Multivalued Morse theory, Asymptotic Analysis, and Mirror symmetry

Kenji Fukaya

Department of Mathematics Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto Japan. ^{∗}

March 1, 2002

Abstract

In this article, we describe a series of conjectures which, if proved, will imply various conclusions of mirror symmetry conjecture assuming the existence of dual torus fibration. We present various evidence of our conjectures and prove some parts of the conjecture.

Contents

1 Semi-flat Mirror symmetry - review - 5 1.1 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Complex structure . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Kähler and Calabi-Yau structures . . . . . . . . . . . . . . . 7 1.4 Mirror symmetry without quantum effect . . . . . . . . . . 9

2 A model - pseudoholomorphic curve - 11 2.1 Multi-valued function . . . . . . . . . . . . . . . . . . . . . 11 2.2 Free field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Interaction and/or Scattering . . . . . . . . . . . . . . . . . 13 2.4 Singular locus. . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Feynman diagram description of pseudoholomorphic disks

CONTENTS 2

- closed string version - . . . . . . . . . . . . . . . . . . . . . 20 3 Including Lagrangian submanifolds 23

∗Partially supported partially by Grant in-Aid for Scientific Research 13852001 and by NSF Grant DMS 9729992.

3.1 Multi-valued function . . . . . . . . . . . . . . . . . . . . . 23 3.2 Free field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Open string vs Open string scattering - review - . . . . . . . 26 3.4 Open string vs Closed string scattering. . . . . . . . . . . . 28 3.5 Caustics, elementary catastrophe, and bifurcation of pseu-

doholomorphic disks. . . . . . . . . . . . . . . . . . . . . . . 29 3.6 Feynman diagram description of pseudoholomorphic disks

- open string version - . . . . . . . . . . . . . . . . . . . . . 34

4 B model - asymptotic analysis - 37 4.1 Free Field - Fourier expansion of complex structure - …. 38 4.2 Super symmetry and Morse theory, revisited. . . . . . . . . 40 4.3 Quantum effect to complex structure - a plan - . . . . . . . 46

5 Asymptotic analysis on mirror bundle 50 5.1 Semiflat homological mirror symmetry - review - . . . . . . 50 5.2 Free field - the case of vector bundle - . . . . . . . . . . . . 53 5.3 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4 Monodromy around caustics and orientation twist. . . . . . 61 5.5 Feynman diagram description of holomorphic structure of

vector bundles. . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.6 Homoloical mirror symmetry . . . . . . . . . . . . . . . . . 68

Introduction

In this paper we are going to discuss how various (many of) conclusions of mirror symmetry are expected to follow in case mirror manifold is obtained as a dual torus fibration, as was proposed by Strominger-Yau-Zaslow [72].

We start with a pair of manifolds M,M^{∨} which are compactifications of dual torus fibrations M_{0} → B_{0}, M_{0}^{∨} → B_{0}, whose fibers are Lagrangian submanifolds. In the case when they are globally a fiber bundle (with torus fiber) then we can construct complex structure on M^{∨} from symplectic structure on M easily. However it seems that only a torus is an example where such fiber bundle structure exists globally. So to study the case other than torus, we necessary need to include singular fibers. The symplectic structure of M_{0} is extended to the singular fiber but the complex structure of M_{0}^{∨} does not extend. We need to include “quantum correction” or “instanton correction” to the complex structure on M_{0}^{∨} so that it will be extended to the singular fibers. Such “instanton correction” is expected to be described by the (pseudo)holomorphic curves in M.

This story then is expected to imply the most famous conclusion of mirror symmetry, that is the coincidence of Gromov-Witten potential and Yukawa coupling, since the former is a generating function of the number of holomorphic disks and the later is obtained from the “quantum effect” to the complex structures on M_{0}^{∨}.

In this article, we study asymptotic behavior of the instanton effect. Namely we study it when the diameter of the fiber is small.

The main idea of this article is to relate both of them, one is pseudoholomorphic curve on M and the other is asymptotic behavior of the “quantum effect” to the complex structure on M_{0}^{∨}, to another object that is Morse homotopy of a multivalued function on B. We explain the procedure how to reduce both problems to the Morse theory on the base space in this article. However there are several points which the author can not make precise yet.

