proof. The first statement follows straightforwardly from Corollary 5.6 and the first item of Lemma 5.7.As for the second statement, since H ( fGC con j +4 ) vanishes for all j ≥ , there is no obstruction to the prolongationof the loop cocycle L j +5 into a Maurer–Cartan element m j +5 ∈ MC ( fGC j +4 ) . Corollary 5.4 then ensures thatthe Maurer–Cartan element m j +5 is mapped via s ∗ Rep ( d ) to a universal deformation of the graded Lie algebra ( T ( n ) poly , [ · , · ] S ) as a Lie ∞ -algebra . As noted in [90], although the cohomology of GC in degree 1 is conjectured to be trivial, a choice of Drinfel’d associator isnecessary in order to convert cocycles of degree 1 in GC into coboundaries of degree 0 so that an iterative procedure can exist butcannot be trivial. If true, the statement only holds in the universal setting, cf. footnote 57 for a statement in the oriented setting. Departing from the stable regime to the oriented regime, we note that the Θ -graph induces the Kontsevich–Shoikhet cocycle in H ( GC or ) whose prolongation to a Maurer-Cartan element is mapped to the Kontsevich–Shoikhet Lie ∞ -algebra structure deformingthe Schouten algebra of polyvector fields, cf. [108, 115]. In the stable setting, universal structures induced from the Grothendieck–Teichmüller algebra grt only occur in dimension d = 2 . However, departing from the stable to the (multi)-oriented setting will allow to generate universal structures from grt indimensions d > [93]. The only non-trivial higher brackets l m of the Lie ∞ -algebra induced by the loop cocycle L j +5 have arities m = p (4 j + 3) + 2 for all p ≥ with l = [ · , · ] S . j = 0 ): Example 5.9.
Let ( V , ω ) be a NP -manifold of degree 3 coordinatised by (cid:26) x µ , ψ α , χ α , p µ (cid:27) , cf. e.g. [65, 83, 54].The pentagon graph L :=
321 5 4 can be promoted to a formal Maurer–Cartan element m ∈ MC ( fGC ) reading m := (cid:15) L + (cid:15) m (8)5 + · · · + (cid:15) p +2 m (3 p +2)5 + · · · which induces a Lie ∞ -algebra structure on the shiftedgraded algebra of functions T (3) poly := C ∞ ( V ) [3] with non-vanishing brackets l , l , l , . . . , l p +2 , p ≥ such that,for all X i ∈ T (3) poly : • l ( X , X ) = s ∗ Rep (4) (cid:0) Γ (cid:1) ( X , X ) = [ X , X ] S • l ( X , . . . , X ) = s ∗ Rep (4) (cid:0) L (cid:1) ( X , . . . , X ) = s − µ (cid:16) ¯∆ ¯∆ ¯∆ ¯∆ ¯∆ (cid:0) s ( X ) , . . . , s ( X ) (cid:1)(cid:17) where ¯∆ ij = ∂∂x µ ( i ) ∂∂p ( j ) µ + ∂∂p ( i ) µ ∂∂x µ ( j ) + ∂∂ψ α ( i ) ∂∂χ ( j ) α + ∂∂χ ( i ) α ∂∂ψ α ( j ) ... • l p +2 ( X , . . . , X p +2 ) = s ∗ Rep (4) (cid:0) m (3 p +2)5 (cid:1) ( X , . . . , X p +2 ) Up to now, our attention has been focused on NP -manifolds. We will now consider additional structures ongraded manifolds. In particular, we will focus on NPQ -manifolds i.e. symplectic graded manifolds endowedwith a Hamiltonian structure – cf.
Section 3 – and discuss how the previously developed machinery can be usedin order to generate universal Hamiltonian deformations.
Universal Hamiltonian deformations
Consider a NP -manifold ( V , ω ) of arbitrary degree n ∈ N with d = n + 1 . The set of Hamiltonian functions on ( V , ω ) will be denoted Ham . Also, we will denote MC ( T ( n ) poly ) the set of formal Maurer–Cartan elements forthe associated graded Lie algebra (cid:0) T ( n ) poly , [ · , · ] S (cid:1) i.e. m ∈ MC ( T ( n ) poly ) ⇔ m ∈ (cid:15) T ( n ) | poly [[ (cid:15) ]] and [ m , m ] S = 0 . (5.16)Note that any Hamiltonian function H ∈ Ham defines a canonical Maurer–Cartan element m H := (cid:15) s − ( H ) where s : T ( n ) poly → C ∞ ( V ) denotes the suspension map of degree n .This fact suggests to define a notion of Hamiltonian deformation in the following sense:
Definition 5.10 (Hamiltonian deformation) . Let H ∈ Ham be a Hamiltonian function on V .A Hamiltonian deformation of H is a formal power series H ∗ ∈ (cid:15) C ∞| d ( V )[[ (cid:15) ]] such that1. H ∗ is nilpotent with respect to the graded Poisson bracket i.e. (cid:8) H ∗ , H ∗ (cid:9) ω = 0 .2. The first order of the expansion of H ∗ in terms of the formal parameter coincides with H or equivalently (cid:15) H ∗ | (cid:15) =0 = H .The set of Hamiltonian deformations of a given Hamiltonian function H will be denoted FHam H . A map Ham → FHam which assigns to each H ∈ Ham an element in
FHam H will be referred to as a Hamiltoniandeformation map . We let (cid:15) be a formal parameter. Recall from Section 3 that a Hamiltonian function is a function H ∈ C ∞| d ( V ) such that (cid:8) H , H (cid:9) ω = 0 . For any (graded) vector space V , we will denote V [[ (cid:15) ]] the (graded) vector space of formal series in the formal parameter (cid:15) withcoefficients in V . m := s − ( H ∗ ) is a formalMaurer–Cartan element of the graded Lie algebra (cid:0) T ( n ) poly , [ · , · ] S (cid:1) i.e. m ∈ MC ( T ( n ) poly ) . This fact, combined withCorollary 5.6, yields the following Proposition: Proposition 5.11.
