s.a. merkulov- de rham model for string topology

8/3/2019 S.A. Merkulov- De Rham model for string topology

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arXiv:math/0

309038v3[math.A

T]25May2004

De Rham model for string topology

S.A. Merkulov

0. Introduction. Let M be a simply connected closed oriented n-dimensional manifold,and LM := MapC(S

1, M) the associated free loop space. String topology of Chas and Sul-livan [CS] deals with an ample family of algebraic operations on the ordinary and equivarianthomologies of LM, the most important being a graded commutative associative product,

: Hp(LM) Hq(LM) Hp+q(LM),

on the shifted ordinary homology1, H(LM) := H+n(LM). The product is compatible with(in general, non-commutative) Pontrjagin product,

: Hp(M) Hq(M) Hp+q(M),

in the homology of the subspace, M, of LM consisting of loops based at some fixed point inM, and the same product is also an extension of the classical intersection product in H(M):the natural chain of linear maps

(H(M), )constant

loops (H(LM),)

intersection withthe subspace M

(H(M), )

is a chain of homomorphisms of algebras.

Let (M =n

i=0 iM, d) be the De Rham differential graded (dg, for short) algebra of M.

With the help of simplicial techniques it was established in [J] that

H(LM) = ExtM

op

M(M,

M) , ()

i.e. that morphisms in the derived category, D(M), of M-bimodules between the two objects,M and its dual

M := HomR(M,R), have a nice free loop space interpretation. As M is

closed oriented, the Poincare duality asserts that M and M[n] are isomorphic in D(M) sothat () can be rewritten equivalently as

H(LM) = ExtMopM(M, M) .

Moreover, as it was shown in [CJ, C] using ring spectra and simplicial methods, the latterisomorphism is actually an isomorphism of algebras,

(H(LM),) =

ExtMopM(M, M) , Yoneda product

, ()

1In this paper all homologies are assumed to be over R.

1
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i.e. the Chas-Sullivan product is the same thing as the composition in the endomorphism ring ofthe object M in the derived category of M-bimodules. The isomorphism () was also studiedin the thesis of Tradler [Tr].

This paper offers new geometrically transparent (and down-to-earth) proofs of the string

topology main theorems which are based on the theory of iterated integrals [Ch]. In particular,the isomorphism () gets incarnated as the holonomy map. Which combined with the Thomclass of M explains why () is an isomorphism of algebras. As a by-product, the resultingDe Rham model for string topology provides us with new algorithms for computing both thefree loop space homology and the Chas-Sullivan product; see also [CJY, FTV, R] for otherapproaches.

In fact, our approach to string topology easily generalizes to its brane version. Letf : Z M be a smooth map from a compact oriented p-dimensional manifold to a compactsimply connected manifold M, and let Lf be defined by the associated pullback diagram,

Lf

GG LM

Z

fGG M .

The shifted homology, H(Lf) := H+p(Lf), is naturally an associative (but, in general, non-commutative) algebra with respect to the obvious analogue, , of the Chas-Sullivan product.

IfZ is a point, then Lf M and (H(Lf),) = (H(M), ). Chen [Ch] has found a nicemodel for the algebra (H(M), ) as the homology of the free differential algebra (R[X],),where R[X] := H(M)[1] is the tensor algebra generated by the reduced homology (withshifted degree) ofM and is a differential on R[X] which can be computed by a certain iterativeprocedure (see Lemma 2.1 below). In this paper we generalize Chens model of (H(M), ) to

the case of the Chas-Sullivan algebra, (H(Lf),), for an arbitrary smooth map f : Z M.More precisely, we establish two isomorphisms of algebras,

(H(Lf),) = (Z R[X], df) ,

and(Z R[X], df) = (Hoch

(M, Z) , Hochschild product) ,

where Z is the De Rham algebra of Z and df is a certain twisting of the natural differentiald + on Z R[X]. In the case f = Id : M M the above two results imply (). In thecase f : point M the same results reproduce the Chen model for the homology algebra of thebased loop space.

The paper is organized as follows: In Sections 1 and 2 we review basic facts of Chens theoryof iterated integrals and of a formal power series connection. In Section 3 we use the latter togive a suitable model for the Hochschild cohomology of a dg algebra with finite dimensionalcohomology and then illustrate its work in two standard examples. In Section 4 we use thetheory of iterated integrals to give a new proof of the isomorphism theorem (). The sections6-10 are devoted to a new proof of the isomorphism of algebras theorem (). In the final section11 we discuss the de Rham model for the string topology on a brane f : Z M.

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1. Iterated integrals. Let M be a smooth n-dimensional manifold and P M the space ofpiecewise smooth paths : [0, 1] M. For each simplex, k = 0, 1, 2, . . . ,

k :=

(t1, . . . , tk) Rk | 0 t1 t2 . . . tk 1

,

consider a diagram,P M k

evk GG

p

Mk+2

P M

where p is just the projection on the first factor and evk is the evaluation map,

evk : P M k Mk+2

(t1, . . . , tk) ((0), (t1), . . . , (tk), (1)).

