The minimal model for the Batalin-Vilkovisky operad

THE MINIMAL MODEL FOR THE BA T ALIN-VILKO VISKY OPERAD GABRIEL C. DR UMMOND-COLE AND BR UNO V ALLE TTE A B S T R A C T . The purpose of thi s paper is to exp lain and to genera lize, in a homotopi cal way , the result of Baranni ko v-Kontse vich and Manin which states that the underlying homology groups of some Batal in-V ilk o- visky algebra s carry a Frobenius manifold structure . T o this extent , we ﬁrst make the minimal model for the operad encoding BV -algebras explicit . Then we prove a homotopy transfer theorem for the associated notion of homotop y BV -algebra. The ﬁnal result provide s an extensi on of the action of the homology of the Deligne- Mumford-Knudsen moduli space of genus 0 curves on the homology of some BV -a lgebras to an action via higher homotopica l operations organized by the cohomology of the open moduli space of genus zero curves. Applica tions in Poisson geometry and Lie algeb ra cohomology and to the Mirror Symm etry conjecture are gi ven. C O N T E N T S Introd uction 1 1. Recollection on homotopy BV -algebras 4 2. The homolo gy of B V ¡ as a defor mation retract 6 3. The minimal model of the operad B V 15 4. Skeletal homotopy B V -algeb ras 19 5. Homotopy bar-cobar adjunction 21 6. Homotopy transfer theorem 24 7. From BV -algeb ras to homotopy Frobenius manifolds 28 References 33 I N T RO D U C T I O N The notion of a Batalin-V ilko visky algebr a , or BV -algebra for short, is made up of a commutative prod- uct, a Lie bracket and a unary op erator, which satisfy some relations. This notion n ow appe ars in many ﬁelds of mathematics like ⋄ A L G E B R A : V ertex (op erator) alge bras [ Bor86 , L Z93 ], Che valley-Eilenberg coh omolog y o f L ie algebras [ K o s85 ], bar construction of A ∞ -algebras [ TTW11 ], ⋄ A L G E B R A I C G E O M E T RY : Gro mov-W itten inv a riants and modu li spaces of cu rves (qu antum co- homolo gy , Froben ius manif olds) [ BK98 , Man 99 , LS07 ], chiral algebras (geometr ic Langland s progr am) [ BD04 , FBZ04 ], ⋄ D I FF E R E N T I A L G E O M E T RY : the sheaf o f po lyvector ﬁelds of an orien table (resp. Poisson or Calabi-Y au) manifold [ K os8 5 , Ran97 , K o n03 ], the differential forms o f a manifo ld (Hod ge de- composition in the Riemannian case) [ BK98 , TT00 , Sul10 ], Lie algebroids [ KS95 , Xu99 , Rog09 ], Lagrang ian (resp. coisotrop ic) i ntersections [ BF09 , BG10 ], ⋄ N O N C O M M U T A T I V E G E O M E T RY : the Hosch child coho mology of a symmetric alg ebra [ TT00 , T r a08 , Gin06 , Men0 9 ] and the cyclic Delig ne conjecture [ Kau 08 , TZ 06 , Cos07 , KS09 , BB09 ] , non-co mmutative differential opera tors [ GS10 ], ⋄ T O P O L O G Y : 2-fold loop spaces on topological spaces carrying an action o f the circle [ Get94a ], topolog ical conformal ﬁeld theories, Rieman n surfaces [ Get94a ], string topology [ CS99 ], ⋄ M A T H E M A T I C A L P H Y S I C S : BV quantization (gauge theory) [ BV81 , W it90 , Sch93 , Rog09 ], BRST cohomo logy [ LZ93 , Sta98 ], string theory [ W it92 , WZ92 , Zwi93 , PS94 ], topolog ical ﬁeld theor y [ Get94a ], Renormaliza tion theory [ CG11 ]. 1 2 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE Nearly all the examples of BV -algeb ras appearing in the afore mentioned ﬁelds actually hav e some homol- ogy g roups as u nderlyin g spaces. T herefo re th ey are some shadow of a higher structure: that o f a homotopy BV -algebr a. Algebra an d homotopy theories do not mix well tog ether a prior i. The study of the homotopy prop erties of algeb raic structures of ten introduce s inﬁnitely many new higher operation s of higher arity . So one u ses the operadic calculus to encode them. The stud y o f the homotopy pr operties of algeb raic structu res o ften in troduces inﬁnitely ma ny new high er operation s of higher arity . This is the case f or Batalin-V ilko visky algebras, which do not h av e ho motopy inv arian ce properties, like the transfer of stru cture under h omotopy e quiv ale nces, see [ L V10 , Section 10 . 3 ]. T o solve this, we have deﬁned, in [ GCTV09 ], a n otion of h omotopy Ba talin-V ilkovisk y algebra with th e requir ed homoto py proper ties. T o do so, we ha ve constructed a quasi-f ree, thu s coﬁbran t, resolutio n of the opera d B V enco ding Batalin-V ilkovisky algebras, using the inhomo geneou s K o szul duality theory . While quite “small”, this resolutio n ca rries a non-tr ivial in ternal differential; so it is not min imal in the sense of D. Sulliv an [ Sul77 ]. The pu rpose of the presen t pap er is to go even furth er an d to prod uce the minimal model of the operad B V , that is, a resolution as a quasi-free operad with a d ecompo sable differential and a c ertain grading on the space of generator s. The ﬁrst main r esult o f this paper is the following computation of the homo logy groups of the b ar construction for the operad B V as a d eformatio n retract. Theorem ( 2.1 ) . The va rious maps deﬁned in Section 2 form the follo wing deformation r etract in the cate- gory of differ ential graded S -modules B V ¡ := (q B V ¡ , d ϕ ) % % / / ( H • (B B V ) ∼ = T c ( δ ) ⊕ S − 1 Grav ∗ , 0) . o o wher e δ d enotes a una ry op erator of de g r ee 2 , wher e S − 1 is the operadic desuspension an d where Gr av is the operad Gravity isomorphic to the homology H • ( M 0 ,n +1 ) of the moduli spa ce o f gen us 0 curves with marked points. This result provides the space of generator s for the m inimal m odel of the op erad B V . But, on the opposite to the Koszul du ality theory , this g raded S - module is n ot en dowed with a cooper ad structur e but with a homotopy coop erad structu re. This means that th ere are higher decom position maps which split elements, not only into 2 but also into 3 , 4 , etc. Finally , the differential of th e m inimal model is made u p of these decompo sition m aps. Let u s r ecall th at the prob lem of making minim al mo dels explicit in algebraic topology is related to the following notio ns. Sulliv an models [ Sul77 ] are dg com mutative alg ebras generated by the (d ual o f the) ration al homotopy group s π • X ⊗ Q of a top ological space X , wher e the differential is given by the Whitehead produc ts. Qu illen mo dels [ Qui69 ] a re d g L ie algebras gen erated by the (dual of the) rationa l homolo gy gr oups H • ( X, Q ) of a topolog ical spac e X , where the dif ferential is gi ven by the Masse y prod- ucts. T he Steenrod algeb ra is an inhomoge nous K oszul algebra, w hose K o szul dual dg algeb ra is the Λ algebra, see [ Pri70 ]. T he minimal model of the Steenrod algebra is generated by the underlyin g homo logy group s of the Λ algebr a and the differential is related to the Adam s spectral seq uence [ BCK + 66 , W an 67 ]. Recall th at th e Λ algebr a is the ﬁrst page o f the Ad ams spectral sequ ence, which c omputes th e homotopy group s of s pheres. S U L L I V A N Q U I L L E N S T E E N RO D B A TA L I N - V I L K OV I S K Y fr ee commutative algebra S ( − ) Lie algebr a L ie ( − ) associativ e algebra T ( − ) operad T ( − ) generators π • X ⊗ Q H • ( X, Q ) Λ H • ( M 0 ,n +1 ) ⊕ ¯ T c ( δ ) differ ential Whitehead brackets Massey products differentials of Adams spectral sequence homoto py cooper ad The operad B V b ehaves exactly in the same way as the Steenrod algeb ra with respect to the inhom o- geneou s Koszul du ality theory , see [ GCTV09 ]. B ut, in contrast to the Steenrod algebra, we ar e able to compute, in th is pap er , the u nderly ing ho mology grou ps of its K o szul dual (co)oper ad toge ther with its THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 3 algebraic structur e. W e also provid e a to polog ical explana tion for ou r result. It was shown by E. Ge tzler [ Get94a ] th at th e operad B V is th e ho molog y of the framed little discs ope rad. I ts m inimal m odel is gen - erated by a h omoto py cooperad extension of th e co operad H • ( M 0 ,n +1 ) b y a free resolu tion T c ( δ ) of th e circle S 1 . W e call the algebras over the minimal model of the oper ad B V skeletal ho motopy Batalin-V ilkovisky algebras , since they in volve fewer generating operation s than the notion o f a homo topy BV -algebra gi ven in [ GCTV09 ]. W e provide new formulae for the h omotopy transfer theorem for algebras over a quasi-free operad on a h omotopy cooperad. T o prove them , we have to introduce a new operadic method , based on a reﬁned b ar-cobar adjunction , since the classical metho ds of [ L V10 , 1 0 . 2 ] (classical bar-cobar adjunction ), and of A. Berglund [ Ber0 9 ] (h omolog ical perturbatio n lem ma) failed to apply . This giv es the homotopy transfer theorem for skeletal homotopy BV -algebras. The d ∆ -condition, also called the d ¯ d -lemma or dd c -lemma in [ DGMS75 ], is a particular condition , coming from K ¨ ahler geom etry , between the two unary op erators: the u nderlyin g differential d and the BV -oper ator ∆ . Un der this cond ition, S. Barannikov and M. Kontse vich [ BK98 ], and Y .I. Manin [ Man99 ] proved th at the under lying homology groups of a dg BV -a lgebra carry a Froben ius manifold structure. Such a structu re is encoded by the ho molog y operad H • ( M 0 ,n +1 ) of the Deligne-Mu mford -Knudsen moduli space o f stable gen us 0 cur ves. Its is als o called an hyper commu tati ve algebra (with a a compatible non-d egenerate p airing). Tree fo rmulae fo r such a structure ha ve b een g iv en by A. Losev a nd S. Shad rin in [ LS07 ]. W e show that th ese re sults are actually a conseq uence of the afor ementione d homotopy transfer theorem. This allows u s to p rove them und er a weaker and op timal condition , called the non- commutative Hodge- to-de Rham condition. W e recover the Losev-Shadrin formu lae and thereb y explain their p articular form. Moreover our appro ach gives hig her n on-trivial operatio ns, which ar e necessary to recover the hom otopy type of the original dg BV -algeb ra. Theorem ( 7.8 ) . Let ( A, d, • , ∆ , h , i ) be a dg BV -a lgebra with non-commutative Hod ge-to-de Rham degen- eration data. The un derlying homo logy gr o ups H ( A, d ) ca rry a homotop y hyp er co mmutative algebra structure , which extends the hypercommutative algebras of M. K o ntsevich and S. Barannikov [ BK98 ] , Y .I. Ma nin [ Man99 ] , A. Losev an d S. Shadrin [ LS07 ] , an d J.-S. P ark [ Par07 ] , and such that the r ectiﬁed dg BV -a lgebra Rec( H ( A )) is homotop y equ ivalent to A in the ca te go ry of dg BV -algebras. In geometrica l terms, this lifts the action of th e operad o f modu li space of genus 0 stable curves (coho- mological ﬁeld theor y) into a ce rtain a ction of the cooper ad of the ope n m oduli space of genus 0 cur ves (extended cohomolog ical ﬁeld theory) : H • +1 ( M 0 ,n +1 ) / /   End H ( A ) H • ( M 0 ,n +1 ) . [ B K − M − LS − P ] 7 7 n n n n n n n n n n n n W e conclude this paper with app lications to th e Poisson geo metry , Lie algebr a c ohomo logy and the Mir- ror Symmetry co njecture. T o co nclude, this paper d ev elops the homo topy theory f or d g BV -algebr as (ho - motopy skeletal BV -algebr as, ∞ - quasi-isomo rphisms) necessary to stud y the Mirr or Symmetry conjectu re, in the same way as the h omotopy theor y of d g L ie algeb ras was u sed to prove the deformation-q uantization of Poisson manifold s by M. Kontse vich in [ K o n03 ]. Some of th e results of the present paper were ann ounced in [ DCV09 ]. While we were ty ping it, V . Dotsenko and A. Khoroshkin computed i n [ DK09 ] the homolog y of the bar construction of the operad B V without the action of th e symmetric gr oups. They u sed the ind ependen t m ethod of Gr ¨ obner basis for shufﬂe operad s d ev eloped in [ DK10 ]. The paper is o rganized as follo ws. W e begin by recalling the Koszul resolution of the oper ad B V g iv en in [ GCTV09 ]. In the seco nd section, we c ompute th e h omolo gy of th e K oszul du al dg coop erad B V ¡ and we write it as a defor mation retract of B V ¡ . In the third section, we reca ll the notio n of homotopy coo perad with its ho motopy proper ties: the homoto py tr ansfer the orem fo r h omotopy coo perads. W ith these tools 4 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE in h and, we p roduce the minimal mo del of the opera d B V at the end of Section 3 . In a fo urth section, we describe th e associated no tion of algebra, called skeletal h omotopy BV -a lgebras. Section 5 de als with a generalization of the bar-cobar adjun ction between operad s and homotopy coop erads. The last sectio n contains the homo topy tran sfer theorem for skeletal homo topy BV - algebras and the extension of the result of Barannikov-Kontse vich an d Manin. The read er is suppo sed to be familiar with the notion of an operad and operadic hom ological algebra, for wh ich we ref er to the book [ L V10 ]. In th e present paper , we use the sam e no tations as used in this referenc e. W e work ov e r a ﬁeld K of characteristic 0 and all the S -m odules M = { M ( n ) } n ∈ N are reduced, that is, M (0) = 0 . 1. R E C O L L E C T I O N O N H O M OT O P Y B V - A L G E B R A S In this section, we recall th e main results of [ GCTV09 ] needed in the rest of the text. In lo c.cit., we made explicit a resolution of the o perad B V u sing the Koszul du ality theory . It is gi ven by a qua si-free operad on a dg cooper ad, which is smaller th an the bar construction of B V . 1.1. BV - algebras. Deﬁnition 1.1 (Batalin-V ilkovisky algebras) . A differ ential graded B atalin-V ilkovisk y algebra , or dg B V - algebra for short, is a d ifferential gra ded vector s pace ( A, d A ) en dowed with ⊲ a sym metric binary product • of d egree 0 , ⊲ a sym metric bracket h , i of degree +1 , ⊲ a u nary operator ∆ of degree +1 , such that d A is a deriv ation with respect to each of them an d such that ✄ the pro duct • is associative, ✄ the bracket satisﬁes the Jacobi identity hh , i , i + hh , i , i . (123 ) + hh , i , i . (321) = 0 , ✄ the pro duct • an d the bracket h , i satisfy the Leibniz relation h - , - • - i = ( h - , - i • - ) + ( - • h - , - i ) . (12) , ✄ the unar y o perator ∆ satisﬁes ∆ 2 = 0 , ✄ the bracket is the obstruction to ∆ be ing a deriv ation with respect to the produ ct • h - , - i = ∆( - • - ) − (∆( - ) • - ) − ( - • ∆( - )) , ✄ the oper ator ∆ is a graded deriv ation with respect to the bracket ∆( h - , - i ) + h ∆( - ) , - i + h - , ∆( - ) i = 0 . The operad encoding BV -algebras is the opera d de ﬁned by generators and relations B V := T ( V ) / ( R ) , where T ( V ) den otes the free operad on the S -module V := K 2 • ⊕ K 2 h , i ⊕ K ∆ , with K 2 being the trivial represen tation of the symm etric grou p S 2 . Th e space of relations R is the sub- S - module of T ( V ) generated by th e relatio ns ‘ ✄ ’ g iv en above. The basis elements • , h , i , ∆ are of degree 0, 1, and 1. Since the relations a re homogen eous, the oper ad BV is graded by this degree , termed the homological de gree . W e den ote by C om the o perad generated by the sym metric product • and the associati vity relation. W e deno te by Lie 1 the o perad g enerated by the sym metric bra cket h , i and th e Jaco bi relation; it is the operad encodin g Lie algebra structures on the s uspension of a space. The operad G governing Gerstenhabe r algebras is deﬁned similar ly . Its un derlying S -mod ule is isomorph ic to C om ◦ Lie 1 , on wh ich the operad structure is giv en by means of distrib utiv e la ws, see [ L V10 , Section 8 . 