There is a canonical map H ( fGC d ) → ( Ham → FHam ) mapping cocycles in the zerothgraph cohomology to universal Hamiltonian deformation maps.Proof. Recall from Corollary 5.6 that to each cocycle γ ∈ H ( fGC d ) , one can associate a Lie ∞ -automorphism U Γ = exp (cid:0) s ∗ Rep ( d ) ( γ ) (cid:1) of (cid:0) T ( n ) poly , [ · , · ] S (cid:1) . The latter induces a bijective map between equivalence classes offormal Maurer–Cartan elements of (cid:0) T ( n ) poly , [ · , · ] S (cid:1) as ˆ U Γ : MC ( T ( n ) poly ) / ∼ ∼ −→ MC ( T ( n ) poly ) / ∼ : [ m ] (cid:55)→ [ ˆ U Γ ( m )] where ˆ U Γ ( m ) := (cid:80) ∞ k =1 1 k ! U Γ k ( m ⊗ k ) . Let H ∈ Ham be a Hamiltonian function. We will denote m H ∈ MC ( T ( n ) poly ) the canonical formal Maurer–Cartan element defined as m H = (cid:15) s − ( H ) . The latter is mapped via ˆ U Γ to ˆ U Γ ( m H ) = (cid:80) ∞ k =1 (cid:15) k k ! U Γ k ( s − ( H ) ⊗ k ) . It follows from the above reasoning that ˆ U Γ ( m H ) is a formal Maurer–Cartan element. Finally, we define H ∗ := s (cid:0) ˆ U Γ ( m H ) (cid:1) . Since ˆ U Γ1 is the identity of T ( n ) poly , then (cid:15) H ∗ | (cid:15) =0 = H and hence H ∗ is a Hamiltonian deformation of H . We conclude that the map Ham → FHam : H (cid:55)→ H ∗ is auniversal deformation map.The denomination universal is to be understood in the sense of Definition 4.2. Also in this context, it reflectsthe fact that the deformation maps induced by graph cocycles are valid on any graded symplectic manifold ofarbitrary dimension.Remember from Section 3 that Hamiltonian functions on NP -manifolds naturally induce geometric structureson the fibration 3.2, referred to as symplectic Lie n -algebroids , e.g. Poisson manifolds ( n = 1 ), Courantalgebroids ( n = 2 ), etc. Proposition 5.11 can thus be interpreted as mapping graph cocycles towards universaldeformations of symplectic Lie n -algebroids. We will develop this line of reasoning in Section 6 by focusing onfirst order deformations, or Hamiltonian flows on the space of symplectic Lie n -algebroids. Universal Hamiltonian flows
Let ( V , ω ) be a NP -manifold of arbitrary degree n ∈ N and denote d = n +1 . By Hamiltonian flow, we will mean amap Ham → H d ( C ∞ ( V ) | Q ) where H • ( C ∞ ( V ) | Q ) refers to the cohomology of the complex induced on C ∞|• ( V ) by the homological vector field Q := (cid:8) H , · (cid:9) ω . In other words, a Hamiltonian flow maps any Hamiltonianfunction H ∈ Ham to a (non-trivial) cocycle, denoted ˙ H ∈ C ∞| d ( V ) , and satisfying (cid:8) H , ˙ H (cid:9) ω = 0 . Inthe case when ˙ H is constructed in terms of H through universal formulas, the Hamiltonian flow is called universal . The set of universal Hamiltonian flows will be denoted Hflow . The following statement can be seenas a linearisation of Proposition 5.11:
Proposition 5.12.
There is a canonical map H ( fGC d ) → Hflow . Explicitly, cocycles γ ∈ H ( fGC d ) with N vertices are mapped to universal flows on the space of Hamiltonianfunctions as maps H (cid:55)→ ˙ H where ˙ H := Rep ( d ) N ( γ )( H ⊗ N ) . The latter satisfies (cid:8) H , ˙ H (cid:9) ω = 0 as a consequenceof (cid:8) H , H (cid:9) ω = 0 . Proof.
The cocycle condition δ γ = 0 ensures that s ∗ Rep ( d ) ( γ ) is a degree 0 Chevalley–Eilenberg cocycle for thegraded Lie algebra (cid:0) T ( n ) poly , [ · , · ] S (cid:1) . Explicitly, denoting U Γ N the component of s ∗ Rep ( d ) ( γ ) acting on N inputs ,the Chevalley–Eilenberg cocycle condition can be expressed as: (cid:2) [ · , · ] S , U Γ N (cid:3) NR = (cid:88) σ ∈ Sh − ( N, ( − | σ | ( − − N Σ N +1 (cid:0) [ · , · ] S ◦ U Γ N (cid:12)(cid:12) σ ) − (cid:88) σ ∈ Sh − (2 ,N − ( − | σ | Σ N +1 (cid:0) U Γ N ◦ [ · , · ] S (cid:12)(cid:12) σ ) = 0 . (5.17) Where we denoted
Γ := exp( γ ) . Or equivalently the first non-trivial Taylor coefficient, beside the identity, of the
Lie ∞ -automorphism U Γ , with Γ := exp( γ ) , cf. the proof of Proposition 5.11. s − ( H ) ⊗ N +1 , the second term vanishes due to (cid:8) H , H (cid:9) ω = 0 and (cid:104) s − ( H ) , U Γ N ( s − ( H ) ⊗ N ) (cid:105) S = 0 .Denoting ˙ H := s (cid:0) U Γ N ( s − ( H ) ⊗ N ) (cid:1) = Rep ( d ) N ( γ )( H ⊗ N ) leads to the flow equation (cid:8) H , ˙ H (cid:9) ω = 0 .Proposition 5.12 generalises to all n Kontsevich’s construction of universal flows on the space of Poisson manifoldsfrom cocycles in H ( GC ) , cf. Section 5.3 in [78] and Section 6.1 below.
