archive.ymsc.tsinghua.edu.cnarchive.ymsc.tsinghua.edu.cn/.../65/9122-multivalued_morse_theory.… ·...

FUKAYA’S CONJECTURE ON WITTEN’S TWISTED

PRODUCTS BY RATIONAL CLOSED 1-FORMS

ZIMING NIKOLAS MA

Abstract. Wedge product on deRham complex of a Riemannian man-ifold M can be pulled back to H∗(M) via explicit homotopy, constructedusing Green’s operator, to give higher product structures. Our ear-lier work [3] prove Fukaya’s conjecture relating these higher productstructures to the operators defined by counting gradient flow trees withrespect to Morse functions. We prove a generalization of Fukaya’s con-jecture by including circle-valued Morse functions in this paper. Thesegeneralization is interesting since many naturally arised Morse functionsin Mathematical Physics or Mirror Symmetry are circle-valued.

1. Introduction

Morse theory, a beautiful method to capture the homotopy type of a(compact oriented) smooth manifold M using smooth Morse functions f ’s,can be generalized by allowing closed Morse 1-forms α’s which is first in-troduced by Novikov in [11, 12]. More precisely, a Novikov-Morse complex(NM∗α, δα) over the Novikov ring Λ (see Notations 2 for the definition), hasbeen introduced to study topology of M . Many works have been done inthis direction, for examples [2, 5, 13, 14, 16] and many others.

In the influential paper [18], Witten suggested a differential geometric ap-proach toward Morse theory by deforming the exterior differential operatord with a Morse function f as

dλ,f := e−λfdeλf = d+ λdf∧,

and considering the space of eigenforms with eigenvalues < 1 as λ → ∞,denoted by Ω∗(M)<1. The idea of Witten deformation is first carried out by[9] and later by [19] using a different approach. This story can be generalizedto the case of closed Morse 1-form α, by replacing df with α and introducinga complex parameter s = λ + iµ, giving the differential ds,α = d + sα∧.The relation between two differentials ds,α and δα are proven by [2], via anintegration map (see Definition 20) identifying

Ξ : Ω∗(M)<1 → NM∗α

and using the technique of [19].In the paper [7], Fukaya suggested a beautiful construction of Witten’s

deformation by Morse functions to incorporate wedge product structure ondifferential forms, resulting in an A∞ structure mk(λ)k≥1 (in this case,

1

2 ZIMING NIKOLAS MA

we restrict ourself to µ = 0) using homological perturbation lemma [10]. Hefurther put forward a conjecture relating the asymptotic expansion of thisA∞ structure (as λ→∞) with the one mMorse

k k≥1 in the Morse category,first defined in [6] (reader may also see [1] for details) by counting gradientflow trees. The conjecture of Fukaya is proven in our earlier work [3], whichsays that there is an asymptotic expansion of the form

mk(λ) = mMorsek (1 +O(λ−1/2))

as λ→∞.It is natural to ask whether Fukaya’s construction can be extended to

the case of closed 1-forms. We prove this natural extension using rationalclosed 1-forms in this paper, under the Morse-theoretical generic assumptionin Definition 10, and a natural assumption concerning well-definedness ofWitten’s map Ξ in Assumption 14 .

Like in our earlier work [3], this will be an enhancement from the originalWitten deformation concerning homology to one concerning real homotopytype as these informations are captured in the differential graded algebra(Ω∗(M), d,∧) suggested by [15, 17].

This machinery plays an important role in understanding SYZ transfor-mation of open strings datum and providing a geometric explanation forKontsevich’s Homological Mirror Symmetry (Abbrev. HMS) as pointed outby Fukaya in [7]. The generalization to rational closed 1-forms is necessarysince the naturally arised Witten’s twisted differential in Mirror Symmetryis often by non-exact rational closed 1-forms, such as Multivalued Morsefunctions in [7].

To be more precise, we consider categories consisting of closed ratio-nal 1-forms α’s in order for the twisted differential to satisfy the Leibnizrule. This leads to the notation of the differential graded (dg) categoryDRs(M), with objects being rational 1-forms αi on M . The correspondingmorphism complex relating αi to αj is given by the Witten twisted com-plex Ω∗ij(s) = (Ω∗(M), dij := d + sαij∧) (Here αij = αj − αi). The finite

dimensional subcomplex Ω∗ij(s)<1 ⊂ Ω∗ij(s) is a homotopy retract under ex-plicit homotopy involving twisted Green operator. We can pull back thewedge product in the deRham category DRs(M) via the homotopy, mak-ing use of homological perturbation lemma in [10], to give a deformed A∞(pre)1-category DRs(M)sm with A∞ structure mk(s)k≥1.

Fukaya’s conjecture in this case says that the A∞ structure mk(s)k≥1,expressed explicitly in terms of Witten twisted Green operator and wedgeproduct, has leading order given by mNM

k k∈Z+ , the Novikov-Morse A∞

1Roughly speaking, an A∞ pre-category allows morphisms and A∞-operations onlydefined for a subcollection of objects, usually called a generic subcollection, and requiringA∞ relation to hold once it is defined. Algebraic construction can be used to constructan honest A∞ category consisting of essentially the same amount of informations, andtherefore we will restrict ourself to A∞ pre-category.

FUKAYA’S CONJECTURE FOR RATIONAL CLOSED 1-FORMS 3

structures defined by counting gradient flow trees, via the isomorphism ξ =Ξ−1 sending a critical point p to the corresponding eigenform ξ(p).

Theorem 1. Under the assumption that the invariant ρ([αij ], g) < ∞ forevery αij, given a generic sequence of 1-forms ~α = (α0, . . . , αk), with cor-responding sequence of critical points ~p = (p01, p12, . . . , p(k−1)k), namely, pijis a critical point of αij, we have

(1.1) Ξ(mk(s)(ξ(~p))) = mNMk (~p)(1 +O(Re(s)−1/2)),

for Re(s) large enough.

The key observation is that the terms on both sides of equation (1.1) isperiodic in µ (using the identification in Notation 2), and admits a Fourier

series expansion in e−ls/n for some n depending on ~α and ~p, where theFourier modes on the R.H.S. of equation (1.1) are related to energy (seeRemark 12) of the gradient flow trees. Therefore the precise statement ofTheorem 1 is understood as an asymptotic expansion as Re(s) = λ→∞ for

the Fourier coefficients of e−ls/n for each fixed l (Notice that the convergencerate depends on l).

The case for k = 1 follows from the work of [2], and there is no error term

O(Re(s)−1/2) in this case. In view of the analysis in [9, 19] and our previouswork [3], what have to be done is to adopt these analysis to the case of ra-tional 1-form. The key idea is to consider a Z-covering πij : Mij →M whereαij = αj − αi can be lifted as a Morse function fij , and consider the cor-responding lifting of eigenforms ξ(pij) and Witten’s twisted Green operatorGαij (Definition 30) to Mij . We observe that this covering transformationis related to different homotopy classes of gradient flow trees appearing inR.H.S. of the equation (1.1). Once a Fourier mode e−ls/n is fixed, it deter-mines a finite set of possible gradient flow trees contributing to this mode,and we choose cut off from infinity on Mij containing these set of gradienttrees for applying the earlier analysis in [3, 9, 19]. The heart of the proofis to argue that, cut off from infinity can be chosen such that the remain-der will be contributing to an error of order at least O(Re(s)−1/2), and theleading order terms is exactly a summation over these finite set of gradienttrees, with each of them exactly contributing ±1 determined by orientation.

This paper is organized as follows. In section 2 we review the definitionof Witten’s deRham category and Novikov-Morse category, and give theprecise statement of the Main theorem 15. Section 3 is devoted to studythe estimate of the remainder of the cut off from infinity on Mij for a fixedαij . Section 4 contains the proof which argue that the cut off from infinitycan be chosen for each fixed Fourier mode, such that the remainder will notcontribute to the leading order term (in Re(s)−1 order) of its fixed Fouriercoefficients appearing in equation (1.1).

4 ZIMING NIKOLAS MA

Acknowledgement

The author would like to thank Kowkwai Chan for various useful conver-sations when drafting this paper. The work of the author was supported byYau Mathematical Sciences Center, TsingHua University and Departmentof Mathematics, The Chinese University of Hong Kong.

2. Witten-Morse A∞ structures

In this section, we introduce the necessary notations and definitions tostate our main theorem 15. Throughout the paper, we will write s = λ+ iµand let Hρ = s ∈ C|Re(s) > ρ and further denote the ring of contin-uous functions and holomorphic functions on Hρ by C∞(Hρ) and O(Hρ)respectively. We would like to interpret the univerisal Novikov ring Λ andits subring Λ0, defined by

Notations 2.

Λ = ∞∑i=1

aiqλi | ai ∈ C, λi ∈ R, lim

i→∞λi = +∞

and Λ0 =

∞∑i=1

∈ Λ | λi ≥ 0

respectively, as formal series of holomorphic functions formally via the for-mula qλi = e−sλi .

2.1. Witten’s twisted deRham category.

2.1.1. deRham category of circle-valued Morse functions. Given a compactoriented Riemannian manifold (M, g), we can construct the deRham cate-gory DRs(M) depending on a complex parameter s with Re(s) > 0. Objectsof the category are Q-valued closed 1-forms α’s. For any two objects αi andαj , we define the space of morphisms between them to be

Hom∗DRs(M)(αi, αj) = (Ω∗(M ;C), dij),

where the differential dij = d+ sαij∧ with αij := αj −αi. The compositionof morphisms is defined to be the wedge product of differential forms onM . This composition is associative and hence the resulted category is adg-category. For simplicity, we will denote the complex corresponding toHom∗DRs(M)(αi, αj) by Ω∗ij(s).

2.1.2. Witten’s deformation of DRs(M). Fixing two 1-forms αi and αj , theWitten Laplacian is given by

∆ij = dijd∗ij + d∗ijdij ,

where d∗ij = d∗ + sια∨ij . We will restrict our attention to those αij satisfying

the following Morse-Smale condition.


Definition 3. αij is said to be Morse if the zero set Z(αij) is finite and ∇αat each critical point p in Z(αij) is non-degenerated.

It is said to be Morse-Smale if V+(p) and V−(q) intersects transversallyfor all p, q ∈ Z(αij), where

V±(p) = y| limt→∓∞

σt(y) = p

are the stable and unstable submanifolds associated to p with respect to flowσ of α∨ij respectively.

When αij satisfies the Morse-Smale condition, Witten-Helffer-Sjostrandtheory (see section 3 and [19] for brief discussion) suggests that there areρ > 0 and constants C1, c2 > 0 such that ∆ij has no eigenvalues in the

interval [C1e−c2λ, c2λ

1/2] for λ = Re(s) > ρ. We denote the eigensubcomplexconsisting of eigenforms with eigenvalues less than 1 by Ω∗ij(s)<1 ⊂ Ω∗ij(s)

for Re(s) > ρ.Similar to our previous paper [3], Witten’s approach indeed produces an

A∞ category, denoted by DRs(M)sm, with A∞ structure mk(s)k∈Z+ . Ithas the same class of objects as DRs(M). However, the space of morphismsbetween two objects αi, αj is taken to be Ω∗ij(s)<1, with m1(s) being the

restriction of dij to the eigenspace Ω∗ij(s)<1.

The natural way to define m2(s) for any three objects α0, α1 and α2 isthe operation given by

Ω∗12(s)<1 ⊗ Ω∗01(s)<1∧−−−−→ Ω∗02(s)

P02−−−−→ Ω∗02(s)<1,

where Pij : Ω∗ij(s)→ Ω∗ij(s)<1 is the orthogonal projection. General mk(s)’sare constructed similarly using the homological perturbation lemma from[10], where the precise definition can be found in [3], giving an A∞ (pre)-category.

2.2. Novikov-Morse category. The Novikov-Morse category Morse(M)has the same class of objects as the deRham category DRs(M), with thespace of morphisms between two objects αi and αj given by

Hom∗Morse(M)(αi, αj) = NM∗αij =⊕

p∈Z(αij)

Λ · p,

graded by the Morse index, denoted by deg(p). We follow [1, 2, 6, 11] forthe definition of the differential under the assumption that their differenceαij is Morse-Smale.

Definition 4. Fixing p, r ∈ Z(αij) and a homotopy class β ∈ P(p; r) ofpath from p to r, we define the moduli space of gradient line from p to r ofclass β to be

Mβij(p; r) = γ : R→M | lim

t→−∞γ(t) = p, lim

t→+∞γ(t) = r, γ′ = α∨ij , [γ] = β,

6 ZIMING NIKOLAS MA

which is a smooth manifold of dimension deg(r) − deg(p) + 1 under theMorse-Smale condition.

Definition 5. For a critical points p ∈ Z(αij), we define

δij(p) =∑

r∈Z(αij)

( ∑β∈P(p;r)

#(Mβij(p; r)/R)qαij(β)

)· r,

where #(Mβij(p; r)/R) 6= 0 only if deg(r) = deq(p) + 1, which is a signed

count of finitely many points in that case.

Remark 6. The differential is well defined due to the compactness lemmaproven by Novikov in [11], saying that for each fixed R ∈ R, any sequenceγkk∈N of flow line from p to q with bounded energy αij(γk) ≤ R has aconvergence subsequence to a broken flow line.

The Novikov-Morse categoryMorse(M) is anA∞ (pre)-category equippedwith higher products mNM

k for every k ∈ Z+, or simply denoted by mk,which are given by counting gradient flow trees. To describe that, we willuse the same terminologies about directed trees following [1].

Definition 7. A k-leafed tree, or simply a k-tree, is a planar trivalent di-rected tree T (every vertex is required to be trivalent, having two incomingedges and one outgoing edge) with k semi-infinite incoming edges and 1semi-infinite outgoing edge.

Given a k-tree, by fixing the anticlockwise orientation of R2, we havecyclic ordering of all the semi-infinite edges. We can label connected com-ponents of R2 \ T by integers 0, . . . , k in anticlockwise ordering, inducing alabelling on edges such that edge e labelled with ij will be lying between com-ponents i and j with the unique normal to e pointing in component i. Thelabelling can be fixed uniquely by requiring the outgoing edge to be labelledby 0k.

Definition 8. Denoting the set of finite edges by E(T ), a metric k-tree Tis a k-tree together with a length function l : E(T )→ (0,+∞).

Given a metric k-tree T , we use |T | to denote the underlying topologi-cal space with internal edge e parametrized by interval [0, l(e)] and infiniteincoming (outgoing resp.) edges parametrized by (−∞, 0] ([0,∞) resp.).

The space of metric k-trees has finite number of components, with eachcomponent corresponding to a topological type T . The component corre-sponding to T , denoted by S(T ), is a copy of (0,+∞)|E(T )|, where |E(T )| isthe number of internal edges and equals to d− 2.

Given a Morse sequence ~α = (α0, . . . , αk) (i.e. αij = αj − αi is Morse-Smale for all i, j) with a sequence of critical points ~p = (p01, . . . p(k−1)k, p0k)

and a k-tree T , we let PT (~α, ~p) be the homotopy class of maps |T | → Mhaving infinite end with label ij limit to corresponding critical point pij .


Definition 9. Fixing β ∈ PT (~α, ~p), we define the moduli space of gradienttree of class β to be

Mβ(~α, ~p) =

γ | γ : |T | →M, [γ] = β, γ′|eij = α∨ij

limt→−∞ γ|e(i−1)i= p(i−1)i, limt→+∞ γ|e0k = p0k

,

where T is metric k-tree of type T .

A gradient tree can be treated as a point in intersections of submanifoldsin∏k+1M as follows. For each critical point pij , there is an Z-choice of

stable submanifolds since the 1-form αij lifts to Morse function on an Z-covering of M . We fix a choice V+(pij) for each incoming critical point pijand a choice V−(p0k) of the outgoing critical point p0k. For an incomingcritical point p(i−1)i, with corresponding stable submanifold V+(p(i−1)i), wedefine a map

fT,(i−1)i : V+(p(i−1)i)× S(T )→M

as follows. Fixing a point x in V+(p(i−1)i) together with a metric tree T ∈S(T ), we need to determine a point in M . First, suppose v is the vertexconnected to the edge labelled (i−1)i, there is a unique sequence of internaledges (e1, . . . , ek−2) connecting v to the unique vertex vr connecting to theoutgoing semi-infinite edge. To determine the image of x under our function,we apply gradient flow with respect to 1-forms associated to ej ’s for timel(ej) to x consecutively according to the sequence (e1, . . . , ek−2).

The maps are then put together to give a map

(2.1) fT : V−(p0k)× V+(p(k−1)k)× · · · × V+(p01)× S(T )→k+1∏

M,

where we use the embedding ι : V−(p0k) → M for the first component. Wewill always assume the following generic assumption to ensure the regularityof the moduli space.

Assumption 10. Given a sequence of 1-forms ~α = (α0, . . . , αk) such thattheir difference αij are all Morse-Smale, it is said to be a generic sequence iffor any order preserving subsequences ~α′ of ~α, the image of f ′T (associated to

~α′) intersect transversally with the diagonal submanifold ∆ ∼= M →Mk′+1,for any sequence of critical point ~q′ associated to ~α′, any topological type Tand any choice of stable and unstable submanifolds.

Under the above assumption, the moduli spaceMβ(~α, ~p) can be partiallycompactified into manifold with corners by allowing edges degenerating intobroken gradient flow lines, using a Gromov-compactness [8] type argument.For details, readers may see [1, 6]. Given a generic Morse sequence ~α, Wecan define the Novikov-Morse A∞ structure mk as follows. First, we letm1 = δ to be the Novikov-Morse differential defined above. For k ≥ 2, wehave the following definition.

8 ZIMING NIKOLAS MA

Definition 11. For a sequence of critical points p01, . . . , p(k−1)k, we define

mNMk (p01, . . . , p(k−1)k) =

∑p0k∈Z(α0k)

∑T

∑β∈PT (~α,~p)

(#(Mβ(~α, ~p))qA(β)s)p0k,

where A(β) =∑

eij :edges

∫eijαij and #(Mβ(~α, ~p)) is signed count of points

which is nonzero only if the space is zero dimensional, i.e. when

k∑i=1

deg(p(i−1)i) + 2− k = deg(p0k).

Remark 12. For the above definition to make sense, we need to show thatfor a given area A, there will only be the finitely many gradient trees Γ’swith area A. This is because for each piecewise smooth map γ : |T | → M ,we can define the energy of the map to be

E(γ) :=∑eij

∫eij

|γ(s)|2.

For a gradient flow tree Γ, we have A(Γ) = E(Γ). For a fixed energy E, aGromov type compactness result for the gradient trees with energy less thanE, together with the fact that the moduli space is zero dimensional will givethe finiteness property.

At the point of writing this paper, we did not find a reference for thecompactness result exactly fix into our setting, while we believe the originalargument in [1, 6] with little modification, can be adopted to give the desiredwell-definedness of the operator mNM

k k∈Z+.

Remark 13. To define the sign in the counting #(Mβij(p; r)/R) and #(Mβ(~α, ~p)),

we first need to choose an orientation of V±(pij) for each critical points suchthat [V−(pij)]⊕ [V+(pij)] = [TMpij ] as orientation, and then inductively de-fine the orientation along the gradient trees. Readers may see [1, 3] fordetails.

2.3. Witten deformation and Novikov-Morse category. The two A∞structures mk(s)k∈Z+ and mNM

k k∈Z+ are closely related to each other,we will formulate and prove our main theorem 15 in this section, roughlysaying that the limit of Re(s)→ +∞ of mk(s) will give mNM

k .

Assumption 14. Starting from now on, we will assume that the invariantρ([α], g) <∞ as in definition 20 being finite for making sense of the followingisomorphism obtained from the results of [2, 9, 19].

Ξ : Ω∗ij(s)<1 → CM∗αij =⊕

pij∈Z(αij)

C∞(Hρij ) · pij ,

when Re(s) > ρij for some large enough ρij. We will denote its inverse byξ = Ξ−1.


Given a generic sequence of 1-forms α0, . . . , αk, with p(k−1)k, . . . , p01 becorresponding critical points, we have the following theorem.

Theorem 15 (Main Theorem). For Re(s) large enough, we will have theasymptotic expansion

Ξ(mk(s)(ξ(p(k−1)k), . . . , ξ(p01)))

=∑

p0k∈Z(α0k)

∑T

∑βT∈PT (~α,~p)

(#(MβT (~α, ~p))e−A(βT )s)(1 +RβT (s))p0k,

with |RβT (s)| ≤ C0,βT λ−1/2. In the other words, we have

mk(s) = mNMk (1 +O(λ−1/2))

in the asymptotic expansion when λ large enough, under the identificationΞ.

The case for m1(s) follows from the work of [2], while the case for othermk(s)’s is a generalization for our earlier work [3].

Remark 16. The constants C0,βT and ρ0 will also depend on the formsα0, . . . , αk. In general, we cannot choose fixed constants that the above state-ment holds true for all mk(s) and all sequences of functions.

3. Witten Morse complexes

We fix α = αij , and recall The Witten-Helffer-Sjostrand theory [9, 18]for a single 1-form α relates the Witten twisted complex (Ω∗(M), ds,α) tothe Novikov-Morse complex (NC∗, δα). We begin with recalling some of thenotations and results from [2, 3, 9, 13], follows by giving some modificationsof these results. We will drop the dependence of α in our notations if thereis no confusion.

3.1. Witten deformation and Novikov-Morse complexes. Given aQ-valued 1-form α, integration over 1-cycles gives a map

(3.1) Γα : H1(M,Z)→ Q.

There is a covering π : M → M of with free abelian covering group H1 :=H1(M,Z)/Tor, such that π∗(α) = dfα for some function fα. The fact thatα being Q-valued suggests that there is a Z-covering πα : Mα → M whichfactors as

(3.2) Mπ−−−−→ Mα

πα−−−−→ M,

and fα descends to Mα. We fix the generator t = tα of the covering groupof πα such that fα(t · x) > fα(x).

We consider the Witten twisted deRham complex (Ω∗(s), ds) by α as insection 2.1.1 (i.e. with ds = d + sα∧), and treat it as family of complexes

10 ZIMING NIKOLAS MA

over the half plane Hρ = s ∈ C∗|Re(s) > ρ for some ρ > 0. From [2, 9],we know that there is a gap in the spectral of Witten Laplacian

∆ = dsd∗s + d∗sds

= ∆ + sL∗α∨ + sLα∨ + |s|2|α|2,which says that there is no eigenvalues in the interval [C1e

−c2λ, c2λ1/2] for

λ = Re(s) > ρ0 for some ρ0 large enough. We will fix ρ0 later in Definition 20and restrict ourself to a family of finite dimensional complexes (Ω∗<1(s), ds)over the half plane Hρ with ρ > ρ0.

We further introduce an Z-action on the family Ω∗<1(s) over Hρ given bythe formula

(k · ϕ)(s+ 2πiknα) = e−2πinαfαϕ(s),

where nα ∈ Q+ is defined such that nαα takes values 1 on the generator tof the covering map πα in equation (3.2), which makes e−2πinαfα is a globalfunction on M . This action is well defined on small eigenforms Ω∗<1(s)because there is a relation

∆s+2πiknα = e−2πinαfα∆e2πinαfα

between the Witten’s Laplacians. We will continue the denote those sectionsof Ω∗<1(s) which are invariant under this Z-action by Ω∗<1(s) from now on,by abusing of notations.

Notations 17. We will write C∗α := Hρ/2πinαZ, since we are interested inthe sections of Ω∗<1(s) which are invariant under this Z-action.

This family of vector space can be identified with the trivial bundle⊕p∈Z(α)C · p over C∗α via the integration map, if we assume that the fol-lowing invariant ρ([α], g) defined in [2] depending only on cohomology class[α] and metric g is finite.

Definition 18. For each critical point p ∈ Z(α), we choose a lifting of

p ∈Mα (or equivalently in M) and the unstable submanifold V−(p) ∼= Rdeg(p)to Mα (or equivalently in M) centered at p. We let

ρ([α], g) = infλ ∈ R+ |∫V−(p)

(eλ(fα−fα(p))) <∞ ∀p ∈ Z(α).

From now on, we will assume the following.

Assumption 19. We assume that ρ([α], g) <∞ defined in [2] is finite, fordefining the following Witten’s map.

Definition 20. We define the integration map Ξ : Ω∗<1(s)→ ⊕p∈Z(α)C∞(C∗α)·p by

(3.3) Ξ(ϕ) =∑

p∈Z(α)

(∫V−(p)

(es(fα−fα(p))ϕ))· p

for ρ > maxρ([α], g), ρ0.


From the result in [2], it is an isomorphism which gives a trivialization ofthe bundle Ω∗<1(s) over C∗α for ρ large enough. We fix such ρ large enoughand denote the inverse of Ξ by ξ.

Remark 21. The map Φ is independent of the choice of lifting of p to Mα

and the lifting fα on Mα.

Remark 22. The condition ρ([α], g) < +∞ is necessary for well-defineness

of the integral over V−(p) ∼= Rdeg(p), which is conjectured to hold even for αirrational in [2].

3.1.1. The line bundle Lα associated to α. For each fixed α, we can considera line bundle Lα defined over C∗α ×M as follows, which is closely related tothe deck transformation of the Z-cover πα : Mα → M . We consider an Zaction on the trivial bundle C over C∗α ×Mα given by

(k · ϕ)(s, tk · x) = eikα(t)µϕ(s, x),

equipped with the standard metric h over C∗α ×Mα. We let (Lα,∇α, hα) tobe the Z quotient of trivial bundle (C, d, h), over C∗α ×M . The holonomy

of the flat line bundle Lα is given by Hol∇α(γ) = eiα(γ)µ along a loop γ inπ1(M).

Taking a section ϕ of Ω∗<1(s), we treat it as differential forms along Mover the product Hρ ×M . Upon multiplying the pull back π∗α(ϕ) by the

function eiµfα on Hρ×Mα, we found that their product descends as sectionof Lα. Furthermore, the operator ds is identified with the operator dλ,α =(d+ λα) with its adjoint d∗s being identified with d∗λ,α = (d∗ + λια∨) acting

on Ω∗M (C∗α ×M,Lα), and therefore the Witten’s Laplacian ∆s is identifiedwith

(3.4) ∆α = dλ,αd∗λ,α + d∗λ,αdλ,α

acting on Γ(Lα). The integration map Ξ and the inverse ξ (we using thesame notations here by abusing of notations) will also be identified as

Ξ(ϕ) =∑

p∈Z(α)

(∫V−(p)

eλfα−sfα(p)ϕ)· p,

for ϕ ∈ Ω∗M (C∗α ×M,Lα).

Notations 23. In the rest of the proof, we will be considering the modifica-tion Ψ of integration map Ξ which is given by

Ψ(ϕ) =∑

p∈Z(α)

(∫V−(p)

eλfαϕ)· p,

which will depends on both lifting fα and p to Mα. The inverse of Ψ isdenoted by ψ.

Remark 24. We prefer to use the operators (dλ,α, d∗λ,α,∆α) with the cor-

responding twisted Green’s operator Gα and projection Pα, the integration


map Ψ and its inverse ψ. The reason is that it will be more convenience forus to apply the previous result from [9, 3] in this settings. Our main theorem15 will be reformulated as in Theorem 4 accordingly (for the case of m3).

Writing C∗α = R>ρ × S1α, we choose a normalized volume element volµ =

1nαdµ on S1

α for the purpose of Fourier series. The result of [2] for m1 canbe restated as follows according to the identification used in Remark 24.Fixing a lifting p for each p ∈ Z(α), we let P0(p; r) to be the homotopyclasses of paths joining p to r on Mα. This gives an identification P(p, r) =∐k∈Z P0(p; tk · r) via the covering transformation acting on r. Such choice

will also fix the ambiguity in the definition of Ψ and ψ.

Theorem 25 (Burghelea-Haller [2]). We have

(Ψ)(dλ,αψ(p)) =∑

r∈Z(α)

(∑t∈Z

∑β∈P0(p;tk·r)

e−kα(t)s#(Mβ(p; r)/R))· r,

= (es(fα(p)−fα(r))δα)(p),

where we make use of the natural inclusion of holomorphic function O(C∗α) →Λ given by taking the Laurent series and identifying e−kα(t)s ↔ qkα(t) in thesecond equality.

3.2. Estimates for circle-valued Morse function. In this section, wewill give the estimates for the Witten’s twisted operator introduced in equa-tion (3.4) which is necessary for proof of our theorem. These results aremodified version of those from [19] and [9], for circle-valued Morse function.

We consider Ω∗M (C∗α×M,Lα) with the corresponding Witten’s Laplaciangiven by Equation (3.4). We will investigate its resolvent (∆α − z)−1 andeigenforms corresponding to small eigenvalues. This is done by comparingthe Witten’s Laplacian ∆α acting on Ω∗(Mα) on Mα and ∆α on Ω∗M (C∗α ×M,Lα).

If we take a L2-integrable section ϕ of ∧∗T ∗M on Mα, then the Fourierseries

(3.5) ϕ(x) =∑k∈Z

e−ikα(t)µϕ(tk · x)

will be invariant under the Z-action on S1α × Mα defining Lα, and hence

descends as section of Lα on S1α ×M (which can also be treated as section

of Lα over C∗α ×M). Furthermore, the L2 norm can be identified as

(3.6) ‖ϕ‖2L2(S1×M) = ‖ϕ‖2L2(Mα).


This can be seen by taking a fundamental domain Mα,0 of the covering Mα,and compute the L.H.S. as

‖ϕ‖2L2(S1×M) =

∫S1×Mα,0

|π∗α(ϕ)|2volµ ∧ volg,

=∑k∈Z

∫Mα,0

|ϕ(tk · x)|2volg,

= ‖ϕ‖2L2(Mα).

This allows us to interchange between the Witten’s Laplacian ∆α on Mα

and ∆α on S1α ×M using the Fourier series

(3.7) (∆αϕ)(x) =∑k∈Z

e−ikα(t)µ(∆αϕ

)(tk · x),

for ϕ and ϕ related as in equation (3.5).Making use of the spectral gap theorem in Witten-Helffer-Sjostrand the-

ory [9, 18] applying to ∆α acting on µ × M for each fixed µ and theabove equation (3.6) and (3.7), we have the following lemma for the Witten

Laplacian ∆α acting on L2-forms on Mα.

Lemma 26. There is λ0 > 0 and constants c, C > 0 such that we have

Spec(∆α) ∩ [ce−cλ, Cλ1/2) = ∅,

for λ > λ0.

Proof. We notice from the result from Witten-Helffer-Sjostrand theory in [9]

(Lemma 1.6 and Proposition 1.7), we have a uniform spectral gap [ce−cλ, Cλ1/2)for the ∆α acting on µ ×M for all µ ∈ S1

α. Therefore we obtain the cor-

responding result for ∆α using equation (3.7).

3.2.1. Resolvent Estimate. From the spectral gap theorem, we notice thatthe resolvent (∆α − z)−1 is well defined for z : c1 < |z| < c2 when λ islarge enough.

Definition 27. In view of the equation (3.6), we define the resolvent (∆−z)−1 on Mα using the operator (∆ − z)−1 acting on S1

α ×M via Fourierseries

(∆α − z)−1(ϕ)(x) =∑k∈Z

e−ikα(t)µ((∆α − z)−1ϕ

)(tk · x).

Therefore, we can estimate the resolvent on Mα using the Agmon distancefunction ρ defined as follows, as in [9].

Definition 28. The Agmon distance dα is the distance function with respectto the degenerated Riemannian metric gα = |α|2g, where g is the backgroundmetric. We also let ρα(x, y) = dα(x, y)− (fα(y)− fα(x)) for x, y ∈Mα.


Notice that ρα(x, y) ≥ 0 with equility holds iff there is a generalizedbroken gradient flow line from x to y. The following lemma says that theresolvent (∆α − z)−1 will have a decay of order O(e−λ(dα(x,y)−ε)).

Lemma 29. For any bounded subset K ⊂ Mα, any j ∈ Z+ and ε > 0,there is lj ∈ Z+, Cj,K,ε > 0 and λ0 = λ0(K, ε) > 0 such that the followingholds. For any two points x0, y0 ∈ K, there exist neighborhoods V and U(depending on ε) of x0 and y0 respectively such that

(3.8) ‖∇j((∆α − z)−1u)‖C0(V ) ≤ Cj,K,εe−λ(dα(x0,y0)−ε)‖u‖W lj ,2(U)

,

for all λ > λ0 and u ∈ C0c (U), where W k,p refers to the Sobolev norm.

Proof. This is essentially a consequence of the corresponding statement forclosed manifold with boundary in proposition 6.5 in [4] (notice that theoperator there is slightly different from our Witten’s Laplacian while theargument for obtaining this estimate are the same), together with the fol-lowing cut off estimate. W.L.O.G., we assume that the bounded subset Kto be K = f−1

α ([−(L − 2), (L − 2)]). Within the compact set K, there isa constant CK = supx,y∈K dα(x, y) > 0. We take choose a cut off func-

tion χ ≥ 0 compactly supported in f−1α ((−L,L)) satisfying χ ≡ 1 on

f−1α ([−(L − 1), (L − 1)]), and another cut off function χ compactly sup-

ported in f−1α ((−(R + 1), (R + 1))) satisfying χ ≡ 1 on f−1

α ([−R,R]). Wechoose R large enough such that dα(supp(∇χ), supp(1 − χ)) ≥ CK + 1. Ifwe choose R+ 1 to avoid critical points of fα, f−1

α ([−(R+ 1), (R+ 1)]) is a

manifold with boundary and we consider the Witten’s Laplacian ∆α,0 on itwith Dirichlet boundary condition.

We will first prove the Lemma for L2-norm and then obtain the resultsfor the derivatives. First, we have the equation

(∆α − z)−1 = χ(∆α,0 − z)−1χ+ (∆α − z)−1(1− χ)(3.9)

−(∆α − z)−1(1− χ)[∆, χ](∆α,0 − z)−1χ,(3.10)

From the result of [9] for (∆α,0 − z)−1 on closed manifold with boundary,

the operator χ(∆α,0 − z)−1χ satisfies the desired estimate

‖(χ(∆α,0 − z)−1χu)‖L2(V ) ≤ CK,εe−λ(dα(x0,y0)−ε)‖u‖L2(U)

and therefore what we need is to estimate rest of the terms in the aboveequation.

First, we have supp(1 − χ) ∩ U = ∅ for small enough ε and therefore

(∆α − z)−1(1 − χ)u = 0. Using the result on (∆α,0 − z)−1 again and thechoice of supports of χ and χ, we have

‖([∆, χ](∆α,0 − z)−1χu

)‖L2(Mα) ≤ Cj,K,εe−λ(CK)‖u‖L2(U).

Together with the fact that (∆α−z)−1(1−χ) being a bounded operator, we

have a decay of order Cj,K,εe−λ(CK)‖u‖L2(U) to the last term. This complete

the proof of the L2-norm estimate.


Next we consider the estimate for the L2-norm of the derivatives ∇jv,where we write v = (∆α − z)−1u. We use the elliptic estimate on a smallerV ′ ⊂ V and get

‖∇jv‖L2(V ′) ≤ Cj,V ′,V(‖(∆α − z)d

j2ev‖L2(V ) + λj+1‖v‖L2(V )

).

We then apply the above proven estimate for L2 norm by replacing u with(∆α − z)ju to obtain the estimate

‖(∆α − z)dj2ev‖L2(V ) ≤ CK,εe−λ(dα(x0,y0)−ε)‖(∆α − z)d

j2e−1u‖L2(U)

≤ Cj,K,εe−λ(dα(x0,y0)−ε)λj+1‖u‖W j−1,2(U).

Combining the above we have desired estimate

‖∇jv‖L2(V ′) ≤ Cj,K,ε(e−λελj+1)e−λ(dα(x0,y0)−2ε)‖u‖W j−1,2(U)

≤ C ′j,K,εe−λ(dα(x0,y0)−2ε)‖u‖W j−1,2(U)

by choosing larger C ′j,K,ε if necessary.

Finally, we further choose V ′′ ⊂ V ′ such that V ′ is ε-ball centered at y0

and V ′′ is the ε/2-ball centered at y0. The desire C0 norm estimate can beobtained by using interior Sobolev embedding on V ′′, which is given by

‖∇jv‖C0(V ′′) ≤ Cε‖v‖W lj ,2(V ′).

Definition 30. We define the Witten’s twisted Green operator as

Gα = (I − Pα)∆−1α ,

consider to be a family of operator over C∗α. We also let Gα to be its Fouriercoefficient, which is a twisted Green operator acting on Mα defined by theFourier series

(Gαϕ)(x) =∑k∈Z

e−ikα(t)µ(Gαϕ

)(tk · x),

where ϕ and ϕ are related by equation (3.5).

The twisted Green operator will be related to the original resolvent bythe formula

(3.11) Gα(u) =

∮γz−1(z −∆α)−1udz

where γ ⊂ z : c1 < |z| < c2 is an counter-clockwise embedded curve.Therefore, using the above Lemma 29 we obtain the following estimate forthe twisted Green operator Gα.

Lemma 31. For any bounded subset K ⊂ Mα, any j ∈ Z+ and ε > 0,there is lj ∈ Z+, Cj,K,ε > 0 and λ0 = λ0(K, ε) > 0 such that the following


holds. For any two points x0, y0 ∈ K, there exist neighborhoods V and U(depending on ε) of x0 and y0 respectively such that

(3.12) ‖∇j(Gαu)‖C0(V ) ≤ Cj,K,εe−λ(dα(x0,y0)−ε)‖u‖W lj ,2(U)

,

for all λ > λ0 and u ∈ C0c (U), where W k,p refers to the Sobolev norm.

3.2.2. Estimate for ψ(p). In this subsection, we will give a decay estimatefor the eigenfrom ψ(p). We fix once for all a lifting p on Mα for each criticalpoint p and the map ψ depends on this choice. By lifting ψ(p) to the coverMα, we have a Fourier series

π∗α(ψ(p))(x) =∑k∈Z

e−ikα(t)µψ(p)(tk · x),

for some L2 section ψ(p) on Mα. We will prove the following lemmas con-

cerning the decay of ψ(p).

Lemma 32. There is a constant C depending only on the values of fα(p)’s,such that for each j ∈ Z, there is a λ0(j) > 0 such that when λ > λ0, wewill have

‖ψ(p)‖Cj(S1α×M) ≤ Dje

Cλ,

for all the eigenforms ψ(p).

Lemma 33. For any compact subset K ⊂ Mα, and ε > 0, there existsλ0 = λ0(K, ε) > 0 such that for λ > λ0, we have

(3.13) ∇jψ(p) = Oj,K,ε(e−λ(dα(p,·)+fα(p)−ε)).

Here Oj,K,ε refers to the dependence of the constant on j, K and ε.

This is done by giving an approximated inverse to the map Ψ. We take Rto be chosen later and take a cut off function χ ≡ 1 on f−1

α ([−R,R]) which iscompactly supported in f−1

α ((−(R+ 1), (R+ 1))). Again, we take R genericsuch that Mα,R = f−1

α ([−(R+ 1), (R+ 1)]) will be a smooth manifold with

boundary and we consider the Witten Laplacian ∆α,0 acting on it as in proofof Lemma 29. We may assume that all the lifting p’s of critical points lyingin Mα,R by taking R large enough.

Definition 34. We take τp,0 to be the eigenform on Mα,R (using Dirichletboundary condition) satisfying∫

V−(p)

(eλfα τp,0

)= 1,∫

V−(q)

(eλfα τp,0

)= 0, for p 6= q ∈Mα,R.

The eigenform τp,0 will satisfy the estimate in lemma 33, and therefore

we have to estimate the difference between ψ(p) and it. We Let

(3.14) τp,0 =∑k∈Z

e−ikα(t)µt−k · (χτp,0),


where t−k ·(χτp,0)(x) = χτp,0(tk ·x), and τp,0 descends to S1α×M as a smooth

form, as χτp,0 is compactly supported.

Definition 35. We define the approximation τ to ψ, by

τ(p) = Pα(τp,0),

which is treated as a map τ :⊕

p∈Z(α) C∞(C∗α) · p→ Ω∗M (C∗α ×M,Lα)<1.

Lemma 36. There is an AR which can be chosen arbitrarily large by choos-ing large enough cut off radius R, such that we have the estimate

‖(τ(p)− τp,0)‖Cj(S1α×M) ≤ CR,je−λAR ,

for some constant CR,j.

Proof. Using the estimate for eigenforms in equaiton (1.39) in [9], we learn

that there is an exponential decay for τp,0 of order OR,ε(e−λ(dα(p,·)+fα(p)−ε)).We let µp,0 to be the eigenvalue of the eigenform τp,0 which is exponen-tially small. We have the equation (∆α − µp,0)τp,0 = rp,0 where rp,0 =∑

k∈Z e−ikα(t)µt−k · ([∆α, χ]τp,0) and therefore we observe that we have an

estimate

‖rp,0‖W j,2(S1α×M) ≤ CR,j(e−λAR)

where AR = infp∈Z(α)dα(p, supp(∇χ)) + fα(p) − 1 (we may choose largeenough R such that AR is always positive).

Therefore by writing

τ(p)− τp,0 =

∮γ(z − µp,0)−1(z −∆α)−1rp,0dz

where γ ⊂ z : c1 < |z| < c2 is an counter-clockwise embedded curve, wehave the decay estimate

(3.15) ‖(τ(p)− τp,0

)‖L2(S1

α×M) ≤ CR,j(e−λAR).

Furthermore, applying the above estimate for L2-norm to the term∇j1∂∂µ

(∆α−

z)j2(τ(p)− τp,0

)as in the proof of above Lemma 29 we obtain an estimate

for the derivatives of(τ(p)− τp,0

). The result in this Lemma again follows

from Sobolev embedding.

Viewing τ as a map approximating ψ, we estimate the error between themin the following lemma.

Lemma 37. There is a constant cR > 0 and λ0(R), which can be arbitrarylarge by choosing large enough R, such that we have

Ψ τ = I − r(λ, µ),

where |r(λ, µ)| = OR(e−cRλ) as function on R>ρ × S1α for λ > λ0(R).


Proof. First of all, we consider the term τ(p)− τp,0 which should contributeto the error term r. From the C0-norm estimate on S1

α ×M for τ(p)− τp,0in the above Lemma 36, we have∫

V−(q)

(eλfα(τ(p)− τp,0)

)≤∫V−(q)

eλ(fα−fα(q)) supV−(q)

(eλfα(q)(τ(p)− τp,0)

)≤ DRe

(fα(q)−AR)λ,

using the assumption that ρ([α], g) < ∞ in Definition 18. By choosing R

large enough, we can absorb the term efα(q)λ to get a decay of O(e−cRλ)(notice that we fix once for all a lifting q for all q). Notice that AR andhence cR can be arbitary large by choosing large enough R.

The next step is to estimate the term integration of the term τp,0 by

estimating the integral of each eλfαt−k · (χτp,0) on V−(q). Since (1− χ)τp,0has a decay of order OR(e−λAR) similar to the term τ(p) − τp,0, the same

argument as above shows that the term∑

k∈Z e−ikα(t)µt−k · ((1 − χ)τp,0)

will contribute an error term of order O(e−cRλ). Therefore we can simplyestimate the integral for t−k · (τp,0).

Let us consider the case q 6= p with deg(p) = deg(q). We will use theequality ∫

V−(q)eλfα(t−k · τp,0) =

∫V−(tk·q)

eλ(fα−kα(t))τp,0,

to compute the integral. Notice that when k < 0 or tk · q ∈ Mα,R, theintegral will be 0 by our choice of τp,0 in Definition 34.

Otherwise, we must have k ≥ k0(R) in order to achieve tk · q /∈ Mα,R,with k0(R) can be chosen arbitrarily large by choosing large enough R.We recall that for fixed λ0 > ρ = ρ([α], g) defined in 18, we learn that∫V−(tk·q) e

λ0(fα−kα(t)) =∫V−(q) e

λ0fα ≤ D < ∞ for all q ∈ Z(α). We notice

that the term τp,0 has a decay of order OR,ε(e−λ(dα(p,·)+fα(p)−ε)) from (3.15).Combining the above and making use of the fact that dα(p, x) + fα(p) ≥fα(x), we obtain

|∫V−(tk·q)

eλ(fα−kα(t))τp,0| = |∫V−(tk·q)

eλ0(fα−kα(t))e(λ−λ0)(fα−kα(t))τp,0|

≤ CR

∫V−(tk·q)∩Mα,R

eλ0(fα−kα(t))e(λ−λ0)(fα−kα(t))e−λ(fα−ε)

≤ CRe−kα(t)(λ−λ0)+λεe

−λ0(inf x∈Mα,R fα)

≤ CRe−kα(t)(λ−λ0)+λε.

From there, we can obtain

|∫V−(q)

eλfα(τp,0)| ≤ DRe−λcR ,

where cR can be chosen large enough by choosing R large enough and hencewe have k0(R) large enough.


For the case q = p, we compute the term∫V−(tk·p) e

λ(fα−kα(t))τp,0 instead.

For k 6= 0, the argument is the same as above which contribute to an errorof order O(e−cRλ). For k = 0, the integral will equal to 1 by our choice ofτp,0.

From the above lemma, we can construct the inverse of ψ using the ap-proximation τ when λ is large enough, of the form

(3.16) ψ = τ (I +∑l>0

rl),

where the remainder∑

l>0 rl has a decay order of O(e−cRλ) (possibly a dif-

ferent cR). Combining this with Lemma 36, we obtain the following Lemma.

Lemma 38. There is an cR which can be chosen arbitrarily large by choosinglarge enough cut off radius R, such that we have the estimate

‖(ψ(p)− τp,0)‖Cj(S1α×M) ≤ DR,je

−λcR ,

for some constant DR,j.

The next step is to obtain a detail estimate of ψ(p) in the fixed compactset K using the above formula terms by terms.

Proof of Lemma 32. To obtain a global estimate for ∇jψ(p), it suffices toobtain a L2 estimate for the term ∇jψ(p) on S1

α ×M . We use the Lemma38 and consider the required estimate for τp,0. We learn from the result foreigenforms in equaiton (1.39) in [9] that there is an exponential decay for τp,0of order OR,ε(e−λ(dα(p,·)+fα(p)−ε)), and we obtain estimate for the C0-normas

‖∇j τp,0‖C0(Mα) ≤ Dj,ReCλ.

Since we have τp,0 is compactly support on Mα which means that on thefundamental domain Mα,0 ⊂Mα, the Fourier series in (3.14) has only finitelymany non-trivial terms. Therefore we obtain the required estimate

‖∇jτp,0‖C0(S1α×M) ≤ Dj,Re

Cλ.

Proof of Lemma 33. We will simply consider the estimate in the fixed com-pact subsetK = f−1

α ([−(L−1), (L−1)]) and letAK = supp∈Z(α) supx∈Kdα(p, x)+

fα(p) + 1. Therefore any terms with a decay greater than e−λAK will beconsidered as error terms. We choose R in Lemma 38 large enough suchthat we have cR ≥ AK . By doing so we can prove the Lemma by showingthe corresponding estimate for τp,0.

Again it is enough to obtain the L2-norm estimate of the derivatives

∇jψ(p) since the result of Cj-norm can be obtain by using Sobolev embed-ding in K. Making us of Lemma 38 we consider the estimate of L2-normof the term∇j τp,0. From estimate for eigenforms in equaiton (1.39) in [9], we


learn that there is an exponential decay for∇j τp,0 of orderOj,R,ε(e−λ(dα(p,·)+fα(p)−ε)).

Passing this estimate to ∇jψ(p) using Lemma 38 we obtain the proof.

3.2.3. WKB approximation. In this section we briefly recall the result onWKB approximation for eigenforms from [9], and the WKB approximationfor Witten’s twisted Green operator from our earlier work [3]. Since WKBapproximation is a local approximation along a compact neighborhood of agradient line, we can directly apply it to the Witten Laplacian ∆α on Mα

regardless the fact that Mα being non-compact.Similar to the previous section, we fix a lifting p for each critical point p

to the cover Mα. We start with the approximation for ψ(p). Restricting ourattention to a small enough neighborhood W containing V−(p)∪ V−(p), the

above decay estimate of eigenforms ψ(p) can be improved from an error oforder Oε(eελ) to O(λ−∞) (this is a modification of Proposition 1.3 in [9]).

Lemma 39. There is a WKB approximation of the eigenform ψ(p) of theform

(3.17) eλfα(p)ψ(p) ∼ λdeg(p)

2 e−λdα(p,·)(ωp,0 + ωp,1λ−1/2 + . . . ),

in any compact subset K ⊂ W . More precisely, it means that for any com-pact K ⊂ W and N large enough, there exists λK,j,N,0 > 0 such that forλ > λK,j,N,0 we have(3.18)

‖eλdα(p,·)∇jeλfα(p)ψ(p)−λdeg(p)

2 e−λdα(p,·)(N∑i=0

ωE,iλ−i/2)‖2L2(K) ≤ CK,j,Nλ

−N−1+2j .

Remark 40. We notice that the form τp,0 satisfies the same estimate if τp,0is taken to be eigenform on f−1

α ([−R,R]) and K ⊂ f−1α ([−R,R]), using the

techinque of WKB approximation in [9]. The above lemma on ψ(p) can beproven using this known result in [9] on WKB approximation for τp,0’s andapplying Lemma 38.

Furthermore, the integral of the leading order term ωp,0 in the normaldirection to the stable submanifold V+(p) is computed in [9].

Lemma 41. Fixing any point x ∈ V+(p) and χ ≡ 1 around x compactlysupported in W , we take any closed submanifold (possibly with boundary)NV+,x(p) of W intersecting transversally with V−(p) at x as shown in Figure3.2.3. We have

λdeg(p)

2 e−λfα(p)

∫NV+,x(p)

e−λ(dα(p,·)−fα)χωq,0 = 1 +O(λ−1).

Similarily, we also have

λdeg(p)

2 eλfα(p)

‖ψ(p)‖2

∫NV−,y(p)

e−λ(dα(p,·)+fα)χ(∗ωq,0) = 1 +O(λ−1),


for any point y ∈ V−(p), with NV−,y(p) intersecting transversally with V−(p)at y.

Figure 1. Neighborhood W of p

Remark 42. Following the discussion in Section 2 in [9],We write g+p (x) =

dα(p, x)−fij(x)+fij(p), which is a nonnegative function with zero set V+(p)that is smooth and Bott-Morse in a neighborhood W of V+(p)∪V−(p). Sim-ilarly, we write g−p (x) = dα(p, x) + fij(x) − fij(p) which is a nonnegative

smooth Bott-Morse in W function with zero set V−(p).

Besides these estimate for eigenforms, we also have the WKB approxi-mation for the Witten’s twisted Green operator Gα, improving the previousLemma 31. We let γ(t) be a flow line of α∨ starts at γ(0) = xS ∈ Mα

and γ(T ) = xE ∈ Mα for a fixed T > 0. The following figure shows theneighborhoods of the corresponding end points of γ.

We consider an input form ζS defined in a neighborhood WS ⊂ Mα ofxS . Suppose we are given a WKB approximation of ζS in WS , which is anapproximation of ζS according to order of λ of the form

(3.19) ζS ∼ e−λψS (ωS,0 + ωS,1λ−1/2 + ωS,2λ

−1 + . . . ),

satisfying

‖eλψS∇j(ζS − e−λψS (

N∑i=0

ωS,iλ−i/2))‖2L∞(WS) ≤ Cj,Nλ

−N−1+2j ,

with Cj,N depending on j,N . We further assume that gS = ψS − fij is anonnegative Bott-Morse function in WS with zero set VS .


Figure 2. Neighborhoods of end points of γ

We consider the equation

(3.20) ∆αζE = (I − Pα)(d∗λ,α)(χSζS),

where χS is a cutoff function compactly supported in WS , and Pα is theorthogonal projection to eigenspaces associated to small eigenvalues on Mα.We have a WKB approximation of ζE as follows.

Lemma 43. For supp(χS) small enough, there is a WKB approximation ofζE in a small enough neighborhood WE of E, of the form

(3.21) ζE ∼ e−λψEλ−1/2(ωE,0 + ωE,1λ−1/2 + . . . ),

in the sense

‖eλψE∇jζE − e−λψEλ−1/2(

N∑i=0

ωE,iλ−i/2)‖2L2(K) ≤ Cj,Nλ

−N+2j .

Furthermore, the function gE := ψE − fα is a nonnegative function which isBott-Morse in WE with zero set VE = (

⋃−∞<t<+∞ σt(VS)) ∩WE which is

closed in WE (Here σt is the time t flow of α∨).

Moreover, the integral of the leading order term ωE,0 over local submani-fold N(VE)vE transversal to VE at vE , is related to that of ωS,0 over N(VS)vStransversal to VS as follows.

Lemma 44. Using same notations in lemma 43 and suppose χS and χE arecut off function supported in WS and WE respectively, suppose ωS,0(xS) ∈∧topN(VS)∗xS , then we have ωE,0(xE) ∈

∧topN(VE)∗xE and furthermore therelation between the integrals

(3.22)

∫N(VE)vE

e−λgEχEωE,0 = (

∫N(VS)vS

e−λgSχSωS,0)(1 +O(λ−1/2))

along the normal submanifolds.


4. proof

After giving the necessary Lemmas concerning a single 1-form α, we areready to give the proof of our main theorem 15. For simplicity, we only givethe proof for the case of m3(s) as we learn from [3] that the proof of highermk(s) is only more complicated in notations.

To begin with, we fixed a generic sequence (defined in Definition 10) of1-forms α0, α1, α2, α3, with a fixed choice of critical point p01, p12, p23, p03

such that pij is a critical point of αij = αj − αi. We may assume that thedegree of the critical points pij are related by

(4.1) deg(p01) + deg(p12) + deg(p23) = deg(p03) + 1,

otherwise both product m3(s) and mNM3 will be trivial. Futhermore, there

is two combinatorial 3-trees T1 and T2 as shown in the following figure.Since we can decompose both operators as m3(s) = mT1

3 (s) + mT23 (s) and

mNM3 = mNM,T1

3 + mNM,T23 , we fix a combinatorial tree, say T1, and only

have to prove the relation between mT13 (s) and mNM,T1

3 .

Figure 3. two different types of 3-trees

Notations 45. From now on, we will consider the operators dij := d +λαij and d∗ij := d∗ + λια∨ij with the associated modified Witten’s Lapla-

cian ∆ij := ∆αij mentioned in Remark 24, together with the correspondingtwisted Green’s operator Gij := Gαij and projection Pij := Pαij and the mod-ified Witten’s map Ψij and ψij introduced in Section 3.1.1. In this setting,the m3(s) defined on Ω∗M (C∗ij ×M,Lij)’s need to be modified accordingly,

namely using the homological perturbation lemma from [10] with the homo-topy operators Hij = d∗ijGij and projection Pij to small eigenforms insteadof those in Section 2.1.1.

We fix a lifting q’s for all critical points of the αij ’s to correspondingcover Mij := Mαij . Such a choice of lifting fixes a subset of homotopy class


PT1(~α, ~p) having incoming ends at p(i−1)i’s and outgoing end at p03. There

is a natural identification of PT (~α, ~p) = Z3× PT1(~α, ~p) given by moving thelifting of the incoming end points p(i−1)i’s by the deck transformation. TheTheorem 15 in this setting can be restated as the follows.

Theorem (m3(s) case for Theorem 15). We will have the asymptotic ex-pansion ∫

M(mT1

3 (s)(ψ(p23), ψ(p12), ψ(p01)) ∧ ∗ψ(p03)

‖ψ(p03)‖2)

=∑

β0∈PT1 (~α,~p)

∑γ∈Z3

(#(Mγ·β0(~α, ~p))e−~α(γ)s)(1 +Rγ(s)),

with |Rγ(s)| ≤ C0,γλ−1/2, for λ > λ0,γ with some λ0,γ. Here we have ~α(γ) =

α23(γ23) + α12(γ12) + α01(γ01).

First, we notice that we can treat ~α : Z3 → Q using the above formula for~α(γ) in the theorem. There is a smallest integer n ∈ Z+ such that n~α has

image in Z, and hence we can treat the expansion in e−~α(γ)s = e−~α(γ)(λ+iµ)

as Fourier series with basis eiµ/n on a S1~α := R/2πnZ. Fixing k ∈ Z and

consider the terms contributing to e−ks/n in the above expansion, we willprove that there is a λk,0 such that when λ > λk,0, we have

(4.2)

∫S1~α

eks/n( ∫

M(mT1

3 (s)(ψ(p23), ψ(p12), ψ(p01))) ∧ ∗ψ(p03)

‖ψ(p03)‖2)dµ

=∑

β0∈PT1 (~α,~p)

∑~α(γ)=k/n

(#(Mγ·β0(~α, ~p)))(1 +Rk(s)),

with |Rk(s)| ≤ C0,kλ−1/2.

The summation∑

~α(γ)=k/n(#(Mγ·β0(~α, ~p))) is essentially a finite sum

since fixing ~α(γ) = k/n is the same as fixing the area of a gradient tree andhence the finiteness follows from the discussion in Remark 12, meaning onlya finite number of γ’s with ~α(γ) = k/n will have non-trivial #(Mγ·β0(~α, ~p)).

The proof will consist of three steps. The first step is to cut off the

Fourier coefficients ψ(pij)’s of eigenforms ψ(pij)’s, Gij ’s of the Green’s op-

erator Gij ’s involved to f−1ij ([−R,R]) ⊂ Mij by choosing large enough R

and applying results from section 3.2. The second step is to further cut offthe integral in Equation (4.2) to neighborhoods of gradient flow trees, usingthe resolvent estimate and estimate for ψ(p) from Section 3.2. The last stepwill be applying the WKB approximation results in Section 3.2 to explicitlycompute the contribution associated to each gradient tree to the first order.

4.1. Apriori Estimates: cut off from infinity. For convenience, we willfix a lifting fij = fαij ’s on the H1 = (H1(M,Z)/Tor) covering π : M → Msuch that they satisfy fij+fjk = fik. Notice that each of the fij will descend


as a function on a Z-covering πij : Mij = Mαij → M . Similar to before,we will consider the modified Witten’s map Ψij : Ω∗M (C∗ij ×M,Lij)<1 →⊕p∈Z(αij)C∞(C∗ij) · p and its inverse ψij defined in Remark 24 instead.

Definition 46. For each element ϕ ∈ Ω∗M (C∗ij ×M,Lij), we can considerits Fourier series on πij : Mij →M given by

(4.3) π∗ij(ϕ) =∑k∈Z

e−ikαij(tij)µt−kij · ϕ,

where tij generates the Z-action for the covering Mij → M , and ϕ ∈L2(Mij). We will call ϕ the Fourier coefficient of ϕ.

Remark 47. In order to work with the wedge product which involve differentMorse-functions fik = fij + fjk, we will also work on the common coveringπijk : Mijk →M and identify Ω∗M (C∗ij ×M,Lij) with the space of invariantsections

ψ | γ−1 · ψ = eiαij(γ)µψ, ψ ∈ Ω∗M (C∗ij ×Mijk),

with γ ∈ H1/(Ker(αij)∩Ker(αjk)) acting on Mijk. The Witten’s operators∆ij = ∆αij and Gij = Gαij can be thought of lifting to the space of invariantsections as well.

Since we only interested in leading order term in λ−1 in the integral∫S1~α

eks/n( ∫

M(mT1

3 (s)(ψ(p23), ψ(p12), ψ(p01))) ∧ ∗ψ(p03)

‖ψ(p03)‖2)dµ,

we begin with replacing the eigenforms ψ(pij)’s with their approximationsτpij ,0 constructed in Definition 34, having compactly supported Fourier co-efficient τpij ,0. First of all, we notice that from the Lemma 32, the inputterms ψ(p(i−1)i)’s have a bound

‖∇jψ(p(i−1)i)‖C0(S1×M) ≤ DjeλC(i−1)i .

Recall that we have construct an approximation τ(p(i−1)i) to ψ(p(i−1)i) in

(3.16) in Section 3.2.2, with an error term of order O(e−cRλ). By choos-ing R large enough and using Lemma 38, we may simply consider the termmT1

3 (s)(τp23,0, τp12 , τp01) and τp03 instead. Therefore, what we have to con-sider is the term

mT13 (s)(τp23,0, τp12,0, τp01,0)∧

∗τp03,0

‖τp03,0‖2= H13(τp23,0∧τp12,0)∧τp01,0∧

∗τp03,0

‖τp03,0‖2.

Furthermore, each of the eigenform τpij ,0 can be lifted to the Z-coveringMij , together with the Fourier series decomposition

(4.4) τpij ,0 =∑k∈Z

e−ikαij(tij)µt−kij · (χij τpij ,0).


We consider the term τ13,S := τp23,0 ∧ τp12,0 upon lifting to the Z-coveringM13, together with its Fourier series decomposition

τ13,S =∑k∈Z

e−ikα13(t13)µt−k13 · τ13,S .

From the fact that both τp23,0 and τp12,0 has compact supports on M23 andM12 respectively, we have the following lemma.

Lemma 48. The form τ13,S has compact support on M13.

Proof. W.L.O.G., we may assume that α23 and α12 are linearly independent.We consider the lifting to the common covering M123 as in Remark 47 suchthat we have the lifting of the Fourier series decomposition on M123. We maychoose t23 ∈ Ker(α12) and t12 ∈ Ker(α23) such that there are m23,m12 ∈Z>0 with m23t23 = t23 ∈ H1/Ker(α23) and m12t12 = t12 ∈ H1/Ker(α12)respectively, where t12 and t23 are generators of the covering map M12 →Mand M23 →M respectively. We further let t13 be a primitive element in the(Z · t23 ⊕ Z · t12)/Ker(α13) with α13(t13) > 0. Notice that we have t13 =m13t13 ∈ H1/Ker(α13) for some m13 ∈ Z>0 with t13 being the generator ofthe covering transformation of M13 →M .

Writing tij = mijtij , we can regroup the Fourier series as

τpij ,0 =∑k∈Z

e−ikαij(tij)µt−kij · τpij ,0,

where

τpij ,0 =

mij−1∑l=0

e−ilαij(tij)µt−lij · (χij τpij ,0)

which is supported in f−1ij ([−L,L]) ⊂ M123 for large enough L. Therefore

we can write

τ13,S =∑k,l

e−iα13(kt23+lt12)µ(t−k23 · τp23,0) ∧ (t−l12 · τp12,0)

=∑k,l

e−iα13(kt23+lt12)µ(t−k23 t−l12) · (τp23,0 ∧ τp12,0)

=∑m∈Z

e−imα13(t13)µt−m13 ·( ∑kt23+lt12∈Ker(α13)

(t−k23 t−l12) · (τp23,0 ∧ τp12,0)

).

Finally, we notice that the term∑

kt23+lt12∈Ker(α13)(t−k23 t−l12) · (τp23,0 ∧ τp12,0)

has support in f−113 ([−2L, 2L]) (f13 on M123 is invariant under action of

Ker(α13)), which complete the proof.

Next we consider the term τ13,E := H13(τ13,S) = d∗13G13(τ13,S). To do

that, we consider the resolvent (∆13 − z)−1 acting on τ13,S and argumentthat the resolvent can be replaced by the approximation in Equation (3.9)as in the proof of Lemma 29. Similar to the proof of Lemma 29, we choose Land R and cut off functions χ and χ accordingly, such that τ13,S is supported


in f−113 ([−(L − 1), (L − 1)]). Assuming that c1 < |z| < c2, we consider the

difference

R13(z) =∑k∈Z

e−ikα13(t13)µt−k13 ·(((∆13 − z)−1 − χ(∆13,0 − z)−1χ)τ13,S

),

which descends to S113 ×M , and have the following lemma.

Lemma 49. There is a cR > 0 which can be arbitrary large when R largeenough, and λ0(R) > 0 such that when λ > λ0(R), we have

‖∇jMR13(z)‖C0(S113×M) ≤ CR,je−λcR ,

where ∇M refers to derivatives along M .

Proof. First we recall that we have the Equation (3.9), which the term

(∆α13 − z)−1(1 − χ)(τ13,S) = 0. Therefore, what we have to estimate isthe term

(∆13 − z)−1(1− χ)[∆, χ](∆13,0 − z)−1χ(τ13,S).

Similar to the proof of Lemma 29, due to the choice of supports of χ and χ,we have

‖∇j([∆, χ](∆13,0 − z)−1χu

)‖L2(M13) ≤ Cj,Re−λ(cR)‖u‖

W lj ,2(M13),

for any u supported in f−113 ([−(L − 1), (L − 1)]). Therefore we obtain the

estimate by passing to the norm on S113 ×M and using Sobolev embedding

as in the proof of Lemma 36.

As a result, we may replace the operator H13 by an approximated operatorH13,A := d∗13G13,A where

(4.5) G13,A(ϕ) =∑k∈Z

e−ikα13(t13)µt−k13 ·( ∮|z|=c

z−1(χ(z − ∆α13,0)−1χ)ϕdz)

is defined using Fourier series. Therefore, what we need to compute is theintegral

(4.6)

∫MH13,A(τp23,0 ∧ τp12,0) ∧ τp01,0 ∧

∗τp03,0

‖τp03,0‖2,

where the Fourier coefficients of each terms will be compactly supported.To conclude, we have prove the following lemma.

Lemma 50. There is a cR which can be arbitrary large depending on Rwhich is the size of cut off of the Fourier coefficients, such that if we con-struct the approximation τpij ,0 as in Definition 34 and H13,A as in Equation(4.5), then we have

(4.7)∣∣ ∫

M(mT1

3 (s)(ψ(p23), ψ(p12), ψ(p01))) ∧ ∗ψ(p03)

‖ψ(p03)‖2−∫

MH13,A(τp23,0 ∧ τp12,0) ∧ τp01,0 ∧

∗τp03,0

‖τp03,0‖2∣∣ ≤ CRe−cR

for some constant CR.


4.2. Apriori Estimates: localized to gradient trees. The next stepwill be further cutting off the above integral (4.6) using partition of unityon Mij ’s, and argue that essential contribution only supported in a neigh-borhood of a gradient flow tree of homotopy class parametrized by γ suchthat ~α(γ) = k/n. The fact that Fourier coefficients of each terms appearingin last line of equation (4.7) being compactly supported shown in previousSection 4.1 allows us to use finitely many partition of unity, which is takenas follows.

For each of the αij ’s, we consider the Z-covering πij : Mij → M . Foreach approximated eigenform τpij ,0, there is a Fourier series on the coveringπij : Mij → M given by (4.4). From the discussion in the previous Section

4.1, we notice that the Fourier coefficient τpij ,0 is supported in f−1ij ([−R,R])

for some fixed R which is fixed to be large enough. We take a partition of

unity %ijl l∈Iij of f−1ij ([−R,R]) on Mij with %ijl support in ball with radius

ε/2 containing a point xijl ∈ Mij , where ε to be chosen small enough. Wewill consider the cut off

(4.8) τpij ,%l =∑k∈Z

e−ikαij(tij)µt−kij · (%ijl χij τpij ,0)

of the Fourier series on Mij . We learn from the Lemma 33 and the corre-sponding statement for τpij ,0 from [9] that we have the decay estimate

(4.9) %ijl χij τpij ,0 = OR,ε,ε(e−λ(dij(pij ,xijl )+fij(pij)−ε−ε)).

For the internal edge labelled by 13, recall that we will be considering theoperator in Equation (4.5) with Fourier coefficient

G13,A(ϕ) :=

∮|z|=c

z−1(χ(z − ∆α13,0)−1χ)ϕdz,

where χ is supported in f−113 ([−R,R]). We take two partition of unity %13,S

l and %13,E

m of f−113 ([−R,R]) on M13, support in ball with radius ε/4 con-

taining x13,Sl and x13,E

m respectively, to cut off the Fourier coefficient as

%13,Em (d∗13,λG13,A)%13,S

l of the homotopy operator H13,A. We let

H13,%l,m(ϕ) =∑k∈Z

e−ikα13(t13)µt−k13 ·((%13,Em (d∗13,λG13,A)%13,S

l )(ϕ)).

Therefore we learn from Lemma 31 that we have

(4.10) ‖∇j((%13,Em (d∗13,λG13,A)%13,S

l )(u))‖

≤ Cj,R,ε,εe−λ(d13(x13,Sl ,x13,E

l )−ε−ε)‖u‖W lj ,2(supp(%13,S

l )).

Starting from now on, we fix a sequence of cut offs ~% = (%01, %12, %23, %03, %13,S , %13,E)as in the beginning of this Section 4.2 (we will omit the subscript l in %l since

they are fixed), and consider the operation mT13 (s)~% defined by applying the


cut off %i(i+1)’s to the inputs τpi(i+1),0’s, %03 to the output ∗τp03,0, %13,S and

%13,E to the homotopy operator H13 as above. There are totally finitelymany cut offs and we estimate them one by one.

In the tree T1 shown in Figure 4, there are two vertices v and vr wherevr is the root vertex connecting to the unique outgoing edge. We first focuson the wedge product τp23,% ∧ τp12,% which corresponding to the operationassociated to the vertex v. There are two cases, depending on whetherthe classes [α23] and [α12] are linearly independent or not. We considerthe situation that they are linearly independent and the treatment for thesecond case will be similar, or even simpler.

In the first case, there exist a common Z2 ∼= H1/(Ker(α23) ∩Ker(α12))covering π123 : M123 → M such that both f23, f12 and f13 = f12 + f23 canbe defined on M123. Upon lifting to M123, we may write

(4.11) τpij ,% =∑γ∈Z2

e−iαij(γ)µγ−1 · τpij ,%,

for ij = 23 or 12. Here we denote (%ijχij τpij ,0) by τpij ,% on Mij , and we

notice that τpij ,% is supported in ε/2-ball containing xij on Mij . We treat itas compactly supported form on M123 in the above equation (4.11) such thatit push forward to give (%ijχij τpij ,0) on Mij , and denote it also by τpij ,% byabusing of notations. We further take a compactly supported lifting of thecut off %13,S to M123 (which will also be denoted by %13,S on M123), whichupon pushing forward to M13 is supported in ε/4 ball centered at x13,S onM13.

By choosing small enough ε such that supports of %ij ’s and %13,S are smallenough, we can force that there is a unique (if any) γ12, γ23 ∈ Z2 such that%13,Sγ−1

23 · τp23,% ∧ γ−112 · τp12,% 6= 0 on M123 (non-trivial term exist only if the

images under the covering maps π13, π12 and π23 of supp(%13,S), supp(%12)and supp(%23) have non-empty intersection in M). Therefore we write

τ13,S,% = e−i(α12(γ12)+α23(γ23))µ∑γ∈Z2

e−iα13(γ)µγ−1·(%13,Sγ

−123 τp23,%∧γ−1

12 ·τp12,%

),

which can be descends to M13. Since we would like to extract the terme−i(α12(γ12)+α23(γ23))µ later on, we will denote the Fourier coefficients of theterm ∑

γ∈Z2

e−iα13(γ)µγ−1 ·(%13,Sγ−1

23 τp23,% ∧ γ−112 · τp12,%

)by τ13,S,% instead.

From the decay estimate in equation (4.9) and (4.10), we notice that thereis a exponential decay

(4.12) e−λ(d23(γ−123 p23,x23)+d12(γ−1

12 p12,x12)+f23(p23)+f12(p12)−2ε−2ε)

of the term τ13,S,% (here γ−123 p23 and γ−1

12 p12 are treated as points in M23

and M12 respectively), where we further require π12(x12) = π23(x23) for thepoints x12 ∈M12 and x23 ∈M23 under their covering maps π12 : M12 →M


and π23 : M23 → M respectively (Notice that we simply require xij and

supp(%ijl ) lying in the same ball of radius ε/2 at the beginning of this Section4.2, and therefore we are free to choose another point within the same ε/2-ball to achieve π12(x12) = π23(x23)).

Remark 51. The treatment for the case [α23] and [α12] being linearly inde-pendent is easier, because in that case M23 = M12 = M13 and we do not haveto further lift the terms to a common covering M123 for taking the product.

Next we consider the decay estimate when applying the cut off homotopyoperator H13,% to τp23,% ∧ τp12,%, giving a cut off τ13,E,% of τ13,E . Explicitly,we will be considering the term

τ13,E,% = e−i(α12(γ12)+α23(γ23))µ∑k∈Z

e−ikα13(t13)µt−k13 ·((%13,E(d∗13G13,A))(τ13,S,%)

)and write %13,E(d∗13G13,A))(τ13,S,%) = τ13,E,% on M13. From the Lemma 31,we notice that there is a decay

e−λ(d23(γ−123 p23,x23)+d12(γ−1

12 p12,x12)+d13(x13S ,x

13E )+f23(p23)+f12(p12)−3ε−4ε),

for the term τ13,E,%. Similar as before, we further require π13(x13S ) = π12(x12) =

π23(x23) in the above equation for simplicity.Finally, we take care of the wedge produce τ13,E,%∧τ01,%∧∗τ03,% associated

to the root vertex vr. Similarly, we assume that [α13] and [α01] are linearlyindependent and therefore we have a Z2 covering M013. Again by taking εsmall enough, we have unique (if any) γ13, γ01 ∈ Z2 such that we can write

(4.13) τ13,E,% ∧ τp01,% ∧ ∗τp03,% = e−i(α01(γ01)+α12(γ12)+α23(γ23)+α13(γ13))µ∑γ∈Z2

γ−1 ·(γ−1

13 · τ13,E,% ∧ γ−101 · τp01,% ∧ ∗τp03,%

)= e−i(α01(γ01)+α12(γ12+γ13)+α23(γ23+γ13))µ∑

γ∈Z2

γ−1 ·(γ−1

13 · τ13,E,% ∧ γ−101 · τp01,% ∧ ∗τp03,%

),

on a common Z2 covering M013 of M . Notice that here the term

(4.14)∑γ∈Z2

γ−1 ·(γ−1

13 · τ13,E,% ∧ γ−101 · τp01,% ∧ ∗τp03,%

)descend to M as a top differential form (with value in trivial line bundleover S1

~α×M) due to the fact the ∗τp03,0 takes value in the conjugate bundle

L03 over S103 ×M .

Using the Lemma 33 again, we learn that the term γ−113 τ13,E,%∧γ−1

01 ·τp01,%∧· ∗ τp03,%

‖τp03,0‖2will have an exponential decay

e−λ(~dT1(~x)+~f(~p)−5ε−6ε),


where

(4.15) ~dT1(~x) = d23(γ−123 p23, x

23) + d12(γ−112 p12, x

12)

+ d13(γ−113 · x

13S , x

13E ) + d01(γ−1

01 p01, x01) + d03(p03, x

03),

and~f(~p) = f23(p23) + f12(p12) + f01(p01)− f03(p03)

such that we further require π13(x13E ) = π01(x01) = π03(x03) under their

covering map as before.

From the generic assumption 10 and the properties of Agmon distancedij in Appendix 2 of [9], we will see that the following continuous function

in variable ~x satisfies

dT1(~x) + ~f(~p)− ~α(γ) ≥ 0,

where ~α(γ) = α23(γ23 + γ13) + α12(γ12 + γ13) + α01(γ01). The minimum is

attended if and only if when ~x being interior vertices of a gradient flow treeΓ with class γ · β0.

For each fixed γ, we recall that in Remark 12, there are only finitelymany gradient tree of the class γ · β0. For each gradient tree Γ, there isa unique lifting of its edges eΓ,ij to the Z-covering Mij , with end points(γ′23)−1p23, (γ

′12)−1p12, (γ

′01)−1p01, for some (γ′23, γ

′12, γ

′01) ∈ Z3 and p03 on

M23,M12,M01 and M03.From the above discussion, unless we find some such gradient tree Γ of

class γ · β0 with γ′23 = γ13γ23, γ′12 = γ23γ12, γ′01 = γ01, and γ−113 x

13S and x13

Elying in ball of small radius r centered at interior vertices xΓ,v and xΓ,vr of

some Γ on M13 respectively, we will have a lower bound by dT1(~x) + ~f(~p) +~α(γ) ≥ cr,γ > 0 by continuity. Therefore, we can fix a r which gives us alower bound cr,γ , we further choose our errors ε and ε small enough such that

error O(e5ελ) and O(e6ελ) can be absorbed into the decay e−λcr . Therefore,we conclude that

(4.16) es~α(γ)

∫M

(mT13 (s)~%(τp23,0, τp12,0, τp01,0)) ∧ ∗τp03,0

‖τp23,0‖2= O(e−cr,γλ),

for λ large enough if γ−113 x

13S and x13

E lying outside r-ball centered at interiorvertices xΓ,v and xΓ,vr of some Γ of class γ · β0.

Notice that the expression for τ13,E,% ∧ τp01,% ∧ ∗τp03,% in Equation (4.13)

consist of the term e−i(~α(γ))µ which is a (periodic)-function in µ. As we are

going to integrate the whole term against eks/n over S1~α, we notice that only

if ~α(γ) = k/n, we will possibly obtain a non-trivial contribution. Therefore,only those cut offs supported near interior vertices of gradient trees of type~α(γ) = k/n (again there are finitely many such trees from Remark 12) canpossibly contribute to non-constant term in the integral (4.7) as λ→∞.


As a conclusion for this subsection, we label the gradient trees having~α(γ) = k/n by Γll. For each of the gradient tree Γl, there is a uniquelifting of each edge eΓl,ij (recall that each edge is labeled by a pair of integerij) to Mij with outgoing point p03 on M03. We further choose a small enoughneighborhood WΓl,v of the corresponding lifting of the interior vertex xΓi,v

on Mij . Up to possibly regrouping the partition of unity ~%’s, we can assumethere is only one sequence of cut offs ~%Γl associated to each tree Γl. It ischosen such that %Γi,z ≡ 1 near xΓi,v on Mij , and having support in WΓl,v

(for z = (eij , v)). The following Figure 4.2 showing two gradient treeswith their corresponding neighborhoods of the interior vertices, ending indifferent liftings of the critical points pi(i+1).

Figure 4. Two gradient trees with neighborhoods of cut offs

We will denote the operation after applying the cut offs associated to Γlby mT1

3 (s)Γl . We have proven the following lemma in the subsection.


Lemma 52. There is a constant c > 0, such that we can write

(4.17)

∫S1~α

eks/n( ∫

M(mT1

3 (s)(ψ(p23), ψ(p12), ψ(p01))) ∧ ∗ψ(p03)

‖ψ(p03)‖2)dµ =

∑Γl

∫S1~α

eks/n( ∫

M(mT1

3 (s)~%Γl(τp23,0, τp12,0, τp01,0)) ∧ ∗τp03,0

‖τp23,0‖2)dµ+O(e−cλ)

=

∫S1~α

eks/n∑Γl

( ∫Mτ13,E,%Γl

∧ τp01,%Γl∧ ∗

τp03,%Γl

‖τp23,0‖2)dµ+O(e−cλ)

which cut off the integral to neighborhood of gradient flow trees Γl’s withhomotopy type ~α(γ) = k/n.

4.3. WKB approximation. In this section, we introduce the WKB methodwhich allows us to compute the explicitly the leading order contribution inmT1

3 (s). We fix a lifting of edges of gradient tree Γ of type T1 as in theprevious section, with outgoing end at p03 on M03, and with interior ver-tices xv and xvr on M13 mapping to xv and xvr on M . We continue to usethe notations from previous Section 4.2 by writing the incoming ends of Γas (γ13γ23)−1p23, (γ13γ12)−1p12, γ

−101 p01 on M23, M12, M01 respectively. We

take neighborhoods Wv and Wvr in M of xv and xvr respectively, with cutoff functions %’s as in Equation (4.17) chosen to be supported in Wv andWvr respectively.

Notice that we will be interested in computing the equation in the lastline of Equation (4.17). Recall that from Lemma 39, we may assume that

each of the τpij ,0 has a WKB approximation as (it holds for both ψ(pij) andτpij ,0)

γ−1τpij ,0 ∼ λdeg(pij)

2 e−λψij (ωij,0 + ωij,1λ−1/2 + . . . )

holds true in Wv for ij = 12, 23 and Wvr for ij = 01, 03, with γ = γ13γijfor ij = 12, 23, γ = γ01 for ij = 01 and γ = 0 for ij = 03. Here we writeψij(x) = dij(γ

−1pij , x) + fij(pij).Next we need to obtain a WKB approximation for the term τ13,E . We

apply lemma 43 to the Witten’s twisted operator G13, input form ζS =γ−1

13 ·(%13,Sγ−1

23 · τp23,% ∧ γ−112 · τp12,%

), where we have

τ13,S,% = e−i(α12(γ12)+α23(γ23))µ∑γ∈Z2

e−iα13(γ)µγ−1·(%13,Sγ−1

23 ·τp23,%∧γ−112 ·τp12,%

)as in the previous Section 4.2. Taking xS = xv and xE = xvr in M13, ψS =ψ23 +ψ12 while applying the lemma, we obtain the WKB approximation ina neighborhood of xvr as

γ−113 · τ13,E,% ∼ λ

deg(p23)+deg(p12)−12 e−λψ13(ω13,0 + ω13,1λ

−1/2 + . . . ).

Here we have ψE = ψ13 and ωE,i = ω13,i as in the lemma.


In order to compute∫S1

eks/n( ∫

Mτ13,E,%Γ

∧ τp01,%Γ ∧ ∗τp03,%Γ

‖τp23,0‖2)dµ

up to an error of order O(λ−1/2), we can simply replace the integrant byleading order term in its WKB approximation

(4.18) λdeg(q23)+deg(q12)+deg(q01)−1

2

∫M(%13,E

Γle−λψ13ω13,0) ∧ (%01

Γle−λψ01ω01,0)∧

∧ (λ−deg(q03)

2

%03Γle−λψ03 ∗ ω03,0

‖φ03‖2)

=1

‖φ03‖2

∫M(e−λ(ψ13+ψ01+ψ03)(%13,E

Γlω13,0) ∧ (%01

Γlω01,0) ∧ ∗(%03

Γlω03,0)).

We first take a look on the exponential decay factor e−λ(ψ13+ψ01+ψ03) inthe integral. We define g13, g+

01 and g−03 by

ψ13 = g13 + f13 + α23(γ23 + γ13) + α12(γ12 + γ13),

ψ01 = g+01 + f01 + α01(γ01),

ψ03 = g−03 − f03.

Therefore we have exponential decay being

e−λ(g13+g+01+g−03)e−λ~α(γ)

of the integrand. The term e−λ~α(γ) will be canceled upon integrating against

esk/n over S1~α. Therefore, we investigate the exponential decay term e−λ(g13+g+

01+g−03).

We recall in remark 42 that g+01 = g+

γ−101 p01

, g+(γ13γ12)−1p12

, g+(γ13γ23)−1p23

and g−03 = g−p03are Bott-Morse with absolute minimums on V+(γ−1

01 p01),

V+((γ13γ12)−1p12), V+((γ13γ23)−1p23) and V−(p03) respectively. Notice thatsince g+

(γ13γ12)−1p12and g+

(γ13γ23)−1p23intersect transversally near xv so we

have the sum g+(γ13γ12)−1p12

+ g+(γ13γ23)−1p23

being Bott-Morse with minimum

at their intersection near xv. Applying Lemma 43 with gS = g+(γ13γ12)−1p12

+

g+(γ13γ23)−1p23

we obtain gE = g13 will be a Bott-Morse in Wvr with absolute

minimum denoted by V13 = (⋃−∞<t<+∞ σt(V13,S)) ∩Wvr flowed out from

V13,S = V+((γ13γ12)−1p12)∩V+((γ13γ23)−1p23)), under the flow of α∨13 whichis denoted by σt. The following Figure illustrates the situation.

The generic assumption 10 of the sequence ~α indicates that xvr =

V13∩V+(γ−101 p01)∩V−(p03) transversally at xvr which means e−λ(g13+g+

01+g−03)

concentrating at xvr as λ → ∞. The leading order contribution dependsonly on the value of ω13,0 ∧ ω01,0 ∧ ∗ω03,0 at the point xvr from the Lemmaof semi-classical expansion appearing in [4], which will be computed usingLemma 41 as follows.


We use the normal bundle NV13 ⊕ NV+(γ−101 p01) ⊕ NV−(p03) at xvr to

parametrize a neighborhood of xvr . Making use of the Lemma of semi-classical expansion (e.g. see Lemma 51 in Section 4 of [3]), we can split theintegral as follows for computing leading order contribution. We have

1

‖ψ(p03)‖2

∫M(e−λ(g13+g+

01+g−03)(%13,EΓl

ω13,0) ∧ (%01Γlω01,0) ∧ ∗(%03

Γlω03,0))

= ±(

∫NV13,xvr

e−λg13%13,EΓl

ω13,0)(

∫NV+(γ−1

01 p01)xvr

e−λg+01%01

Γlω01,0) ·

(

∫NV−(p03)xvr

e−λg−03%03

Γl

∗ω03,0

‖ψ(p03)‖2)(1 +O(λ−1)),

where the sign depends on whether the orientations ofNV13⊕NV+(γ−101 p01)⊕

NV−(p03) and TM at the point xvr match or not. Applying Lemma 41 andLemma 44, we obtain that

1

‖ψ(p03)‖2

∫M(e−λ(g13+g+

01+g−03)(%13,EΓl

ω13,0)

∧ (%01Γlω01,0) ∧ ∗(%03

Γlω03,0)) = ±(1 +O(λ−1)),

and hence the result∫M

(mT13 (s)~%Γ

(ψ(p23), ψ(p12), ψ(p01)))∧ ∗ψ(p03)

‖ψ(p03)‖2= ±e−~α(γ)s(1+O(λ−1/2)),

where the± sign agree with the corresponding sign associated to the gradienttree Γ. This complete the proof of our main theorem 15 for m3(s). The prooffor mk(s) will be exactly the same but only more complicated in notations,readers may see [3] for details about situation for higher mk’s.

References

1. M. Abouzaid, Morse homology, tropical geometry, and homological mirror symmetryfor toric varieties, Geom. Topol. 10 (2006), 1097–1157.

2. D. Burghelea and S. Haller, On the topology and analysis of a closed one form. I(Novikov’s theory revisited.), Arxiv preprint math 0101043 (2001).


3. K.-L. Chan, N. C. Leung, and Z. N. Ma, Fukaya’s conjecture on Witten’s twisted A∞structures, preprint, arXiv:1401.5867.

4. M. Dimassi and J. Sjostrand, Spectral asymptotics in the semi-classical limit, LondonMathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999.

5. M. Farber, Exactness of the Novikov inequalities, Funktsional. Anal. i Prilozhen 19(1985), 49–59.

6. K. Fukaya, Morse homotopy, A∞-category, and Floer homologies, 18 (1993), 1–102.MR 1270931 (95e:57053)

7. , Multivalued Morse theory, asymptotic analysis and mirror symmetry, 73(2005), 205–278. MR 2131017 (2006a:53100)

8. M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones mathe-maticae 82 (1985), no. 2, 307–347.

9. B. Helffer and J. Sjostrand, Puits multiples en limite semi-classique IV - Etdue ducomplexe de Witten, Comm. in PDE 10 (1985), no. 3, 245–340.

10. M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations,(2001), 203–263. MR 1882331 (2003c:32025)

11. S. P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory,24 (1981), no. 2, 222–226.

12. , The Hamiltonian formalism and a multi-valued analogue of Morse theory,Russian Math. Surveys 37 (1982), no. 5, 1–56.

13. A. V. Pazhitnov, On the Novikov complex for rational Morse forms, 4 (1995), no. 2,297–338.

14. , Rationality and exponential growth properties of the boundary operators inthe Novikov complex, Mathematical Research Letters 3 (1996), 541 – 548.

15. D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205–295.16. A. A. Ranicki, Finite domination and Novikov rings, Topology 34 (1995), 619 – 632.17. D. Sullivan, Infinitesimal computations in topology, Publications Mathematiques de

l’IHES 47 (1977), no. 1, 269–331.18. E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4,

661–692 (1983). MR 683171 (84b:58111)19. W. Zhang, Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts

in Mathematics, vol. 4, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.MR 1864735 (2002m:58032)

Department of Mathematics, The Chinese University of Hong Kong, HongKong

Email address: [email protected]

archive.ymsc.tsinghua.edu.cnarchive.ymsc.tsinghua.edu.cn/.../65/9122-multivalued_morse_theory.… ·...

Documents