According to Chen [Ch], the space of iterated integrals, Ch(P M), on P M is a subspace ofthe de Rham space, PM, of smooth differential forms on P M. By definition, Ch(P M) is theimage of the following composition of pull-back and push-forward maps,

k=0

p evk :

k=0

M kM M PM.

Chen reserved a special symbol,w1w2 . . . wk := p ev

k(1 w1 w2 . . . wk 1),

for the restriction of p evk to the subspace 1

kM 1 M

kM M. The latter is a

differential form on P M of degree |w1|+. . .+|wk|k where |wi| stands for the degree of the form

wi M, i = 1, . . . , k. If we denote the composition of evk with the projection, Mk+2

M, tothe first (respectively, last) factor by 0 (respectively, 1), then we can write,

Ch(P M) = span

0w0

w1w2 . . . wk

1wk+1

,

where {wi}0ik+1 are arbitrary differential forms on M.

The Stokes theorem,dp = pd + p |k ,

implies [Ch]

dw1w2 . . . wk = ki=1

(1)|w1|+...+|wi1|i w1 . . . d wi . . . wk

k1i=1

(1)|w1|+...+|wi|i

w1 . . . (wi wi+1) . . . wk

0w1

w2 . . . wk

+(1)|w1|+...+|wk1|k+1

w1 . . . wk1 1wk.

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The wedge product of iterated integrals is again an iterated integral [Ch],w1 . . . wk

wk+1 . . . wk+l =

Sh(k,l)

(1)

w(1)w(2) . . . w(k+l),

where the summation goes over the set, Sh(k, l), of all shuffle permutations of type (k, l), and(1) is the Koszul sign computed as usually but with the assumption that the symbols wi haveshifted degree |wi| + 1.

This all means that Ch(P M) is a dg subalgebra of the de Rham algebra PM. In fact, thissubalgebra (together with variants, Ch(LM) := Ch(P M) |LM and Ch(M) := Ch(P M) |M)encodes much of the essential information about the free path space P M and its more importantsubspaces, LM and M, of free loops and based loops respectively:

1.1. Fact [Ch]. If M is simply connected, then the cohomology of Ch(M) (respectively,Ch(LM)) equals the de Rham cohomology H(M) (respectively, H(LM)).

2. Formal power series connection and holonomy map. Let (A, d) be a unitaldg algebra over a field k with finite dimensional cohomology, say dim H(A) = n + 1. Let{[1], [ei]}1in be a homogeneous basis of the graded vector space H

(A), and {1, xi}1inthe associated dual basis of the shifted dual vector space Hom k(H

(A), k)[1]. Let kX :=kx1, . . . , xn be the free graded associative algebra generated by non-commutative

2 indeter-minates x1, . . . , xn, and kX := kx1, . . . , xn its formal completion. Finally, let {e

i} bearbitrary lifts of the cohomology classes {[ei]} to cycles in A.

2.1. Lemma [Ch]. There exists a degree one element in the algebra A kX and adegree one differential, , of the algebra kX such that

I I2, where I is the maximal ideal in kX;

mod A I2 = ni=1 ei xi; the Maurer-Cartan equation,

d + + = 0,

is satisfied.

The proof goes by induction in the tensor powers of I (cf. [Ch, H]).

2.2. Remark. If (A, d) is a non-negatively graded connected and simply connected (i.e.H0(A, d) = k and H1(A, d) = 0) differential algebra, then the above result holds true with A kX A kX. This case is of major interest to us as it covers the particularexample when A is the De Rham algebra M of a connected and simply connected compactmanifold. A geometric significance of the differential for A = M was discovered by Chen inthe following beautiful

2.3. Fact [Ch]. LetM be a compact simply connected manifold. Then there is a canonicalisomorphism of algebras,

(H(M), Pontrjagin product ) = H(RX, ).

2As opposite to kx1, . . . , xn, the free graded commutative associative algebra generated by homogeneousindereterminants x1, . . . , xn will be denoted by k[x1, . . . , xn].

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Chens proof of this statement employs the notion of the holonomy map,

Hol : (C(M), ) (RX,)

simplex :

k

M (1)

|| k (T),from the complex of (negatively graded) smooth singular chains in the based loop space Mto the complex (RX,). The centerpiece is the restriction to M of the degree zero element,T PM kx1, . . . , xn, given by the following iterated integral

T := 1 +

+

+

+ . . . .

This element is called the transport of a formal power series connection from Lemma 2.1, andhas two useful properties [Ch]:

dT + T + 0() T T 1() = 0, and

given any two smooth simplices, 1 : k P M and 2 : l P M, such that for anyv k, w l the path 2(w) begins where the path 1(v) ends (so that the Pontrjaginproduct 1 2 :

k l P M makes sense), thenkl

(1 2)(T) =

k

1(T)

l

2(T).

As T restricted to M satisfies dT = T, it is now obvious that Hol : (C(M), ) (RX,) is a morphism of dg algebras. Some extra work involving Adams cobar constructionshows that Hol is a quasi-isomorphism; see [Ch] for details.

2.4. Remark. The dual of the finitely generated dg algebra (kX,) is a dg coalgebra,i.e. an A-structure on the vector space H

(A). The latter is precisely a cominimal3 modelfor the original dg algebra (A, d) and Chens formal power series connection A kX isnothing but an A morphism

(H(A), cominimal A structure) (A, d).

