SéminairedeTopologiealgébriqueduLAGA18Mars2021 ThibautMazuir n -multiplihedra Higheralgebraof A -algebrasandthe

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Higher algebra of A

_∞

-algebras and the n -multiplihedra

Thibaut Mazuir

IMJ-PRG - Sorbonne Université

Séminaire de Topologie algébrique du LAGA 18 Mars 2021

Thibaut Mazuir Higher algebra ofA_∞-algebras and then-multiplihedra

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The results presented in this talk are taken from my two recent papers : Higher algebra ofA∞andΩBAs-algebras in Morse theory I (arXiv:2102.06654) and Higher algebra ofA∞andΩBAs-algebras in Morse theory II (arxiv:2102.08996).

The talk will be divided in three parts : recollections on

A∞-algebras andA∞-morphisms ; denition of higher morphisms betweenA∞-algebras, or n−A∞-morphisms, and their properties ; denition of then-multiplihedra, which are new families of

polytopes generalizing the standard multiplihedra and which encode n−A∞-morphisms betweenA∞-algebras.

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1 A∞-algebras andA∞-morphisms

2 Higher algebra ofA∞-algebras

3 Then-multiplihedra

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The material presented in this rst part is standard and was drawn from [LV12], [Val14] and [LH02].

Suspension : LetAbe a graded Z-module. We denotesA, the suspension ofA to be the gradedZ-module dened by

(sA)ⁱ :=Aⁱ⁻¹. In other words, fora∈A,|sa|=|a| −1.

For instance, a degree 2−n map A^⊗n→Ais equivalent to a degree+1 map(sA)^⊗n→sA.

Cohomological conventions : dierentials will have degree +1.

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1 A∞-algebras andA∞-morphisms A∞-algebras

A∞-morphisms

The operadic viewpoint

Homotopy theory ofA∞-algebras

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Denition

LetAbe a dg-Z-module with dierentialm₁. AnA∞-algebra structure onAis the data of a collection of maps of degree 2−n

mn:A^⊗n−→A, n>1, extendingm₁ and which satisfy

[m₁,mn] = X

i₁+i₂+i₃=n 26i26n−1

±m_i₁_+1+i₃(id^⊗i¹⊗m_i₂⊗id^⊗i³).

These equations are called theA∞-equations.

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Representingm_n as ¹² ⁿ , these equations can be written as

[m₁,

1 2 n

] = X

h+k=n+1 26h6n−1 16i6k

± ¹ ^k ^d¹

1 d₂

.

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In particular,

[m₁,m₂] =0 ,

[m₁,m₃] =m₂(id⊗m₂−m₂⊗id) ,

implying thatm₂ descends to an associative product on H^∗(A). An A∞-algebra is thus simply a correct notion of a dg-algebra whose product is associative up to homotopy.

The operationsm_n are the higher coherent homotopies which keep track of the fact that the product is associative up to homotopy.

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A∞-algebras can also be dened using the shifted bar construction.

The reduced tensor coalgebra, or bar construction, of a graded Z-moduleV is dened to be

T V :=V ⊕V^⊗²⊕ · · · endowed with the coassociative comultiplication

∆_{T V}(v₁. . .v_n) :=

n−1

X

i=1

v₁. . .v_i ⊗v_i+₁. . .v_n .

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Then for a gradedZ-moduleA, we can check that there is a one-to-one correspondence







collections of morphisms of degree 2−n mn:A^⊗n→A, n>1,

satisfying theA∞-equations







←→

coderivations D of degree+1 ofT(sA) such that D²=0

.

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Denotingbn: (sA)^⊗n→sA the degree+1 maps which determine the coderivationD :T(sA)→T(sA) (universal property of the bar construction), check that the restriction ofD to the(sA)^⊗n summand is given by

X

i₁+i₂+i₃=n

±id^⊗i¹⊗bi₂⊗id^⊗i³ ,

and that the equationD² =0 is equivalent to theA∞-equations for the mapsb_n.

This one-to-one correspondence yields an equivalent denition for A∞-algebras.

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A∞-morphisms

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Denition

AnA∞-morphism between two A∞-algebras AandB is a

dg-coalgebra morphismF : (T(sA),DA)→(T(sB),DB)between their shifted bar constructions.

As previously, one-to-one correspondence

collections of morphisms of degree 1−n f_n:A^⊗n→B , n>1,

←→

morphisms of graded coalgebras F :T(sA)→T(sB)

.

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Component ofF mapping(sA)^⊗n to (sB)^⊗s given by X

i₁+···+is=n

±f_i₁⊗ · · · ⊗fis .

A coalgebra morphism preserves the dierential if and only if X

i₁+i₂+i₃=n

±f_i₁₊₁_+i₃(id^⊗i¹ ⊗m^A_i₂⊗id^⊗i³)

= X

i₁+···+is=n

±m^B_s(f_i₁⊗ · · · ⊗f_i_s). (?)

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This yields an equivalent denition with operations for A∞-morphisms :

Denition

AnA∞-morphism between two A∞-algebras AandB is a family of mapsf_n:A^⊗n→B of degree 1−n satisfying

[m₁,f_n] = X

i₁+i₂+i₃=n i₂>2

±f_i₁₊₁_+i₃(id^⊗i¹⊗m_i₂⊗id^⊗i³)

+ X

i₁+···+is=n s>2

±m_s(f_i₁ ⊗ · · · ⊗f_i_s) .

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Representing the operationsf_nas , the operationsm^A_n in red and the operationsm^B_n in blue, these equations read as

h

∂, i

= X

h+k=n+1 16i6k

h>2

±¹ ⁱ ^k

1 h

+ X

i₁+···+is=n s>2

±

1 i_s i₁

1

.

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We check that[∂,f₂] =f₁m^A₂ −m^B₂(f₁⊗f₁).

AnA∞-morphism between A∞-algebras induces a morphism of associative algebras on the level of cohomology, and is a correct notion of morphism which preserves the product up to homotopy.

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Given two coalgebra morphismsF :T V →T W and

G :T W →T Z, the family of morphisms associated to G ◦F is given by

(G◦F)_n:= X

i₁+···+is=n

±g_s(f_i₁⊗ · · · ⊗f_i_s) .

This formula denes the composition ofA∞-morphisms. Hence, A∞-algebras together withA∞-morphisms form a category,

denoted A∞−alg. This category can be seen as a full subcategory of dg−cog using the shifted bar construction viewpoint.

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A∞-morphisms

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LetC be one the two following monoidal categories : the category (dg−Z−mod,⊗) or the category of polytopes(Poly,×). Denition

AC-operad P is the data of a collection of objects {P_n}_n>1 ofC together with a unit elemente ∈P₁ and with compositions

P_k ⊗Pi₁ ⊗ · · · ⊗Pi_k −→

ci1,...,ik

Pi₁+···+i_k

which are unital and associative.

The objectsPn are to be thought as spaces encoding arityn operations while the compositionsc_i₁_,...,i_k dene how to compose these operations together.

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LetAbe a dg-Z-module andn>1. The space of graded maps Hom(A^⊗n,A) is a graded Z-module, and the collection of spaces Hom(A) :=Hom(A^⊗n,A) can naturally be endowed with an operad structure (compositions are dened as one expects).

ForP be a (dg−Z−mod)-operad, a structure ofP-algebra onA is dened to be the datum of a morphism of operads

P −→Hom(A) .

In other words, of a way to interpret each operation ofP_n in Hom(A^⊗n,A), such that abstract composition inP coincides with actual composition in Hom(A).

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Consider{P_n}_n>1 and{Q_n}_n>1 two operads, and {R_n}_n>1 a collection of spaces of operations.

Denition

A(P,Q)-operadic bimodule structure onR is the data of action-composition maps

Rk ⊗Qi₁ ⊗ · · · ⊗Qik −→

µ_i₁_,...,_ik Ri₁+···+ik , P_h⊗R_j₁⊗ · · · ⊗R_j_h −→

λj1,...,jh

R_j₁+···+j_h ,

which are compatible with one another, with identities, and with compositions inP andQ.