Conformal Hamiltonian flows
Let us emphasise that the map H ( fGC d ) → Hflow is not surjective in general. We will show this explicitlyby exhibiting a new class of universal Hamiltonian flows which are not induced by elements in the zerothcohomology. Such Hamiltonian flows will be called conformal as they pair any Hamiltonian function H ∈ Ham with a function Ω( H ) on the base manifold M – the “conformal factor” – such that Ω( H ) ∈ H ( C ∞ ( V ) | Q ) which ensures that ˙ H := Ω( H ) · H satisfies (cid:8) H , ˙ H (cid:9) ω = 0 . The subset of conformal Hamiltonian flows willbe denoted cHflow ⊂ Hflow . Proposition 5.13.
There is a canonical map H − d ( fGC d ) → cHflow . The universal Hamiltonian flows in the image of this map are conformal with conformal factor defined as Ω( H ) := Rep ( d ) N ( γ )( H ⊗ N ) where γ ∈ H − d ( fGC d ) is a cocycle with N vertices. Proof.
We start by pointing out that the necessary condition ensuring that a graph ¯ γ induces a universalHamiltonian flow is given by (cid:2) [ · , · ] S , U ¯ γ • (cid:3) NR ( s − ( H ) ⊗• +1 ) = 0 . The latter is thus weaker that the cocyclecondition and is ensured by: δ ¯ γ = Γ ∪ γ (cid:48) (5.18)with γ (cid:48) an arbitrary graph of degree | γ (cid:48) | = | ¯ γ | − d and where we denoted ∪ the concatenation of two graphsin fGC d into a single (disconnected) graph. It remains to be shown that elements of H − d ( fGC d ) define We will denote ∪ : Gra d ( N ) ⊗ Gra d ( N (cid:48) ) → Gra d ( N + N (cid:48) ) the concatenation of two graphs into a single (disconnected) graph,as in the following example: 1 23 i iiiii ∪ i = i iiiii iv . (5.19)The morphism Rep ( d ) preserves the concatenation product as: Rep ( d ) N + N (cid:48) ( γ ∪ γ (cid:48) )( f ⊗ · · · ⊗ f N + N (cid:48) ) = Rep ( d ) N ( γ )( f ⊗ · · · ⊗ f N ) · Rep ( d ) N (cid:48) ( γ (cid:48) )( f N +1 ⊗ · · · ⊗ f N + N (cid:48) ) . (5.20)The concatenation product can be checked to be associative and to satisfy the “commutation” relation: γ ∪ γ (cid:48) = ( − kk (cid:48) (1 − d ) Σ N + N (cid:48) ( γ (cid:48) ∪ γ | σ ) (5.21)with γ ∈ gra N,k , γ (cid:48) ∈ gra N (cid:48) ,k (cid:48) and where the permutation σ ∈ S N + N (cid:48) is defined as σ := (cid:18) · · · N + N (cid:48) N (cid:48) + 1 · · · N + N (cid:48) · · · N (cid:48) (cid:19) so that σ − = (cid:18) · · · N + N (cid:48) N + 1 · · · N + N (cid:48) · · · N (cid:19) . (5.22)We will denote with the same symbol ∪ the corresponding concatenation operation of two graphs in fGC d . The latter can be shownto be:1. of degree d
2. graded commutative i.e. γ ∪ γ (cid:48) = ( − ( γ + d )( γ (cid:48) + d ) γ (cid:48) ∪ γ
3. associative i.e. ( γ ∪ γ (cid:48) ) ∪ γ (cid:48)(cid:48) = γ ∪ ( γ (cid:48) ∪ γ (cid:48)(cid:48) ) .Furthermore, the differential δ satisfies the Leibniz rule δ ( γ ∪ γ (cid:48) ) = δ γ ∪ γ (cid:48) + ( − | γ | + d γ ∪ δ γ (cid:48) . (5.23) γ ∈ H − d ( fGC d ) , one defines ¯ γ = Γ ∪ γ where Γ stands for the onlygraph in gra , . The fact that δ Γ = Γ together with the Leibniz rule (5.23) ensures that ¯ γ is indeeda solution of (5.18). Assuming that the graph γ possesses N vertices, property 5.20 ensures that ˙ H := Rep ( d ) N +1 (¯ γ )( H ⊗ N +1 ) = Rep ( d )1 (Γ )( H ) · Rep ( d ) N ( γ )( H ⊗ N ) . Denoting Ω( H ) := Rep ( d ) N ( γ )( H ⊗ N ) , the latter isof degree | Ω( H ) | = | γ | + dN where | γ | stands for the degree of γ in Gra d i.e. | γ | = k (1 − d ) . Since γ is of degree − d in fGC d , then d ( N −
1) + k (1 − d ) = − d and thus | γ | = k (1 − d ) = − dN so that | Ω( H ) | = dN − dN = 0 and Ω( H ) is hence a function on the base manifold M of V . Furthermore, since γ is a cocycle in fGC d , then (cid:8) H , Ω( H ) (cid:9) ω = 0 . Finally, Rep ( d )1 (Γ )( H ) = H so that ˙ H = Ω( H ) · H satisfies (cid:8) ˙ H , H (cid:9) ω = 0 as aconsequence of (cid:8) H , H (cid:9) ω = 0 .Similarly as before, one can make use of the results reviewed in Section 4.3 in order to classify universal conformalHamiltonian flows in the stable setting. In fact one can check that H − d ( fGC con d ) = ∅ for all d (cid:54) = 3 . It follows thatstable conformal Hamiltonian flows only appear in dimension where the conformal factor is induced fromtrivalent graphs modulo IHX relations (cid:0) see Figure 6 and eq.(4.15) (cid:1) . Explicit examples of universal conformalflows on the space of Courant algebroids will be displayed in Section 6.2. n -algebroids The aim of the present section is to illustrate some of the machinery developed in Section 5 to the case of
NPQ -manifolds of degrees 1 and 2. As recalled in Example 3.1, the associated geometric notions ( i.e. symplectic Lie , -algebroids) identify with the one of Poisson manifolds and Courant algebroids, respectively. n = 1 ) As shown in [99], NP -manifolds V of degree are in bijective correspondence with ordinary smooth manifolds M via the identification of V with the shifted cotangent bundle T ∗ [1] M . The tower of fibrations 3.2 thusreduces to the vector bundle structure M ← T ∗ [1] M . The graded Poisson algebra of functions on T ∗ [1] M isisomorphic to the Gerstenhaber algebra of polyvector fields T poly and Hamiltonian functions are in bijectionwith Poisson bivectors on M .The representation morphism Rep (2) : Gra (cid:44) → End C ∞ ( T ∗ [1] M ) of the 2-dimensional graph operad Gra on thespace of functions of the shifted cotangent bundle was first introduced by M. Kontsevich in [78, Section 5.2] andreads as (5.9) with ∆ given by (cid:0) cf. eq.