We also include the case when there are a Lagrangian submanifold L on M and a holomorphic vector bundle on M^{∨}. In this case, we are studying homological mirror symmetry conjecture due to Kontsevich [59]. (See [32, 33] for more precise statement on homological mirror symmetry.) We then also have a similar story as follows. In case when the restriction of L of the projection M → B is a covering map, then the story is very much similar to the case when there is no singular fiber. Namely we can construct a mirror bundle rather easily. (This part was worked out by the author in [32].) New phenomenon occurs when L → B is not a covering map somewhere. We then need to study caustics or Lagrangian singularity which is extensively studied in symplectic geometry. Let the caustics, S(L) ⊂ B, be the subset where L → B is not a covering map. In the mirror side, the mirror bundle E(L) with “classical” holomorphic structure can be defined on π_{M^{∨}}^{−1}(B\S(L)). This holomorphic structure does not extend to S(L), the caustics. Hence we need to include again “instanton effect”, which is described by a pseudoholomorphic disks in M which bounds L. Our idea in this article is to reduce both, the asymptotic behavior of holomorphic structure of the mirror bundle and holomorphic disks which bounds our Lagrangian submanifold L to the Morse theory of multivalued function on B.

Relation of Morse theory to pseudoholomorphic curve is classical and goes back to Floer’s invention [22] of Floer homology of Lagrangian intersection. In [36] the author with Y.G. Oh generalized Floer’s story. Namely Floer studied the case when there are two Lagrangian submanifolds (and one Morse function) and described the pseudoholomorphic strip in terms of gradient line. In [36] we discussed the case when there are several Lagrangian submanifolds and several Morse functions and described pseudoholomorphic disks in terms of Morse homotopy. In [36], the case when M is a cotangent bundle was studied. The result of [36] was used by the author in [32] to study homological mirror symmetry of complex torus. Recently Kontsevich-Soibelman [61] showed that the result of [36] can be generalized to the case when M is a torus T^{2n} (and the fiber is T^{n} instead of ℝ^{n}) and use it to give an alternative proof of a part of the results of [32]. (Kontsevich-Soibelman in [61] also proposed to use rigid analytic geometry to study mirror symmetry. We use in this paper asymptotic analysis, which is somewhat similar to their proposal.)

Both in [36] and [61], the case when there are no caustics are discussed. The new point in this article is that we include caustics. We also study closed string and singular fiber. Then we study multivalued function on B, where in [36] we studied several functions simultaneously. Namely instead of studying several functions we need to study several branches of a multivalued function. Caustics and singular locus plays a role of the branched locus and critical points of the function.

Relation of Morse theory to asymptotic analysis is also classical. Especially it was discussed in a famous paper by Witten [76]. In [76] Witten deformed De Rham complex “by hand” by using a Morse function. In this article we consider Dolbault operator \overline{∂} on M^{∨} and study asymptotic behavior of the solution of the Maurer-Cartan equation \overline{∂}B+B∧B = 0 when the fiber shrinks to a point. For this purpose we develop B as a Fourier series along fibers, (torus). The idea to use Fourier series along fibers to study mirror symmetry appeared in Arinkin-Polishchuk [5] and was used by several authors (see for example by [61]). Then, in each Fourier mode, the equation \overline{∂}B = 0 is equivalent to deformed De Rham operator. Here the deformation is by the same multivalued function which appears in the study of the pseudoholomorphic disks of the mirror manifold and is similar to Witten’s. Thus if we follow Witten’s idea, the solution of the linearized equation \overline{∂}B = 0 is described by the Morse theory of our multivalued function. We however need to study nonlinear term. The way we do so is actually the same as one due to Feynman himself. Namely we use Feynman diagram to include the effect of the nonlinear terms.

We will define the multivalued function which plays the central role in our story in §2.1 and §3.1. Roughly speaking a branch of the function at x ∈ B corresponds to an element of the fundamental group of the fiber M → B at x. The pseudoholomorphic sphere which is described by a branch is one which wraps a loop representing the element of the fundamental group corresponding to the branch.

