morse homotopy and chern-simons perturbation theory

Commun. Math. Phys. 181, 37-90 (1996) Communications in Mathematical

Physics �9 Springer-Verlag 1996

Morse Homotopy and Chern-Simons Perturbation Theory

Kenji Fukaya 1

Department of Mathematics, Faculty of Sciences, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, Japan. E-mail: [email protected]

Received: 1 June 1995/Accepted: 2 February 1996

Abstract: We define an invariant of a three manifold equipped with a flat bundle with vanishing homology. The construction is based on Morse theory using several Morse functions simultaneously and is regarded as a higher loop analogue of various product operations in algebraic topology. There is a heuristic argument that this invariant is related to perturbative Chern-Simons Gauge theory by Axelrod-Singer, etc. There is also a theorem which gives a relation o f the construction to open string theory on the cotangent bundle.

Contents

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 I. Statement o f the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2. Transversality and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3. Independence o f combinatorial propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4. Independence o f the Morse functions I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5. Independence o f the Morse functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6. Compactification of configuration space and transversality at diagonal . . . . . . . 73 7. Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

0. Introduction

Let M be a 2n dimensional manifold and N be its n dimensional submanifold. We consider a current TN such that TN(O)) = fU ~0. We try to justify the integral

f T N /k T N . (0.1) M

(We remark that TN/X TN itself is not well defined.) One way to do so is to take a perturbation N ~ of N so that N ~ and N are transversal to each other and consider

tPartially supported by Grants-in-Aid for Scientific Research on Priority Areas 231 "Infinite Analysis".

38 K. Fukaya

fM TN A TN,. It is easy to see that TN A TN, is a delta current supported at the intersection N f] N I. Therefore fM TN A TN, is the intersection number N �9 N I = N �9 N. What is important here is that the integral fM TN A TN, is independent of the perturbation N ~.

There is an alternative way to get the same answer. Namely we choose a harmonic n form hu which represents the De-Rham cohomology class of N. Then again fM hu A hu is well defined and gives the self intersection number N �9 N.

We can continue in a similar way to define the secondary invariant as follows. Let M = A C M 2 be the diagonal. Assume that Tz is an exact current. (In fact this never happens. But we can find this kind of situation by working with a local coefficient.) We choose an n - 1 form co on M 2 such that de) = TA. Assume that M is three dimensional. Then the integral

f co A co A co (0.2) M 2

is a number. By modifying this integral fM2 co A co A co, an invariant of a 3 manifold is discovered by Axelrod-Singer [AS], Bar-Natan [Ba], Guadagnini-Martinelli- Mintchev [GMM], Kontsevich [Ko], etc. We discuss their result a bit more in Sect. 8. Their construction is based on harmonic theory and hence is an analogy of the second method we mentioned above to justify (0.1).

On the other hand, we can also imitate the first approach. Namely we can use an appropriate intersection number to justify (0.2). To perform this kind of construction is the purpose of this paper.

To do so, we need to find a cycle X such that aX = A. Such an X can be found as follows. Choose a Morse function f . Let M(f) be the set of all pairs (x, y) c M 2 such that x, y lie on the same gradient line of f . One can easily find that a connected component of the boundary of M(f) is a diagonal A. Hence the intersection number M(fl )�9 M ( f 2 ) � 9 M(f3) should be related to (0.2).

Let us here review various results related to the contents of this paper. First it was discovered by [As, Ba, GMM, Ko] that an integral like f~/: co A co A co gives the second term of the expansion of Witten's invariant [W2, Kh, ReT] that is a Chern- Simons gauge theory. The first term of this expansion is basically the Analytic torsion of Ray-Singer and was discussed by Witten [W2]. Axelrod-Singer, etc. defined a higher term also.

In fact it is not yet proved that the construction of Axelrod-Singer, etc. really gives the expansion. What they did is to give an argument of physical level of rigor to show that this is an expansion of the Wit-ten invariant and also they proved rigorously that the invariant is independent of the various choices involved.

Witten in [W3] found that Chern-Simons Gauge theory on a 3-manifold M is equivalent to the open string field theory of its cotangent bundle T*M. However to make the latter rigorous in a mathematical sense is not easy and remains yet an open question. (See Sect. 8 for more discussion about it.)

Floer studied the pseudo-holomorphic disk with Lagrangian boundary condition to define his celebrated Floer homology for Lagrangian intersection [Fll]. In the case of the cotangent bundle he proved that Floer homology between 0 section and its Hamiltonian perturbation is equal to the usual homology of the manifold. The problem to define Floer homology for Lagrangian intersection in a more general situation is studied by Oh [Oh]. The author introduced an A~-structure on Floer homology [Ful, Fu2]. In the case of the cotangent bundle T'M, it is proved by

Morse Homotopy and Chem-Simons Perturbation Theory 39

Oh and the author that the A~-structure of Floer homology is equivalent to one for M [FO]. The latter is described by using Morse theory. The main idea to do so is to use several Morse functions at once. This idea is due to the author [Fu2, Fu3] and to M. Betz and R. Cohen [BC] independently. Roughly speaking we consider the moduli space of maps from a graph to M such that each edge is a gradient line of some Morse function. The A~-structure then corresponds to the case when the graph is a tree. The result of Oh and the author says that the case of a tree is equivalent to the 0-loop amplitude of open string field theory on T*M.

Betz-Cohen studied also the graph which is not a tree. They announced that characteristic classes of a manifold are described in that way. The author in [Fu3] also studied the case when the graph is not a tree. However the discussion there was not yet satisfactory.

The main point of this paper is that the Morse theory for a graph which is not a tree gives Chern-Simons perturbation theory. The author does not yet prove that they really coincide. What is proved in this paper is that (for the 2 loop amplitude) there is a well defined invariant based on Morse theory.

We can generalize our result with Oh to this case and show that our invariant is related to open string theory (see Sect. 8). Thus in a sense our construction is a rigorous mathematical definition of a 2 loop open string amplitude on the cotangent bundle of a 3 manifold.

It is remarkable that our construction is similar to the construction of Chern- Simons Perturbation theory in many points. Also in Sect. 8 we discuss some heuristic argument which suggests that our invariant coincides to Chem-Simons Perturbation theory. This idea is closely related to Witten's work on Morse theory [Wl].

Note that the invariant we introduced here is a secondary invariant to the (co)homology group (with cup product). Also here the homology group is studied from the point of view of singular theory (since Morse function gives a cell decomposition.) With this respect Reidemeister torsion is another natural secondary invariant of the (co)homology theory with local coefficient. In Chern-Simons perturbation theory analytic torsion of Ray-Singer [RS] appeared as the first invariant. The coincidence of Reidemeister and Analytic torsion was established by Cheeger [Ch] and M/iller [M/i]. Our conjecture that the invariant in this paper coincides with one by Axelrod-Singer, etc. may be regarded as the higher genus analogue of this theorem of Cheeger-Miiller.

The organization of this paper is as follows:

In Sect. 1 we define our invariant. Roughly speaking it is obtained by counting the order of the set of solutions of an appropriate ordinary differential equation, with appropriate weight.

The proof that this number is independent of various choices is based on the study of compactification of the moduli space of the solution of an ordinary differential equation. There are basically two points to clarify. To define an intersection number M ( f l ) �9 M(f2) �9 M(f3) , there are two problems. One is that the boundary of M ( f l ) is not in fact equal to the diagonal A. There is another boundary that is the set of pairs (x, y), where x is in an unstable manifold of grad f with respect to a critical point p of f and y is in the stable manifold of the same critical point p. We need to add various correction terms to settle this problem.

In Sect. 2, we discuss the point mainly related to this problem. Based on it the well-definedness of our invariant is proved in Sects. 3-5.

40 K. Fukaya

In Sect. 6, we discuss another problem with the transversality of M(f l ) �9 M(f2) �9 M(f3). Namely these submanifolds are not transversal to each other at the diagonal, even if we choose f i generic. We need to use an appropriate blow up of the diagonal of M 2. This construction is similar to the argument of compactification of configuration space due to Fulton-Macpherson [FM] and Kontsevich [Ko], which was used also by [AS] and [Ko].

In Sect. 7 we discuss orientation of our moduli space. In Sect. 8 we describe some ideas related to the problem that our invariant

is equal to both Chem-Simons perturbation theory and open string theory of the cotangent bundle.

The result of this paper was announced in [Fu4] without proof. During the preparation of this paper, the author visited Maryland University,

Hong Kong University of Science and Technology, Newton Institute, International Center of Theoretical Physics and Stanford University. The author would like to thank these universities and institutes for their hospitalities.

The author would also like to thank Professor Hiraku Nakajima for his excellent explanation of Chem-Simons perturbation theory to the author. He also would like to thank T. Gocho who pointed out an error on sign in the preliminary version.

1. Statement of the Result

Let M be a compact oriented 3 manifold and ~ be a flat vector bundle on it. In this section we define a number Z2(M;fl,f2,f3; 4) by fixing a metric on M and three generic functions f l , f2, f3 on M. The proof that it is invariant of these choices will be given in Sects. 2-7.

The number Zz(M; f l , f2, f3, 4) is a sum of Zo(M; f l , f2, f3; 4) and correction terms. We first define the leading term.

For a function f on M, let q~}: M --~ M be the one parameter group of diffeo- morphisms associated to grad f . Namely it satisfies

{ = x 0~) (x ) grad f(~b}(x)) T t=to=

Let f l , f2, f3 C M. We put

J/[o(fl , f2,f3) = {(x,y;tl,t2,t3) E M 2 • R 3 I ~ ( x ) = y,i = 1,2,3}. (1.1)

Lemma 1.2. For 9eneric f l , f2 , f3 , the space Jgo(f l , f2 , f3) is an oriented man# fold of dimension 3 - dim M.

Lemma 1.3. I f dimM = 3, then for 9eneric f l , f2 , f3 , the space dgo(f l , f2, f3) is compact.

These lemmas are proved in Sect. 2. For h,tz, t3 > 0, we define a metric on O as follows. Let ej, j = 1,2,3 be the

edges &leng th tl,t2,t3 respectively. We take two vertices vl, Vz and attach each of the edges of ej to vl, v2.


V ~ e2 ~ ' 1 ) 2

e3

Fig. 1.

41

For an element (x,y;h,tz, t3) of ~lo( f l , f 2 , f3 ) we associate a map I : O ~ M as follows. We put I ( v l ) = x , I ( v 2 ) = - y , and on the edge ej we define I ( t )= O~j(x). (Here we identify ej = [0, tj].) By the definition of Jgo( fb f z , f3 ) this map is continuous at v2. Hereafter we regard an element of dge( f l , f z , f3) as such a map.

We next associate a weight Z(/, 4) to each element I : O ~ M and a flat vector bundle ~ as follows. We define element 7i E ~1(O), i = 1,2,3,4 by

71 = e 2 1 o e l ,

'~2 = e31 0 e2 ,

~;3 = e l 1 o e 3 ,

~4 : e2 1 oe3 oe l 1 oe 2 oe31 oel .

Roughly speaking, 71,72,73 are boundary loops of one of the ribbon structures of the O-graph, while ~4 is the boundary loop of another ribbon structure of the O-graph. Now we put

3 Z(/, 4) = - I - [ Tr(Uolr + Tr(Holr (1.4)

i=1 Here Hole({) is the holonomy homomorphism of the flat bundle ~ along the loop ~. Now we define

Definition 1.5.

Zo(f l , f2 , f3;~) = ~ ~;IZ(I,~). I E J # o ( f b f 2 , f 3 )

Before going to the next step, let us rewrite our weight by using Lie algebra bundle g = Ad~. More generally we consider a flat bundle g over M, whose fibre has a structure of semi-simple Lie algebra compatible with the fiat structure. By using a canonical invariant inner product, we have an inner product on ~ which is also compatible with the flat structure. Let I C s//lo(fl,f2,f3). We choose an orthonormal basis el,...,edim0 of ~x for x = / (Vl ) . We put

dim 0 Z(I , r ~ ([ei, ej],ek)([Pel(ei),Pe~(ek)],Pe2(ej)). (1.6)

i , j , k= l

Here Pei : gx ~ ~y is the parallel transport along the path I(ei).

42 K. Fukaya

Lemma 1.7. I f ~ is a C 2 bundle with fiat su(2) structure and/ f r = Ad~, then we have

2Z(/, 4) = Z( I, g).

Proof We embed su(2) --~ gl(2; C) and choose e0 so that e0 . . . . . e3 is an orthonormal basis of gl(2; C) and el, e2, e3 is an orthonormal basis of su(2). Then since e0 is in the center, we have

3 X ( I , g ) : ~ ([ei, ej],ek)([Pel(ei),Pe3(ek)],Pe2(ej)). (1 .8 )

i,j,k=O

We then find that (1.8) is equal to

3 -2 ~ (eiey, ek)(Pel(ei)Pe2(ej),Pe3(ek))

i,j,k=O 3

+2 ~ (eiej, ek)(Pe~(ej)Pel(ei),Pe3(ek)). (1.9) i,j,k=O

(Here eiej etc. denotes the product of matrix.) It is straightforward to see that (1.9) is independent of the choice of orthonor-

mal basis. So we change our basis and take f i | j, i , j= 1,2. Here f l , f : is an orthonormal basis of ix and fl, f2 is its dual basis. Using this basis the first term of (1.9) is

2 2 E

i,j,k=l ((f/ | fY)(fy | fk),(fk | fi))(Pe~(fi | fJ)P~(fy | fk),Pe3(fk | fi))

2 = 2 ~ (Pe,(fi),Pe3(fi))(Pel(fj),Pe2(fj))(Pe:(fk),Pe3(fk)).

i,j,k=l 3

= 2 H Tr(H~162 �9 i=1

Similarly we find that the second term of (1.9) is equal to 2Tr(Hol~(?4)). The proof of Lemma 1.7 is now complete.

We define Zo(A,f2,f3;g) = ~ -4-Z(I,r

ICJdo(fl,f2,f3)

for a flat Lie algebra bundle r Here 4- is determined by the orientation of dgo(f~, f2,f3) , which is discussed in Sect. 7.

We put 1

Zo(fl , f2,f3;q) = gai~lZO(alfl,~2f2,~3f3;g). (1.I0)

We thus defined the leading term of our invariant. To define correction terms, we first introduce the Witten complex [Wl] with local coefficient. Let f : M --+ R be a Morse function. We put

Ck(M;f;r (~ Cp. pECr(f) ~(p)=k

Morse Homotopy and Chern-Simons Perturbation Theory 43

Here C r ( f ) is the set of critical points of f and q(p) is its Morse index. We define the boundary operator O:C k(M ; f ; q)---+ C k - l ( M ; f ; q) as follows. Let q(p) = q(q) + 1. We are going to define the component (~pq " q p ~ qq of ~. Fol- lowing the definition of the usual Witten complex, we use the moduli space

dg(p,q) = { f : R ---+ M

= -grad f

, r i m + ( t ) = p .

t+lim f ( t ) = q

In case the Morse function f is generic and q(p) --- ~/(q) + 1, this space is an oriented manifold of dimension one. The group R acts on it by translation of parameter. Let rig(p, q) be the quotient space of this action. We then define Opq : qp ~ r by

OPq = 2 -~ P f . +C~+(p,q)

Here P+ is the parallel transportation along the arc f, and the sign is determined by the orientation of the moduli space ~ ' (p ,q ) .

Lemma 1.11. <~<9 = 0, H.(C.(M; f ; q), ~) = H.(M; q).

Here the right-hand side is the homology with local coefficient. The proof of Lemma 1.11 is a straightforward analogue of the result in the case when q is trivial. The proof in that case is given by various authors. See for example [Sc].

Now we go back to our problem to define correction terms. We make use of the assumption H ( M ; q ) = 0. By Lemma 1.11 it follows that the chain complex (C. (M;f ;q) ,O) is acyclic. Hence so is End(C.(M;f;q) ,O). The element id c End(C.(M; q), ~) is a cycle. So there exists an element gf, q E End(C.(M; f ; q)) of degree one such that @f,~ = id. (Namely ~ o gf, c + gf, c o ~ = id.) We call such gf, c a combinatorial propagator. We remark that combinatorial propagator is not unique. But we prove later that the invariant we define using it is independent of the choice of combinatorial propagator.

Now let ci E EndgC.(M;f i ;q) ) . We are going to define Zo(Cl,/Z/,fZ/;fl ,f2, f3; q) etc. We put

Cl = ~ c(p, q)[p] | [q]*. q(p)=r/(q)+ 1

Here c (p ,q )E Hom(qq,~p). For r / ( p ) = r / (q)+ 1, we define a moduli space ggO(1,O,O)(P, q; f l , f2, f3 ) as follows:

f

Jgo(1,0,0)(P, q; f b f2, f3) = ~ (x, y; t2, t3)

L E M 2 x Re+

l lim @t(x)=p } t---++o~ Jl

lira @S(y)=q t - - + - - o o J l

@~(x) = y,i = 2,3

As in the previous case, an element of the moduli space Jgo(1,,o,o)(p,q; f l , f 2 , f 3 ) can be regarded as a map from a graph to M. Namely let O(1, 0, 0) be the graph obtained by cutting the first edge el of the O-graph. Then O(1,0,0) has 4 edges, e~, e 2, e2, e3. For an element (x, y; t2, t3), we let e~ be mapped to the gradient line of f~ joining x to p, e~ is mapped to the gradient line of f l joining q to y. e2, e3 will be mapped to the gradient line of f2, f3 joining x to y respectively.

44 K. Fukaya

Let I : O(1,0,0) ~ M be a map identified to an element of Jgoo,o,o)(P,q;fl, fz, f3). We define a weight Z(I,q) r Hom(gp,r as follows. Let u E gp, v E gq. Let e i be an orthonormal basis of qx. We then put:

(z(l,g)(u),v) = ~ ([Pe{(U),ej],ek)([Pe~V, Pesek],Pe2(ej)). (1.12) j,k

Now we define

Zo(c ,~ ,~ ; f l , f 2 , f3;g ) = ~ +Tr(c(p,q) oz(Lr (1.13) Ic~loO,o,o)(P,q;fl,f2,f3)

Here the sign is defined by making use of the orientation of the moduli space. (see Sect. 7.) Using it we define

Zo(fl, f2, f3; g) = zo(U1, f2,/3; r - Zo(gA, ~ , .@; f l , f2, f3; r

- Z o ( ~ , g f2, ~; f l , f2, f3; g) -- Zo(~, ~ , g f3; f b f2, f3; g)

+Zo(gf,,gf2, ~2~; fl, f2, f3; r + Zo(gf,, ~,gf3; fl , f2,f3; ~)

-t-Zo(f,~,gf2,gf3; fb f2, f3; r - Zo(gf~,~tf~,gf3; fl , f2, f3; r .

(1.14)

We then put

Zo(f l , f2, f3; ~) = -~ civil lo( ,~l f l,e2f2,/33/3; ~). (1.15)

(See Remark 1.17 the reason why we take 1/4 in place of 1/8.) The numbers defined in (1.14), (1.15) may depend on the choice of combinatorial propagators.

To obtain a number independent of the choice of functions f i and the Riemannian metric g of M, we need to do a similar construction for the graph A in Fig. 3. We first consider

~/a(A,f2, f4)= (x ,y; t l , t2 , t4)cM2xR 3 ~ } ~ ( Y ) = Y �9

~4(x ) = y

u

x ,&ej e2 , ~ y

Fig. 2.

Morse Homotopy and Chem-Simons Perturbation Theory

Fig. 3.

45

However, in fact, it turns out that this moduli space is empty, for generic j~. The reason is that there is no periodic orbit of gradient flow except the trivial loop at critical point.

Thus the leading term corresponding to the graph A is zero. But the correction term may be nonzero, which we are now going to define. Let Pi, q~ be critical points of f/ for i = 1,2. Assume that rl(pi ) -~ tl(qi ) + 1. We put

JZA(I,O, 1)(Pl,ql, P3, q3; fl,f4, f3) = {(x, y, t4) E M 2 • R+

lim Ot(x) = pl } t----~ -~OO J l

lim ~ (x) = ql t - - ~ - - O0 J l

lira (PL (y) = P3 t ---~ ~ -O~ .]3

lira ~ t (y) = q3 t - - -* - - Oo ./3

q~-4(x) = y

We let A(1,0, 1) be the graph obtained by cutting A at two points (Fig. 4). A(1,0, 1) consists of 4 edges el , el,e3,Z 1 e 2, e4. We define Z(I, ~) : qpl | qP3 --+ gql | qq3 by

(z(I, ~)(u | v),u' | v') = ~ ([Pel(U),P~(v)],e,)([Pe~(u'),&(e,)],P4(v')). i

Here ei is an orthonormal basis of qx. We then define, for C i E E n d l ( C . ( M , f , q ) )

ZA(CI, Sg, c3; UI, f 4, U3 ) = E E Pbqb P3,q31EJgA(kO, l)(pl,qb p2,qz; fl,f4,f3 )

-t-Tr((cl (PI , ql ) | c3(P3, q3 )) o Z(/, q)) .

We define ZA(CI,C4, C3"~fl,f4, f3 ) in a similar way. Using them we define

2A( f l , f4 , f3 ) = ! ~ ZA(1,o, 1)(gelfl,~,gssf3,~31fl,~33f3,~34f4 ) 8 ei=•

1 ZAO,l,1)(g~lf,ge4f4g~gf3; e l f l ,~4f 4 ,e3f 3 ) . 8 ei=~l

46

e e12 ei

e4

U' V'

Fig. 4.

K. Fukaya

We remark here that C,(M; - f ; q) = C.(M; f ; q)*. Hence we have End(C.(M; - f ; g ) ) = End(C.(M; f ; g)*). We identify 9f and 9 - f by this isomorphism.

We finally define

Zz ( f l , f 2 , f 3 , f 4 ;q ) = Z o ( f l , f 2 , f 3 ; q ) - ZA( f l , f z , f4;q)

- -Za( f l , f 4 , f 3 ; g ) - Za(f4, fz , f3;g) . (1.16)

Remark 1.17. We remark that the order of the group of isomorphisms of the tg- graph is 12 and one of graph A (graph in Fig. 3) is 8. Hence

(terms related to O ) - g(terms related to A)

1 = ~(2( ( te rms related to O) - 3(terms related to A))

is the natural way to sum up contributions from two different graphs. This is the reason why we take 1/4 in place of 1/8 in (1.15). Since we take 3 different choices {fi , f / } E { f l , f 2 , f 3 } , we did not put the factor 3 in (1.16).

Now our main result of this paper is as follows.

Main Theorem I. Let g, g' be the fiat Lie algebra bundles with the same Lie algebra 9 as their fibres. Assume H*(M; q) = H*(M; g/) = O. Then the difference, Z2( f l , f2 , f3, f 4 ; q ) - 2 2 ( f l , f 2 , f3 , f4;q ' ) is independent o f the choice of the Riemannian metric and functions f l,. . . , f4 and combinatorial propagators.

We next assume that M is a homology sphere. Then the Witten complex for the trivial flat bundle is almost acyclic. More precisely we consider the augmentation homomorphism C . ( M , f ; 9 ) ~ 9 for �9 = 0,3. Here we regard 9 as a trivial flat bundle. Let c.ed(M,f; 9) be its kernel. This complex is acyclic. Therefore we can perform the same construction to obtain Z2(fl, f2, f3, f4; 9). In fact in this case the weight is independent of the element of moduli space and depends only on 9. This


is because all holonomy is trivial. Let ng be this number. (nsu(2) = 23 - 2 = 6 for example.) Then z2(fl,f2,f3,f4; g) = n~Z2(fl,f2,f3,f4; 1). We now have:

Main Theorem II. Let ~ be the flat Lie algebra bundle with the same Lie algebra g as their fibres. Assume H*(M; ~) = 0. Then the difference, Zz(fl , f2, f3, f4; g) - ngZ2(fl,f2,f3,f4; 1) is independent of the choice of the Riemannian metric and functions f l , . . . , f4 and combinatorial propagator.

We remark that our invariant is well defined also for Lie algebra bundle of positive characteristic (if cohomology vanishes) provided the characteristic is not 2 or 3.

Since the correction terms we put look rather complicated at first sight, the author would like to add here some heuristic discussion to show where they come from.

Let us go back to the idea explained in introduction. Namely we consider the set

~ ' ( f ) = {(x,y) E M2[3t > 0 ~tf(x) = y} .

The boundary of this set is, roughly speaking, the union of the diagonal and

J//(f, id)= U {(x,y) r M Z l l i m ~ t f ( x ) = P = lim ~/i~(y)}. pGCr(f ) t---+ cx~ t--*-- oo

(see Lemma 2.4.) We put

J/g(f; p , p ) = { ( x , y )6M2l t f im ~tr(x) = P = t~-~lim q~r(Y)} �9

As explained in introduction, if we take r ig(f) such that O ~ ( f ) is the diagonal, then

#(d/[(f1 ) n J/{(fs) n J//(f3))

is the invariant we want to define. To find such a Jg ( f ) , we need to find JC"(f) such that 0 ~ " ( f ) --- U ~ ' ( f ; p, p). Our correction term comes from these components.

To explain the reason why the contribution of J # ' ( f ) is calculated by the terms we add, let us consider the following situation. We assume that Cr ( f ) -- {Pi, qi : i = 1 . . . . . N}, such that ~](Pi)= ~l(qi)§ 1, #Jg(Pi, qi)---- 1, and that Jg(x,y) is empty for any other pair x, y with q(x) = q(y) + 1. In fact this assumption is never satisfied because the homology group (with trivial coefficient) of M is always nontrivial. We put this assumption only to simplify the explanation.

We then have that our combinatorial propagator is ~i[Pi] | [qi]*. We now put

~ / ' ( f ) - - y. { ( x , y ) E M2lt~m ~ ( x ) = Pi, t---~-cclim ~tr(y)-----, qi}" We then find that & # ' ( f ) = U ~t ' (f ; pi, p;) u U J g ( f ; qi, qi) as required. Also

we see that

#(dC'(fl ) N J/r n J/~(f3 )) = Ze(gA, (ZJ, ~ ; f l , f2, f3; trivial).

This explains the origin of our correction terms related to O. To understand why we need to consider the other graph A, we need to consider

the transversality at diagonal. One important remark is that for any choice of J} two

48 K. Fukaya

submanifolds .//{(fl) and Jg( f2) are never transversal to each other at diagonal. In fact

d i m ( ~ ' ( f l ) n ~ ' ( f 2 ) ) > dim of diagonal = 3,

while dim ~t ' ( f l ) § dim J / ( f 2 ) - d imM 2 = 0.

Therefore we need to study carefully what happens in the neighborhood of diagonal. We remark that this is related to the framing of our manifold M, since the framing of M gives a way to perturb the diagonal. This is probably related to the fact that Witten's invariant [W2] depends on the choice of framing. However in this paper we do not use framing directly. Maybe framing is introduced in some sense implicitly when we choose our Morse functions.

We remark that in our situation the transversality at diagonal is related to the compactification of the moduli space d{(f l , f2, f3) we introduced. As we discuss in next section, the end of this moduli space occurs in two ways. One is the case when ti --+ co . This end is related to the boundary component J / ( f ; p, p ) we discussed above. The other case is when ti ~ O. This is related to the transversality we are discussing now.

In Chem-Simons Perturbation theory, two constructions are used to handle the problem related to the transversality of diagonal. One is to consider another graph A and the other is to cancel anomaly by introducing another term. These two phenomena both have an analogy in our approach.

First we consider also another graph A. These terms are used to cancel the effect of ends such that one of ti --* O. (see the end of Sect. 3.)

The other is related to the case when all of ti --~ O. This causes some anomaly term also. We kill them by taking the difference of invariant related to two different fiat bundles. In place of taking the difference of an invariant determined by different fiat bundles there may be an alternative way in the case of homology spheres. We will discuss it informally in Sect. 8.

2. Transversality and Compactness

We begin by proving Lemma 1.2 The idea of the proof is similar to one used in various topological field theories (see for example [FU]) and is described as follows. We first consider the union of moduli spaces UjScc~(M)Jg(fl , f2, f3). We then prove that this space is an infinite dimensional manifold. It then follows that J/{(fl, f2, f3 ) is a manifold for generic f l , f2, f3.

We are going to write f in place of ( f l , f2, f3) for simplicity. We define ~ f :

M x R3+ --~M by

�9 f ( p ; t,, t2, t3 ) = (~1 (p), ~)22 (p ) ' ~)3 3 ( p ) ) .

Let A ~ M C M 3 be the diagonal. By definition we have J / / o ( f ) = ~71(A). By

moving f and also moving the metric we have

qb : M x R3+ x Met x ( C ~ ( M ) ) 3 ~ M 3 .

Here Met is the set of all metrics of M.


Lemma 2.1. The map �9 is transversal to A.

Proof Let (p;tl,tz, t3; f ,g) be an element of #-X(A). It suffices to show that the differential of �9 is surjective at this point. We choose disjoint open subsets U~-, i =

1,2,3 of M such that Ui N [,.Jt~[o, td q)tf'(P) +;g and U~ A Ut~to, tj] #t~(p) = ~ for

i4=j. By restricting # to {(p;tl,t2,t3)} • C~(U1) • C~(Uz) • C~(U3) x {9}, we obtain ev : C~(U1) • C~(U2) • C~(U3) ~ M 3. (Here C ~ is the set of smooth fimction of compact support.) Namely

eV(hl,h2,h3) tl ~t2 , t3 = (~f'l+hl(P), f2+,2(P) ~fa3+h3(P))"

Let Vg E Tcb~(p)m, i = 1,2,3 be arbitrary tangent vectors. By our choice of U; we

can find hi such that

d*~+~h,(P) ds s=~ = V/.

It follows that the differential of �9 at (p; t l , t2 , t3 , fb f2 , fa ,g) is surjective. The lernma follows.

We remark that we use the fact ti > 0 in the proof of the lemma. I f we generalize the map �9 so that it is defined at the point tl = t2 -- t3 = 0, then the lemma may not hold. (We discuss this point a bit later in this section, when we study the compactness, and we discuss it in more detail in Sect. 6.)

By Lemma 2.1, the set ~/i-~(A) is an infinite dimensional manifold. We consider the projection rc : ~ - I ( A ) ~ Met x ( C ~ ( M ) ) 3. Then by a simple counting argument, we obtain the following:

Lemma 2.2. ~ : ~ - I (A) - -+ Met • (C~176 3 is a Fredholm map of index 3 - dimM.

We remark that we do not assume that d imM --- 3 up to this point. We now assume it. Then by Lemma 2.2 and Sard-Smale theorem [Sm], the space Jgo( f ) = ~ j l ( A ) is 0 dimensional for generic choice of f and metric. We thus proved the first

statement of Lemma 1.2 We postpone the discussion about orientation until Sect. 7. We next tum to the proof of the compactness. Before going there we remark that the following one parameter family version of Lemma 1.2 holds. (This is used in a proof of well-definedness in later sections.)

Let (J~l), gl),(~2),g2) be two elements of Met x ( C ~ ( M ) ) 3 such that transversality holds for them. We consider a path L joining them. Then Lemma 2.2 and the Sard-Smale theorem again implies that:

Lemma 2.3. For generic L the space, ~ - I ( L ) = U(Lo)CL ~o(f.g) is one dimen-

sional manifold with boundary = -dgo(J~l) , gi ) tO ~'o(f(2), g2).

We hereafter put ~/o(K) = U(~0)eK JC/o(f, g). Now we begin the discussion of the compactness. First of all we remark that if

we fix (tl,tz, t3) then the space

{(p;tl,t2, t3; f ,g)](p;h,t2,t3) E d/ lo(f ,g) , ( j~g) E K}

50 K. Fukaya

is compact (if K is compact.) This is a consequence of the fact that our moduli space is the set of solutions of an ordinary differential equation. The analysis one needs to prove this compactness is fairly easy. (In various kinds of topological field theories (such as Yang-Mills theory or pseudo holomorphic curve) one proves a similar compactness theorem using the a priori estimate of partial differential equations.)

Therefore to study the compactness we only need to consider the case when one of ti goes to zero or infinity. This is similar to the bubbling phenomenon which appears in various kinds of topological field theories.

We first study the case when ti --~ c~. In case there is only one Morse function, the study of "bubble" of this kind was used in the proof of the fact that the Witten complex is a complex (namely ~ = 0.) The argument used there can be applied to our situation without change. (See [Sc] for the detailed account of this argument.) Here we recall the result we need for our purpose. Let ~. : [0, ti] --+ M be a sequence of lines such that

{d~ = grad f

ti -~ c~

Assume also that Pi = ~ ' ( 0 ) converges to p E M and qi = ~ . ( t i ) converges to q C M. We assume also that p ,q d~Cr(f). Then we have:

Lemma 2.4. There exists Xl . . . . . X k E C r ( f ) and gradient lines mj joining xj and Xj+l. Also there exists a gradient line mp, mq joining p, xk to Xl, q respectively, such that the image o f ~- : [0, ti] --~ M converges to the union o f mj and mp, mq with respect to the Hausdorff distance.

We are going to apply this lemma to study the compactification of our moduli space. We first remark that Cr(y~)n Cr()~) : t :~ at the only codimension 3 subset of (C~176 3. Since we are considering only at most a 1 dimensional family, we always assume C r ( ~ ) N Cr(y~)= ~ for i + j . Hereafter we write (C~ for such a subset for simplicity.

Now w e suppose 1(i ) E ~o( f ( i ) , g i ) , and that l i m i ~ ( f ( i ) , g i ) = ( f , g ) . We also assume that I(i) is identified to (xi, Yi, tl,(i), t2,( i) , t3,( i)) such that tj,(i ) > C for some positive C independent of i.

We assume that the sequence I(i) diverges. Then at least one of tj,(i) converges to infinity (for some j = 1,2,3.) Let us assume for simplicity that tl,(i) --+ ~ and the other two are bounded. In fact by a counting argument (similar to those we are going to discuss) we find that in a 0 or 1 dimensional family of ( f , g) only one of tj,(i) can go to infinity.

Now we apply Lemma 2.4 to the restriction of I(i ) t o the first edge. The assumption that xi = I(i)(O) converges to x E M and Yi = l(i)(ti) converges to y E M is satisfied (after taking a subsequence), since the restriction to I(i) to the other two edges converges. The other assumption that x, y r ) is also satisfied since x 4: y. Thus the image of the first edge el by the map l(i) splits into the union of several gradient lines. Again by a counting argument (which we are soon going to discuss) we may assume that the limit will be the union of 2 gradient lines both of which


contain a critical point p of f l . We then define the following moduli space:

dCo(1 ,o ,o ) (p ,p ; f i , fE , f3 ) = {(x, y;t2, t 3 ) E M 2 • 2

lim #fj1- -(x) = pp } t - - - + + o o J 1

lim cbf ( y ) = t---+ - - o o

�9 ~(x) = y, j = 2 , 3

51

By moving f and the metric in a set K C (C~(M))3 x Met, we define Jgo(1,o,o)(id, K ) as follows:

~/o(1,o,o)(id, K ) = U [.] ~'o(1,o,o)(p, p; f l , f2, f3 ). (f,o)EK pGCr(f, )

We then can prove that J#oO,o,o)( id , (CO~ 3 • Met) is an infinite dimensional manifold in a way similar to the proof of Lemma 2.1. We find that the projection ~loO,o,o)(id, (CO~ 3 • Met) ---+ (C~176 3 • Met is a Fredholm map of index -1 . The calculation of the index is done as follows. In the definition of /go(1,0,0)(P, P; f ) there are 2 variables moving on M and two positive numbers. For the equation, the first equation puts t/(p) conditions. The second equation puts 3 - t l (p) equation. The two equations in the third line puts 3 conditions each. In total the index is 3 • 2 + 2 - t l (p) - (3 - t l (p ) ) - 6 = - 1 .

Thus using the Sard-Smale theorem again, we find that, in the case when we fix a generic f there is no such an end. And in case f moves in a generic one dimensional set there are finitely many points like that.

Next we turn to the case when one of tj,(i), say tl,(i) goes to zero. Namely

we assume that (xi, yi; tl,(i), t2,(i), t3,(i) = I(i) E J[/[o(~i), Oi), limi~oo tt,(i) -- 0,

l i m i ~ ( f i ) , g i ) = ( f , g ) . By assumption l imi-- ,ootl , ( i)= 0 we have (by taking a subsequence if necessary) that l i m i ~ xi = limi--+oo Yi = x. Since Cr(f2) A Cr(f3) = ~ , we may assume without loss of generality that x ~Cr(f3). Then there are two cases.

Case 1. x ~Cr(f2). In this case we have limi~oo t2,(i) -- lim/~oo t3,(i) = 0. We can

then use the fact ~ i l i l ( x i ) = Yi to show that three vectors grad ]~(x), j = 1,2,3 are

parallel to each other.

Case 2. x E Cr(f2). In this case we find gradJ)(x) , j = 1,2 are parallel. Since grad f 2 ( x ) = 0, we may say that three vectors grad J)(x), j = 1,2, 3 are parallel also in this case.

Here we say that gradfi.(x) and grade(x) are parallel if there exists a non- negative number ri, rj such that rigrad f i ( x ) = rjgrad f j ( x ) and (ri, r j )@(O , 0) . (This notation is a bit different from usual one, since we avoid, for example grad f,-(x) = -grad fj.(x).)

Now we put

R ( f ) = (x E M[grad J)(x) j = 1,2,3 are parallel to each other}.

For K C_ ( C ~ ( M ) ) 3 x Met we put

R ( K ) = { ( ( f , g ) , x ) l ( f , g ) �9 K , x ~ R ( f ) } .

52 K. Fukaya

Lemma 2.5. R( ( CC~(M ) ) 3 • Met) is an infinite dimensional manifold. And the map rc : R((C~176 3 x Met) ~ (C~176 3 x Met is a Fredholm map o f index -1 .

The proof of the first statement is a standard transversality argument and the proof of the second statement is a simple dimension counting.

Lemma 2.5 and Sard-Smale theorem imply that for generic ( f , g), the set R ( f ) is empty. Thus we have proved the following:

Lemma 2.6. For generic ( f , 9) the space Jgo( f , g) is compact.

This completes the proof of Lemma 1.2 except orientation�9 We turn to study Jge(L) for the generic one dimensional space L C (CC~(M)) 3 x

Met. We already discussed most of the phenomena which can occur�9 One is described by the space Jge(1,o,0)(L) U ~'e(0j,0)(L) U Jge(0,0,1)(L) (which corresponds to the case when tj ~ ~ ) the other is described by the set R(L). For each element of Jge(1,0,0)(L)U J/go(0,1,0)(L)U ~/e(0,0,1)(L) we can construct an end of Jgo(L) homeomorphic to [0,oo). This fact is based on a converse to Lemma 2.4 which was also proved and used in the construction of the Witten complex.

We can prove also that, for each element of R(L), there exists an end of J//e(L) which is homeomorphic to [0, co), if L is generic�9 The proof of this fact is more delicate and will be given in Sect. 6. We observe that elements of Jge(L) which are in an end corresponding to an element of R(L) are of small diameter�9 Hence they are contractible. This fact is important since it follows that the weight associate to such an element is independent of the fiat bundle ~. Therefore, since we consider the difference of invariants obtained from 2 fiat bundles, the contribution from this kind of end cancels automatically�9 We summarize our result as follows�9

Lemma 2.7. Let L C (C~(M))3 x Met be a generic arc joining two generic elements o f ( C ~ ( M ) ) 3 x Met. Then there exists a one dimensional compact man- ifoM cgJgo(L) with boundary such that J/le(L) is its interior and ~ [ e , L ( f ) -

Jge ,L( f ) -- ~e(1,0,0)(L) U Jge(0,1,0)(L) U d/e(0,0,1)(L) U R(L).

In later sections, we use this and similar lemmas to prove the well-definedness of our invariant.

3. Independence of Combinatorial Propagator

To prove our main theorem (independence of invariant of various choices) we first consider, in this section, the independence of the combinatorial propagator. We assume that we have two choices gA, g'fl of the combinatorial propagator for the first Morse function f l and assume that other combinatorial propagators gfi, i -- 2, 3, 4 are the same. And we prove that the resulting invariants are the same.

Since the Witten complex of local coefficient are acyclic by our assumption, �9 ' . We put there is h ~ End2(C.(M; f l , q) such that ~h = gA - 9 A

h = ~ h(pl,q2)[p]] | [ql]* �9 ~/(Pl )=r/(ql )+2

We are going to study the moduli space J//eO,0,0)(Pl,ql) in the case r/(pl ) = r/(ql) + 2. (Since we fix J} here we omit them in the above notation.) In this case


the moduli space is one dimensional. (One needs transversality to prove it. It can be proved in the same way as in the last section.) We need to know the ideal boundary of this moduli space JC/oo,o,o)(Pbql ).

Lemma 3,1. ,/ddo(1,o,o)(pl,ql ) is an interior o f a compact oriented one dimensional manifold cdJgo(1,o,o)(Pl, ql ) such that

8(K//goO,o,o)( Pl , ql ) =- U ~ ( p l , p~) x :go(~,o,o)(p'l, ql ) r/(ptl ) = q ( p l ) - - I

- - !

U U J/Zo(1,o,o)(Pa,q~) x ~/(ql,ql) q(q', )=r/(ql )+1

LJ U Jdo(1,1,o)(Pl, ql; P2, P2) ,02 E Cr(f~)

U U ,~goo,o,1)(pl,ql;p3, p3) U 5ao(t,o,o)(pi,ql). P3 E Cr(f3 )

Let us explain the notations above. We define

J/'~(1,0,1)(Pl, ql; P3, P3) ---- / (x ,y ; t2) E M 2 X R+

A(x) = Pl,

lim Ct (y) = ql

,.m ~;,(x)= p3,

�9 ; ; (x ) = y

and

5Po(1,o,o)(f,g;pl,ql) = x G M lira ~t (x)-=ql t___.+_b O O J l

grad f2(x) is parallel to grad f3(x)

Other notations are defined in a similar way.

P3 P3 x

Fig, 5.

54 K. Fukaya

Proof. The proof of Lemma 3.1 goes almost in the same way as Sect. 2.

Let (xi, Yi, t~ i), t~ i)) E J/O(1,O,O)(Pl, ql ) be a divergence sequence. Then by taking a subsequence if necessary, we may assume that one of the following happens.

(1) t~ i) -'+ ~ ,

(2) t~ i) --+ ~ , (3) Xi converges to a point in the boundary of the unstable manifold of pm, (4) Yi converges to a point in the boundary of the stable manifold of ql, (5) t~ i) "--* 0 or t~ i) --* O.

It is straightforward to see that cases (1 )~ (5 ) correspond to the 1,-5 th term of the right-hand side of Lemma 3.1. It is easy to see that, for each element in 1,-~4 th term of the right-hand side of Lemma 3.1, there exists an end of dgo(1,o,o)(pl,ql) corresponding to it. We postpone to Sect. 6, the proof of the fact that there is also an end of M//oO,o,o)(Pl, ql ) corresponding to each element of 5~oO,0,0)(f, 9; Pl, ql ). We here prove that 5eoO,0,0)(j, g; pl, ql ) is zero dimensional.

We prove this fact again by using transversality. We put

r176 ~--- U U ~0(1 ,0 ,0) ( / , g, p l , ql ) �9 (~9)E(C~(M)) 3 • Pl,ql Ecr(fi)

q(Pl )=~(ql )+2

We then have:

Lemma 3.2. The space Sfo(1,0,0),2 is an (infinite dimensional) manifold and the projection: 5a00,0,0),2 ~ (C~ 3 • Met is a Fredholm map of index O.

Lemma 3.2 is a consequence of the usual dimension counting and transversality. It follows immediately from the lemma that 5~o(1,0,0)(f, g; Pl, ql ) is zero dimen-

sional for r/(pl) = r/(ql) + 2. The proof of Lemma 3.1 is now complete. In order to deduce identities from Lemma 3.1, we are going to define a num-

ber Zo(1,1,0)(el, c2, (~) for Cl E End2(C.(M; f l ; q)), e2 E Endl(C.(M; f2; q)). Let us write

c j = ~ ej(pj, qj)[pj]| j = 1,2.

An element of the moduli space ~'oO,l,0)(pbql; P2,q2) is regarded as a map I : O(1, 1,0)--+ M. Here O(1, 1,0) is a graph obtained by cutting the O graph at two points, one on the first edge and the other on the second edge. We define the weight Z(I) : qp, @ r/p2 ---+ qq, | ~/q2 as we did in Sect. 1. We now put

/O(l,l,0)(Cl, C2, ~ ) = E E • Tr(z(I ) o @1 (Pb ql ) | c2(P2, q2 ))) . pl,ql,p2,q2 IEdCo(pbqbp2,q2)

To study the fifth term we define

ZSo(1,o,o)( C1, f~, ~ ) -~" ~ E 5: Tr(z(I ) o (el(p~, ql ) ) . pl,ql 1C Sfe(1, o,o)( pb ql )

Here Z(1) �9 rlp --+ ~q is a parallel transport along the gradient line of f l containing x. (Here we regard x E M.)

Now Lemma 3.l implies:


Lemma 3.3.

55

Z ! O(1,O,O)(g fi, f~, ~ ) -~ Zo(1,o,o)(g f~, f~J, ~ ) -- Zo(1,1,o)( h, id, ~ )

-Zo(1,o,1)(h, ~ , id) - ZSo(1,o,o)( h, ~ , fZJ) .

We can also prove the following formula:

Zo(1,1,1)(gtf~, gf2, g f3 ) = Zo(1,1,1)(gf~, gA, g f3 ) - Zo(1,1,1)(h, t~gf2, g f3 )

-Zo(1,1,1)( h, g f2, (~g f3 ) " (3.4)

Equation (3.4) follows from the following description of the boundary of moduli spaces:

0 c ~ o ( 1 , 1 , 0 ) ( P l , q l ; P2, q2; P3 , q3 )

--= U ~ ( P l , P ~ ) • J/lo(1,1,1)(P~l,ql;p2, q2;p3,q3) ~l(ptl )=r/(pl )-- 1

tA U -~(qt l ,ql) • "/go(1,1,1)(Pbqtl; P2, q2; P3,q3) q(q'l )=r/(ql )+ 1

U U "~(P2, P~) • ~[O(l l l ) (Pl ,q l ' " , , ; P2, q2, p3,q3) r/( p~)=r/( P2 ) - 1

U U - - t �9 ! d/[(q2, q2 ) • ./gO(I,1,1 ) (P l , ql, P2, q2; P3, q3 ) r/(q~ )=~/(q2 )+ 1

U U ~ ( P 3 , P~) x Jgo(1,1,1)(Pl,ql; pz, q2; P3,q3) ~/(p~)=~/(p3)-- 1

U U ~(q~,q3) • Jgo(1,1,1)(pl,ql; p2,q2; p3,q~). q(q~)=~l(q3)+l

We can prove this equality in a way similar to Lemma 3.1. We remark that we do not need the term 5aeO,o,o)(Pl, ql ).

To consider the term Zoo:,o)(gfl , gA, ~ ) and like that, we need to study another kind of degeneration. To describe it we use the following moduli space:

d/[O(1,1,X)(Pl,ql; P2,q2) = {x E M t~_moo tJ~t ~__ A(x) ql

"

t limor O:~(x) ---- q2

(This moduli space is the case when t3 = 0.) We can assume that unstable and stable manifolds of f l , f 2 , f 3 are transversal to each other. So in case q ( P l ) = q ( q l ) + 2 , q ( P 2 ) = q(q2)+ 1, the space dgo(1,1,x)(pl,ql;p2,q2) is 0-dimensional and compact. Using this space we can describe the boundary of the moduli space

56 K. Fukaya

J/40(1,1,0)(pl, ql ; p2, q2) as follows:

~d~/oO,l,0)(pl, ql; P2, q2)

= U ~ ( p l , p ~ ) x ~e(1,1,0)(p'l,ql;p2,q2) q(P'l )=~(Pl )- I

U U ~(qt l ,q l ) • d/[O(1,1,O)(Pl,q'l; P2, q2) rl(qtl )=q(ql )+1

t J U (p2,p2) X Jgo(1J,O)(Pl,ql;P2, q2). q(p':)=~l(p2)- 1

U U - - t , , J//(q2,q2) • J/oO 10)(Pl,ql; P2,q~) r/(q~)=q(q2)+l

U U /gOO,l,1)(pl,ql; p2,q2; p3, p3) P3 E Cr(j~ )

U J4o(1,1,x)(pl, ql; p2, q2). (3.5)

We define the number Zoo,Lx)(h, gf2, ~ ) using the moduli space JgO(Ll,X)(Pb ql; P:,q2) in a similar way. Then (3.5) implies the following:

Z t O(1,1,O)(gfl, g f2, ~J) = Zo(1,O,O)(gfl, g f2, ~J) + Zo(1,1,0)(h, Ogf2, ~ )

+Zo(1,1,o)(h, g f:, id) + Zo(1,1,x)(h, g f2, ~J)" (3.6)

Now we use the fact that ~3g~ = id. Then Lemma 3.3, (3.4), (3.6) imply:

Lemma 3.7.

2 ~ ( f , g) -- Z o ( f , g) ---- Zo(1,1,x)(h, g f2, ~ ) + Zo(1,x,1)(h, (~J, g~ )

+ ZSoo,i,o)( h, KS, ;ZJ) .

We next sum them up for signs ej of J~. We then remark the following:

Lemma 3.8.

ZSo(1,0,0)(fl, f2, f3; h A , ~ , ~ ) = -ZSo(1,0,0)(-fl, f2, f3; h-Ji, ~ , ~ ) .

Pl ql

Fig. 6.

)


Proof. We first remark that the equality between moduli spaces Jgso(Pl, q l ; f l , f2, f3) = J/so(q1, p t ; - f l , f z , f 3 ) . The orientation is opposite as we will see in Sect. 7. We remark also that the isomorphism End(C , (M; f ;g ) )= End(C,(M; - f ; g)) is obtained by c[p] | [q]* ~-+ c[q] | [p]*. The lemma then follows immediately from the definition.

Using Lemmas 3.7 and 3.8 we have

Lemma 3.9.

42'o(f, g) - 4 Z o ( f , g) = E Zo(1,1,X)(glfl,/32f2, g3f3; hell1, Qe2f2, ~ ) 81,g2, ~3=-4-1

+ ~ Zo(1,x,1)(elfl,e2f2, e3f3;hexf~,f~J, ge3f3). ~l,g2,~3=q- 1

The proof that this last term cancels with the corresponding term related to the other graph A uses the Jacobi identity and is similar to an argument in Chem- Simons Perturbation theory. We are going to explain it here.

We perform the same construction using the other graph A and obtain the equality

8(2~1 -- ZA) = ~ ZA(1,1,x)(helA, 9e2f2, ~)( /~l f l , g2f2, g4f4) 81, g2, g4=::[=l

+ E ZA(l,x,m)(h,,Ii,~,O,3i,)(elfl,e4f4,e3f3). ~1,82,g4=-~ 1

Here the number ZA(1,1,x)(h~IA, 9~2f2, ~ ) ( e l f l , e2f2, e4f4) is defined by using moduli space ,///[A(I,I,x)(Pl,ql, P2, q2)(glfl , 82f2, g4f4), which is a component of the boundary of .///[a(1,1,0)(pl,ql,pz, q2)(elfl,e2fz, e4f4) corresponding to t4 = 0.

