Algebraic string bracket as a Poisson bracket

Algebraic string bracket as a Poisson bracket

Hossein Abbaspour, Thomas Tradler and Mahmoud Zeinalian

Abstract.In this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman’s results using string topology operations.The main result can be applied to the de Rham complex of a smooth manifold as well as to the Dolbeault resolution of the endomorphisms of a holomorphic bundle on a Calabi–Yau manifold.

Mathematics Subject Classification(2010).55P35, 53D20, 14A22, 14A20.

Keywords.Free loop space, cyclic homology, symplectic reduction.

1. Introduction

Goldman’s original work [Go] on the Lie algebra of free homotopy classes of ori- ented closed curves on an oriented surface was extensively generalized through the introduction of String Topology by Chas and Sullivan [CS]. In particular, they gener- alized this Lie bracket to one on the equivariant homology of the free loop space of a compact and oriented manifoldM. From the beginning, it was clear that this bracket had a deep relation to the holonomy map on a vector bundle; see [Go], [CFP], [CCR], [CR]. This relation is the subject of a paper, [AZ], by the first and third author. It is shown there that using Chen’s iterated integral one obtains a map of Lie algebras from the equivariant homology of the free loop space to the space of functions on a space of generalized flat connections.

Algebraic analogues of string topology Lie algebra have also been considered in recent years. Jones [J] had shown that for a simply connected topological space X the equivariant homology of the free loop space is isomorphic to the negative cyclic cohomology of the algebra of cochains onX. Using this, and Connes’ long exact sequence relating negative cyclic cohomology and Hochschild cohomology, together with the BV-algebra on Hochschild cohomology, Menichi [Men] deduced a Lie bracket on the negative cyclic cohomology in a way similar to the one in string topology [CS], Section 6.

The starting point for this work was to obtain a generalization of the results in [AZ] and place it in a more algebraic setting where the equivariant homology of the

loop space is replaced by negative cyclic cohomology. A suitable setting for this is to consider a unital differential graded algebraA over a fieldk D R orC, with a reasonable trace TrW A!k. Using the results of [T], the above assumptions imply an isomorphism HH.A; A/ Š HH.A; A/of the Hochschild cohomologies ofA with values inA and its dualA such that the cup product on HH.A; A/and the dual of Connes’B-operator on HH.A; A/make these spaces into a BV-algebra.

This BV-algebra, together with Connes’ long exact sequence between the Hochschild cohomology HH.A; A/ and negative cyclic cohomology HC.A/, imply a Lie algebra structure on HC.A/by a theorem of Menichi [Men], Proposition 7.1, which is based on a similar marking/erasing result of Chas and Sullivan [CS], Theorem 6.1.

Now, using work of Gan and Ginzburg in [GG], we may look at the moduli space MC D fa 2A^oddjdaCaaD0g= (1) of Maurer–Cartan solutions. Since we only consider odd elements, the trace induces a symplectic structure! onMC, and thus one can define a Poisson bracket on the function ring O.MC/of MC. More details of this construction will be given in Section3.

We may connect the two sides of the above discussion via a canonical map fa 2 A^odd j daCaa D 0g ! HC.A/,a 7! P

n01˝a^˝n, and dualizing this gives a mapW HC.A/!O.MC/. We may now compare the two Lie alge- bras from above. Our main result then states that the brackets are indeed preserved.

Theorem 1. W HC².A/!O.MC/is a map of Lie algebras.

In a special case considered in [AZ] this map becomes the generalized holonomy map from the equivariant homology of the free loop space of M to the space of functions on the moduli space of generalized flat connections on a vector bundle E!M. In fact one has a commutative diagram

HC².A/ ^//O.MC/

H₂^S1.LM /;

‰

88r

rr rr rr rr

r ffMMMMM

MMMMM (2)

where‰ is the generalized holonomy discussed in [AZ] and comes from Chen’s iterated integral map, as described in Section5. In particular, for dimM D 2, this recovers Goldman’s results on the space of flat connections on a surface.

Another motivation for this work was the study of the algebraic structure of the deformation complex of a CY manifold. We were interested in the Hochschild and cyclic complexes of A D ^0;.M;End.E//, the Dolbeault resolution of the en- domorphisms of a strong generatorE of the category of perfect complexes on an algebraic CY manifold. The discussion of this paper, once done at the chain level, relates to the algebraic structure of the deformation complex.