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LetAandB be two dg-Z-modules. We have seen that they each determine an operad, Hom(A) and Hom(B) respectively. The collection of spaces Hom(A,B) :={Hom(A^⊗n,B)}_n>1 in dg-Z-modules is then a (Hom(B),Hom(A))-operadic bimodule where the action-composition maps are dened as one could expect.

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The structure ofA∞-algebra is governed by the operadA∞ : the quasi-free dg−Z−mod-operad

A∞:=F( , , ,· · ·)

generated in arityn by one operation ¹² ⁿ of degree 2−n and whose dierential is dened by

∂(

1 2 n

) = X

h+k=n+1 26h6n−1 16i6k

±¹ ^k ^d¹

1 d₂

.

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A∞-morphisms betweenA∞-algebras are encoded by the (A∞,A∞)-operadic bimodule A∞−Morph, dened to be the (A∞,A∞)-operadic bimodule

A∞−Morph=F^A^∞^,A^∞( , , , ,· · ·) ,

generated in arityn by one operation of degree 1−n and ...

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... whose dierential is dened by

∂( ) = X

h+k=n+1 16i6k

h>2

±¹ ⁱ ^k

1 h

+ X

i₁+···+is=n s>2

±

1 i_s i₁

1

,

where the generating operations of the operadA∞ acting on the right in blue and the ones of the operadA∞ acting on the left in red .

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A∞-morphisms

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The category A∞−alg provides a framework that behaves well with respect to homotopy-theoretic constructions, when studying homotopy theory of associative algebras. See for instance [Val20]

and [LH02].

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It is because this category is encoded by the two-colored operad A²_∞:=F( , , ,· · · , , , ,· · ·, , , , ,· · ·) . It is a quasi-free object in the model category of two-colored operads in dg-Z-modules and a brant-cobrant replacement of the two-colored operadAs², which encodes associative algebras with morphisms of algebras,

A²_∞−→As˜ ² .

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Theorem (Homotopy transfer theorem)

Let(A, ∂_A) and(H, ∂_H) be two cochain complexes. Suppose that H is a homotopy retract ofA, that is that they t into a diagram

(A, ∂A) (H, ∂H) ,

h

p i

where idA−ip = [∂,h]andpi =idH. Then if (A, ∂A) is endowed with an associative algebra structure,H can be made into an A∞-algebra such thati andp extend to A∞-morphisms, that are thenA∞-quasi-isomorphisms.

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Our goal now : study the higher algebra ofA∞-algebras.

Higher algebra is a general term standing for all problems that involve dening coherent sets of higher homotopies (also called n-morphisms) when starting from a basic homotopy setting.

Considering twoA∞-morphisms F,G, we would like rst to determine a notion giving a satisfactory meaning to the sentence

"F andG are homotopic". Then,A∞-homotopies being dened, what is now a good notion of a homotopy between homotopies ? And of a homotopy between two homotopies between homotopies ? And so on.

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2 Higher algebra ofA∞-algebras A∞-homotopies

Higher morphisms betweenA∞-algebras The HOM-simplicial sets HOM_A∞−alg(A,B)•

A simplicial enrichment of the category A∞−alg ?

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Start with a notion of homotopy. Drawn from [LH02].

TakeC andC⁰ two dg-coalgebras,F andG morphisms C →C⁰ of dg-coalgebras. A(F,G)-coderivation is a mapH:C →C⁰ such that

∆_C⁰H = (F ⊗H+H⊗G)∆_C .

The morphismsF andG are then said to be homotopic if there exists a(F,G)-coderivationH of degree -1 such that

[∂,H] =G −F .

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∆∆∆¹:=Z[0]⊕Z[1]⊕Z[0<1], with dierential∂^sing

∂^sing([0<1]) = [1]−[0] ∂^sing([0]) =0 ∂^sing([1]) =0 , and coproduct the Alexander-Whitney coproduct

∆_∆_∆_∆1([0<1]) = [0]⊗[0<1] + [0<1]⊗[1]

∆_∆_∆_∆1([0]) = [0]⊗[0]

∆_∆_∆_∆1([1]) = [1]⊗[1].

The elements[0]and[1]have degree 0, and the element[0<1] has degree−1.

