EELLIPSITOMIC ASSOCIATORS
DAMIEN CALAQUE AND MARTIN GONZALEZ
Abstract.
We develop a notion of ellipsitomic associators by means of operad theory.We take this opportunity to review the operadic point-of-view on Drinfeld associatorsand to provide such an operadic approach for elliptic associators too. We then show thatellipsitomic associators do exist, using the monodromy of the universal ellipsitomic KZBconnection, that we introduced in a previous work. We finally relate the KZB ellipsitomicassociator to certain Eisenstein series associated with congruence subgroups of SL ( Z ) , andto twisted elliptic multiple zeta values. Contents
Introduction 11. Background material on operads and groupoids 42. Operads associated with configuration spaces (associators) 103. Modules associated with configuration spaces (elliptic associators) 214. The module of parenthesized ellipsitomic braids 335. Ellipsitomic chord diagrams and ellipsitomic associators 386. The KZB ellipsitomic associator 457. Number theoretic aspects: Eisenstein series 52Appendix A. An alternative presentation for
PaB Γ e(cid:96)(cid:96) Introduction
The torsor of associators was introduced by Drinfeld [16] in the early nineties, in the contextof quantum groups and prounipotent Grothendieck–Teichmuller theory. Since then, it hasproven to have deep connections with several areas of mathematics (and physics): numbertheory [32], deformation quantization [21, 31, 36], Chern–Simons theory and low-dimensionaltopology [30], algebraic topology and the little disks operad [37], Lie theory and the Kashiwara–Vergne conjecture [1, 2] etc... In this paper we are mostly interested in the operadic and alsonumber theoretic aspects. For instance,(a) The torsor of associators can be seen as the torsor of isomorphisms between twooperads in (prounipotent) groupoids related to the little disks operad, denoted
PaB and
PaCD (for parenthesized braids and parenthesized chord diagrams ). These can be a r X i v : . [ m a t h . QA ] A p r DAMIEN CALAQUE AND MARTIN GONZALEZ understood as the Betti and de Rham fundamental groupoids of an operad of suitablycompactified configuration spaces of points in the plane. See Section 2 for more details,and accurate references.(b) It is expected that associators can be seen as generating series for (variations onmotivic) multiple zeta values (MZVs), as was observed for the KZ associator [32] andthe Deligne associator [9].The first example of an associator was produced by Drinfeld as a renormalized holonomyof a universal version of the so-called Knizhnik–Zamolodchikov (KZ) connection [16], whichis defined on a trivial principal bundle over the configuration space of points in the plane.The defining equations of an associator, that are reminiscent of the braid group relations,for that braid groups appear as fundamental groups of configuration spaces, can be deducedfrom intuitive geometric reasonings about paths on configuration spaces, and they lead torepresentations of braid groups.Almost twenty years after Drinfeld’s original work, Enriquez, Etingof and the first author[12] introduced a universal version of an elliptic variation on the KZ connection (known asKnizhnik–Zamolodchikov–Bernard, or KZB, connection, as the extension to higher genus isdue to Denis Bernard [5, 6]). It is a holomorphic connection defined on a non trivial principalbundle over configuration spaces of points on an elliptic curve. They showed that ● The holonomy of the universal KZB connection along fundamental cycles of an ellitpiccurve satisfy relations which lead to representations of braid groups on the (2-)torus. ● They also satisfy a modularity property, that is a consequence of the fact that the(universal) KZB connection extends from configuration spaces of points on an ellipticcurve to moduli spaces of marked elliptic curves (see also [33] for when there are atmost 2 marked points).Enriquez later introduced the notion of an elliptic associator [18], and proved that the holonomyof the universal elliptic KZB connection does produce, for every elliptic curve, an example ofelliptic associator. The class of elliptic associators that are obtained via this procedure arecalled
KZB associators . In another work [19], Enriquez defined and studied an elliptic versionof MZVs; he showed that KZB associators are generating series for elliptic MZVs (eMZVs).In a recent paper [13] we introduced a generalization of the universal elliptic KZB connection:the universal ellipsitomic
KZB connection. It is defined over twisted configuration spaces,where the twisting is by a finite quotient Γ of the fundamental group of the elliptic curve.When Γ = define ellipsitomic associators in an operadic way, and to sketch the rudiments ofan ellipsitomic Grothendieck–Teichm¨uller theory.(b) Then we show that holonomies of the universal ellipsitomic KZB connection alongsuitable paths produce examples of ellipsitomic associators, and are generating series LLIPSITOMIC ASSOCIATORS 3 for elliptic multiple polylogarithms at Γ-torsion points, that are similar to the twistedelliptic MZVs (teMZVs) studied in [8] by Broedel–Matthes–Richter–Schlotterer.Our work fits in a more general program that aims at studying associators for an orientedsurface together with a finite group acting on it. We summarize in the following table thecontributions to this program that we are aware of:gen. group associators operadicapproach Universal connection /existence proof coefficients0 trivial [16] [4, 23] rational KZ [16] / ibid.
MZVs [32]0 Z / N Z cyclotomic as-sociators [17] [14] trigonometric KZ [17] / ibid. coloredMZVs [17]0 fin. ⊂ P SU ( C ) unknown unknown [34] / unknown unknown1 trivial elliptic associ-ators [18] this paper(Sec. 3) elliptic KZB [12] / [18] eMZVs [19]1 Z / M Z × Z / N Z ellipsitomicassociators(this paper) this paper(Sec. 4 &5) ellipsitomic KZB [13] /this paper (Sec. 6) this paper(Sec. 7) > Description of the paper.
The first section is devoted to some recollection on operads andoperadic modules, with some emphasis on specific features when the underlying category isthe one of groupoids. Section 2 also recollects known results, about the operadic approach to(genuine) associators and to various Grothendieck–Teichm¨uller groups. The main results westate are taken from the recent book [23].The main goal of section 3 is to provide a similar treatment of elliptic associators, usingoperadic modules in place of sole operads. We show in particular that (a variant of) theuniversal elliptic structure
PaB e(cid:96)(cid:96) (resp. its graded/de Rham counterpart G PaCD e(cid:96)(cid:96) ) from[18] carries the structure of an operadic module in groupoids over the operad in groupoid
PaB (resp. G PaCD ). We provide a generators and relations presentation for
PaB e(cid:96)(cid:96) (Theorem3.3), and deduce from it the following
Theorem (Theorem 3.15) . The torsor of elliptic associators from [18] coincides with the torsorof isomorphisms from (a variant of )
PaB e(cid:96)(cid:96) to G PaCD e(cid:96)(cid:96) that are the identity on objects.Similarly, the elliptic Grothendieck–Teichm¨uller group (resp. its graded version) is isomorphicto the group of automorphisms of
PaB e(cid:96)(cid:96) (resp. of G PaCD e(cid:96)(cid:96) ) that are the identity on objects.
The fourth section introduces a version of
PaB e(cid:96)(cid:96) , with an additional labelling/twistingby elements of Γ (recall that Γ is the group of deck transformations of a finite cover of thetorus by another torus). We give a geometric definition of the operadic module
PaB Γ e(cid:96)(cid:96) ofparenthesized ellipsitomic braids, and then provide a presentation by generators and relationsfor it (Theorem 4.5). In the fifth section we define an operadic module of ellipsitomic chorddiagrams, that mixes features of PaCD e(cid:96)(cid:96) from section 3, and of the moperad of cyclotomic
DAMIEN CALAQUE AND MARTIN GONZALEZ chord diagrams from [14]. This allows us to identify ellipsitomic associators, which we definein purely operadic terms, with series satisfying certain algebraic equations (Theorem 5.9).Section 6 is devoted to the proof of the following
Theorem (Theorem 5.11) . The set of ellipsitomic associators over C is non-empty. The proof makes crucial use of the ellipsitomic KZB connection, introduced in our previouswork [13], and relies on a careful analysis of its monodromy. We actually prove that one canassociate an ellisitomic associator with every element of the upper half-plane (Theorem 6.1).In the last section we quickly explore some number theoretic and modular aspects of thecoefficients of the “KZB produced” ellipsitomic associators from the previous section.Finally, in an appendix we provide an alternative presentation for
PaB Γ e(cid:96)(cid:96) . Acknowledgements.
Both authors are grateful to Adrien Brochier, Benjamin Enriquez,Richard Hain, Nils Matthes, and Pierre Lochak for numerous conversations and suggestions.The first author has received funding from the Institut Universitaire de France, and fromthe European Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (Grant Agreement No. 768679).This paper is extracted from the second author’s PhD thesis [26] at Sorbonne Universit´e, andpart of this work has been done while the second author was visiting the Institut Montpelli´erainAlexander Grothendieck, thanks to the financial support of the Institut Universitaire de France.The second author warmly thanks the Max-Planck Institute for Mathematics in Bonn andUniversit´e d’Aix-Marseille, for their hospitality and excellent working conditions.1.
Background material on operads and groupoids
In this section we fix a symmetric monoidal category (C , ⊗ , ) having small colimits andsuch that ⊗ commutes with these.1.1. S -modules. An S -module (in C ) is a functor S ∶ Bij → C , where
Bij denotes thecategory of finite sets with bijections as morphisms. It can also be defined as a collection ( S ( n )) n ≥ of objects of C such that S ( n ) is endowed with a right action of the symmetricgroup S n for every n ; one has S ( n ) ∶= S ({ , . . . , n }) . A morphism of S -modules ϕ ∶ S → T isa natural transformation. It is determined by the data of a collection ϕ ( n ) ∶ S ( n ) → T ( n ) of S n -equivariant morphisms in C .The category S -mod of S -modules is naturally endowed with a symmetric monoidal product ⊗ defined as follows: ( S ⊗ T )( n ) ∶= ∐ p + q = n ( S ( p ) ⊗ T ( q )) S n S p × S q . Here, if H ⊂ G is a group inclusion, then (−) GH is left adjoint to the restriction functor fromthe category of objects carrying a G -action to the category of objects carrying an H -action.We let the reader check that the symmetric sequence ⊗ defined by ⊗ ( n ) ∶= ⎧⎪⎪⎨⎪⎪⎩ if n = ∅ else LLIPSITOMIC ASSOCIATORS 5 is a monoidal unit.There is another (non-symmetric) monoidal product ○ on S -mod, defined as follows: ( S ○ T )( n ) ∶= ∐ k ≥ T ( k ) ⊗ S k ( S ⊗ k ( n )) . Here, if H is a group and X, Y are objects carrying an H -action, then X ⊗ H Y ∶= coeq ⎛⎜⎝ ∐ h ∈ H X ⊗ Y h ⊗ id —→—→ id ⊗ h X ⊗ Y ⎞⎟⎠ . We let the reader check that the symmetric sequence ○ defined by ○ ( n ) ∶= ⎧⎪⎪⎨⎪⎪⎩ if n = ∅ elseis a monoidal unit for ○ .1.2. Operads. An operad (in C ) is a unital monoid in ( S -mod , ○ , ○ ) . The category of operadsin C will be denoted Op C .More explicitly, an operad structure on a S -module O is the data: ● of a unit map e ∶ → O( ) . ● for every sets I, J and any element i ∈ I , of a partial composition ○ i ∶ O( I ) ⊗ O( J ) —→ O ( J ⊔ I − { i }) satisfying the following constraints: ● if we have sets I, J, K , and elements i ∈ I , j ∈ J , then the following diagram commutes: O( I ) ⊗ O( J ) ⊗ O( K ) id ⊗○ j (cid:15) (cid:15) ○ i ⊗ id (cid:47) (cid:47) O ( J ⊔ I − { i }) ⊗ O( K ) ○ j (cid:15) (cid:15) O( I ) ⊗ O ( K ⊔ J − { j }) ○ i (cid:47) (cid:47) O ( K ⊔ J ⊔ I − { i, j })● if we have sets I, J , J and elements i , i ∈ I , then the following diagram commutes: O( I ) ⊗ O( J ) ⊗ O( J ) (○ i ⊗ id )( ) (cid:15) (cid:15) ○ i ⊗ id (cid:47) (cid:47) O ( J ⊔ I − { i }) ⊗ O( J ) ○ i (cid:15) (cid:15) O ( J ⊔ I − { i }) ⊗ O( J ) ○ i (cid:47) (cid:47) O ( J ⊔ J ⊔ I − { i , i })● if we have sets I, I ′ , J , i ∈ I and a bijection σ ∶ I → I ′ , then the following diagramcommutes: O( I ) ⊗ O( J ) ○ i (cid:15) (cid:15) O( σ ) (cid:47) (cid:47) O( I ′ ) ⊗ O( J ) ○ σ ( i ) (cid:15) (cid:15) O ( J ⊔ I − { i }) O( id ⊔ σ ∣ I −{ i } ) (cid:47) (cid:47) O ( J ⊔ I ′ − { σ ( i )}) DAMIEN CALAQUE AND MARTIN GONZALEZ ● if we have a set I and i ∈ I , then the following diagrams commute: ⊗ O( I ) ≃ (cid:39) (cid:39) e ⊗ id (cid:47) (cid:47) O({ }) ⊗ O( I ) ○ (cid:15) (cid:15) O( I ) O( I ) ⊗ ≃ (cid:15) (cid:15) id ⊗ e (cid:47) (cid:47) O( I ) ⊗ O({ }) ○ i (cid:15) (cid:15) O( I ) i ↦ ≃ (cid:47) (cid:47) O ( I ⊔ { } − { i }) Example 1.1.
Let X be an object of C . Then we define, for any finite set I , the setEnd ( X )( I ) ∶= Hom C ( X ⊗ I , X ) . Composition of tensor products of maps provide End ( X ) withthe structure of an operad in sets.Given an operad in sets O , an O -algebra in C is an object X of C together with a morphism ofoperads O →
End ( X ) .1.3. Example of an operad: Stasheff polytopes.
To any finite set I we associate theconfiguration space Conf ( R , I ) = { x = ( x i ) i ∈ I ∈ R I ∣ x i ≠ x j if i ≠ j } and its reduced versionC ( R , I ) ∶= Conf ( R , I )/ R ⋊ R > . The Axelrod–Singer–Fulton–MacPherson compactification C ( R , I ) of C ( R , I ) is a disjointunion of ∣ I ∣ -th Stasheff polytopes [35], indexed by S I . The boundary ∂ C ( R , I ) ∶= C ( R , I ) − C ( R , I ) is the union, over all partitions I = J ∐ ⋯ ∐ J k , of ∂ J , ⋯ ,J k C ( R , I ) ∶= k ∏ i = C ( R , J i ) × C ( R , k ) . The inclusion of boundary components provides C ( R , −) with the structure of an operad intopological spaces (where the monoidal structure is given by the cartesian product).One can see that C ( R , I ) is actually a manifold with corners, and that, considering onlyzero-dimensional strata of our configuration spaces, we get a suboperad Pa ⊂ C ( R , −) that canbe shortly described as follows: ● Pa ( I ) is the set of pairs ( σ, p ) with σ is a linear order on I and p a maximal parenthe-sization of ●⋯●– ∣ I ∣ times , ● the operad structure is given by substitution.Notice that Pa is actually an operad in sets, and that Pa -algebras are nothing else than magmas .1.4. Modules over an operad: Bott-Taubes polytopes. A module over an operad O (in C ) is a right O -module in ( S -mod , ○ , ○ ) . Notice that any operad is a module over itself. Welet the reader find the very explicit description of a module in terms of partial compositions,as for operads. We are using the differential geometric compactification from [3], which is an analog of the algebro-geometricone from [24].
LLIPSITOMIC ASSOCIATORS 7
To any finite set I we associate the configuration space Conf ( S , I ) = { x = ( x i ) i ∈ I ∈ ( S ) I ∣ x i ≠ x j if i ≠ j } and its reduced versionC ( S , I ) ∶= Conf ( S , I )/ S . The Axelrod–Singer–Fulton–MacPherson compactification C ( S , I ) of C ( S , I ) is a disjointunion of ∣ I ∣ -th Bott–Taubes polytopes [10], indexed by S I . The boundary ∂ C ( S , I ) ∶= C ( S , I ) − C ( S , I ) is the union, over all partitions I = J ∐ ⋯ ∐ J k , of ∂ J , ⋯ ,J k C ( S , I ) ∶= k ∏ i = C ( R , J i ) × C ( S , k ) . The inclusion of boundary components provides C ( S , −) with the structure of a module overthe operad C ( R , −) in topological spaces.One can see that C ( S , I ) is actually a manifold with corners, and that, considering onlyzero-dimensional strata of our configuration spaces, we get Pa ⊂ C ( S , −) , which is a moduleover Pa ⊂ C ( R , −) , which we already knew (as every operad is a module over itself).1.5. Convention: pointed versions.
Observe that there is an operad
U nit defined by
U nit ( n ) = ⎧⎪⎪⎨⎪⎪⎩ if n = , ∅ elseBy convention, all our operads O will be U nit -pointed and reduced, in the sense that they willcome equipped with a specific operad morphism
U nit → O that is an isomorphism in arity ≤ O( n ) ≃ if n = ,
1. Morphisms of operads are required to be compatible with this pointing.Now, if P is an O -module, then it naturally becomes a U nit -module as well, by restriction.By convention, all our modules will be pointed as well, in the sense that they will comeequipped with a specific
U nit -module morphism
U nit → P that is an isomorphism in arity ≤ ● For operads and modules, we want to have “deleting operations” O( n ) → O( n − ) that decrease arity. ● For modules, we want to be able to see the operad “inside” them, i.e. we want to havedistinguished morphism
O → P of S -modules.1.6. Group actions.
Let G be a ∗ -module in group, where ∗ is the terminal operad: thepartial composition ○ i is a group morphism G ( n ) → G ( n + m − ) . Example 1.2.
Let Γ be a group, we consider the S -module in groups Γ ∶= { Γ n / Γ diag } n ≥ ,where Γ diag denotes the normal closure of the diagonal subgroup in each Γ n . It is equippedwith the following ∗ -module structure: the i -th partial composition is given by the partial DAMIEN CALAQUE AND MARTIN GONZALEZ diagonal morphism Γ n / Γ —→ Γ n + m − / Γ [ γ , . . . , γ n ] z→ [ γ , . . . , γ i − , γ i , . . . , γ i ·„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„¶ m times , γ i + , . . . , γ n ] Given an operad O in C , we say that an O -module P carries a G -action if ● for every n ≥
0, there is an S n -equivariant left action G ( n ) × P( n ) → P( n ) . ● for every m ≥ n ≥
0, and 1 ≤ i ≤ n , the partial composition ○ i ∶ P( n ) ⊗ O( m ) —→ P( n + m − ) is equivariant along the above group morphism G ( n ) → G ( n + m − ) .A morphism P → Q of O -modules with G -action is said G -equivariant if, for every n ≥
0, themap P( n ) → Q( n ) is G ( n ) -equivariant.Given a group Γ, we say that an O -module P carries a diagonally trivial action of Γ if itcarries a Γ-action.The quotient G /P of an O -module P with a G -action is defined as follows: ● For every n ≥ ( G ( n )/P)( n ) ∶= G ( n )/P( n ) ; ● The equivariance of the partial composition ○ i tels us that it descends to the quotient ( G ( n )/P( n )) ⊗ O( m ) —→ G ( n + m − )/P( n + m − ) . Semi-direct products and fake pull-backs.
Let
Grpd be the category of groupoids.For a group G , we denote G − Grpd the category of groupoids equipped with a G -action.There is a semi-direct product functor G − Grpd —→ Grpd / G P z→ P ⋊ G where the group G is viewed as a groupoid with a single object, and where P ⋊ G is defined asfollows: ● Objects of
P ⋊ G are just objects of P ; ● In addition to the arrows of P , for every g ∈ G , and for every object p of P , we havean arrow g ⋅ p g → p ; ● These new arrows multiply together via the group multiplication of G ; ● For every morphism f in P , and every g ∈ G , we have gf g − = g ⋅ f .The semi-direct product functor has a left adjoint Grpd / G —→ G − Grpd (Q ϕ → G ) z→ G( ϕ ) that one can describe as follows: ● The G -set of objects of G( ϕ ) is the free G -set generated Ob (P) ; ● A morphism ( g, x ) → ( h, y ) in G( ϕ ) is a morphism x f → y in Q such that gϕ ( f ) = h . LLIPSITOMIC ASSOCIATORS 9
Example 1.3.
It is easy to check that G( B n → S n ) is the colored braid groupoid CoB ( n ) from [23, § Remark 1.4.