Thus Chens Lemma 2.1 is tantamount to saying that the cohomology, if finite dimensional,of a dg k-algebra has an induced structure of an A-algebra. A fact proved later in greatergenerality and independently by Kadeishvili [K].

2.5. Remark. A formal power series connection as described in Lemma 2.1 is not

canonical but depends on a number of choices. There is, however, a canonical object underlyingthis power series which is best described using the language of non-commutative differentialgeometry.

First we replace the pair (kx1, . . . , xn,) by a germ of the n-dimensional formal non-commutative smooth graded manifold X equipped with a degree one smooth vector field .

3An A structure on a vector space V, that is a codifferential Q of the free coalgebra V[1], is called

cominimal if the restriction of Q to the linear bit V[1] V[1] vanishes. Any A algebra over a field ofcharacteristic zero can be represented as a direct sum of a cominimal and a contractible ones, the former beingdetermined uniquely up to an isomorphism.

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This means essentially that we enlarge the automorphism group of (kx1, . . . , xn,) fromGL(n, k) (a change of basis xi

aijxj in H(A)[1]

) to arbitrary formal diffeomorphisms(xi

aijxj + aijkxjxk + . . . ), i.e. we forget the flat structure in H(A)[1]

.

Next we observe that defines a flat connection, + d + , along the vector field in the trivial bundle over X with typical fibre A. It is this flat -connection which is defined

canonically and invariantly. Chens formal power series is nothing but a representative of in a particular coordinate chart (x1, . . . , xn) on X.

This whole paper (more precisely, everything starting with Theorem 3.1 below) should havebeen written using the geometric language of flat -connections4 on the non-commutativedg manifold (X, ). After some hesitation we decided not to do it and work throughout ina particular coordinate patch (x1, . . . , xn); this makes our formulae more transparent but at aprice of using a non-canonical, a choice of coordinates dependent object . Of course, ultimatelynothing depends on that choice.

3. A model for Hochschild cohomology. Let (A, d) be a unital dg k-algebra with finitedimensional cohomology. Lemma 2.1 says that there is a canonical (see the remark just above)

deformation, d + + , of the differential d in A kX. We shall need below its Lie bracketversion,

da := da + a + [, a], a A kX.

Clearly, the Maurer-Cartan equation implies d2 = 0. If (M, d) is a dg bi-module over (A, d),then formally the same expression as above defines a differential dM in M kX.

The following result should be well-known though the author was not able to trace areference.

3.1. Theorem. (i) The Hochschild cohomology Hoch(A, A) is canonically isomorphic asan associative algebra to the cohomology H(A kX, d).

(ii) The Hochschild cohomology Hoch(A, M) is canonically isomorphic asa vector space to the cohomology H(M kX, dM ).

Proof. We show the proof of (i) only (the proof of statement (ii) is analogous). The ideais simple: we shall construct a continuous morphism of topological dg algebras,

C(A, A) := n1Homk(An, A)[n], Hoch, dHoch

(A kX, product product, d) ,

which induces an isomorphism in cohomology.

Usually C(A, A) gets identified with the vector space of coderivations of the bar construc-tion on A as it nicely exposes the dg Lie algebra structure. We are interested, however, in thedg associative algebra structure on C(A, A) and thus have to look for something else.

Let A1 and A2 be A-algebras, that is, a pair of co-free codifferential coalgebras(B(A1), , Q1) and (B(A2), , Q2), where B stands for the bar construction,

B(Ai) :=n1

(Ai[1])n,

4Let (X,) be a smooth dg manifold, that is, a graded manifold X equipped with a degree one smooth vectorfield such that [, ] = 0. Let E be a vector bundle over X which we understand as a locally free sheaf ofOX -modules, OX being the structure sheaf on X. An -connection on E is a linear map : E E such that(f e) = (1)

|f|f(e) + (f)e for any f OX , e E. It is called flat if 2 = 0.

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for the coproduct,

: B(Ai) B(Ai) B(Ai)

[a1|a2| . . . |an]

n1k=1[a1, . . . , ak] [ak+1| . . . |an],

and Qi : B(Ai) B(Ai) for a degree 1 coderivation of (B(Ai), ) satisfying the equationQ2i = 0. For example, if, say, A2 is just a dg algebra (which we now assume from now on)with the differential d : A2 A2 and the product 2 : A2 A2 A2, then the associatedcodifferential is given by

Q2[a1|a2| . . . |an] :=n

k=1

(1)|a1|+...+|ak1|k[a1| . . . |dak| . . . |an]

n1k=1

(1)|a1|+...+|ak|k[a1| . . . |2(ak, ak+1)| . . . |an].

The vector space of all coalgebra maps,

Comap(A1, A2) := HomCoalg(B(A1), B(A2)),

is naturally isomorphic to Homk(B(A1), A2[1]) = n1Homk(An1 , A2)[1 n]. Then one can

turn Comap(A1, A2)[1] into an associative algebra with the product given by (see e.g. [HMS])

f g := 2 (f g) , f, g Homk(B(A1), A2).