Finally, we remark that the above discussion generalizes in a straight forward way to the case of a cyclicA₁-algebraA. This will be the topic of the last Section6. In fact, by the same reasoning as above, we obtain the Lie bracket on the negative cyclic cohomology HC.A/. Also, by symmetrization we may associate anL₁-algebra to A, which induces a Maurer–Cartan space similar to (1). We find that the canonical mapis still well defined such that Theorem1also remains valid in this generalized setting.

Notation. For a map F of complexes, F (resp.F) denotes the induced map in homology (resp. cohomology).

Acknowledgments. The authors would like to thank Victor Ginzburg and Luc Menichi for useful discussions and correspondence on this topic. The authors were partially supported by the Max-Planck Institute in Bonn and the second author warmly thanks the Laboratoire Jean Leray at the University of Nantes for their invitation through the Matpyl program.

2. The Lie algebra HC.A/

In this section, we recall the Lie algebra structure of the negative cyclic cohomology HC.A/, for a dga.A; d;/with a trace TrW A!k. The Lie bracket comes from the long exact sequence that relates negative cyclic (co-)homology to Hochschild (co-)homology. For simplicity, we will work in the normalized setting.

Definition 2. Let.A D L

i2ZAⁱ; dW Aⁱ ! A^iC1; /be a differential graded as- sociative algebra over a field k, and let M D L

i2ZMⁱ be a differential graded A-bimodule. The (normalized) Hochschild chain complex is defined as

Cx.A; M /´ Q

n0M ˝ NA^˝n; (3)

whereANDA=kandsdenotes shifting down by one. The boundaryıW xC.A; M /! CxC1.A; M /is defined by,

ı.a₀˝a₁˝ ˝a_n/

´ Pⁿ

iD0

.1/ⁱa₀˝ ˝d.a_i/˝ ˝a_nCⁿ¹P

iD0

.1/ⁱa₀˝

˝.a_ia_iC1/˝ ˝a_n.1/ⁿ⁰.a_na₀/˝a₁˝ ˝a_n1; wherea₀ 2M,a₁; : : : ; a_n2A,₀ D ja₀j,_i D.ja₀j C C ja_i1j Ci 1/, and _n⁰ D .ja_nj C1/.ja₀j C C ja_n1j Cn1/. Note that the differential is well defined; see [L]. Similarly, the (normalized) Hochschild cochain complex is defined by

Cxⁿ.A; M /´ ff WsAN^˝n!M jf .a₁˝ ˝a_i˝ ˝a_n/D0ifa_i D1g: (4)

Here the differentialıW xC.A; M /! xC¹.A; M /is given by .ıf /.a₁˝ ˝a_n/

´ Pⁿ

iD1

.1/^jfjCif .a₁˝ ˝da_i˝ ˝a_n/Cd.f .a₁˝ ˝a_n//

Cⁿ¹P

iD1

.1/^jfjCif .a₁˝ ˝.a_ia_iC1/˝ ˝a_n/ C.1/^jfj.ja1jC1/a₁f .a₁˝ ˝a_n/

C.1/^jfjCnf .a₁˝ ˝a_n1/a_n: The respective (co-)homology theories are denoted by

HH.A; M /DH.Cx.A; M /; ı/; HH.A; M /DH.Cx.A; M /; ı/:

Denoting byADHom.A; k/the graded dual ofA, we see that the dual ofCx.A; A/

is given byCx.A; A/. Recall furthermore that there is a cup productYonCx.A; A/

defined by

.f Yg/.a₁˝ ˝a_mCn/´f .a₁˝ ˝a_m/g.a_mC1˝ ˝a_mCn/:

Next, we define the (normalized) negative cyclic chains CC.A/ofAto be the vector spaceCx.A; A/ŒŒu, whereuis of degreeC2, and with differentialıCuB, whereBW xC.A; A/! xC₁.A; A/is Connes’ operator,

B.a₀˝a₁˝ ˝a_n/´ Pⁿ

iD0

.1/ⁱ1˝a_i˝ ˝a_n˝a₀˝ ˝a_i1; (5) where_i D .ja_ij C C ja_nj CniC1/.ja₀j C C ja_i1j Ci 1/. Thus, every element of CC_n.A/ is an infinite sum P₁

iD0a_iuⁱ 2 xC.A; A/ŒŒu, where a_i 2 xC_n2i.A; A/,ıacts ona_i 2 xC.A; A/, anduB acts as

x^uB C.A; A/u^{2 uB} xC.A; A/u x^uB C.A; A/: (6) Dually, define the (normalized) negative cyclic cochains CC.A/ofAby taking CC.A/ D xC.A; A/˝ kŒv; v¹=vkŒv, where v is an element of degree 2.