Given an object q of Q , Aut G( ϕ ) ( g, q ) is the kernel of the morphism Aut Q ( q ) → G for every g ∈ G .These constructions still make sense for modules over a given operad O whenever G is anoperadic ∗ -module in groups.Let P , Q be two operads (resp. modules) in groupoids. If we are given a morphism f ∶ Ob (P) → Ob (Q) between the operads (resp. operad modules) of objects of P and Q , then(following [23]) we can define an operad (resp. operad module) f ⋆ Q in the following way: ● Ob ( f ⋆ Q) ∶= Ob (P) , ● Hom ( f ⋆ Q)( n ) ( p, q ) ∶= Hom Q( n ) ( f ( p ) , f ( q )) .In particular, f ⋆ Q , which we call the fake pull-back of Q along f , inherits the operad structureof P for its operad of objects and that of Q for the morphisms. Remark 1.5.
Notice that this is not a pull-back in the category of operads in groupoids.1.8.
Prounipotent completion.
Let k be a Q -ring. We denote by CoAlg k the symmetricmonoidal category of complete filtered topological coassociative cocommutative counital k -coalgebras, where the monoidal product is given by the completed tensor product ˆ ⊗ k over k . Let Cat ( CoAlg k ) be the category of small CoAlg k -enriched categories. It is symmetricmonoidal as well, with monoidal product ⊗ defined as follows: ● Ob ( C ⊗ C ′ ) ∶= Ob ( C ) × Ob ( C ′ ) . ● Hom C ⊗ C ′ (( c, c ′ ) , ( d, d ′ )) ∶= Hom C ( c, d ) ˆ ⊗ k Hom C ′ ( c ′ , d ′ ) .All the constructions of the previous subsection still make sense, at the cost of replacingthe group G with its completed group algebra ̂ k G (which is a Hopf algebra) in the semi-directproduct construction.Considering the cartesian symmetric monoidal structure on Grpd , we have a symmetricmonoidal functor
Grpd —→ Cat ( CoAlg k )G z→ G( k ) defined as follows: ● Objects of P( k ) are objects of P . ● For a, b ∈ Ob (P) , Hom P( k ) ( a, b ) = ̂ k ⋅ Hom P ( a, b ) . Here k ⋅ Hom P ( a, b ) is equipped with the unique coalgebra structure such that theelements of Hom P ( a, b ) are grouplike (meaning that they are diagonal for the coproductand that their counit is 1), and the “ ̂ ” refers to the completion with respect to the topology defined by the sequence ( Hom I k ( a, b )) k ≥ , where I k is the category havingthe same objects as P and morphisms lying in the k -th power (for the composition ofmorphisms) of kernels of the counits of k ⋅ Hom P ( a, b ) ’s. ● For a functor F ∶ P → Q , F ( k ) ∶ P( k ) → Q( k ) is the functor given by F on objectsand by k -linearly extending F on morphisms.Being symmetric monoidal, this functor sends operads in groupoids to operads in Cat ( CoAlg k ) . Example 1.6.
For instance, viewing Pa as an operad in groupoid (with only identities asmorphisms), then Pa ( k ) is the operad in Cat ( CoAlg k ) with same objects as Pa , and whosemorphisms are Hom Pa ( k )( n ) ( a, b ) = ⎧⎪⎪⎨⎪⎪⎩ k if a = b k being equipped with the coproduct ∆ ( ) = ⊗ (cid:15) ( ) = G ∶ Cat ( CoAlg k ) —→ Grpd , that we can describe as follows: ● For C in Cat ( CoAlg k ) , objects of G ( C ) are objects of C . ● For a, b ∈ Ob (G) , Hom G ( C ) ( a, b ) is the subset of grouplike elements in Hom C ( a, b ) .Being right adjoint to a symmetric monoidal functor, it is lax symmetric monoidal, and thus itsends operads (resp. modules) to operads (resp. modules).We thus get a k -prounipotent completion functor G ↦ ˆ G( k ) ∶= G (G( k )) for (operads andmodules in) groupoids. Remark 1.7.
Let ϕ ∶ G → S be a surjective group morphism, and assume that S is finite. Onecan prove that the prounipotent completion ˆ G( ϕ )( k ) of the construction from the previoussubsection is isomorphic to G( ϕ ( k )) , where ϕ ( k ) ∶ G ( ϕ, k ) → S is Hain’s relative completion[28]. This essentially follows from that, when S is finite, the kernel of the relative completionis the completion of the kernel.2. Operads associated with configuration spaces (associators)
Compactified configuration space of the plane.
To any finite set I we associate aconfiguration space Conf ( C , I ) = { z = ( z i ) i ∈ I ∈ C I ∣ z i ≠ z j if i ≠ j } . We also consider its reduced versionC ( C , I ) ∶= Conf ( C , I )/ C ⋊ R > . We then consider the Axelrod–Singer–Fulton–MacPherson compactification C ( C , I ) of C ( C , I ) .The boundary ∂ C ( C , I ) = C ( C , I ) − C ( C , I ) is made of the following irreducible components: LLIPSITOMIC ASSOCIATORS 11 for any partition I = J ∐ ⋯ ∐ J k there is a component ∂ J , ⋯ ,J k C ( C , I ) ≅ C ( C , k ) × k ∏ i = C ( C , J i ) . The inclusion of boundary components provides C ( C , −) with the structure of an operad intopological spaces. One can picture the partial operadic composition morphisms as follows: ◦ A presentation for the pure braid group.
The pure braid group PB n is generatedby elementary pure braids P ij , 1 ≤ i < j ≤ n , which satisfy (at least) the following relations: ( P ij , P kl ) = { i, j } and { k, l } are non crossing , (PB1) ( P kj P ij P − kj , P kl ) = i < k < j < l , (PB2) ( P ij , P ik P jk ) = ( P jk , P ij P ik ) = ( P ik , P jk P ij ) = i < j < k . (PB3)In this article we will the generator P ij in the following two equivalent ways:11 ii ...... jj nn ←→ ∢ ni j i,j conjugated to P i,j . We can represent two incarnationsof the generator P i,j in the following way11 ii ...... jj nn ←→ ∢ i j n Indeed, one can define O ij ∶= P i ( i + ) P i ( i + ) . . . P ij . In other words, P ij = O − i ( j − ) O ij . And wedefine P ij ∶= O ij O − i ( j − ) = O i ( j − ) P ij O − i ( j − ) .2.3. The operad of parenthesized braids.
We have inclusions of topological operads Pa ⊂ C ( R , −) ⊂ C ( C , −) . Then it makes sense to define
PaB ∶= π ( C ( C , −) , Pa ) , which is an operad in groupoids. Example 2.1 (Description of
PaB ( ) ) . Let us first recall that Pa ( ) = S , and that C ( C , ) ≃ S . Besides the identity morphism in PaB ( ) going from ( ) to ( ) , we have an arrow R , in PaB ( ) going from ( ) to ( ) which can be depicted as follows :12 21 21Two incarnations of R , We will denote ˜ R , ∶= ( R , ) − . Example 2.2 (Notable arrows in
PaB ( ) ) . Let us first recall that Pa ( ) = S ×{(●●)● , ●(●●)} and that C ( R , ) ≅ S × [ , ] . Therefore, we have an arrow Φ , , (the identity path in [ , ] )from ( ) ( ) in PaB ( ) . It can be depicted as follows: (
11 2 ) ( ) , , The following result is borrowed from [23, Theorem 6.2.4], even though it perhaps alreadyappeared in [4] in a different form. We actually have another arrow, that can be obtained from the first one as ( R , ) − according to thenotation that is explained after Theorem 2.3, and which can be depicted as an undercrossing braid. LLIPSITOMIC ASSOCIATORS 13
Theorem 2.3.
As an operad in groupoids having Pa as operad of objects, PaB is freelygenerated by R ∶= R , and Φ ∶= Φ , , together with the following relations: Φ ∅ , , = Φ , ∅ , = Φ , , ∅ = Id , ( in Hom PaB ( ) ( , )) , (U1) R , Φ , , R , = Φ , , R , Φ , , ( in Hom PaB ( ) (( ) , ( ))) , (H1) ˜ R , Φ , , ˜ R , = Φ , , ˜ R , Φ , , ( in Hom PaB ( ) (( ) , ( ))) , (H2) Φ , , Φ , , = Φ , , Φ , , Φ , , ( in Hom PaB ( ) ((( ) ) , ( ( )))) . (P)We now briefly explain the notation we have been using in the above statement, which isquite standard. Notation 2.4.
In this article, we write the composition of paths from left to right (and wedraw the braids from top to bottom). If X is an arrow from p to q in PaB ( n ) , then ● for any r ∈ Pa ( k ) , the identity of r in PaB ( k ) is also denoted r , ● for any r ∈ Pa ( k ) , we write X ,...,n for r ○ X ∈ PaB ( n + k − ) , ● we write X ∅ , ,...,n ∈ PaB ( n + k − ) for the image of X ,...,n by the first braid deletingoperation, ● for any σ ∈ S n + k − we define X σ ,...,σ n ∶= ( X ,...,n ) ⋅ σ , ● for any r ∈ Pa ( k ) , X r ,k + ,...,k + n − ∶= X ○ r ∈ PaB ( n + k − ) , ● we allow ourselves to combine these in an obvious way.We let the reader figuring out that this notation is unambiguous as soon as we specify thestarting object of our arrows.For example, the pentagon (P) and the first hexagon (H1) relations can be respectivelydepicted as (( ) ) ( ( )) = (( ) ) ( ( )) (P)and (
12 2 ) ( ) = (
12 2 ) ( ) (H2)or, as commuting diagrams (giving the name of the relations) ( )( ) Φ , , (cid:36) (cid:36) ( ) Φ , , (cid:47) (cid:47) ( R , ) − (cid:122) (cid:122) ( ) ( R , ) − (cid:36) (cid:36) ( ( )) Φ , , (cid:58) (cid:58) Φ , , (cid:15) (cid:15) (( ) ) ( ) Φ , , (cid:36) (cid:36) ( ) Φ , , (cid:122) (cid:122) (( ) ) Φ , , (cid:47) (cid:47) ( ( )) Φ , , (cid:79) (cid:79) ( ) ( R , ) − (cid:47) (cid:47) ( ) The operad of chord diagrams.
The holonomy Lie algebra of the configuration spaceConf ( C , n ) ∶= { z = ( z , . . . , z n ) ∈ C n ∣ z i ≠ z j if i ≠ j } of n points on the complex line is isomorphic to the graded Lie C -algebra t n ( k ) generated by t ij , 1 ≤ i ≠ j ≤ n , with relations t ij = t ji , (S) [ t ij , t kl ] = { i, j, k, l } = , (L) [ t ij , t ik + t jk ] = { i, j, k } = . (4T)In [4, 23] it is shown that the collection of Kohno-Drinfeld Lie k -algebras t n ( k ) definedin the introduction is provided with the structure of an operad in the category grLie k ofpositively graded finite dimensional Lie algebras over k , with symmetric monoidal strucure isgiven by the direct sum ⊕ . Partial compositions are described as follows: ○ k ∶ t I ( k ) ⊕ t J ( k ) —→ t J ⊔ I −{ i } ( k )( , t αβ ) z→ t αβ ( t ij , ) z→ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ t ij if k ∉ { i, j }∑ p ∈ J t pj if k = i ∑ p ∈ J t ip if j = k Even though the author of [4] does not use the concept of an operad.
LLIPSITOMIC ASSOCIATORS 15
Observe that we have a lax symmetric monoidal functorˆ
U ∶ grLie k —→ Cat ( CoAlg k ) sending a positively graded Lie algebra to the degree completion of its universal enveloppingalgebra, which is a complete filtered cocommutative Hopf algebra, viewed as a CoAlg k -enrichedcategory with only one object.We then consider the operad of chord diagrams CD ( k ) ∶= ˆ U( t ( k )) in Cat ( CoAlg k ) . Remark 2.5.
This denomination comes from the fact that morphisms in CD ( k )( n ) can berepresented as linear combinations of diagrams of chords on n vertical strands, where the chorddiagram corresponding to t ij can be represented as i j n ni j and the composition is given by vertical concatenation of diagrams. Partial compositions caneasily be understood as “cabling and removal operations” on strands (see [4, 23]). Relations(L) and (4T) defining each t n ( k ) can be represented as follows: j ki li lj k = j ki li lj k (L) i j ki j k + i j ki j k = i j ki j k + i j ki j k (4T)2.5. The operad of parenthesized chord diagrams.
Recall that the operad CD ( k ) hasonly one object in each arity. Hence we can define the operad PaCD ( k ) ∶= ω ⋆ CD ( k ) of parenthesized chord diagrams , where ω ∶ Pa = Ob ( Pa ( k )) → Ob ( CD ( k )) is the terminalmorphism. Here is a self-explanatory example of how to depict a morphism in PaCD ( k )( ) : f ⋅ ( i j ) ki ( k j ) where f ∈ CD ( k )( ) . Example 2.6 (Notable arrows of
PaCD ( k ) ) . We have the following arrows in
PaCD ( k )( ) : H , ∶= t ⋅ Id , = t ⋅
11 22 =: 11 22 X , = ⋅
12 21We also have the following arrow in
PaCD ( k )( ) : a , , = ⋅ (
11 2 ) ( ) Remark 2.7.
The elements H , , X , and a , , are generators of the operad PaCD ( k ) andsatisfy the following relations: ● X , = ( X , ) − , ● a , , a , , = a , , a , , a , , , ● X , = a , , X , ( a , , ) − X , a , , , ● H , = X , H , ( X , ) − , ● H , = a , , ( H , + X , ( a , , ) − H , a , , X , )( a , , ) − .In particular, even if PaCD ( k ) does not have a presentation in terms of generators andrelations (as is the case for PaB ), one can show that
PaCD ( k ) has a universal propertywith respect to the generators H , , X , and a , , and the above relations (see [23, Theorem10.3.4] for details).2.6. Drinfeld associators.
Let us first introduce some terminology that we use in thisparagraph, as well as later in the paper: ● Let
Grpd k denote the (symmetric monoidal) category of k -prounipotent groupoids(which is the image of the completion functor G ↦ ˆ G( k ) ); ● For C being Grpd , Grpd k , or Cat ( CoAlg k ) , the notationAut + Op C ( resp . Iso + Op C ) LLIPSITOMIC ASSOCIATORS 17 refers to those automorphisms (resp. isomorphisms) which are the identity on objects.In the remainder if this section we recall some well known results on the operadic point ofview on associators and Grothendieck-Teichm¨uller groups, which will be useful later on. Eventhough the statements and proofs of all the results in this subsection can be found in [23], it isworth mentionning that a “pre-operadic” approach was initiated by Bar-Natan in [4].
Definition 2.8. A Drinfeld k -associator is an isomorphism between the operads ̂ PaB ( k ) and G PaCD ( k ) in Grpd k , which is the identity on objects. We denote by Assoc ( k ) ∶= Iso + Op Grpd k (̂ PaB ( k ) , G PaCD ( k )) the set of k -associators. Theorem 2.9.
There is a one-to-one correspondence between the set of Drinfeld k -associatorsand the set Ass ( k ) of couples ( µ, ϕ ) ∈ k × × exp ( ˆ f ( k )) , such that ● ϕ , , = ( ϕ , , ) − in exp ( ˆ t ( k )) , ● ϕ , , e µt / ϕ , , e µt / ϕ , , e µt / = e µ ( t + t + t )/ in exp ( ˆ t ( k )) , ● ϕ , , ϕ , , ϕ , , = ϕ , , ϕ , , in exp ( ˆ t ( k )) ,where ϕ , , = ϕ ( t , t ) is viewed as an element of exp ( ˆ t ( k )) via the inclusion ˆ f ( k ) ⊂ ˆ t ( k ) sending x to t and y to t . Three observations are in order: ● The free Lie k -algebra f ( k ) in two generators x, y is graded, with generators havingdegree 1, and its degree completion is denoted by ˆ f ( k ) ; ● The k -prounipotent group exp ( ˆ f ( k )) is thus isomorphic to the k -prounipotent com-pletion ̂ F ( k ) of the free group F on two generators; ● The quotient ˆ¯ t ( k ) of the Lie algebra ˆ t ( k ) by its center, generated by t + t + t , isisomorphic to ˆ f ( k ) . Thus, the second relation in the above theorem is equivalent to ϕ , , e µy / ϕ , , e µz / ϕ , , e µx / = ( ˆ f ( k )) , where x, y, z are variables subject to relation x + y + z = PaB from Theorem2.3. In particular, a morphism F ∶ ̂ PaB ( k ) —→ G PaCD ( k ) is uniquely determined by ascalar parameter µ ∈ k and ϕ ∈ exp ( ˆ f ( k )) such that we have the following assignment in themorphism sets of the parenthesized chord diagram operad PaCD : ● F ( R , ) = e µt / ⋅ X , , ● F ( Φ , , ) = ϕ ( t , t ) ⋅ a , , ,where R and Φ are the ones from Examples 2.1 and 2.2. Example 2.10 (The KZ Associator) . The first associator was constructed by Drinfeld withthe help of the monodromy of the KZ connection and is known as the KZ associator Φ KZ .It is defined as the the renormalized holonomy from 0 to 1 of G ′ ( z ) = ( t z + t z − ) G ( z ) , i.e., Φ KZ ∶= G + G − − ∈ exp ( ˆ t ( C )) , where G + , G − are the solutions such that G + ( z ) ∼ z t when z → + and G − ( z ) ∼ ( − z ) t when z → − . We haveΦ KZ ( V, U ) = Φ KZ ( U, V ) − , Φ KZ ( U, V ) e π i V Φ KZ ( V, W ) e π i W Φ KZ ( W, U ) e π i U = , where U = t ∈ f ( C ) ≃ ¯ t ( C ) ∶= t ( C )/( t + t + t ) , V = t ∈ ¯ t ( C ) and U + V + W =
0, andΦ , , Φ , , = Φ , , Φ , , Φ , , , hence ( π i , Φ KZ ) is an element of Ass ( C ) .2.7. Grothendieck–Teichmuller group.Definition 2.11.
The
Grothendieck–Teichm¨uller group is defined as the group GT ∶= Aut + Op Grpd ( PaB ) of automorphisms of the operad in groupoids PaB which are the identity of objects. Onedefines similarly its k -pro-unipotent version ̂ GT ( k ) ∶= Aut + Op Grpd k (̂ PaB ( k )) . There are also pro- (cid:96) and profinite versions, denoted GT (cid:96) and ̂ GT , that we do not consider inthis paper.We can also characterize elements of GT and ̂ GT ( k ) as solutions of certain explicit algebraicequations. This characterization proves that the above operadic definition of GT coincideswith the one given by Drinfeld in his original paper [16]. In this article we will focus on the k -pro-unipotent version of this group in genus 0 and 1, and twisted situations. Definition 2.12.
Drinfeld’s Grothendieck–Teichm¨uller group ̂ GT ( k ) consists of pairs ( λ, f ) ∈ k × × ̂ F ( k ) which satisfy the following equations: ● f ( x, y ) = f ( y, x ) − in ̂ F ( k ) , ● x ν f ( x , x ) x ν f ( x , x ) x ν f ( x , x ) = ̂ F ( k ) , ● f ( x x , x ) f ( x , x x ) = f ( x , x ) f ( x x , x x ) f ( x , x ) in ̂ PB ( k ) ,where x , x , x are 3 variables subject only to x x x = ν = λ − , and x ij is the elementarypure braid P ij from the introduction. The multiplication law is given by ( λ , f )( λ , f ) = ( λ λ , f ( x λ , f ( x, y ) y λ f ( x, y ) − ) f ( x, y )) . Theorem 2.13.
There is an isomorphism between the groups ̂ GT ( k ) and ̂ GT ( k ) . This was first implicitely shown by Drinfeld in [16]. An explicit proof of this theorem canbe found for example in [23, Theorem 11.1.7]. In particular, one obtains the couple ( λ, f ) from LLIPSITOMIC ASSOCIATORS 19 an automorphism F ∈ ̂ GT ( k ) as follows. We have F ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
12 21 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
11 22 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ν ⋅
12 21 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
12 21 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ν + (1) F ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ (
11 2 ) ( ) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = f ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ( ( ) ) , ( ( ) ) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⋅ (
11 2 ) ( ) (2)In other words, if we set λ = ν +
1, we get the assignment ● F ( R , ) = ( R , ) λ , ● F ( Φ , , ) = f ( x , x ) ⋅ Φ , , .Next, one obtains the composition law of ̂ GT ( k ) from the composition of automorphisms F and F inAut + Op Grpd k (̂ PaB ( k )) as follows: the associated couples ( λ , f ) and ( λ , f ) in k × × ˆ F ( k ) satisfy ( F F )( R , ) = ( F ○ F )( R , ) = ( R , ) λ λ , and ( F F )( Φ , , ) = ( F ○ F )( Φ , , ) = F ( f ( x , x ) Φ , , )= F ( f ( x , x )) F ( Φ , , )= f ( F ( x ) , F ( x )) f ( x , x ) Φ , , = f ( x λ , f ( x , x ) x λ f ( x , x ) − ) f ( x , x ) Φ , , . Remark 2.14.