We next remark the following obvious facts:

�9 /[gO(1,1,y)(Pl, ql , P2, q2 )(f l , f2, + f3 ) = #///O(1,1,x)(ql, Pl, P2, q2 ) ( - f l , f2, +f3 )

= ,A/[O(1,1,X)(Pl, ql, q2, P2 ) ( f l , --f2, + f3 )

= JgO(1,1,x)(ql, Pl, q2, P2)( - - f l , --f2, + f 3 )

= "-/~A(1,1,x)(pl, ql, P2, q2)(f l , f2, + f 4 )

= #/{A(1,1,x)(Pl, ql, q2, Pz) ( f l , --f2, + f 4 )

= J[//a(1,1,x)(ql, P l , P2, q2)(--f l , f2, + f 4 )

= "~/[A(1,1,x)(ql, Pl , q2, P2)(- - f l , --f2, -+-f4), (3.10)

and also a similar formula with P2, q2, f2 replaced by P3, q3, f3 holds. To show the cancellation we need to discuss the weights.

The moduli space in Formula (3.10) is also described as the set U(pl )N S(ql )fq U(p2) A S(q~). Here U(.) is the unstable manifold and S(.) is the stable manifold. Hence for each point x of U ( p l ) N S ( q l ) N U(pE)AS(q2), there are 4 gradient lines joining it to pl ,ql , p2,q2 respectively. Let Pp,,x : gp, --~ gx, Pq,x : gq, --~ gx be the parallel translation along these paths. The weight of x gives a map gpl | ~ql --~

gp2 | gq2- But the weight does depend on the moduli space whose dement x is

58 K. Ftakaya

regarded as. Let us regard x E ,////o(1,1,x)(pl, ql, p2, q2; f l , fe , • In other words, x is regarded as a map from the O-graph such that the third edge is mapped to the point x. Therefore, for u E r v C gin, u' E qql, v' E qq2, the weight is given by

E ([Ppl,xU, Pp2,xV], ei) ([ei, PqE,xvt], Pql,xU'> = - ([ [ep,,xU, ep2,xV], eq,,xU'], PqE,xv'>. i

(See Fig. 7a.) The weight as an element of Mo(1,1,x)(ql, Pl, p2 ,q2 ; - f l , f 2 , :t:f3) is

([ [eql ,x u ' , epz,x V], ep, ,xU], eq2,x V') - ~ - -- < [[ep2,x v, eql ,xU '], P~I, Xu], nqE,x vt) .

(This sign in the formula is related to the orientation and is explained in Sect. 7.) For M~9(1,1,x)(pz,qt,q2, P 2 ; f i , - f 2 , 4 - f 3 ) the weight is

<[[ep,,xU, eq2,xV'],Pq,,xUt],Pp2,xV > = -([Pp,,xu, Pq2,xV'], [Pp2,xv, Pqj,xU'])

: "-}-<[Ppl,xU,[Pp2,xv, Pq,,xd]],Pq2,xVt >

= -([[ep:,xv, Pq,,xU'],Ppl,xul,Pq2,xV' ) .

And for Mo(1,1,x)(ql, Pl, q2, P2; - f l , - f 2 , +f3) we have

-([[Ppl,xu, Pp>xV],Pql,xu'],Pq2,xV') .

For another graph .//IAO,l,x)(qb Pl, P2, q2; f l , f2, ~ f4 ) the weight is

([Pq,,xU', Pp,,xu], e,) ([ei, Pp2,xV], Pq2,xv') = ([[Pq~,xU', Pp,,xU], Pp2,xV], Pq>xV') i

(Fig. 7b). One can check that the weight for JgAO,l,x)(qb Pl,q2,P2; f l , - - f 2 , 4-f4), etc. is also ([[Pq,,xu',Pp>xu],Pp:,xv],Pq2,xV').

Then by the Jacobi identity

[[Pp,,xU, Pp2,xv],Pq,,xu t] q- [[Ppa,xv, Pq,,xut],Ppl,xU ] -3- [[Pq,,xU',Ppl,xu],Pp>xV ] = O. (3.11)

Pl ql Pl qt

Fig. 7a, b.


Those terms cancel. To see this we show that in the formula

(el,e2,e3=_4_l Zo(1,1,X)(Slf l, ~2f2, e3f3; helfl, ge2f2, ~ )

+ ~ Zo(1,x,1)(elfl, g2f2, 83f3; helfl, ~ge3f3 )) gl, 82~83::~1

1 ( ~ ZA(1,1,x)(h~i~,O~2~,~)(e~f~,e2f2,e4f4 ) 8 el,g2, s

+ ~ ZA(1,x,1)(he,f,, (2~, 9~3f, )(el f l , e4f4, 83f3 ) ) gl, 82, 84:nt-1

the coefficient of each of the terms in (3.11) is - 1 / 2 . In fact, the term [[Pp.xu, Pp2,xv],Pq.xU' ] appeared twice in Zo(l,l,x)(" ") or Zo(1,x,1)(" "), hence its coefficients is - 1 / 2 . One can show that the coefficient of [[Pm,xv, Pq.xU'],Pp~,xU]. On the other hand [[Pql,xU',Pp.xu],Pp2,xv ] appeared 4 times in ZAO,I,X)(" "') or ZAO,x,1)('" "). Therefore its coefficient is again - 1 / 2 . Thus we proved the independence of our invariant of the combinatorial propagator.

4. Independence of Morse Functions I

We next consider a one parameter family (fs, gs) E ( C ~ ( M ) ) 3 x Met(M)and show

that (f0,90) and ( f l , g l ) give the same invariant. Unfortunately the proof is rather long and technically complicated. It seems that one needs to develop some kinds of homotopical algebra to get rid of the mess in this and the next sections. The reader who is interested in only a basic idea and not so much technical detail may skip the next section.

Before starting the proof let us give an informal explanation of well definedness. One typical way how the Morse function changes is death or birth of critical points. Namely for example we can consider a family

UI,s(XI,X2,X3 ) =

Then a pair o f critical points will die at that t / (p) + t/(q) = 1. We then find that a component [p] | [q]*.

+ s x , - x 2 - x 2

s = 0. Let p, q be these critical points such the combinatorial propagator for s < 0 has

Let us assume that there is a family of elements in (xs, Ys, t2,s, t3,s) E dgo(1,0,0)(p,q : f l ,~, f2, f3), and try to find what happens when s = 0. Let 's assume moreover that (t2#,t3,~) are bounded. Then when s = 0 we have an element of J/{oo,o,o)(0, O:f l ,0 , fe , f3). One can expect after passing zero a new element of dgoo,o,o)(f l#,f2,f3) will be born. Hence our invariant will be the same. To make this argument rigorous is the purpose o f this and the next sections.

The proof of invariance of the Morse function is done in two steps. First we assume that Witten complexes are the same for (f~,9~), s C [0, 1], (Sect. 4). Next we conisder the case when the Witten complex changes.

To simplify the notation we assume that Cr(J},~), i = 1,2,3, s c [0, 1] are independent o f s. (We can always do it by using an appropriate diffeomorpbism of M, since we are assuming that the Witten complex is independent o f s.)

60 K. Fukaya

We consider the moduli space Jgo(L), where L = {(f~, g~)ls C [0, 1]}. In Sect. 2, we gave a compactification of this moduli space such that ~r J go (L )= ~'oO,0,0)(L) tO dgo(0j,0)(L)U ~/o(0,0,1)(L). Here J//o(a,0,0)(L) is a moduli space of maps from the graph O(1, 0, 0) to M, such that the two exterior vertices go to the same point in Cr(fl,s). This moduli space is of zero dimension. By exactly the same formula we used to define the weight for elements of JgoO,o,o)(P,q;fl,fz, f3), we define Z(1,~) : ~p ~ ~p for each element I C J/o(1,0,0)(L). We put

• Tr(z(I, r IEd4e(i,o,o)(L)

Zo(1,o,o)(L; ~; id, ~ , ~ ) =

Then Lemma 2.7 implies that

Lemma 4.1.

(zo(f ; - z o ( f o ; - (zo(f ; - z o ( f o ; { ' ) )

= Zo(1,o,o)(L; r id, ;~, ~ ) - Ze(l,0,0)(L; r id, ;~, (ZJ)

+ Zo(oj,o)(L; ~; ~ , id, (~j) - Zo(0,i,0)(L; r ~ , id, ;ZJ)

+Zo(o, oj)(L; ~; ~ , ;3, id) - Ze(0,0,1)(L; ~'; ~ , ~ , id) .

Proof Lemma 2.7 implies that cg~e (L) gives a cobordism between d/ lo( fo)U - ~ / e ( f l ) and Jgo(1,0,0)(L)U Jge(0j,0)(L)U Jge(0,0,1)(L). Since bundles are flat the weights are compatible. The space cddge(L) is noncompact but for elements in the end the weights are the same for ~, ~'. The lemma follows.

We next study the change of the correction terms. First we remark that since we assume that the Witten complex is the same, we can choose the same combinatorial propagator for fs, s E [0, 1].

For p l , q l e C r ( f l ) with q ( p l ) = q ( q l ) + l and p E C r ( f 2 ) , we define JgoO,l,o)(Pl,ql, p, p ) for each fs, s c [0, 1] in a way similar to Sect. 3. We take its union over all s and denote it by Jgo(1,1,o)(L; Pl, qbP, P). By a similar transversality argument as before we find that J//o(1,1,0)(L; Pl,ql, P, P) is a 0 dimensional compact oriented manifold. Also each element I of ~/oO,1,0/(L; Pl, ql, P, P) induces a map Z(L r ~p | ~p~ ~ ~p | ~q~. We then define

Zoo,l ,o)(L;~;gf , , id ,~) = ~ + Tr ( z (L~ )o id | IE.fflo(t, l ,o) (L;pl ,qbp)

Here O(Pl, ql) is a coefficient of combinatorial propagator. We define

t----~+oo Jt,s

5~o(l,o,o)(L; p b q l ) = ( x , s ) E M • [0,1] lim q)t ( x ) = q l t-'+--O0 Jl,s

[grad fe,s(x) is parallel to grad f3,s(x)

In case when r/(pl ) = q(ql ) + 1 this is a compact oriented 0 dimensional manifold. (In Lemma 3.2 it was q(Pl ) -- q(ql ) + 2. But here we put an extra parameter s.) Using this moduli space we define

SZeo,o,o),(L;q;gfl,fZj, fZj ) = ~ + T r ( z ( I , r IE 5~ )

Now in a way similar to Lemma 3.3 we have the following:


Lemma 4.2.

Zo(1,o,o)(fl; ~; O'j~, ~ , ~ ) - Zo(1,o,o)(fo; ~; gjq, ~ , ~ )

= Zo(1,1,o)(L; q; Ogf~, ~J, f~J) + Zo(1,1,o)(L; q; gf~, id, ~ )

+ Zo(1,o,1)(L; q; g f~, ~ , id) + SZoo,o,o)(L; q; Oft, ~ , ~J) .

Since tgfl = id. we find that the first term of fight-hand sides of Lemma 4.2 cancels with the first term of the right-hand side of Lemma 4.1.

To discuss the change of Zoo,o,o)(fl; q;gA,gf~,(3) we need to define

#~O(1,1,x)(L; Pl, ql; p2, q2 )

{ lim ~ t ( x ) l i m ~ t (x) } t-~+oo jl,s = Pl , = ql l_.+__O~ Jl,s = ( x , s ) ~ M x [ O , 1] lim ~ ( x ) = p 2 , lim ~ 9 ~ ( x ) = q 2 "

t--~+00 j2,s t..._~_o: 2 J2,

If ~l(Pi) = ~l(qi) + 1, this space is of 0 dimensional. Using it we define the number Zoo, 1,x)(L; r g f,, gfz, (3) in a way similar to Sect. 3. Then we have

ZoO, 1,o)(fl, ~; gfa, gf2, ~ ) - ZoO, 1,o)(fo; ~; gfl, g f2, ff~)

= Zo(1,1,0)(L, q; Ogf,, gf:, ~ ) -}- Zo(1,1,0)(L, q; g j-q, ~3gf2~)

+Ze(1,1,o(L, r gA, g f2, id) + Zoo, 1,x)(L, g; gA, gA, (3) .

Thus using Lemmas 4.1 and 4.2 we can prove the following lemma in a way similar to Lemma 3.9.

Lemma 4.3.

4(Zo(jT1, g; r - Zo(fo, g; g)) - 4(2o(371, g; q') - Zo(fo, g; q'))

= E (Zo(1,1,X)( ~-'~L; ~; g~'lfl' g~2f2' ~ ) -}- Zo(1,X, l)( ~L; ~; g'~lfl' ~r gg3f3 ) gl ,,~2, 83 =::[:: 1

+ZSo(1,1,0)(~'L; ~; ~ , ge2f2, ge3f3 )) - E ( Z o o , 1,x)(~-'L; ~t; Oe,f,, gee f2, ~ ) gl, 82, ~3 =::1= 1

+Zoo,x, 1)(5; r g~,y,, ~ , ge3y~ ) + ZSe(1,1,0)(~; r ~ , g~:r g~3y3 )) .

Here we denote ~-Z = {(elfl,s, e2f2,~,e3fs,~,gs)lS E [0, 1]}. Now the rest of the proof is completely parallel to one of Sect. 3. Namely using the Jacobi identity we find that the right-hand side of Lemma 4.3 cancels with a similar contribution from another graph A. Thus we find

.Z2(?l, g, ~) -- Z 2 ( L , g; ~) = Z2(? l , g; ~t) __ 2 2 ( L , ~; ~t)

in the case when the Witten complex does not change for s c [0, 1].

62 K. Fukaya

5. Independence of Morse Functions II

Next we consider the case when the critical point set and the boundary operator change. Let L = {(f~,9~) E ( C ~ ( M ) ) 3 • Met(M)[s E [0, 1]} be a generic one parameter family of metrics and functions.

For simplicity we assume that f,,~ is independent of s for i = 2,3,4. Let C,(M; g; fl,~) be the Witten complex for fl,s.

We first recall the proof of the fact that C,(M; q; fl,0) is chain homotopy equivalent to C,(M; q; f1,1). (In fact they are both acyclic in our case. But we use the explicit chain homotopy equivalence which (in the general ease) is used to show that a chain homotopy type of Witten complex is independent of the choice of Morse function.) Let p E Cr(fl,0), q E Cr(fl,1) such that ~/(p) = q(q). We define a moduli space ~lL(P,q;fks) as follows. We choose and fix a smooth function Z : R ~ [0, 1] such that

Z ( S ) = 0 f o r s < - 1 ,

and

~/z(p ,q ; f l ,~) = {E

We put

Z ( s ) = l for s > 1.

: R - -~M

~t (t) = -gradx(t)fl,x(t)(f(t)), }

tfimoo r(t) = p,

[tfi_moo f(t) = q

Here gradx(t ) is the gradient with respect to the metric gz(t). We omit this suffix from now on, since the metric 9x(t) is always used to take the gradient of fl,x(t).

J/lL(p,q;fl,s) is a 0-dimensional compact oriented manifold. We define q)p,q" gp ~ Cq by

(PP, q = E 4- Pt �9 IE~L(p,q;f l ,s)

We thus obtain ~o �9 C,(M; q; fl,0) ~ C,(M; g; fl,1).

Lemma 5.1. ~tA, ~ o ~ = q~ o c~A, 0.

Proof Let q ( p ) = q ( q ) + 1. We obtain a one dimensional moduli space JgL(p,q; fl,s) in a similar way. Its boundary is described as:

t t X - - / . O~/L(p,q; f l ) = (-2(p,p ;fl ,0) x -2r(p',q;fl))U (~L(P,q ; f l ) d//(q ,q;f l ,1))

The lemma follows. We can prove that q~ is a chain homotopy equivalence as follows. Using an

equation d( ~-[(t) = -grad fkz(-t)(~(t))

in a similar way, for p ECr(fl,o),qECr(fkl), we define a moduli space JtL(q,p;fl,s) and a chain map r : C,(M;q;fl,1) ~ C,(M;g;fl,o). Chain homotopy between r and the identity is obtained as follows. Choose a sufficiently


large positive number S. We consider the equation

63

~t (t ) ( - g r a d fl,z(t+s)(f(t)) t < - S + 1 = ~ - g r a d f l , l ( f ( t ) ) - S - t - 1 < t < S 1 .

( - g r a d f t , x ( s _ t ) ( f ( t ) ) t > S - 1

Using this equation we have a 0-dimensional moduli space Jg (p , q; fl,~; S) for p, q E Cr(fl ,0), with q (p) -- t/(q). We find

( ~1 ~O ) pq ~- E 4- P < . (5.2) EcJg(p,q;f~,s;s)

The proof of (5.2) is a gluing argument which is now standard in topological field theory. (See [Sc].)

For r E [0, 1 ] we consider the equation

( -g radf l , rz ( t+s ) (E( t ) ) t < - S + 1 (t) = ~ - g r a d f l , r ( f ( t ) ) - S + 1 < t < S - 1 .

( - g r a d fl,rz(S-t)(f(t)) t > S - 1

Let r / ( p ) = q ( q ) - 1. Using the above equation we obtain a moduli space �9 ///[para(P,q;fl,s,S) of 0 dimension. (In fact we need to perturb a bit to achieve transversality. But we omit the discussion about it for simplicity.) Using it we obtain Hpq = EECdgp~(p,q;fl,s;S) 4- pg. This is a map of degree +1. We have

L e m m a 5.3. H o •A,0 + dA,o o H = 1 - ~ (0.

We omit the proof. Now let gJi,o be the combinatorial propagator for C.(M;fl ,o; ~). We consider

the following moduli space for p,q c Cr(fl ,0):

Jgo(1,0,0)(f; P, q; S) I : O(1, O, O) --+ M, t2, t3 > 0

= grad f2 on e2 ------- [0, t2]

= ' (I, t2, t3)

dI

d l ~-~ = grad f3 on e 3 ~ [0, t3]

dl f gradfl,z(t-s) i f t E [S - 1, ~ ) C [0, ~ ) ~ el, 1 = ]. grad fl,0 if t E [0,S - 1) C [0, cx~) ==- el,1

dI (gradf l ,z( -s- t ) if t E ( - o 0 , - S + 1] _C (-cx~, 0Z = ~ grad fl,0 if t E [ - S + 1, 0])

_C ( - ~ , O] -~ el,2 lira I ( t ) = p t E [ 0 , cx~)~el,1

t---+§162 lim l ( t ) = q t c [O,c~) -~el,2

t - - - - + - - ( X )

txO ~---e12 )

I f t / ( p ) = t / (q )+ 1 then dgoO,o,o)(f;p,q;S) is a compact 0 dimensional space.

Using it and the moduli space Jgo(1,o,o)(fs;P,q) (which we defined in Sect. 1), we define homomorphisms W(O(1,O,O);S) : Oqp ~ ( ~ q and W(Os(1,O,O)) :

64

@qp -+ @r as follows:

Wp, q(O(1,O,O),S) = ~ ~ Z(I, ~) , I E.,Cr o, o)(f;P, q;S)

Wp, q(OS(1,O,O)) = ~ 4- Z(1, q). IEJr

K. Fukaya

Let gJq.l ---- Y'~gfl, I(P,q)[P] | [q]* : | --~ | be a combinatorial propagator for fl,1. We put

Zo(1,0,0)(?l; ~; S; gfl,,, ~ , ~ ) = Tr(gfl,, o W(O(1, 0, 0 ) , S ) ) .

We recall

Zo(1,0,0)(?l; q; gfl,,, ~ , ~ ) -- Tr(gA., o W(O0(1, 0, 0 ) ) ) .

L e m m a 5.5. For large S, we have

mp, q (O(1 , O, O ) , S ) = E gOp, p, o mp, ,q , (Oo( l , O, 0 ) ) o ~lq,,q. p~,q~

The lemma follows from the following homeomorphism, which is proved by a

U , /g- i~(p,p ' ; f l ,D pt,q' C Cr(J], o)

# . x JgoO,o,o)(f; p ' , q ' ) x #s q, f l , , ) �9

usual gluing argument:

J /oo ,0 ,0) ( f ; P, q; S)

By Lemma 5.5 we have

Zo(1,0,0)(?; ~; S; g f1,1, ~ , ~ ) = Tr(gf,,, o W(O(1, O, 0), S))

= Tr(gf,,~ o q) o W(O(1, O, 0)) o ~ )

= Tr(~9 o gf,,, o ~o o W(O(1, 0, 0 ) ) ) . (5.6)

We next remark that

L e m m a 5.7. ~9 o gA,l o ~o + H is a combinatorial propagator o f fl,0.

Proof By Lemmas 5.1 and 5.3 we have

0S,,0 o ( 0 o gS~,, o 40 + H ) + ( 0 o gs,,, o ~o + H ) o ~S,,0

= I/J ~ (6~J],, ~ gfl,, + gi,,, ~ 0 A , , ) ~ 4~ ~ 1 7 6 1

as required. Since our invariant is independent of the combinatorial propagator, we may

choose gA,o = ~ o gA,~ o qo +H. We thus have

Zo(1,o,o)(fl; ~; S; g fl,,, ~ , ~ ) = Zo(1,o,o)(fo; ~; g fl,o, ~ , ~r

-Tr (H o W(O0(1, 0, 0))). (5.8.1)


para ~. . Next, in a way similar to define J / /~( l ,o ,0)(f ,P,q ,S) , we define ,.~/'O para ( f - (a, Lo)~J, Pl, ql; P2, q:; S) as follows:

dr 1,0)(f; Pl , ql; P2, q2; S)

= ( h t 3 )

I : O ( 1 , 1 , O)--+M, t2, t3 > 0, r E [0,1]

t = gradf2 on e l l , e2,2

t~+mo I ( t ) = p2 onez,1

t ~ m I( t ) = q 2 on e2,2

d[ ~ - = gradf30n e3 ~ [0, t3]

dI f grad f l z(t-s) if t E [S - 1, co) C [0, oc) ~ e1,1 ~ -- [ gradfli0 i f t E [ O , S - 1 ) C [ O , oc)~e l ,1 dl ( gradfLz(_s_t ) if t E ( - c x ~ , - S + 1] C_ (-(x~,0] -~ eL2 d t = / grad fl,0 if t E [ - S + 1, 0]

C_ (--c~,0] ~ el,2 t]imooI(t) = Pl t E [0, e~) ~ el,1

lim l ( t ) = q l t E ( - ~ , O ] ~ e l , 2 t--+-- OO

Taking the combinatorial propagator gA and using the moduli space

Jd/oO, l,o)(ft; pt,ql; p2,q2;S), we define W(1,gf2,0 ) such that

(W(1,gA,O)(u),v) = E E E Pl,ql,P2,q2 ICJr i, o)(fl ;pl,ql ;p2,q2;S) i

(g f2(P2, q2) o Z(I, q) o (Upz | ei), Vq~ | el) .

Here ei is an orthonormal basis of qm" Similarly we obtain W(O(1,gf~,gf3)) and W(O(1,O, gf3)). We can define

Zo(a,l,O)(j~; G S; gA.~, g f2, ;~) etc. in a similar way. Namely we put

Zoo,1,0)(Ym; q; S; gA,~, gf~, ~ ) = Tr(gA,l o W(O(1, g f2,0))) ,

and so on. In a way similar to the proof o f (5.8.1) we have:

Zo(l,l>0)(fl; G S; gA,,, gf2, ~ ) = Zo(IA>o)(fo; q; gA,o, gf2, ;3)

- -Tr (H o W(O(1,gf2,0))), (5 .8 .2 )

Zoo,0,1)(fl; G S; gA.l, ~ , gf~ ) = Zo(1,1,0)(?0; ~; gfl,0, ~ , g f3 )

- -Tr (H o W(O(1,O, gf3))), (5.8.3)

Zo(I, 1, O(f l ; q; S; gA.,, gf2, gs3 ) = ZooJ,o)(fo; q; gA,o, Of 2, g f3 )

- T r ( g o W(O(1,gf2,gf~))) , (5.8.4)

66 K. Fukaya

etc. We thus found a relation between Zo(. . . ) ( f l ;S; ' . . ) and Zo(...)(jT0;...). Summing up we got

- -~ E(Zo(1 ,o ,o ) (~ f l, ~; S; ge,fl,,, ~ , ~ ) -- /O(1,O,O)(~*fo ; ~, ge,fl,o, ~J, ~ ) )

1 + -~(Zo(1,1,o)(gfl; q; S; g~lJ~,l, ge:f2, ~ ) -- Zo(1,1,0)(~*?0; q; ge,fl,0, ge2f2, ~ ) )

1 +~ ~(Zo(1,O, 1)(gfl ; q; S; gelfl,1, ~ , ge3f3 ) - go(l,0,1)(~'fl ; ~; ff~lJ],o, ~ , ge3f3 ))

-- 4 S ( Z o ( I , 1,1)(~*fl; ~; S; ge~j],l, ge2f2, ae3f3 ) - Zo(l, 1,1)(E*/I ; q; ~e,fl,o, ae2f2, ae3f3 ))

1 +~ ~(ZA(1,o, l)(k'fl ; q; S; geaA,,, ~ , g ' 3 f ' ) - - /A(1, O, 1 )(~fl ; ~; ge,A,O' ;ZJ, g'3f3 ))

1 + ~ E(ZA(1,1,0)(g'fl ; g; S; gelfi,,, ge2f2, ~J) - ZA(1,1,O)(g?I; ~; g.,fi,o, g~2S2, (,~J))

1 Z - - - 8 E ( A(1,1,1)(gfl;~;S;ge,fi,l'g~2f2,ge4f4) -- ZA(1,1, m)(Ffl;~;gelA,o'ge2f2,ge,4f4))

1 ~ E(ZA(1,1, D(~'fl ; g; S; ge,fl,1, geaf3, g~af4 ) - ZA(1,1, D(~'fl; q; ge,fl,o, 0~3f3, ge4f4 ))

---- + ~ - ~ T r ( H o W~-f(O(i, 0, 0 ) ) ) - ~ T r ( H o W~.f(O(1,g~2f:,O)))

E T r ( H o W~.f(O(1, 0, es ) ) ) -~ ~ E T r ( H 0 ~ - f ( O ( 1, et;2f2' e"3f3 )))

1 - ~ T r ( H o W.7(A(1 , O, ge3f3)) ) - - g ~ T r ( H o W.~(A(1,g,2f2,O)) )

1 + ~ T r ( n o Wcf(A(1, g~4f4,ge3f3))) + ~ T r ( m o W~-f(A(1,O,2f2,g,f4))).(5.9)

We next are going to find a relation between Zo(1,0,0)(fl; q;#ft,l,~,~) etc. and Zo(l ,o,o)(f l;g;gfa,t ,~,~) etc. For this purpose we use the following moduli space.

0 (u,t)

S-1 S S+1 u-S-1 u-S u-S+l

Fig. 8.

I ] , t U


para ~ , . . para ~ . Hereafter we write Zg(1,0,0)(efl, S, zd, ~ , ~5) = Zo(1,0,0)(e~fl, q,S; id, (,?J, (2J) -

para ~ . t . . Z~(t,o.0)(e~cl, q , S; td, ~ , ~ ) , etc.

,/~'0 para ( 3" 0 ,0 ,0 )~ ' P'q; S)

(/, t2, t'3, r )

I : O(1, 0, 0) ---* M, tz, t3 > 0 , r E [ 0 , 1 ] dI

= grad f2 on e2 ~ [0, t2]

dI d-f = grad f3 on e3 ~ [0, t3]

{ gradfl,z(t-S)+r(1-z(t-s)) if t E I S - 1,oo) dI __. [0, o(3) ---- el,1 -~ = gradfl,r if t E [0, S - 1)

C [0, oo) -- el,1 "

{ gradfl,z(t+s)+r(t-z(t+s)) if t E ( - c ~ , - S + 1] dI C ( - 0 0 , 0 ] ~ e1,2 d-t = gradfL~ if t E [ - S + 1, 0]

( - 0 0 , 0] -~ el,2 lim I ( t ) = p t E [0,00) ~ el, l

t - -++oo lim l ( t ) = q t E (--oo, 0] TM e l , 2

t---+ - -o0

I f r / ( p ) = q ( q ) + 1 this moduli space is one dimensional. Its boundary is given as

para ~. ~ , , , , OJ/4'~)(1,O,O)(f, p, q; S) Jr'O(1 0 o)(f ; P, q; S) tJ -,~/o(1 o o)(/o; P, q)

It implies

- - j , /para [ 3 . ["J U , /~(fl , l ; p, p ' ) x ~ooo,o,o)~J, p ' ,q ;S )

pt para ~. - - t

ul.JJgo(1,o,o)(f , p ,q ' ;S ) x J [ ( f l , 1 ; q ,q) q~

para ~ . . U d/loo,l,0)(f, p, q, p2, p2; u S)

P2 ECr(~)

para ~. . U [3 J/ /oo,o,1)(f ,P,q, P3, P3;S) .

p2ECr( f~ )

- Z o ( 1 , 0 , 0 ) ( f ; ~; S ; ~ f l , t , ~ , ~ ) ~ - Z o ( 1 , 0 , 0 ) ( J 0 ; q ; gfl,1, ~ , ~ )

~ p a r a r "7 para = LoO,0,o)ty; q; S; id, ~ , f3) + Zo(1,1,o)(f; q; S; gf~,~, id, ~ )

para ~, +Z~o,l,o)(f , g; S; gf~.l, ;25, id) .

Here we put for example

pala ~ . Z~(1j,o)(f, S; gl,1, id, QS) --- U 4- Tr((gl,1 | id) o Z(I, q ) ) .

para 1E J~ oO,I,o) ( f ;P,q: P2,p2;S)

68 K. Fukaya

Now we perform a similar cancellation argument to Sect. 3 and obtain the following formula:

1 -~ ~(Zoo,o,o)(gfl;S; g ~ l f l . a , ~ , ~ ) - - Zoo,o,o)(~fl;g~lA,1, ;25, (~3))

1 E ( Zo(1,1,o)( ~ f l ; S; gel f,.i, g~2f 2, ~ ) - Zo(1,1,0)(g'fl ; QelJ].~, ge2f2, J~5))

1 -- -~ ~ ( Zo(1,O, 1)(gfl; S; ga13q,1, ~ , ge3f 3 ) -- Zo(l,O, 1)(~'fl ; gelfl,,, ~ , gg3f3 ))

§ 1,1 )(g*fl, S; ge,f,.a, ge2f2, ge3f3 ) - Zo(l, 1, l)gfl ; Oglfl,,' gE2f2, g83f3 ))

+~E(Za(1,o,l)(efl;S;g,,f, ,1,1~J,g,,f,) - ZAo,o, 1)(gf~;g~.,A,,,f3, g , , f , ) )

1 +~ ~(ZA(1,1,O)(~f l; S; Oe,fl,1' gg2f2' ~ ) -- lA(l,l,O)(~?l; ge,fl,l, gezf2, ~ ) )

1 Z - - B E ( A(1,1,1)(gjl;S;gelf,,, O'g2f2,~]e3f3) -- gA(1,1,1)(gfl;ge, ft,t,ge2f2,ge3f3))

-- ~Ezga~l,O,O)(gy,;S;id, f~ j ,~ ) + ~ E Z ~ l , 0 ) ( g / , ; S ; i d , g,af2,~J)

§ Eg~rla O, 1)(8"Yl ; S'~ id, J~J,g~,f,) - ~ ~ZPT~,,, ,)(gf, ; S; id, g,2fa,g~,f3)

§ ZPA{rlal,o)(g~fl;S;id, gezfz" ~ ) § ! E ~para . . 7 8 ~ ZAO'O' 1)(~J1 ; S; id, ~ , gg3f3 ) �9

1- - para + ! E --para .+-~ ~ (ZSao, 1,1) (g f l; S; id, Or:A, g*4I, ) - 8 e z ~ ~ ~' 1)(gYl; S; id, g, f4, g,3 f3 )" (5.10)

We next study the right-hand side of (5.10). To study the first term, we define the moduli space X l o ( f l ; S , r ) . We choose a function 0 : {(u,t)[t E [0,u],u > 0} --+ [0, 1] such that if u > 2S + 10 we have (Fig. 8)

t - s ,

Z(u - S - t) 0

if t E [ 0 , S - 1 ] if t c [ S - 1 , S + l ] i f t c [ S + l , u - 2 S - 2 ] if t ~ [ u - S - l , u - S + l ] if t c [ u - S + l , u ]

(We define 0 in the case u < 2S + 10 also. However the above condition is not assumed in the case u < 2S + 10.) We then put:

I : O--+ M, tl,t2, t3 > 0 , dI d-7 = grad f2 on e2 ~ [0, tz]

~ / o ( f ; S , r ) = (Ltl,t2, t3,r) dI ~- = grad f3 on e3 --- [0, t3]

dl -~ = gradfl,l-~+~(t.t) if t E e~ ~ [O, tl]

Morse Hornotopy and Chem-Simons Perturbation Theory 69

This moduli space has boundaries and ends. The boundary corresponds to r = 1, 0. I f r = 0, this space is equal to ddO(fl ).

On the other hand, there are ends o f the moduli space Ure[0,1] J~o(fi ; S, r) which comes from bubbling o f the first edge, namely the case when tl --+ o~. They correspond one to one to the moduli space we used to define ZP~(r~,o,o)(fi;S;id, f25, f25 ). (We remark that the first edge splits between S + 1 and tl - S - 1.)

There are other ends corresponding to t2 ~ oo, t3 ---* o~ or ti ---* O. But they will cancel to each other if we consider the other terms. (see Sect. 3 ,4 and (5.11).)

Using the moduli space o f the case r = 1, namely Jr 1), we define:

zo(f~;s,1;~,25,~) = ~ + z ( I , O �9 lc~r ;s,1)

We define Zo(o,l,o)(j71;S, l ; ~ , g f 2 , ~ ) etc. in a similar way. Then we obtain

zPoa(r~,o,o)(e'fl; S; td, ~5, ~ ) - 4 ~. Z~176 S, ld, Oe2f2, ff~5)

~ZPT~,O,l)("fl;g;id,~,,~3f3)§ j~zPa~,l.l)(~fl;S, id,'eJ2,'e3f3)

§ g.zf2,e) § ~ZPA~:O,l)(~Zl,S;id, e,O,3f3)

_j~Z:~,l.l,(~Zl;S;id, oe2f2,O,f4)_ I~ para .'.. �9 8 ~. z ]O ' 1, l)(efl , S, id, g,,f4, g.3f3 )

g

1 (~Zo(o,l,O)(g.fl;S, 1 ; ~ . g . j 2 , ~ ) - ~Zo(o,l,O)(~il;~,ge2f2,~))

__14 (~ ZO(O'O'l)(~fl;S'- I ; J~' ~' ffe3f3 ) -- ~ Z~176176 1)(8"iI; J~' ~' '~3/3 ) ) g ,

) +~ Zo(o,i,1)(g'f,;S, 1;~,g.,f2,g.3f3)- ~Zo(o,i,i)(~fl;~,g,2f2,g.3f3) g 1 Z +~Za(o,l,O)(~'fl;S, 1;~,ge2f2,~)+ -~"~.~ ~ A(O'~

__Lg (~ZA(o,l,l,(~fl;S,l;~,g,zf2,..4f4) _ ~ ZA(0, l,,)(.il ; j~, ffe.2/.~, ..4j~) ) ~ .

8

(5.11)

70 K. Fukaya

We remark terms ~gZa(o,o, 1)(g f l; S, 1; ~ , ~Z~, 9~3f3 ) etc. may be nonzero while ~gZA(o,o, 1)(gJ71; ~ , ~ , 9~3f3 ) must be zero. (The equation we put on the first edge in the definition of the moduli space we use to define ~gZa(o,O, l)(gfl; S, 1; ~Z~, ~Z~, ~c3f3 ) is a time dependent gradient flow equation. So it may have a nontrivial closed orbit.)

Our next step is to compare the left-hand side of (5.11) to terms related to Tr(H o W(O(1, 0, 0))), etc. In the following we write Wg(O(1, 0, 0)), etc. in place of W(O(1, 0, 0)), etc. in case we use (elfl#,e2fz, e3f3) to define them.

Lemma 5.12.

Zo(gfl; S, 1; ~J, (2~, (2~) - ~ ~-]~Zo(o,l,O)(gfl; S, 1; ~,9~2f~, ~ ) - g

1 ~ 1 - - Y ~ Zo(o 0 1)(gfl; S, 1; JZ~, ~,9~3s + ~Zo(o,l,1)(gfl;S, 1; ~J,O~2f2,9~3f,)

4 g ' ' _ g

---g~ZA(O,I,I)(~fI;S, 1; 25, 0~2S~, 0~S~)

4

l(~Zo(o,l,O)(gfo;f2~,9e=f~,f2~)+Y~Tr(H~ 4 - g

l(~gZo(o,o,1)(gfo;f2~,f25, ge3f3)+~_,Tr(H~ O, 9~3f3))))

+ - Zo(o, ~)(~'fo; N, ) + 2 rr(H o N(O(1, ))) g

+ ~ : r r ( H o N(A(1,0~2s2,0)))+ 1-2"rr(H8g o W~(A(1, 0, 0<6 )))

1 1 Tr(H o W~(A(1, 0~2f2, O,,~f,, ))) - 5 ~ Tr(H o Wg(A(1, 9~4y,~, 9~3f3 ))).

8 7

Before proving Lemma 5.12 we remark that (5.9), (5.10), (5.11) and Lemma 5.12 imply that f0 and f l give the same invariant.

To prove Lemma 5.12 we use the following moduli space which is similar to but different from Jgo(fl;S, r). We choose a function ~" : {(u, t)]t E [0, u], u > 0 } ~ [ 0 , 1 ] such that i f u > S + T + 1 0 w e have

0 ~ ( t - r )

E(u, t) = 1 z ( T + S - I - t ) 0

if t E [ 0 , S - 1 ] if t E [ T - 1, T + 1] if t E [ S + I , 2 S - 1 ] if t E [ T + S - 1 , T + S + I ] if t E [ T + S + 1,u]


~o(f; s, r , r ) = (/, tl, t2, t3, r )

I : 6) ~ M, h,t2,t3 > O, )

} dI ~-~ = gradf2 on e2 ~ [0, t2]

dI = grad f3 on e3 ~ [0, t3] dt d I --~ = gradfl,~z(t~,t) i f t E el ~ [0,fi]

The space (-J~c[o,l] JC 'e( f ;S , T,r) has boundaries and ends. The boundary is

- , / /go( f ;S ,T ,O)UU~io( f ;S ,T , 1). The set J~/o(Y;S,T,O)is equal to J//o(fo). We next prove that the ends of (.J,.~to,~]J/[o(f;S,T,r) corresponding to the case when

h ~ ec is equal to Up, qccr(j],0)~/[o(1,o,o)(p, q , / 0 ) X dgpara(q,p; f l , s ,S) if S and T are sufficiently large. Here we recall that J/par~(q, P; fl,s, S) is the moduli space we used to define H .

To show the behaviour of the end of Ure[0,1] J/ /o( f ; S, T, r) described above, we first remark that the ends in question correspond to the following set. (We remark that the first edge splits between S + T + 1 and tl .)

(I, t2, t3,r, p)

~_~_ : O(1 ,0 ,0 ) --~ M, t2,t3 > O,p C Cr(fl ,0)

= gradf2 dI on e2 ~ [0, t2]

~_ grad f3 dI = ~-~ on e3 ~ [0, t3]

gradfl,o if t c [S + T + 1, ee] C e1,1

dI gradfl,rz(-t+s+r) if t E [S + T - 1,S + T + 1] _C el,1

~-~-- gradfl ,r i f t c [ T + l , S + T - 1 ] C e l , 1

gradfl,rz<t-s) if t C [T - 1, T + l] C_ e1,1

gradfl,o if t E [0, T - 1] C_ el,1

-~/t--gradfl,o on el,2 (-cx~,0]

t im I ( t ) = p , i f t E e~,l ~= [0,oo)

trim I ( t ) = p, i f t E el,2 ~ ( - c ~ , 0 ]

E (u,t)

/ s; S + T - 1 S + T + I

Fig. 9.

u

72 K. Fukaya

We consider the behavior of this moduli space when T ~ c~. We then find, by a gluing argument, that this space is homeomorphic to J/o(1,o,o)(P,q;fo)x �9 //gpara(q, p; fl,s, S). Therefore by counting elements of the end of

Urc[0,1] d/lo(f; S, T,r) corresponding to the case when tl ~ oc with weight, we obtain Tr(H o We(O(1,0, 0))).

Thus we obtain the following:

Zo(~fl;S;T; 1 ; ~ , ~ , ~ ) = T r ( H o W~(O(1, 0, 0))) + Z o ( g J T 0 ; ~ , ~ , ~ ) . (5.13)

We have a similar formula for graph O(1, 1, 0), etc. Therefore, to complete the proof of Lemma 5.12, it suffices to prove the following Lemma 5.13. We put

Zo(f1;S,T, 1; (25 ,~ ,~) = ~ -4-X(I,r IE~#o(fl;S,T,1)

Lemma 5.1,:1. I f S and T are sufficiently large, then we have

1 1 ~Zo(~fa;S,T, 1; ,Q~,,Q~,,Q~) - ~ ~Zo(o,I,o)(~fl;S,T, 1; (~,g~2fi,,Q~)

1 ~ T 1 - ~ Z o r , 1; ~ , ~ , g~i,) + 4 ~Zo~o11~(~fS;s, T, 1; ~ , g,:i~, o ~ ) _ g g.

1 1 -}- ~ ~ ZA(0,1,0)(~J~I ; S, T, 1; ~ , g~zf2, ~ ) + ~ ~ ZA(o,o,i)(gf; S, T, 1; ~ , ~ , gesf3 )

1 _1 E ZAr ; S, ~r, 1; ~ , a~,I4, a,~i~) - x -ZAr176 (~'~ ~ ~ - 8

1 o ~1 ~Zo(o,l,O)(gfl;S ' 1; (2~,ge~fz, (~) = ~ Z o ( U f ; S , 1 ; ~ , ~ , ~ ) -

1 -l~mo(o,o,1)(g'fl;S, 1; ~,~,gesf3) + -~Zo(o,l,l)(e,'fl,S, 1; (d~,g~2f~,gesf3) 1 1

+g ~']~ZAfo,I,0)(g'fl; S, 1; ~,g~f: , (~) + -~ ~ mA(o,o,1)(~fl,S, 1; ~ , ~ , g~f3 )

1 -~- EZAr 1; ~ , g,:~, g~4~4) -- ~ EZAr 1; ~,g~4~, ,a~) . 8 ~

Proof. We write ~s,T, Os to specify their dependence of S, T. We may assume that they depend smoothly on S. We define a three parameter family of functions Rs, r~ for S > 0, T > S, A ~ [0, 1] such that

Rs, v,o = ~s,r , (5.15.1 )

Rs, r,1 = Os �9 (5.15.2)

We put B1 = A T + ( 1 - A ) S , B z = A ( S + T ) + ( 1 - A ) ( u - S ) . Then for u > T + S + 10 we have

0 if t E [ 0 , B I - 1 ] Z ( t - B 1 ) if t E [B1 - 1,B1 + 1]

Rs, v,A(u,t) = 1 if t c [B1 + 1,B2 - 1] . z(B2 - t) if t c [B2 - 1,B2 + 1] 0 if t E [ B 2 + l , u ]

(5.15.3)


We now put l( > o, = grad f2 on e2 ~ [0, t2]

d//O(f;S,T,A) = I,h,t2, t3) dI = grad f3 on e3 ~ [0, t3]

dI -d-[ = gradf'ss'r'A(tl't) i f t E el ~ [0, h]

This is again a 0 dimensional space for each fixed S, T and A. Moving A E [0, 1] we obtain a one dimensional manifold. By (5.14.1) and (5.14.2) its bound- arT is the union J/ /o( f ;S , 1 ) U - J g o ( f ; S , T, 1). These moduli spaces are the one we used to define right-and left-hand sides of Lemma 5.13. So to complete the proof of Lemma 5.13 we only need to show that the contribution from the ends o f

U,~[o,1] J g o ( f ; S, T,A) cancels.

To study the end of UA~[0,1] ~ o ( f ; S, T,A) we use the following:

Lemma 5.16. There exists So, To, C such that i f S > So then there is no element OfUAE[O,1]~o(y;S,T,A ) such that h > C, and C > So + To + 10.

Proof I f Lemma 5.16 is false we have a sequence of elements (Ii, h,i, t2,i, t3,i) of

Jgo( f ;S i , Ti,Ai) such that Si --+ c~z, Ti ~ oo, h,i ~ ~ and h,i > Si + Ti + 10. By dimension counting it suffices to consider the case when Ai converges and t2,i, t3,i converges to some positive number. Then by (5.13.3) Ii splits into 2 gradient lines of time dependent gradient vector flow and an element o f ~'od,0,0)(J~, P, q) for

some p,q. Therefore we have an element of Jged,0,0)(J~, P,q)xJ//---L(q,r;f,s)x ~ - L ( r , p; f , s ) . (Remark that the parameter A does not play a role here. Hence we loose one freedom this way. This is the basic reason why the lemma holds.) This space is empty by simple dimension counting. We get a contradiction.

Lemma 5.16 implies that if we choose S sufficiently large, then there is no end

of UAE[0,1] ~{O(j; S~ T,A) corresponding to ti ~ c~. Now we are ready to repeat the argument we did many times. Namely the end

coming from t2 ~ ~ , t3 ~ ~ or t2 ~ 0, t3 ~ 0 cancels after summing them up with other similar terms. The proof of Lemma 5.14 is now complete.

We thus completed the proof of Main Theorems I, lI, assuming some lemmas on transversality and orientation.

6. Compactification of Configuration Space and Transversality at Diagonal

In this section we discuss the moduli space 5ao(1,o,o)(f,g; p l ,q l ) we introduced in Sect. 3 and similar moduli spaces which appeared when a non-simply connected subgraph degenerates to a point. To clarify the idea, we consider a more general situation. Probably one can use these ideas to prove the well-definedness of higher loop amplitude.

Let F be a graph. We assume that each vertex of F has more than three edges or has only one edge. The vertex with one edge is called an exterior vertex and otherwise it is called an interior vertex. An edge is called an exterior edge i f it

74 K. Fukaya

contains an exterior vertex. Otherwise it is called an interior edge. We also fix an orientation to each edge.

Let 3-r be the set of all possible ways to assign a positive number te to each interior edge e.

We associate a Morse function fe to each edge of F. And also we choose a point Pv to each exterior vertex of F. We assume that pv is a critical point of the function fe, where e is the unique edge containing v.

We consider the following moduli space:

~ ( r ; 2 7 , / 3 ) = ( ( te ) , I )

(te) E J'r ) I : F - ~ M } ~ J ~ = gradfe on an interior edge e [0, te]

dlie = gradfe on an exterior edge e ~ ( - o % 0 ]

lim lle(t ) = pv v E e is an exterior vertex !---+-- O0

There is a natural projection ~ ' ( F ; 27, /3) --* J r . For example if F = O then Jr is the moduli space J g o ( f l , f 2 , f 3 ) we studied before.

We put J / ( r ; V,/3) = Uyev J/g(F; 27,/3). Here V is a sufficiently small neighbor-

hood of 27 in ( C ~ ( M ) ) k and/~ = (Pv)v:exterior vertex. (Here k is the number of interior edges.) We choose V so small that the critical point set o f f [ where ( f [ , f J , f~) E V is identified to the critical point set o f J~. Then the expression ~ ' ( F ; 27,/3) makes sense. We have again a map ~ ' ( F ; / 3 ) --+ ~-r.

Lemma 6.1. JC/(F; V,/3) is a C~-man i foM and rcr : J/f (F; /3) ~ ~ r is a submersion.

Lemma 6.1 is verified by a straightforward transversality argument similar to one in Sect. 2, where we proved the corresponding transversality result already for the graph we use.

The transversality at the point where one of the numbers is 0 is of different problem. In fact transversality does not hold in the most naive sense. To see this, let us consider

{ e : interior vertex } ~ - r = (te) te >= 0 ~ RC-->0"

This is a (partial) compactification of ~ r . We define (gdi'(F; f , / 3 ) in a way similar to JC/(F; 2 7,/3) but requiring I ( e )=po in t if te = 0. There is a natural map rc : qf/J/(F; 27,/3)--* Cf~r. (g J r is a manifold with comers. In other words, (g~r is a stratified set such that each stratum is a manifold. (We put (gk~r = {(te)C (~ r l#{e l t e = 0} = k}.) One may ask if each cycle L C_ (g~r is transversal to rc : ( f Jg(F; 27,/3) --* (f~--r. However this is not the case. This is a problem related to the anomaly we discussed�9 Let us give an example in the case when the dimension of the manifold is 2 (Probably a similar problem occurs in higher dimension�9


We put f l (x, y ) = xe -y ,

f 2 ( x , y ) = x ,

f3(x, y ) = xe y.

One can easily see that the moduli space c g J g ( O ; f ) is a line R and the map dr f ) ~ Yo ~ R3__>o hits the origin. (See Fig. 10.)

This picture is quite stable. Namely by perturbing f l , f2, f3 we still find a similar situation that is the map J g ( O ; f ) ~ <g~--o ~ R3_>_o still hits the origin. Thus

L = {(0,0,0)} is not transversal to the map n : cgJg(O; f ) ~ c~--o. As we discussed already, to handle a similar problem, we can consider the

subsets of the points where gradient lines are parallel, to analyze this problem. To do it more systematically we are going to use compactification of the configuration space similar to one in [FM, Ko]. However, there is a small difference, caused by the fact that we break the symmetry (isomorphism of the graph) by introducing different Morse functions to different edges.

Roughly speaking, we take the real blow up of CgYo ~ R3=>o along each

<gk~--r, k < m, and obtain ~--r. More precisely we proceed as follows. We choose and fix a maximal tree T C F. For each edge e C_ F - T we choose one of its vertices and attach e to T at this vertex. We thus obtain a tree T and a surjection J : T ---+ F. Let Co,int(/') be the set of all vertices of T which is mapped to an interior vertex of F and let Cl,int(/') be the set of all interior edges of F. We regard an element of Cl,int(/') as an edge of T as well. We say that A C Cl,int(/') is connected if there exists a connected subset of T such that A is the set of all edges contained in it. Our space ~'-r is a stratified set whose stratum corresponds one to one to the following set X ( F ) . A n element ~ of X ( F ) is a set of subsets of Cl,int(-F) such that

I f A, B c ~ A n B + f2~ then A C B or B C_ A. (6.2.1)

I f A E ~e then A is connected (6.2.2)

).

).

) .

grad fl grad f2

Fig. 10.

grad f3

76 K. Fukaya

Let A E ~e. We consider the set o f all e E Cl, int( / ' ) such that e E A and that if e E B _C A,B E ~ then B = A. Let V(A) denote this set.

For each ~e E X(F) we define J ( ~ 1 e) as follows:

j _ ( ~ ) = Map(Cl,int(F) ~ R+) t',,.~ Z

We define ~ as follows. For t E Map(C l , i n t ( / ' ) -+ R+), r E R+, and A E ~( we consider

t'(e) = { t(e) if e ~ V(A) . rt(e) if e E V(A)

Then we put t ~ t ~. We let ~z be the equivalence relation generated by this relation. ~-r is by definition the union of all J - (~f) . I f ~e = ~ then ~ - ( ~ ) = ~--r. Hence ~-r D J r . The map n : J-r -- ' cgJ r is de-

fined as follows:

rfft)(e) = { ~(e) otherwiseif there is no A E ~ such that e E V(A)

The topology of ~-r is defined as follows. Let ti E Jr . We suppose that ti(e) converges to an element o f R=>0 for each e. We say that d -4 e ifti(e')/ti(e) is bounded. We say e ~ d if e -< d and d -< e. We assume furthermore that l imi-~ti(d)/ti(e) converges to a positive number for each e N d .

We define ~ o = {{e E cl,~nt(r)Ie -~ e0}te0 E C~,int(F)}. We define ~ so that an element of it is a connected component of an element of ~(0. For each A E ~3Y, we choose e0 E Cl,int(F) such that

A = A~ 0 = the connected component o f {e E Co(F)[e -4 eo}containing eo E ~Sf.

Let e E Cl,int(F). Take eo such that e E Aeo and e ~ eo. (In other words, e E V(Aeo).) We put t ~ ( e ) = limi~ti(e)/ti(eo). And we let limi--,ooti be the "~z equivalence class o f too. We remark that this element (up to equivalence) is independent of the choices o f eo.

We thus defined the limit o f the sequence o f elements o f J r . The limit of the elements to J-r is defined in a similar way. Let us describe this set J-c explicitly in the case we need.

First we consider the graph 0 (0 ,0 , 1). Then the tree T is as in Fig. 12. X ( O ( 1 , 0 , 0 ) ) consists o f 6 elements ~ ,{{e l}} ,{{e2}} ,{ebe2} ,{{e l} ,

{el,e2}},{{e2},{el,e2}}. The set j ( ~ e ) for each of this four is R~_,R+,R+,

o (o,o,1)

Fig. 11.


/ A

T

Fig. 12.

Fig. 13.

77

{(t(el)'t(e2))lt(el)'t(e2)>O} ~ (0, 1), one point and one point, respectively. Gluing them R+ together we obtain the 2 manifold with comers in Fig. 13.

Then 3-o(0,0,1) --+ J-o(0,0,1) is an isomorphism outside (0,0), and 7r-1(0, 0) is identified with an arc.

Next we consider F = O. Our tree T is given by Fig. 14. The space ~-o is given by Fig. 15. In other words, 7r-l(0,0, t) is an arc and

1r-1(0, 0, 0) is given by Fig. 15. We go back to the general case and will construct a space M(F) for each

manifold M. This space M(F) is a compactification of the following configuration space:

{p:Co,int(T)---+ M[p(v)~p(vt), if there exists an edge of T joining v and v~}.

M(F) is a stratified set. The strata is indexed again by the same set X(F). Here we regard X(F) as the set of all ~ consisting of subsets of C0,int(T) satisfying the following conditions:

IfA, B C "r A AB4=(2~ then A C_ B or B C A. (6.3.1)

If A E ~U then A is connected. Here we say that A is connected if there exists a connected subcomplex of T such that A is the set of all interior vertices of T.

(6.3.2)

{v} C "OK for each v E C0,int(T). (6.3.3)

Let us verify that the set of all such ~tU is equal to the set of all Z satisfying (6.2). Let ~tK satisfy (6.3). Let A c ~ with #A > 2. Let BA be the set of all interior edges e such that both of the vertices of e is contained in A. We put ~(~/U)

78

Vl 81

V o e 2

8 3

v 3

Fig. 14.

. . . . . 7C::,

Fig. 15.

V 2

K. Fukaya

= {B.~[A E W,#A > 2}. It is easy to see that ~/U ~ Z(~/r) is the required one to one co, rrespondence.

Now for ~/U c X ( F ) we associate a space M(~U) as follows. We say that an element A c ~ is maximal if it is maximal with respect to the inclusion. An element of M(~f ~) consists of p : {all maximal elements of ~4/"} ~ M and u(A) for each element A E "/U. We describe u(A) later. The condition for p is as follows. Let T ( ~ ) be the tree obtained from T by shrinking each maximal element to a point. (Here we regard each element A E =/U as a connected subcomplex.) Then p associates a point of M to each vertex of T ( ~ ) . We assume that p ( v ) + p ( v ~) for each of the vertices of T(g/ ') which are joined by an edge.

We next describe u(A). First we consider the case when A is maximal. We consider all elements A(~/U) of ~q~ contained in A. (A ~ A(~/U).) Let us take all elements of A(~/U) which are maximal among elements of A ( ~ ) . Let A m ( ~ ) be this set. Then

Map(Am(~f') ~ Tp(a)M) u(A)

Here we say u l ( A ) ~ u2(A) if there exists r E R+ and u c Tp(A)M such that ui(A)(B) = ru:(A)(B) + u for each B C Am(~gr).

Next let B E Am(~r We define B(~/r) and B m ( ~ ) in the same way. Then

u(B) Map(Bm(~U) ---+ Tu(~)(B)(Tp(A)(M)))

Here N is defined in a similar way. We remark that if ul(A) ~ u2(A) then there is a canonical isomorphism T,,I(A)(B)(Tp(A)(M)) ) ~ T,~(A)(B)(Tp(~)(M))).

We continue in the same way and define u(C) for each C E ~ . The space M(~/U) is the correction of all such (p,u). We remark that if ~ = {{v}iv E


Fig. 16.

Co,int(F)} then

M(fr = { p : Co,int(T) ~ Mlp(v)4=p(v'),

if there exists an edge o f T joining v and v~}.

We have an obvious topology on M(fr r ) so that M ( f r is a smooth manifold. We put M(F) = U r We define a topology on it as follows. For

simplicity we define a limit o f the sequence Pi E M({{v}lv ~ c0, i . t ( r )}) only. We assume that pi(v) converges for each v E C0,int(/'). Let poo(v) be its limit.

We first define fCF c X(F). For each v E Co,int(F) we consider the set o f all v p E C0,int(F) such that po~(v) = po~(v'). Let Av be the connected component of this set containing v. A maximal element o f for is Av for some v C C0,int(F).

We identify Av with a connected subcomplex of F. Let Cl,int(Av) be the set o f all interior edges o f this subcomplex. Let e,e ~ E Cl,int(Av). Let vl(e) and v2(e) be the vertices o f this edge. We say e -< e ' if d(pi(vl(e)), pi(v2(e))/d(pi(vl(e')), pi(v2(e')) is bounded as i ~ oo. Here d is a distance function for a Riemannian metric o f M. (We fix it but the construction is independent o f it.) We let Av,e be the connected component o f {e ~ E Cl,~.t(Av)le' -< e} containing e. Av,e may be regarded as a connected subcomplex of A~. We define A~(fg') by

A v ( ~ ) = {Av,ele E Cl,int(Av)} U {{v'}lv' EAv}.

We thus defined ~/U. We define p ~ : {all maximal elements o f fCF} ---. M by po~(Av) = p~(v) .

Let Av be a maximal element o f for. We next define u(Av):(Av)m('r162 Tpo~(~)(M ). We remark that po~(Av)= p~(v) . We fix elements vo, v E Av, such that if v, Vo E A p, A ~ E ~///" then A ~ = Av. For v ~ E B,B E (Av)m('r162 we set

(exp~l(v) (pi(v')) u(Av)(B) = ili rn \ d(pi(v), pi(vo)) ) ~ Tp=~v)(M)

in case the right-hand side converges. We remark that u(Av)(B) is independent o f the choice of v ~ E B and the , - ~ equivalence class of u(Ao) is independent o f the choice o f v and v0.

Next let B C (Av)m('r We define u(B) :Bm(~) - -~ Tu(B)(Tp~v)M). We fix v', v~ C B such that v', v~ 6 B' B' E ~//", B' C B imply B' = B. Let C E Bm('Cg'). We

80 K. Fokaya

choose v" E C and put

•i(C) = (D expp~(v))-l(expp~Iv,)(v")).

Here exp~l(v,)(v ") E Tp~(v,)(M), and Dexpp~(v) : Texp~(p~(v,))(Tp~(v)(M)) --+ Tp~(v,)M

is a differential o f the exponential map. Hence ~i(C) = (Dexpp~(v))-l(expp~Iv,)(v")) is an element o f Texp~(p~(v,))(Tp~(v)M ). Its norm is almost equal to d(pi(v~),

pi(v")). We put

( , u(B)(C) : i l im \ d ( p i ( ~ ( V o ) ) j c Texp;,l~)(p~(v,))(Tp,(v)M )

if the right-hand side converges. Again this limit is independent o f v" c C. And the equivalence class o f u(B) is independent o f v t, v~.

We continue in this way and define u for each element o f ~/C. We thus described the topology of M(F) = U~rcx(r)M(~/~) �9 We can prove that

M(F) is a smooth manifold with comers. We are going to define a map cb : M • ~-r x (C~(M) ) m • Met ~ M(F). This

map is a generalization o f one we defined at the beginning of Sect. 2. Here m is the number of interior edges. We remark that both M(F) and ~-r are stratified sets. Their strata both correspond one to one to the set X(F). Our map �9 respects this stratification. We are going to define �9 for each stratum.

In case Z -- ~Z~ which corresponds to Y~ = {{v}[v c C0,int(F)} our definition is as follows. We first recall

M(~/U) = { p : C0,int(T) ~ Mlp(v)+p(v ' ) ,

if there exists an edge o f T joining v and v ~}

and ~--(~e) : Map(Cl,int(r) --~ R+)

in this case. We fix an interior vertex v0. Then for each v C Co,int(T) there is a unique minimal path #v joining it to v0. We put fv = el,v U . . . U ek(v),v such that Vo 6 el,v, ei, v N ei+l,v = one point, and v E ek(v),v. Then for t E j - (~e ) : Map(Cl,int(F) --'+ R+) we put

~(p, t , ( fe) ,g)(v) = (~)t(ek(v)'v) . o ~t(el,v)] ( p ) . \ fek(v),v o . . fel, v ]

(We recall that ~t(el,v) is the exponential map of the vector field grad fel,~.) fel,v We next consider the case when ~ -- {Ai} such that Ai NA 7 = ~Z~ for i+ j . This

corresponds to the case ~/U = {Ai} U {{V}[V C C0,int(F)}. (Here Ai is a connected subcomplex of T, which is regarded both as a subset o f Cl,int(F) and C0,int(T).) Let J - ( ~ O be the tree obtained by shrinking each of the elements of ~ = {Ai} to a point. An element t o f 3"(Z) determines an element of Map(Cl,int(~-(~()) ---+ R+). Then in the same way as above we obtain p : {all maximal elements of~U} -+ M. (Remark that the vertex o f Tg corresponds to a maximal element of "~'.) Next we determine u(Ai), t also determines an element o f ti : Cl , in t (Ai) ---+ R+ (up to an equivalence ti ~ r �9 ti). Since T is a tree there is a unique vertex vi of Ai which is closest from v0. For v C Co,int(Ai ) there is a unique minimal path fi,v in Ai joining


v to vi. Let ~i ,v = ei, l ,v U �9 . . U ei, kt(v),v such that vi C ei, l,v, ei,j,v I"1 e i , j + l , v = one point, v E ei,ki(v),v. We put

u(Ai)(v) = ~ t(ei, j,v)grad feij,v. J

We then define ~(p , t , ( f e ) ,9 ) = (t,u). The definition o f @ for general strata is similar.

^

Lemma 6.4. The map @ : M • Y r x (CC~(M)) m • Met ~ M(F) is a smooth map between manifolds with corners. Its restriction to each stratum is a submersion.

The lemma is a direct consequence of the definition and the straightforward transversality argument.

Now we can use Lemma 6.4 to construct a compactification o f the moduli space JZ(F; 97/7). We first consider the submanifold o f

M ( ~ ) = {p : Co, int(T) --~ Mlp(v):~(v') ,

if there exists an edge o f T joining v and v~}.

We fix Morse functions fe for each exterior edge e o f F. And we fix a critical point Pv for each exterior vertex v. We consider the set Y of all elements o f M ( ~ ) such that

I f v ~ is an interior vertex joined by e to a exterior vertex v. Then p(v ~) is contained in the unstable manifold o f pv o f the gradient vector field grad fe

(6.5.1)

I f v, v t E C0,int(T) such that J(v) = J(vt), then p(v) = p(v~). (6.5.2)

The following lemma is immediate from the definition.

Lemma 6.6. @ - l ( y ) N (M • ~ r • {(fe)} x {9}) is equal to ~ ' ( F ; f , /7). Here the

component o f 97 is fe in the above formula for interior edoe e and is fe in (6.5.1) for exterior edge e.

We next use the following:

Lemma 6.7. For 9eneric fe, the closure Y o f Y in M ( F ) is a manifoM with corners. For each strata M(~Cr) o f M ( F ) the codimension of M ( ~ ) n Y depends only on the Morse index of p~ and the combinatorial type of the 9raph and is independent o f ~/U.

Proof Let (p, u) E M(~#r). We describe the condition for this element in Y. Let A be a maximal element o f ~/g'. The condition for p(A) E M is

I f v ~ is an interior vertex joined by e to an exterior vertex v and if v ~ E A, then p(A) is contained in the stable manifold o f pv of the gradient vector field grad fe.

(6.8.1)

Next we consider a pair v, v ~ o f vertices of T such that for J(v) = J(v ~) we put conditions for such pairs. Let us describe those conditions. Let A,A ~ E 71K be the maximal element such that v E A and v ~ E A ~. I f A ~=A ~ then we assume

p(A) = p(A') . (6.8.2.1)

82 K. Fukaya

I f A = A' we need to put conditions on u. Let B,B ' be the maximal element of A(r162 such that v E B and v' E B'. I f B :t:B' then we put

u(B) = u(B') . (6.8.2.2)

I f B : B I we choose the maximal elements of B(~r containing v, v'. We continue in this way. It is straightforward to see that these conditions describe Y.

The co-dimension for each stratum is found from these descriptions. Namely we put dim M conditions for each pair v, v' o f vertices of T such that J(v) = J(v'). (Note that condition (6.8) is applied once for each such pair).

The number of conditions corresponding to each exterior edge is the Morse index and is independent of the stratum.

The proof of Lemma 6.7 is complete. Using Lemma 6.7 we obtain:

Lemma 6.9. ~ - l ( y ) is a manifold with corners. The projection ~ - l ( y ) __~ J-r • (C~ m • Met is a Fredholm map. This projection respects stratification. The index o f the restriction to each stratum is the same.

The tamsversality we used in earlier sections follows immediately from Lemma 6.9.

7. Orientation

In this section we define an orientation of our moduli spaces and verify compatibility of them in various contexts. We first define an orientation of J lo ( f l , f 2 , f 3 ) . We regard it as an intersection of three submanifolds of dimension 4 in M 2, as we discussed in introduction. Namely we put

M ( f ) = {(p, cb}(p)) [ p C M, t > 0} _C M 2 .

We regard it as an image of the map ~f : M x R+ ~ M 2, (p , t) ~ (p , ~}(p)) . Then an orientation of M induces one on M ( f ) . We recall d g o ( f l , f 2 , f 3 ) = M ( f l ) N M ( f 2 ) r i M ( f 3 ) . Hence we define a sign for each point on Jgo( f l , f z , f3) induced from the orientation of M 2 and M(J}). Thus ~ / e ( f l , f2 , f3) is an oriented 0-dimensional manifold.

We remark that there is a diffeomorphism z : J g o ( f l , f2, f3 ) ~ J l o ( - f l , - f z , - f 3 ) , ( p , q ; q , t z , t3) ~ (q, p;ta,t2,t3).

Lemma 7.1. z is an orientation preservin9 diffeomorphism.

We recall that we took the sum over J/lo(elfl ,ezfz, eaf3). Hence if z were orientation reversing then our invariant would be always 0.

Proof o f Lemma 7.1. Let J : M x R+ --~ M x R+ be an orientation preserving diffeomorphism (p , t ) ~-+ (~f- l (p) , t ) . And R : M x M --* M x M, (p ,q ) ~-+ (q ,p) be an orientation reversing diffeomorphism. We remark that ~ _ f = R o ~f o J . Hence the lemma is a consequence of the following:

Sublemma 7.2. Let Vi, i = 1, 2, 3 be oriented linear subspaces of an oriented vector space V and R : V -+ V be an orientation reversin9 isomorphism. Orientations o f


Vi induces one on R(Vi) . Then RIv1NVzNV 3 : V 1 0 V2 N V 3 ---+ R ( V 1 ) f ' ) R ( V 2 ) f ' ) R ( V 3 ) is orientation preserving.

Proof One finds that RIv~nv 2 : V1 n V2 ~ R ( V 1 ) n R ( V 2 ) is orientation reversing. Hence RIv, nv=nv~ is orientation preserving, as required.

We next turn to the orientation of ~go(1,o,o)(p,q;fl,fa, f3), etc. We consider stable and unstable manifolds. Namely:

We remark that

U(P)= { x E M [ tlimc qb](x)= P} ,

S(P) = {x c M [ tfimoo q~(x) = P } �9

�9 /ge(1,o,o)(p,q;fbf2,f3) = (S(p) x U(q)) n M(f2) r i M ( f 3 ) .

Therefore orientations of stable and unstable manifolds determines orientation of ~e(1,0,0)(P, q; f l , f2, f3). The most natural way to define orientation of stable and unstable manifolds is to modify the definition of the Witten complex a bit as follows:

qp'[P,~] pECr(f),~(p)=k

e is an orientation of S(p) Ck(M; f ; q) =

Here [p,e] ~ - [ p , - e ] . Then the orientation of ~([p , ep],[q, eq]) is automatically fixed. Namely the orientation e of S(p) and the orientation of M determines an orientation of U(p). Therefore the orientation of dg([p, ep], [q, eq]) = S(p) A U(q) will be fixed. Using this orientation and induced orientation on ~([p, ep],[q, eq]) we can determine the sign in the definition of Opq.

Also we can determine the orientation of ~leO,o,o)([p, ep],[q, eq];fbfz, f3) = (S(p) • U(q)) n M(f2) A M(f3).

We next discuss the compatibility of this orientation to one for d//o(fl, f2, f3). Let us consider a one parameter family L of ( f , g). We fix an orientation of L. Then Jgo(L) = [-J(fl,f~,f3;o)EL ~ / e ( f l , f2, f3) is an oriented one dimensional manifold. One of its boundary components is given as U(~,s2,s3;g)eLJgoo,o,o)(p,p;fl,f2,f3). As be-

pE Cr(f 1 )

fore we can define orientation of [,] (Sx,S2,s3;,)eL Jgoo,0,0)(P, P; f l , f2, f3). To prove that p E C r ( f 1 )

these orientations are compatible we only need to prove the following:

Lemma 7.3. U(Ji,s2s3;~)eL S(p) • U(p) is a boundary component of p E Cr(f 1 )

U(fi,A,f3;g)eL M(f) . Their orientations are compatible.

Proof. The problem is local so we only need to work in a neighborhood of a critical point p of fl,s. Then by using the Morse lemma, we may assume that f is a standard quadratic function. In that case the lemma is easy to verify by looking at the spaces directly.

We need to verify compatibility of the orientation in various contexts. First compatibility of orientation in Lemma 3.1 is an immediate consequence of Lemma 7.3.

84 K. Fukaya

More interesting is the statement on orientation for Lemma 3.8. Let us state it more precisely.

We consider two moduli spaces U(fl,f~,f3;o)~L Jgo(1,0,0)(p, q; f l , f2, f3) and U(fl,f2,f3;o)~L JP/o(1,0,0~(q, p; - f l , f2, f3). Their closures both contain

5e(1,0,0)(pl ,ql ; f l , f 2 , f 3 ) = 5e(1,0,0)(ql, pl; - f l , f 2 , f 3 ) .

We are going to compare orientations attached to each term of the element of 5a(1,O,O)(pa,ql;fl,f2,f3) = 5 e ( 1 , 0 , 0 ) ( q l , P l ; - f l , f 2 , f 3 ) .

Lemma 7.4. The orientations o f 5r and 5a(1,0,0)(ql ,pl; - f l , f2, f3) are opposite to each other.

Proof. We put

5e(fz, f3) = {(x,x) E M2 I grad f2(x) is parallel to grad f3(x)) �9

5e(f2, f3) is a component of the boundary of M ( f 2 ) n M(f3), and hence is an oriented manifold. We remark that 5~(f2, f3) is contained in the diagonal. We have

~(1,O,O)(pl ,ql ; f l , f2, f3) = (SfI(Pl) • Ufl(ql)) n ,5~'(f2, f3 ) ,

5P(1, 0, 0)(ql, Pl; - f l , f2, f3 ) = (S-A (ql) x U_fl (P l ) ) n 5P(f2, f3 ) .

We remark that S A (ql) is the same manifold as U_ A (ql). We may assume that their orientations coincide. SA(pl)• UA(ql)--+ UA(ql)•215 U_A(pl ) is an orientation preserving diffeomorphism. The lemma then follows from the fact that if R: V ~ V is orientation reversing, and R: Vi--~ R(V~-) is orientation preserving, then R[VlnV2 : V1 n V2 ~ R(V1) A R(V2) is orientation reversing.

We next turn to the compatibility of the orientation we used at the end of Sect. 3. Namely we prove:

Lemma 7.5. d/[O(1,1,x)(Pl , q l , P2, q2)(fl, f2, f3) and ~'O(1,1,x)(ql, Pl, P2, q2)(--fh f2, f3) has an opposite orientation to each other.

Proof Let A C M 2 be the diagonal. Then

Jgo(1,1,y)(Pl, q~, P2, q2 )(f l , f2, f3 ) = (Sfl (Pl ) • Ufl (q l ) )n( Uf2(p2 ) • Sf;(q2 ) )NA ,

J[C[o(1,1,x)(ql, pl, p2, q2)(fl, f2, f3 ) = (S-fl (ql) x U_ A (Pl ) ) n(uj5 (p2) x Sf2(q2))nA .

The lemma therefore follows from the same argument as Lemma 7.1. There are many other cases for which we need to verify the compatibility of

orientations. But the argument for them are the same as those we discussed in this section already.


8. Concluding Remarks

85

As we mentioned in the introduction, we conjecture that the invariant discussed in this paper coincides with Chern-Simons Perturbation theory by Axelrod-Singer, etc. We here discuss a heuristic argument to "show" this conjecture.

We first recall the definition of the 2-loop amplitude of Chem-Simons Pe~ar- bation theory briefly. Let q be a flat g bundle on a compact oriented 3 manifold M such that H*(M; q ) = 0, Let Qk(M; q) be the set of all smooth k-forms on M with a g coefficient and let dk : (2k(M; q) ~ Ok+l(M; q) be the exterior derivative. Since our bundle ~ is flat we have dk+ldk = 0. The assumption H*(M; q ) = 0 is then equivalent to ker dk = I m dk- 1.

We fix a Riemannian metric on M and consider an orthonormal complement (kerdk) • of kerdk = Imdk-l. The restriction of dk to (kerdk) • is an isomorphism: (kerdk) • --+ kerdk+l. Let dk -1 be its inverse.

We consider the bundle Hom(nTg , n~q) on M 2. Here ni " M 2 ---+ M is the projection to the ith component.

Definition 8.1. A section P E F(MZ\A;Hom(n~q,n~g)| AZ(M2)) is said to be a propagator if

dZl(u)(x)= f P(x,y) Au(y)dy yEM

holds for each u E kerdk+l.

We next define the O-trace: End(g) | ~ R for each semi simple Lie algebra ~. Let ei be an orthonormal basis of g. We put

Tro(ul | U2 | U3) ~--- ~ ([ei, e j ] , e k ) ( [ u l ( e i ) , u 2 ( e j ) ] , u 3 ( e k ) ) . i,j,k

Now the leading term of the 2 loop amplitude of Chern-Simons Perturbation theory is given by

f Tro(P(x, y) A P(x, y) A P(x, y) ) . (8.2) (x,y)CM 2

We recall here that P(x, y) A P(x, y) A P(x, y) C A~x,y)(M 2) | Hom(qx, gy)| Hence

Yro(P(x, y) A P(x, y) A P(x, y)) E A~x,y)(M2). So (8.2) gives a number. It is proved in [AS] that (8.2) together with correction terms (terms related to the other graph A, etc.) is independent of the Riemannian metric, etc.

We conjecture that this invariant coincides with ours. We remark that the validity of this conjecture implies an integrability or rationality theorem of Chern-Simons Perturbation theory.

In [W1] Witten introduced a perturbation of De-Rham complex and Laplace operator using a Morse function. Let us propose to use this perturbation to verify the above conjecture as follows.

Let G(t;x, y) C Hom(qx, ~y) | (E~)i A / | A~) be the Green kemel of the Laplace operator. Namely if we put

u(t,x)= f (G(t;x,y),u(y))VolM, yEM

86 K. Fukaya

then

{ = d u _ -d-{ - - A u

In other words G(t;x, y) is the Schwartz kernel for e -tA. We recall the formula

O<3

f e-t'~dt = 2 -1 . 0

Since there is no harmonic form, it follows that

o ~

A -1 = f e-tAdt. 0

We remark that d -1 = 83 -1 . Therefore we have:

o ~

P = (8 @ *) f G( t ;x ,y)d t . 0

We put

We then have

P(t;x, y) = (6 | * )G(t;x, y) .

f f d S Tro(P(x, y) A P(x, y) A P(x, y ) ) M 2

o o ~ o o

= f f f f dtldt2dt3Tro(P(tl;x, y) A P(t2;x, y) A P(t3;x, y ) ) . M 2 0 0 0

(8.3)

We now try to perturb the Laplace operator using the Morse function j~. Let us consider the norm

(u, v)f,,, = f (u(x), v(x))ef4X)/~dS . M

Let 6 e.~ be the dual operator to d with respect to this norm. We put A ~J5 = 8~J5 o d + d o 82. Then

A ~ = A + --tL, gradf, . f,

Here Lx is the Lie derivative by the vector field X. We let G,,f,(t;x,y) be the Green function of A * Using it we define f,.

P~,f~(t;x, y) = (6~,f | * )G~,f,(t;x, y) ,

and

o ~ oG o ~

f dxdy f f f dtldt2dt3Tre(P~,f~ (tl;X, y ) A P~,A(t2; x, y ) A Pe,~5 (t3; x, y ) ) . M 2 0 0 0

(8.4)

We expect that after suitable normalization this integral also gives the leading term of the 2 loop amplitude o f Chern-Simons Perturbation theory.

Let us then see what happens when e goes to zero. We remark the leading term 1L is the Lie derivative. Hence after suitable normalization we of A~ = A + ~ egradj~


1L find that e - t ( A+ I L ~ ) ~ e - 7 ~g~f~ ,~ ( ~Tt/e ) *. Therefore G~,f~( t; x, y) will be a delta

measure supported on M(J~). Thus the limit of (8.4) is an integral of the current supported at the set M ( f l ) n M ( f 2 ) N M(f3) . This shows that this limit is equal to our leading term Z2(M; f l , f 2 , f3 ; g). However to show that our invariant coincides with Chem-Simons Perturbation theory, we need to discuss how the correction term occurs. The author does not know how to do it.

Let us turn to the other topic, that is the relation of our Morse theory invariant to open string theory. (The discussion here is rigorous unless otherwise specified.) For this purpose we consider a domain D with genus 0 and 3 boundaries. Let Ai, i = 1,2, 3 be Lagrangian submanifolds in T*M. We choose an almost complex structure J on T*M tamed by the standard symplectic structure. We put

~l(A1, A2, A3 ) =

J : TD ~ TD, j2 = _ 1, /

(J,~o) ~o:D~ T*M, Ao=~oJ q~(OiD) C_ Ai

Here (J, ~0) ~ (~*J,~o o ~u). And ~3A = OlA tO O2A t,.J ~3A is the decomposition to connected components. Now we choose functions f l , f2, f3 E Coo(M) and put Ai = AT = {(x, dfi(x ) Ix E M)} .

Lemma 8.5. For generic fm,f2,f3 E C~176 the space ~(AA,Af2,Af3 ) is a smooth manifoM of dimension 3 - dim M. I f M is 3-dimensional this space is compact also.

We omit the proof. The main point is that the moduli space of complex structures on D is real 3 dimensional.

Now imitating the proof of [FO] we can prove the following:

Theorem 8.6. I f ~ is sufficiently small and f l , f 2 , f 3 c Coo(M) is generic then J / / ( A e f l , A~ f : , A e f 3 ) is homeomorphic to Jgo( f l - f2, f2 - f3, f3 - f l ).

We do not prove this theorem here since its proof is completely parallel to [FO]. We next mention some trouble to use this idea to show that the open string on

T*M is equivalent to the Morse homotopy of M. We remark that in Theorem 8.6 we find d//o(fl - f z , f2 - f 3 , f 3 - f l ) . This space is a bit special compared to the general ~/'O(gl, 02, g3). Namely g5 + g2 + g3 = 0 is satisfied automatically. This causes some trouble for our construction, since we took ///o(elgl, e292, e393) for ei = 4-1. The condition gl + g2 + g3 = 0 is not preserved by this symmetry. This symmetry was necessary for the cancellation argument.

Namely to cancel the effect from, for example, the degeneration of the second edge, we need to sum up two terms, one from gl,g3 and the other from -91,93. (Here/72 can be arbitrary.) So to preserve the relation gl + g2 + 93 = 0 we need to put g2 H g2 + 291. This corresponds to the matrix (~ ~). The other two symmetries we need are ( -o 1~ 1 1 ) and (0 ~)" These matrices generate an infinite subgroup of GL(2; Z). (The symmetry we used in this paper is a finite group (Z/2Z)3.) Then there is trouble to take the average.

There is another trouble. From the point of the open string it is natural to consider the ribbon graph rather than the graph. In the case of the O-graph this

88 K. Fukaya

corresponds to consider another Riemann surface as well. Namely we need to take D ~ = T 2 - D i s k . We can try to imitate the construction Jg(A1, A2, A3) and define

~[1,1 (A) =

I J : T D I ~ TD I, j 2 = _ l ,

(J, q0 ~p : D' --~ T*M, J~p = qJJ

(o(OD') cc_ A

(Here 1, 1 in the suffix means that our Riemann surface D ~ is of genus 1 and 1 boundary component. In that sense one we used before is Jgo,3(Au1,Af~,Af3).)

However in fact for this moduli space transversality never holds. In fact we can easily see that elements of Jgl,l(A) are constant maps to the origin. Hence the actual dimension is dimM, while the formal dimension is (3 - d imM)(2g + k - 2). (Here g is the genus and k is the number of boundary component. In the case of J/C'I,I(A) the formal dimension is hence 3 - d i m M . )

This fact is parallel to a similar problem in closed string theory. Namely we consider the moduli space of holomorphic map ~o from the genus one (or higher) (closed) Riemann surface Z to a symplectic manifold X such that q~*(cl(X)) = 0, then we have a moduli space homeomorphic to X, that is the set of constant maps. The virtual dimension of this moduli space is 2g(3 - d i m M ) and is different from the actual dimension. This problem was studied by Ruan-Tian [RUT] using the inhomogeneous perturbation. Maybe there is an approach to our open string setting also using inhomogeneous perturbation.

This problem will cause a trouble to generalize Theorem 8.6 also. Namely the homeomorphism in this case should be

J[i,l(A~f) L JLo( f - f , f - f , f - f ) .

We can not perturb Jdo( f - f , f - f , f - f ) = ~ ' e (0 , 0, 0) ! to achieve transversality in a similar way discussed in this paper.

The author yet does not know how to overcome these problems, define the open string amplitude and prove that it coincides to our invariant in the case of the cotangent bundle.

Finally we discuss another way to handle the anomaly of the Morse theory version of Chern-Simons perturbation theory. Our argument here is not rigorous, so we put .... in the Lemmas below and mention 8.9 as a conjecture. We recall that the reason we need to take two flat bundles and take the difference is the degeneration of the moduli space parametrized by

R ( f ) = {x c M ] grad J ) ( x ) j = 1,2,3 are parallel to each other}

R(L) = { ( ( f , g ) , x ) l ( f , g ) ~ L,x C R ( f ) } .

(See Sect. 2.) Here L is a one dimensional subspace of ( C ~ ( M ) ) 3 x Met. We are going to discuss another way to cancel this degeneration. For two functions f , f~, we consider the space

S ( f , f ' ) = {x E M ] grad f ( x ) is parallel to grad f ' ( x ) } ,

and put S ( f , f ' ) = -S(f , f ' ) U S ( - f , f ' ) .


(We recall that our notation that the two vector is parallel was a bit unusual.) We have:

S / " L e m m a 8.7". For generic f , f~ the space ( f , f ) is a compact one dimensional oriented manifold.

Proof The proof that S( f , f ' ) is one dimensional at a regular point is a counting argument we discussed many times. It is also easy to see that S ( f , f ' ) is smooth outside the critical point o f f or H . Let us consider a critical point p of f . Let OB~(p) be the boundary of a small metric ball o f radius e centered at p. Then the restriction of grad f to ~B~(p) is regarded as a map from S n-I to Rn\{0}. The degree o f this map is • according to the parity o f the Morse index. By choosing e small grad f t is almost constant here. Therefore S ( f , H ) N OB~(p) and S ( - f , f t ) N ~B~(p) both consists o f one point. The lemma follows immediately.

Now we consider S( fb f2 ) and S(f2, f3). For generic f~ they intersect only at Cr(f2). There is exactly 2 #cr(A) choices to perturb S ( f b f 2 ) so that it does not intersect to S(f2,f3). Let S(f l , f2;e) be those perturbations. We put

Lk(fl, f2; A, f3) = 2 -#cr~ ~ Lk(S(A, A; e), s(A, f3)).

Here Lk denotes the linking number. To make it symmetric we take

Lk( fb f2 , f3 ) = ~ (Lk( f l , f2; f2,f3) + Lk(f2,f3; f3, f l ) + Lk(f3,f l; f l , f 2 ) ) .

They we have the following:

" L e m m a 8.8". For a 9eneric path L joinin9 (j7,9) and ( j7 ' ,9 ' ) we have

#R(gL) = 4(Lk(f ' ) - Lk( f ) ) .

"Proof". The linking number Lk( f l , f2; f2 , f3) changes if and only if S( f l , f2) intersects with S(f2,f3) outside Cr(f2). This intersection is exactly the point of R(f). The lemma follows.

These " lemmas" may suggest

Conjecture 8.9. Under the assumption of Main Theorem II, the number Zz(f l , f2, f3, f4; ~) -- n, TLk( f l , f2 , f3) is an invariant of (M; g).

References

[At] Atiyah, M.: The geometry and physics of Knots. Cambridge: Cambridge University Press, 1990

[AS] Axelrod, S., Singer, I.: Chern Simons perturbation theory I. In: Proceeding of the XXth DGM Conf. S. Catto and A. Roeha eds., Singapore: World Scientific, 1992, II. J. Diff. Geom. 39, 173-213 (1994)

[BC] Betz, M., Cohen, R.: Graph moduli spaces and Cohomology Operations. Turkish J. Math. 18, 23-41 (1995)

[Ba] Bar-Natan, D.: Perturbative aspects of the Chern-Simons topological field theory. PhD thesis, Princeton University, 1991

90

[Bi]

[Ch] [Fll] [F12]

[FU]

[Ful]

[Fu2]

[Fu3]

[Fu4] [FO]

[FM]

[GMM]

[Kh]

[Ko]

[Mfi]

[Oh]

[RS]

[ReT]

[RUT] [Sc] [Sm]

[Wl] [W2]

[w3]

K. Fukaya

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Communicated by R.H. Dijkgraaf

morse homotopy and chern-simons perturbation theory

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