Explicitly, the degree n part CCⁿ.A/ is represented by finite sums P_k

iD0a_ivⁱ wherea_i 2 xCⁿ²ⁱ.A; A/. The differential is given byıCvB, whereıacts on Cx.A; A/, andvBacts as follows.

! x^vB C.A; A/v^{2 vB}

! x C.A; A/v^{1 vB}

! x C.A; A/:

Note that ifC.A; A/is finite dimensional in each degree, then the graded dual of CCⁿ.A/is isomorphic to the chain complex CC_n.A/DHom.CCⁿ.A/; k/; see also [HL], Lemma 3.7. It is easy to see that B² D ıB CBı D 0, and we define the associated (co-)homology theories by

HC.A/DH.CC.A/; ıCuB/; HC.A/DH.CC.A/; ıCvB/:

Lemma 3. If H.A; A/ is bounded from below, then both Cx.A; A/Œu and Cx.A; A/ŒŒuwith differentialıCuB calculate negative cyclic homologyHC.A/.

This lemma follows from a spectral sequence argument for the inclusion Cx.A; A/Œu ,! xC.A; A/ŒŒu, similarly to [HL], Lemma 3.6. Note that our sign convention is opposite to the one from [HL] but in agreement with [GJP] since our differentialıW xC.A; A/! xCC1.A; A/is of degreeC1.

From now on, we additionally assume that we also have a suitable trace map.

Definition 4. Let TrWA !k be a trace map satisfying Tr.da/D 0and Tr.ab/D .1/^jajjbjTr.ba/for alla; b2A. Assume furthermore that the map!WA!A,

!.a/.b/´Tr.ab/, is a bimodule map which induces an isomorphism on homology H.A/!H.A/. By abuse of language, we will also view!as a map!WA˝A!k,

!.a; b/DTr.ab/. In this case,Ais also called asymmetric algebra.

Notice that !W A ! A induces a morphism of the Hochschild complexes

!_]W xC.A; A/ ! xC.A; A/ via composition !_].f / ´ ! Bf, which is an iso- morphism on homology !_]W H.A; A/ ! H.A; A/. We may thus transfer the cup productY on H.A; A/to a productt on HH.A; A/ by settingf tg ´

!_]..!_]/¹f Y.!_]/¹g/. Define furthermore the operator W HH.A; A/ ! HH.A; A/as the dual ofBon homology. Then we assume that.HH.A; A/;t; / is a BV-algebra, i.e.,tis a graded associative, commutative product, ² D 0, and the bracketfa; bg ´.1/^jaj.atb/.1/^jaj.a/tbat.b/is a derivation in each variable.

Recall from Menichi [Men] that this BV-algebra induces a Lie algebra on the negative cyclic cohomology HC.A/using the long exact sequences of Hochschild and negative cyclic cohomology. The inclusion CC.A/ ^u! CC.A/ given by multiplication byuhas cokernelCx.A; A/. We thus obtain a short exact sequence

0! CC.A/!^u CC.A/! xC.A; A/!0; (7) which induces Connes’ long exact sequence of homology groups:

!HH_n.A; A/!^B HC_n1.A/!HC_nC1.A/!^I HH_nC1.A; A/! ^B : (8) Here the projection to theu⁰termIW CC.A/! xC.A; A/induces the mapI, and the connecting map Bis induced by the compositionCx.A; A/ ! x^B C.A; A/ !^inc CC.A/. Notice that unlike inc B BW xC.A; A/ ! CC.A/, the inclusion incW xC.A; A/! CC.A/is not a chain map.

Dually, we have the short exact sequence

0! xC.A; A/! CC.A/! CC.A/!0;

inducing Connes’ long exact sequence of cohomology groups !HHⁿ.A; A/ ^I

! HCⁿ.A/HCⁿ² .A/ ^B

! HHⁿ¹.A; A/ ^I

! : (9)

Notice that the composition

BDBBI (10)

is exactly theoperator of our BV-algebra on HH.A; A/so that we may obtain an induced Lie algebra from [Men], Lemma 7.2, much like the marking/erasing situation in [CS].

Proposition 5(L. Menichi [Men]). The bracketfa; bg ´I.B.a/tB.b//induces a Lie algebra structure onHC.A/.

We end this section with some examples of the above definitions.

Examples 6. LetM be a smooth, compact and oriented Riemannian manifold.

– A first example is obtained by takingAD .M /the de Rham forms onM, d Dd_DRthe exterior derivative onA, and Tr.a/´R

Ma.