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We check that there is a one-to-one correspondence between (F,G)-coderivations and morphisms of dg-coalgebras

∆

∆∆¹⊗C −→C⁰. Denition

For twoA∞-algebras(T(sA),D_A) and(T(sB),D_B) and two A∞-morphisms F,G : (T(sA),D_A)→(T(sB),D_B), an A∞-homotopy fromF to G is dened to be a morphism of dg-coalgebras

H: ∆∆∆¹⊗T(sA)−→T(sB) ,

whose restriction to the[0]summand is F and whose restriction to the[1]summand is G.

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can be rephrased in terms of operations.

Denition

AnA∞-homotopy between two A∞-morphisms (fn)_n>1 and(gn)_n>1 is a collection of maps

hn:A^⊗n−→B , of degree−n, satisfying

[∂,hn] =gn−fn+ X

i₁+i₂+i₃=m i₂>2

±h_i₁₊₁_+i₃(id^⊗i¹⊗mi₂⊗id^⊗i³)

+ X

i₁+···+is+l +j₁+···+jt=n

s+1+t>2

±m_s+₁_+t(fi₁ ⊗ · · · ⊗fis ⊗h_l⊗gj₁⊗ · · · ⊗gjt) .

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In symbolic formalism, [∂, [0<1] ] =

[1] −

[0] +X

±

[0<1]

+X

±^[0] ^[0] ^[0^<^1] ^[1] ^[1]

[0] [1]

,

where we denote _[₀_] , _[₀_<₁_] and _[₁_] respectively for thefn, the hn and the gn.

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The relation beingA∞-homotopic on the class of A∞-morphisms is an equivalence relation. It is moreover stable under composition.

These results cannot all be proven using naive algebraic tools, some of them require considerations of model categories.

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Move on ton-morphisms betweenA∞-algebras.

Dene∆∆∆ⁿ the gradedZ-module generated by the faces of the standardn-simplex∆ⁿ,

∆∆∆ⁿ= M

06i1<···<i_k6n

Z[i₁<· · ·<i_k] . The grading is|I|:=−dim(I) forI ⊂∆ⁿ.

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It has a dg-coalgebra structure, with dierential

∂_∆∆∆ⁿ([i₁ <· · ·<ik]) :=

k

X

j=1

(−1)^j[i₁<· · ·<bij <· · ·<ik], and coproduct the Alexander-Whitney coproduct

∆_∆_∆_∆ⁿ([i₁ <· · ·<i_k]) :=

k

X

j=1

[i₁ <· · ·<i_j]⊗[i_j <· · ·<i_k] .

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Denition ([MS03])

LetI be a face of ∆ⁿ. An overlapping partition of I to be a sequence of faces(I_l)_16`6s ofI such that

(i) the union of this sequence of faces isI, i.e. ∪_16`6sI_l =I ; (ii) for all 16` <s, max(I`) =min(I`+1).

An overlapping 6-partition for[0<1<2]is for instance [0<1<2] = [0]∪[0]∪[0<1]∪[1]∪[1<2]∪[2].

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Overlapping partitions are the collection of faces which naturally arise in the Alexander-Whitney coproduct.

The element∆_∆∆∆ⁿ(I)corresponds to the sum of all overlapping 2-partitions ofI. Iteratings times ∆_∆_∆_∆ⁿ yields the sum of all overlapping(s+1)-partitions of I.

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We have seen thatA∞-morphisms correspond to the set Hom_dg−cog(T(sA),T(sB))

andA∞-homotopies correspond to the set

Hom_dg−cog(∆∆∆¹⊗T(sA),T(sB)),

Denition ([Maz21b])

We dene the set ofn-morphisms betweenA andB as

HOM_A∞−alg(A,B)n:=Hom_dg−cog(∆∆∆ⁿ⊗T(sA),T(sB)).

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Using the universal property of the bar construction,n-morphisms admit a nice combinatorial description in terms of operations.