6 ]. THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 5 1.2. Q uadratic analogue. W e consider the ho mogen eous quadr atic analog ue q B V of th e opera d B V . This operad is deﬁned by the same spaces of generato rs V and relation s except fo r the inhomog eneous re lation • ∆ − • ∆ − • ∆ − h , i , which is change d into the ho mogen ous relation : • ∆ − • ∆ − • ∆ . W e d enote this h omog enous quad ratic space of relation s by q R . This operad q B V = T ( V ) / (q R ) is also given by means of distributi ve laws on the S -mod ule q B V ∼ = G ◦ D ∼ = C om ◦ L ie 1 ◦ K [∆] / (∆ 2 ) , where D := K [∆] / (∆ 2 ) is the algebra of dual numb ers, s ee [ GCTV09 , Proposition 3 ]. 1.3. Koszul dual coopera d o f the opera d q B V . W e d enote by s th e homo logical suspension , which shifts the h omolog ical degree by +1 . Recall that the K oszul dual co operad of a qua dratic op erad T ( V ) / (q R ) is deﬁned as th e sub-c oopera d C ( sV , s 2 q R ) ⊂ T c ( sV ) cog enerated by the suspension sV of V with corelators in the d ouble suspen sion s 2 q R o f q R , see [ L V10 , Cha pter 7 ]. Namely , it is the “smallest” sub-coo perad o f the cofree cooperad on sV wh ich contains the corelators s 2 q R . W e denote by S c := End c K s − 1 = { Hom(( K s − 1 ) ⊗ n , K s − 1 ) } n ∈ N the suspension co operad of endo mor- phisms of th e o ne dimensional vector space s − 1 K concen trated in degree − 1 . The desusp ension S c C of a cooper ad C is the co operad deﬁned by the aritywise tensor p roduc t, called the Hadamard ten sor pro duct, ( S c C )( n ) = ( S c ⊗ H C )( n ) := S c ( n ) ⊗ C ( n ) . The u nderlyin g S - module of the K oszul d ual coo perad of q B V is equ al to q B V ¡ ∼ = T c ( δ ) ◦ S c C om c 1 ◦ S c Lie c , where T c ( δ ) ∼ = K [ δ ] ∼ = D ¡ is the coun ital cof ree coalgebra on a degree 2 generator δ := s ∆ , wh ere Lie c ∼ = Lie ∗ is the coo perad en coding Lie coalgeb ras and wh ere C om c 1 ∼ = C om ∗ − 1 is the coo perad encodin g cocommu tativ e coalgeb ra structures on the suspension of a space, see [ GCTV09 , Corollary 4 ] . The degree of the elements in K δ m ⊗ S c C om c 1 ( t ) ⊗ S c Lie c ( p 1 ) ⊗ . . . ⊗ S c Lie c ( p t ) ⊂ q B V ¡ is n + t + 2 m − 2 , fo r n = p 1 + · · · + p t . 1.4. Koszul dual dg cooperad of the operad B V . W e consider the map ϕ : q R → V deﬁned b y • ∆ − • ∆ − • ∆ 7− → h , i and 0 on the other relations of q R , so that the graph of ϕ is equal to the space of re lations R . The induced map q B V ¡ → sV extends to a square-zer o co deriv ation d ϕ on the c ooperad q B V ¡ , see [ GCTV09 , Lemma 5 ]. T he dg cooper ad B V ¡ := (q B V ¡ , d ϕ ) is called the K oszul du al dg cooperad of the inhomog eneous q uadratic operad B V . W e use th e notatio n ⊙ for the ‘symmetric’ tensor p roduc t, that is, the quotien t of the tensor pro duct under the permu tation of ter ms. In particular, we denote by δ m ⊗ L 1 ⊙ · · · ⊙ L t a generic element of T c ( δ ) ◦ S c C om c 1 ◦ S c Lie c with L i ∈ S c Lie c , for i = 1 , . . . , t ; the elements o f S c C om c 1 being implicit. Under these notation s, the coderi vation d ϕ is explicitly gi ven by d ϕ ( δ m ⊗ L 1 ⊙ · · · ⊙ L t ) = t X i =1 ( − 1) ε i δ m − 1 ⊗ L 1 ⊙ · · · ⊙ L ′ i ⊙ L ′′ i ⊙ · · · ⊙ L t , (1) where L ′ i ⊙ L ′′ i is Sweedler-type notation for the im age of L i under the binary part S c Lie c → S c Lie c (2) ⊗ ( S c Lie c ⊗ S c Lie c ) ։ S c Lie c ⊙ S c Lie c 6 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE of the deco mposition map of th e cooperad S c Lie c . The sign, giv en by the Koszul rule, is equal to ε i = ( | L 1 | + · · · + | L i − 1 | ) . The image o f d ϕ is equal to 0 when m = 0 o r when L i ∈ S c Lie c (1) = K I for all i . Remark. Let us denote the linear d ual of δ by ~ := δ ∗ . This is an element o f hom ological d egree − 2 . The K o szul dual op erad is deﬁned by q B V ! := S q B V ¡ ∗ = S ⊗ H q B V ¡ ∗ , wh ere S stand s for the endo - morph ism o perad S := E nd c K s − 1 . Up to a d egree shift, the K oszul d ual dg operad B V ! := (q B V ! , t d ϕ ) , when v iewed as a c ohomo logically gr aded differential K [[ ~ ]] -o perad, correspo nds to the Beilinson- Drinfeld operad [ BD04 , CG11 ]. 1.5. Koszul resolution of the o perad B V . W e denote by B V ∞ the quasi-free o perad g iv en b y th e c obar construction on B V ¡ : B V ∞ := Ω B V ¡ ∼ = ( T ( s − 1 q B V ¡ ) , d = d 1 + d 2 ) , where d 1 is the u nique derivation which e xtends the inter nal differential d ϕ and whe re d 2 is the u nique deriv a tion wh ich extends the inﬁnitesimal (or partial) coprodu ct of th e coop erad q B V ¡ , see [ L V10 , Sec- tion 6 . 5 ]. The total deriv ation d = d 1 + d 2 squares to zero and faithf ully enco des the algebraic struc- ture of the d g cooperad on B V ¡ . The space o f generator s of this quasi-free operad is isomorph ic to T c ( δ ) ◦ S c C om c 1 ◦ S c Lie c , up to coaug mentation and desuspension. Theorem 1.2. [ GCTV09 , Theorem 6 ] The operad B V ∞ is a r eso lution of the operad B V B V ∞ = Ω B V ¡ =  T ( s − 1 q B V ¡ ) , d = d 1 + d 2  ∼ − → B V . It is called the K oszul r esolution of B V . Notice that it is much smaller than the bar-cobar r esolution ΩB B V ∼ − → B V . The K oszul resolutio n and the bar-cobar resolution are bo th quadratic. But they are not minimal reso lutions: they are both quasi-free operads with a differential which is the sum of a qu adratic term ( d 2 ) and a non- trivial linear ter m ( d 1 ). Algebras over the operad B V ∞ are c alled h omotop y BV -a lgebras . For an explicit de scription o f this algebraic notion together with its homotopy properties, we refer the reader to [ GCTV09 ]. 1.6. H omotopy transfer theorem for homotopy BV -algebra s. W e c onsider the data ( A, d A ) h % % p / / ( H, d H ) i o o of two chain complexes, where i and p are cha in ma ps and where h has d egree 1 . I t is called a homoto py r etract when id A − ip = d A h + hd A and when, e quiv ale ntly , i or p is a quasi-isomo rphism. I f, mor eover , the composite pi is e qual to id H , then it is called a deformation r etract . Theorem 1.3. [ GCTV09 , Theorem 3 3 ] An y ho motopy BV -algebra structure on A transfers to H thr ough a homotop y retr act such that i extends to an ∞ -quasi-isomorphism. 2. T H E H O M O L O G Y O F B V ¡ A S A D E F O R M A T I O N R E T R AC T The p urpose of th is section is to co nstruct a n explicit contr acting hom otopy for the ch ain co mplex B V ¡ := (q B V ¡ , d ϕ ) . This is a necessary in gredient for the constructio n of the min imal m odel of the operad B V given in th e n ext sectio n. As a bypro duct, this compu tes the hom ology o f the bar constru ction of the operad B V in term s of the homolo gy of the mo duli space M 0 ,n +1 of genus 0 curves. Th e main result of this section is the following theorem. Theorem 2.1. The various maps deﬁned in this section form the following deformation r etract: (q B V ¡ ∼ = T c ( δ ) ⊗ G ¡ , d ϕ ∼ = δ − 1 ⊗ d ψ ) δ ⊗ H & & / / ( T c ( δ ) ⊗ I ⊕ 1 ⊗ G ¡ / Im d ψ ∼ = T c ( δ ) ⊕ S − 1 Grav ∗ , 0 ) . o o THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 7 2 1 4 3 5 10 4 6 7 9 11 8 12 4 5 6 8 2 1 3 7 F I G U R E 1 . Examp le of a planar representation of a tre e with ordered vertices 2.1. T rees. A r educed r oo ted tr ee is a r ooted tree wh ose vertices hav e at least one input. W e consider the category o f reduced rooted trees with leav es labeled bijectively from 1 to n , den oted b y T ree . The trivial tree | is c onsidered to be part of T ree . Since the trees are reduc ed, there are on ly trivial isom orphisms of trees. So we identify th e isomorp hism classes o f tree s with the trees themselves; see [ L V10 , Ap pendix C] for more details. W e conside r the planar repr esentation of red uced trees provided by shufﬂe trees, s ee E. Hoffbeck [ Hof10 , 2 . 8 ], V . Dotsenko and A. Khoroshkin [ DK10 , 3 . 1 ], and [ L V10 , 8 . 2 ]. W e d eﬁne a total ord er o n the vertices of a tree by reading its planar rep resentation from leaf 1 to the root by follo win g t he internal ed ges without crossing them. See Fig ure 1 for an example. 2.2. Free opera d and co free cooperad. Th e under lying S -modu le of the f ree operad T ( V ) on an S - module V is given by the direct sum L t ∈ T ree t ( V ) , where t ( V ) is the treewise tenso r modu le obtained by labeling e very vertex of th e tree t with an eleme nt o f V according to th e arity and the action of the symmetric groups. The operadic composition map is giv e n th e the graf ting of trees. Dually , the und erlying S -modu le of th e conilpotent cofree cooperad T c ( V ) is eq ual to the same direct sum over trees and its decomp osition map is g iv en b y cutting the trees horizontally ; see [ L V10 , Chapter 5 ] for more details. The subcategory of trees with n vertices is denoted by T ree ( n ) . The number of vertices end ows the free operad T ( V ) ∼ = L n ∈ N T ( V ) ( n ) and the co nilpoten t cofree cooperad T c ( V ) ∼ = L n ∈ N T c ( V ) ( n ) with a weight grading . W e represent a labeled tree by t ( v 1 , . . . , v n ) , using th e aforementio ned total order on vertices. 2.3. Co derivations on the cofree cooperad. Coder iv ations on cof ree cooperad s are character ized by their projection onto the space of generato rs. In other words, Lemma 2.2 . Let η b e a h omogeneous morphism T c ( M ) → M of g raded S modules. Then the r e exists a unique coderivation d η on T c ( M ) extending η , given on an element of T c ( M ) repr esented by a decorated tr ee b y applying η to any subtr ee. This is a classical gene ralization of the character ization of coderivation fo r cofree coalgeb ras. Here a re two simple but u seful e xamples, for more details see [ L V10 , Section 6 . 3 . 14 ] . (1) If η factors throu gh the projection T c ( M ) ։ T c ( M ) (1) = M , then d η is g iv en on a dec orated tree as a signed sum over the vertice s of the tree. The summand corresp onding to a vertex v is th e same tree with η applied to the de coration of v and all other decor ations the same. The sign is the K o szul sign. (2) If η factors through the projectio n T c ( M ) ։ T c ( M ) (2) , then d η is given on a d ecorated tree as a signed sum over the internal ed ges o f the tree. The summand correspon ding to an ed ge e ha s the edge contrac tion along e of the original tree as its und erlying tree. The deco rations a way from the contraction vertex are the same; the decoration on the contraction vertex is g iv en b y applying η to the two deco rated vertices in volved in th e contra ction, vie wed as a two-vertex de corated tree in T c ( M ) (2) . Th e sign is the K o szul sign. 8 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE 2.4. A contra cting homotopy for a cofree coo perad. Let M b e the S -mod ule which is the linear span of elem ents µ and β , bo th o f ar ity two, in d egrees 1 and 2 respec ti vely , b oth with trivial symmetr ic grou p action. M := K 2 s • |{z} µ ⊕ K 2 s h , i | {z } β Let ψ den ote the degree on e morp hism of graded S -m odules ψ : T c ( M ) → M wh ich ﬁrst projects T c ( M ) to the cogenerators M and t hen takes µ to β and β to zero. ψ can be extended uniquely to a de gree one cod eriv atio n d ψ of T c ( M ) b y Lemma 2.2 . W e will construct a de gree − 1 cha in h omoto py H of g raded S -modu les on T c ( M ) so that d ψ H + H d ψ is th e iden tity o utside arity 1 and th e zero map on arity 1 ( which is one dimension al, span ned by a representative of the coimag e of the counit map). T o d o this, we will need a combinato rial factor . Deﬁnition 2.3. Le t T be a binar y tre e. T he vertex v has some numb er o f lea ves m v above one of its incoming ed ges, and ano ther numb er n v above t he o ther (we need not conc ern ourselves which is w hich). Let the weight ω ( v ) b e their produc t m v n v . For an illustration, see Figure 2 . ω = 6 ω = 2 ω = 1 ω = 1 F I G U R E 2 . A binary tree with the weight ω indicated at each vertex J.-L. Loday used this weight function to describe a parameteriz ation of the Stasheff associahedra. Lemma 2.4. [ Lod04 ] The sum of the weights of all the vertices of a binary tr ee with n vertices is  n +1 2  . Deﬁnition 2.5. Let h : M → M be the d egree − 1 morp hism of graded S -modules given by tak ing β to µ and µ to 0 . W e will u se h to deﬁne the contr acting homotopy H . Let the homotopy H be deﬁned on a decorated tre e with n vertices in T c ( M ) as a sum o ver the vertices. For the vertex v , the contribution to the su m is ω ( v ) ( n +1 2 ) times the d ecorated tree o btained by a pplying h to v (includin g the K oszul sign ). So it has a similar ﬂav or to extending h as a cod eriv ation , but also include s combinato rial factor s. Lemma 2.6. Th e map d ψ H + H d ψ is zer o in arity one and the identity in all other arities. Pr oof. First, applied to the coidentity sub space of T c ( M ) , the d egree zero p art o f T c ( M )(1) , this sum is clearly zero. Next, consider a tree in T c ( M ) with at least one vertex. The map H a cts on it by taking the signed and weighted sum o f replacin g each β with a µ ; the c oderiv ation d ψ acts by takin g the signed sum over all th e µ a nd replacing it with a β . T o act ﬁrst with one and then with the oth er means that either THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 9 (1) Th e maps H and d ψ act on two distinct vertices of the tree, or (2) they act on the same vertex, changin g it ﬁr st from β to µ or v ice versa an d then back , endin g u p with the same tree, with a combina torial f ac tor . The ﬁrst type come in pairs, on e fr om d ψ H and one fro m H d ψ , with the same combin atorial factors. T hey have the oppo site sign, because the sign c onv e ntions fo r H and d ψ are the same, an d in one of the cases, there is o ne more or fewer µ than th e other in a position that ind uces a sign. This means all o f these term s cancel. For the second typ e, note ﬁr st of a ll that the induce d signs from H and d ψ will be th e same sign acting on whichever vertex we h av e chosen, and every v ertex will be acted on by precisely one of H d ψ and d ψ H nontrivially , d ependin g on whether it begins decora ted by β o r µ , so the ﬁnal result of acting in t his w a y on ev ery vertex will be the sum over all v ertices of the underlying tree T : X v ω ( v )  n +1 2  T = T . By Lemma 2.4 , the sum of the coef ﬁcients over all the vertices is exactly one , wh ich yields the desired result.  2.5. Cha racterizing t he K o szul dual of the Gerstenhaber operad. Con sider the operad G governing Gerstenhaber algebras. Th is operad has a presentation as T ( s − 1 M ) / ( R ) , whe re R is a set of quad ratic relations in • = s − 1 µ a nd h , i = s − 1 β . The Koszul dual cooperad (See [ L V10 , Section 7 . 3 ]) G ¡ is a graded sub S -module of T c ( M ) ⊂ T c ( s G ) , characterized by being th e intersection of T c ( M ) with the kernel of the degree − 1 co deriv ation d 2 on B G := T c ( s G ) indu ced by the inﬁnitesimal composition map γ (1) : T c ( G ) (2) → G . Applying d 2 to a d ecorated tre e in T c ( M ) gives a sum of trees, each of which has one sp ecial 4 - valent vertex decorated by an element of s G (2) obtained by contr action of one edge an d compo sition o f the associated two opera tions. The r est o f the vertices ar e triv alent and deco rated with a n element o f s G (1) = M . Because M is o ne dimensional in each degree, we can specify that e ach tr iv alent vertex is decorated by either µ o r β , with an overall coe fﬁcient on the dec oration of the spe cial vertex. Then i n order that two separate terms be in t he same summan d o f T c ( s G ) so that they might cancel, the underlying trees must be the same and the decoratio ns on eac h tri valent vertex must be the same. Deﬁnition 2.7. A contraction tr ee is a tree with one u ndecor ated 4 -valent vertex and all other vertices triv alent and decora ted b y either µ o r β . Note that we can indu ce a ﬁxed order on the leaves of the special 4 -valent vertex o f a con traction tree by using the order o n the leav e s o f th e whole tr ee; order the le av es o f th e special vertex by the smallest number of a tree leaf above each o ne, see [ L V10 , Section 8 . 2 ] A sum of decorated trees P c T T ∈ T c ( M ) is in the kern el o f d 2 if an d only if o nce we sum over all possible edge co ntractions, any summand s that have the same u nderlyin g contr action tree cancel with ea ch other . Therefo re it is impo rtant to know which d ecorated bin ary trees T can hav e the same contraction tree. There are precisely three un derlying trees that can give rise to a gi ven con traction trees by an edge contrac- tion, correspo nding to th e three distinct binary trees with two vertices: 10 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE 1 2 3 1 2 1 3 2 1 2 1 2 3 2 1 Deﬁnition 2.8. There are twelve d istinct two-vertex b inary trees with vertices decorated b y µ and/or β , which form a K -linear basis f or the twelve d imensional space T c ( M ) (2) . W e will ref er to these basis trees with th e no tation t i ( a, b ) wher e a and b are each one of the symbols µ and β and i is one of 1 , 2 , and 3 . The numbe rs correspo nd, respectiv ely , to the trees pictu red above, while a an d b are decor ations of the two vertices, following the vertex o rder co n vention established in sub section 2.1 . If S is a con traction tree, then S ( t i ( a, b )) is the de corated b inary tre e ob tained by replacing the 4 -valent vertex with t i ( a, b ) and S [ t i ( a, b )] is th e decora ted tre e obtained by decorating the 4 -valent vertex with d 2 ( t i ( a, b )) . Up to scale, t here are precisely twelve decor ated trees that can contract to y ield the contraction tree S if each vertex is gi ven either µ or β as a d ecoration ; these a re th e trees S ( t i ( a, b )) . These twelve are clear ly in correspon dence with the set { t i ( a, b ) } . W e h av e sho wn: Lemma 2.9. Th e following ar e equ ivalent: (1) The eleme nt P c T T is in G ¡ . (2) F or ea ch co ntraction tr ee S , X c S ( t i ( a,b )) S [ t i ( a, b )] = 0 where the sum runs over the twelve decorated tr ees S ( t i ( a, b )) th at can yield S as a con traction. (3) F or each con traction tree S , X c S ( t i ( a,b )) d 2 ( t i ( a, b )) = 0 where the sum runs over the twelve decorated tr ees S ( t i ( a, b )) th at can yield S as a con traction. In words, a sum of decorated trees c an o nly be in G ¡ if the re is local cance llation for every possible contraction tree. Global cancellatio n is sufﬁcient but local cancellatio n is necessary ( this means that they are equiv a lent, but it will be easier to use local cancellation to check global cancellatio n in the s equel). 2.6. Restr icting the homotopy to G ¡ . Lemma 2.10. The h omotopy H : T c ( M ) → T c ( M ) r e stricts to G ¡ . Pr oof. Let P c T T be a su m of d ecorated trees in G ¡ . T hen fo r every contra ction tree S , the sum over the twelve basis elements P c S ( t i ( a,b )) d 2 ( t i ( a, b )) is zero. Let u s co nsider ap plying H to P c T T . By deﬁnition, this is a sum over e very vertex of th e d ecorated tree T . T o sho w that the resultant sum is in G ¡ , we then apply d 2 and dem onstrate that we get zero. Applying d 2 in volves applyin g the desuspen sion of the in ﬁnitesimal composition ma p γ (1) on ea ch set of two ad jacent vertices, su mming over all suc h p airs. W e will confuse such sub sets with inter nal ed ges, with which th ey are in bijection, as described in Section 2.3 . In total, to ap ply H and the n d 2 to a decorated tree T inv olves summ ing over all choices of a vertex and e dge of T ; eac h in dividual summand is the ap plication of ﬁrst a weigh ted multiple o f h to th e ch osen vertex, and then inﬁn itesimal composition γ (1) to the chosen edge. This sum splits into those pairs of vertex and edge which a re distinct, and th ose pairs where th e chosen edge is inciden t on the ch osen vertex. W e will show that each of the se two co nstituent sum s is zero individually . If the vertex and edge are distinct, then, up to sig n, the application o f h on the vertex and inﬁnitesi- mal comp osition on the edge commute. For a gi ven co ntraction tr ee and cho ice of tri valent vertex on the contraction tree, th e overall sign of co mmuting the shifted inﬁnitesimal com position o n the edge of on e THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 11 of the twe lve d ecorated trees wh ich yields the g iv en contraction tree an d h on th e correspo nding vertex will be inde penden t of the particular ch oice of de corated tree within the twelve. Let S v [ t i ( a, b )] be ob - tained f rom S [ t i ( a, b )] b y applying the a pprop riate weighted multiple o f h to the vertex v . Since the sum P c S ( t i ( a,b )) S [ t i ( a, b )] over the twelve correspond ing decorate d tree s wi thout any application of h is zero, for each choice of vertex, the sum P c S ( t i ( a,b )) S v [ t i ( a, b )] is also zero . The other case to consider is wh en the edg e inv olved in the contr action is incident on the vertex where h is ap plied. Let u s ﬁx a con traction tree S ; because th e origin al sum is in G ¡ , it is tru e that P c S ( t i ( a,b )) S [ t i ( a, b )] is zero, or , eq uiv alen tly , P c S ( t i ( a,b )) d 2 ( t i ( a, b )) is zero. W e will replace d 2 ( t i ( a, b )) with the weig hted sum of applying h to the top an d b ottom vertex of t i ( a, b ) , followed by d 2 , and show that the result is still zero. The tw o -vertex co mponen t o f th e K o szul d ual co operad to a quadratic oper ad is isomorph ic to the space of relations, up to a degree shift. In this case, we have: Lemma 2.1 1. The kernel of d 2 on the linea r span of t i ( a, b ) is six dimensional, spa nned by th e shifted Gerstenhaber r e lations: (1) the two dimen sional space of associativity r elations t i ( µ, µ ) − t j ( µ, µ ) , (2) the th r ee dimensio nal space of Leibniz r e lations spanned by L 1 = t 1 ( µ, β ) + t 2 ( β , µ ) + t 3 ( µ, β ) , L 2 = t 1 ( β , µ ) + t 2 ( µ, β ) + t 3 ( µ, β ) , and L 3 = t 1 ( β , µ ) + t 2 ( β , µ ) + t 3 ( β , µ ) (note that the signs are d iffer ent than in the usual Leibn iz r elatio n becau se of the shift, and th at the pr esen tation is not symmetric in our basis), and (3) the o ne-dimen sional spac e of the J acobi r elatio n t 1 ( β , β ) + t 2 ( β , β ) + t 3 ( β , β ) . For the weightin g o f h , it is necessary to look at the shape and decoration s of the contractio n tree S . Choose a representative so th at the two vertices inv olved in the con traction edge are adjacent in the total orderin g o f vertices. Let the numb er o f lea ves above the edge i of the contraction vertex be n i . · · · n 1 · · · n 2 · · · n 3 b a F I G U R E 3 . The top part of S ( t 3 ( a, b )) Then up to an overall sign a nd overall factor of 1 ( n +1 2 ) , the we ighted sum of apply ing h to b oth vertices of each of the basis elements is given by: 12 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE (1) t i ( µ, µ ) 7→ 0 , (2) t 1 ( µ, β ) 7→ n 3 ( n 1 + n 2 ) t 1 ( µ, µ ) , t 2 ( µ, β ) 7→ n 2 ( n 1 + n 3 ) t 2 ( µ, µ ) , t 3 ( β , µ ) 7→ n 1 ( n 2 + n 3 ) t 3 ( µ, µ ) , (3) t 1 ( β , µ ) 7→ − n 1 n 2 t 1 ( µ, µ ) , t 2 ( β , µ ) 7→ − n 1 n 3 t 2 ( µ, µ ) , t 3 ( µ, β ) 7→ − n 2 n 3 t 3 ( µ, µ ) , (4) t 1 ( β , β ) 7→ n 3 ( n 1 + n 2 ) t 1 ( β , µ ) + n 1 n 2 t 1 ( µ, β ) , t 2 ( β , β ) 7→ n 2 ( n 1 + n 3 ) t 2 ( β , µ ) + n 1 n 3 t 2 ( µ, β ) , an d t 3 ( β , β ) 7→ n 1 ( n 2 + n 3 ) t 3 ( µ, β ) + n 2 n 3 t 3 ( β , µ ) . Applying these formu lae to th e kernel described abov e gi ves: (1) t i ( µ, µ ) − t j ( µ, µ ) 7→ 0 , (2) L 1 7→ n 2 n 3 ( t 1 ( µ, µ ) − t 3 ( µ, µ ) ) + n 1 n 3 ( t 1 ( µ, µ ) − t 2 ( µ, µ )) L 2 7→ n 1 n 2 ( t 2 ( µ, µ ) − t 1 ( µ, µ ) ) + n 2 n 3 ( t 2 ( µ, µ ) − t 3 ( µ, µ )) L 3 7→ n 1 n 2 ( t 3 ( µ, µ ) − t 1 ( µ, µ )) + n 1 n 3 ( t 3 ( µ, µ ) − t 2 ( µ, µ ) ) , (3) and : t 1 ( β , β ) + t 2 ( β , β ) + t 3 ( β , β ) 7→ n 1 n 2 ( t 1 ( µ, β ) + t 2 ( β , µ ) + t 3 ( µ, β )) + n 1 n 3 ( t 1 ( β , µ ) + t 2 ( µ, β ) + t 3 ( µ, β )) + n 2 n 3 ( t 1 ( β , µ ) + t 2 ( β , µ ) + t 3 ( β , µ ) ) . So the k ernel of d 2 in this twelve dimension al spac e is st able und er the weighted app lication of h , no matter the particular tr ees th at deﬁne th e w eights. Th is m eans th at f or ea ch contraction tree S , the sum obtained from P c S ( t i ( a,b )) d 2 ( t i ( a, b )) by replacing d 2 ( t i ( a, b )) with the weighted sum of applying h to the top an d bottom vertex o f t i ( a, b ) , followed by d 2 , is still zero, as desired.  2.7. Proo f of Theorem 2.1 . Lemma 2.1 2. Let O = T ( N ) / ( R ) be a q uadratic operad with K oszul d ual coo perad O ¡ ⊂ T c ( sN ) . Let d b e a coderivation of T c ( sN ) . If the compo sition O ¡ / / / / T c ( sN ) d / / T c ( sN ) / / / / T c ( sN ) (2) / O ¡ (2) is zer o , then d r estricts to be a coderivation of O ¡ . Pr oof. Since the Koszul dual cooperead O ¡ = C ( sN , s 2 R ) is a quadratic coop erad, this proof is dual to the proof that a der iv ation o f th e f ree o perad T ( N ) passes to the qu otient T ( N ) / ( R ) , w ith R ⊂ T ( N ) (2) , if the composite R / / / / T ( N ) d / / T ( N ) / / / / T ( N ) / ( R ) is zero.  Corollary 2.13. The coderiva tion d ψ deﬁned on T c ( M ) restricts to G ¡ . W e will r efer to the r estriction w ith the same notatio n. Pr oof. In o rder to check this, we n eed check o nly that e lements of G ¡ which d ψ takes into T c ( M ) (2) land in G ¡ (2) . For d egree reasons, such eleme nts must belo ng to G ¡ (2) , whic h is described b y Lemma 2.11 . A direct calculation veriﬁes that d ψ takes an associati vity relation to a d ifference of two Leib niz relations, takes each Leibniz relation to the Jacobi relation, and takes the Jacobi relation t o zero.  Proposition 2.14. The counit map ( G ¡ , d ψ ) → (I , 0) o f the differ e ntial graded cooperad ( G ¡ , d ψ ) , the coaug mentation ( I , 0 ) → ( G ¡ , d ψ ) , and the homo topy H fo rm the following deformation r etract: ( G ¡ , d ψ ) H % % / / (I , 0 ) o o THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 13 Pr oof. This is a d irect corollary of Lemma 2.6 , Lemma 2.10 , and Corollary 2.13 .  Remark. One can easily check that the dual of the chain complex ( G ¡ , d ψ ) is isomorph ic to both t he K o szul complex L ie ¡ ◦ κ Lie ( see [ L V10 , Section 7 . 4 ]) and the Chev alley-E ilenberg co mplex of the free Lie algebr a. This isom orphism along with the preceding propo sition implies as a corollary the well-k nown facts that the Lie and comm utative o perads are K oszul, and that, equ iv alently , the Chev alley-Eilen berg h omolo gy of the free Lie algebra is tri vial. Deﬁnition 2.15. W e de ﬁne a map of S -modules θ : T c ( δ ) ⊗ T c ( M ) → T c ( M ⊕ K δ ) as fo llows. W e will describe the im age o f δ m ⊗ x wher e x has under lying tree T . Let λ range over assignments of a non-n egati ve integer to each edge of T so that the sum of all the integers is m . Then the imag e o f δ m ⊗ x has underly ing tree T ′ which is obtain ed fro m T by inserting λ ( e ) biv alen t vertices on each edge e , labeled by δ . Lemma 2.1 6. The r estriction of θ to T c ( δ ) ⊗ G ¡ , still denoted θ , is the invers e to the distributive isomor- phism ρ : q B V ¡ → T c ( δ ) ⊗ G ¡ . Pr oof. First, let x ∈ G ¡ . W e will verify that θ ( δ n ⊗ x ) is in q BV ¡ by check ing th at d 2 θ is zero on T c ( δ ) ⊗ G ¡ (here d 2 is the differential induced by composition in q BV ). The map θ inserts vertices decor ated by δ , and d 2 composes pairs of adjacent vertices. The sum inv o lved in applying d 2 includes compositions in volving 0 , 1 , and 2 vertices decorated by δ . Each of these vanishes for a different reason. (1) Th e insertion o f a vertex decorated with δ com mutes up to sign with composition s that do not in volve it, s o inser ting m vertices deco rated with δ and th en contr acting an edge whose v ertices are decorated by µ o r β is the same a s c ontracting the e dge ﬁrst and th en inserting vertices decorated with δ . But since d G ¡ 2 coincides with d q BV ¡ 2 on the δ 0 compon ent of q B V ¡ , and we are starting in the kernel of d G ¡ 2 to begin with, this summand is zero. (2) Contr acting an edge whose vertices are both decor ated by δ giv e s a bi valent v ertex whose decora- tion is s (∆ ◦ ∆) , whic h is zero since ∆ ◦ ∆ = 0 in q B V . (3) Finally , consider co ntracting an edge between a vertex v decorated by a µ or β an d an ad jacent vertex deco rated b y a δ . L et λ ′ be a map from th e edges of T to the natural nu mbers so that the sum of th e im ages ad ds to m − 1 . There are p recisely th ree ch oices of λ with a δ adjacent to v which can be forgotten to yield an elemen t who se un derlying tree is T with vertices inserted accord ing to λ ′ . The sum of the three contraction s with v associated to λ ′ together make up a relation of q B V . Now con sider ρθ ( δ m ⊗ x ) . Because ρ ﬁrst decomposes an d then pro jects, it is zero on any tree deco rated by β , δ , and µ unless all of the vertices decorated by δ are below a ll of the o ther v ertices. There is precisely one su mmand in the sum de ﬁning θ which satisﬁes this condition . That is the sum mand c orrespon ding to the partition λ with λ o f the ou tgoing edge of th e root eq ual to m an d λ of e very other edge eq ual to zero. The map ρ splits this into two le vels and then projects; the only way f or the projection to be n onzero is for it to split with δ m as th e bo ttom lev el; th en ρθ ( δ m ⊗ x ) = ( δ m ⊗ x ) . Because ρ is an isomo rphism, a one-sided in verse is an inverse.  Lemma 2.17. Und er the above isomorphism θ , the differ entia l δ − 1 ⊗ d ψ is sent to d ϕ : (q B V ¡ , d ϕ ) ∼ = ( T c ( δ ) ⊗ G ¡ , δ − 1 ⊗ d ψ ) Pr oof. It is eno ugh to prove it on the lev e l of the cofree co operad s. W e show that the following diag ram is commutative T c ( δ ) ⊗ T c ( M ) θ / / δ − 1 ⊗ d ψ   T c ( M ⊕ K δ ) ˜ d ϕ   T c ( δ ) ⊗ T c ( M ) θ / / T c ( M ⊕ K δ ) , 14 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE where ˜ d ϕ is the unique coderivati on of the cofree cooperad T c ( M ⊕ K δ ) which extends the map ϕ . Since δ − 1 ⊗ d ψ is a co deriv ation, it is enoug h to prove it by pro jecting onto the sp ace of cogen erators M ⊕ K δ . W e co nclude by showing that t he only non-trivial component is µ δ  / / µ δ − µ δ − µ δ _   _   β  / / β .  Proposition 2.18. Unde r the isomorphism of Lemma 2.17 , the chain complex (q BV ¡ , d ϕ ) a dmits a de gr ee given by the powers δ m of δ , for which: H • (q B V ¡ ) ( m ) =  one dimensiona l, spa nned by ( δ m ⊗ I) : m > 0 isomorphic to 1 ⊗ G ¡ / Im d ψ : m = 0 . Pr oof. Write the chain com plex as · · · / / δ m ⊗ G ¡ δ − 1 ⊗ d ψ / / δ m − 1 ⊗ G ¡ / / · · · / / G ¡ / / 0 The homology is then one dimensional by Propo sition 2.14 e very where except at 1 ⊗ G ¡ , where everything is in the kernel of the dif ferential so the homolog y is just the quo tient by the image of d ψ .  Pr oof of Theorem 2.1 . W e p rove that the fo llowing data ( T c ( δ ) ⊗ G ¡ , δ − 1 ⊗ d ψ ) δ ⊗ H & & p / / ( T c ( δ ) ⊕ Im ( H d ψ ) , 0 ) . o o form a def ormation retract, where the projection map p is th e sum of the projection onto T c ( δ ) and the projection onto G ¡ composed with H d ψ . Assume th at x is in the coaug mentation c oideal G ¡ . Since H is a co ntracting ho motopy for d ψ , ( d ψ H + H d ψ ) x = x . Then d ψ H d ψ x = − H d ψ 2 x + d ψ x = d ψ x so ( x − H d ψ x ) is closed un der d ψ . Since G ¡ is co ntractible and x is in the coau gmentatio n coid eal, this means that x − H d ψ x is in the image o f d ψ , therefore in the image of d ϕ . T his shows that H d ψ x is in the sam e homolo gy class as x . It is indepen dent of choice o f representative beca use it gi ves zero on all of Im d ψ . A quick calculation veriﬁes tha t d ϕ ( δ ⊗ H ) − ( δ ⊗ H ) d ϕ giv es th e pr ojection onto δ m ⊗ G ¡ except on the rightmost factor, where it gives id − H d ψ . T his co ncludes the proo f of the theo rem, with the exception of the rightmost identiﬁcation with the dual to the Gravity operad g iv en in the next section.  2.8. The homology of B V ¡ in terms of the moduli space o f curves and the Gra vity operad. Let u s recall fr om E. Getzler’ s p apers [ Get94b , Get9 5 ] the d eﬁnition of the quad ratic operad Grav encodin g gravity algebras . It is generated by ske w-symm etric operation s [ x 1 , . . . , x n ] of degree 2 − n for any n ≥ 2 , which satisfy the following relations: X 1 ≤ i 0 , 0 for l = 0 . The sign is the K oszu l sign coming from the permutation of the elements. W e conside r the m oduli spa ce M 0 ,n +1 of gen us 0 curves with n + 1 m arked poin ts. The g luing alo ng two points and the Poincar ´ e residue map induce an operad structure on the suspension sH • ( M 0 ,n +1 ) o f its homolo gy , see [ Get95 , Section 3 . 4 ]. Let S − 1 denote both the desu spension operad and cooper ad structure on End K s − 1 . Proposition 2.19 ([ Get94b ]) . The g ravity operad is related to th e ho mology of the moduli sp ace of genus 0 cu rves by the following isomorphism of operads: S − 1 Grav ∼ = sH • ( M 0 ,n +1 ) . THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 15 Proposition 2.20. The quo tient G ¡ / Im d ψ is a cooperad is omorphic to S − 1 Grav ∗ . Pr oof. This is the cooperadic du al of Theorem 4 . 5 o f [ Get94a ]. The aritywise linear dual o f th e differential graded quadr atic coop erad ( G ¡ , d ψ ) , with degree 1 coderiv ation, is a d ifferential grad ed q uadratic o perad, with degree 1 deriv atio n. (W e consider the opp osite ho mologica l degree on the linear dual) . By [ GJ94 , Theorem 3 . 1 ], the under lying operad is isomor phic to ( G ¡ ) ∗ ∼ = S 2 G := End K s 2 ⊗ H G , wh ich admits th e same quadratic p resentation as the operad G except fo r th e − 2 degree shift of th e gen erators s − 2 • and s − 2 h , i . By th e un iv ersal prop erty of q uadratic op erads, the deriv a tion t d ψ is ch aracterized by the images of the se gener ators, th at is s − 2 • 7→ s − 2 h , i and s − 2 h , i 7→ 0 . Ther efore, up to the degree shif t, th e deriv a tion t d ψ is equa l to the d eriv atio n ∆ o n G deﬁned in [ Get94a ]. Theorem 4 . 5 of lo c. cit. states that S − 1 Grav ∼ = Ker ∆ . Dually , it gives G ¡ / Im d ψ ∼ = S − 1 Grav ∗ .  This concludes the proof of Theorem 2.1 . Theorem 2.21. Ther e e xist isomorphisms of graded S -modules H • (B B V ) ∼ = H • (q B V ¡ , d ϕ ) ∼ = T c ( δ ) ⊕ S − 1 Grav ∗ . Pr oof. The ﬁrst isomorph ism is a g eneral fact about K oszul oper ads. In the case of an inhomo genou s K o szul operad P , it is proved as follows. Th e degree − 1 m ap q P ¡ ։ sV → V ֌ P is a twisting mor- phism κ : P ¡ = (q P ¡ , d ϕ ) → P ∈ Tw ( P ¡ , P ) , see [ GCTV09 , App endix A] o r [ L V10 , Section 7 . 8 ]. By the general proper ties of the b ar-cobar ad junction [ L V10 , Section 6 . 5 ], it induces a morphism of dg cooperad s f κ : P ¡ → B P , which is equal to the fo llowing comp osite: P ¡ = q P ¡ ֌ T c ( sV ) → T c ( s P ) = B P . On the right- hand side, the op erad P comes eq uipped with a ﬁltration; we co nsider th e induce d ﬁltratio n on the bar construction. On the left-h and side, we con sider the ﬁltration given by th e weig ht g rading on the coop erad q P ¡ . The coderiv atio n d ϕ lowers th is ﬁltration by 1 and the m orphism f κ preserves the re - spectiv e ﬁltrations. By the Poincar ´ e-Birkh off-W itt theorem [ GCTV09 , Th eorem 3 9 ], gr P ∼ = q P , th e ﬁrst page ( E 0 , d 0 ) of the right hand-side spectral sequence is isomorphic to B q P . So the map f κ induces the map f ¯ κ : (q P ¡ , 0 ) → B q P , on the level of the ﬁrst pag es of the spectra l sequences, where ¯ κ is the twisting morp hism associated to the homogen eous quadratic operad q P . Since it is K oszul, the m orphism f ¯ κ is a qu asi-isomorph ism and we conclu de by th e con vergence theo rem of spectral sequences a ssociated to bou nded below and exhaustiv e ﬁltrations [ ML95 , Chapter 11 ]. Th e second isomo rphism follows f rom Theorem 2.1 .  Remarks. ⋄ While we were writing this paper , V . Dotsenko an d A. Khoroshkin in [ DK09 ] p roved, with another method (Gr ¨ obner bases for shufﬂe operad s), th e second isomorphism on th e level of graded N - modules, i.e. without the action of the symmetric groups. ⋄ Th e cooperad G ¡ with the action o f d ψ is the K oszul dual cooperad of the op erad G with the ac tion of ∆ is th e sense of Koszul duality theory of operads over Hopf alg ebras, see the Ph. D. Th esis of O. Bellier [ Bel11 ] for more details. 3. T H E M I N I M A L M O D E L O F T H E O P E R A D B V In this section, we recall the notion o f a homotopy cooperad, and we de velop a transfer theo rem for such structur es across hom otopy equi valences. W e apply this result to the defor mation retract giv en in the previous section. This allows u s to make the minimal model of the operad B V explicit. 3.1. H omotopy coo perad. W e recall fr om [ VdL02 ] th e n otion of a h omoto py cooperad, studied in mo re detail in [ MV09a , Section 4 ]. Deﬁnition 3 .1 (H omotopy coope rad) . A h omotopy co operad structure on a grad ed S -modu le C is the datu m of a squ are-zero d egree − 1 d eriv ation d on the free o perad T ( s − 1 C ) wh ich respects the augm entation m ap. An ∞ -morph ism C D o f homo topy coo perads is a m orphism of augmented dg operad s between the associated quasi-free operad s ( T ( s − 1 C ) , d ) → ( T ( s − 1 D ) , d ′ ) . W e den ote this category by ∞ - c oop ∞ . W e co nsider the isomorphism of S -modules T ( s − 1 C ) ∼ = T ( C ) given by t ( s − 1 c 1 , . . . , s − 1 c n ) 7→ ( − 1) ( n − 1) | c 1 | +( n − 2) | c 2 | + ··· + | c n − 1 | t ( c 1 , . . . , c n ) . 16 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE Since th e map d is a d eriv ation o n a free op erad, it is co mpletely chara cterized by its image on gen erators ∆ : C → T ( C ) , under the above isomor phism. T he substitution of a tree t at th e i th vertex by a tree t ′ is denoted by t ◦ i t ′ , see [ L V10 , Section 5 . 5 ] for more details. Proposition 3.2 ([ MV09a ], Proposition 24 ) . Th e data of a ho motopy co operad ( T ( s − 1 C ) , d ) is equivalent to a family of morphisms of S -module s { ∆ t : C → t ( C ) } t ∈ T ree such that ⋄ ∆ | = 0 , ⋄ th e degr ee of ∆ t is equa l to the n umber of vertices of t min us 2, ⋄ fo r every c ∈ C , the number of non-trivial ∆ t ( c ) is ﬁn ite, ⋄ fo r every c ∈ C , X ( − 1) i − 1+ k ( l − i ) t ◦ i t ′ ( c 1 , . . . , c i − 1 , c ′ 1 , . . . , c ′ k , c i +1 , . . . , c l ) = 0 , wher e th e sum runs over the elements t ( c 1 , . . . , c l ) a nd t ′ ( c ′ 1 , . . . , c ′ k ) such that ∆( c ) = X t ∈ T ree ∆ t ( c ) = X t ∈ T ree t ( c 1 , . . . , c l ) and ∆( c i ) = X t ′ ∈ T ree ∆ t ′ ( c i ) = X t ′ ∈ T ree t ′ ( c ′ 1 , . . . , c ′ k ) . A ho motopy cooperad structur e on a graded S -modu le C with v a nishing maps ∆ t = 0 for trees t ∈ T ree ( ≥ 3) with more than 3 vertices is equivalent to a c oaugmen ted dg cooperad structure on C := C ⊕ I . In this case, the d eﬁnition in terms of a squ are-zero derivation on the free o perad is equivalent to the differential of the co bar construction Ω C . In th e sam e way , the datum o f an ∞ -m orphism F : ( T ( s − 1 C ) , d ) → ( T ( s − 1 D ) , d ′ ) is eq uiv alen t to a morph ism o f S -m odules f ∞ : C → T ( D ) , that is , a family of morphisms { f t : C → t ( D ) } t ∈ T ree , satis fying some relations. An in terpretation in terms of Maurer-Cartan elements is g iv e n in [ MV09a , Section 4 . 7 ]. The projection C → T ( C ) ։ C o f d o n the grad ed S -m odule C endows it with a d ifferential den oted by d C , which is equal to the sum d C = P ∆ t over the corollas t . The images on corollas of any ∞ -mor phism deﬁne a morp hism of dg S -modules ( C , d C ) → ( D , d D ) . When this latter map is a quasi-isomor phism, the ∞ -morph ism is called an ∞ -q uasi-isomorph ism . 3.2. H omotopy transfer theorem for homotopy cooperads. Theorem 3.3. Let ( C , { ∆ t } ) be a homo topy coo perad. Let ( H , d H ) be a dg S -mo dule, which is a ho motopy r etract of the dg S -modu le ( C , d C ) : ( C , d C ) h % % p / / ( H , d H ) . i o o Ther e is a homo topy coope rad structu r e on th e dg S -module ( H , d H ) , which extends the transferr ed com- position maps t ( p ) ◦ ∆ t ◦ i and such that the map p extends to an ∞ -quasi-isomorphism. Pr oof. For any cor olla t , the tran sferred structure map e ∆ t on H is gi ven by the differential d H . For any tre e t ∈ T ree with at least 2 vertices, we consider all the possible ways of writting it by succe ssi ve substitutions of trees with at least 2 vertices: t = ((( t 1 ◦ j 1 t 2 ) ◦ j 2 t 3 ) · · · ) ◦ j k t k +1 . The transferre d struc ture map e ∆ t : H → t ( H ) is then g i ven by e ∆ t := X ± t ( p ) ◦  (∆ t k +1 h ) ◦ j k ( · · · (∆ t 3 h ) ◦ j 2 ((∆ t 2 h ) ◦ j 1 ∆ t 1 ))  ◦ i , where the notation (∆ t ′ h ) ◦ j ∆ t means here the composite of ∆ t with ∆ t ′ h at the j th vertex o f the tree t . The extension of th e map p : C → H into an ∞ -mo rphism p ∞ : C → T ( H ) is giv e n by the same kind of formula. On corollas, it is gi ven by the map p , and for any tree t ∈ T ree ( ≥ 2) with at least 2 vertices, it is giv en by p t := X ± t ( p ) ◦  (∆ t k +1 h ) ◦ j k ( · · · (∆ t 3 h ) ◦ j 2 ((∆ t 2 h ) ◦ j 1 ∆ t 1 ))  ◦ h . When C is a dg cooper ad, these formulae ar e the exact duals to the ones given by [ Gra07 ] for dg (pr)op erads. The rest of the proof is straightforward, following the ideas of loc. cit.  THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 17 3.3. The homo topy cooperad structure on H (B B V ) . Let us deno te the graded S -module H := T c ( δ ) ⊕ S − 1 Grav ∗ ∼ = H • ( B B V ) ∼ = H • ( q B V ¡ , d ϕ ) . Theorem 2.1 provides us with the follo wing deformation retract in the category o f dg S -modules: (q B V ¡ ∼ = T c ( δ ) ⊗ G ¡ , d ϕ ∼ = δ − 1 ⊗ d ψ ) h := δ ⊗ H & & p / / ( H ⊕ I , 0) . i o o Corollary 3.4. The graded S -mo dule H := T c ( δ ) ⊕ S − 1 Grav ∗ is en dowed with a homotop y cooperad structur e and with a n ∞ -quasi-isomorph ism fr om th e dg cooperad BV ¡ = (q B V ¡ , d ϕ ) . Pr oof. This is a d irect application of the Homotopy T r ansfer Theorem 3.3 for homoto py cooper ads.  3.4. The minimal model o f the operad B V . Deﬁnition 3.5. A min imal operad is a quasi-f ree dg operad ( T ( X ) , d ) ⋄ with a dec omposab le differential, that is d : X → T ( ≥ 2) ( X ) , and ⋄ such that the generating degree g raded S -mo dule admits a d ecompo sition in to X = L k ≥ 1 X ( k ) satisfying d ( X ( k +1) ) ⊂ T ( L k i =1 X ( i ) ) . A minimal mod el o f a dg op erad P is the data of a m inimal operad ( T ( X ) , d ) together with a qua si- isomorph ism of dg o perads ( T ( X ) , d ) ∼ / / / / P , which is an ep imorph ism. (This last co ndition is always satisﬁed when the differential o f P is trivial). The gen eralization of the notion of a minimal m odel fro m dg com mutative alg ebras [ DGMS75 , Sul7 7 ] to dg o perads was in itiated by M. Markl in [ Mar96 ], see also [ M SS02 , Section II . 