– More generally, ifE !M is a finite dimensional complex vector bundle over M, with a flat connectionr, then we may takeAD.M;End.E//with the usual differentiald_r. Similarly, the trace is given by a combination of integration and trace in End.E/. The cyclic property of the trace guarantees that this induces an injective bimodule map!W A!Athat is a quasi-isomorphism.

– Both of the above examples are special cases of elliptic Calabi–Yau space as defined in [C]. By definition, this means that we have a bundle of finite dimensional associativeCalgebras overM, whose algebra of sections is denoted byA. Further- more, there is a differential operatordW A ! A, which is an odd derivative with d²D0makingAinto an elliptic complex, aClinear trace TrW A!C, a hermitian metricA˝A!C, and a complex antilinear,C¹.M;R/linear operatorWA!A satisfying certain natural conditions. It can be seen that this example satisfies the above assumptions. The details and other examples of elliptic Calabi–Yau spaces can be found in [C] and [DT].

3. Maurer–Cartan solutions

In this section we define the moduli space of Maurer Cartan solutions for a symmetric algebraA D L

i0Aⁱ and then explain its symplectic nature. The main reference for this section is the paper [GG] by Gan–Ginzburg together with Section 4 of [AZ].

Let us assumekDRorC.

Fora; b2Adefine the Lie bracketŒa; b´ab.1/^jajjbjbaand the bilinear form!.a; b/´Tr.ab/. The first remark is that.ADA^odd˚A^even; d; Œ;; !/is a cyclic differential graded Lie algebraas it is defined in Section 4 of [AZ]. Therefore all results in [GG] apply here to define the Maurer–Cartan solutions.

Definition 7. We define the Maurer–Cartan moduli stack as

M C ´ fa2A^oddjdaC¹₂Œa; aDdaCaaD0g;

MC ´M C =;

where the equivalence is generated by the infinitesimal action ofA⁰onA. Here for a2A⁰, the vector field_xonAis defined by

_x.a/DŒx; adx:

Recall that ! is a symplectic form and the infinitesimal action is Hamiltonian.

Moreover, the map Wa7!_a 2.A^even/, where .x/D!.daC¹₂Œa; a; x/

is the moment map corresponding to the Hamiltonian action above. One should think of the tangent spaceT_ŒaMCat a classŒaas the3-term complexT_ŒaMC:

T_Œa¹MC ´A^even !^.a/ T_Œa⁰ MC ´T_aA^oddDA^odd

0a

!T_Œa¹ MC´A^even; (11) graded by1,0and1. Here .a/is the mapx 7!_x.a/. The kernel of ⁰ is the Zarisky tangent space toM C, and the image of .a/accounts for the tangent space of the action orbit. Ideally, when0is a regular value for and the infinitesimal action ofA^evenonM C D ¹.0/is free, this complex is concentrated in degree zero, and the Zarisky tangent space toMC atŒais the cohomology groupH⁰.T_ŒaMC/ D H.A^odd; d_a/whered_abDdbCŒa; b.

Note that the3-term complex (11) is self-dual, where the self-duality at the middle term is given by the symplectic form

!.X_a; Y_a/´Tr.X_aY_a/2k: (12) By assumption from the previous section, ! is non-degenerate. This gives rise to an isomorphismT_ŒaMC !^' .T_ŒaMC/and equips.T_ŒaMC/with a symplectic form given by (12). In the case of a nonsingular pointŒathis is the usual pairing on H⁰.T_ŒaMC/DH.A^odd; d_a/induced by!.

The function spaceO.MC/is defined to be the subspace ofO.M C /invariant un- der the infinitesimal action. The symplectic form allows us to define the Hamiltonian vector fieldX of a function 2O.MC/via

!.X_a; Y_a/Dd _a.Y_a/´lim

t!0 d

dt .aCt Y_a/ for allY_a2T_Œa¹ MC: We then define the Poisson bracket onO.MC/by

f ; g ´!.X ; X/DTr.X X/:

4. The induced Lie map

In this section we define a mapW HC².A/ !O.MC/and prove that it respects the brackets. We start by defining a mapPWM C ! xC.A; A/, and in turn the map RW M C ! CC.A/which factors throughP. DualizingRwill induce the wanted map.

Definition 8. Recall thatM C D fa 2 A^odd j daCa a D 0gand Cx.A; A/ D Q

n0A˝ NA^˝n. Let thenPW M C ! xC.A; A/be given by the expression P .a/´ P

i0

1˝a^˝i D.1˝1/C.1˝a/C.1˝a˝a/C :

Note thatı.P .a//DP1˝a˝ ˝da˝ ˝aCP1˝a˝ ˝.aa/˝ ˝aD0 fora 2M C, due to the relationdaCaa D0inM C. Thus, we obtain in fact a Hochschild homology classŒP .a/2HH.A; A/.

Next define the mapR´incBP as the compositionRW M C ! x^P C.A; A/!^inc CC.A/. Just as above, we have thatı.R.a//D0, and since we are in the normalized setting, we see thatB.R.a//D0and so.ıCuB/.R.a//D0. The induced negative cyclic homology class is again denoted byŒR.a/2HC.A/. It is immediate to see that under the long exact sequence (8), we have thatI.R.a//DP .a/.

Using the pairing between negative cyclic homology and negative cyclic coho- mology,h ; iW HC.A/˝HC.A/!k, we define the mapby

W HC.A/!O.MC/; .Œ˛/.Œa/´ hŒ˛; ŒR.a/i D h˛; R.a/i;

forŒ˛2HC.A/,Œa2MC. To simplify notation, we will also write.˛/instead of.Œ˛/.

Lemma 9. is well defined.

Proof. We need to show that the value.Œ˛/.Œa/D h˛; R.a/iis independent of the choice of the representativeŒa2 fx2A^oddjdxCxxD0g=. Infinitesimally, this amounts to showing thatL_X.b/.˛/.a/D0, whereL_X.b/is the Lie derivative along a vector field in the directionX.b/_a DdbCŒa; b2T_ŒaM C, for anyb2A^even. To see this, note that

L_X.b/.˛/.a/D.i_X.b/Bd CdBi_X.b//.˛/.a/

Di_X.b/Bd..˛//.a/

D h˛;_dt^dj_tD0R.aCtX.b/_a/i:

Now, for anyY_a 2T_ŒaM C, we have

d

dtj_tD0R.aCt Y_a/D1˝Y_aC1˝Y_a˝aC1˝a˝Y_aC

DB.Y_aC.Y_a˝a/C.Y_a˝a˝a/C /; (13)

where we used Connes’ operator BW xC.A; A/ ! CC.A/ from the long exact sequence (8) applied toY_aC.Y_a˝a/C.Y_a˝a˝a/2 xC.A; A/. Thus, setting Y_aDX.b/_aDdbCŒa; bin the above expression, we obtain

L_X.b/.˛/.a/D h˛;B

dbCŒa; bCdb˝aCŒa; b˝a Cdb˝a˝aCŒa; b˝a˝aC

i D h˛;BBı.bC.b˝a/C.b˝a˝a/C /i D h˛; ıBB.bC.b˝a/C.b˝a˝a/C /i D hı˛;B.bC.b˝a/C.b˝a˝a/C /i

D0:

We are now ready to prove our main theorem.

Theorem 1. W HC².A/!O.MC/is a map of Lie algebras.

Proof. We saw in (13) that_dt^dj_tD0R.aCt Y_a/DB.Y_aC.Y_a˝a/C /2 CC.A/, where.Y_aC.Y_a˝a/C.Y_a˝a˝a/C /2 xC.A; A/forY_a2T_ŒaMC. Therefore,

.d.˛//_a.Y_a/D h˛;_dt^dj_tD0R.aCt Y_a/i

D h˛;B.Y_aC.Y_a˝a/C.Y_a˝a˝a/C /i D hB˛; Y_aC.Y_a˝a/C.Y_a˝a˝a/C i D.B˛/.1CaCa˝aC /.Y_a/;

where˛ 2 CC.A/;B˛ 2 xC.A; A/, and thus.B˛/.P

i0a^˝i/ 2A. Now, using the isomorphism!_]W HH.A; A/!HH.A; A/from Definition4, we apply its inverse to obtain an elementŒf_˛ ´.!_]/¹BŒ˛2HH.A; A/. We then claim that the Hamiltonian vector fieldX_a^.˛/may be expressed as

X_a^.˛/Df_˛ P

i0

a^˝i

2T_ŒaMC: (14) This should be compared with [AZ], Lemma 7.2, and [Go], Proposition 3.7. To this end, first note that the relation 0 D .ıf /.P

i0a^˝i/ D d_a.f .P

i0a^˝i//, for f 2 xC.A; A/, shows thatX_a^.˛/ given by equation (14) represents a well-defined class inT_ŒaMC. We show (14) by applying the non-degeneracy of!in the following equation, which is valid for anyY_a2T_ŒaMC:

!.f_˛.P

a^˝i/; Y_a/DTr.f_˛.P

a^˝i/Y_a/ D.!_]f_˛/.P

a^˝i/.Y_a/ D.B˛/.P

a^˝i/.Y_a/ D.d.˛//_a.Y_a/ D!.X_a^.˛/; Y_a/:

Now, calculating the Lie bracket gives .f˛; ˇg/.a/D hfŒ˛; Œˇg; ŒR.a/i

D hI.BŒ˛tBŒˇ/; ŒR.a/i

D hI!_]..!_]/¹BŒ˛Y.!_]/¹BŒˇ/; ŒR.a/i D h!_].Œf_˛YŒf_ˇ/; IŒR.a/i

D h!_].Œf_˛YŒf_ˇ/; ŒP .a/i:

To evaluate this expression, note that for f_˛W NA^˝m ! A and f_ˇW NA^˝n ! A,

!_].Œf_˛YŒf_ˇ/is represented by the composition

AN^˝mCn!^f˛˝fˇ A˝A! A!^! A:

The first arrow withf_˛˝f_ˇ applied toP .a/D1C.1˝a/C.1˝a˝a/C 2 Q

i0A˝ NA^˝i then gives an expression, where we applyato all possible inputs in AN^˝nCm. To this, we then apply the product inA, and apply!with input12Asince P .a/D1˝. : : : /. We thus obtain

.f˛; ˇg/.a/DTr.f_˛.1CaCa˝aC /f_ˇ.1CaCa˝aC /1/

(14)D Tr.X_a^.˛/X_a^{.ˇ /}/D!.X_a^.˛/; X_a^{.ˇ /}/D f.˛/; .ˇ/g.a/:

This is the claim of the theorem.

5. Comparison with generalized holonomy

In this section we compare the mapwith the generalized holonomy map‰studied in [AZ]. The relationship may be summarized in diagram (2). This shows how a special case the result of this paper relates to the main theorem of [AZ]. The map TrW A ! C is induced by the trace function ong GL.n;C/and integration of forms onM; see Example6.

Our model ofS¹-equivariant de Rham forms ofLM is..LM /Œu; d Cu/

whereuis a generator of degree2andW .LM /!¹.LM /is the map induced by theS¹-action onLM; see [GJP]. This model is quasi-isomorphic to the small Cartan model.inv.LM /Œu; dCi_Xu/for theS¹-action, whereXis the fundamental vector field generated by the natural action ofS¹. The quasi-isomorphism is given by the averaging map.LM / !_inv.LM /. More explicitly, for! 2.LM /, .!/is given by

.!/DZ

fibre

ev.!/2¹.LM /:

S¹LM ^{ev //}

LM

LM

(15)

Chen’s iterated integral map and the trace map ong(see (6.3) [AZ], and Theorem A in [GJP]) yields a map, which we denote by

SW .Cx.A; A/; ı/!..LM /; d /:

The map S induces the map S^HHW HH.A; A/ ! H.LM / on homology, and, after applying the pairing between homology and cohomology groups, we get

H.LM /

!HH HH.A; A/:

ExtendingS byu-linearity, we obtain a map, which we denote by abuse of notation by the same letter,

SW .Cx.A; A/Œu; ıCuB/!..LM /Œu; d Cu/:

Since, by Lemma3,.Cx.A; A/Œu; ıCuB/and.Cx.A; A/ŒŒu; ıCuB/are quasi- isomorphic in our setting, we obtain the induced mapS^HCW HC.A/!H_S1.LM / on homology. ComposingS^HC with the mapRW M C ! CC.A/D xC.A; A/Œu from Section4, we get

M C !^R HC.A/ ^S

!HC H_S1.LM /:

Thus by duality, and using Lemma9, we have H^S1.LM / ^D

!HC HC.A/! O.MC/:

The compositionB is the generalized holonomy map‰discussed in [AZ].

HC.A/ ^//O.MC/

H^S1.LM /

‰

88r

rr rr rr rr

r ffLLLLL

LLLLL (16)

It was proved in [AZ] that‰ is the morphism of Lie algebras. We will shortly see how this is a consequence of Theorem1. We first recall the following theorem.

Theorem 10(S. Merkulov [Mer]). The Chen integral induces a map of algebras .H.LM /;/!.HH.A; A/;Y/.

Thus, by definition, ^HHW .H.LM /;/ ! .HH.A; A/;t/is also a map of algebras. With this, we can now prove the following statement.

Theorem 11. The mapW .H^S1.LM /;f ; g/!.HC.A/;f ; g/induced by the Chen iterated integrals is a map of Lie algebras. Here, thefirst bracket is the string bracket and the second one is defined in the statement of Proposition5.

Proof. The brackets on H^S1.LM / and HC.A/ are determined by the products onH.LM / and HC.A; A/, together with the maps in the corresponding Gysin long exact sequences. By Theorem10, it thus remains to show that the long exact sequences correspond to each other, i.e., that the following diagrams commute:

: : : //H^S1.LM /

m //HC1.LM /

^HH

e //

H^S_C1¹.LM /

//H^S₁¹.LM / //: : : : : : //HC.A/ ^{B //}HH^C1.A; A/ ^I

//HC^C1.A/ ^//HC¹.A/ ^//: : : Equivalently, we need to show the commutativity of the dual sequence:

: : : //HC.A/ ^{I //}

S^HC

HH.A; A/ ^{B //}

S^HC

HC₁.A/ ^//

S^HH

HC₁.A/ ^//

S^HC

: : :

: : : //H_S1.LM / ^e //H.LM / ^{m //}H_S¹₁ .LM / ^//H_S¹₁ .LM / ^//: : : The top long exact sequence is induced by the short exact sequence (7), while the bottom one is induced by the short exact sequence

0!..M /Œu; dCu/!^u ..M /Œu; d Cu/!^j .M /!0; (17) where j.P

a_iuⁱ/ D a₀, cf. [GS], [Ma]. In this picture, m corresponds to the connecting map of the long exact sequence (17). By a diagram chasing argument one finds thatmD.iB/, whereiW .M / ,!.M /Œucorresponds toBD .incBB/ using Chen iterated integrals as corollary of Theorem A in [GJP]. Note thatiis not a chain map, whereasiBis a chain map sinced Ddand² D0, (cf. [GJP]).

6. A1-generalization

The previous sections, dealing with the case of dgas.A; d;/with invariant inner product!WA˝A!k, generalize in a straightforward way to the setting of cyclic A₁-algebras. In this section, we recall the relevant definitions (cf. [T]), and adopt the above to this situation.

Definition 12. AnA₁-algebra onAconsists of a sequence of mapsf _ng_n1, where

nWA^˝n!Ais of degree.2n/, such that, for alln1, P

kClDnC1 rD0;:::;nl

.1/^lr _k.a₁˝ ˝ _l.a_rC1˝ ˝a_rCl/˝ ˝a_n/D0;

where^r_l D.l1/.ja₁jC Cja_rjr/. A unit is an element12k A⁰such that

2.a; 1/D ₂.1; a/D a, and _n. ˝1˝: : : / D0forn¤2. Again, we write ANDA=k. We define the Hochschild chain complex ofAwith values inAorAto be the vector spacesCx.A; A/andCx.A; A/from equation (3) with the differentials modified as follows:

ıW xC.A; A/! xC.A; A/; ı.a₀˝ ˝a_n/ DP˙a₀˝ ˝ _k.: : : /˝ ˝a_n

CP˙ _k.a_s˝ ˝a₀˝ ˝a_r/˝a_rC1˝ ˝a_s1; ıW xC.A; A/! xC.A; A/; ı.a₀˝ ˝a_n/

DP˙a₀˝ ˝ _k.: : : /˝ ˝a_n CP˙

k.a_s˝ ˝a₀˝ ˝a_r/˝a_rC1˝ ˝a_s1; where _k.a_s˝ ˝a₀ ˝ ˝a_r/2Ais given by

k.a_s˝ ˝a_n˝a₀˝a₁˝ ˝a_r/.a/´ ˙a₀. _k.a₁˝ ˝a_r˝a˝a_s˝ ˝a_n//:

Here, the signs are given by the usual Koszul rule, where a factor of .1/⁰ is introduced whenever elements of degreeand⁰are being commuted. For an explicit discussion of the signs; see e.g. [T]. Similarly,Cx.A; A/andCx.A; A/are defined by the spaces from (4) with the modified differentials

ıW xC.A; A/! xC.A; A/;

ıf .a₁˝ ˝a_n/DP˙f .a₁˝ ˝ _k.: : : /˝ ˝a_n/ CP˙ _k.a₁˝ ˝f .: : : /˝ ˝a_n/;

ıW xC.A; A/! xC.A; A/;

ıf .a₁˝ ˝a_n/DP˙f .a₁˝ ˝ _k.: : : /˝ ˝a_n/ CP˙ _k.a₁˝ ˝f .: : : /˝ ˝a_n/:

Sinceı² D0,.ı/² D0in all the above cases, we obtain the associated homologies and cohomologiesH.A; A/,H.A; A/,H.A; A/, andH.A; A/.