(5.10) (cid:1) ∆ ij = ∂∂x µ ( i ) ∂∂p ( j ) µ + ∂∂p ( i ) µ ∂∂x µ ( j ) . (6.1)Following the leitmotiv of Section 5, the representation morphism Rep (2) can be used in order to induce universalstructures on M . In particular, using the isomorphism H ( GC ) (cid:39) grt , it follows from Corollary 5.6 that theGrothendieck–Teichmüller group GRT := exp( grt ) acts via Lie ∞ -automorphisms on the Schouten algebra T poly , see [113, 66].At the linear level, Proposition 5.12 ensures that cocycles in H ( GC ) yield universal flows on the space ofPoisson bivectors. In other words, given a manifold M and a cocycle γ ∈ H ( GC ) with N vertices, one candefine a map π (cid:55)→ ˙ π mapping Poisson bivectors π ∈ Γ (cid:0) ∧ T M (cid:1) (thus satisfying [ π, π ] S = 0 ) on M to universal Lichnerowicz cocycles i.e. bivectors ˙ π ∈ Γ (cid:0) ∧ T M (cid:1) satisfying δ π ˙ π := [ π, ˙ π ] S = 0 . Concretely, this is done byfirst defining the function H = π µν ( x ) p µ p ν – which can be checked to be Hamiltonian ( i.e. (cid:8) H , H (cid:9) ω = 0 )as a consequence of the fact that π is Poisson – and then define the function ˙ H := Rep (2) N ( γ )( H ⊗ N ) – satisfying Again, allowing for (multi)-oriented graphs will generate a number of conformal Hamiltonian flows in various dimensions d (cid:54) = 3 , cf. [93]. H , ˙ H (cid:9) ω = 0 as a consequence of δγ = 0 and (cid:8) H , H (cid:9) ω = 0 . Finally, one defines ˙ π as the principal symbolof the function ˙ H i.e. ˙ H = ˙ π µν p µ p ν .The simplest example of the previous construction is given by the tetrahedral flow introduced in [78, Section 5.3]and further studied in [16, 15]. The latter is induced by the tetrahedron graph γ ∈ H ( GC ) ( cf. Proposition 4.9and Figure 5). Explicit expression for the map π (cid:55)→ ˙ π can be obtained by first using the orientation morphism(4.18) on γ as in Figure 7, yielding a linear combination of four directed graphs. Decorating vertices withcopies of the Hamiltonian function H and interpreting edges as differential operators ¯∆ (cid:0) see eq.(5.5) (cid:1) , the firsttwo graphs vanish since they include vertices with more than two outgoing edges. The two remaining graphsyield the following local expression for the Lichnerowicz cocycle ˙ π associated with the Poisson bivector π : ˙ π µν = ∂ (cid:15) π αβ ∂ α π γδ ∂ γ π (cid:15)λ ∂ βδλ π µν + 6 ∂ (cid:15) π αβ ∂ α π γδ ∂ γλ π (cid:15) [ µ ∂ βδ π ν ] λ . (6.2)Furthermore, it follows from Proposition 5.11 that the cocycle ˙ π can be promoted to a full Maurer-Cartanelement in ( T poly , δ π , [ · , · ] S ) thus yielding a universal formal Poisson structure π ∗ = π + (cid:15) ˙ π + · · · such that [ π ∗ , π ∗ ] S = 0 .Note that the tetrahedral flow is only the first and simplest example of an infinite set of universal flows onthe space of Poisson bivectors provided by elements in the Grothendieck–Teichmüller algebra grt . We refer inparticular to [22] and [20] for results regarding the flows associated with the pentagon γ and heptagon graphs γ , respectively. Relation to quantization
Before concluding with the n = 1 case, we recall known results regarding the deformation quantization problemfor Poisson manifolds. Our emphasis will be on the classification problem for formality morphisms and how theabove results regarding universal deformations of Poisson structures can be used to shed light on the matter.Such considerations will hopefully provide guiding lines in order to address cases for which the deformationquantization problem is less well understood ( cf. Section 6.2 for a related discussion on Courant algebroids).First, recall from Section 1 that Kontsevich’s solution to the deformation quantization problem for Poissonmanifolds involves a formality morphism (1.1) i.e. a quasi-isomorphism of
Lie ∞ -algebras between the Schoutenalgebra on T poly and the Hochschild dg Lie algebra of multidifferential operators D poly . As emphasised earlier,Kontsevich’s formality morphism is stable in a precise sense introduced in [32]. The set of (homotopy classes of) stable formality morphisms of the form (1.1) will be denoted SQI . A first incarnation of the Grothendieck–Teichmüller group as playing a classification rôle for
SQI stems from a construction due to D. Tamarkin in hisformulation of an alternative proof to Kontsevich’s formality theorem [109, 57]. The latter provides a bijectivemap U : DAss ∼ −→ SQI where
DAss stands for the set of Drinfel’d associators. As mentioned earlier, the set
DAss is a
GRT -torsor thus providing an (implicit) action of the Grothendieck–Teichmüller group on SQI . However,Tamarkin’s map is far from being explicit making it difficult to precisely characterise the corresponding
GRT -action on quantization procedures. The situation has been clarified by V. A. Dolgushev who showed in [32] thatthe set SQI is naturally endowed with a regular action of the pro-unipotent group exp (cid:0) H ( GC ) (cid:1) . This result,combined with T. Willwacher’s isomorphism H ( GC ) (cid:39) grt [113] defines a regular GRT -action on SQI , so thatboth sides of Tamarkin’s map U : DAss ∼ −→ SQI are