A remarkable fact is that every A morphism, i.e. every element of the set

HomA(A1, A2) := { HomCoalg(B(A1), B(A2)) : || = 0, Q1 = Q2 } ,

makes (Comap(A1, A2)[1], ) into a dg algebra. Indeed, the defining equation Q1 = Q2 for (understood as a degree 1 element of Homk(B(A1), A2)) takes the form of a twisting cochainequation,

d + Q1 + D + = 0,

where D is the natural differential in Homcomplexes((B(A1), Q1), (A2, d)).

Then (Comap(A1, A2)[1], , ) with the differential defined by

(f) := Df + f (1)|f|f

is obviously a topological dg algebra.

For example, for any dg algebra A, (Comap(A, A)[1], , Id ) is nothing but the Hochschilddg algebra of A.

Now we assume, in the above notations, that A2 is the dg algebra of Theorem 3.1 andA1 is its cohomology H(A, d) equipped with an induced A-structure Q. As H(A, d) is finite-dimensional, the dualization of the standard completion of (B(H(A, d)), Q) is a free topologicaldg algebra which is nothing but (kX,). Thus the dg algebra,

(A kX, product product, d),

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is canonically isomorphic to (Comap(H(A, d), A)[1], , ). On the other hand, the composi-tion with the A-morphism Comap(H(A, d), A) induces a natural continuous map

: Comap(A, A)[1] Comap(H(A, d), A)[1].

A straightforward calculation shows that

is actually a continuous morphism of topologicaldg algebras,

(Comap(A, A)[1], , Id ) (Comap(H(A, d), A)[1], , ),

that is, a continuous morphism of the topological dg algebras

(C(A, A), Hoch, dHoch)

(A kX, product product, d).

Being continuous, this morphism preserves the decreasing filtrations, Cr(A, A) and A Ir, Ibeing the maximal ideal in kX, and, as it is easy to see, induces an algebra isomorphism ofthe E1 terms of the associated spectral sequences. As filtrations are complete and exhaustive,

the spectral sequences converge to Hoch

(A, A) and H

(A kX, d) respectively. Then theComparison Theorem establishes the required isomorphism. 2

3.2. Remark. If A is connected and simply connected, then kX can be replaced inthe above theorem by kX.

3.3. Example. Let M be an n-dimensional sphere Sn. Let A = R[] be a dg subalgebraof the de Rham algebra M generated over R by a volume form . As A is quasi-isomorphic toM, Hoch

(A, A) = Hoch(M, M).

Clearly, the data = 0, = x, |x| = 1 n, is a solution of the Maurer-Cartan equationfor A,

d + + = 0.

Hence, by Theorem 3.1, one gets immediately

Hoch(M, M) =Ker : [ x, ] : R[] R[x] R[] R[x]

Im : [ x, ] : R[] R[x] R[] R[x]

=

R[, x]/(2), || = n, |x| = 1 n, if n is odd,R[,,]/(2, 2,,), || = n, || = 1, || = 2 2n if n is even.

Which is in accordance with the earlier calculations [CJY, FTV]. The key to the n even case is

x, 1 x2.

3.4. Example. Let M be an n-dimensional complex projective space CPn. Let A =R[h]/hn+1 be a dg subalgebra of the de Rham algebra M generated over R by the standardKahler form h. As A is quasi-isomorphic to M, Hoch

(A, A) = Hoch(M, M).

The data,

=n

i=1

hi xi, |xi| = 1 2i,

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and

= n

i=2

i1j=1

xjxij

xi,

give a solution of the Maurer-Cartan equations for A. Plugging these into Theorem 3.1, one gets

after a straightforward inspection,Hoch(M, M) = R[h,,]/(h

n+1, hn, hn), |h| = 2, || = 1, || = 2n.

This is in agreement with a spectral sequence calculation made earlier in [CJY]. The key to theanswer is

h h 1, n

i=1

ihi xi,

i+j=n+1

1 xixj.

4. Holonomy map for the free loop space. Let M be a compact smooth manifold,and M RX a formal power series connection of the de Rham algebra (M, d) (seeLemma 2.1 and Remark 2.4). As M is naturally a dg bi-module over (M, d), the connection gives rise to a differential, d

= d++[ , . . . ], on HomR(

M,RX) (cf. section 3). Explicitly,

for any f HomR(M,RX) and any (. . . ) M,

(df)(. . . ) = (1)|f|f(d . . . ) + f(. . . ) +

l

(1)|f||l|[plX, f(wl . . . )],

where we used a decomposition of a formal power series connection RX M into asum of tensor products, =

l plX wl.

The transport T of when restricted to the free loop subspace, LM, of the path spaceP M satisfies the equation,

dT + T + ()T T () = 0,

where : LM M is the evaluation map 0

|LM

= 1

|LM

.

Let (C(LM), ) be the complex of (negatively graded) smooth singular chains in the freeloop space.

4.1. Theorem. The map

Hol : (C(LM), ) (HomR(M,RX), d)

simplex : LM Hol() : M RX

(. . . ) (1)||

((. . . ) T) ,

is a morphism of complexes. Moreover, it induces an isomorphism in cohomology if M is simplyconnected.