3 . 10 ]. Notice howe ver that the aforemen tioned deﬁnition is strictly mo re g eneral than lo c. cit. and includes the cru cial case of dg associative algebras, since we do not requ ire that X (1) = 0 here. (A min imal oper ad in the sense of Markl is minim al in the p resent sense: the extra grading is given by the arity grad ing X ( k ) := X ( k + 1) ). The present deﬁnition faithfully fo llows Sulliv an ’ s ideas: the increasing ﬁltration F k := L k i =1 X ( i ) is the S ullivan trian gulation assumption . Th e extra g rading X ( k ) is c alled th e syzygy degree. No tice that any no n-negatively graded q uasi-free o perad with d ecompo sable differential is minimal; o ne o nly has to consider X ( k ) := X k − 1 . The fo llowing lemm a co mpares the two appr oaches of Qu illen ( coﬁbrant) an d Sulliv an (minimal) of homoto pical algebra. Lemma 3.6. A minima l operad is coﬁbant i n the model category given by V . Hin ich [ Hin97 ] . Pr oof. This is a p articular case of [ MV09b , Corollary 40 ].  Since the deﬁnition is different, one nee ds a more general proof for the uniqueness of minimal models. Proposition 3 .7. Let P be a dg operad. When it exists, the minimal mod el of the o perad P is unique up to isomorphism. Pr oof. W e work with the mod el category stru cture on d g ope rads deﬁned by V . Hin ich in [ Hin97 ]. Let M and M ′ be two minimal m odels o f th e gr aded op erad P . T hey are coﬁbran t operads b y the preced ing propo sition. Since the qu asi-isomorph ism M ′ ∼ / / / / P is an epimo rphism, it is a trivial ﬁb ration. By the lifting prop erty of a model category , there exists a qua si-isomorph ism f : M = ( T ( X ) , d ) ∼ − → M ′ = ( T ( X ′ ) , d ′ ) of d g operads. It induces a qu asi-isomorph ism of dg S -modules between the sp ace of g enera- tors ( X , d X ) ∼ − → ( X ′ , d X ′ ) by [ MV09a , Propo sition 43 ]. Sin ce the differentials are decomposab le, we get d X = 0 and d X ′ = 0 . So the aforemen tioned quasi-isomor phism is actually an isomorph ism of graded S -modu les X ∼ = X ′ . Therefor e, the map f is an isomorp hism of dg operads.  Theorem 3.8. The data of Cor ollary 3.4 pr ovide us with the minimal model of the operad BV :  T ( s − 1 ( T c ( δ ) ⊕ S − 1 Grav ∗ )) , d  ∼ − → B V , wher e th is quasi-isomorphism is deﬁned by s − 1 δ 7→ ∆ a nd by s − 1 µ 7→ • . 18 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE Pr oof. First, the quasi-free operad  T ( s − 1 ( T c ( δ ) ⊕ S − 1 Grav ∗ )) , d  is minimal since i t is non-negativ e ly graded with the deco mposable differential coming from the transfer red homotopy coop erad structure on H . Then, the ∞ -q uasi-isomor phism p ∞ : q B V ¡ H of Corollary 3.4 indu ces a mor phism of dg operads P : Ω B V ¡ → ( T ( s − 1 H ) , d ) . I t is a quasi-isomorph ism by the following argument. W e co nsider the ﬁltration F • on Ω B V ¡ , and respectively F ′ • on ( T ( s − 1 H ) , d ) , given by the numbe r of vertices o f the underly ing tree: F − k := M t ∈ T ree ( ≥ k ) t ( s − 1 q B V ¡ ) and F ′ − k := M t ∈ T ree ( ≥ k ) t ( s − 1 H ) . The ﬁrst ter ms of the respective associated sp ectral sequences are ( E 0 , d 0 ) ∼ = ( T ( s − 1 q B V ¡ ) , d 1 ) and ( E ′ 0 , d ′ 0 ) ∼ = ( T ( s − 1 H ) , 0) . Th e mor phism of dg operads P preserves th e afo remention ed ﬁltrations. Moreover , it satisﬁes E 0 ( P ) = T ( s − 1 p ) . So it is a q uasi-isomor phism by the K ¨ unneth fo rmula. The two ﬁltrations are obviou sly exhaustive. At ﬁxed arity , they are bounded below: for a ﬁxed degree, the n umber of vertices is limited since the g enerator of arity one h ave degree gr eater or equal to 1 . W e conclude the argument by means of the classical con vergence theo rem for spectral sequences [ ML95 , Chapter 11 ]. Finally , we deﬁne a morphism of operads F : T ( s − 1 H ) → B V by s − 1 δ 7→ ∆ , s − 1 S − 1 Grav ∗ (2) ∼ = s − 1 Im H d ψ (2) ∼ = K s − 1 µ → K • , and the rest b eing sent to 0 . W e now check th e commu tativity of the differentials o n the gener ators. It is straightfor ward on s − 1 δ m . The only elements of Im H d ψ whose image un der d are tr ees with vertices labeled only by µ and δ are in Im H d ψ (3) . In deed, l et t b e an element of Im H d ψ ( n ) , which is the sum of trees with k vertices labeled by µ and with n − 1 − k vertices labele d by β . T o g et trees labeled only b y µ and δ , one has to apply h = δ ⊗ H a total of n − 1 − k times. This introduces the n − 1 − k power o f δ an d applies the coproduct of the coop erad T c ( δ ) ⊗ G ¡ a total of n − k times. In the end , we get trees labe led by n − 1 copies of µ and n − 1 − k copies of δ sp lit n − k times. T o get totally split trees, we s hould ha ve n − k = 2 n − 3 − k , which implies n = 3 . The on e-dimension al spac e Li e ¡ 1 (3) , generated by the Jacobi relation, liv es in Im d ψ = Ker d ψ . The image under d o f the corresp onding elem ent in H Lie ¡ 1 (3) is a sum of 7 trees with 3 vertices ( d 3 ), wh ose image in the operad B V is th e 7 -term relation ∆( - • - • - ) + (∆( - • - ) • - ) . (id +(123 ) + (3 2 1)) + (∆( - ) • - • - ) . (id +(123 ) + (3 21)) = 0 , which is a co nsequenc e of the d eﬁnition of th e opera d B V . T he two-dimen sional space C om ¡ (3) is gen - erated b y (th e suspensio n of) the associators of • . The compo site H d ψ acts on it as the identity . Its image under d is equ al to d 2 , w hich produces the associativity relation in the oper ad B V . So the map F : ( T ( s − 1 H ) , d ) → B V is a morp hism of dg operads. It remains to show that the following diagram is commutative Ω B V ¡ ∼ / / ∼ P ' ' N N N N N N N N N N N B V ( T ( s − 1 H ) , d ) , F 8 8 q q q q q q q q q q q to co nclude that F is a quasi-isomo rphism. I t is enoug h to check it on the genera tors, which is equivalent to the commu tativity of the following diag ram (q B V ¡ , d ϕ ) κ / / p ∞   B V ( T ( s − 1 H ) , d ) / / / / ( T ( K s − 1 δ ⊕ K s − 1 µ ) , 0 ) . F O O It is easily checked on δ , µ , and β . Both maps v an ish on the rest o f q B V ¡ by the same arguments as above: the on ly element which pro duces a non -trivial elemen t in T ( K s − 1 δ ⊕ K s − 1 µ ) u nder p ∞ is β , wh ich conclud es the p roof.  THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 19 W e d enote by d n : s − 1 H → T ( s − 1 H ) ( n ) the part of the differential d , w hich splits elements into n pieces. T he compon ent d 2 coincides with the decomp osition map on th e cooperad S − 1 H y c ¡ ∗ . Proposition 3.9. The differ ential of the minimal model of the operad B V has the following shape: d 2 : s − 1 δ m 7→ X m 1 + m 2 = m s − 1 δ m 1 ⊗ s − 1 δ m 2 and d n : s − 1 δ m 7→ 0 , fo r n ≥ 3 . Up to the de suspension s − 1 , the image of an elem ent of degr ee k o f S − 1 Grav k ∗ under the map d n is a sum of tr ees with n vertices labeled by elemen ts o f S − 1 Grav ∗ and of T c ( δ ) , such th at the tota l degr ee of the eleme nts fr om S − 1 Grav ∗ is eq ual to k − n + 2 a nd such tha t the total weig ht, i.e. the total p ower , of the elements coming fr om T c ( δ ) is equ al to n − 2 . F o r instance, this induces d n ( S − 1 Grav k ∗ ) = 0 for n > k + 2 . Pr oof. By dire ct inspection of the v arious formulae.  W e d enote the minimal model of the operad B V by B V ∞ :=  T ( s − 1 ( T c ( δ ) ⊕ S − 1 Grav ∗ )) , d  . Remarks. ⋄ Th e results a bout TCFT , two-fold loop sp aces, and the cyclic Delig ne conjectur e, obtain ed in [ GCTV09 ] using the co ﬁbrance pro perty o f the Koszul resolution of the o perad B V ho ld as well with this minimal mo del. The p roof of the Lian-Zu ckerman con jecture with this minimal mo del requires furth er w o rk and will be the subject of another paper . ⋄ Th e same method can be app lied to [ HL11 ] to make explicit the minimal model of the inhomo ge- neous quad ratic operad H 0 ( S C ) , where the operad S C is K ontsevich S wiss-cheese operad. 4. S K E L E TA L H O M OT O P Y B V - A L G E B R A S W e ca ll algebras over the minim al mo del of the operad B V skeletal homo topy BV -alge bras. W e m ake this notion explicit and we g i ve a description in terms of Maurer-Cartan elements in a homoto py Lie algebra. 4.1. Seco nd deﬁnition of homotopy BV -algebras. Deﬁnition 4.1 . A skeletal homoto py B atalin-V ilkovisk y algebra is an algebra over the minimal op erad B V ∞ . Recall that a skeletal homo topy BV - algebra struc ture on a dg mo dule ( A, d A ) is the datum o f a morp hism of dg op erads B V ∞ → E nd A . Th e differential ∂ A of the o perad End A is equal to ∂ A ( f ) := d A ◦ f − ( − 1) | f | P n i =1 f ◦ i d A . W e den ote e µ th e image of an element µ of B V ∞ into End A . Proposition 4.2. A skeletal homotopy BV -a lgebra is a chain complex ( A, d A ) en dowed with operations ∆ m : A → A, of degr e e 2 m − 1 , for m ≥ 1 , and e µ : A ⊗ n → A, o f de gree | µ | + n − 2 , for an y µ ∈ Grav ∗ ( n ) , such that ∂ A (∆ m ) = X m 1 + m 2 = m ∆ m 1 ◦ ∆ m 2 , for m ≥ 1 , and ∂ A ( e µ ) = X ± e µ 1 ◦ i e µ 2 + X t ( e ν 1 , . . . , e ν k , ∆ m 1 , . . . , ∆ m l | {z } ≥ 1 ) , wher e th e ﬁ rst sum runs over the d ecompo sition pr odu ct o f the coo perad structur e on Gr av ∗ , ∆ Gr av ∗ ( µ ) = P µ 1 ◦ i µ 2 , and wher e the second sum corr esp onds to composites of at least thr ee op erations with at least one ∆ m . Pr oof. This is a d irect corollary of Proposition 3.9 .  20 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE A BV -algeb ra is a skeletal h omotopy BV -algebr a with v anishing oper ations ∆ m , for m ≥ 2 , an d e µ , f or µ ∈ Gr av ∗ ( n ) , n ≥ 3 . The aforem entioned quasi-isomorphism P : B V ∞ := Ω B V ¡ ∼ − → B V ∞ := ( T ( s − 1 H ) , d ) shows h ow a skeletal homotopy BV -algebr a carries a ho motopy BV -algebra. Theore m 4 . 7 . 4 of [ Hin97 ] implies the functor P ∗ : s kelet al homotop y BV -algebras → homotop y BV -algebras induces equiv alen ces of the ass ociated homo topy catego ries Ho ( homotop y BV -algebras ) ∼ = Ho ( skel etal homotopy BV - algebras ) ∼ = Ho ( BV -algebras ) . Recall that a h yper commutative algebra [ KM94 , Get95 ] is a ch ain comp lex equip ped with a totally symmetric n -ary opera tion ( x 1 , . . . , x n ) o f degree 2( n − 2 ) for any n ≥ 2 , which satisfy X S 1 ⊔ S 2 = { 1 ,...,n } (( a, b , x S 1 ) , c, x S 2 ) = X S 1 ⊔ S 2 = { 1 ,...,n } ( − 1) | c || x S 1 | ( a, ( b , x S 1 , c ) , x S 2 ) , for any n ≥ 0 . W e den ote the associated o perad by H y perC om . It is iso morph ic to the homolog y operad of the Deligne-Mu mford- Knudsen compactiﬁcation o f th e modu li space of gen us 0 curves H • ( M 0 ,n +1 ) . It is K o szul dual to the operad Gravity: H y p erC om ! ∼ = Grav . Proposition 4.3. A skeletal homo topy BV -algebra with vanish ing operators ∆ m , for m ≥ 1 , is a homotopy hypercommutative algebra. Pr oof. This is a direc t cor ollary of Pro position 4.2 together with the fact that the operad H y perC om is K o szul, that is Ω( S − 1 Grav ∗ ) ∼ − → H y perC om , see [ Get95 ].  In operad ic terms, this mea ns that  s − 1 T c ( δ )  ֌ B V ∞ ։ H y pe rC om ∞ is a short exact sequen ce of dg o perads, w here  s − 1 T c ( δ )  is the ideal of B V ∞ generated by s − 1 T c ( δ ) . Equiv alently , the short sequen ce of homotopy cooperads T c ( δ ) ֌ H ։ S − 1 Grav ∗ ∼ = sH • ( M 0 ,n +1 ) is exact, i.e. H is an extension of the (non-u nital) coope rads T c ( δ ) = H • ( S 1 ) ¡ and sH • ( M 0 ,n +1 ) = H • ( M 0 ,n +1 ) ¡ . Theorem 4.4. The operad H y per C om is a repr esentative of the homo topy quotient of the operad B V b y ∆ in the ho motopy cate gory of dg operads. Pr oof. Let D ∞ denote T ( s − 1 T c ( δ )) . T his is the minimal resolution of the algebra of dual num bers D . The pushou t of I ← D ∞ ֌ B V ∞ giv es a representative o f the hom otopy qu otient of B V by ∆ since D ∞ ֌ B V ∞ is a coﬁbratio n and since all th e o perads in the diagram a re c oﬁbrant, see [ Hir03 , Chapter 15 ]. A m ap fro m this diagram to an operad is the same thing as a map of the ge nerators o f H y per C om ∞ that respects th e differentials; since th e augm entation ideal o f D ∞ vanishes in any map from this d iagram, th e differentials coincide with tho se of H y per C om ∞ . So the image of H y perC om in the homotopy category of dg operad s gi ves the homotopy quotient of B V by ∆ .  W e refer the read er to [ Mar09 , DK09 , KMS11 ] fu rther studies o n th is topic. T his resu lt on th e level of homolo gy allows us to conjecture that the hom otopy quo tient o f the frame d little disk by the circle is the compactiﬁed modu li s pace of genus zero stable curves M 0 ,n +1 . This will be the subject of another paper . Remark. Since the g enerator s of H y per C om ∞ form a co operad , one can deﬁne the notion of ∞ -morphism of homo topy hypercommutative algebras using [ L V10 , 10 . 2 ]. In the case of the operad B V ∞ , to deﬁne the n otion o f ∞ -mo rphism o f skeletal homotopy B V - algebras , one has to reﬁne the arguments, using the homoto py pullback of en domor phism opera ds for instance. With these deﬁnitions, Proposition 4.3 shows that the category of ho motopy hy percomm utative alge bras with ∞ -morph isms is a sub category of the category of sk eletal homo topy BV -algeb ras with ∞ -morphisms, b ut not a full subcategory . THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 21 4.2. Maurer-Cartan interpretation. Recall fr om [ MV09a , Theorem 28 ] th at the mo dule of mo rphisms of S -modu les Hom S ( H , End A ) := Y n ≥ 1 Hom S n ( H ( n ) , End A ( n )) carries an L ∞ -algebra structure, { ℓ n } n ≥ 1 , gi ven in terms o f the homo topy coop erad structure on H by the formu la: ℓ n ( f 1 , . . . , f n ) := X t ∈ T re e ( n ) σ ∈ S n ± γ End A ◦ t ( f σ (1) , . . . , f σ ( n ) ) ◦ ∆ t , where γ End A is the co mposition m ap o f o peration s of E nd A and w here th e sign is the K o szul sign due to the permutation of the graded elements { f i } . The solutions to the (generalize d) Maur er-Cartan eq uation X n ≥ 1 1 n ! ℓ n ( α, . . . , α ) = 0 , with | α | = − 1 , in this conv olu tion L ∞ -algebra ar e called the ( generalized ) twisting morphisms and denoted by Tw ∞ ( H , End A ) . Proposition 4.5. Ther e is a natural bijection Hom dg op ( B V ∞ , E nd A ) ∼ = Tw ∞ ( H , End A ) . Pr oof. This fo llows fr om Theorem 54 of [ MV09a ].  This result gives an interp retation o f skeletal homo topy BV -algeb ra s tructures in term of Maurer-Cartan elements in an L ∞ -algebra. W e d enote by (Im H d ψ ) [ k ] the subspace of Im H d ψ spanned by the tree monomials w ith k vertices labelled by µ . Lemma 4.6. Th e isomorphism of Theor em 2.21 pr eserves the respective gr a dings: S − 1 Grav ∗ ( k ) ∼ = (Im H d ψ ) [ k ] . Pr oof. By dire ct inspection.  This result allows us to organ ize the operation s of a skeletal hom otopy BV - algebra into strata. The ﬁrst stratum is described as follows. Since S − 1 Grav ∗ (1) ∼ = H L ie ¡ 1 , in weight 1 , is equal to the tri vial representatio n o f S n , then we get Hom S ( Grav ∗ (1) , E nd A ) ∼ = Hom( S c ( ≥ 2) ( A ) , A ) , u p to suspension. ∆ t lands entirely in weight one only when t h as precisely tw o vertices. So a twisting elem ent α vanishing outside the weight 1 part o f Grav ∗ actually satisﬁes th e truncated Maurer-Cartan equation ∂ α + 1 2 ℓ 2 ( α, α ) = 0 . This c orrespon ds to the deﬁnition o f a Frob enuis manifold in terms of a hyperco mmutative algebr a structure, see Y .I. Man in [ Man99 ]. 5. H O M OT O P Y B A R - C O BA R A D J U N C T I O N In this section , we introd uce a new bar-cobar ad junction between the category of augmen ted dg operads and the category of homotopy cooper ads. T his ba r constru ction relies on the notion of a cof ree hom otopy cooper ad, which we m ake explicit in ter ms of nested trees. 5.1. Co free homotopy coo perad. W e consider now the category of h omotopy cooperad s with (strict) morph isms. Deﬁnition 5.1. A morphism f : ( C , { ∆ t } ) → ( D , { ∆ ′ t } ) of homotopy co operad s is a morph ism of grade d S -modu les C → D which commute s with the structure maps. A morph ism of hom otopy cooperads is an ∞ -morphism with vanishing com ponen ts C → T ( D ) ( n ) for n ≥ 2 . Th e associated category is denoted by coop ∞ . Th ere is a forgetful functor U : coop ∞ → dg- S - Mod , ( C , { ∆ t } ) 7→ ( C , d C ) , which retains only the underlyin g dg S -module struc ture of a homotopy cooperad. 22 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE Deﬁnition 5.2. A nested tr e e is a tree t ∈ T ree \ {|} equ ipped wit h a set of subsets o f vertices { T i } i , called nests , such that: ⋄ eac h nest T i correspo nds to a sub tree of the tree t , ⋄ eac h nest T i has at least two elements, ⋄ if T i ∩ T j 6 = ∅ , then T i ⊂ T j or T j ⊂ T i , and ⋄ th e full subset correspo nding to the tree t is a nest as long as t has mor e than one vertex. The associated category is denoted by NestedT ree . See Figur e 4 for an e x ample. 2 1 4 3 5 10 4 6 7 9 11 8 12 4 5 6 8 2 1 3 7 F I G U R E 4 . Examp le of a nested tree W e co nsider the following total order on nests. The inn ermost n ests are the largest on es. W e co mpare them using their minimal element. The n we forget abo ut th ese nests and proceed in the same way until reaching the full nest, which is the minimal nest. I n the example of Figure 4 , it gives T 1 = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } < T 2 = { 1 , 2 , 3 } < T 3 = { 2 , 3 } < T 4 = { 4 , 8 } < T 5 = { 6 , 7 } . T o any d g S -modu le ( V , d V ) , we associate th e S -module sp anned b y nested tr ees with vertices labeled by elements of V . It is den oted by N T ( V ) := M t ∈ NestedT ree t ( V ) . Using the o rder on vertices gi ven in Sectio n 2.1 and th e above ord er on nests, we write a simple element o f N T ( V ) b y t ( T 1 , T 2 , . . . , T N ; v 1 , v 2 , . . . , v n ) . Its homologica l degree is equa l to P n k =1 | v k | + N − n + 1 . So the degree of a lab eled corolla t ( v ) is equal to | v | . T wo nests T j ( T i are ca lled co nsecutive if T j ⊂ T k ⊂ T i implies eith er T k = T i or T k = T j . W e deﬁne a differential d N by d N ( t ) := X consecutive pair s T j ( T i ± t ( T 1 , . . . , T i , . . . , b T j , . . . , T N ; v 1 , . . . , v n ) , where the notation b T i means that we forget the nest T i . T he sign is g iv en as usual by the K oszul rule as follows. T o e very nest T i , we associate the tree t i obtained fr om the subtree of t deﬁned by T i after contracting all its p roper subnests. Each vertex thereb y ob tained is lab eled by the least ele ment of the contracted nest. The degree of a nest T i is equ al to | T i | := 2 − # t i , where # t i stands for the number of vertices of the tree t i . (I n the example o f Figure 4 , one has | T 1 | = − 2 .) If T j ( T i , then i < j . So we ﬁrst permu te T j with the nests T j − 1 , . . . , T i +1 to b ring it next to T i . T hen we apply the differential to the p air ( T i , T j ) , that is we fo rget a bout the nest T j . T his com es with a sign equal to ( − 1) to the power THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 23 # t i + # t j + k + des ( t j , t i ) , where k is the numb er of vertices of t i smaller than th e smallest vertex of t j and wher e des ( t j , t i ) is the num ber of descen ts, that is the numbe r of pairs ( a, b ) of vertices of t j and t i respectively s uch th at a > b . But th e d ifferential has to “jump over” the nests T 1 , . . . , T i − 1 . In the e nd, it produ ces the sign ( − 1 ) ε , with ε := | T 1 | + · · · + | T i − 1 | + | T j | ( | T i +1 | + · · · + | T j − 1 | ) + # t i + # t j + k + des ( t j , t i ) . W e con sider the d ifferential on N T ( V ) g iv en by th e sum over all the vertices of the imag e of the labeling element of V un der d V . By a slight ab use of notation, it is still denoted d V : d V ( t ) := n X i =1 ( − 1) N − n +1+ | v 1 | + ··· + | v i − 1 | t ( T 1 , . . . , T N ; v 1 , . . . , d V ( v i ) , . . . , v n ) . W e consider ma ps { ∆ t : N T ( V ) → t ( N T ( V )) } t ∈ T ree ( ≥ 2) deﬁned as f ollows. Let τ be a simple element o f N T ( V ) . W e con sider the aforemen tioned tre e t 1 associated to the full n est T 1 , which is obtain ed by co ntracting all the sub trees co rrespond ing to the in terior nests. If t 6 = t 1 , then ∆ t ( τ ) := 0 . Oth erwise, if t = t 1 , the image of τ und er ∆ t is equa l to the tree t 1 with vertices labeled by th e n ested trees obta ined from τ by forgetting its full nest. Proposition 5.3. F or any dg S -modu le ( V , d V ) , the data ( N T ( V ) , d V + d N , { ∆ t } t ∈ T ree (2) ) form a homo- topy coo perad. Th is deﬁn es a functo r N T : dg - S - Mo d → co o p ∞ which is right adjoint to the for getful functor U : co op ∞ → dg - S - Mo d . Pr oof. The three ﬁr st p oints o f th e eq uiv alen t deﬁnition o f a homo topy cooperad g i ven in Proposition 3.2 are trivially s atisﬁed by N T ( V ) . The last po int is straightforward to check. Let C be a homo topy c oopera d. W e consider the morphism of S -modules ∆ iter : C → N T ( C ) deﬁne d as follows. For any tre e t , the extra data giv en by the nests t ( T 1 , . . . , T N ) is equiv alen t to the d ecompo sition of t into successiv e s ubstitutions t = ((( t 1 ◦ i 1 t 2 ) ◦ i 2 t 3 ) · · · ) ◦ i N − 1 t N , where the trees { t i } are associated to the ne sts { T i } as d eﬁned above. The image of the ma p ∆ iter on a nested tree t ( T 1 , . . . , T N ) is d eﬁned by ∆ iter t := ∆ t N ◦ i N − 1 ( · · · (∆ t 3 ◦ i 2 (∆ t 2 ◦ i 1 ∆ t 1 ))) . Let V b e a d g S - module. T o any morph ism of dg S -m odules f : U ( C ) → V , we associate a morp hism F : C → N T ( V ) deﬁn ed by the composite F := C ∆ iter − − → N T ( C ) N T ( f ) − − − − → N T ( V ) . The map F is a m orphism of homotopy cooperads which satisﬁ es the following uni versal proper ty V N T ( V ) o o o o C , f c c F F F F F F F F F ∃ ! F O O which conclud es the p roof.  Hence the homoto py cooper ad N T ( V ) is called the co fr ee homo topy cooperad on V . Remarks. ⋄ Th e endofu nctor U ◦ N T in dg - S - Mo d can be endowed with a comonad structure: decomp ose a nested tree in to all the po ssible ways of seeing it as a nested tree of n ested subtrees. Prop osition 5.3 and its proof are equivalent to saying th at the category of h omotopy c ooperad s is the category of coalgebr as o ver t he comonad U ◦ N T . ⋄ Recall that the notion of an A ∞ -algebra ca n b e enc oded geom etrically by the Stasheff polytopes, also called the associahe dra. In the same way , the n otion of a homoto py coo perad can be encoded by a family of polyto pes, d eﬁned b y by means of graph associahedra la belled b y nested trees as introdu ced by M. P . Carr and S.L. Devadoss in [ CD06 , DF0 8 ]. No tice tha t th is notio n g eneralizes 24 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE the nested sets of C. De Conc ini an d C. Pr ocesi [ DCP95 ]. For in stance, the c hain subcomp lex of nested trees with ﬁxed underlyin g tree t should be isomorph ic to the cochain complex ( N T t , d N ) ∼ = C • ( graph associahedro n associated to t ) This surely deserves further study , which we leave to a f uture work or to the interested reader . 5.2. H omotopy bar -co bar adjunction. Deﬁnition 5.4. Let ( P , γ , d P ) be an augmented dg operad. The underlyin g S -mod ule of the bar construc- tion B π P is given by the cofree hom otopy coo perad N T ( s P ) o n the suspen sion of the augmentation ideal of P . W e d eﬁne the differential d γ by d γ ( t ( T 1 , . . . , T N , s µ 1 , . . . , sµ n )) := X innermost T i = { i 1 ,...,i k } ± t ( T 1 , . . . , b T i , . . . , T N ; s µ 1 , . . . , sγ ( t i ( µ i 1 , . . . , µ i k )) , . . . , d sµ i 2 , . . . , d sµ i k , . . . , sµ n ) . W e co nsider B π P := ( N T ( s P ) , d P + d N + d γ , { ∆ t } t ∈ T ree ( ≥ 2) ) . Proposition 5.5. The data ( N T ( s P ) , d P + d N + d γ , { ∆ t } t ∈ T ree ( ≥ 2) ) fo rm a homotopy cooperad. Pr oof. Checkin g this is a straightforward calculation.  Deﬁnition 5 .6. The cobar co nstruction Ω π C o f a ho motopy coopera d C is the aug mented dg operad Ω π C := ( T ( s − 1 C ) , d ) . Theorem 5.7. Ther e are natural bijec tions Hom dg op (Ω π C , P ) ∼ = Tw ∞ ( C , P ) ∼ = Hom coop ∞ ( C , B π P ) . In plain wor d s, the pair of functo rs Ω π and B π ar e adjoint and this ad junction is r epr esen ted by the twisting morphism bifunctor . Pr oof. The ﬁrst n atural bijection is given by [ MV09a , Theor em 54 ]. The second one is describ ed as follows. Proposition 5.3 already provides us with a natural bijection Hom coop ∞  C , ( N T ( s P ) , d N , { ∆ t } t ∈ T ree ( ≥ 2) )  ∼ = Hom S ( C , s P ) , F 7→ f . Under this b ijection, a m orphism o f S -m odules f : C → s P induces a mor phism of homo topy coopera ds F : C → B π P if and only if the following diagram commutes C F / / d C   N T ( s P ) d P + d γ / / N T ( s P )     C f / / s P . This last co ndition is equivalent to f d C = d P f + P t ∈ T ree ( ≥ 2) γ ◦ t ( f ) ◦ ∆ t , which is exactly the Maurer- Cartan equation X n ≥ 1 1 n ! ℓ n ( s − 1 f , . . . , s − 1 f ) = 0 satisﬁed by s − 1 f in the con volution L ∞ -algebra Hom S ( C , P ) .  Remark. Th e u niv ersal opera dic twisting morphism π : B( S As ) → S As induces a pair of adjo int fu nctors B π and Ω π between the categor y o f dg associa ti ve algebras and th e category of hom otopy coa lgebras by [ GJ94 ], see also [ L V1 0 , Chap ter 1 1 ]. One can prove that it coincid es with the r estriction o f th e above bar and cobar constructio ns B π and Ω π to S -mod ules concentrated in a rity one, which explains the notation. 6. H O M OT O P Y T R A N S F E R T H E O R E M In this section, we prove the homo topy tr ansfer theo rem and the re ctiﬁcation theo rem for skeletal ho- motopy BV -algebras. THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 25 6.1. Universal morphism of homoto py coopera ds. Let ( H, d H ) be a ho motopy retract of a cha in co m- plex ( A , d A ) : ( A, d A ) h % % p / / ( H, d H ) . i o o Recall that th e ho motopy transfer theore m for ho motopy algebras over a K o szul o perad of [ GCTV09 , Append ix B . 3 ] and of [ L V10 , Section 10 . 3 ] relies on the classical bar-cobar adjunction Hom dg op (Ω P ¡ , E nd A ) ∼ = Tw( P ¡ , E nd A ) ∼ = Hom dg coo p ( P ¡ , B E nd A ) and on the quasi-isomo rphism o f dg coopera ds Ψ : B E nd A ∼ − → B E nd H introdu ced by P . V an de r Laan in [ VdL03 ], see also [ L V10 , Section 1 0 . 3 . 3 ]. Such a m ap is characterized by its projection B End A = T c ( s End A ) → s E nd H onto the space of genera tors. T he V an der L aan map Ψ is explicitly given by labeling the leaves of e very tree by the map i , the root by the map p an d the interior edges by the homo topy h . W e co nsider the map G (End A ) : B E nd A → B π End A deﬁned, for any t ∈ T ree , by t ( sf n , . . . , sf 1 ) ∈ T c ( s End A ) 7→ X ± t ( T 1 , . . . , T n − 1 ; s f n , . . . , sf 1 ) ∈ N T ( s End A ) , where the sum runs over all the m aximal nesting s, that is the ones with a maxim al numb er of nests. Since the bar construction B End A is a cooperad, it carries a hom otopy coo perad structure; the map G (End A ) is a quasi-isomo rphism o f homotopy cooperads. Proposition 6 .1. Let ( H, d H ) be a homo topy r etract of a chain complex ( A, d A ) . There exists a qua si- isomorphism of homotop y co operads Φ : B π End A ∼ − → B π End H such that the following diagr am, made up of quasi-isomorphisms of homotopy cooperads, is commu tative, B End A G (End A ) / / Ψ   B π End A Φ   B End H G (End H ) / / B π End H . Pr oof. Let us ﬁrst give the proo f in arity 1 ; so here End A = Hom( A, A ) . W e consider the qua si- isomorph ism of co operads G : As c ∼ − → B ( S As ) . The map G (End A ) is eq ual to G (End A ) = G ◦ id : As c ◦ κ ′ S As ◦ S A s s End A ∼ − → B ( S As ) ◦ π S As ◦ S A s s End A , where κ ′ := S κ : As c = S As ¡ → S As is the K o szul morphism co ming from th e K oszul duality of the o perad As . By the Compar ison Lemma [ L V10 , Lemma 6 . 4 . 13 ], the quasi-isomorp hism G indu ces a quasi-isomo rphism id ◦ G ◦ id : S As ◦ κ ′ As c ◦ κ ′ S As ∼ − → S As ◦ π B ( S As ) ◦ π S As of quasi-free left S As -mod ules (or e quiv ale ntly of quasi-free anti-associative a lgebras in the category of S -modu les). By the left lifting property , it admits a homotopy in verse quasi-isomorphism F : S As ◦ π B ( S As ) ◦ π S As ∼ − → S As ◦ κ ′ As c ◦ κ ′ S As . Under the bar-cobar adjunc tion, the q uasi-isomor phism of coopera ds Ψ is equiv alen t to the q uasi-isomor phism of operad s e Ψ : ΩB E nd A ∼ − → End H . Finally , we deﬁne th e morp hism of h omotopy co operad s Φ : B π End A ∼ − → B π End H to be the map correspon ding to the q uasi-isomor phism of op erads Ω π B π End A F ◦ S As s End A − − − − − − − − → ΩB E nd A e Ψ − → E nd H under the homoto py bar-cobar adjunction. One exten ds these arguments to higher ar ity b y u sing the colored K oszul operad o f [ VdL03 ], which encodes operads, instead of the K o szul (non-sym metric) op erad As which enco des associati ve algebras. 26 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE By deﬁnition, the following d iagram is commutative B π End A F ◦ S As s End A / / Φ ) ) ΩB End A e Ψ / / End H B End A , G (End A ) O O 5 5 5 5 k k k k k k k k k k k k k k k Ψ 5 5 which conclud es the p roof.  