GRT -torsors. It has furthermore been shown in [34] thatTamarkin’s map is equivariant with respect to the action of GRT i.e. U is a bijection of GRT -torsors. Underthis bijection, the (homotopy class of) Kontsevich’s morphism with standard (or harmonic) propagator [79]is mapped to the Alekseev–Torossian associator [5, 106] (see [34]) while the (homotopy class of) Kontsevich’smorphism with logarithmic propagator [80, 3] is mapped [97] to the Knizhnik–Zamolodchikov associator [40].In this picture, one can argue that the map assigning to each group element in exp (cid:0) H ( GC ) (cid:1) (cid:39) GRT a Lie ∞ -automorphism of T poly ( cf. [113, 66] and Proposition 5.6 for its generalisation to all d ) constitutes a useful Although the two terms of eq.(6.2) already appeared in [78], the relative factor was only recently obtained in [16, 15]where it was also shown to constitute the unique choice allowing for the cocycle property to hold. We refer to [8] for results regarding the Lichnerowicz cohomology associated with universal deformations π ∗ of Poisson manifolds. We refer to [31] for a definition of the notion of homotopy equivalence between
Lie ∞ -morphisms. Φ ∈ DAss be a Drinfel’d associator and denote U Φ : T poly ∼ −→ D poly the formalitymorphism associated with Φ through Tamarkin’s procedure. Let furthermore Γ ∈ exp (cid:0) H ( GC ) (cid:1) (cid:39) GRT .The following diagram commutes: T poly U Γ (cid:15) (cid:15) U Φ · Γ (cid:35) (cid:35) U Φ (cid:47) (cid:47) D poly U D (Φ , Γ) (cid:15) (cid:15) T poly U Φ (cid:47) (cid:47) D poly (6.3)where: • U Φ · Γ : T poly ∼ −→ D poly denotes the formality morphism associated with the Drinfel’d associator Φ · Γ . • U Γ : T poly ∼ −→ T poly denotes the Lie ∞ -automorphism of T poly associated with the element Γ ∈ GRT through Proposition 5.6 (for d = 2 ). • U D (Φ , Γ) : D poly ∼ −→ D poly denotes the Lie ∞ -automorphism of D poly associated with the pair (Φ , Γ) anddefined through U D (Φ , Γ) = U Φ ◦ U Γ ◦ U − .The regularity of the action of GRT on SQI can be restated as follows: for any pair of inequivalent stableformality morphisms U Φ and U Φ (cid:48) , there exists a unique element Γ ∈ GRT such that U Φ (cid:48) = U Φ ◦ U Γ . The space SQI of stable formality morphisms can then be fully explored by composition with
Lie ∞ -automorphisms of T poly induced from GRT . Such a reasoning can also be shown to hold at the level of quantization maps. Indeed, eacharrow appearing in Diagram 6.3 is a Lie ∞ quasi-isomorphism and thus induces a bijection between (equivalenceclasses of) Maurer–Cartan sets ( cf. footnote 2) as: FPoiss ˆ U Γ (cid:15) (cid:15) ˆ U Φ (cid:47) (cid:47) ˆ U Φ · Γ (cid:39) (cid:39) Star ˆ U D (Φ , Γ) (cid:15) (cid:15) FPoiss ˆ U Φ (cid:47) (cid:47) Star [ π ] (cid:95) ˆ U Γ (cid:15) (cid:15) (cid:31) ˆ U Φ (cid:47) (cid:47) (cid:11) ˆ U Φ · Γ (cid:37) (cid:37) [ ∗ ] (cid:95) ˆ U D (Φ , Γ) (cid:15) (cid:15) [ π (cid:48) ] (cid:31) ˆ U Φ (cid:47) (cid:47) [ ∗ (cid:48) ] (6.4)Mimicking the above reasoning allows to span the whole space of universal quantization maps ˆ U Φ by compositionwith universal deformation maps ˆ U Γ induced from GRT , cf. Proposition 5.11. The resulting characterisationof the action of
GRT on universal quantization maps in terms of universal deformations has the merit to makecertain features relatively explicit. As an example, it follows from the previous reasoning that formal Poissonstructures [ π ] which are insensitive to universal deformations admit canonical quantizations i.e. their quantumclass is unique . Straightforward reasoning on the number of derivatives involved in universal deformations (cid:0) see e.g. eq.(6.2) (cid:1) entails that Poisson bivectors whose local description is at most quadratic in coordinates admita unique local quantization. This is in particular the case for constant Poisson bivectors (and in particular forsymplectic manifolds in Darboux coordinates) which are uniquely (locally) quantized by the Groenewold–Moyalstar product [52, 94], cf. eq.(5.14). Slightly less trivial is the Kostant–Souriau–Kirillov Poisson bracket – definedon the dual of any Lie algebra – which is linear in coordinates. The latter admits two known quantizations,namely the Gutt [56, 41] and Kontsevich [79] star products. According to the previous reasoning, these twostar products must belong to the same equivalence class. However, they do not coincide, as shown for examplein [79, 68, 107, 30, 9]. Rather, they are related via an isomorphism given by the Duflo map ( cf. e.g. Theorem14 in [43]). Contrarily to its counterpart U Γ , the family of Lie ∞ -automorphisms U D does depend on the existence of a Drinfel’d associator(although the explicit choice does not matter due to the equivariance relation U D (Φ · Γ (cid:48) , Γ) = U D (Φ , Ad Γ (cid:48) Γ) with Ad Γ (cid:48) Γ =Γ (cid:48) · Γ · Γ (cid:48)− ). That is, such that ˆ U Γ ([ π ]) = [ π ] for all Γ ∈ GRT . In other words, the associated (class of) star products [ ∗ ] = ˆ U Φ ([ π ]) does not depend on the choice of Drinfel’d associator Φ . .2 Courant algebroids ( n = 2 ) The present section applies the results of Section 5 to symplectic Lie -algebroids. The latter notion identifieswith the one of Courant algebroids that we now review, following the presentation à la Dorfman ( cf. e.g. [67]for details and [81] for a historical account).