Proof. That Hol commutes with the differentials is nearly obvious,

Hol() = (1)||1

((. . . ) T)

= (1)||1

d ((. . . ) T)

= (1)||

((d . . . ) T + (. . . ) (T + ()T T ())

= dHol().

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For the second part, consider two increasing filtrations,

FpHomR(M,RX) :=

f HomR(M,RX) | f(w) = 0 w pM

and

FpC(LM) := simplexes LM whose composition with

land in the skeleton Mp

for an appropriate smooth triangulation of M

Clearly, the map Hol preserves the filtrations. The Leray-Serre spectral sequence associatedwith the filtration FpC(LM) has

E2, = H(M) H(M).

As M is simply connected, the formal power series connection w =

l pl(x1, . . . , xn) wlhas all wl

2M . Then using Fact 2.3, it is easy to compute the E

2 term of the spectral sequenceassociated to FpHomR(M,RX),

E2

, = (H

(M))

H(M).

Thus Hol provides an isomorphism of the E2 terms. Then the standard spectral sequencescomparison argument finishes the proof. 2

The above Theorem in conjunction with Theorem 3.1(ii) immediately implies the Jonesisomorphism [J]:

4.2. Corollary. If M is simply connected, the holonomy map induces the isomorphism,

H(LM,R) = ExtMM (M,

M) .

5. BV operator and an odd holonomy map. The holonomy map Hol can also beunderstood as a degree 0 morphism of complexes,

Hol : (M, d) LM RX, d() := d + + [(), . . . ] () T

Consider now a degree -1 linear map

: M RX LM RX

n=0

ni=0 . . . i . . . ni

5.1. Lemma. For any M RX,

[T, ()] = d() () + (d).Proof is a straightforward calculation of the r.h.s.

Note that (M 1, d) is naturally a subcomplex of (M RX, d).

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5.2. Corollary. The map 0 := |M1 induces a degree 1 morphism of complexes,(M, d)

LM RX, d()

,

and, through pairing with chains, a degree 1 morphism of complexes,

: (C(LM), ) (HomR(M,RX), d)

simplex : LM () : M RX

(. . . )

0(. . . ) .

There is a natural action of the circle group S1 on the free loop space,

Rot : LM S1 LM,

by rotating the loops, Rot : ((t), s) (t + s). This action induces a degree 1 operator on

(negatively graded) chains,

BV : C(LM) C+1(LM)

: LM BV() : S1 LM

(z, s) Rot((z), s).

This operator commutes with the differential and hence induces an operator on singularhomology, BV : H(LM) H+1(LM), which satisfies the condition

2 = 0 as its iterationk2BV increases the geometric dimension of the input simplex only by one [CS].

5.3. Proposition. Hol BV = .

Proof. Let q : LM S1 LM be the natural projection. It was shown in [GJP] that theS1 rotation affects an arbitrary iterated integral as follows,

q Rot

(w0)

w1w2 . . . wk

=i=0

(1)r

wi . . . wkw0w1 . . . wi1,

where r = (|w0| + |w1| + . . . + |wi1| i)(|wi| + . . . + |wk| k + i. Applying this formula to(. . . ) T, (. . . ) M, one immediately obtains the required result. 2

6. Poincare duality. Here is a deformed version of the classical duality:

6.1. Theorem. For any smooth compact oriented manifold M the linear map

P : (M RX, d) (HomR(M,RX), d)

P() : M X,

(. . . ) M

(. . . ) ,

is a degree n quasi-isomorphism of complexes.

Proof. Checking commutativity with differentials is just an elementary application of theStokes theorem. The map P obviously preserves natural filtrations of both sides by powers of

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the maximal ideal in RX. At the E, 1 level of the associated spectral sequences this mapinduces, by the classical Poincare duality, a degree n isomorphism,

P : H(M) RX HomR(H(M),R) RX.

Hence the spectral sequences comparison theorem delivers the required result. 2

6.2. Corollary. If M is simply connected, then, for any cycle : LM in(C(LM), ), there exists a cycle (M RX, d) such that,

(1)||

(T (. . . )) =

M

(. . . ) + dH(. . . ),

for all (. . . ) M and some H HomR(M,RX).

Our next task is to find an explicit expression of such a cycle in terms of iteratedintegrals and a formal power series connection.

7. Thom class and a suitable homotopy. Let i : M M M be the diagonalembedding, and [U] Hn(M M, M M \ i(M)) the associated Thom class. For any tubularneighbourhood, T ub, of i(M) in M M one can represent [U] by a closed n-form U MMsuch that

support U T ub, and

pr(U) = 1, where pr : M M M is the projection to the first factor and 1 is theconstant function on M.

Let us consider consider a commutative diagram of maps,

M LMpy

77uuu

uuuuuuu

py

((WWW

WWWW

WWWW

WWWWW

px

LM

M M

where px and py are natural projections to, respectively, the first and second factors, and let usintroduce a list of notations,

T ub := (px py)

1(T ub) M LM,

Ty := ( py)T LMM RX,

Ux,y := (px py)(U) nMLM,

x := (px) LMM RX,

y := (py) LMM RX,

(so that a symbol with the hat always stands for the object living on M LM). A point in T ubis a pair (x, ly) consisting of a point x M and a loop ly in LM based in y := (l) such that(x, y) T ub.

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Let be an arbitrary smooth connection on M. For any two sufficiently close to each otherpoints x, y M there is a unique vector V TxM such that y = expx V. For any s [0, 1]define the path

Psx,y : [0, 1] M

t expx((1 s)tV + sV)

which is just a [0, 1]-parameterized geodesic from s x := expx(sV) to y = expxV. Such a pathallows us in turn to introduce a smooth map

F : T ub [0, 1] T ub(x, ly) s

x, (Psx,y l) P

1sx,y

where is the Pontrjagin products of paths.

x

s x

y

Note that F1 = F|Tub1 is the identity map T ub T ub, while F0 = F|Tub0 factors throughthe based diagonal,

.

T ub M LM LM

Id M LM

(x, ly) (Px,y l) P1x,y

In particular, the pullback map F0 : Tub Tub vanishes on any differential form which has afactor of the type py(. . . ) p

x(. . . ), (. . . ) M.

Next we decompose, as usual, the pullback map,

F = F F,

into a bit, F, containing ds and a bit, F, not containing ds and introduce a homotopyoperator,

h : Tub 1

Tub(. . . )

10

F(. . . ),

such thatId = F0 + dh + hd.

There are lots of homotopy operators satisfying the above equation, but the one we describedabove is particularly suitable:

7.1. Fact. h : Tub 1Tub is a morphism of px(M)-modules.13


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Extending by zero differential forms with compact support in T ub to the whole M LMwe can summarize our achievements in this section as follows.

7.2. Fact. There exists a morphism of px(M)-modules,

H = (1)nUx,y h : MLM

+n1

MLM

,

such that, for any (. . . ) M and any MLM ,

Ux,y py(. . . ) p

x(. . . )

= dH

py(. . . ) p

x(. . . )

+(1)nH

d

py(. . . ) p

x(. . . )

+(1)n+||H

py(d . . . ) p

x(d . . . )

In particular, extending H by RX-linearity to MLM RX, taking = Ty and

using the equation, dTy + Ty + [y(), Ty] = 0, we get

Ux,y Ty py(. . . ) px(. . . ) = (d + )HTy py(. . . ) px(. . . )+(1)n1H

[y, Ty]

py(. . . ) p

x(. . . )

+(1)nH

Ty

py(d . . . ) p

x(d . . . )

= (1)nH

[y x, Ty]

px(. . . )

+(d + )H

Ty py(. . . ) p

x(. . . )

+(1)n1H

[py( . . . ) p

y( . . . ), Ty]

+(1)nH

Ty

py(d . . . ) p

x(d . . . )

8. An inversion of the deformed Poincare duality map. We assume from now onthat M is a compact simply connected manifold. For any cycle : LM in (C(LM), )there is associated a diagram of maps,

M

Id

88xxx

xxxx

xxxx

p

zz

M M LM

Then, for any (. . . ) M, we have

(1)||Hol()(. . . ) =

(T (. . . ))

=

M

(Id )

Ux,y Ty py(. . . )

=

M

(Id )

Ux,y Ty px(. . . )

+

M

(Id )

Ux,y Ty (py(. . . ) p

x(. . . ))

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Hence, setting

= p (Id )

Ux,y Ty + (1)nH

[y x, Ty]

M RX

and,

H : M R

X(. . . )

M H

Ty

py(. . . ) p

x(. . . )

we get,

(1)||Hol()(. . . ) =

M

(. . . ) + dH(. . . ),

where we have used a calculation at the end of 7.2.

We almost proved the following

8.1. Theorem. The degree n linear map

(C(LM), ) (M RX, d)

(1)||

is a quasi-isomorphism of complexes whose composition with P induces on cohomology the mapHol:

H(LM)

HolAA

GG H+n(M RX, d))

P

H(HomR(M RX, d)

Proof. We need only to check that the map is well defined, i.e. that is a cycle. Thelatter can be suitably represented as

= p (Id )

Ux,y

Ty + h([y x, Ty])

where we understand Ty + h([y x, Ty]) as a differential form on T ub and Ux,y (Ty + h([y x, Ty])) as a differential form on M LM (extended by zero from T ub).

As opposite to dy Ty = 0, we have,

dx

Ty + h([y x, Ty ])

=

y x, Ty

+ (d + )h([y x, Ty])

x, h([y x, Ty])

= h(d[y x, Ty]) + h([y x, Ty]) + h(x, [y x, Ty])= h([yy xx, Ty]) h([y x, [y, Ty]]) h(

x, [y x, Ty])

= 0.

Pushing down this equation along : LM finally gives d = 0. 2

9. Inverse Poincare map revised. As the calculation in beginning of Section 8 shows,

the class = p (Id )

Ux,y

Ty + h([y x, Ty])

provides us with a natural lift of

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the holonomy map [Hol] : H(LM) H(HomR(M RX, d

) along the deformed Poincare

duality map, [P] : H(MRX, d) H(HomR(MRX, d

). However, it is not unique.

We shall use below another geometrically transparent lift of [P]1 [Hol] from the coho-mology to the chain level. Note in this connection that has to be specified only up to a

d-exact term implying that the formTy + h([y x,

Ty]) on T ub has to be specified only upto a dx-exact term.

Recall that for a tubular neighbourhood, T ub, of the diagonal in M M, the subspaceT ub M LM consists of pairs (x, ly), x being a point x M and ly a loop in LM based aty := (ly), such that (x, y) T ub. Let Px,y be the geodesic from x to y for some fixed affineconnection on M. Consider a map,

ix,y : T ub LM(x, ly) (Px,y ly) P

1x,y .

9.1. Proposition. Ty + h([y x, Ty]) = ix,yT mod Im dx .

Proof. Comparing the map ix,y with the homotopy F in Section 7, one notices that F0 isprecisely the composition of ix,y with the natural embedding,

emb : LM M LM

l ((l), l).

Then applying the equality of maps, Id = ix,y emb + dh + hd, to Ty one obtains,

ix,yT = Ty dh(Ty)) h(dTy))

= Ty + h([y x, Ty) dxh(Ty).

2

9.2. Corollary. The degree n linear map

Hol : (C(LM), ) (M RX, d)

(1)|| := p (Id )

Ux,y ix,yT

is a quasi-isomorphism of complexes making the following diagram

H+n(M RX, d))

P

H(LM)Hol

GG

Hol

SSjjjjjjjjjjjjjjj

H(HomR(M RX, d)

commutative.

10. Compatibility with products. Now we have all the threads to show a new proofof a remarkable result of [CJ, C, Tr] which establishes an isomorphism of algebras,

(H(LM), Chas-Sullivan product ) =

ExtMM (M, M) , Yoneda product

.

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It follows immediately from Theorem 3.1(i) and the following

10.1. Theorem. The degree 0 linear map

Hol : (C+n(LM), ) (M RX, d)

(1)||

induces on cohomology a quasi-isomorphism of algebras,

(H(LM),) H(M RX, d).

Proof. We shall use small Latin letters, coordinate variables, to distinguish variouscopies of M and LM, e.g. Mx, My, LMy, etc.

Let y : LMy and z :

z LMz be any two cycles in (C+n(LM), ) such that

their projections, y y : y My and z z :

z Mz intersect transversally at a cycle

y y z z : yz M, where

yz := (y y z z)1 y y(y) z z(z) {diagonal in My Mz} .Chas and Sullivan [CS] define the product chain, y z, as

1 2 : xy LM

z 1(z) 2(z).

The theorem will follow if we show that

y z = yz mod Im d. ()

In the obvious notations associated with the commutative diagram,

Mx y

z

Idyz

AA

p

xxqqqqqqqqqqq

Mx Mx LMy LMz

we can represent the l.h.s. of () as follows

y z = p (Idx y z)

Ux,y ix,yT Ux,z i

x,zT

.

Using homotopy 7.2 we can transform . . . Ux,z . . . into . . . Uy,z . . . modulo dx-exactterms:

Ux,y Ux,z ix,yT i

x,zT = Ux,y Uy,z i

x,yT i

x,zT + dHx,y

(Ux,z Uy,z) ix,yT ix,zT+H

(Ux,z Uy,z) d(i

x,yT i

x,zT)

= Ux,y Uy,z Tx,y Tx,z + dxHx,y

(Ux,z Uy,z) i

x,yT i

x,zT

.

Hence

y z = p (Id y z)

Ux,y Uy,z ix,yT i

x,zT

mod Im d.

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Let T uby,z be a tubular neighbourhood of the diagonal in My Mz and let T aby,z be itspre-image under the natural evaluation projection,

y z : LMy LMz My Mz.

Similarly, let T ubx,y,z be a tubular neighbourhood (supporting the form Ux,y Uy,z) of the smalldiagonal in Mx My Mz and let T abx,y,z be its pre-image under the natural map,Idx y z : Mx LMy LMz Mx My Mz.

The form Uy,z ix,yT i

x,zT is supported in Mx

T aby,z Mx LMy LMz. As T uby,zis homotopy equivalent to My, there exists a one parameter family of maps,

G : T uby,z [0, 1] T uby,zwith the property that G1 = G|

Tuby,z1

is the identity map while G0 = G|

Tuby,z0

factors through

the composition T uby,z r LMy My LMy LMy LMzfor a smooth map r. Then one applies the associated to Idx G homotopy equality of linearmaps,

Id = Idx (G0 + dhG + hGd) : Tubx,y,z Tubx,y,z,

to the form ix,yT ix,zT and gets

ix,yT ix,zT = i

x,y (q

1T q

2T) mod Im dx ,

where q1 and q2 are the natural projections,

LMx Mx LMxq2

99yyy

yyyy

yyyyq1

wwooooooooooo

LMx LMx

and ix,y is given by (cf. with ix,y in the beginning of Sect. 9)

ix,y : T ubx,y,z LMx Mx LMx(x, ly, lz) (Px,y r(ly, lz)) P

1x,y .

There is a natural Pontrjagin product map

LMx Mx LMx

LMx,

and the basic property of the transport of a formal power series connection says that

T = p1(T) p2(T).

Hence we can eventually write

Ux,y Ux,z ix,yT i

x,zT = Ux,y Uy,z i

x,yT mod Im dx ,

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which implies

y z = p (Idx y z)

Ux,y Uy,z ix,yT

mod Im d.

Recalling the definition of y z and using transversality of py y and pz z to get rid of

the Thom class Uy,z through z-integration one finally obtains

y z = p (Idx y z)

Ux,y ix,y(T)

mod Im d.

Which is precisely the desired result (). 2

11. Loops based at a brane. If f : Z M is a smooth map from a compact orientedp-dimensional manifold to a compact simply connected manifold M, then one can consider aspace, Lf, defined by the pullback diagram,

Lf

GG LM

Zf

GG M .

The shifted homology, H(Lf) := H+p(LM), is obviously an associative algebra with respectto the Chas-Sullivan product (see [CS, S]). If f is an embedding, then Lf is a subspace ofLM consisting of loops in M which are based at Z M.

The pull-back map on differential forms, f : M Z, is naturally extended to the tensorproduct M RX as f

Id and is denoted by the same symbol f. The map

f : (M RX, d)

Z RX, df()

is obviously a morphism of dg algebras.The proofs of Theorems 4.1, 6.1 and 10.1 apply with trivial changes to their brane

versions:

11.1. Theorem. The map

Hol : (C(Lf), )

HomR(Z,RX), df()

simplex : Lf Hol() : Z RX

(. . . ) (1)||

{(. . . ) (T)} ,

is a quasi-isomorphism of complexes.

11.2. Theorem. The linear map

P :

Z RX, df()

HomR(Z,RX), df()

P() : Z RX,

(. . . ) Z

(. . . ) ,

is a degree p quasi-isomorphism of complexes.

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11.3. Theorem. The degree p linear map

Hol : (C(Lf), ) (Z RX, df())

(1)|| p (f )

Ux,y i

x,yT

associated with the diagram,Z

f

99yyy

yyyy

yyyy

p

{{wwwwwwwww

Z M LM ,

is a morphism of complexes inducing on cohomology an isomorphism of algebras,

(H(Lf),) H(Z RX, df()).

Moreover, if f is an embedding, then the pull-back map f : H(M RX, d) H(Z

RX, df()) corresponds to the intersection map [Lf] : H(LM) H(Lf).

11.4. Example. Let Z be the projective space CPm and let f : Z M = CPn, m < n,be a linear embedding. A straightforward calculation using Theorem 11.3 gives,

(H(Lf,) = H

R[h]/hm+1 Rx1, . . . , xn, df() = +

m

i=1

hi xi, . . .

= R[h,,x]/hm+1, |h| = 2, || = 2n, |x| = 1,

where is the same as in Example 3.4. (The key to the r.h.s. is x =

m+1i=1 ih

i1 xi,

i+j=n+1 1 xixj.) Moreover the intersection mapH(LCPn)

[Lf]GG H(Lf)

R[h,,]/(hn+1, hn, hn) GG R[h,x,]/hm+1

is given on generators as follows (see Example 3.4 again for the notations used in the l.h.s.),

h h hx.

Acknowledgement. It is a pleasure to thank Ralph Cohen, Thomas Tradler and especially Dennis

Sullivan for valuable correspondences and remarks.

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References

[CS] M. Chas and D. Sullivan, String topology, preprint math.GT/9911159.

[Ch] K.-T. Chen, Bull. Amer. Math. Soc. 83 (1977), 831-879.

[C] R.L. Cohen, Multiplicative properties of Atiyah dual, preprint 2003.

[CJ] R.L. Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology, Math. Ann. 324(2002), 773-798.

[CJY] R.L. Cohen, J.D.S. Jones and J. Yan, The loop homology algebra of sphere and projective spaces,preprint math.AT/0210353.

[FTV] Y. Felix, J.-C. Thomas and M. Vigue-Poirrier, Structure of the loop algebra of a closed manifold,preprint math.AT/0203137.

[GJP] E. Getzler, J.D.S. Jones and S. Patrick, Differential forms on loop spaces and the cyclic barcomplex, Topology 30 (1991), 339-371.

[H] R. Hain, Iterated integrals and homotopy periods, Mem. Amer. Math. Soc. 291 (1984), 1-98.

[HMS] D. Husemoller, J.C. Moore and J. Stasheff, Differential homological algebra and homogeneousspaces, J. Pure Appl. Alg. 5 (1974), 113-185.

[J] J.D.S. Jones, Cyclic homology and equivariant homology, Inv. Math. 87 (1987), 403-423.[K] T.V. Kadeishvili, The algebraic structure in the cohomology ofA()-algebras, Soobshch. Akad. Nauk

Gruzin. SSR 108 (1982), 249-252.

[R] Jan-Erik Roos, Homology of free loop spaces, cyclic homology and nonrational Poincare-Betti seriesin commutative algebra, Lecture Notes in Math., 1352 (1988), 173-189, Springer, Berlin, 1988.

[S] D. Sullivan, Open and closed string field theory interpreted in classical algebraic topology, preprintmath.QA/0302332.

[Tr] T. Tradler, Ph. D. thesis, CUNY, (2002).

Department of Mathematics

Stockholm University10691 StockholmSweden

s.a. merkulov- de rham model for string topology

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