The mor phism of homotopy cooper ads Φ : B π End A ∼ − → B π End H is co mpletely char acterized by its projection onto the space of cogener ators, which we d enote by φ : N T ( s End A ) → s End H . 6.2. H omotopy transfer theorem for skeletal homotopy BV -alg ebras. Theorem 6 .2. Let A be a skeletal h omotopy BV -algebra an d let ( H , d H ) be a homotopy retr a ct of t he chain complex ( A, d A ) : ( A, d A ) h % % p / / ( H, d H ) . i o o Ther e is a skeletal homoto py BV -algebra on ( H , d H ) , which e xtends the transferr ed operations p e µi ⊗ n , for any µ ∈ H . If we denote by α ∈ Tw ∞ ( H , End A ) th e skeletal homo topy BV - algebra structu r e on A , such a transferr ed skeletal homotopy BV -a lgebra structure on H is give n by H ∆ iter − − → N T ( H ) N T ( sα ) − − − − − → N T ( s End A ) s − 1 φ − − − → End H . Pr oof. W e ap ply the bar-cobar adju nction of Theorem 5.7 to Hom dg op (Ω π H , End A ) ∼ = Tw ∞ ( H , End A ) ∼ = Hom coop ∞ ( H , B π End A ) . So a skeletal h omotopy BV -algebr a stru cture α : H → E nd A on A is equiv alently g iv en b y a mor phism of hom otopy cooperads F α : H → B π End A . Th e transferred skeletal hom otopy B V -algeb ra on H is then obtained by pushing along the morp hism Φ : Φ ◦ F α : H → B π End A → B π End H , which is equiv alent to th e following twisting mo rphism H ∆ iter − − → N T ( H ) N T ( sα ) − − − − − → N T ( s End A ) s − 1 φ − − − → End H .  Remark. W e pr oved th e ho motopy transfer theo rem for homoto py B V -algeb ras, i.e. for the Koszul model B V ∞ of the operad B V in [ GCTV09 , T heorem 33 ]. Since the S -module of generators of the m inimal model B V ∞ forms a homotopy cooper ad and no t a cooperad, we cannot app ly the arguments of [ GCTV09 , Append ix B . 3 ] and of [ L V10 , Section 10 . 3 ] based on the classical bar-cobar adjunction . Neither can we use the homo logical perturbation lem ma of [ Ber09 ]. No tice that the existence of the homotopy tr ansferred structure fo llows fro m mode l categor y arguments by [ Rez96 , BM03 ]. But we need here an explicit fo rmula for the application to Frobeniu s manifolds in th e next section. Needless to say th at the Hom otopy Transfer Theorem 6.2 hold s f or any algebra s over a qua si-free o p- erad generated by a h omotopy coo perad. In the case of a quasi-free op erad gen erated by a dg coop erad, K o szul models or bar-cobar resolutions for instance, we recov er the formulae of [ GCTV09 ] and of [ L V10 , Chapter 10 ] as follows. Proposition 6.3. Let P be a K o szul ope rad, eventually in homogeneous. Let A be a h omotop y P -algebra and let ( H, d H ) b e a homotopy r etract of the chain comple x ( A, d A ) . The transferr ed homo topy P -algebra structure on H given by [ GCTV09 , Theorem 4 7 ] and by [ L V10 , Theorem 10 . 3 . 6 ] is equal to the transferr ed homoto py P -algebra structure on H given by Theor em 6.2 . THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 27 Pr oof. The pro of relies on the followi ng diagram being commutative: Tw( P ¡ , E nd A ) ∼ = / / Hom dg coo p ( P ¡ , B E nd A ) G (End A ) ∗   Ψ ∗ / / Hom dg coo p ( P ¡ , B E nd H ) G (End H ) ∗   ∼ = / / Tw ( P ¡ , E nd H ) Tw ∞ ( P ¡ , E nd A ) ∼ = / / Hom coop ∞ ( P ¡ , B π End A ) Φ ∗ / / Hom coop ∞ ( P ¡ , B π End H ) ∼ = / / Tw ∞ ( P ¡ , E nd H ) .  The two homotopy transfer th eorems fo r h omoto py BV -a lgebras an d skeletal homotopy BV -alge bras commute un der the functor P ∗ : skeletal homotop y BV -algebras → homotopy BV -algebras as fol- lows. Proposition 6.4. Let ( H , d H ) be a homoto py r etract o f a cha in complex ( A, d A ) . Consider a skeletal ho- motopy BV -algebra structur e o n A . The associated ho motopy BV - algebra s tructur e P ∗ ( A ) on A transfer s to a h omotop y BV -a lgebra to H b y Theorem 33 of [ GCTV09 ] . This homo topy BV -algebra structur e o n H is eq ual to th e ho motopy BV -algebra associated, un der P , to the transferr ed skeletal BV -a lgebra given by Theor em 6.2 . Pr oof. The pro of relies on the commutativity of the fo llowing diagra m: Hom dg op (Ω π H , End A ) ∼ =   P ∗ & & Hom dg op (Ω π H , End H ) P ∗ x x Hom coop ∞ ( H , B π End A ) Φ ∗ / /     Hom coop ∞ ( H , B π End H )     ∼ = O O Hom ∞ - co op ∞ ( H , B π End A ) Φ ∗ / / p ∗ ∞   Hom ∞ - co op ∞ ( H , B π End H ) p ∗ ∞   Hom ∞ - co op ∞ ( B V ¡ , B π End A ) Φ ∗ / / Hom ∞ - coop ∞ ( B V ¡ , B π End H ) Hom coop ∞ ( B V ¡ , B π End A ) O O O O Φ ∗ / / Hom coop ∞ ( B V ¡ , B π End H ) O O O O Hom dg coop ( B V ¡ , B E nd A ) O O G (End A ) ∗ O O Ψ ∗ / / Hom dg coo p ( B V ¡ , B E nd H ) ∼ =   O O G (End H ) ∗ O O Hom dg op (Ω B V ¡ , E nd A ) ∼ = O O Hom dg op (Ω B V ¡ , E nd H ) .  6.3. Rect iﬁcation theorem for skeletal homot opy BV -alge bras. W e pr oved in [ GCTV09 , Pro position 32 ] the following Rectiﬁcation Theo rem: for any homoto py BV -algeb ra A , th ere is an ∞ -qu asi-isomorph ism A ∼ Ω κ B ι A o f ho motopy BV -alg ebras, wh ere Ω κ B ι A := B V ( B V ¡ ( A )) is a dg BV -algebra . W e refer to loc. c it. and to [ L V10 , Chapter 11 ] for mor e details. T o every sk eletal homotopy BV - algebra H , we deﬁn e its r ectiﬁ ed d g BV -algebr a by Rec( H ) := Ω κ B ι P ∗ ( H ) . Theorem 6. 5. Let ( H , d H ) be a homotopy r e tract of a chain complex ( A, d A ) . W e con sider a d g B V - algebra structu r e o n A together with the transferr ed skeletal homotopy B V - algebra o n H given by Th eo- r em 6.2 . The dg BV - algebra R ec( H ) is ho motopy equivalent to A in the cate gory of dg BV - algebras. 28 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE Pr oof. By Propo sition 6.4 , the hom otopy BV -algeb ra structur e P ∗ ( H ) is equal to the one produced b y the homoto py transfer theor em for h omotopy BV -algebr as [ GCTV09 , Th eorem 33 ]. Hence, there exists an ∞ -quasi-isom orphism of h omoto py BV -algeb ras A ∼ P ∗ ( H ) by Th eorem 10 . 4 . 7 of [ L V10 ]. The Rectiﬁcation Theorem for homotopy BV -algebra s provid es us with an ∞ -quasi-isomorp hism P ∗ ( H ) ∼ Ω κ B ι P ∗ ( H ) . Finally , the two dg B V -algeb ras A ∼ ← − • ∼ − → • · · · • ∼ ← − • ∼ − → Ω κ B ι P ∗ ( H ) = Rec( H ) are linked by a zig-zag of quasi-isomorp hism of dg BV -alg ebras by Theorem 11 . 4 . 14 of [ L V10 ].  This theore m gives hom otopy co ntrol of the tra nsferred structu re. It play s a ke y role in the inter pretation of the main result in the next section. 7. F RO M B V - A L G E B R A S T O H O M OT O P Y F RO B E N I U S M A N I F O L D S W e apply the H omotopy T r ansfer theorem to endow the underly ing homology of a dg BV -algeb ra with Massey p roducts. When the in duced action of ∆ is tri v ial, we recover and extend up to hom otopy the Barannikov-Kontse vich -Manin Frobeniu s manifold structure. Applications of th is gen eral result are g iv en in Poisson geometry and Lie algebra cohomo logy and to th e Mirror Symmetry conjecture. 7.1. Massey products. W orking over a ﬁeld K , on e can always write the underly ing h omolog y ( H • ( A, d A ) , 0 ) of a dg BV -algebra A as a defor mation r etract of ( A, d A ) . Deﬁnition 7.1 . W e call Masse y-Batalin- V ilkovisky pr oducts the oper ations co mposing the transfer red skele- tal ho motopy BV -alg ebra structure on the h omolog y H ( A ) of a d g BV -algebra given by the Homotopy T r ansfer Theorem 6.2 . Recall that the homo logy of any dg commutative (associative) algebra carries partial Massey pr oducts , see [ Mas58 ]. For instance, the partial Massey triple-pro duct h x, y , z i is deﬁned for three homology classes x, y , z ∈ H ( A ) such that xy = 0 = y z as follows. Let ¯ x, ¯ y, ¯ z ∈ A be cycles which represent x , y , and z respectively and let a, b ∈ A such th at ¯ x ¯ y = da, ¯ y ¯ z = db . Th en the ch ain a ¯ z − ( − 1 ) | ¯ x | ¯ xb is a cycle. So it deﬁnes an ele ment h x, y , z i in H ( A ) / ( xH ( A ) + H ( A ) z ) . When the p artial Massey prod ucts are deﬁned, they are gi ven by the same formulae as the (uniform) Mass ey products, see [ L V10 , Sections 9 . 4 and 10 . 3 ]. For dg Lie algebras, partial Massey products were deﬁned by V .S. Retakh in [ Ret93 ]. The present Ma ssey- Batalin-V ilkovisky produ cts generalize both the p artial commutative and Lie Massey products. Theorem 6.5 shows that the data of the Massey products allo w one to reconstru ct the homotopy type of the initial dg BV -algeb ra. 7.2. T r ivialization of the action of ∆ . Proposition 7.2. Let A be a dg BV - algebra. If there exists a h omotop y retr ac t to the ho mology , which satisﬁes p (∆ h ) m − 1 ∆ i = 0 , for m ≥ 1 , then the transferr ed skeletal h omotop y BV -algebra o n homology forms a homotop y hyp er co mmutative algebr a Pr oof. The transferred o perations ∆ m under T heorem 6.2 are given by ∆ m := p (∆ h ) m − 1 ∆ i . The n, one conclud es with Pro position 4.3 .  A mixed chain complex is a g raded vector spac e A equipp ed with two an ti-commu ting square-zero operator s d and ∆ of respe cti ve de gree − 1 and 1 . Deﬁnition 7.3. Let ( A, d, ∆) be a m ixed chain complex. Non-commutative Hodge-to- de Rham de genera- tion data consists of a defor mation re tract ( A, d ) h % % p / / ( H ( A ) , 0) , i o o such that p (∆ h ) m − 1 ∆ i = 0 , for m ≥ 1 . THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 29 Deﬁnition 7.4. Th e compatibility relation Ker d ∩ Ker ∆ ∩ (Im d + Im ∆) = Im d ∆ = Im ∆ d between the operato rs d and ∆ of a mixed chain complex is called the d ∆ -condition . Lemma 7.5. [ DGMS75 , Proposition 5 . 17 ] A mixed c hain comple x ( A • , d, ∆) satisﬁes the d ∆ -conditio n if and only if ther e exist two sub-graded modules H • and S • of A • such that A n ∼ = H n ⊕ S n ⊕ dS n +1 ⊕ ∆ S n − 1 ⊕ d ∆ S n , wher e d H n = 0 , ∆ H n = 0 , and wher e the map s of the following commutative diagram are isomorp hisms S n ∆ ∼ = / / ∼ = d   ∆ S n ∼ = d   dS n ∼ = − ∆ / / d ∆ S n . A dg BV -alge bra, whic h satisﬁes this con dition, is c alled a Hodge dg BV - algebra by A. Losev and S. Shadrin in [ LS07 ]. (In this case, the obviou s homoto py h , which contracts A to its h omolog y H , is such that [ h, ∆] = h ∆ + ∆ h = 0 .) Deﬁnition 7.6. [ Par07 ] A m ixed cha in complex is c alled semi-classical if ev e ry homology class h as a representative in the kernel of ∆ . Proposition 7.7. Let ( A, d A , ∆) be a mixed chain. The following implications hold ( d ∆ -condition ) = ⇒ ( semi-classical ) = ⇒ ( NC Hodge-to-de Rham de generation data ) . Pr oof. The ﬁrst a ssertion is given by Lemma 7.5 . T o prove the second one, it is enoug h to write the homolo gy H ( A ) as a d eformatio n retract of A , with repr esentativ es in K er ∆ . In this case, ∆ i = 0 , which conclud es the p roof.  The existence o f NC Hodg e-to-de Rham degeneration data is th erefore the most general c ondition that naturally suppor ts th is notion of the trivialization o f the action of ∆ on the homo logy of a dg BV -alg ebra Examples. ⋄ Le t M be a compact K ¨ ahler manifold, with comp lex stru cture denoted by J . The space of differen- tial forms (Ω • ( M ) , d DR , ∆ := J d DR J ) f orms a dg BV -alg ebra which satisﬁes the d ∆ -condition , see P . D eligne, P . Grifﬁths, J. Morgan and D. Sullivan [ DGMS75 ]. (No tice that here the o perator ∆ h as order less than 1 ). ⋄ Le t M be a Calabi-Y au manifold. The Do lbeault complex o f anti-ho lomorp hic dif fer ential forms with coefﬁcients into holomorp hic polyvector ﬁelds (Γ( M , ∧ • ¯ T ∗ M ⊗ ∧ • T M ) , d := ¯ ∂ , ∧ , ∆ := div , h , i S ) is a dg BV -algebra satisfying the d ∆ -cond ition, see S. Barannikov and M. K ontsevich [ BK98 ]. This is an exten sion, fro m vector ﬁelds to p olyvector ﬁelds, of the K odaira-Spen cer d g Lie algebr a [ KS58 , KS 60 ], which encode s th e complex structures of a manifold. ⋄ Le t ( M , w ) be a Poisson manifo ld. The space o f differential fo rms (Ω • ( M ) , d DR , ∧ , ∆ := [ i w , d DR ]) form a dg BV -algebra , see [ K o s85 , Bry88 ]. When ( M , ω ) is a compact symplectic manifold of dimensio n n , O. M athieu proved in [ M at95 ] that M satisﬁes th e hard L efschetz con- dition, i.e. th e cup product [ ω k ] : H n − k ( M ) → H n + k ( M ) is an iso morph ism, for k ≤ n/ 2 , if and only if this dg BV -algebr a is semi-classical. S. Merkulov further proved th at this is equiv alen t to the d ∆ -conditio n in [ Mer98 ]. Th is is the case when M is a K ¨ ahler man ifold, see [ Bry88 ]. ⋄ Le t V b e ﬁnite dimensional vector s pace with b asis { v i } 1 ≤ i ≤ n . W e consider the free com mutative algebra A := S ( V ⊕ s − 1 V ∗ ) of function s on the cotangent bundle of V ∗ , equipped with the orde r 2 and degree 1 operator ∆ := P n i =1 ∂ ∂ v i ∂ ∂ v ∗ i . These data deﬁne the pr ototypical examp le o f BV - algebras, see [ BV81 ]. Any element w o f degre e − 2 such th at ∆( w ) = h w, w i = 0 gives rise to a dg BV - algebra ( A, d w := h w, −i , • , ∆ , h , i ) . One can ﬁnd dg BV - algebras of this typ e e quipped with NC Ho dge to de Rham degeneration data but which does no t satisfy the d ∆ -condition , see [ Par07 , Example 9 ] and [ T er08 , Section 3 . 2 ]. 30 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE 7.3. H omotopy Frobenius manifold. Theorem 7.8. Let ( A, d, • , ∆ , h , i ) be a d g BV -algebra with n on-co mmutative Hodge-to-de Rha m deg en- eration data. The un derlying homo logy gr o ups H ( A, d ) ca rry a homotop y hyp er co mmutative algebra structure , which extends the hypercommutative algebras of M. K o ntsevich and S. Barannikov [ BK98 ] , Y .I. Ma nin [ Man99 ] , A. Losev an d S. Shadrin [ LS07 ] , an d J.-S. P ark [ Par07 ] , and such that the r ectiﬁed dg BV -a lgebra Rec( H ( A )) is homotop y equ ivalent to A in the ca te go ry of dg BV -algebras. Pr oof. The transfe rred skeletal homotopy BV -algebra structure on homo logy gi ven by Theorem 6.2 forms a homo topy hyp ercommu tativ e alge bra by Proposition 7.2 . W e make exp licit the various construction s of [ BK98 ] as follows. When a dg BV -alg ebra satisﬁes th e d ∆ -condition , the re is a zig-zag of quasi-isomor phisms of dg Lie algeb ras (smooth formality) ( A, d, h , i ) (Ker ∆ , d, h , i ) ∼ o o ∼ / / ( H • ( A, ∆) ∼ = ( H • ( A, d )) , 0 , 0) . By [ L V10 , Th eorem 1 0 . 4 . 