Definition 6.1 (Courant algebroid) . A Courant algebroid is a quadruplet ( E, (cid:104)· , ·(cid:105) E , D , [ · , · ] E ) where: • The pair ( E, (cid:104)· , ·(cid:105) E ) is a pseudo-Euclidean vector bundle i.e. – E → M is a vector bundle over the smooth manifold M . We will denote ( C ∞ ( M ) , · ) the commuta-tive associative algebra of functions on M and ∗ : C ∞ ( M ) ⊗ Γ ( E ) → Γ ( E ) the module structure onfibers of E . The latter satisfies the associativity relation f ∗ ( g ∗ X ) = ( f · g ) ∗ X for all f, g ∈ C ∞ ( M ) and X ∈ Γ ( E ) . – The map (cid:104)· , ·(cid:105) E : Γ ( E ) ∨ Γ ( E ) → C ∞ ( M ) satisfies the following conditions:1. C ∞ ( M ) -bilinear i.e. (cid:104) f ∗ X, Y (cid:105) E = (cid:104) X, f ∗ Y (cid:105) E = f · (cid:104) X, Y (cid:105) E
2. symmetric i.e. (cid:104)
X, Y (cid:105) E = (cid:104) Y, X (cid:105) E
3. non-degenerate i.e. (cid:104)
X, Y (cid:105) E = 0 for all Y ∈ Γ ( E ) ⇔ X = 0 .A bilinear form satisfying these conditions will be referred to as a fiber-wise metric . • The pair ( D , [ · , · ] E ) is a Courant–Dorfman structure on ( E, (cid:104)· , ·(cid:105) E ) i.e. – [ · , · ] E : Γ ( E ) ⊗ Γ ( E ) → Γ ( E ) is a K -bilinear form on the fibers of E called the Dorfman bracket . – D : C ∞ ( M ) → Γ ( E ) is a K -linear derivation i.e. D ( f · g ) = f ∗ D f + g ∗ D f for all f, g ∈ C ∞ ( M ) .The derivation D defines a C ∞ ( M ) -linear map ρ : Γ ( E ) → Γ ( T M ) called the anchor as ρ X [ f ] = (cid:104) X, D f (cid:105) E for all f ∈ C ∞ ( M ) , X ∈ Γ ( E ) .such that the following conditions are satisfied:1. The Dorfman bracket satisfies the Jacobi identity in its Leibniz form: [ X, [ Y, Z ] E ] E = [[ X, Y ] E , Z ] E + [ Y, [ X, Z ]] E for all X, Y, Z ∈ Γ ( E ) (6.5)so that the pair (Γ ( E ) , [ · , · ] E ) is a K -Leibniz algebra.2. The symmetric part of the Dorfman bracket is controlled by the derivation D as: [ X, Y ] E + [ Y, X ] E = D (cid:104)
X, Y (cid:105) E for all X, Y ∈ Γ ( E ) . (6.6)3. The fiber-wise metric (cid:104)· , ·(cid:105) E is compatible with the Courant–Dorfman structure ( D , [ · , · ] E ) , i.e. (cid:104) X, D (cid:104)
Y, Z (cid:105) E (cid:105) E = (cid:104) [ X, Y ] E , Z (cid:105) E + (cid:104) Y, [ X, Z ] E (cid:105) E = 0 for all X, Y, Z ∈ Γ ( E ) . (6.7)Introducing a basis { e a } a =1 ,..., dim E of the space of sections Γ ( E ) allows to provide a component expression ofthe Courant algebroid maps as follow: • In components, the fiber wise metric reads (cid:104)
X, Y (cid:105) E = κ ab X a Y b where the constant matrix κ satisfies:1. κ is symmetric i.e. κ ab = κ ba .2. κ admits an inverse κ − such that κ ac κ cb = δ ab with δ the Kronecker delta.32 The component expression for the Courant–Dorfman structure ( D , [ · , · ] E ) is captured by a pair ( ρ aµ , T abc ) ,where T abc is totally skewsymmetric. Explicitly, we have: – D -map: D f = κ ab ρ bµ ∂ µ f e a – Anchor: ρ X [ f ] = X a ρ aµ ∂ µ f – Dorfman bracket: [ X, Y ] E = (cid:0) ρ X [ Y a ] − ρ Y [ X a ] − T bca X b Y c + κ ab ρ bµ ∂ µ X c κ cd Y d (cid:1) e a where indices are raised and lowered with κ .It can be checked that the defining conditions of a Courant algebroid are satisfied if and only if the pair ( ρ aµ , T abc ) satisfies the set of conditions:1. C µν := ρ aµ κ ab ρ bν = 0 C µab := ρ cµ κ cd T dab + 2 ρ [ aλ ∂ λ ρ b ] µ = 0 C abcd := T e [ ab κ ef T cd ] f + ρ [ aµ ∂ µ T bcd ] = 0 .Comparing this set of constraints with (3.10)-(3.12) allows to relate Courant algebroids with symplectic Lie -algebroids (or NPQ -manifolds of degree 2). The precise nature of this relation is articulated in the followingtheorem:
Theorem 6.2 (D. Roytenberg [99]) . NP -manifolds of degree 2 are in bijective correspondence with pseudo-Euclidean vector bundles.2. NPQ -manifolds of degree 2 are in bijective correspondence with Courant algebroids.