It is important to notice that the profinite, pro- (cid:96) , k -pro-unipotent versions ofthe Grothendieck–Teichm¨uller group do not coincide with the profinite, pro- (cid:96) , k -pro-unipotentcompletions of the “thin” Grothendieck–Teichm¨uller group GT which only consists of thepairs ( , ) and (− , ) . There are morphisms GT —→ ̂ GT ↠ GT (cid:96) ↪ ̂ GT ( Q (cid:96) ) and GT —→ ̂ GT ( k ) . Graded Grothendieck–Teichmuller group.Definition 2.15.
The graded Grothendieck–Teichm¨uller group is the group
GRT ( k ) ∶= Aut + Op Grpd k ( G PaCD ( k )) of automorphisms of G PaCD ( k ) that are the identity on objects. Remark 2.16.
When restricted to the full subcategory
Cat ( CoAlg connk ) of CoAlg k -enrichedcategories for which the hom-coalgebras are connected, the functor G leads to an equivalencebetween Cat ( CoAlg connk ) and Grpd k . Hence there is an isomorphism GRT ( k ) ≃ Aut + Op Cat ( CoAlg k ) ( PaCD ( k )) . Again, the operadic definition of
GRT ( k ) happens to coincide with the one originally givenby Drinfeld. Definition 2.17.
Let GRT be the set of elements in g ∈ exp ( ˆ f ( k )) ⊂ exp ( ˆ t ( k )) such that ● g , , = g − and g , , g , , g , , =
1, in exp ( ˆ t ( k )) , ● t + Ad ( g , , )( t ) + Ad ( g , , )( t ) = t + t + t , in ˆ t ( k ) , ● g , , g , , g , , = g , , g , , , in exp ( ˆ t ( k )) ,One has the following multiplication law on GRT : ( g ∗ g )( t , t ) ∶= g ( t , Ad ( g ( t , t ))( t )) g ( t , t ) . Drinfeld showed in [16] that GRT is stable under ∗ , which defines a group structure on it,and that rescaling transformations g ( x, y ) ↦ λ ⋅ g ( x, y ) = g ( λx, λy ) define an action of k × of GRT by automorphisms. Theorem 2.18.
There is a group isomorphism
GRT ( k ) ≅ k × ⋊ GRT =∶ GRT ( k ) . This was first implicitely shown by Drinfeld in [16]. An explicit proof of this theorem canbe found for example in [23, Theorem 10.3.10]. In particular, we obtain the couple ( λ, g ) froman automorphism G ∈ GRT ( k ) by the assignment ● G ( X , ) = X , , ● G ( H , ) = e λt H , , ● G ( a , , ) = g ( t , t ) ⋅ a , , .The composition of automorphisms G and G in Aut + Op ˆ G ( G PaCD ( k )) is given as follows:the associated couples ( λ, g ) and ( µ, g ) in k × × exp ( ˆ¯ t ( k )) satisfy ( G G )( H , ) = ( G ○ G )( H , ) = λµH , , ( G G )( a , , ) = ( G ○ G )( a , , ) = g ( µt , g ( t , t )( µt ) g ( t , t ) − ) g ( t , t ) ⋅ a , , . Bitorsors.
Recall first that there is a free and transitive left action of ̂ GT ( k ) on Ass ( k ) ,defined, for ( λ, f ) ∈ ̂ GT ( k ) and ( µ, ϕ ) ∈ Ass ( k ) , by (( λ, f ) ∗ ( µ, ϕ ))( t , t ) ∶= ( λµ, f ( e µt , Ad ( ϕ ( t , t ))( e µt )) ϕ ( t , t )) , where Ad ( f )( g ) ∶= f gf − , for any symbols f, g .Recall that there is also a free and transitive right action of GRT ( k ) on Ass ( k ) defined asfollows: for ( λ, g ) ∈ GRT ( k ) and ( µ, ϕ ) ∈ Ass ( k ) , given by (( µ, ϕ ) ∗ ( λ, g ))( t , t ) ∶= ( λµ, ϕ ( λt , Ad ( g )( λt )) g ( t , t )) . The two actions commute making (̂ GT ( k ) , Ass ( k ) , GRT ( k )) into a bitorsor. Theorem 2.19.
There is a torsor isomorphism (3) ( ̂ GT ( k ) , Assoc ( k ) , GRT ( k )) —→ (̂ GT ( k ) , Ass ( k ) , GRT ( k )) LLIPSITOMIC ASSOCIATORS 21
Proof.
On the one hand, in [23, Theorem 10.3.13] it is shown that the natural free and transitiveleft action of ̂ GT ( k ) on Assoc ( k ) coincides with the action of GT ( k ) on Ass ( k ) via thecorrespondence of Theorem 2.13. On the other hand, in [23, Theorem 11.2.1], it is shown thatthe natural free and transitive right action of GRT ( k ) on Assoc ( k ) coincides with the actionof GRT ( k ) over Ass ( k ) via the correspondence of Theorem 2.18. (cid:3) Modules associated with configuration spaces (elliptic associators)
Compactified configuration space of the torus.
Let T be the topological (2-)torus.To any finite set I we associate a configuration spaceConf ( T , I ) = { z = ( z i ) i ∈ I ∈ T I ∣ z i ≠ z j if i ≠ j } . We also consider its reduced versionC ( T , I ) ∶= Conf ( T , I )/ T . We then consider the Axelrod–Singer–Fulton–MacPherson compactification C ( T , I ) of C ( T , I ) .The boundary ∂ C ( T , I ) = C ( T , I ) − C ( T , I ) is made up of the following irreducible components:for any partition I = J ∐ ⋯ ∐ J k there is a component ∂ J , ⋯ ,J k C ( T , I ) ≅ C ( T , k ) × k ∏ i = C ( C , J i ) . The inclusion of boundary components provide C ( T , −) with the structure of a module overthe operad C ( C , −) in topological spaces. = 53 1 4 2 21 73 4321 4 6 The pure braid group on the torus.
The reduced pure braid group PB ,n with n strands on the torus (that is the fundamental group of C ( T , n ) ) is generated by paths X i ’sand Y i ’s ( i = , . . . , n ), which can be represented as follows nY j X i i j Moreover, the following relations are satisfied in PB ,n : ( X i , X j ) = = ( Y i , Y j ) , for i < j , (T1) ( X i , Y j ) = P ij and ( X − j , Y − i ) = P ij , for i < j , (T2) ( X , Y − ) = P n ⋯ P , (T3) ( X i , P jk ) = = ( Y i , P jk ) , for all i, j < k , (T4) ( X i X j , P ij ) = = ( Y i Y j , P ij ) , for i < j , (T5) X ⋯ X n = = Y ⋯ Y n , (TR)There are also the following relations, satisfied in the fundamental group B ,n of C ( T , n )/ S n :(N) X i + = σ − i X i σ − i , Y i + = σ − i Y i σ − i , where σ i are the generators of the braid group B n with geometric convention as follows: ni i + 1 i The PaB-module PaB e(cid:96)(cid:96) of parenthesized elliptic (or beak) braids.
In a similarmanner as in § Pa ⊂ C ( S , −) ⊂ C ( T , −) . Then it makes sense to define
PaB e(cid:96)(cid:96) ∶= π ( C ( T , −) , Pa ) , which is a PaB -module in groupoids.As said in subsection 1.5, there is a map of S -modules PaB —→ PaB e(cid:96)(cid:96) and we abusivelydenote R , and Φ , , the images in PaB e(cid:96)(cid:96) of the corresponding arrows in
PaB . Example 3.1 (Structure of
PaB e(cid:96)(cid:96) ( ) ) . As in Example 2.1 we have an arrow R , going from ( ) to ( ) . Additionnally, we also have two automorphisms of ( ) , denoted A , and B , ,corresponding to the following loops on C ( T , ) : A ; B ;
21 1 2 The second one depends on the choice of an embedding S ↪ T : we choose by convention the “horizontalembedding”, which corresponds to S × {∗} . LLIPSITOMIC ASSOCIATORS 23
By global translation of the torus, these are the same loops as the following A ; B ;
21 1 2
In particular, A , ˜ R , and B , ˜ R , , which are morphisms from ( ) to ( ) , correspond tothe following paths C ( T , ) : A ; R ~ ; B ; R ~ ;
21 2 1
Remark 3.2.
The arrows A , and B , correspond to A ± , in [18, § A , and B , as11 22 + and 11 22 − The images of R , and Φ , , by the S –module morphism PaB —→ PaB e(cid:96)(cid:96) will still bedenoted the same way. One can rephrase [18, Proposition 1.3] in the following way:
Theorem 3.3.
As a
PaB -module in groupoids having Pa as Pa -module of objects, PaB e(cid:96)(cid:96) isfreely generated by A ∶= A , and B ∶= B , , together with the following relations: Φ , , A , ˜ R , Φ , , A , ˜ R , Φ , , A , ˜ R , = Id ( ) , (N1) Φ , , B , ˜ R , Φ , , B , ˜ R , Φ , , B , ˜ R , = Id ( ) , (N2) R , R , = ( Φ , , A , ( Φ , , ) − , ˜ R , Φ , , B , ( Φ , , ) − ˜ R , ) . (E) All these relations hold in the automorphism group of ( ) in PaB e(cid:96)(cid:96) ( ) .Proof. Let Q e(cid:96)(cid:96) be the PaB -module with the above presentation. We first show that thereis a morphism of
PaB -modules Q e(cid:96)(cid:96) → PaB e(cid:96)(cid:96) . We have already seen that there are twoautomorphisms
A, B of ( ) in PaB e(cid:96)(cid:96) ( ) (see Example 3.1). We have to prove that theyindeed satisfy the relations (N1), (N2) and (E).Relations (N1) and (N2) are satistfied: the two nonagon relations (N1) cand (N2) can bedepicted as ( ( ) ) = ( ( ) ) ±±± (N1,N2)It is satisfied in PaB e(cid:96)(cid:96) , expressing the fact that when all (here, three) points move in thesame direction on the torus, this corresponds to a constant path in the reduced configurationspace of points on the torus. The same is true with the second nonagon relation (N2).Relation (E) is satisfied: below one sees the path that is obtained from the right-hand-side ofthe mixed relation (E): ● Φ , , A , ( Φ , , ) − is the path ● ˜ R , Φ , , B , ( Φ , , ) − ˜ R , is the path LLIPSITOMIC ASSOCIATORS 25
One easily sees that the commutator of these loops is homotopic to the pure braiding of thefirst two points in the clockwise direction, that is R , R , , by means of the following picture: Thus, by the universal property of Q e(cid:96)(cid:96) , there is a morphism of PaB -modules Q e(cid:96)(cid:96) → PaB e(cid:96)(cid:96) ,which is the identity on objects. To show that this map is in fact an isomorphism, it sufficesto show that it is an isomorphism at the level of automorphism groups of objects arity-wise, asall groupoids are connected. Let n ≥
0, and p be the object (⋯(( ) )⋯⋯) n of Q e(cid:96)(cid:96) ( n ) and PaB e(cid:96)(cid:96) ( n ) . We want to show that the induced morphismAut Q e(cid:96)(cid:96) ( n ) ( p ) —→ Aut
PaB e(cid:96)(cid:96) ( n ) ( p ) = π ( C ( T , n ) , p ) is an isomorphism.On the one hand, as ¯C ( T , n ) is a manifold with corners, we are allowed to move the basepoint p to a point p reg which is included in the simply connected subset obtained as the image of D n,τ ∶= { z ∈ C n ∣ z j = a j + b j τ, a j , b j ∈ R , < a < a < ... < a n < a + , < b n < ... < b < b n + } in C ( T , n ) , where T = C /( Z + τ Z ) . We then have an isomorphism of fundamental groups π ( ¯C ( T , n ) , p ) ≃ π ( C ( T , n ) , p reg ) .On the other hand, in [18, Proposition 1.4], Enriquez constructs a universal elliptic structure PaB
Ene(cid:96)(cid:96) , that by definition carries an action of the (algebraic version of the) reduced braidgroup on the torus B ,n in the following sense: ● PaB
Ene(cid:96)(cid:96) is a category; ● for every object p of Pa ( n ) , there is a corresponding object [ p ] in PaB
Ene(cid:96)(cid:96) , and [ p ] = [ q ] if p and q only differ by a permutation (but have the same underlying parenthesization); ● there are group morphisms B ,n ˜ → Aut
PaB
Ene(cid:96)(cid:96) ( p ) → S n . We have already done so for the proof of relation (E).
Moreover, one has by constuction of
PaB
Ene(cid:96)(cid:96) that Aut Q e(cid:96)(cid:96) ( n ) ( p ) is the kernel of the mapAut PaB
Ene(cid:96)(cid:96) ([ p ]) → S n . One can actually show that we have a commuting diagramPB ,n ≃ (cid:47) (cid:47) (cid:15) (cid:15) Aut Q e(cid:96)(cid:96) ( n ) ( p ) (cid:47) (cid:47) (cid:15) (cid:15) π ( C ( T , n ) , p ) (cid:15) (cid:15) π ( C ( T , n ) , p reg ) ≃ (cid:111) (cid:111) (cid:15) (cid:15) B ,n ≃ (cid:47) (cid:47) (cid:15) (cid:15) Aut
PaB
Ene(cid:96)(cid:96) ( p ) (cid:47) (cid:47) (cid:15) (cid:15) π ( C ( T , n )/ S n , [ p ]) (cid:15) (cid:15) π ( C ( T , n )/ S n , [ p reg ]) ≃ (cid:111) (cid:111) (cid:15) (cid:15) S n S n S n S n where all vertical sequences are short exact sequences. Thus, in order to show that the mapAut Q e(cid:96)(cid:96) ( n ) ( p ) → π ( C ( T , n ) , p ) is an isomorphism, we are left to show thatB ,n —→ π ( C ( T , n ) , p reg ) is indeed an isomorphism. But this map is nothing else than a conjugate of the map constructedin [7, Theorem 5], identifying the algebraic and topological versions of the braid group on thetorus. (cid:3) The CD ( k ) -module of elliptic chord diagrams. For any n ≥
0, recall that t ,n ( k ) isdefined as the bigraded Lie k -algebra freely generated by x , . . . , x n in degree ( , ) , y , . . . , y n in degree ( , ) (for i = , ..., n ), and t ij in degree ( , ) (for 1 ≤ i ≠ j ≤ n ), together with therelations (S), (L), (4T), and the following additional elliptic relations as well: [ x i , y j ] = t ij for i ≠ j , (S e(cid:96)(cid:96) ) [ x i , x j ] = [ y i , y j ] = i ≠ j , (N e(cid:96)(cid:96) ) [ x i , y i ] = − ∑ j ∣ j ≠ i t ij , (T e(cid:96)(cid:96) ) [ x i , t jk ] = [ y i , t jk ] = { i, j, k } = , (L e(cid:96)(cid:96) ) [ x i + x j , t ij ] = [ y i + y j , t ij ] = i ≠ j . (4T e(cid:96)(cid:96) )The ∑ i x i and ∑ i y i are central in t ,n ( k ) , and we also consider the quotient¯ t ,n ( k ) ∶= t ,n ( k )/(∑ i x i , ∑ i y i ) . Example 3.4. ¯ t , ( k ) is equal to the free Lie k -algebra f ( k ) on two generators x = x and y = y .Both t ,n and ¯ t ,n are acted on by the symmetric group S n , and one can show that the S -modules in grLie k t e(cid:96)(cid:96) ( k ) ∶= { t ,n ( k )} n ≥ and ¯ t e(cid:96)(cid:96) ( k ) ∶= { ¯ t ,n ( k )} n ≥ actually are t ( k ) -modules in grLie k . Partial compositions are defined as follows: for I a finiteset and i ∈ I , ○ k ∶ t ,I ( k ) ⊕ t J ( k ) —→ t ,J ⊔ I −{ i } ( k )( , t αβ ) z→ t αβ ( t ij , ) z→ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ t ij if k ∉ { i, j }∑ p ∈ J t pj if k = i ∑ p ∈ J t ip if j = k ( x i , ) z→ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ x i if k ≠ i ∑ p ∈ J x p if k = i ( y i , ) z→ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ y i if k ≠ i ∑ p ∈ J y p if k = i We call t e(cid:96)(cid:96) ( k ) , resp. ¯ t e(cid:96)(cid:96) ( k ) , the module of infinitesimal elliptic braids , resp. of infinitesimalreduced elliptic braids .We finally define the CD ( k ) -module CD e(cid:96)(cid:96) ( k ) ∶= ˆ U( ¯ t e(cid:96)(cid:96) ( k )) of elliptic chord diagrams .Similarly to the genus 0 situation, morphisms in CD e(cid:96)(cid:96) ( k )( n ) can be represented as chordson n vertical strands, with extra chords correponding to the generators x i and y i as in thefollowing picture: + ii and − ii The elliptic relations introduced above can be represented as follows, analogously as for thegenus 0 case: −+ ii jj − +− ii jj = +− ii jj − −+ ii jj = ii jj (S e(cid:96)(cid:96) ) ±± ii jj = ±± ii jj (N e(cid:96)(cid:96) ) +− ii − A − + ii = − ∑ j ; j ≠ i i ji j (T e(cid:96)(cid:96) ) ± ii jj kk = ± ii jj kk (L e(cid:96)(cid:96) ) ± ii jj + ± ii jj = ± ii jj + ± ii jj (4T e(cid:96)(cid:96) ) Remark 3.5.
The relation between (a closely related version of) CD e(cid:96)(cid:96) ( k ) and the ellipticKontsevich integral was studied in Philippe Humbert’s thesis [29].3.5. The PaCD ( k ) -module of parenthesized elliptic chord diagrams. As in the genuszero case, the module of objects Ob ( CD e(cid:96)(cid:96) ( k )) of CD e(cid:96)(cid:96) ( k ) is terminal. Hence we have amorphism of modules ω ∶ Pa = Ob ( Pa ( k )) → Ob ( CD e(cid:96)(cid:96) ( k )) over the morphism of operads ω from § PaCD ( k ) -module PaCD e(cid:96)(cid:96) ( k ) ∶= ω ⋆ CD e(cid:96)(cid:96) ( k ) , in Cat ( CoAss k ) , of so-called parenthesized elliptic chord diagrams . Recall that
PaCD ( k ) is defined as ω ⋆ CD ( k ) . LLIPSITOMIC ASSOCIATORS 29
Example 3.6 (Notable arrows in
PaCD e(cid:96)(cid:96) ( k )( ) ) . We have the following arrows X , e(cid:96)(cid:96) , Y , e(cid:96)(cid:96) in PaCD e(cid:96)(cid:96) ( k )( ) : X , e(cid:96)(cid:96) = x ⋅
11 22 = +
11 22 Y , e(cid:96)(cid:96) = y ⋅
11 22 = −
11 22
Remark 3.7.
As said in subsection 1.5, there is a map of S -modules PaCD ( k ) —→ PaCD e(cid:96)(cid:96) ( k ) and we abusively denote X , , H , and a , , the images in PaCD e(cid:96)(cid:96) ( k ) of the corresponding arrows in PaCD ( k ) . The elements X , e(cid:96)(cid:96) , Y , e(cid:96)(cid:96) are generators of the PaCD ( k ) -module PaCD e(cid:96)(cid:96) ( k ) and satisfy the following relations in End PaCD e(cid:96)(cid:96) ( k )( ) ( ) : ● X , e(cid:96)(cid:96) + X , X , e(cid:96)(cid:96) ( X , ) − = ● Y , e(cid:96)(cid:96) + X , Y , e(cid:96)(cid:96) ( X , ) − = PaCD e(cid:96)(cid:96) ( k )( ) (( ) ) : ● X , e(cid:96)(cid:96) + a , , X , X , e(cid:96)(cid:96) ( a , , X , ) − + X , ( a , , ) − X , e(cid:96)(cid:96) ( X , ( a , , ) − ) − = ● Y , e(cid:96)(cid:96) + a , , X , Y , e(cid:96)(cid:96) ( a , , X , ) − + X , ( a , , ) − Y , e(cid:96)(cid:96) ( X , ( a , , ) − ) − = ● H , = [ a , , X , e(cid:96)(cid:96) ( a , , ) − , X , a , , Y , e(cid:96)(cid:96) ( a , , ) − ( X , ) − ] .3.6. Elliptic associators.
Let us introduce some terminology, complementing the one of § C for the category of pairs (P , M) , where P is an operad and M is a right O -module, in C . A morphism (P , M) → (Q , N ) is a pair ( f, g ) , where f ∶ P → Q is a morphismbetween operads and g ∶ M → N is a morphism of P -modules.The superscript “ + ” still indicates that we consider morphisms that are the identity onobjects. Definition 3.8.
An elliptic associator over k is a couple ( F, G ) where F is a k -associator and G is an isomorphism between the ̂ PaB ( k ) -module ̂ PaB e(cid:96)(cid:96) ( k ) and the G PaCD ( k ) -module G PaCD e(cid:96)(cid:96) ( k ) which is the identity on objects and which is compatible with F : Ell ( k ) ∶= Iso + OpR
Grpd k ((̂ PaB ( k ) , ̂ PaB e(cid:96)(cid:96) ( k )) , ( G PaCD ( k ) , G PaCD e(cid:96)(cid:96) ( k ))) . The following theorem identifies our definition of elliptic associators with the original oneintroduced by Enriquez in [18].
Theorem 3.9.
There is a one-to-one correspondence between the set
Ell ( k ) and the set Ell ( k ) of quadruples ( µ, ϕ, A + , A − ) , where ( µ, ϕ ) ∈ Ass ( k ) and A ± ∈ exp ( ˆ¯ t , ( k )) , such that: (4) α , , ± α , , ± α , , ± = , where α ± = ϕ , , A , ± e − µ ( t + t )/ , (5) e µt = ( ϕA , + ϕ − , e − µt / ϕ , , A , − ( ϕ , , ) − e − µt / ) . All these relations hold in the group exp ( ˆ¯ t , ( k )) .Proof. An associator F corresponds uniquely to a couple ( µ, ϕ ) ∈ Ass ( k ) and an isomorphism G between ̂ PaB e(cid:96)(cid:96) ( k ) and G PaCD e(cid:96)(cid:96) ( k ) sends the arrows A , and B , of End ̂ PaB e(cid:96)(cid:96) ( k )( ) ( ) to A + ⋅ X , e(cid:96)(cid:96) and A − ⋅ Y , e(cid:96)(cid:96) with A ± ∈ exp ( ˆ¯ t , ( k )) (recall that ˆ¯ t , ( k ) is the completed free Liealgebra over k in two generators). The image of relations (N1), (N2) and (E) are precisely therelations (4) and (5). (cid:3) Example 3.10 (Elliptic KZB Associators) . Let us fix τ ∈ h . Recall that the Lie algebra ¯ t , ( C ) is isomorphic to the free Lie algebra f ( C ) generated by two elements x ∶= x and y ∶= y . Wedefine the elliptic KZB associators e ( τ ) ∶= ( A ( τ ) , B ( τ )) as the renormalized holonomies from0 to 1 and 0 to τ of the differential equation(6) G ′ ( z ) = − θ τ ( z + ad x ) ad xθ τ ( z ) θ τ ( ad x ) ( y ) ⋅ G ( z ) , with values in the group exp ( ˆ¯ t , ( C )) More precisely, this equation has a unique solution G ( z ) defined over { a + bτ, for a, b ∈] , [} such that G ( z ) ≃ (− π i z ) −[ x,y ] at z →
0. In this case, A ( τ ) ∶= G ( z ) G ( z + ) − , B ( τ ) ∶= G ( z ) G ( z + τ ) − e − π i x . These are elements of the group exp ( ˆ¯ t , ( C )) . More precisely, Enriquez showed in [18] thatthe element ( π i , Φ KZ , A ( τ ) , B ( τ )) is in Ell ( C ) .3.7. Elliptic Grothendieck–Teichm¨uller group.Definition 3.11.
The ( k -prounipotent version of the) elliptic Grothendieck–Teichm¨uller group is defined as the group ̂ GT e(cid:96)(cid:96) ( k ) ∶= Aut + OpR
Grpd k (̂ PaB ( k ) , ̂ PaB e(cid:96)(cid:96) ( k )) . Again, we now show that our definition coincides with the original one defined by Enriquezin [18]. Recall that the set ̂ GT e(cid:96)(cid:96) ( k ) is the set of tuples ( λ, f, g ± ) , where ( λ, f ) ∈ ̂ GT ( k ) , g ± ∈ ̂ F ( k ) such that, in ̂ ¯B , ( k ) ,(7) ( f ( σ , σ ) g ± ( A, B )( σ σ σ ) − λ − σ − σ − ) = , (8) u = ( g + , u − g − u − ) , where u = f ( σ , σ ) − σ λ f ( σ , σ ) and g ± = g ± ( A, B ) .For ( λ, f, g ± ) , ( λ ′ , f ′ , g ′± ) ∈ ̂ GT e(cid:96)(cid:96) ( k ) , we set ( λ, f, g ± ) ∗ ( λ ′ , f ′ , g ′± ) ∶= ( λ ′′ , f ′′ , g ′′± ) , where g ′′± ( A, B ) = g ± ( g ′+ ( A, B ) , g ′− ( A, B )) . This gives ̂ GT e(cid:96)(cid:96) ( k ) a group structure. Moreover,for ( λ, f, g + , g − ) ∈ ̂ GT ell ( k ) and ( µ, ϕ, A + , A − ) ∈ Ell ( k ) , we set ( λ, f, g + , g − ) ∗ ( µ, ϕ, A + , A − ) ∶= ( µ ′ , ϕ ′ , A ′+ , A ′− ) , where A ′± ∶= g ± ( A + , A − ) . In [18], it is shown that this defines a free and transitive left actionof ̂ GT e(cid:96)(cid:96) ( k ) on Ell ( k ) . Proposition 3.12.
There is an isomorphism ̂ GT e(cid:96)(cid:96) ( k ) —→ ̂ GT e(cid:96)(cid:96) ( k ) such that the bijection Ell ( k ) ˜ —→ Ell ( k ) becomes a torsor isomorphism. LLIPSITOMIC ASSOCIATORS 31
Proof.
Suppose that we have an automorphism ( F, G ) of (̂ PaB ( k ) , ̂ PaB e(cid:96)(cid:96) ( k )) which is theidentity on objects. We already know (see § F is determined by a pair ( λ, f ) ∈̂ GT ( k ) , and that any such pair determines an F . Moreover, the images of the two generators A , , B , ∈ Aut ̂ PaB e(cid:96)(cid:96) ( k )( ) ( ) = ̂ ¯PB , ( k ) are G ( A , ) = g + ( A , , B , ) and G ( B , ) = g − ( A , , B , ) , with g ± ∈ ̂ F ( k ) ≃ ̂ ¯PB , ( k ) . It therefore follows from Theorem 3.3 that ( λ, f, g ± ) satisfiesrelations (7) and (8) if and only it determines an automorphism ( F, G ) .Let us then prove that the bijective assignement ( F, G ) ↦ ( λ, f, g ± ) that we just describedis a group morphism. For this we show that the composition of automorphisms corresponds tothe group law of GT e(cid:96)(cid:96) ( k ) . We already know (see § ̂ PaB ( k ) corresponds to the group law in GT ( k ) . Now, given automorphisms ( F , G ) and ( F , H ) , and there respective images ( λ , f , g ± ) and ( λ , f , h ± ) , we have that ( H ○ G )( A ) = H ( g + ( A, B )) = g + ( H ( A ) , H ( B )) = g + ( h + ( A, B ) , h − ( A, B )) , and, likewise, ( H ○ G )( B ) = g − ( h + ( A, B ) , h − ( A, B )) .We finally prove the equivariance statement. Let ( F, G ) ∈ GT e(cid:96)(cid:96) ( k ) , with image ( λ, f, g pm ) ∈ GT e(cid:96)(cid:96) ( k ) , and let ( K, H ) ∈
Ell e(cid:96)(cid:96) ( k ) , with image ( µ, ϕ, A ± ) . It is known (see § K ○ F is sent to ( µ, (cid:36) ) ∗ ( λ, f ) . It remains to compute: ( H ○ G )( A ) = H ( g + ( A, B )) = g + ( H ( A ) , H ( B )) = g + ( A + , A − ) , and, similarly, ( H ○ G )( B ) = g − ( A + , A − ) . (cid:3) Graded elliptic Grothendieck–Teichm¨uller group.Definition 3.13.
The graded elliptic Grothendieck-Teichm¨uller group is the group
GRT e(cid:96)(cid:96) ( k ) ∶= Aut + OpR
Cat ( CoAlg k ) ( PaCD ( k ) , PaCD e(cid:96)(cid:96) ( k )) . Notice that there is an isomorphismAut + OpR
Cat ( CoAlg k ) ( PaCD ( k ) , PaCD e(cid:96)(cid:96) ( k )) ≃ Aut + OpR
Grpd k ( G PaCD ( k ) , G PaCD e(cid:96)(cid:96) ( k )) . As before, our goal in this paragraph is to show that our definition coincides with theone of Enriquez [18]. Recall that he defines GRT ell ( k ) as the set of triples ( g, u + , u − ) ∈ GRT ( k ) × ( ˆ¯ t , ( k )) × , satisfying(9) Ad ( g , , )( u , ± ) + Ad ( g , , )( u , ± ) + u , ± = , (10) [ Ad ( g , , )( u , ± ) , u , ± ] = , (11) [ Ad ( g , , )( u , + ) , Ad ( g , , )( u , − )] = t , as relations in ˆ¯ t , ( k ) . He defines a group structure as follows: ( g, u + , u − ) ∗ ( h, v + , v − ) ∶= ( g ∗ h, w + , w − ) , where w ± ( x , y ) ∶= u ± ( v + ( x , y ) , v − ( x , y )) . The group k × acts on GRT ell ( k ) by rescaling: c ⋅ ( g, u ± ) ∶= ( c ⋅ g, c ⋅ u ± ) , where c ⋅ g is as before,and ● ( c ⋅ u + )( x , y ) ∶= u + ( x , c − y ) , ● ( c ⋅ u − )( x , y ) ∶= cu − ( x , c − y ) .We then set GRT e(cid:96)(cid:96) ( k ) ∶= GRT ell ( k ) ⋊ k × .Moreover, there is a right action of GRT ell ( k ) on Ell ( k ) : for ( g, u ± ) ∈ GRT ell ( k ) and ( µ, ϕ, A ± ) ∈ Ell ( k ) , we set ( µ, ϕ, A ± ) ∗ ( g, u ± ) ∶= ( µ, ˜ ϕ, ˜ A ± ) , where˜ A ± ( x , y ) ∶= A ± ( u + ( x , y ) , u − ( x , y )) and, for c ∈ k × , we set ( µ, ϕ, A ± ) ∗ c ∶= ( µ, c ∗ ϕ, c ♯ A ± ) , where ( c ♯ A ± )( x , y ) ∶= A ± ( x , cy ) . In[18] this action is shown to be free and transitive. Notice that ˜ A ± = θ ( A ± ) , where θ ∈ Aut ( ˆ¯ t k , ) is defined by x ↦ u + ( x , y ) and y ↦ u − ( x , y ) . Proposition 3.14.
There is an injective group morphism
GRT e(cid:96)(cid:96) ( k ) → GRT e(cid:96)(cid:96) ( k ) . More-over, the bijection Ell ( k ) → Ell ( k ) from Theorem 3.9 is equivariant along this morphism.Proof. For every ( G, U ) ∈
GRT e(cid:96)(cid:96) ( k ) , we have ● G ( X , ) = X , , ● G ( H , ) = λH , , ● G ( a , , ) = f ( t , t ) a , , , ● U ( X , e(cid:96)(cid:96) ) = u + ( x, y ) Id , ● U ( Y , e(cid:96)(cid:96) ) = u − ( x, y ) Id ,where ( λ, g ) ∈ GRT ( k ) and u ± ∈ ˆ¯ t , ( k ) . In light of relations of Remark 3.7, we obtain that ( λ, g, u ± ) satisfies relations (9), (10) and (11). The assignment ( G, U ) ↦ ( λ, g, u ± ) defines aninjective map GRT e(cid:96)(cid:96) ( k ) → GRT e(cid:96)(cid:96) ( k ) .We now show that this map is a group morphism. The proof is the same as one of theanalogous statement in Proposition 3.12: for two automorphisms ( G , U ) and ( G , V ) , wealready know that the composition G ○ G corresponds to the product in GRT ( k ) , and wecompute: ( V ○ U )( X , e(cid:96)(cid:96) ) = V ( u + ( x , y ) Id ) = u + ( v + ( x , y ) , v − ( x , y )) Id , and, likewise, ( V ○ U )( X , e(cid:96)(cid:96) ) = u − ( v + ( x , y ) , v − ( x , y )) Id .Finally, the equivariance of the bijection is proven in a similar way. (cid:3) Bitorsors.
Summarizing the results we have proven so far, we get that the bijection
Ell ( k ) —→ Ell ( k ) from Theorem 3.9 has been promoted to a bitorsor isomorphism. Indeed,we know (by definition) that ( ̂ GT e(cid:96)(cid:96) ( k ) , Ell ( k ) , GRT e(cid:96)(cid:96) ( k )) is a bitorsor, and (from [18]) that (̂ GT e(cid:96)(cid:96) ( k ) , Ell ( k ) , GRT e(cid:96)(cid:96) ( k )) is a bitorsor as well. Theorem 3.15.
There is a bitorsor isomorphism (12) ( ̂ GT e(cid:96)(cid:96) ( k ) , Ell ( k ) , GRT e(cid:96)(cid:96) ( k )) ˜ —→(̂ GT e(cid:96)(cid:96) ( k ) , Ell ( k ) , GRT e(cid:96)(cid:96) ( k )) . LLIPSITOMIC ASSOCIATORS 33
Proof.
This is a summary of most of the above results: ● There is a group isomorphism between ̂ GT e(cid:96)(cid:96) ( k ) and ̂ GT e(cid:96)(cid:96) ( k ) that is such that thebijection from Theorem 3.9 is a torsor isomorphism (Proposition 3.12). ● There is an injective group morphism
GRT e(cid:96)(cid:96) ( k ) → GRT e(cid:96)(cid:96) ( k ) such that the bijectionfrom Theorem 3.9 is equivariant (Proposition 3.14).Knowing from Example 3.10 that Ell ( k ) is non-empty, we obtain that GRT e(cid:96)(cid:96) ( k ) → GRT e(cid:96)(cid:96) ( k ) is an isomorphism. (cid:3) The module of parenthesized ellipsitomic braids
All along this Section, Γ denotes the abelian group Γ = Z / M Z × Z / N Z where M, N ≥ ∶= ( ¯0 , ¯0 ) .4.1. Compactified twisted configuration space of the torus.
Let T be the topologicaltorus, and consider the connected Γ-covering p ∶ ˜ T → T corresponding to the canonical surjectivegroup morphism ρ ∶ π ( T ) = Z → Γ senging the generators of Z to their correspondingreduction in Γ. To any finite set I with cardinality n we associate the Γ-twisted configurationspace Conf ( T , I, Γ ) ∶= { z = ( z , . . . , z n ) ∈ ˜ T I ∣ p ( z i ) ≠ p ( z j ) if i ≠ j } , and let C ( T , I, Γ ) ∶= Conf ( T , I, Γ )/ ˜ T be its reduced version.The inclusion(13) Conf ( T , I, Γ ) ↪ Conf ( ˜ T , I × Γ ) sending ( z i ) i ∈ I to ( γ ⋅ z i ) ( i,γ )∈ I × Γ induces an inclusionC ( T , I, Γ ) ↪ C ( ˜ T , I × Γ ) ↪ C ( ˜ T , I × Γ ) , which allows us to define C ( T , I, Γ ) as the closure of C ( T , I, Γ ) inside C ( ˜ T , I × Γ ) . The boundary ∂ C ( T , I, Γ ) = C ( T , I, Γ ) − C ( T , I, Γ ) is made up of the following irreducible components: forany partition J ∐ ⋯ ∐ J k of I there is a component ∂ J , ⋯ ,J k C ( T , I, Γ ) ≅ k ∏ i = ( C ( C , J i )) × C ( T , k, Γ ) . The inclusion of boundary components provides C ( T , − , Γ ) with the structure of a module overthe operad C ( C , −) in topological spaces.On the one hand, the left action of Γ on ˜ T gives us an action of Γ I , resp. Γ I / Γ, onConf ( ˜ T , I × Γ ) , resp. C ( ˜ T , I × Γ ) . On the other hand, Γ I also acts on Conf ( ˜ T , I × Γ ) andC ( ˜ T , I × Γ ) in the following way: ( α ⋅ z ) ( i,γ ) ∶= z i,γ + α . The inclusion (13) is Γ I -equivariant, so that one gets a diagonally trivial Γ-action on C ( C , −) ,in the sense of § The Pa-module of labelled parenthesized permutation.
For every finite set I , wehave a Γ I / Γ-covering map φ I ∶ C ( T , n, Γ ) —→ C ( T , n ) which extends to a continuous map¯ φ I ∶ C ( T , I, Γ ) —→ C ( T , I ) , everything being natural (with respective to bijections) in I . This defines a morphism ¯ φ ofC ( C , −) -modules from C ( T , − , Γ ) to C ( T , −) .Recall from § Pa ⊂ C ( S , −) ⊂ C ( T , −) over Pa ⊂ C ( R , −) ⊂ C ( C , −) . We define the S -module Pa Γ ∶= ¯ φ − Pa , which carriesa Pa -module structure. Indeed, it is a fiber product Pa Γ ∶= Pa × C ( T , −) C ( T , − , Γ ) in the category of Pa -modules in topological space.The Pa -module Pa Γ admits the following algebraic description. First of all, one can easilycheck that it happens to be discrete (i.e. spaces of operations are discrete sets). Then, anelement of Pa Γ ( n ) is a parenthesized permutation of 1 . . . n together with a label function { , . . . , n } → Γ that is defined up to a global relabelling (i.e. the labelling is an element ofΓ n / Γ). For instance, 2 γ = − γ belongs to Pa Γ ( ) for every γ ∈ Γ. In geometric terms,having the label [ γ , . . . , γ n ] means that, in our configuration of points, the (− γ i ) ⋅ z i ’s are onthe same parallel of the torus. Here is a self-explanatory example of partial composition: ( γ ) δ ○ ( ) = ( (( γ γ ) γ )) δ . Finally, Pa Γ is acted on by Γ in the following way: for n ≥
0, Γ n only acts on the labellings, via the group law of Γ. For instance, if [ α ] ∈ Γ n / Γ and γ ∈ Γ n , then γ ⋅ [ α ] ∶= [ γ + α ] .In other words, according to the terminology of § § Pa Γ is identified with G( Pa → Γ ) .4.3. The PaB-module of parenthesized ellipsitomic braids.
We define
PaB Γ e(cid:96)(cid:96) ∶= π ( C ( T , − , Γ ) , Pa Γ ) , which is a PaB -module (in groupoids), that also carries a diagonally trivial action of Γ. Themorphism ¯ φ induces a PaB -module morphism
PaB Γ e(cid:96)(cid:96) → PaB e(cid:96)(cid:96) . Example 4.1 (Notable arrows in
PaB Γ e(cid:96)(cid:96) ) . Recall the following notable arrows in
PaB e(cid:96)(cid:96) : ● A , and B , are automorphisms of 12 in PaB e(cid:96)(cid:96) ( ) . ● R , goes from 12 to 21 in PaB e(cid:96)(cid:96) ( ) . ● Φ , , goes from ( ) ( ) in PaB e(cid:96)(cid:96) ( ) .All are represented by paths which, apart from the endpoints that are in Pa , remain in theopen part C ( T , n ) of the configuration spaces ( n = , α ∶= ( ¯1 , ¯0 ) and β ∶= ( ¯0 , ¯1 ) .Since we have covering maps C ( T , n, Γ ) —→ C ( T , n ) , LLIPSITOMIC ASSOCIATORS 35 then these paths admits unique lifts, with starting point being the same parenthesized permu-tation with the trivial labelling (the one being constantly equal to ). These lifts are denotedthe same way: ● The lift A , goes from 1 to 1 α = − α in PaB Γ e(cid:96)(cid:96) ( ) . ● The lift B , goes from 1 to 1 β = − β in PaB Γ e(cid:96)(cid:96) ( ) . ● etc...Here is a drawing of paths representing A , and B , : z ; z z z z ; z ; We may chose to alternatively depict them as diagrams representing elliptic pure braids(i.e. loops in the base configuration space) together with appropriate base points (whichuniquely determines the lift in the covering twisted configuration space): A , = α + and B , = β − Remark 4.2.
It is important to observe that, the action of Γ being diagonally trivial, onecan shift the global labelling of the indexed points, and thus A , and B , can also berepresented as follows: A , = − α + and B , = − β − As for R , and Φ , , , they are depicted in the usual way: R , = 1 and Φ , , = ( )( ) Actually, every morphism in
PaB e(cid:96)(cid:96) can be uniquely lifted to
PaB Γ e(cid:96)(cid:96) , once the lift ofthe source object has been fixed; all other lifts are obtained by the Γ-action. Moreover,all morphisms can be obtained like this. This shows that the PaB -module
PaB Γ e(cid:96)(cid:96) has analternative simple algebraic descrition that we explain now. First observe that the PaB -module
PaB e(cid:96)(cid:96) comes with a morphism π to the ∗ -module Γ, which is the composition of theabelianization morphism to Z with the projection Z → Γ.In terms of the presentation from Theorem 3.3, we have π ( A ) = α = [( ¯1 , ) , ] and π ( B ) = β = [( , ¯1 ) , ] , where we adopt the following Notation 4.3.