7 ] , the re exists an ∞ -qu asi-isomorph ism of d g Lie alge bras H ∼ Ke r ∆ , ex- plicitly g iv en by sums of binary trees with vertices labelled by • and with ed ges an d ro ot labelled by h ∆ . Normalizing each sum of tr ees of arity n by a factor 1 n ! , this p rovides a solution γ to the Maurer-Cartan equation in the dg Lie algebra Ho m( ¯ S c ( H ) , Ker ∆) , where ¯ S c stands for the non-co unital cofree co com- mutative coalgebr a. Th e twisted data (Hom( ¯ S c ( H ) , A ) , d γ := d + h γ , − i , • , ∆ , h , i ) fo rm a dg BV -algebra over the ring of fo rmal power series b S ( H ∗ ) with out con stant term. I ts hom ology with re spect to d γ is equal to Hom( ¯ S c ( H ) , H ) ∼ = b S ( H ∗ ) ⊗ H . The transferred comm utativ e pr oduct on homology b S ( H ∗ ) ⊗ H pro- vides us with the desired hyper commutative algebr a structure on H , see [ Man99 , Chapters 0 and 3 ] for the v arious equ iv alent deﬁnitions of a formal Fr obenius manifold. T rac ing through the aforeme ntioned construction s, one can see that the associated poten tial is given b y the same kind o f sums of labelled trees but with a nor malizing coefﬁcient given by th e n umber of autom orphisms of the trees. W e recover the explicit for mula of [ LS07 ]. Manin [ Man99 ] and P ar k [ P ar 07 ] use obstruction theor y , for which choices can be made to produ ce the above structure. The ﬁrst s tratum of operations composing the transferred homotopy hyp ercommu tativ e algebr a is equ al to the tree form ulae o f Losev-Shadrin as fo llows. Le mma 4.6 shows that the weigh t 1 part of Grav ∗ is isomorph ic to H Lie ¡ 1 . For any n ≥ 2 , the space Li e ¡ 1 ( n ) is one dimension and gener ated by th e elem ent, which in T c ( β ) is the su m of all b inary tree with vertices labe led by β . The image of such trees und er the formu la of Theore m 6.2 is m ade up of binary tre es with each vertex labelled by • , one leaf la belled by ∆ , and with edge s labelled by h . (One can see that the image of a maxim al n esting under the map Φ is given by labeling all in terior edges by h .) Unde r the d ∆ -conditio n, the relations p ∆ = ∆ i = h ∆ + ∆ h = ∆ 2 = 0 make many trees can cel and this produces the aforemen tioned Lo sev-Shadrin form ulae. The last assertion is a direct corollary of Theorem 6.5 .  Remarks. ⋄ First, this theo rem co nceptually explains the result of Barannikov-K ontsevich, Man in, Losev- Shadrin, and Park in term s of the hom otopy transfer th eorem, thereb y answering a question asked by the referee of [ Par07 , Section 5 ]. ⋄ Sinc e ther e is no differential on ho mology , the ﬁrst stratum of oper ations of this homoto py hy - percomm utative algebra satisﬁes the re lations of a n h ypercom mutative algebra. So Theor em 7.8 proves the existence of such a stru cture un der a weaker con dition (NC Hod ge-to-d e Rham d egen- eration data) than in [ BK98 , Man99 , Par07 ] ( d ∆ -cond ition, semiclassical). ⋄ Un like the f ramework of Frobenius man ifolds, we do not work here with cyclic unital BV -algeb ras. First, a cyclic BV -algebra is equ ipped with a n on-degene rate biline ar fo rm which forces its d imen- sion to be ﬁnite. The present method works in the inﬁnite dim ensional case. Then, the op erad which encodes BV -alg ebras with unit is not augmented, so it does not admit a minimal m odel. T o make a coﬁbran t replaceme nt explicit, one would need to use the more g eneral Koszul duality theory developed by J. Hirsh and J. Mill ` es in [ HM1 0 ]. ⋄ Fina lly , Th eorem 7.8 provides high er structur e on h omolog y , which is shown to be necessary to recover the homotopy type of the origin al dg BV -alg ebra and not to lose an y homoto py data when passing to hom ology , see also Ex ample 7.4 below . THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 31 In geometrica l term s, we have lifted the action of the Deligne-Mu mford -Knud sen moduli space of gen us 0 cu rves to an action of the open moduli space of genus 0 cur ves as f ollows. H • +1 ( M 0 ,n +1 ) α / / κ   End H ( A ) H • ( M 0 ,n +1 ) . f 7 7 n n n n n n n n n n n n The map f is the morp hism of o perads given by [ BK98 , Man9 9 , LS07 , Par07 ]. The ma p κ is the twisting K o szul morp hism f rom the cooper ad H • +1 ( M 0 ,n +1 ) given in [ Get95 ]. It sends th e coho mologica l class correspo nding to H 0 ( M 0 ,n +1 ) to the fun damental class o f M 0 ,n +1 . The c onstruction given in Theo rem 7.8 correspo nds to the map α , which is a twisting morp hism f rom the c oopera d H • +1 ( M 0 ,n +1 ) . Th e map κ vanishes ou tside the to p dimensional classes and the restr iction of the map α to these top dimensional classes is equal to the com posite f ◦ κ . Such a morp hism of operads f deﬁnes the genus zero part of wha t K o ntsevich-Manin call a Coh omolog ical Field Theo ry in [ KM94 ]. Deﬁnition 7.9. An genus 0 extended cohomological ﬁeld theory is a gr aded vector space H equ ipped with an operadic twisting morph ism H • +1 ( M 0 ,n +1 ) → End H . 7.4. An ex ample. L et us conside r the following n on-un ital dg commutativ e algeb ra A gener ated by the 5 generato rs x 3 , y 3 , z 7 , u 7 , an d v 8 , where the subscript indicates the homolog ical degree , satis fying the relation s A := ¯ S ( x, y , z , u , v ) / ( xu, y u, z u, xv , y v , z v , uv , v 2 ) . (The product by u an d by v is eq ual to zero.) Th e differential map is deﬁned on the generators by dz := xy , dv := u , and by 0 otherwise. The alg ebra A is ﬁnite dimension al and spanned by the 9 elements: x, y , xy , z , u , v , xz , y z , xy z . Its under lying ho mology H • ( A, d ) is ﬁve dimensional and spanned by the classes of: x, y , xz , y z , xy z . W e d eﬁne the degree +1 o perator ∆ on the aforeme ntioned elements by ∆( xy ) := u , ∆( z ) := − v , and by 0 otherwise. Proposition 7.10. The dg co mmutative algebra ( A, d, ∆) is a dg BV -algebra, which satisﬁes the d ∆ - condition . Pr oof. It is straigh tforward to see that ∆ commu tes with d , that it has o rder less th an 2 (but not less than 1 ) and that it squares to 0 . A decomposition such as the one of Lemma 7.5 is given by H • := K x ⊕ K y ⊕ K xz ⊕ K y z ⊕ K xy z an d S • = K z . Therefo re, this dg BV - algebra satisﬁes the d ∆ -condition .  The ﬁrst Ma ssey product in the second stra tum of th e transferr ed homoto py hypercommu tati ve algebra structure is the ﬁrst ho motopy in the ass ociated C ∞ -algebra structure, since S − 1 Grav ∗ (2) (3) ∼ = C om ¡ (3) . In the presen t examp le, this prod uct is not trivial since it is equal to − y z o n the elements x, y , y . So this provides an example of a dg BV -alg ebra, whic h satisﬁ es the d ∆ -condition, the stron gest condition, an d for which the Barannikov-Kontse vich -Manin structure of a Froben ius manifold o n homolog y is n ot enough to recover the orig inal homotopy type of the dg BV -alg ebra. 32 GABRIEL C. DRUMMOND-COLE AND BR UNO V ALL ETTE 7.5. Application to Poisson geometr y and Lie algebra cohomology . L et M b e an n -d imensional man- ifold. W e consider the Ger stenhaber algebra of polyvector ﬁelds A := Γ( M , Λ • T M ) on M , equippe d with the Scho uten-Nijen huis b racket h , i S N . Recall fr om J.-L. Koszul [ K o s85 , Proposition (2 . 3 ) ] that any torsion-fr ee conn ection ∇ o n T M which induce s a ﬂat co nnection on Λ n T M giv es rise to a square-zer o order 2 op erator D ∇ making ( A, ∧ , D ∇ , h , i S N ) in to a BV -algebr a. For instance, this is the case when M is orientable with volume form Ω o r when M is a Riema nnian manifold with the Levi-Ci vita connection . Moreover , if M carries a Poisson structure, i.e. w ∈ Γ( M , Λ 2 T M ) satisfying h w, w i S N = 0 , such that the inﬁn itesimal automo rphism D ∇ ( w ) = 0 vanishes, then th e twisted d ifferential d w := h w, −i S N induces a dg BV -algeb ra (Γ( M , Λ • T M ) , d w , ∧ , D ∇ , h , i S N ) . For in stance, this is the case when M is orientable with unimodula r Poisson stucture, i.e. D Ω ( w ) = 0 . The h omolog y group s associated to th e d ifferential d w form the P o isson cohomology of the manifold M , see [ Lic77 ]. (For similar constructio ns in non-commutative geom etry , we re fer the reader to [ GS10 ]). Proposition 7.11. [ Kos85 ] When M is a symp lectic manifold , the contraction with the symplectic form ω induces an isomorphism of dg BV -algebras (Ω • ( M ) , d DR , ∧ , ∆ , h , i ) ∼ = (Γ( M , Λ • T M ) , d w , ∧ , D , h , i S N ) , wher e D := [ i ω , d w ] . Recall that the h omolog y groups associated to the differential ∆ on the left- hand side fo rm the P oisson homology of the manifold M . The Poisson homolo gy and co homolo gy are proved t o be isomorphic under the weaker condition that the Poisson manifold is orientable and unimodular, see P . Xu in [ Xu99 ]. Theorem 7.12. The de Rham c ohomology of a P o isson man ifold M carries a skeletal hom otopy BV - algebra, who se r ectiﬁed dg BV -algebra is h omotopy equivalen t to the d g BV -algebra (Ω • ( M ) , d DR , ∧ , ∆) . The P o isson co homology of an orientable P oisson manifold M c arries a skeletal homoto py BV -algeb ra, whose r ectiﬁed d g BV -a lgebra is homotop y equ ivalent to th e dg BV -algebra (Γ( M , Λ • T M ) , d w , ∧ , ∆ , h , i S N ) . The de Rh am coho mology an d the P oisson cohomology of a symplectic manifold are isomorphic skeletal homotop y BV -algebras. When the manifold M is co mpact and sa tisﬁes th e har d Lefsechtz co ndition, th is isomorphism r ed uces to an isomorphism of homotopy hyper co mmutative algebr as. Pr oof. This is a d irect corollary of Theorem 7.8 and Proposition 7.11 .  Let us now describe the linear case. Un der the same n otations as in the last example o f Section 7.2 , when V = g ∗ is the linear dual of a ﬁnite dimension al Lie algebra, the transpose of the bracket produces a degree − 2 elem ent w in g ⊗ Λ 2 g ∗ satisfying h w , w i = 0 , by the Jacobi relation . In this case, the twisted differential d w is equal to the Che valley-Eilenberg d ifferential on A ∼ = S ( g ) ⊗ Λ ( g ∗ ) ⊂ C ∞ ( g ∗ ) ⊗ Λ( g ∗ ) , which compu tes the cohomo logy o f g with co efﬁcients in S ( g ) a nd the adjoin t action. If the L ie algebra g is unimodular, th at is Tr ( h x, −i ) = 0 , for any x ∈ g , then ∆( w ) = 0 a nd the C hev alley-Eilen berg com plex ( S ( g ) ⊗ Λ( g ∗ ) , d w , • , ∆ , h , i ) is a dg BV -algebr a. Theorem 7.13 . The Chevalley-Eilenber g coh omology H • C E ( g , S ( g )) of a ﬁnite dimensional u nimodu lar Lie algebr a g , with coefﬁcients in S ( g ) with adjoint a ction, carries a s keletal homotopy BV -algebra, whose r ectiﬁed d g BV - algebra is homo topy equivalent to the dg BV -algebra ( S ( g ) ⊗ Λ( g ∗ ) , d w , • , ∆ , h , i ) . Remark. It would b e now in teresting to study the relationship with the Du ﬂo isomor phism, the analog ue of the space of differential forms, and the s ymplectic and the hard Lefschetz conditio n, in this line ar case. 7.6. Application to Mirror Symmetry. Theorem 7. 14. The D olbeault cohomo logy of a Calabi- Y au manifold carries a ho motopy hyper commu- tative algebra structu r e, w hich extends th e hypercommutative algebra structur e of [ BK98 ] an d whose r ecti- ﬁed d g BV -algebra is ho motopy equivalen t to the Dolbea ult comp lex (Γ( M , ∧ • ¯ T ∗ M ⊗∧ • T M ) , ¯ ∂ , ∧ , div , h , i S ) . The mod uli space M of Ma urer-Cartan elements associated to the Dolbeault comp lex is an extension of the mod uli space M classical associated to th e K o daira-Spen cer dg Lie subalg ebra, which enco des defo rma- tions of comp lex s tructures. The n otion of generalized complex g eometry was introduc ed by N. Hitchin in THE MINIMAL M ODEL FOR THE BA T ALIN-VILKO VISKY OPERAD 33 [ Hit03 ] and then developed by h is students M. Gualtieri [ Gua0 4 ] and G.R. Ca valcanti [ Ca v05 ] as a fr ame- work which enco mpasses both comp lex an d symplectic g eometries. In this sen se, the moduli space M was s hown b y Gualtieri to co rrespond to deformations of gener alized co mplex s tructures. Several v ersions of the d ∆ - condition were shown to hold in this setting, see [ A G0 7 , Ca v07 ]. Finally the d g BV algebra structure of [ Li05 ] allo ws us to apply the same argument wh ich produ ces a version of Theorem 7.14 i n the context of generalized complex g eometry . S. Barannikov gene ralized in [ Bar02 ] the notions of periods and v ar iations o f Hodge structure from M classical to M . He showed, for in stance, that th e image o f these generalized p eriods on H • ( M , C ) coincide with th e Gromov-Witt en inv ariants. Th is is based o n the fact that the Dolbeault coho mology admits not o ne but a family of Frob enius manifold structures. This remark coin cides with the presen t approa ch: there are many choices in the Homotopy Transfer theorem . M oreover , the various transferr ed structures are related b y the grou p of ∞ -isomo rphisms, see [ L V10 , T heorem 10 . 3 . 15 ]. In the c ase o f homoto py BV - algebras, this group should be related to the Gi vental group [ Giv01a , Gi v 01b ]. The Mir ror Symm etry conjectu re [ K on 95 ] claims that the Fuk aya A ∞ -category of L agrang ian sub- manifold s of a Calabi-Y au ma nifold M (A-side) should b e equiv alen t to th e bo unded de riv ed category of coheren t sh eav es on a dual Calab i-Y au man ifold f M (B-side). Th e tangen t space of the moduli space of A ∞ - deform ations of the Fukaya category is co njectured to be given b y the de Rham coh omolog y H • DR ( M , C ) of X . By the Kontse vich for mality [ K o n03 ], the A ∞ -defor mations o f the latter categor y are encod ed by the Dolbeau lt complex. So the de Rh am coho mology equipped with the Gr omov-Wit ten inv ariants should be isomor phic to th e Dolbeau lt cohomolo gy H • ( f M , Λ • T f M ) as Fro benius manifo lds. T he following con- jecture of Cao-Zh ou [ CZ01 ], similar to Prop osition 7. 11 , giv es a way to study this q uestion: there is a quasi-isomo rphism of dg BV -algebra s (Ω n −• ( M ) , d DR , ∧ , ∆ , h , i ) ∼ − → (Γ( M , ∧ • ¯ T ∗ f M ⊗ ∧ • T f M ) , ¯ ∂ , ∧ , div , h , i S ) . The results o f the present pa per show th at it is ac tually enou gh to prove the existence of an ∞ -quasi- isomorph ism of dg BV -algebr as to get the aforem entioned isomorp hism on the cohomolo gy lev el and to relate the two associated deformation functors. A C K N OW L E D G E M E N T S W e a re grateful to Damien Calaque, Cl ´ ement Dupon t, Vladimir Dotsen ko, Jean-Lo uis Lod ay and Jim Stasheff for the ir useful comments on the ﬁrst version of this paper . B.V . would like to expr ess his deep gratitude to the Max -Planck Institute f ¨ ur Math ematik in Bon n for the lo ng term invitation and fo r the excellent work ing condition s. 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E V A N S T O N , I L 6 0 2 0 8 - 2 7 3 0 , U S A E-mail addr ess : gabriel@ math.northwest ern.edu M A X - P L A N C K - I N S T I T U T F ¨ U R M A T H E M AT I K , V I V AT S G A S S E 7 , 5 3 1 1 1 B O N N , G E R M A N Y E-mail addr ess : brunov@u nice.fr

The minimal model for the Batalin-Vilkovisky operad

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