The Poisson algebra of functions associated with a given NP -manifold V of degree 2 (cid:0) or equivalently the -Schouten algebra T (2) poly = C ∞ ( V ) [2] (cid:1) was referred to as the Rothstein algebra in [69]. The latter can beinterpreted as the deformation complex of Hamiltonian functions on V . Via the second point of Theorem 6.2,this can be rephrased as saying that the Rothstein algebra controls the deformation theory of Courant–Dorfmanstructures ( D , [ · , · ] E ) on the pseudo-Euclidean vector bundle ( E, (cid:104)· , ·(cid:105) E ) – where E is defined by the fibration(3.2) as M ← E [1] ← V – according to the following sequence of bijective correspondences: ( D , [ · , · ] E ) ⇔ ( ρ aµ , T abc ) ⇔ H = ρ aµ ξ a p µ + 16 T abc ξ a ξ b ξ c (6.8)where the right-hand side makes use of the local set of coordinates (cid:110) x µ , ξ a , p µ (cid:111) ( cf. Example 3.1).The supergeometric interpretation of Courant algebroids provided by Theorem 6.2 will allow us to apply theresults of Section 5 in order to generate new universal deformation formulas for Courant–Dorfman structures ( D , [ · , · ] E ) on a given pseudo-Euclidean vector bundle ( E, (cid:104)· , ·(cid:105) E ) . As noted in Lemma 5.7, the zeroth coho-mology of the connected part of the full Kontsevich graph complex in d = 3 is one dimensional and spannedby the triangle class i.e. H ( fGC con ) = K (cid:104) L (cid:105) , cf. Figure 4. This result ensures (cid:0) cf.
Proposition 5.8 (cid:1) thatthere exists a unique universal deformation of Courant algebroids in the stable setting that we now explicitlycharacterise.Letting ( E, (cid:104)· , ·(cid:105) E ) be a pseudo-Euclidean vector bundle, we use the bijective correspondence (6.8) in order toassociate to each Courant–Dorfman structure ( D , [ · , · ] E ) on ( E, (cid:104)· , ·(cid:105) E ) the corresponding Hamiltonian function H = ρ aµ ξ a p µ + T abc ξ a ξ b ξ c with associated homological vector field Q := (cid:8) H , · (cid:9) ω . Via Proposition 5.12, theuniversal Hamiltonian flow associated with L is defined as: ˙ H = Rep (3)3 (cid:0) L (cid:1) ( H ⊗ ) = µ (cid:0) ∆ ∆ ∆ ( H ⊗ ) (cid:1) (6.9) In this sense, the procedure does not deform the full
Courant algebroid structure since the bilinear form (cid:104)· , ·(cid:105) E remainsundeformed. Rep (3)3 follows (5.9) with ∆ given by (cid:0) cf. eq.(5.11) (cid:1) : ∆ ij = ∂∂x µ ( i ) ∂∂p ( j ) µ − ∂∂p ( i ) µ ∂∂x µ ( j ) + ∂∂ξ a ( i ) κ ab ∂∂ξ b ( j ) . (6.10)Explicitly, the triangle Hamiltonian flow maps any Hamiltonian function H towards the associated Rothsteincocycle ˙ H ∈ H ( C ∞ ( V ) | Q ) defined as ˙ H = ˙ ρ aµ ξ a p µ + ˙ T abc ξ a ξ b ξ c where: • ˙ ρ aµ = Rep (3)3 (cid:0) L (cid:1) ( H ⊗ ) aµ = ρ • ρ • µ ρ a + ρ • ρ • µ T a •• • ˙ T abc = 6 Rep (3)3 (cid:0) L (cid:1) ( H ⊗ ) abc = ρ a ρ b ρ c − ρ a ρ b ρ c − T a •• T b •• T c •• + 3 ρ • ρ a T bc • + 3 ρ • T a •• T bc • + skewsym. ( a − b − c ) Here, the directed arrows stand for space-time derivatives while undirected arrows represent contractions offiber indices with the non-degenerate symmetric bilinear form κ . The local expression of the Hamiltonian flowinduced by the triangle cocycle can be equivalently expressed in components as: • ˙ ρ aµ = ρ bλ ∂ λ ρ aν ∂ ν ρ b | µ + ρ bλ ∂ λ ρ cµ T abc (6.11) • ˙ T abc = ∂ µ ρ aν ∂ ν ρ bλ ∂ λ ρ cµ − ∂ µ ρ aλ ∂ ν ρ bµ ∂ λ ρ cν − T ade T bdf T cef +3 ρ dµ ∂ µ ρ [ aν ∂ ν T bc ] d +3 ρ dµ T [ ade ∂ µ T bc ] e (6.12)where indices are raised and lowered with κ . Consistently with Proposition 5.12, it can be checked that (cid:8) H , ˙ H (cid:9) ω = 0 modulo the relations (3.10)-(3.12) coming from (cid:8) H , H (cid:9) ω = 0 .It should be emphasised that the situations corresponding to d = 2 and d = 3 are drastically different. Inthe case d = 2 , the zeroth cohomology is the “dominant” degree i.e. contains an infinite number of non-trivialclasses leading to infinitely many universal deformations of Poisson manifolds, cf. Section 6.1.On the contrary, for d = 3 , the zeroth cohomology is one-dimensional and thus yields a unique universaldeformation of Courant–Dorfman structures given by (6.11)-(6.12). The “dominant” degree of fGC being − (cid:0) cf. Section 4.3 (cid:1) , it would be desirable to find a construction mapping elements of H − ( fGC ) to universalCourant–Dorfman deformations. Although it remains unclear how to perform such a construction in the fullcase, it can be realised at the infinitesimal level using conformal Hamiltonian flows. Universal conformal Hamiltonian flows on the space of Courant algebroids
Recall from Section 4.3 that H − ( fGC ) is spanned by trivalent graphs modulo IHX relations (cid:0) see Figure 6and eq.(4.15) (cid:1) . Proposition 5.13 ensures that each element γ ∈ H − ( fGC ) is mapped to a universal conformalHamiltonian flow on the space of Courant algebroids. Such a conformal flow maps Hamiltonian functions H to conformal factors Ω( H ) ∈ H ( C ∞ ( V ) | Q ) (cid:39) Ker D so that ˙ H := Ω( H ) · H is a Rothstein cocycle.The explicit local expression of the conformal factor Ω( H ) associated to a given graph γ ∈ H − ( fGC ) with N vertices is given by Ω( H ) := Rep (3) N ( γ )( H ⊗ N ) . We now exemplify this construction by displaying the conformalfactors associated to the simplest trivalent graphs.The simplest example of trivalent graph is given by the “ Θ ” graph being the only connected trivalent graphwith N = 2 vertices. The latter yields the following conformal factor Ω Θ ( H ) := Rep (3)2 (Θ)( H ⊗ ) = T ••• T ••• + 6 ρ • ρ • = T abc T abc + 6 ∂ ν ρ aλ ∂ λ ρ a | ν (6.13)and the equality D Ω Θ = 0 follows from (cid:8) H , H (cid:9) ω = 0 thus ensuring that ˙ H Θ := Ω Θ ( H ) · H is indeed aRothstein cocycle. It can be checked