For γ ∈ Γ and 1 ≤ i ≤ n , then we write γ i ∶= [ , . . . , , γ i , , . . . , ] ∈ Γ n / Γ Proposition 4.4.
There is an isomorphism G( PaB e(cid:96)(cid:96) → Γ ) ˜ —→ PaB Γ e(cid:96)(cid:96) of PaB -modules with a Γ -action, which is is the identity on objects.Proof. We first describe the morphism: ● It is the identity on objects; ● Given two labelled parenthesized permutations ( p , γ ) and ( q , δ ) , it sends a the classin PaB e(cid:96)(cid:96) of a path f ∶ p → q such that γ + π ( f ) = δ to the class of the unique lift of f that starts at the base point determined by ( p , γ ) .As we have already seen, to show that this morphism is in fact an isomorphism, it suffices toshow that it is an isomorphism at the level of automorphism groups of objects arity-wise. Thisis indeed the case, as on both sides, in arity n , the automorphism group of an object is thekernel of the morphism PB ,n → Γ n / Γ sending X i to ( ¯1 , ¯0 ) i and Y j to (¯0 , ¯1 ) j . (cid:3) The universal property of PaB Γ e(cid:96)(cid:96) . We are now ready to provide an explicit presen-tation for the
PaB -module
PaB Γ e(cid:96)(cid:96) . As before, we keep the convention that α = ( ¯1 , ¯0 ) and β = ( ¯0 , ¯1 ) . Theorem 4.5.
As a
PaB -module in groupoids with a diagonally trivial Γ -action and having Pa Γ as Pa -module of objects, PaB Γ e(cid:96)(cid:96) is freely generated by A ∶ → α and B ∶ → β , together with the following relations, satisfied in Aut
PaB Γ e(cid:96)(cid:96) ( )⋊( Γ / Γ ) (( ) ) : (tN1) Φ , , A , ˜ R , Φ , , A , ˜ R , Φ , , A , ˜ R , = Id ( ) LLIPSITOMIC ASSOCIATORS 37 (tN2) Φ , , B , ˜ R , Φ , , B , ˜ R , Φ , , B , ˜ R , = Id ( ) (tE) R , R , = ( Φ , , A , ( Φ , , ) − , ˜ R , Φ , , B , ( Φ , , ) − ˜ R β , α ) where A ∶= Aα and B ∶= Bβ . Remark 4.6.
The above relations are clearer when stated within the semidirect product, eventhough they can be written within
PaB Γ e(cid:96)(cid:96) itself. For instance, (tN1) can be written asΦ , , A , α ⋅ ( ˜ R , Φ , , A , α ⋅ ( ˜ R , Φ , , A , )) ˜ R , = Id ( ) . We let the reader check that the expression for (tE) becomes unpleasant to write.
Proof of the Theorem.
Let Q Γ e(cid:96)(cid:96) be the PaB -module with the above presentation, and let Q e(cid:96)(cid:96) be the PaB -module with the presentation in Theorem 3.3. Our goal is to prove that there isan isomorphism
G(Q e(cid:96)(cid:96) → Γ ) ˜ —→Q Γ e(cid:96)(cid:96) of PaB -modules with a Γ-action, which is is the identity on objects. The result will then followfrom Proposition 4.4.By definition we have a morphism Q e(cid:96)(cid:96) —→ Q Γ e(cid:96)(cid:96) ⋊ Γ, which sends A to A , and B to B .Moreover, when we compose this morphism with the projection Q Γ e(cid:96)(cid:96) ⋊ Γ → Γ, we get back themorphism π ∶ Q e(cid:96)(cid:96) → Γ from the previous section, that sends A to α and B to β .By the adjunction from § G(Q e(cid:96)(cid:96) → Γ ) —→ Q Γ e(cid:96)(cid:96) . of PaB -modules with a Γ-action. It is surjective on morphisms, because both generatorsof Q Γ e(cid:96)(cid:96) have preimages. Finally, as we have already seen, to show that this is in fact anisomorphism, it suffices to show that it is an isomorphism at the level of automorphism groupsof objects arity-wise, and it is sufficient to do it for a single object in every arity.Let n ≥ p be the object (⋯(( ) )⋯) n of Q Γ e(cid:96)(cid:96) ( n ) and G(Q e(cid:96)(cid:96) ( n ) → Γ n / Γ ) ,which lifts p = (⋯(( ) )⋯) n in Q e(cid:96)(cid:96) ( n ) . We have a commuting diagram1 (cid:47) (cid:47) Aut G ( Q e(cid:96)(cid:96) ( n )→ Γ n / Γ )( ˜ p ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) Aut Q e(cid:96)(cid:96) ( n ) ( p ) (cid:47) (cid:47) Γ n / Γ (cid:47) (cid:47) Q Γ e(cid:96)(cid:96) ( n ) ( ˜ p ) (cid:54) (cid:54) where the horizontal sequence is exact. Therefore the vertical morphism is injective, and weare done. (cid:3) Ellipsitomic Grothendieck–Teichm¨uller groups.Definition 4.7.
The ( k -pro-unipotent version of the) ellipsitomic Grothendieck–Teichm¨ullergroup is defined as ̂ GT Γ e(cid:96)(cid:96) ( k ) ∶= Aut + OpR
Grpd k (̂ PaB ( k ) , ̂ PaB Γ e(cid:96)(cid:96) ( k )) Γ , where, as usual, the superscript Γ means that we are considering the subgroup of Γ-equivariantautomorphisms.Let M ′ , N ′ ≥
1, and assume we are given a surjective group morphism ρ ∶ Γ ↠ Γ ′ ∶= Z / M ′ Z × Z / N ′ Z . This gives a (surjective) map between the corresponding covering spaces of the torus, whichcan be used to construct a morphism of C ( C , −) -modules C ( T , − , Γ ) —→ C ( T , − , Γ ′ ) . Following the construction of subsections 4.2 and 4.3, we get a morphism of
PaB -modules
PaB ρe(cid:96)(cid:96) ∶ PaB Γ e(cid:96)(cid:96) —→ PaB Γ ′ e(cid:96)(cid:96) . The morphism
PaB ρe(cid:96)(cid:96) is Γ-equivariant, and has a straightforward algebraic description: ● On objects, it consists in applying ρ to the labelling, keeping the underlying parenthe-iszed permutation unchanged; ● It sends the generating morphisms A and B in PaB Γ e(cid:96)(cid:96) to their counterparts(which are denoted the same way) in PaB Γ ′ e(cid:96)(cid:96) .As a consequence, the PaB -module
PaB Γ ′ e(cid:96)(cid:96) can be obtained as the quotient of PaB Γ e(cid:96)(cid:96) byker ρ . We therefore obtain a group morphism ̂ GT Γ e(cid:96)(cid:96) ( k ) —→ ̂ GT Γ ′ e(cid:96)(cid:96) ( k ) .5. Ellipsitomic chord diagrams and ellipsitomic associators
Infinitesimal ellipsitomic braids.
In this paragraph and the next one, ( Γ , , +) canbe any finite abelian group.For any n ≥ t Γ1 ,n ( k ) to be the bigraded k -Lie algebra with generators x i (1 ≤ i ≤ n )in degree ( , ) , y i (1 ≤ i ≤ n ) in degree ( , ) , and t γij ( γ ∈ Γ, i ≠ j ) in degree ( , ) , andrelations t γij = t − γji , (tS e(cid:96)(cid:96) [ x i , y j ] = [ x j , y i ] = ∑ γ ∈ Γ t γij , (tS e(cid:96)(cid:96) [ x i , x j ] = [ y i , y j ] = , (tN e(cid:96)(cid:96) ) [ x i , y i ] = − ∑ j ∶ j ≠ i ∑ γ ∈ Γ t γij , (tT e(cid:96)(cid:96) ) [ t γij , t δkl ] = , (tL e(cid:96)(cid:96) [ x i , t γjk ] = [ y i , t γjk ] = , (tL e(cid:96)(cid:96) [ t γij , t γ + δik + t δjk ] = , (t4T e(cid:96)(cid:96) [ x i + x j , t γij ] = [ y i + y j , t γij ] = , (t4T e(cid:96)(cid:96) ≤ i, j, k, l ≤ n are pairwise distinct and γ, δ ∈ Γ. We will call t Γ1 ,n ( k ) the k -Lie algebraof infinitesimal ellipsitomic braids . Observe that ∑ i x i and ∑ i y i are central in t Γ1 ,n . Thenwe denote by ¯ t Γ1 ,n ( k ) the quotient of t Γ1 ,n ( k ) by ∑ i x i and ∑ i y i , and the natural morphism t Γ1 ,n ( k ) → ¯ t Γ1 ,n ( k ) ; u ↦ ¯ u . LLIPSITOMIC ASSOCIATORS 39
There is an alternative presentation of t Γ1 ,n ( k ) and ¯ t Γ1 ,n ( k ) : Lemma 5.1.
The Lie k -algebra t Γ1 ,n ( k ) (resp. ¯ t Γ1 ,n ( k ) ) can equivalently be presented with thesame generators, and the following relations: (tS e(cid:96)(cid:96) , (tS e(cid:96)(cid:96) , (tN e(cid:96)(cid:96) ) , (tL e(cid:96)(cid:96) , (tL e(cid:96)(cid:96) , (t4T e(cid:96)(cid:96) , and, for every i ∈ I , (tC e(cid:96)(cid:96) ) [∑ j x j , y i ] = [∑ j y j , x i ] = (resp. ∑ j x j = ∑ j y j = ).Proof. If x i , y i and t αij satisfy the initial relations, then [∑ j x j , y i ] = [ x i , y i ] + [∑ j ≠ i x j , y i ] = − ∑ j ∶ j ≠ i ∑ γ ∈ Γ t γij + ∑ j ∶ j ≠ i ∑ γ ∈ Γ t γij = . Now, if x i , y i and t αij satisfy the above relations, then relations [∑ j x j , y i ] = [ x j , y i ] =∑ γ ∈ Γ t γij , for i ≠ j , imply that [ x i , y i ] = − ∑ j ∶ j ≠ i ∑ γ ∈ Γ t γij . Now, relations [∑ k x k , y j ] = [∑ k x k , x i ] = [∑ k x k , ∑ γ ∈ Γ t γij ] =
0. Thus, as [ x i , t γjk ] = { i, j, k } = , we obtainrelation [ x i + x j , t γij ] =
0, for i ≠ j . In the same way we obtain [ y i + y j , t γij ] =
0, for i ≠ j . (cid:3) Both t Γ1 ,n ( k ) and ¯ t Γ1 ,n ( k ) are acted on by the symmetric group S n , we get that t Γ e(cid:96)(cid:96) ( k ) ∶= { t Γ1 ,n ( k )} n ≥ and ¯ t Γ e(cid:96)(cid:96) ( k ) ∶= { ¯ t Γ1 ,n ( k )} n ≥ define S -modules in grLie k . They are actually t ( k ) -module in grLie k , where partial composi-tions are defined as follows : for I a finite set and k ∈ I , ○ k ∶ t Γ1 ,I ( k ) ⊕ t J ( k ) —→ t Γ1 ,J ⊔ I −{ k } ( k )( , t uv ) z→ t uv ( t γij , ) z→ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ t γij if k ∉ { i, j }∑ p ∈ J t γpj if k = i ∑ p ∈ J t γip if j = k ( x i , ) z→ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ x i if k ≠ i ∑ p ∈ J x p if k = i ( y i , ) z→ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ y i if k ≠ i ∑ p ∈ J y p if k = i We call t Γ e(cid:96)(cid:96) ( k ) , resp. ¯ t Γ e(cid:96)(cid:96) ( k ) , the module of infinitesimal ellipsitomic braids , resp. of infinitesimalreduced ellipsitomic braids . When k = C we write t Γ1 ,n ∶= t Γ1 ,n ( C ) and ¯ t Γ1 ,n ∶= ¯ t Γ1 ,n ( C ) . We give the formulæ for t Γ e(cid:96)(cid:96) ( k ) . Formulæ for ¯ t Γ e(cid:96)(cid:96) ( k ) are the exact same. Both t ( k ) -modules are acted on by Γ: any element γ = ( γ i ) i ∈ I ∈ Γ I acts as γ ⋅ x i = x i ( i ∈ I ) ,γ ⋅ y i = y i ( i ∈ I ) ,γ ⋅ t δij = t δ + γ i − γ j ij ( δ ∈ Γ and i ≠ i ) . We also have the following functoriality in Γ, with respect to surjections:
Proposition 5.2.
Let ρ ∶ Γ ↠ Γ be a surjective group morphism, and let a, b, c, d ∈ k suchthat ad − bc = ∣ ker ρ ∣ . We have Γ I -equivariant surjective comparison morphisms t Γ ,I ( k ) → t Γ ,I ( k ) and ¯ t Γ ,I ( k ) → ¯ t Γ ,I ( k ) , defined by x i ↦ ax i + by i , y i ↦ cx i + dy i , t γij ↦ t ρ ( γ ) ij . These are morphisms of t ( k ) -modules in grLie k .Proof. This follows from direct computations. (cid:3)
Actually, these morphisms exhibit t Γ ,I ( k ) , resp. ¯ t Γ ,I ( k ) , as the quotient of t Γ ,I ( k ) , resp. ¯ t Γ ,I ( k ) ,by ( ker ρ ) I . Remark 5.3.
Whenever Γ i = Z / M i Z × Z / N i Z , there is a natural choice for the scalars a, b, c, d .Indeed, if ρ ∶ Γ → Γ is surjective, then there exists elements ( a, b ) and ( c, d ) in the lattice M Z × N Z that generate the sublattice M Z × N Z . Hence, in particular, the determinant ad − bc equals M N M N = ∣ ker ρ ∣ .5.2. Horizontal ellipsitomic chord diagrams.
In this paragraph we define the CD ( k ) -module CD Γ e(cid:96)(cid:96) ( k ) of ellipsitomic chord diagrams .We first consider the CD ( k ) -module ˆ U( ¯ t Γ e(cid:96)(cid:96) ( k )) . Morphisms in ˆ U( ¯ t Γ e(cid:96)(cid:96) ( k )) can be given apictorial description, which mixes the features of the horizontal N -chord diagrams from [11](see also [14]) together with the elliptic chord diagrams from § x i and y j are, respectively, + ii = + ii γ − γ and − jj = − jj γ − γ and the one corresponding to t γij = t − γji is i ji jγ − γ = i ji j γ − γ LLIPSITOMIC ASSOCIATORS 41
Relations can be depicted as follows:(tS e(cid:96)(cid:96) ∓± ii jj − ±∓ ii jj = ∑ γ ∈ Γ ii jjγ − γ (tN e(cid:96)(cid:96) ) ±± ii jj = ±± ii jj (tT e(cid:96)(cid:96) ) +− ii − −+ iii = − ∑ j ; j ≠ i ∑ γ ∈ Γ ii jjγ − γ (tL e(cid:96)(cid:96) j ki li lj kγ − γ δ − δ = j ki li lj kγ − γ δ − δ (tL e(cid:96)(cid:96) ± ii jj kkγ − γ = ± ii jjγ − γ kk (t4T e(cid:96)(cid:96) i j ki j kγδ − γ − δ + i j ki j kγ − γ δ − δ = i j ki j kγ + δ − δ − γ + i j ki j kδ − δγ − γ (t4T e(cid:96)(cid:96) ± ii jjγ − γ + ± ii jjγ − γ = ± ii jjγ − γ + ± ii jjγ − γ (tC e(cid:96)(cid:96) ) ∑ i ± ii = on each strand of all the above diagrams.We are now ready to define the CD ( k ) -module CD Γ e(cid:96)(cid:96) ( k ) . ● In arity n , objects of CD Γ e(cid:96)(cid:96) ( k ) are just labellings { , . . . , n } → Γ up to a global shift:Ob ( CD Γ e(cid:96)(cid:96) ( k ))( n ) = Γ n / Γ; ● The ∗ -module structure is given as follows on objects: for every i , ○ i ∶ Γ n → Γ n + m − isthe partial diagonal ( α , . . . , α n ) z→ ( α , . . . , α i − , α i , . . . , α i ·„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„¶ m times , α i + , . . . , α n ) ; ● Given two objects [ α ] = [ α , . . . , α n ] and [ β ] = [ β , . . . , β n ] in arity n , the k -vectorspace of morphisms from [ α ] to [ β ] in CD Γ e(cid:96)(cid:96) ( k ) is the vector space of horizontalΓ-chord diagrams such that, for every i , the sum of labels on the i -th strand is β i − α i ; ● The CD ( k ) -module structure on morphisms is the exact same as the one for ˆ U( ¯ t Γ e(cid:96)(cid:96) ( k )) .Moreover, CD Γ e(cid:96)(cid:96) ( k ) carries an action of Γ, by translation on the labelling of objects.For every surjective group morphism ρ ∶ Γ ↠ Γ ′ , the morphism of Proposition 5.2 gives riseto a Γ -equivariant surjective morphism CD Γ e(cid:96)(cid:96) ( k ) → CD Γ ′ e (cid:96)(cid:96) ( k ) which exhibits CD Γ ′ e (cid:96)(cid:96) ( k ) as the quotient ker ( ρ )/ CD Γ e(cid:96)(cid:96) ( k ) . LLIPSITOMIC ASSOCIATORS 43
Example 5.4 (Notable arrows in CD Γ e(cid:96)(cid:96) ( k )( ) ) . In addition to the arrows of ˆ U( ¯ t Γ1 , ( k )) , wealso have, in CD Γ e(cid:96)(cid:96) ( k )( ) , I , e(cid:96)(cid:96) = α α and J , e(cid:96)(cid:96) = β β recalling that α = ( ¯1 , ¯0 ) and β = ( ¯0 , ¯1 ) .Notice that, by definition, we have for γ = ( ¯ p, ¯ q ) ∈ Γ,Ad (( I , e(cid:96)(cid:96) ) p ( J , e(cid:96)(cid:96) ) q )( t ) = t γ , where this identity takes place in CD Γ e(cid:96)(cid:96) ( k )( ) ⋊ ( Γ / Γ ) , and I , e(cid:96)(cid:96) ∶= I , e(cid:96)(cid:96) α and J , e(cid:96)(cid:96) ∶= J , e(cid:96)(cid:96) β are automorphisms of 1 . Notation 5.5.
For later purposes, we also introduce the notation X , e(cid:96)(cid:96) = x ⋅ Id = + and Y , e(cid:96)(cid:96) = y ⋅ Id = − Parenthesized ellipsitomic chord diagrams.
There is a Γ-equivariant morphism ofmodules ω ∶ Pa Γ → Ob ( CD Γ e(cid:96)(cid:96) ( k )) , which forgets the parenthesized permutation (and onlyremembers the labelling), over the terminal operad morphism ω ∶ Pa → ∗ = Ob ( CD ( k )) from § PaCD ( k ) -module PaCD Γ e(cid:96)(cid:96) ( k ) ∶= ω ⋆ CD Γ e(cid:96)(cid:96) ( k ) of parenthesized ellipsitomic chord diagrams , which is still acted on by Γ. Remark 5.6.
As explained in subsection 1.5, there is a map of S -modules PaCD ( k ) —→ PaCD Γ e(cid:96)(cid:96) ( k ) and we keep the same symbol for the image in PaCD Γ e(cid:96)(cid:96) ( k ) an arrows in PaCD ( k ) . X , = ⋅ H , = t ⋅ a , , = ⋅ ( ) ( ) They satisfy the following relations, in End
PaCD Γ e(cid:96)(cid:96) ( k )( )⋊( Γ / Γ ) ( ) : ● ( I , e(cid:96)(cid:96) ) M = Id , ● ( J , e(cid:96)(cid:96) ) N = Id , ● ( I , e(cid:96)(cid:96) , J , e(cid:96)(cid:96) ) = Id , These relations allows to unambiguously define, for every γ = ( ¯ p, ¯ q ) ∈ Γ, a morphism K , γ ∶ → γ by K , γ ∶= K γ γ = ( I , e(cid:96)(cid:96) ) p ( J , e(cid:96)(cid:96) ) q , so that the assignement γ ↦ K γ is multiplicative.We also have the following relations in End PaCD Γ e(cid:96)(cid:96) ( k )( )⋊( Γ / Γ ) (( ) ) :0 = X , e(cid:96)(cid:96) + Ad ( a , , X , ) ( X , e(cid:96)(cid:96) ) + Ad ( X , ( a , , ) − ) ( X , e(cid:96)(cid:96) ) , = Y , e(cid:96)(cid:96) + Ad ( a , , X , ) ( Y , e(cid:96)(cid:96) ) + Ad ( X , ( a , , ) − ) ( Y , e(cid:96)(cid:96) ) , = I , e(cid:96)(cid:96) + Ad ( a , , X , ) ( I , e(cid:96)(cid:96) ) + Ad ( X , ( a , , ) − ) ( I , e(cid:96)(cid:96) ) , = J , e(cid:96)(cid:96) + Ad ( a , , X , ) ( J , e(cid:96)(cid:96) ) + Ad ( X , ( a , , ) − ) ( J , e(cid:96)(cid:96) ) , ∑ γ ∈ Γ Ad ( K , γ )( H , ) = [ Ad ( a , , ) ( X , e(cid:96)(cid:96) ) , Ad ( X , a , , ) ( Y , e(cid:96)(cid:96) )] . Definition 5.7.
The graded ellipsitomic Grothendieck-Teichm¨uller group is defined as
GRT Γ e(cid:96)(cid:96) ( k ) ∶= Aut + OpR
Cat ( CoAlg k ) ( PaCD ( k ) , PaCD Γ e(cid:96)(cid:96) ( k ))) Γ Recall that we have an isomorphismAut + OpR
Cat ( CoAlg k ) ( PaCD ( k ) , PaCD Γ e(cid:96)(cid:96) ( k ))) Γ ≃ Aut + OpR
Grpd k ( G PaCD ( k ) , G PaCD Γ e(cid:96)(cid:96) ( k ))) Γ . For every group surjective morphism ρ ∶ Γ → Γ ′ , and every a, b, c, d ∈ k such that ad − bc = ∣ ker ( ρ )∣ ,using the fact that ker ( ρ )/ PaCD Γ e(cid:96)(cid:96) ( k ) ≃ PaCD Γ ′ e(cid:96)(cid:96) ( k ) , we obtain a group morphism GRT Γ e(cid:96)(cid:96) ( k ) —→ GRT Γ ′ e(cid:96)(cid:96) ( k ) . Ellipsitomic associators.
We now fix Γ ∶= Z / M Z × Z / N Z . Definition 5.8.
The set of ellipsitomic k -associators is Ell Γ ( k ) ∶= Iso + OpR
Grpd k ((̂ PaB ( k ) , ̂ PaB Γ e(cid:96)(cid:96) ( k )) , ( G PaCD ( k ) , G PaCD Γ e(cid:96)(cid:96) ( k ))) Γ . Theorem 5.9.
There is a one-to-one correspondence between elements of
Ell Γ ( k ) and thoseof the set Ell Γ ( k ) consisting of quadruples ( µ, ϕ, A, B ) ∈ Ass ( k ) × exp ( ˆ¯ t Γ1 , ( k )) such that, for A ∶= Aα and B ∶= Bβ , we have the following relations in exp ( ˆ¯ t Γ1 , ( k )) ⋊ ( Γ / Γ ) : (14) 1 = ϕ , , A , e µ ( ¯ t + ¯ t )/ ϕ , , A , e µ ( ¯ t + ¯ t )/ ϕ , , A , e µ ( ¯ t + ¯ t )/ , (15) 1 = ϕ , , B , e µ ( ¯ t + ¯ t )/ ϕ , , B , e µ ( ¯ t + ¯ t )/ ϕ , , B , e µ ( ¯ t + ¯ t )/ , (16) e µ ¯ t = ( ϕ , , A , ϕ , , , e − µ ¯ t / ϕ , , B , ϕ , , e − µ ¯ t / ) . Proof.
Let ( F, G ) be an ellipsitomic associator. We have already seen that the choice of theoperad isomorphism F corresponds bijectively to the choice of an element ( µ, ϕ ) ∈ Ass ( k ) .From the presentation of PaB Γ e(cid:96)(cid:96) , we know that G is uniquely determined by the imagesof A , ∈ Hom
PaB Γ e(cid:96)(cid:96) ( k )( ) ( , α ) and B , ∈ Hom
PaB Γ e(cid:96)(cid:96) ( k )( ) ( , β ) . There areelements A, B ∈ exp ( ˆ¯ t Γ1 , ( k )) such that ● G ( A , ) = A ⋅ I , e(cid:96)(cid:96) ; LLIPSITOMIC ASSOCIATORS 45 ● G ( B , ) = B ⋅ J , e(cid:96)(cid:96) .These elements must satisfy relations (14), (15) and (16), that are images of (tN1), (tN2) and(tE). Conversely, if (14), (15) and (16) are satisfied, then G is well-defined. (cid:3) Remark 5.10.
It follows from the alternative presentation of
PaB Γ e(cid:96)(cid:96) (see Theorem A.3) that Ell Γ ( k ) is also in bijection with the set of ( µ, ϕ, A, B ) ∈ Ass ( k ) × ( exp ( ˆ¯ t Γ1 , ( k ))) × satisfying(17) A , = ϕ , , A , ϕ , , e − µ ¯ t / ϕ , , A , ϕ , , e − µ ¯ t / (18) B , = ϕ , , B , ϕ , , e − µ ¯ t / ϕ , , B , ϕ , , e − µ ¯ t / (19) ϕ , , e µ ¯ t ϕ , , = ( A , ϕ , , ( A , ) − ϕ , , , ( B , ) − ) As before, if we are given a surjective group morphism ρ ∶ Γ ↠ Γ ′ ∶= Z / M ′ Z × Z / N ′ Z , then, choosing a, b, c, d ∈ k as in Remark 5.3, we have a bitorsor morphism ( ̂ GT Γ ( k ) , Ell Γ ( k ) , GRT Γ ( k )) —→ ( ̂ GT Γ ′ ( k ) , Ell Γ ′ ( k ) , GRT Γ ′ ( k )) . In Section 6 we prove that ellipsitomic associators do exist. More precisely, the followingresult is a direct consequence of Theorem 6.1:
Theorem 5.11.
The set
Ell Γ ( C ) is non empty. The KZB ellipsitomic associator
Recall from Theorem 5.9 that the set of ellipsitomic associators can be regarded, eitheras the set of Γ-equivariant ̂ PaB ( k ) -module isomorphisms ̂ PaB Γ e(cid:96)(cid:96) ( k ) —→ G PaCD Γ e(cid:96)(cid:96) ( k ) which are the identity on objects, or as quadruples ( λ, Φ , A + , A − ) , where ( λ, Φ ) ∈ Ass ( k ) and A ± ∈ exp ( ˆ¯ t Γ1 , ( k )) , satisfying relations (14), (15), (16).The following result tells us that the set Ell Γ ( C ) is not empty. We write Ell ΓKZB ∶= Ell Γ ( C ) × Ass ( C ) { π i , Φ KZ } . Theorem 6.1.
There is an analytic map h —→ Ell
ΓKZB τ z→ e Γ ( τ ) = ( A Γ ( τ ) , B Γ ( τ )) . In particular, for each τ ∈ h , the element ( π i , Φ KZ , A Γ ( τ ) , B Γ ( τ )) is an ellipsitomic C -associator (i.e. it belongs to Ell Γ ( C ) ). The rest of this section is devoted to the proof of the above theorem.
The pair e Γ ( τ ) . We adopt the convention for monodromy actions of [13, Appendix A].First of all, recall that ¯ t Γ1 , is the Lie C -algebra generated by x ∶= ¯ x , y ∶= ¯ y and t α ∶= ¯ t α , for α ∈ Γ, such that [ x, y ] = ∑ α ∈ Γ t α . We define the KZB ellipsitomic associator as the couple e Γ ( τ ) ∶= ( A Γ ( τ ) , B Γ ( τ )) ∈ exp ( ˆ¯ t Γ1 , ) × exp ( ˆ¯ t Γ1 , ) consisting in the renormalized holonomies along the straight paths from 0 to 1 / M and from 0to τ / N , respectively, of the differential equation(20) J ′ ( z ) = − ∑ α ∈ Γ e − π i ax θ ( z − ˜ α + ad ( x )∣ τ ) θ ( z − ˜ α ∣ τ ) θ ( ad ( x )∣ τ ) ( t α ) ⋅ J ( z ) , with values in the group exp ( ˆ¯ t Γ1 , ) ⋊ Γ.More precisely, for all α ∈ Γ and ˜ α = ( a , a ) ∈ Λ τ, Γ a lift of α , this equation has a uniquesolution J α ( z ) defined over { ˜ α + s M + s N τ, for s , s ∈] , [} such that we have J α ( z ) ≃ ( z − ˜ α ) e − π i a ad ( x ) t α at z − ˜ α →
0. By denoting J ( z ) ∶= J ( z ) we define A Γ ( τ ) ∶= J ( z ) J ( z + M ) − ( ¯1 , ¯0 ) ∈ exp ( ˆ¯ t Γ1 , ) ⋊ Γ . Then the A -ellipsitomic KZB associator A Γ ( τ ) is the exp ( ˆ¯ t Γ1 , ) -component of A Γ ( τ ) : A Γ ( τ ) ∶= A Γ ( τ )(− ¯1 , ¯0 ) = J ( z ) J ( z + M ) − ∈ exp ( ˆ¯ t Γ1 , ) . In the same way, we define B Γ ( τ ) ∶= J ( z ) J ( z + τN ) − e − π i xN ( ¯0 , ¯1 ) , and the B -ellipsitomic KZB associator B Γ ( τ ) is then its exp ( ˆ¯ t Γ1 , ) -component: B Γ ( τ ) ∶= B Γ ( τ )( ¯0 , − ¯1 ) = J ( z ) J ( z + τN ) − e − π i xN ∈ exp ( ˆ¯ t Γ1 , ) . A particular solution of the ellipsitomic KZB system.
More generally, the ellip-sitomic KZB system is ( ∂ / ∂z i ) F Γ ( z ∣ τ ) = K i ( z ∣ τ ) F Γ ( z ∣ τ ) , ( ∂ / ∂τ ) F Γ ( z ∣ τ ) = ∆ ( z ∣ τ ) F Γ ( z ∣ τ ) , where F Γ ( z ∣ τ ) is a function ( C n × H ) − Diag n, Γ ⊃ U → G Γ n . We refer the reader to [13,Subsection 1.5 Subsection 2.3, Subsection 3.3] for the definitions of K i ( z ∣ τ ) , G Γ n and ∆ ( z ∣ τ ) respectively.Now, let us denote z ij = z i − z j and let us compute the expansions of K i ( z ∣ τ ) and ∆ ( z ∣ τ ) inthe region z ij ≪ τ → i ∞ . Here z ij ≪ z ij is close to zero. We LLIPSITOMIC ASSOCIATORS 47 have K i ( z ∣ τ ) = − y i + ∑ j ; j ≠ i ∑ α ∈ Γ ( e − π i a ad ( x i ) θ ( z i − z j − ˜ α + ad ( x i ) ; τ ) θ ( z i − z j − ˜ α ; τ ) θ ( ad ( x i ) ; τ ) − ( x i ) ) ( t αij )= ∑ j ; j ≠ i ∑ α ∈ Γ ( ( x i ) + t αij z i − z j − ˜ α − ( x i ) ) ( t αij ) + O ( )= ∑ j ; j ≠ i ∑ α ∈ Γ t αij z i − z j − ˜ α + O ( ) = ∑ j ; j ≠ i ∑ α ∈ Γ t αij z i − z j − a M + O ( ) . For the expansion of ∆, recall that, if γ ∈ Γ, we have g γ ( x, z ∣ τ ) ∶= ∂ x k γ ( x, z ∣ τ ) and g − γ ( x, ∣ τ ) = ∑ s ≥ A s,γ ( τ ) x s . We then have∆ ( z ∣ τ ) = − iπ ⎛⎝ ∆ + ∑ s ⩾ ∑ γ ∈ Γ A s,γ ( τ ) ⎛⎝ δ s,γ + ∑ i,j ∶ i < j ad ( x i ) s ( t − γij )⎞⎠⎞⎠ + o ( ) , for z ij ≪ τ ∈ H .In Section 7 we will relate A s,γ ( τ ) to Eisenstein-Hurwitz series which have a q N -expansion,where q N = e π i τ / N and we define the normalized version ˜ A s,γ ( τ ) of the twisted Eisensteinseries A s,γ ( τ ) such that A s,γ ( τ ) = a s,γ ˜ A s,γ ( τ ) , and such that we have an expansion ˜ A s,γ ( τ ) = + ∑ l > a kl,γ e π i lτ / N as τ / N → i ∞ .We now determine a particular solution F ( n ) Γ ( z ∣ τ ) of the ellipsitomic KZB system.Let D Γ n ⊂ ( C n × H ) − Diag n, Γ be defined as {( z , τ ) ∈ C n × H ∣ z i = a i + b i τ, a i , b i ∈ R , a < a < ... < a n < a + M , b n < ... < b < b n + N } , which is simply connected. A solution of the ellipsitomic KZB system on this domain is thenunique, up to right multiplication by a constant.Then, by applying Proposition 3 in Appendix A of [12] with u n = z n , u n − = z n − , / z n ,..., u = z / z , u = q N , we obtain a unique solution F ( n ) Γ ( z ∣ τ ) with the expansion F ( n ) Γ ( z ∣ τ ) ≃ z t z t + t ...z t n + ... + t n − ,n n exp ⎛⎝− τ π i ⎛⎝ ∆ + ∑ s ≥ ,γ ∈ Γ a s,γ ⎛⎝ δ s,γ − ∑ i < j ad s ( x i )( t − γij )⎞⎠⎞⎠⎞⎠ in the region z ≪ z ≪ ... ≪ z n ≪ τ / N → i ∞ , ( z , τ ) ∈ D Γ n . The sign ≃ means here thatany of the ratios of both sides is of the form1 + ∑ k > ∑ i,a ,...,a n r i,a ,...,a n k ( u , ..., u n ) , where the second sum is finite with a i ≥ i ∈ { , ..., n } , r i,a ,...,a n k ( u , ..., u n ) has degree k , andis O ( u i ( log u ) a ... ( log u n ) a n ) . Finally, D Γ n is invariant under C (∑ i δ i ) , the solution F ( n ) Γ is invariant under translationby C (∑ i δ i ) and, by [13, Proposition 3.14], induces a unique solution F ([ n ]) Γ with values in G Γ n ⋊ ( Γ n ⋊ S n ) .6.3. A presentation of B Γ1 ,n . Let us define, for z ∈ D Γ n , B Γ1 ,n ∶= π ( Conf ( E τ, Γ , [ n ] , Γ ) , [ z ]) and recall that B ,n = π ( Conf ( E τ, Γ , [ n ]) , [ z ]) . Now, since the canonical surjective mapConf ( E τ, Γ , [ n ] , Γ ) ↠ Conf ( E τ, Γ , [ n ]) defines a Γ-covering, then B Γ1 ,n = ker ( ρ ) , where ρ ∶ B ,n → Γ sends σ i to = ( ¯0 , ¯0 ) , A i to ( ¯1 , ¯0 ) and B i to ( ¯0 , ¯1 ) . If A Mi (resp. B Ni ) is the class of thepath given by [ , ] ∋ t ↦ z − t ∑ nj = i δ i (resp. [ , ] ∋ t ↦ z − tτ ∑ nj = i δ i ), then it follows fromthe geometric description of B Γ1 ,n that A Mi , B Ni ( i = , . . . , n ) and P αij ∶= X pi Y qi P ij Y − qi X − pi σ αi ∶= X pi Y qi σ i Y − qi + X − pi + (for i < j , 1 ≤ p ≤ M , 1 ≤ q ≤ N and α = ( ¯ p, ¯ q ) ) are generators of B Γ1 ,n .We denote again A Mi , B Ni , σ αi and P αij ( α ∈ Γ , i = , ..., n ) for the projections of these elementsto B Γ1 ,n .6.4. The monodromy morphism γ n ∶ B ,n → G Γ n ⋊ ( Γ n ⋊ S n ) . The monodromy of the flat G Γ n ⋊ ( Γ n ⋊ S n ) -bundle ( P Γ , [ n ] , ∇ Γ , [ n ] ) on M , [ n ] induced by combining [13, Proposition 3.13,Proposition 3.14] provides us with a group morphism µ z , Γ , [ n ] ∶ π (( Γ n ⋊ S n )/M Γ1 ,n ) —→ G Γ n ⋊ ( Γ n ⋊ S n ) , where π (( Γ n ⋊ S n )/M Γ1 ,n ) is the mapping class group (i.e. the orbifold fundamental group)associated to ( Γ n ⋊ S n )/M Γ1 ,n . This actually fits into a morphism of short exact sequences1 (cid:47) (cid:47) PB Γ1 ,n (cid:47) (cid:47) (cid:15) (cid:15) B ,n (cid:47) (cid:47) (cid:15) (cid:15) Γ n ⋊ S n (cid:47) (cid:47) (cid:47) (cid:47) MCG Γ1 ,n (cid:47) (cid:47) (cid:15) (cid:15) π (( Γ n ⋊ S n )/M Γ1 ,n ) (cid:47) (cid:47) (cid:15) (cid:15) Γ n ⋊ S n (cid:47) (cid:47) (cid:47) (cid:47) G Γ n (cid:47) (cid:47) G Γ n ⋊ ( Γ n ⋊ S n ) (cid:47) (cid:47) Γ n ⋊ S n (cid:47) (cid:47) , where MCG Γ1 ,n ∶= π (M Γ1 ,n ) is the mapping class group associated to M Γ1 ,n , the top verticalarrows are injections and the bottom first vertical morphism is the monodromy morphism µ z ,n, Γ ∶ MCG Γ1 ,n —→ G Γ n of associated with the flat G Γ n -bundle ( P n, Γ , ∇ n, Γ ) on M Γ1 ,n .Indeed, this comes from the fact that ∇ Γ , [ n ] is obtained by descent, from ∇ n, Γ and using itsequivariance properties of [13, Proposition 3.13]. We denote˜ γ Γ n ∶ PB Γ1 ,n —→ G Γ n and γ Γ n ∶ B ,n —→ G Γ n ⋊ ( Γ n ⋊ S n ) the corresponding vertical composites. LLIPSITOMIC ASSOCIATORS 49
Let F Γ ( z ∣ τ ) be a solution of the ellipsitomic KZB system defined on D Γ n with values in G Γ n ⋊ Γ n . Let us consider the domains H Γ n ∶= {( z , τ ) ∈ C n × H ∣ z i = a i + b i τ, a i , b n < ... < b < b n + N } and V Γ n ∶= {( z , τ ) ∈ C n × H ∣ z i = a i + b i τ, a i , b i ∈ R , a < a < ... < a n < a + M } . Both of these domains are simply connected and invariant. We denote F H Γ ( z ∣ τ ) and F V Γ ( z ∣ τ ) the prolongations of F Γ ( z ∣ τ ) to these domains.Then ( z , τ ) z→ F H Γ ⎛⎝ z − n ∑ j = i δ j M ∣ τ ⎞⎠( z , τ ) z→ e − π i N ( x i + ... + x n ) F V Γ ⎛⎝ z − τ ( n ∑ j = i δ j N )∣ τ ⎞⎠ are solutions of the ellipsitomic KZB system on H Γ n and V Γ n respectively. Let us define A Fi , B Fi ∈ G Γ n ⋊ Γ n by F H Γ ( z ) = A Fi ( ¯1 , ¯0 ) i,...,n F H Γ ⎛⎝ z − n ∑ j = i δ j M ∣ τ ⎞⎠ F V Γ ( z ) = B Fi ( ¯0 , ¯1 ) i,...,n e − π i N ( x i + ... + x n ) F V Γ ⎛⎝ z − τ ( n ∑ j = i δ j N )∣ τ ⎞⎠ . We define A Fi , B Fi to be the ∈ G Γ n -components of A Fi , B Fi . Namely A Fi ∶= A Fi ( ¯1 , ¯0 ) i,...,n B Fi ∶= B Fi ( ¯0 , ¯1 ) i,...,n . We also define σ Fi ∈ S n by means of F Γ ( z ∣ τ ) = σ Fi σ − i F Γ ( σ i ⋅ z ∣ τ ) , where, on the left hand side, F Γ is extended to the universal cover of ( C n × h ) − Diag n, Γ . Noticethat σ i exchanges z i and z i + , z i + passing to the right of z i . Its monodromy is given by e π i t i ( i + ) . We also define ˜ σ i as the path exchanging z i and z i + , z i + passing to the left of z i .Its monodromy is given by e − π i t i ( i + ) .Let us denote X pi ∶= A pi + ( A pi ) − and Y qi ∶= B qi + ( B qi ) − , and recall that α i ⋅ t ij = t αij . Lemma 6.2.
The morphism ˜ γ n ∶ PB Γ1 ,n —→ G Γ n induced by the solution F Γ takes A Mi to ( A Fi ) M , B Ni to ( B Fi ) N . Let us denote, for all α = ( ¯ p, ¯ q ) ∈ Γ , ˜ X pi ∶= ˜ γ n ( X pi ) and ˜ Y qi ∶= ˜ γ n ( X qi ) .Then P αij is sent via ˜ γ n to ˜ P αij = g ( ¯ p, ¯ q ) i e π i t ij ( ¯ − p, ¯ − q ) i g − and σ αi is sent via ˜ γ n to ˜ R αi = g ( ¯ p, ¯ q ) i e π i t i,i + ( ¯ − p, ¯ − q ) i + g − . Proof.