Denition ([Maz21b])

An-morphism from Ato B is dened to be a collection of maps f_I^(m) :A^⊗m −→B of degree 1−m+|I|forI ⊂∆ⁿ andm>1, that satisfy

h

∂,f_I^(m) i

=

dim(I)

X

j=0

(−1)^jf_∂^(m)

jI + X

i₁+···+is=m I₁∪···∪Is=I

s>2

±m_s(f_I⁽ⁱ¹⁾

1 ⊗ · · · ⊗f_I⁽ⁱ^s⁾

s )

+ (−1)^|I| X

i₁+i₂+i₃=m i₂>2

±f_I⁽ⁱ¹⁺¹⁺ⁱ³⁾(id^⊗i¹⊗m_i₂⊗id^⊗i³).

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Equivalently and more visually, a collection of maps ^I satisfying

[∂, I ] =

k

X

j=1

(−1)^j ∂_j^singI + X

I₁∪···∪Is=I

±^I¹ ^I^s

+X

±

I

.

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The dg-coalgebras∆∆∆^• :={∆∆∆ⁿ}_n>0 naturally form a cosimplicial dg-coalgebra.

The sets HOMA∞−alg(A,B)n then t into a HOM-simplicial set HOM_A∞−alg(A,B)•. This HOM-simplicial set provides a

satisfactory framework to study the higher algebra ofA∞-algebras.

Theorem ([Maz21b])

ForAandB two A∞-algebras, the simplicial set HOMA∞(A,B)• is an∞-category.

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Write∆ⁿ the simplicial set realizing the standard n-simplex ∆ⁿ, andΛ^k_n the simplicial set realizing the simplicial subcomplex obtained from∆ⁿ by removing the faces [0<· · ·<n]and

[0<· · ·<kb<· · ·<n]. The simplicial setΛ^k_n is called a horn, and if 0<k <n it is called an inner horn.

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An∞-category is a simplicial set X which has the left-lifting property with respect to all inner horn inclusionsΛ^k_n →∆ⁿ.

Λ^k_n X

∆ⁿ

u

∃ u

The vertices ofX are then to be seen as objects, and its edges correspond to morphisms.

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Theorem ([Maz21b])

ForAandB two A∞-algebras, the simplicial set HOMA∞(A,B)• is an∞-category.

Beware that the points of these∞-categories are the A∞-morphisms, and the arrows between them are the

A∞-homotopies. This can be misleading at rst sight, but the points are the morphisms and NOT the algebras and the arrows are the homotopies and NOT the morphisms.

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Consider an inner hornΛ^k_n→HOMA∞(A,B)•, where 0<k <n. We want to complete the diagram

Λ^k_n HOMA∞(A,B)•

∆ⁿ

.

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This inner horn corresponds to a collection of degree 1−m+|I|

morphisms

f_I^(m):A^⊗m−→B forI ⊂Λ^k_n, which satisfy theA∞-equations.

Filling this horn amounts then to dening a collection of operations f^(m)

[0<···<bk<···<n]:A^⊗m −→B andf_∆^(m)n :A^⊗m −→B ,

of respective degree 1−m−(n−1) and 1−m−n, and satisfying theA∞-equations.

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combinatorics) shows that they are in fact algebraic -categories.

References II

Naruki Masuda, Thibaut Mazuir, and Bruno Vallette, The diagonal of the multiplihedra and the product of

A∞-categories, In preparation.

James E. McClure and Jerey H. Smith, Multivariable cochain operations and littlen-cubes, J. Amer. Math. Soc. 16 (2003), no. 3, 681704. MR 1969208

Naruki Masuda, Hugh Thomas, Andy Tonks, and Bruno Vallette, The diagonal of the associahedra, 2019, arXiv:1902.08059.

Daniel Robert-Nicoud and Bruno Vallette, Higher Lie theory, 2020, arXiv:2010.10485.

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Bruno Vallette, Algebra + homotopy = operad, Symplectic, Poisson, and noncommutative geometry, Math. Sci. Res. Inst.

Publ., vol. 62, Cambridge Univ. Press, New York, 2014, pp. 229290. MR 3380678

, Homotopy theory of homotopy algebras, Ann. Inst.

Fourier (Grenoble) 70 (2020), no. 2, 683738. MR 4105949

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Thanks for your attention !

Acknowledgements : Alexandru Oancea, Bruno Vallette, Jean-Michel Fischer, Guillaume Laplante-Anfossi, Florian Bertuol, Thomas Massoni, Amiel Peier-Smadja and Victor Roca Lucio.