On the other hand, a term of the Fourier expansion along the fiber of the dual fibration M^{∨} → B also corresponds to the fundamental group of the fiber of M → B. (This is a consequence of Pontrjyagin duality.)

Thus in our argument, the cycle over which the closed string wraps will turn out to be a Fourier mode in the mirror. This is exactly the idea of T-duality.

1 SEMI-FLAT MIRROR SYMMETRY - REVIEW - 6

1 Semi-flat Mirror symmetry - review -

The content of this section is not new. There are many articles describing them. We refer, for example, [72, 64, 61, 50, 3, 15, 47, 70, 32, 35].

Let B_{0} be an n dimensional manifold. We do not assume B_{0} is compact. Let π_{Λ} : Λ → B_{0} be a bundle whose fiber is ℤ^{n} and whose structure group is SL(n; ℤ). We put Λ_{x} = π_{Λ}^{−1}x. Let E = Λ ⊗_{ℤ} ℝ, and let π_{E} : E → B_{0} be the projection. E is a real n dimensional vector bundle over B_{0}. We put E_{x} = π_{E}^{−1}x. We may regard Λ as a subbundle of E.

Since the structure group of E is reduced to a discrete group SL(n; ℤ), it follows that E has a flat connection ∇^{GM}. Following [61], we call ∇^{GM} the Gauss-Manin connection. We put M_{0} = E/Λ. M_{0} is a total space of the fiber bundle π_{M_{0}} : M_{0} → B_{0}. We put π_{M_{0}}^{−1}(x) = F_{x} = E_{x}/Λ_{x}. The fiber F_{x} is diffeomorphic to an n dimensional torus T^{n}. The fiber bundle π_{M_{0}} has a flat connection induced from ∇^{GM}. We denote it also by ∇^{GM} and call it the Gauss-Manin connection.

The zero section s_{0} : B_{0} → E induces a section to π_{M_{0}}. We denote it by the same symbol s_{0} : B_{0} → M_{0}.

Our manifold M_{0} has locally a T^{n} action (whose orbits are the fibers). Because of the presence of the monodromy, this local action does not induce a global action of T^{n}. An appropriate language to describe this situation is called T -structure and was introduced by Cheeger-Gromov [17, 18], to study collapsing Riemannian manifolds. We recall its definition here since the story of mirror symmetry is actually related to collapsing Riemannian manifolds at this point.

Definition 1.1. Let M be a 2n dimensional manifold. A pure polarized T structure of dimension n on M consists of the following date :

(1) An open covering M = ∪U_{i}.

(2) An effective action of T^{n} on U_{i}. We assume that the dimension of every orbit is n. We write the action as (g, x) ↦ g ·_{i} x.

(3) If U_{i} ∩ U_{j} ≠ ∅ then U_{i} ∩ U_{j} is invariant of both of the T^{n} actions (the ones on U_{i} and on U_{j}).

(4) An isomorphism I_{i,j} : T^{n} → T^{n} such that g ·_{i} x = I_{i,j}(g) ·_{j} x, for x ∈ U_{i} ∩ U_{j} and g ∈ T^{n}.

Remark 1.1. The definition in Cheeger-Gromov [17, 18] uses more sophisticated language, the action of sheaf of groups. The definition here, (which is equivalent to Cheeger-Gromov’s) is taken from [25].

Obviously, our manifold M_{0} has a pure polarized T structure of dimension n. One can define T structure invariance of tensors in an obvious way.

1.1 Symplectic structure

Proposition 1.1. There exists a bijection between the following two objects (A) and (B).

(A) The ∼ equivalence class of symplectic form ω on M_{0} such that ω|_{F_{x}} ≡ 0 and ω|_{s_{0}(B_{0})} ≡ 0. Here ω ∼ ω′ if and only if there exists a diffeomorphism f : M_{0} → M_{0} such that π ∘ f = π and f^{∗}ω = ω′.

(B) An isomorphism φ : T B_{0} ