The real homo- topy type can be captured by homotopic pull back of the wedge product to Ω∗(M)0, giving a structure of A∞ algebra via the homological perturbation lemma in [12]

WITTEN’S TWISTED A_∞-STRUCTURES

KAILEUNG CHAN, NAICHUNG CONAN LEUNG AND ZIMING NIKOLAS MA

Abstract. Wedge product on deRham complex of a Riemannian man- ifoldMcan be pulled back toH^∗(M) via explicit homotopy, constructed using Green’s operator, to give higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type ofM. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya in [6].

1. Introduction

It is well known that the differential graded algebra (Ω^∗(M), d,∧) on a smooth manifold M determine real homotopy type of M (if π1(M) = 0), a simplified homotopic classification of manifolds founded by Quillen [13]

and Sullivan [14]. If M is compact oriented Riemannian manifold, Hodge decomposition of the Laplacian ∆ enables us to represent the cohomology of M by the finite dimensional kernel Ω^∗(M)₀⊂Ω^∗(M) of ∆. The real homo- topy type can be captured by homotopic pull back of the wedge product to Ω^∗(M)₀, giving a structure of A_∞ algebra via the homological perturbation lemma in [12].

On the other hand, equipping M with a Morse-Smale function f allows us to study the cohomology of the manifold by the Morse complex CM_f^∗, which is a finite dimensional vector space freely generated by critical points of f equipped with the Morse differential δ defined by counting gradient flow lines off. Higher product structures can be introduced to enhance the Morse complex to the MorseA_∞(pre)-category defined as in [1, 5], involving A_∞ products {m^{M orse}_k }k∈Z+ defined by counting gradient trees.

In the paper [6] by Fukaya, he conjectured that theA_∞product structures from these two constructions can be related to each others via technique of Witten deformation, a differential geometric approach suggested in an influential paper [15] by Witten to relate Hodge theory to Morse theory by deforming the exterior differential operatordwith

d_f :=e^−λfde^λf =d+λdf∧,

1

by a Morse functionf with large parameterλ∈R⁺. In this paper, we prove this conjecture by Fukaya.

This machinery plays an important role in understanding SYZ transfor- mation of open strings datum and providing a geometric explanation for Kontsevich’s Homological Mirror Symmetry (Abbrev. HMS) as pointed out by Fukaya in [6].

More precisely, given a Morse-Smale function f, we can define the Wit- ten’s twisted Laplace operator by

(1.1) ∆f :=dfd^∗_f+d^∗_fdf.

Witten argued that if we consider eigenvalues of the operator ∆_f lying inside an interval [0,1), the sum of corresponding eigensubspaces

Ω^∗(M, λ)_sm ⊂Ω^∗(M) could be identified with the Morse complexCM_f^∗ via a linear map

(1.2) ϕ= Φ⁻¹ :CM_f^∗ →Ω^∗(M, λ)_sm.

For any critical pointqoff, the eigenformϕ(q) will concentrate nearqwhen λis large enough. Furthermore, the Witten differentiald_f is also identified with the Morse differential m^{M orse}₁ via ϕ. The original proof can be found in [9, 10, 11] while readers may see [16] for a detailed introduction.

In order to incorporate the product structure, we are forced to consider more than one Morse function as the Leibniz rule associated to twisted differential is given by

d_g+h(α∧β) =dg(α)∧β+ (−1)^|α|α∧d_h(β).

This leads to the notation of the differential graded (dg) categoryDRλ(M), with objects being smooth functions on M. The corresponding morphism complex between two objectsf_i and f_j is given by the Witten twisted com- plex Ω^∗_ij(M, λ) = (Ω^∗(M), d_fij), where fij =fj −fi. When fij satisfies the Morse-Smale condition, we can define Ω^∗_ij(M, λ)sm and a homotopy retrac- tion P_ij : Ω^∗_ij(M, λ) →Ω^∗_ij(M, λ)_sm using explicit homotopy H_ij =d^∗_f

ijG_ij, where Gij is the twisted Green’s operator. We can pull back the wedge product via the homotopy, making use of homological perturbation lemma in [12], to give a Witten’s deformed A_∞ (pre)¹-category DRλ(M)sm with A_∞ structure{m_k(λ)}k∈Z⁺.

For instant, suppose we have smooth functions f₀, f₁, f₂ and f₃ such that their pairwise differences are Morse-Smale, with φij ∈ Ω^∗_ij(M, λ)sm, the higher product

m₃(λ) : Ω^∗₂₃(M, λ)_sm⊗Ω^∗₁₂(M, λ)_sm⊗Ω^∗₀₁(M, λ)_sm→Ω^∗₀₃(M, λ)_sm

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

is defined by

(1.3) m3(λ)(φ23, φ12, φ01)

=P03(H13(φ23∧φ12)∧φ01) +P03(φ23∧H02(φ12∧φ01)).

In general m_k(λ) will be given by a combinatorial formula involving sum- mation over directed planar trees with k inputs and 1 output, with wedge product ∧ being applied at vertices and the homotopy operator Hij being applied at internal edges.

Fukaya’s conjecture says that the A_∞ structure {mk(λ)}k∈Z+, defined using twisted Green’s operators, has leading order given by{m^{M orse}_k }k∈Z⁺, defined by counting gradient flow trees, via the isomorphism ϕ.

Conjecture (Fukaya [6]). For generic (see Definition 6) sequence of func- tions f⃗ = (f0, . . . , f_k), with corresponding sequence of critical points ⃗q = (q₀₁, q₁₂, . . . , q_(k−1)k), namely, q_ij is a critical point of f_ij, we have

(1.4) Φ(mk(λ)(ϕ(⃗q))) =m^{M orse}_k (⃗q) +O(λ^−1/2).

Theorem (Main Theorem). Fukaya’s conjecture is true.

As A_∞ relations of {mk(λ)}k∈Z+ are obvious from their algebraic con- structions while those of {m^{M orse}_k }k∈Z+ require studies for boundaries of moduli spaces of gradient flow trees (see e.g. [1, 5]), we obtain an alterna- tive proof for A_∞ relations of{m^{M orse}_k }k∈Z⁺ as an corollary.

The original proof in [9, 10, 11, 16] is exactly the case k = 1, involving detailed estimate of operator dij along gradient flow lines. Starting from k≥3, our theorem involves the semi-classical analysis of the Witten twisted Green operator which is not needed in the k= 1 case.

Our Main Theorem for k = 2 involves three functions f₀, f₁, f₂, hav- ing q₀₁, q₁₂, q₀₂ being critical points of f₀₁, f₁₂, f₀₂ respectively, and can be proven using the analytical techniques in [9, 11] as the Green’s operatorGij

does not appear in the definition of m₂(λ). We compute the leading order term in the matrix coefficients ofm2(λ), which is essentially the integral (1.5)

∫

M

m₂(λ)(ϕ(q₀₁), ϕ(q₁₂))∧ ∗ϕ(q₀₂)

∥ϕ(q02)∥².

First, we make use of the global a priori estimate of the form ϕ(q_ij) ∼ O(e^λρ(qij,·)) (lemma 17), withρ being the Agmon distance defined in defini- tion 14, to cut off the integrand to neighborhoods of gradient trees appeared in m^{M orse}₂ . After cutting off the integrand, we need to compute the lead- ing order contribution from each gradient tree. The WKB approximation (lemma 20) of the eigenforms ϕ(q_ij) is used to compute the leading order contribution of (1.5).

When k ≥ 3, what we need is an WKB approximation of G_ij along a gradient flow line of f_ij in §4. More precisely, we need to study the local

behaviour of the inhomogeneous Witten Laplacian equation of the form (1.6) ∆_ijζ_E =d^∗_ij(e^−λψSν)

along a gradient flow line segment of fij from xS (starting point) to xE

(ending point), and obtain an approximation ofζ_E of the form ζ_E ∼e^−λψEλ^1/2(ω_E,0+ω_E,1λ^−1/2+. . .).

The key step in our proof is to determineψ_E from ψ_S and detailed con- struction is given in §4. A naive guess ˜ψE(x) := infy(ψS(y) +ρ(y, x)) cap- tures the desired behaviors of ψ_E near x_E but is singular along a hyper- surfaceU_S containing x_S. Singularity of ˜ψ_E prohibits us to solve Equation (1.6) iteratively in order ofλ⁻¹.

In order to solve (1.6) iteratively, we consider the minimal configuration in variational problem associated to inf_y(ψ_S(y)+ρ(y, x)) and find that the point y is forced to lie on U_S, with a unique geodesic joining to x which realizes ρ(y, x), for those xclosed enough to x_E. This family of geodesics {γ_y}y∈US

gives a foliation of a neighborhood of the flow line segment. Therefore we can use ψE(γy(t)) = ψS(y) +t as an extension of ˜ψE across US and solve the Equation (1.6) iteratively.

The analytic results forGij (§4) can be used to give the proof for k= 3 case. The proof of the general case is similar to the k = 3 case, but with more involved combinatorics.

This paper consists of two parts. The first part involves the setup and definitions in§2 and the proof in§3 modulo technical analysis. The second part is a study of Witten twisted Green operator in §4 which is used in previous sections.

2. Setting

In this section, we introduce the definitions and notations we need and state our main theorem. We begin with recalling the definition of Morse category.

2.1. Morse category. The Morse categoryM orse(M) has the same class of objects being smooth functionsf :M →R, with the space of morphisms between two objects given by

Hom^∗_{M orse(M)}(fi, fj) =CM^∗(fij) = ⊕

q∈Crit(fij)

C·eq.

It is the Morse complex which is defined when fij satisfying the following Morse-Smale condition.

Definition 1. A Morse functionfij is said to satisfy the Morse-Smale con- dition if V_p⁺ and V_q⁻ intersecting transversally for any two critical points p̸=q of f_ij.

It is graded by the Morse index of corresponding critical pointq, which is the dimension of unstable submanifoldV_q⁻. The Morse categoryM orse(M) is anA_∞-category equipped with higher productsm^{M orse}_k for everyk∈Z+, or simply denoted by m_k, which are given by counting gradient flow trees.

To describe that, we first need some terminologies about directed trees.

2.1.1. Directed trees.

Definition 2. A trivalent directed k-leafed tree T means an embedded tree in R², together with the following data:

(1) a finite set of vertices V(T);

(2) a set of internal edges E(T);

(3) a set of k semi-infinite incoming edges E_in(T);

(4) a semi-infinite outgoing edge eout.

Every vertex is required to be trivalent, having two incoming edges and one outgoing edge.

For simplicity, we will call it a k-tree. They are identified up to orien- tation preserving continuous map of R² preserving the vertices and edges.

Therefore, the topological class for k-trees will be finite.

Given a k-tree, by fixing the anticlockwise orientation of R², we have cyclic ordering of all the semi-infinite edges. We can label connected com- ponents of R²\T by integers 0, . . . , k in anticlockwise ordering, inducing a labelling on edges such that edge e labelled with ij will be lying between components iand j with the unique normal to e pointing in component i.

The labelling can be fixed uniquely by requiring the outgoing edge to be labelled by 0k. For example, there are two different topological types for 3-tree, with corresponding labelings for their edges as shown in the following figure.

Figure 1. two different types of 3-trees

Notations 3. A pair (e, v), with e being an edge (either finite or semi- infinite) and v being an adjacent vertex, is called a flag. The unique vertex attached to the outgoing semi-infinite edge is called the root vertex.

For the purpose of Morse homology, we need the following notation of metric trees.

Definition 4. A metrick-treeT˜ is ak-tree together with a length function l:E(T)→(0,+∞).

Metrick-trees are identified up to homeomorphism preserving the length functions. The space of metrick-trees has finite number of components, with each component corresponding to a topological type T. The component corresponding to T, denoted by S(T), is a copy of (0,+∞)^|E(T)|, where

|E(T)|is the number of internal edges and equals tod−2. The spaceS(T) can be partially compactified to a manifold with corners (0,+∞]^|E(T)|, by allowing the length of internal edges going to be infinity. In particular, it has codimension-1 boundary

∂S(T) = ⨿

T=T^′⊔T^′′

S(T^′)× S(T^′′),

where the equation T =T^′⊔T^′′ means splitting the treeT intoT^′ and T^′′

at an internal edge.

2.1.2. Morse A_∞ structure. We are going to describe the productm_kof the Morse category. First of all, one may notice that the morphisms between two objects fi and fj is only defined when fij is Morse. Given a sequence f⃗= (f₀, . . . , f_k) such that all the differencef_ij’s are Morse, with a sequence of points ⃗q = (q01, . . . q_(k−1)k, q0k) such that qij is a critical point of fij, we have the following definition of gradient flow tree.

Definition 5. A gradient flow treeΓoff⃗with endpoints at⃗qis a continuous map f : ˜T → M such that it is a upward gradient flow lines of f_ij when restricted to the edge labelled ij, the incoming edge i(i+ 1) begins at the critical pointq_i(i+1) and the outgoing edge 0k ends at the critical point q_0k. We useM(f , ⃗⃗ q) to denote the moduli space of gradient trees (in the case k = 1, the moduli of gradient flow line of a single Morse function has an extra R symmetry given by translation in the domain. We will use this notation for the reduced moduli, that is the one after taking quotient byR).

It has a decomposition according to topological types M(f , ⃗⃗ q) =⨿

T

M(f , ⃗⃗ q)(T).

This space can be endowed with smooth manifold structure if we put generic assumption on the Morse sequence, which will be discibed as follows.

For an incoming critical pointq_i(i+1), with corresponding stable submanifold V_q⁺_i(i+1), we define a map

f_T,i(i+1) :V_q⁺_i(i+1)× S(T)→M.

Fixing a point x in V_q⁺

i(i+1) together with a metric tree ˜T, we need to de- termine a point in M. First, supposev is the vertex connected to the edge labelled i(i+ 1), there is a unique sequence of internal edges (e1, . . . , e_k−2) connecting v to the root vertex v_r. To determine the image of x under our function, we apply Morse gradient flow with respect to Morse function as- sociated to e_j’s for time l(e_j) to x consecutively according to the sequence (e1, . . . , ek−2).

The maps are then put together to give a map

(2.1) fT :V_q⁻_0k×V_q⁺_(k−1)k × · · · ×V_q⁺₀₁× S(T)→ ∏

k+1

M, where we use the embedding ι:V_q⁻_0k →M for the first component.

Definition 6. A Morse sequence f⃗ is said to be generic if the image offT

intersect transversally with the diagonal submanifold ∆∼=M ,→ M^k+1, for any sequence of critical point⃗q and any topological type T.

When the sequence is generic, the moduli spaceM(f , ⃗⃗ q) is of dimension dim_R(M(f , ⃗⃗ q)) = deg(q_0k)−

k−1

∑

i=0

deg(q_i(i+1)) +k−2,

where deg(q_ij) is the Morse index of the critical point. Therefore, we can definem^{M orse}_k , or simply denoted by mk, using the signed count #M(f , ⃗⃗ q) of points in dim_R(M(f , ⃗⃗ q)) when it is of dimension 0. In order to have a signed count, we have to get an orientation of the space M(f , ⃗⃗ q). We will come to that later in definition 33.

We now give the definition of the higher products in the Morse category.

Definition 7. Given a generic Morse sequence f⃗ with sequence of critical points ⃗q, we define

m_k:CM_k(k^∗ ₋₁₎⊗ · · · ⊗CM₀₁^∗ →CM_0k^∗ given by

(2.2) ⟨m_k(q_(k−1)k, . . . , q01), q_0k⟩= #M(f , ⃗⃗ q), when

deg(q_0k)−

k−1∑

i=0

deg(q_i(i+1)) +k−2 = 0.

Otherwise, the m_k is defined to be zero.

One may noticem^{M orse}_k can only be defined when f⃗is a Morse sequence satisfying the generic assumption in definition 6. The Morse category is indeed aA_∞pre-category instead of an honest category. We will not go into detail about the algebraic problem on getting an honest category from this structures. For details about this, readers may see [1, 5].

2.2. Witten’s twisted deRham category. Given a compact oriented Riemannian manifoldM, we can construct the deRham categoryDR_λ(M) depending onλ. Objects of the category are again smooth functions, while the space of morphisms between fi and fj is

Hom^∗_DR

λ(M)(fi, fj) = Ω^∗(M),

with differential d+λdf_ij∧, where f_ij := f_j−f_i. The composition of mor- phisms is defined to be the wedge product of differential forms on M. This composition is associative and hence the resulted category is a dg category.

We denote the complex corresponding to Hom^∗_DR

λ(M)(f_i, f_j) by Ω^∗_ij(M, λ) and the differential d+λdf_ij by d_ij.

To relateDR_λ(M) andM orse(M), we need to apply homological pertur- bation toDR_λ(M). Fixing two functionsf_i andf_j, we consider the Witten Laplacian

∆ij =dijd^∗_ij +d^∗_ijdij,

whered^∗_ij =d^∗+λι_∇fij. We denote the span of eigenspaces with eigenvalues contained in [0,1) by Ω^∗_ij(M, λ)_sm as before.

If the functionfijsatisfying the Morse-Smale assumption 1, it is proven by Laudenbach [4] that the closureVq⁺andVq⁻have a structure of submanifold with conical singularities. Using this result, one can define the following map as in [16]

Φ = Φij : Ω^∗_ij(M, λ)sm →CM^∗(fij) given by

(2.3) Φ(α) = ∑

p∈Crit(fij)

∫

Vp⁻

e^λfijα

which is an isomorphism identifyingd_ij with Morse differentialm₁ when λ large enough.

Remark 8. This identification gives a connection on the family of vector space Ω^∗_ij(M, λ)_sm parametrized byλ by declaring the basis e_p associated to critical point of fij to be flat. Equivalently, it is same as defining

∇^∂

∂λα(λ) =P_ij(e^−λfij ∂

∂λe^λfijα(λ)), for α(λ)∈Ω^∗_ij(M, λ)sm.

It is natural to ask whether the product structures of two categories are related as λ → ∞, and the answer is definite. The first observation is that the Witten’s approach indeed produces an A_∞ category, denoted by DR_λ(M)_sm, with A_∞ structure{m_k(λ)}k∈Z+. It has the same class of ob- jects as DR_λ(M). However, the space of morphisms between two objects f_i,f_j is taken to be Ω^∗_ij(M, λ)_sm, withm₁(λ) being the restriction ofd_ij to the eigenspace Ω^∗_ij(M, λ)_sm.

The natural way to definem2(λ) for any three objectsf0,f1 andf2 is the operation given by

Ω^∗₁₂(M, λ)sm⊗Ω^∗₀₁(M, λ)sm −−−−→∧ Ω^∗₀₂(M, λ) −−−−→^P02 Ω^∗₀₂(M, λ)sm, wherePij : Ω^∗_ij(M, λ)→Ω^∗_ij(M, λ)sm is the orthogonal projection.

Notice that m2(λ) is not associative, and we need a m3(λ) to record the non-associativity. To do this, let us consider the Green’s operator G⁰_ij corresponding to Witten Laplacian ∆ij. We let

(2.4) G_ij = (I−P_ij)G⁰_ij and

(2.5) H_ij =d^∗_ijG_ij.

Then Hij is a linear operator from Ω^∗_ij(M, λ) to Ω^∗−_ij ¹(M, λ) and we have dijHij +Hijdij =I−Pij.

Namely Ω^∗_ij(M, λ)sm is a homotopy retract of Ω^∗_ij(M, λ) with homotopy operatorHij. Suppose f0,f1,f2 and f3 are smooth functions onM and let φ_ij ∈Ω^∗_ij(M, λ)_sm, the higher product

m3(λ) : Ω^∗₂₃(M, λ)sm⊗Ω^∗₁₂(M, λ)sm⊗Ω^∗₀₁(M, λ)sm→Ω^∗₀₃(M, λ)sm

is defined by

(2.6) m₃(λ)(φ₂₃, φ₁₂, φ₀₁) =

P₀₃(H₁₃(φ₂₃∧φ₁₂)∧φ₀₁) +P₀₃(φ₂₃∧H₀₂(φ₁₂∧φ₀₁)).

In general, construction ofmk(λ) can be described usingk-tree. Fork≥2, we decomposem_k(λ) :=∑

T m^T_k(λ), whereT runs over all topological types of k-trees.

m^T_k(λ) : Ω^∗_(k−1)k(M, λ)_sm⊗ · · · ⊗Ω^∗₀₁(M, λ)_sm →Ω^∗_0k(M, λ)_sm is an operation defined along the directed treeT by

(1) applying wedge product∧ to each interior vertex;

(2) applying homotopy operator Hij to each internal edge labelledij;

(3) applying projectionP_0k to the outgoing semi-infinite edge.

The following graph shows the operation associated to the unique 2-tree.

Figure 2. The unique 2-tree and the corresponding assign- ment of operators for definingm₂(λ).

The higher products {mk(λ)}k∈Z+ satisfies the generalized associativ- ity relation which is the so called A_∞ relation. One may treat the A_∞ product as a pullback of the wedge product under the homotopy retract Pij : Ω^∗_ij(M, λ)→Ω^∗_ij(M, λ)sm. This proceed is called the homological per- turbation lemma. For details about this construction, readers may see [12].

As a result, we obtain an A_∞ pre-categoryDR_λ(M)_sm.

Finally, we restate our Main Theorem with the notations from this section.

Theorem 9 (Main Theorem). Given f0, . . . , f_k satisfying generic assump- tion 6, with q_ij ∈ CM^∗(f_ij) be corresponding critical points, there exist λ₀ > 0 and C₀ > 0, such that ϕ = Φ⁻¹ : CM^∗(f_ij) → Ω^∗_ij(M, λ)_sm are isomorphism for all i̸=j when λ < λ₀. Furthermore, then we have

Φ(m_k(λ)(ϕ(q_(k−1)k), . . . , ϕ(q₀₁)))

= m^{M orse}_k (q_(k−1)k, . . . , q01) +R(λ), with |R(λ)| ≤C₀λ^−1/2.

Remark 10. The constants C₀ and λ₀ depend on the functions f₀, . . . , f_k. In general, we cannot choose fixed constants that the above statement holds true for all m_k(λ) and all sequences of functions.

Remark 11. The relation with SYZ Mirror Symmetry is as follows. If we consider the cotangent bundle T^∗M of a manifold M which equips the canonical symplectic form ω_can, and take L_i = Γ_dfi to be the Lagrangian sections. Then a critical poin qij of fij can be identified with qij ∈Li tLj. Applying the theorem of Fukaya-Oh [7], the Morse A_∞ operation m^{M orse}_k is equivalent to Floer theoreticalA_∞operations counting holomorphic disks. In the simplest situation concerning (T^∗M, ωcan), the Witten’s twisted deRham category DR_λ(M)_sm is related to the Floer theory on (T^∗M, ω_can) via our Main Theorem 9 and Fukaya-Oh’s theorem. In more general situation, one expect that these correspondence will be part of the ingredients for realizing HMS geometrically.

3. Proof of Main Theorem

In the proof, we fix a generic sequence f⃗ of k+ 1 functions, with corre- sponding sequence of critical points ⃗q. First of all, we have

deg(m_k(λ)(ϕ(q_(k−1)k), . . . , ϕ(q₀₁)) =

k−1

∑

i=0

deg(q_i(i+1))−k+ 2,

so⟨m_k(λ)(ϕ(q_(k−1)k), . . . , ϕ(q01), ϕ(q_0k)⟩ is non-trivial only when the equal- ity

(3.1)

k−1

∑

i=0

deg(q_i(i+1))−k+ 2 = deg(q_0k)

holds, which is exactly the condition for m^{M orse}_k in the Morse category to be non-trivial. We will therefore assume condition (3.1) and consider the integral ∫

M

⟨mk(λ)(ϕ(q_(k−1)k), . . . , ϕ(q01)), ϕ(q_0k)

∥ϕ(q0k)∥²⟩volg.

Recall that each directed tree T gives an operation m^T_k(λ) and m_k(λ) =

∑

T m^T_k(λ) which is also the case in Morse category. Therefore, we just have to consider each m^T_k(λ) separately.

3.1. Results for a single Morse function. We start with stating the results on Witten deformation for a single Morse functionf_ij. These results come from [16] and [11], with a few modifications to fit our content. We will assume that the Morse function fij we are dealing with satisfy the Morse- Smale assumption 1.

Under this assumption, one can prove the following spectral gap in the twisted deRham complex.

Lemma 12. For each f_ij, there is λ₀ >0 and constants c, C >0 such that we have

Spec(∆ij)∩[ce^−cλ, Cλ^1/2) =∅, for λ > λ0.

Furthermore, we have the following theorem on Witten deformation on the level of chain complexes.

Theorem 13 ([11, 16]). The map Φ = Φij : Ω^∗_ij(M, λ)sm → CM_ij^∗ in equation (2.3)is a chain isomorphism for λlarge enough.

We will denote the inverse by ϕ=ϕ_ij and investigate the asymptotic be- haviour ofϕ(q) for a critical pointqof fij for understanding the asymptotic of the product structure. We will need the following Agmon distanceρ_ij for this purpose.

Definition 14. For a Morse functionf_ij, the Agmon distanceρ_ij, or simply denoted by ρ, is the distance function with respect to the degenerated Rie- mannian metric ⟨·,·⟩fij =|df_ij|²⟨·,·⟩, where ⟨·,·⟩ is the background metric.

Readers may see [8] for its basic properties. The Agmon distance is closely related to the Witten’s Laplacian, or more preciely the corresponding Green’s operator associated to it by the following lemma in [10].

Lemma 15. Let γ ⊂ C to be a subset whose distance from Spec(∆_ij) is bounded below by a constant. For any j ∈ Z+ and ϵ >0, there is kj ∈Z+

and λ₀ = λ₀(ϵ) > 0 such that for any two points x₀, y₀ ∈ M, there exist neighborhoods V and U (depending on ϵ) of x0 and y0 respectively, and C_j,ϵ>0 such that

(3.2) ∥∇^j((z−∆ij)⁻¹u)∥C⁰(V)≤Cj,ϵe^{−λ(ρij(x0,y0)−ϵ)}∥u∥_Wkj ,2(U),

for all λ > λ0 and u∈C_c⁰(U), where W^k,p refers to the Sobolev norm.

We will also need modified version of the resolvent estimate forGij, which can be obtained by applying the original resolvent estimate to the the for- mula

(3.3) Gij(u) =

I

γ

z⁻¹(z−∆ij)⁻¹u.

Lemma 16. For anyj∈Z+andϵ >0, there iskj ∈Z+andλ0 =λ0(ϵ)>0 such that for any two pointsx₀, y₀ ∈M, there exist neighborhoods V and U (depending on ϵ) of x0 and y0 respectively, and Cj,ϵ>0 such that

(3.4) ∥∇^j(G_iju)∥C⁰(V)≤C_j,ϵe^{−λ(ρij(x0,y0)−ϵ)}∥u∥_Wkj ,2

(U),

for all λ < λ0 and u∈C_c⁰(U), where W^k,p refers to the Sobolev norm.

For a critical point q of f_ij, ϕ(q), treated as the eigenform associated to the critical point, has certain exponential decay measured by the Agmon distance from the critical point q.

Lemma 17. For any ϵ, there exists λ₀ = λ₀(ϵ) >0 such that for λ > λ₀, we have

(3.5) ϕ(q) =Oϵ(e^{−λ(ψq(x)−ϵ)}),

and same estimate holds for the derivatives ofϕ_ij(q) as well. HereOϵ refers to the dependence of the constant on ϵ and ψq(x) =ρij(q, x) +fij(q).

Remark 18. We write g_q⁺=ψ_q−f_ij, which is a nonnegative function with zero setV_q⁺that is smooth and Bott-Morse in a neighborhoodW ofV_q⁺∪V_q⁻. Similarly, we writeg_q⁻=ψq+fij which is a nonnegative function with zero set V_q⁻ and is smooth and Bott-Morse in W.

In that case, we can write

ϕ(q) = Oϵ(e^{−λ(gq+−ϵ)}),

∗ϕ(q)/∥ϕ(q)∥² = Oϵ(e^{−λ(gq−−ϵ)}).

Furthermore, we notice that the normalized basisϕ(q)/∥ϕ(q)∥’s are almost orthonormal basis in the following sense.

Lemma 19. There is a C, c > 0 and λ0 such that when λ > λ0, we will have

⟨ ϕ(p)

∥ϕ(p)∥, ϕ(q)

∥ϕ(q)∥⟩=δ_pq+Ce^−cλ.

3.1.1. WKB approximation for eigenforms ϕ(q). Restricting our attention to a small enough neighborhood W containing V_q⁺∪V_q⁻, the above decay estimate of eigenformsϕ(q) from [11] can be improved from an error of order Oϵ(e^ϵλ) to O(λ^−∞).

Lemma 20. There is a WKB approximation of the eigenform ϕ(q) of the form

(3.6) ϕ(q)∼λ^deg(q)2 e^−λψq(ωq,0+ωq,1λ^−1/2+. . .).

Remark 21. The precise meaning of this WKB approximation is given in section 4.6. Roughly speaking, it is in the sense of C^∞ approximation in order of λ on every compact subset ofW.

Furthermore, the integral of the leading order term ω_q,0 in the normal direction to the stable submanifoldV_q⁺ is computed in [11].

Lemma 22. Fixing any point x ∈ V_q⁺ and χ ≡ 1 around x compactly supported in W, we take any closed submanifold (possibly with boundary) N V_q,x⁺ of W intersecting transversally with V_q⁺ atx. We have

λ^deg(q)2

∫

N Vq,x⁺

e^−λgq+χω_q,0= 1 +O(λ⁻¹).

Similarily, we also have λ^deg(q)2

∥ϕ_ij(q)∥²

∫

N Vq,x⁻

e^−λgq−χ∗ω_q,0 = 1 +O(λ⁻¹),

for any point x∈V_q⁻, withN V_q,x⁻ intersecting transversally with V_q⁻. So far we have been considering a fixed Morse functionf_ij. From now on, we will consider a fixed generic sequence f⃗with corresponding sequence of critical points ⃗q as in the beginning of section 3.

Notations 23. We useq_ij to denote a fixed critical point off_ij. The eigen- form ϕ(qij) associated to qij is abbreviated by ϕij.

3.2. Proof ofm₃. We will use the result in the previous section to localize the integral

(3.7)

∫

M

m_k(λ)(ϕ(q_(k−1)k), . . . , ϕ(q01))∧ ∗ϕ(q_0k)

∥ϕ(q_0k)∥²

to gradient flow trees, when the degree condition (3.1) holds. We begin with the m3(λ) case which involves less combinatorics to illustrate the analytic argument.

3.2.1. Apriori estimate for m₃(λ) case. There are two 3-leafed directed trees, which are denoted by T1 and T2. We simply consider m^T₃¹(λ) for T1which is the tree shown in figure 2.1.1 and relate this operation to count- ing gradient trees of typeT₁. T₁ has two interior vertices, which are denoted byvandvr as in the figure. According to the combinatorics ofT1, we define

⃗

ρ_T1 :M^|V(T1)|→R+ which is given by

⃗

ρ_T1(x_v, x_vr)

= ρ₁₃(x_v, x_vr) +ρ₀₁(x_vr, q₀₁) +ρ₁₂(x_v, q₁₂) +ρ₂₃(x_v, q₂₃) +ρ₀₃(x_vr, q₀₃).

Roughly speaking, it is the length of the geodesic tree of typeT1with interior vertices x_v, x_vr and end points of semi-infinite edgese_ij’s laying onq_ij’s as shown in the following figure.

Making use of the fact thatρ_ij(x, y)≥f_ij(x)−f_ij(y) with equality holds iffyis connected toxthrough a (generalized) flow line offij, we notice that

⃗

ρ_T1(x_v, x_vr)≥A=f₀₃(q₀₃)−f₀₁(q₀₁)−f₁₂(q₁₂)−f₂₃(q₂₃) and the equality holds if and only if (xv, xvr) are interior vertices of a gradient flow tree of the type T₁. Here we only have gradient flow trees instead of generalized gradient trees since we assume the sequence of Morse function f⃗ satisfying generic assumption 6.

From the lemma 17 and 15, we notice the integrand (3.8)

∫

M

m^T₃¹(ϕ23, ϕ12, ϕ01)∧ ∗ϕ03

∥ϕ03∥² =

∫

M

H13(ϕ23∧ϕ12)∧ϕ01∧ ∗ϕ03

∥ϕ03∥², can be controlled bye^{−λ(ρT1−A)} in the following sense.

More precisely, fixing two pointsxv, xvr ∈M andϵsmall enough, lemma 16 holds for operatorG₁₃and henceH₁₃withU andV being balls centering at xv and xvr (with respect to ρ13) of radius r1. If we have two cut off functions χand χ_r supported inB(x_v, r₁) andB(x_vr, r₁) respectively, then we have

∥χrH13(χϕ23∧ϕ12)∥L^∞ ≤Cϵe^{−λ(ψq23(xv)+ψq12(xv)+ρ13(xv,xvr)−2r1−3ϵ)} for those large enoughλ. Here the decay factorsψq23(xv) andψq12(xv) come from the a priori estimate in lemma 17 for the input forms ϕ₂₃ and ϕ₁₂ respectively, while the decay factor ρ₁₃(x_v, x_vr) comes from the resolvent estimate lemma 16. Combining with the decay estimates forϕ01and _∥^∗_ϕ^ϕ03

03∥², we obtain

∥χ_rH₁₃(χϕ₂₃∧ϕ₁₂)∧ϕ₀₁∧ ∗ϕ₀₃

∥ϕ₀₃∥²∥L^∞(M)≤C_ϵe^{−λ(⃗ρT1(xv,xvr)−A−4r1−5ϵ)}

Figure 3. Cut off of integral near gradient trees

where x_v, x_vr are the center of balls chosen for taking the cut off χ, χ_r as above.

We assume there are gradient trees Γ₁, . . .Γ_l of the typeT₁. For each tree Γ_i, we take open neighborhoods D_Γi,v and W_Γi,v of interiors vertices x_Γ,v withD_Γi,v⊂W_Γi,v, and similarlyD_Γi,vr andW_Γi,vr forx_Γ,vr. The following Figure 3.2.1 illustrates the situation.

Since ⃗ρ_T1 is a continuous function in (x_v, x_vr) attending minimum value A exactly on internal vertices (xΓ,v, xΓ,vr) of gradient trees Γi’s, we can assume there is a constant C such that ρ⃗T1 ≥ A+C in M^|V(T1)|\ ∪iDΓi, where DΓi = DΓi,v ×DΓi,vr. If B(⃗⃗ x, r1) = B(xv, r1)×B(xvr, r1) is away from the D_Γi, we will have

∥χ_rH₁₃(χϕ₂₃∧ϕ₁₂)∧ϕ₀₁∧ ∗ϕ₀₃

∥ϕ03∥²∥L^∞(M)≤C_ϵe^−λ(C2).

Therefore, we can take cut off functions χΓi,v, χΓi,vr associating to each tree Γ_i, with support in W_Γi,v, W_Γi,vr and equal to 1 on D_Γi,v, D_Γi,vr re- spectively, to get

∫

M

m^T₃¹(ϕ₂₃, ϕ₁₂, ϕ₀₁)∧ ∗ϕ₀₃

∥ϕ03∥²

= ∑

i

∫

M

{χ_Γi,vrH₁₃(χ_Γi,vϕ₂₃∧ϕ₁₂)∧ϕ₀₁∧ ∗ϕ₀₃

∥ϕ03∥²}+O(e^−λ(C2)).

This localizes the integral computingm^T₃¹ to gradient trees of typeT1. Notice that the neighborhood D_Γi and W_Γi can be chosen to be arbitrarily small in the previous argument.