There is a generalized cup productYonH.A; A/induced by .f Yg/.a₁˝ ˝a_n/´ P

k2

˙ _k.a₁˝ ˝f .: : : /˝ ˝g.: : : /˝ ˝a_n/:

Furthermore, equation (5) defines an operatorBW xC.A; A/! xC.A; A/withB²D ıBCBıD0. We define the negative cyclic chains CC.A/ofAto be the vector space

Cx.A; A/ŒŒuwith differentialıCuB, and denote the negative cyclic homology by HC.A/. Dualizing CC.A/, we obtain CC.A/with dual differential and denote the negative cyclic cohomology by HC.A/. For the same reasons as in Section2, we obtain the long exact sequences (8) and (9).

Finally, assume that we have a trace TrW A ! k such that the associated map

!W A˝A!k,!.a; b/DTr. ₂.a˝b//is a quasi-isomorphism, which satisfies

!. _n.a₁˝ ˝a_n/; a_nC1/D ˙!. _n.a_nC1˝a₁˝ ˝a_n1/; a_n/; (18) forn1. In this case,!W A!Ainduces a map of the Hochschild cohomologies H.A; A/!H.A; A/,!_].f /D!Bf, which we assume to be an isomorphism.

Thus, we may transfer the productY onH.A; A/to a product tonH.A; A/.

.HH.A; A/;t; DB/is a BV-algebra, cf. [T], so that we obtain the Lie bracket fa; bg ´I.B.a/tB.b//on HC.A/just as in Proposition5.

Using this setup, we may now also generalize Section3.

Definition 13. Recall that there are maps from then-th symmetric power of a vector space to the n-th tensor power SⁿW A^{^n} ! A^˝n, where Sⁿ.a₁ ^ ^a_n/ D P

2†n.1/.a_.1/˝ ˝a_.n//. Defining_nW A^{^n}!Aas_n´ _nBSⁿ, we obtain anL₁-algebra onA, cf. [LM], Theorem 3.1. Furthermore, from (18), it is immediate to see that, forn1, we have

!._n.a₁^ ^a_n/; a_nC1/D ˙ !._n.a_nC1^a₁^ ^a_n1/; a_n/:

For thisL₁-algebra, recall from [GG], Section 2, that the Maurer–Cartan solu- tions are defined by

M C ´ fa2A^oddj₁.a/C _2Š¹₂.a^a/C _3Š¹₃.a^a^a/C D0g;

MC ´M C =;

where the equivalence is again generated by the infinitesimal action ofA^evenonA^odd and where, fora2A^even, the vector field_xonA^oddis defined by

_x.a/D₁.x/C₂.a^x/C 1

2Š₃.a^a^x/C :

Note that under the above assumptions the tangent spaceT_ŒaMCtoMCatŒais the self-dual3-term complex

T_Œa¹MC ´A^even!^.a/ T_Œa⁰ MC´T_aA^oddDA^odd

0a

!T_Œa¹ MC ´A^even; (19) where

0a.b/D₁.b/C₂.a^b/C 1

2Š₃.a^a^b/C :

The self-duality at the middle term is given by the symplectic form

!.X_a; Y_a/DTr. ₂.X_a˝Y_a//2k:

This can be used to define the Hamiltonian vector fieldX associated to a function 2O.MC/, and thus the Lie bracket onO.MC/via the usual formulaf ; g D

!.X ; X/.

We may now define the mapPW M C ! xC.A; A/by P .a/´P

i01˝a^˝i D.1˝1_AN˝0/C.1˝a/C.1˝a˝a/C ; andR D i nc BPW M C ! CC.A/. As in Definition8, we may again see that ı.P .a//D0, and.ıCuB/.R.a//D0, and we define

W HC².A/!O.MC/; .Œ˛/.Œa/´ hŒ˛; ŒR.a/i D h˛; R.a/i;

forŒ˛2HC.A/,Œa2MC. With this, we have the same theorem as in the previous sections.

Theorem 14. The mapis a well-defined map of Lie algebras.

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Zbl 1144.55012 MR 2353864 Received January 28, 2009

H. Abbaspour, Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France

E-mail: [email protected]

T. Tradler, Department of Mathematics, New York City College of Technology, 300 Jay Street, Brooklyn, NY 11201, U.S.A.

E-mail: [email protected]

M. Zeinalian, Department of Mathematics, Long Island University, C.W. Post College, Brookville, NY 11548, U.S.A.

E-mail: [email protected]