This follows from the geometric description of the generators of B , [ n ] : if ( z , τ ) ∈ D Γ n ,then A i is the class of the projection of the path [ , ] ∋ t ↦ ( z − t ∑ nj = i ( δ j / M ) , τ ) and B i isthe class of the projection of [ , ] ∋ t ↦ ( z − tτ ∑ nj = i ( δ j / N ) , τ ) . Finally, as paths in H Γ n , A M and A ( M ) are homotopic. Likewise, as paths in V Γ n , B N and B ( N ) are homotopic. (cid:3) We will denote by ˜ γ n ∶ PB Γ1 ,n —→ G Γ n the morphism induced by the solution F ( n ) Γ ( z ∣ τ ) and γ n ∶ B ,n → G Γ n ⋊ ( Γ n ⋊ S n ) the one induced by F ([ n ]) Γ .6.5. Expression of γ n ∶ B ,n → G Γ n ⋊ ( Γ n ⋊ S n ) using γ and γ .Lemma 6.3. ˜ γ ( A M ) , ˜ γ ( B N ) and ˜ γ ( P α ) belong to exp ( ˆ t Γ1 , ) ⊂ G Γ2 .Proof. If F Γ ( z ∣ τ ) ∶ H Γ2 → G Γ2 is a solution of the ellipsitomic KZB equation for n =
2, then A F = F H Γ ( z ∣ τ ) F H Γ ( z − δ ∣ τ ) − is the iterated integral, from z ∈ D Γ n to z − δ , of K ( z ∣ τ ) ∈ ˆ t Γ1 , .Thus, A F ∈ exp ( ˆ t Γ1 , ) . Then, as ˜ γ ( A M ) is a conjugate of ( A F ) M , it belongs to exp ( ˆ t Γ1 , ) asexp ( ˆ t Γ1 , ) ⊂ G Γ2 ⋊ S is normal. One proves in the same way that ˜ γ ( B N ) ∈ exp ( ˆ t Γ1 , ) . Finally,as ˜ γ ( P α ) is a conjugate of ˜ P α and thus also belongs to exp ( ˆ t Γ1 , ) . (cid:3) Let Φ , , be the image in exp ( ˆ t Γ1 , ) of the KZ associator Φ , , induced by t ij ↦ t ij anddenote Φ i ∶= Φ ...i − ,i,i + ...n ... Φ ...n − ,n − ,n ∈ exp ( ˆ t Γ1 ,n ) . Proposition 6.4. γ n ( A i )( ¯1 , ¯0 ) i...n = Φ i γ ( A ) ...i − ,i...n ( ¯1 , ¯0 ) i...n ⋅ Φ − i ,γ n ( B i )( ¯0 , ¯1 ) i...n = Φ i γ ( B ) ...i − ,i...n ( ¯0 , ¯1 ) i...n ⋅ Φ − i , ( i = , ..., n ) , and γ n ( σ αi ) = g ( ¯ p, ¯ q ) i e π i t i,i + ( ¯ − p, ¯ − q ) i + g − where i = , ..., n − , and α = ( ¯ p, ¯ q ) .Proof. Let G Γ i ( z ∣ τ ) be the solution of the elliptic Γ-KZB system, such that G Γ i ( z ∣ τ ) = z t ...z t + ... + t ,i − i − , z t i,n + ... + t n − ,n n,i ...z t n − ,n n,n − × exp ⎛⎝− τ π i ⎛⎝ ∆ + ∑ s ≥ ,γ ∈ Γ a s,γ ⎛⎝ δ s,γ − ∑ i < j ad s ( x i )( t − γij )⎞⎠⎞⎠⎞⎠ , when z ≪ ... ≪ z i − , ≪ z n,n − ≪ ... ≪ z n,i ≪ τ / N → i ∞ and ( z , τ ) ∈ D Γ n (we set z ij = z i − z j as before). Then G Γ i ( z ∣ τ ) = γ ( A ) ...i − ,i...n ( ¯1 , ¯0 ) i...n G Γ i ( z − n ∑ j = i δ j ∣ τ ) , because in the domain considered K i ( z ∣ τ ) is close to K ( z , z n ∣ τ ) ...i − ,i...n (where K ( z , z n ∣ τ ) corresponds to the 2-point system). Next, we have F Γ ( z ∣ τ ) = Φ i G Γ i ( z ∣ τ ) , which implies theformula for γ n ( A i ) . The formula for γ n ( B i ) is proved in the same way. Finally, the behaviorof F ( n ) Γ ( z ∣ τ ) for z ≪ ... ≪ z n ≪ LLIPSITOMIC ASSOCIATORS 51 know that the twisted elliptic KZB connection is Γ-equivariant. This implies the formula for γ n ( σ αi ) . (cid:3) Algebraic relations for the ellipsitomic KZB associator.
Let us now finish theproof of Theorem 6.1.
Remark 6.5.
We let the reader check that the results of subsections 6.2, 6.4 and 6.5 remaintrue in the reduced case and we will make use of the same notation as in the previoussubsections.In what follows, we will denote ● ¯˜ γ n ∶ PB ,n → ¯ G Γ n ⋊ Γ n / Γ the reduced monodromy morphism corresponding to theunique solution with values on ¯ G Γ n ⋊ Γ n / Γ induced by F ( n ) Γ ; ● ¯ γ n ∶ B ,n → ¯ G Γ n ⋊ ( Γ n / Γ ⋊ S n ) the reduced monodromy morphism corresponding to theunique solution with values on ¯ G Γ n ⋊ ( Γ n / Γ ⋊ S n ) induced by F ([ n ]) Γ .Let us set ˜ A ∶= ¯˜ γ ( A ) = ¯ γ ( A ) , ˜ B ∶= ¯˜ γ ( B ) = ¯ γ ( B ) . If we denote ( ˜ A, ˜ B ) the exp ( ˆ¯ t Γ1 , ) -components of ( ˜ A, ˜ B ) , then it is a fact that A Γ ( τ ) = ˜ A and B Γ ( τ ) = ˜ B .The image of the relation A − A = σ A − σ by ¯ γ yields(21) ˜ A , = Φ , , ˜ A , ( Φ , , ) − e − π i ¯ t Φ , , ˜ A , ( Φ , , ) − e − π i ¯ t . In the same way, the image of the relation B − B = σ B − σ by ¯ γ yields(22) ˜ B , = Φ , , ˜ B , ( Φ , , ) − e − π i ¯ t Φ , , ˜ B , ( Φ , , ) − e − π i ¯ t . Accordingly, the image by ¯ γ of the relation ( A A − , B − ) = P in B , then givesΦ e π i ¯ t Φ − = ˜ A , Φ ( ˜ A , ) − Φ − ( ˜ B , ) − Φ ˜ A , Φ − ( ˜ A , ) − ˜ B , . (23)One can simply draw the r.h.s. of the first part of this double equation as follows: wesimplify the paths by just neglecting the associators and we suppose that the central portionof the torus corresponds to the -labelled region with respect to the sublattice Λ τ, Γ , where = ( ¯0 , ¯0 ) . Then we enumerate the different movements (read from left to right in the l.h.s of the equation) of the marked points in the twisted configuration space: z ; z z z z ; z ; We can see that the z is only braided with z since the ˜ A and ˜ B moves don’t change thelabelling of the point that moves.This shows that the couple ( ˜ A, ˜ B ) satisfies (17), (18) and (19) so that, by Remark 5.10, thecouple e Γ ( τ ) = ( A Γ ( τ ) , B Γ ( τ )) satisfy (14), (15) and (16).This concludes the proof of Theorem 6.1. (cid:50) Remark 6.6.
If Γ is trivial, we retrieve relations (22), (23), (25) and (26) from [12], up to achange of convention for the monodromy action and the configuration of the marked points.The modularity relations of e Γ ( τ ) , depending on the chosen congruence subgroup of SL ( Z ) ,will be investigated in forthcoming works by the second author.7. Number theoretic aspects: Eisenstein series
For any γ ∈ Γ, and any lift ˜ γ = ( c , c ) ∈ Λ τ, Γ − Λ τ of γ , recall that g γ ( x, z ∣ τ ) ∶= ∂ x k γ ( x, z ∣ τ ) ,where k γ ( x, z ∣ τ ) ∶= e − π i cx θ ( z − ˜ γ + x ∣ τ ) θ ( z − ˜ γ ∣ τ ) θ ( x ∣ τ ) − x . Until now, the terms A s,γ ( τ ) were determined as the coefficients of the expansion g − γ ( x, ∣ τ ) = ∑ s ≥ A s,γ ( τ ) x s . In this section we give an explicit formula of these functions, we show that they are (quasi-)modular forms for the group SL Γ2 ( Z ) and relate them to cyclotomic zeta values. We alsodetermine their normalized variant ˜ A s,γ ( τ ) with constant term equal to 1 on its q N -expansionthat we used to apply [12, Proposition A.3] at the end of Subsection 6.2. LLIPSITOMIC ASSOCIATORS 53
Recall that the Weierstrass function is the function ℘ ∶ C × h —→ C given by ℘( z, τ ) = z + ∑ ( m,n )∈ Z −{( , )} ( ( z + m + nτ ) − ( m + nτ ) ) . In the variable z , this function is even, periodic with respect to the latice Z ⊕ τ Z andmeromorphic with poles of order exactly 2 in ( Z ⊕ τ Z ) .We have the following identities for z ∈ C − ( Z ⊕ τ Z ) : ℘( z, τ ) = − ( θ ′ θ ) ′ ( z, τ ) + c = − ∂ z ( log ( θ ( z, τ ))) + c, for a constant c ∈ C . Next, for z in a suitable punctured neighborhood of z = ℘( z, τ ) = z + ∞ ∑ k = b k z k = z + ∞ ∑ k = ( k + ) G k + ( τ ) z k , where b n = f ( n ) ( )( n ) ! with f ( z ) = ℘( z ) − z . Here G k ( τ ) are the Eisenstein series defined for all k ≥
1, by G k ( τ ) ∶= ∞ ∑ n =−∞ ⎛⎜⎝ ∞ ∑ m =−∞ m ≠ n = ( m + nτ ) k ⎞⎟⎠ = ζ ( k ) + ⋅ ( π i ) k ( k − ) ! ∞ ∑ m = σ k − ( m ) q m , where σ α ( k ) = ∑ d ∣ k d α . We have G k ( τ ) = k is odd. We will also use the normalizedEisenstein series E k ( τ ) , defined for k ≥ E k ∶= G k ( τ ) ζ ( k ) so that, for n ≥
1, we have ( n + ) G n + ( τ ) = ˜ a n E n + ( τ ) , where ˜ a n = −( n + )B n + ( π ) n + /( n + ) !and B n are the Bernoulli numbers given by x /( e x − ) = ∑ r ≥ (B r / r ! ) x r . In particular, theconstant term in the q -expansion of the series E n is equal to 1.Finally, also recall the expansion θ ( x, τ ) = x + π i ∂ τ log η ( τ ) x + O ( x ) .7.1. Eisenstein series for SL Γ2 ( Z ) . First of all, set γ = . We get, as in [12, Section 4.1], g ( x, ∣ τ ) = ( θ ′ / θ ) ′ ( x ) + / x + c = − ∑ k ≥ ˜ a k E k + ( τ ) x k , where ˜ a = π / E ( τ ) = π i ∂ τ log η ( τ ) .Now, let γ ∈ Γ − { } and let ˜ γ = ( c , c ) ∈ Λ τ, Γ − Λ τ be any lift of γ . By using the identity ∂ x f ( x, τ ) = ∂ x ( log ( f ( x, τ ))) × f ( x, τ ) , we get g γ ( x, z ∣ τ ) = ∂ x k γ ( x, z ∣ τ )= ( π i c + ( θ ′ θ ) ( z + x − ˜ γ ∣ τ ) − ( θ ′ θ ) ( x ∣ τ )) e π i cx θ ( z − ˜ γ + x ∣ τ ) θ ( z − ˜ γ ∣ τ ) θ ( x ∣ τ ) + x = ∞ ∑ n = g ( n ) γ ( x, ∣ τ ) n ! z n . Let us determine the coefficients of g − γ ( x, ∣ τ ) = ∑ s ≥ A s,γ ( τ ) x s = ∑ s ≥ g ( s )− γ ( , ∣ τ ) s ! x s .Set F γ ( x ∣ τ ) ∶= e π i cx θ ( ˜ γ + x ∣ τ ) θ ( ˜ γ ∣ τ ) θ ( x ∣ τ ) so that(24) log ( F γ ( x ∣ τ )) = log ( θ ( ˜ γ + x ∣ τ )) − log ( θ ( x ∣ τ )) + π i cx − log ( θ ( ˜ γ ∣ τ )) . We have ∂ x ( log ( F γ ( x ∣ τ ))) = ∂ x ( log ( θ ( ˜ γ + x ∣ τ ))) − ∂ x ( log ( θ ( x ∣ τ )))= ℘( x, τ ) − ℘( ˜ γ + x, τ )= x − ( x + ˜ γ ) + ∑ ( m,n )∈ Z −{( , )} ( ( x + m + nτ ) − ( x + ˜ γ + m + nτ ) )= x + ∑ ( m,n )∈ Z −{( , )} ( x + m + nτ ) − ∑ ( m,n )∈ Z ( x + ˜ γ + m + nτ ) . Now let s ⩾
0. Recall the expansion 1 ( x + y ) = ∑ s ⩾ a s x s y s + , where a s is the generalized binomial coefficient ( − s ) = (− ) s ( + s − s ) = (− ) s ( s + ) . On the one hand, for y = m + nτ , we have H ( x, τ ) ∶= ∑ ( m,n )∈ Z −{( , )} ( ( x + m + nτ ) − ( m + nτ ) )= ∑ s ⩾ ( s + ) G s + ( τ ) x s . LLIPSITOMIC ASSOCIATORS 55
On the other hand, for y = m + nτ + ˜ γ , we obtain H γ ( x, τ ) ∶= ∑ ( m,n )∈ Z ( ( x + ˜ γ + m + nτ ) − ( ˜ γ + m + nτ ) )= ∑ ( m,n )∈ Z ∑ s ⩾ a s x s ( ˜ γ + m + nτ ) s + = ∑ s ⩾ ∑ ( m,n )∈ Z a s x s ( ˜ γ + m + nτ ) s + = ∑ s ⩾ (− ) s ( s + ) G s + ,γ ( τ ) x s , where, for s ≥
3, we define G s,γ ( τ ) = ∑ ( m,n )∈ Z ( m + nτ + ˜ γ ) s . Then, for s ≥
3, we write B s,γ ( τ ) = G s ( τ ) − G s,γ ( τ ) and we have H ( x, τ ) − H γ ( x, τ ) = ∑ s ⩾ (− ) s ( s + ) B s + ,γ ( τ ) x s . and we write ¯ A s,γ ( τ ) = G s ( τ ) + G s,γ ( τ ) = − B s,γ ( τ ) , as H ( x, τ ) and G s ( τ ) are even. If Γ isthe trivial group, ¯ A s,γ ( τ ) reduces to twice the Eisenstein series G s ( τ ) .Notice that G s,γ ( τ ) is not even for the variable x but is even for the variable x + γ i.e. it isinvariant under the transformation x + ˜ γ ↦ − x − ˜ γ . Thus, we obtain G s,γ = (− ) s G s, − γ , whichimplies that ¯ A s,γ = (− ) s ¯ A s, − γ .Now, we define G ,γ ( τ ) by G ,γ ( τ ) = ∞ ∑ n =−∞ ( ∞ ∑ m =−∞ ( ˜ γ + m + nτ ) ) and ¯ A ,γ ( τ ) ∶= G ( τ ) + G ,γ ( τ ) .In conclusion, we obtain ∂ x ( log ( F γ ( x ∣ τ ))) = x + ∑ s ⩾ (− ) s + ( s + ) ¯ A s + ,γ ( τ ) x s , which gives log ( F γ ( x ∣ τ )) = − log ( x ∣ τ ) + ∑ s ⩾ (− ) s + ¯ A s + ,γ ( τ ) s + x s + + lx + m. Thus, following the identification of the above formula with equation (24), we obtain lx + m = π i cx − log ( θ ( ˜ γ ∣ τ )) and F γ ( x ∣ τ ) = x exp (∑ s ⩾ (− ) s + ¯ A s + ,γ ( τ ) s + x s + + π i cx − log ( θ ( ˜ γ ∣ τ )))= xθ ( ˜ γ ∣ τ ) e π i cx exp (∑ s ⩾ (− ) s + ¯ A s + ,γ ( τ ) s + x s + ) . We conclude that g − γ ( x, ∣ τ ) = ∂ x ( F γ ( x ∣ τ )) + x = x − e π i cx x θ ( ˜ γ ) ( x ( π i c + ∑ s ⩾ (− ) s + ¯ A s + ,γ ( τ ) x s + ) − ) exp (∑ s ⩾ (− ) s + ¯ A s + ,γ ( τ ) s + x s + ) . We have g − γ ( x, ∣ τ ) = lim z → (( π i c + ( θ ′ θ ) ( z + x + ˜ γ ) − ( θ ′ θ ) ( x )) e π i cx θ ( z + ˜ γ + x ) θ ( z + ˜ γ ) θ ( x ) ) + x = ( π i c + ( θ ′ θ ) ( x + ˜ γ ) − ( θ ′ θ ) ( x )) e π i cx θ ( ˜ γ + x ) θ ( ˜ γ ) θ ( x ) + x = ( π i c + ( θ ′ θ ) ( ˜ γ ) − x )( + π i cx + π i cx )( x + ( θ ′ θ ) ( ˜ γ )) + x + o ( x )= ( π i c ) − ( θ ′ θ ) ( ˜ γ ) − π i c ( θ ′ θ ) ( ˜ γ ) − π i c + o ( x ) , thus we also define A ,γ ( τ ) ∶= ( π i c ) − ( θ ′ θ ) ( ˜ γ ) − π i c ( θ ′ θ ) ( ˜ γ ) − π i c. Modularity of the Eisenstein-Hurwitz series A s,γ . Consider, for s ≥
2, the function G s,γ ( τ ) ∶= ∞ ∑ n =−∞ ( ∞ ∑ m =−∞ ( m + nτ + ˜ γ ) k ) and denote as above ¯ A s,γ ( τ ) = G s ( τ ) + G s,γ ( τ ) . Proposition 7.1.
Let s ≥ . The function ¯ A s,γ is a modular form of weight s for SL Γ2 ( Z ) .Proof. The proof of this fact is standard but we reproduce it here for sake of completeness.First, we will show the modular quasi-invariance. Then we will show holomorphy at thecusps by characterising holomorphy in terms of a q N -expansion, where q N = e π i τ / N (see [15,Definition 1.2.3]). For s ≥
3, the series ¯ A s,γ ( τ ) converge normally.Let us first show that, if α = ( a bc d ) ∈ SL Γ2 ( Z ) , then ¯ A s,γ ( α ⋅ τ ) = ( cτ + d ) s ¯ A s,γ ( τ ) . Wealready know that the Eisenstein series G s ( τ ) are modular forms of weight s , for s ≥ G ( τ ) =
0. We have G s,γ ( τ ) = ∑ ( m,n )∈ Z ( m + uM + ( n + vN ) τ ) s . Thus, G s,γ ( α ⋅ τ ) = ∑ ( m,n )∈ Z ( m + uM + ( n + vN ) aτ + bcτ + d ) s LLIPSITOMIC ASSOCIATORS 57 = ( cτ + d ) s ∑ ( m,n )∈ Z ( md + nb + ( mc + na ) τ + uM d + vN b + ( uM c + vN a ) τ ) s = ( cτ + d ) s ∑ ( m,n )∈ Z ( m + nτ + ˜ γ ′ ) s , for some lift ˜ γ ′ of γ ∈ Γ. The last line holds by the fact that, since a ≡ M , d ≡ N , b ≡ N and c ≡ M , we have uM d ∈ Z M , vN b ∈ Z , uM c ∈ Z and vN a ∈ Z N . Then we canrewrite the term md + nb + ( mc + na ) τ as m + nτ by applying ( n m ) z→ ( n m ) ( a bc d ) , and we can rewrite the term uM d + vN b + ( uM c + vN a ) τ as m + nτ + ˜ γ ′ by applying ( vN uM ) z→ ( vN uM ) ( a bc d ) , where ( a bc d ) is invertible. Finally, as we already know that ¯ A s,γ does not depend on thechoice of the lift ˜ γ of γ , we obtain G s,γ ( α ⋅ τ ) = ( cτ + d ) s G s,γ ( τ ) . As the function ¯ A s,γ isholomorphic on h , it remains to show that it is also holomorphic at all cusps of the compactifiedmodular curve X ( Γ ) . Let’s start with the cusp at i ∞ .Recall that the Hurwitz zeta function is defined by ζ ( s, z ) ∶= ∑ m ≥ ( m + z ) s , where s, q ∈ C are such that Re ( s ) > ( q ) > Lemma 7.2.