by brute-force computation that the vector space of universal Hamiltonian34ows for N = 3 is of dimension 2 and spanned by the triangle flow (6.11)-(6.12) and the conformal Θ -flowdefined from (6.13).The next to simplest case is given by the graphs A and B from Figure 6 for N = 4 yielding: Ω A ( H ) := Rep (3)4 (cid:0) A (cid:1) ( H ⊗ ) = T abc T abd T cef T def + 4 ∂ µ ρ aν ∂ ν ρ bµ ∂ λ ρ a | ρ ∂ ρ ρ b | λ − ∂ µ ρ aν ∂ ν ρ a | λ ∂ λ ρ bρ ∂ ρ ρ b | µ + 4 ∂ µ ρ a | ν ∂ ν ρ dµ T abc T dbc (6.14) Ω B ( H ) := Rep (3)4 (cid:0) B (cid:1) ( H ⊗ ) = T abc T ade T bdf T cef − ∂ µ ρ aν ∂ ν ρ bλ ∂ λ ρ cµ T abc − ∂ µ ρ aν ∂ ν ρ bλ ∂ λ ρ a | ρ ∂ ρ ρ b | µ . Together with Ω Θ ( H ) (corresponding to the disconnected graph γ = Θ ∪ Θ ), these are the only conformalfactors available for N = 4 . Note however that the trivalent graphs A and B can be related through the IHXrelation (4.15) as A ∼ B , cf. footnote 44. This ensures that their respective conformal factors are relatedvia Ω A ( H ) = 2 Ω B ( H ) where the correspondence can be shown by making use of the constraints (3.10)-(3.12)coming from (cid:8) H , H (cid:9) ω = 0 . Relation to quantization
Remarkably, Kontsevich’s original quantization formula ( i.e. with standard propagator) can be interpreted[79, 24, 25] as the Feynman diagram expansion of a 2-dimensional topological field theory – the
Poisson σ -model – introduced in [61, 64, 102]. As mentioned previously, the Poisson σ -model constitutes the first rung ofan infinite ladder of AKSZ σ -models [6] associating to any symplectic Lie n -algebroid a topological field theoryof dimension d = n + 1 . An interesting open problem concerns the possibility of generalising such interplaybetween deformation quantization results (on the algebraic side) and quantization of AKSZ-type of models (onthe field theoretic side) to higher values of n .For n = 2 , the relevant AKSZ σ -model was constructed by D. Roytenberg in [100] ( cf. [63] for an earlierderivation from consistent deformations of a Chern–Simons gauge theory coupled with a -dimensional BF theory). Such model associates to any Courant algebroid a canonical 3-dimensional topological field theory –the Courant σ -model . From the field theory side, quantization of the Courant σ -model within the Batalin–Vilkovisky formalism [12] has been considered in [60, 59] ( cf. also [62] for a discussion of observables in generalAKSZ σ -models).On the algebraic side, a possible candidate for the quantum notion associated with Courant algebroids is givenby vertex algebroids , as introduced in [50] from truncation of vertex algebras [51]. Indeed, it was shown in[17] that the semi-classicalisation of (commutative) vertex algebroids yields a Courant algebroid. This suggestsa formulation of a deformation quantization problem for Courant algebroids, similar to the one formulated in[14, 13] for Poisson manifolds.Although it is outside of the scope of the present paper to address the quantization problem for Courantalgebroids, we note that some insights can be gained from the classification of graph cocycles in H • ( fGC ) :1. H ( fGC ) = : The existence of universal formality morphisms for Courant algebroids is unobstructed. H ( fGC ) = K : The space of universal formality morphisms for Courant algebroids is of dimension . In other words, the first statement asserts that, given a dg Lie algebra D (2) poly such that H • ( D (2) poly ) is isomorphicto T (2) poly as a graded Lie algebra and a quasi-isomorphism of complexes U : T (2) poly ∼ −→ D (2) poly given by universalformula, then the “HKR-type” map U can always be prolongated to a full universal Lie ∞ quasi-isomorphism U : T (2) poly ∼ −→ D (2) poly . Recall that such reasoning constituted the initial rationale behind the introduction ofthe graph complex fGC in [78]. However, in this case, it is a hard open conjecture that H ( fGC ) = 0 so that M. Kontsevich had to rely on different methods in order to prove his formality theorem for Poissonmanifolds. On the contrary, for Courant algebroids, it is straightforward to show the rigidity of the -Schoutenalgebra ( T (2) poly , [ · , · ] S ) – at least in the stable setting – so that one can use the original Kontsevich approach toprove a formality theorem for Courant algebroids. From the second statement, we learn that such morphismis not unique, but rather that formality morphisms form a -dimensional space. Consequently, there should35xist a one-parameter family of stable universal quantization maps for Courant algebroids . This is againin sharp contrast with the Poisson case for which H ( fGC ) is infinite-dimensional (being isomorphic to theGrothendieck–Teichmüller Lie algebra grt ) and thus the space of formality morphisms (and consequently alsothe one of universal quantization maps for Poisson manifolds) forms an infinite-dimensional space (in bijectivecorrespondence with the space of Drinfel’d associators). It would nevertheless be desirable to witness someincarnation of the Grothendieck–Teichmüller Lie algebra within the “Courant world”. In the companion paper[93], it will be argued that such incarnation can be made possible by replacing the full graph complex fGC withthe subgraph complex fGC or ⊂ dfGC spanned by acyclic or oriented graphs. Moving from the stable to theoriented setting will yield a deformation theory for Courant algebroids closer to the Poisson case in which theGrothendieck–Teichmüller group plays a non-trivial classifying rôle. As in the Poisson case, some Courant algebroids are insensitive to deformations so that their associated quantum class is unique i.e. independent of the parameter. This can be shown to be the case of exact
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