The function G s,γ ( τ ) admits a q N -expansion, where q N = e π i τ / N .Proof. We have G s,γ ( τ ) = ∑ m ∈ Z ( m + ˜ γ ) s + ∑ n ∈ Z ∑ m ∈ Z ( m + nτ + ˜ γ ) s = ∑ m ∈ Z ( m + ˜ γ ) s + ∑ n ∈ Z ∑ m ∈ Z ( m + uM + ( n + vN ) τ ) s = ∑ m ∈ Z ( m + ˜ γ ) s + (− iπ ) s ( s − ) ! ∑ n ∈ Z ∑ r ≥ r s − e π i r ( uM + τ ( n + vN )) = ∑ m ∈ Z ( m + ˜ γ ) s + (− π i ) s ( s − ) ! ∑ n ∈ Z ∑ r ≥ r s − e π i ruM q ( Nn + v ) rN = ∑ m ≥ ( m + ˜ γ ) s + (− ) s ∑ m ≥ ( m − ˜ γ ) s + (− π i ) s ( s − ) ! ∑ n ≥ ∑ r ≥ r s − e π i ruM q ( Nn + v ) rN + ( π i ) s ( s − ) ! ∑ n ≥ ∑ r ≥ r s − e − π i ruM q ( Nn − v ) rN = ζ ( s, γ ) + (− ) s ζ ( s, − γ )+ (− π i ) s ( s − ) ! ∑ n ≥ ∑ r ≥ r s − e π i ruM q ( Nn + v ) rN + ( π i ) s ( s − ) ! ∑ n ≥ ∑ r ≥ r s − e − π i ruM q ( Nn − v ) rN , where ζ ( s, γ ) is the Hurwitz zeta function evaluated at ( s, γ ) . (cid:3) This shows that G s,γ ( τ ) is N -periodic and is holomorphic at i ∞ . Thus, we define, for γ = u / M , a s,γ = ζ ( s, γ ) + (− ) s ζ ( s, − γ ) to be the constant term in this expansion (it also depends on τ but logarithmically). In otherwords, G s,γ ( τ ) has constant term equal to a s,γ if γ = u / M and 0 else.The term a s,γ tends to 0 when τ → i ∞ .We now show that this function is also holomorphic at the remaining cusps of the modularcurve X ( Γ ) . Lemma 7.3.
For all α ∈ SL ( Z ) , the function τ ↦ ( cτ + d ) − s G s,γ ( α ⋅ τ ) has a q N -expansion.Proof. We have ( cτ + d ) − s G s,γ ( α ⋅ τ ) = ∑ ( m,n )∈ Z ( md + nb + ( mc + na ) τ + uM d + vN b + ( uM c + vN a ) τ ) s = ∑ ( m,n )∈ Z (( m + uM ) d + ( vN + n ) b + ( mc + na + uM c + vN a ) τ ) s = ∑ ( m,n )∈ Z ( md + uM d + ( vN + n ) b + ( mc + na + uM c + vN a ) τ ) s = ∑ ( m,n )∈ Z ( d ( m + d ( uM d + ( vN + n ) b + ( mc + na + uM c + vN a ) τ ))) s = d s ∑ ( m,n )∈ Z ( m + d ( uM d + ( vN + n ) b + ( mc + na + uM c + vN a ) τ )) s . By denoting z = d ( uM d + ( vN + n ) b + ( mc + na + uM c + vN a ) τ ) , we have ( cτ + d ) − s G s,γ ( α ⋅ τ ) = d s ∑ ( m,n )∈ Z ( m + z ) s = d s (− π i ) s ( s − ) ! ∑ n ∈ Z ∑ r ≥ r s − e π i rz = d s (− π i ) s ( s − ) ! ∑ n ∈ Z ∑ r ≥ r s − e π i r d ( uM d +( vN + n ) b +( mc + na + uM c + vN a ) τ ) = d s (− π i ) s ( s − ) ! ∑ n ∈ Z ∑ r ≥ r s − e π i r ( m + nbd + uM + vbNd ) e π i rτ ( mc + nad + ucMd + vaNd ) = d s (− π i ) s ( s − ) ! ∑ n ∈ Z ∑ r ≥ r s − e π i r ( m + nbd + uM + vbNd ) e π i rτN ( N ( mc + nad + ucMd )+ vad ) = d s (− π i ) s ( s − ) ! ∑ n ∈ Z ∑ r ≥ r s − e π i r ( m + nbd + uM + vbNd ) q ( N ( mc + nad + ucMd )+ vad ) rN , which concludes the proof. (cid:3) LLIPSITOMIC ASSOCIATORS 59
We conclude that, for all α ∈ SL Γ2 ( Z ) , the function τ ↦ ( cτ + d ) − s ¯ A s,γ ( α ⋅ τ ) is holomorphic at i ∞ , which concludes the proof. (cid:3) Remark 7.4.
From the expression of the function ( cτ + d ) − s ¯ A s,γ ( α ⋅ τ ) , we can notice that ourfunctions ¯ A s,γ will degenerate at all cusps of X ( Γ ) to functions closely related to cyclotomiczeta values. More precisely, the function ∑ γ ∈ Γ −{ } ¯ A s,γ ( τ ) has a q N -expansion such that, if τ → i ∞ , its remaining non zero component is ∑ ≤ u ≤ M − ( ζ ( s, uM ) + (− ) s ζ ( s, − uM )) . Differential equations in τ . Recall that ξ s,γ is the image of δ s,γ in Der ( ¯ t Γ1 , ) . Definethe derivation ε s,γ ∶= ξ s,γ + [( ad x ) s t − γ + (− ad x ) s t γ , −] . As an application of the above subsection, one can retrieve the following
Conjecture 7.5.
We have2 π i ∂∂τ e Γ ( τ ) = e Γ ( τ ) ∗ ⎛⎝− ∆ − ∑ γ ∈ Γ ∑ s ⩾ A s,γ ( τ ) ε s,γ ⎞⎠ . To fully prove this conjecture we need to study the twisted elliptic Grothendieck-Teichm¨ullergroup and a twisted version of the special derivation algebra defined independently by Tsunogaiand Pollack. This will be the subject of subsequent works by the second author.7.4.
Elliptic multiple zeta values at torsion points.
The ellipsitomic KZB associator e Γ ( τ ) ∶= ( A Γ ( τ ) , B Γ ( τ )) has an expression in terms of iterated integrals. Let us denote K Γ ( z ) ∶= − ∑ α ∈ Γ e − π i ax θ ( z − ˜ α + ad ( x )∣ τ ) θ ( z − ˜ α ∣ τ ) θ ( ad ( x )∣ τ ) ( t α ) . By Picard iteration and well-known properties of iterated integrals, we can define I Γ ( τ ) = ( lim t → z − t (( ¯1 , ¯0 ) exp [∫ α ( − tM ) t K Γ ( z ) dz ]) z t ) op and 2 iπJ Γ ( τ ) = ( lim t → z − t (( ¯0 , ¯1 ) exp [∫ β ( τ − tN ) t K Γ ( z ) dz ]) z t ) op where the superscript op denotes the opposite multiplication on the algebra C ⟨⟨ x, t α ; α ∈ Γ ⟩⟩ ,defined by ( f ⋅ g ) op = g ⋅ f . Here we choose the principal branch of the logarithm so that log (± i ) = ± π i / Definition 7.6.
Let n , . . . , n r ⩾ α , α , . . . , α r ∈ Γ. The twisted elliptic multizeta values I Γ A ( n n , . . . , n r α α , . . . , α r ; τ ) and I Γ B ( n n , . . . , n r α α , . . . , α r ; τ ) are defined equivalently: (1) as the coefficients of ad n ( x )( t α ) . . . ad n r ( x )( t α r ) in the renormalized generating seriesof regularized iterated integralslim t → z − t ( ¯1 , ¯0 ) exp [∫ α ( − tM ) t K Γ ( z ) dz ] z t and lim t → z − t ( ¯0 , ¯1 ) exp [∫ β ( τ − tN ) t K Γ ( z ) dz ] z t (2) by means of two functions A Γ ( τ ) and B Γ ( τ ) , closely related to A ( τ ) and B ( τ ) , of theform A Γ ( τ ) = ∑ r ⩾ (− ) r ∑ n ,...,n r ⩾ ∑ α ,...,α r ∈ Γ I Γ A ( n n , . . . , n r α α , . . . , α r ; τ ) ad n ( x )( t α ) . . . ad n r ( x )( t α r ) and B Γ ( τ ) = ∑ r ⩾ (− ) r ∑ n ,...,n r ⩾ ∑ α ,...,α r ∈ Γ I Γ B ( n n , . . . , n r α α , . . . , α r ; τ ) ad n ( x )( t α ) . . . ad n r ( x )( t α r ) Our approach to multiple zeta values at torsion points is somewhat different to that in therecent work of Broedel–Matthes–Richter–Schlotterer [8], and generalizes to the case of anysurjective morphism Z —→ Γ sending the generators of Z to their class modulo M and N ,respectively. More general surjective morphisms could be considered. The relation between thetwisted elliptic multiple zeta values obtained in this paper and that in [8] will be investigatedby the second author and N. Matthes in a forthcoming collaboration. Appendix A. An alternative presentation for
PaB Γ e(cid:96)(cid:96) In this appendix, we provide an alternative presentation for
PaB Γ e(cid:96)(cid:96) , that we use in Section6 when proving that the monodromy of the ellipsitomic KZB connection indeed gives rise toan ellipsitomic associator.A.1. An alternative presentation for PaB e(cid:96)(cid:96) . We first state and prove the result for thetrivial group.The relations (N1bis) and (N2bis). We introduce three new relations, which are satisfied inthe automorphism group of the object ( ) PaB e(cid:96)(cid:96) (this can be seen topologically): A , = Φ , , A , ( Φ , , ) − ˜ R , Φ , , A , ( Φ , , ) − ˜ R , , (N1bis) B , = Φ , , B , ( Φ , , ) − ˜ R , Φ , , B , ( Φ , , ) − ˜ R , , (N2bis) Φ , , R , R , ( Φ , , ) − = ( A , Φ , , ( A , ) − ( Φ , , ) − , ( B , ) − ) . (Ebis)For instance, equations (N1bis) and (N2bis) can be depicted as LLIPSITOMIC ASSOCIATORS 61 ( ( ) ) ± = ( ( ) ) ±± (N1bis,N2bis)The statement. Theorem A.1.
As a
PaB -module in groupoids having Pa as Pa -module of objects, PaB e(cid:96)(cid:96) is freely generated by A ∶= A , and B ∶= B , , together with the relations (N1bis) , (N2bis) , and (Ebis) . The above theorem is a direct consequence of Theorem 3.3 together with the following
Proposition A.2.
Let us consider a
PaB -module in groupoids
PaM , having Pa as Pa -moduleof objects, and let A, B be a automorphisms of . Then (i) Equations (N1) and (N1bis) are equivalent; (ii)
Equations (N2) and (N2bis) are equivalent; (iii) If (N1) and (N2) are satisfied, then equations (E) and (Ebis) are equivalent. A useful observation. Both (N1) and (N1bis) imply A , ˜ R , A , ˜ R , = Id . For both, this is obtained by applying (−) , , ∅ . Similarly, both (N1) and (N1bis) imply B , ˜ R , B , ˜ R , = Id . Proof of (i) and (ii) in Proposition A.2. The following calculation takes place in
PaM ( ) . Forease of comprehension, we put a brace under a sequence of symbols where we use a relation inorder to pass to the next step.Φ , , A , ˜ R , Φ , , ·„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„¶ A , ˜ R , Φ , , A , ˜ R , = Φ , , A , ( Φ , , ) − ˜ R , Φ , , ˜ R , A , ·„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„¶ ˜ R , Φ , , A , ˜ R , = Φ , , A , ( Φ , , ) − ˜ R , Φ , , A , ˜ R , ˜ R , Φ , , ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ A , ˜ R , = Φ , , A , ( Φ , , ) − ˜ R , Φ , , A , ( Φ , , ) − ˜ R , ˜ R , A , ˜ R , ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶= Φ , , A , ( Φ , , ) − ˜ R , Φ , , A , ( Φ , , ) − ˜ R , ( A , ) −
12 DAMIEN CALAQUE AND MARTIN GONZALEZ
We repeatedly used (various forms of) the hexagon equation, and only at the last step we usedthe useful observation from the previous paragraph. This gives that (N1) and (N1bis) are botha consequence of each other. The proof that (N2) and (N2bis) are equivalent is the same. (cid:50)
Another useful fact. One can also show that (N1) and (N1bis) are equivalent to(25) A , Φ , , ˜ R , A , Φ , , ˜ R , A , Φ , , ˜ R , = Id ( ) . Proof of (iii) in Proposition A.2. Relation (N1bis) is equivalent toΦ , , ( A , ) − ( Φ , , ) − A , = ˜ R , Φ , , A , ( Φ , , ) − ˜ R , . Thus, (Ebis) is equivalent toΦ R , R , Φ − = ( ˜ R , Φ , , A , ( Φ , , ) − ˜ R , , ( B , ) − ) . Now, as we have ˜ R , B , = B , ˜ R , , we deduce that (Ebis) is equivalent toΦ R , R , Φ − = ˜ R , Φ , , A , ( Φ , , ) − ( B , ) − Φ , , ( A , ) − ( Φ , , ) − B , ( ˜ R , ) − , which is equivalent to ( Φ , , ) − ( ˜ R , ) − Φ R , R , Φ − ˜ R , ( B , ) − Φ , , = A , ( Φ , , ) − B , Φ , , ( A , ) − . Now, by using ● ( Φ , , ) − ( ˜ R , ) − Φ R , R , Φ − ˜ R , = ˜ R , ( Φ , , ) − R , Φ , , R , ● ( B , ) − = ( R , ) − B , ( R , ) − ● Φ , , = R , ( Φ , , ) − ˜ R , Φ , , ˜ R , , ● ( Φ , , ) − = ˜ R , ( Φ , , ) − ˜ R , Φ , , R , we obtain ˜ R , ( Φ , , ) − R , Φ , , B , ( Φ , , ) − ˜ R , Φ , , ˜ R , = A , ˜ R , ( Φ , , ) − ˜ R , Φ , , B , ( Φ , , ) − ˜ R , Φ , , ˜ R , ( A , ) − . After performing A , ˜ R , = ˜ R , A , in the r.h.s. of the above equation, one can cancelout the ˜ R , terms in both sides of the equation. We obtain, by performing the permutation ( ) ↦ ( ) that ( Φ , , ) − R , Φ , , B , ( Φ , , ) − ˜ R , Φ , , = A , ( Φ , , ) − ˜ R , Φ , , B , ( Φ , , ) − ˜ R , Φ , , ( A , ) − . This is equivalent toΦ , , A , ( Φ , , ) − ˜ R , Φ , , B , ( Φ , , ) − ˜ R , Φ , , ( A , ) − ( Φ , , ) − = R , Φ , , B , ( Φ , , ) − ˜ R , , which is equivalent toΦ , , A , ( Φ , , ) − ˜ R , Φ , , B , ( Φ , , ) − ˜ R , Φ , , ( A , ) − ( Φ , , ) − ( ˜ R , ) − Φ , , ( B , ) − ( Φ , , ) − ( R , ) − = Id ( ) . LLIPSITOMIC ASSOCIATORS 63 As ( R , ) − R , R , = R , = ( ˜ R , ) − , we obtain R , R , = ( Φ , , A , ( Φ , , ) − , ˜ R , Φ , , B , ( Φ , , ) − ˜ R , ) . A.2.
An alternative presentation for PaB Γ e(cid:96)(cid:96) . Below, we borrow the notation from Theo-rem 4.5.
Theorem A.3.
As a
PaB -module in groupoids with a diagonally trivial Γ -action and having Pa Γ as Pa -module of objects, PaB Γ e(cid:96)(cid:96) is freely generated by A and B together with the relations A , = Φ , , A , ( Φ , , ) − ˜ R , Φ , , A , ( Φ , , ) − ˜ R , , (tN1bis) B , = Φ , , B , ( Φ , , ) − ˜ R , Φ , , B , ( Φ , , ) − ˜ R , , (tN2bis) Φ , , R , R , ( Φ , , ) − = ( A , Φ , , ( A , ) − ( Φ , , ) − , ( B , ) − ) . (tEbis)In order to prove Theorem A.3, one can(i) Either deduce it from Theorem 4.5 in a similar manner as we deduced Theorem A.1from Theorem 3.3;(ii) Or deduce it from Theorem A.1 in a similar manner as we deduced Theorem 4.5 fromTheorem 3.3.Both strategies are straightforward to implement; this is left to the reader. List of notation
Operads.PaB:
Operad of parenthesized braids. 12
PaCD ( k ) : Operad of parenthesized chord diagrams. 15
PaB e(cid:96)(cid:96) : PaB -module of elliptic parenthesized braids. 22
PaCD e(cid:96)(cid:96) ( k ) : PaCD ( k ) -module of ellitpic parenthesized chord diagrams. 28 PaB Γ e(cid:96)(cid:96) : PaB -module of ellipsitomic parenthesized braids. 34 PaCD Γ e(cid:96)(cid:96) ( k ) : PaCD ( k ) -module of ellipsitomic parenthesized chord diagrams. 43 Groups. PB n : Pure braid group on the complex plane. 11
GT:
Operadic Grothendieck-Teichm¨uller group. 18 ̂ GT ( k ) : k -pro-unipotent Grothendieck-Teichm¨uller group. 18 GRT ( k ) : Operadic graded Grothendieck-Teichm¨uller group. 19GRT ( k ) : Graded Grothendieck-Teichm¨uler group. 20PB ,n : Reduced pure braid group on the torus. 21 ̂ GT e(cid:96)(cid:96) ( k ) : Operadic k -pro-unipotent elliptic Grothendieck-Teichm¨uller group. 30 ̂ GT e(cid:96)(cid:96) ( k ) : k -pro-unipotent elliptic Grothendieck-Teichm¨uller group. 30 GRT e(cid:96)(cid:96) ( k ) : Operadic graded elliptic Grothendieck-Teichm¨uller group. 31GRT e(cid:96)(cid:96) ( k ) : Graded elliptic Grothendieck-Teichm¨uller group. 32 ̂ GT Γ e(cid:96)(cid:96) ( k ) : Operadic k -pro-unipotent ellipsitomic Grothendieck-Teichm¨uller group. 37 GRT Γ e(cid:96)(cid:96) ( k ) : Operadic graded ellipsitomic Grothendieck-Teichm¨uller group. 44B Γ1 ,n : Γ-decorated braid group on the torus. 48
Spaces. C ( C , I ) : Reduced configuration space of I -indexed points in C . 10C ( C , I ) : Fulton-MacPherson compactification of C ( C , I ) . 10Conf ( C , n ) : Configuration space of n points in C . 14C ( T , I ) : Reduced configuration space of I -indexed points in T . 21C ( T , I ) : Fulton-MacPherson compactification of C ( T , I ) . 21Conf ( T , I, Γ ) : Γ-decorated configuration space of I -indexed points in T . 33C ( T , I, Γ ) : Reduced Γ-decorated configuration space of I -indexed points in T . 33C ( T , I, Γ ) : Fulton-MacPherson compactification of C ( T , I, Γ ) . 33 Lie and associative algebras. t n ( k ) : Rational Kohno-Drinfeld Lie k -algebra. 14 t ,n ( k ) : Elliptic Kohno-Drinfeld Lie k -algebra. 26 t Γ1 ,n ( k ) : Γ-ellipsitomic Kohno-Drinfeld Lie k -algebra. 38 lossary 65 Torsor sets.Assoc ( k ) : Operadic k -associators. 17Ass ( k ) : k -associators. 17 Ell ( k ) : Operadic elliptic k -associators. 29Ell ( k ) : Elliptic k -associators. 29 Ell Γ ( k ) : Operadic ellipsitomic k -associators. 44Ell Γ ( k ) : Ellipsitomic k -associators. 44 Series. Φ KZ : KZ associator. 17 e ( τ ) : Elliptic KZB associator. 30 A Γ ( τ ) : A -ellipsitomic KZB associator. 46 B Γ ( τ ) : B -ellipsitomic KZB associator. 46 G s,γ ( τ ) : Eisenstein-Hurwitz series. 56
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Damien CALAQUEIMAG, Univ Montpellier, CNRS, Montpellier, France & Institut Universitaire de France.
E-mail address : [email protected] Martin GONZALEZI2M, Universit´e d’Aix-Marseille, 39, rue F. Joliot Curie, 13453, Marseille, France.
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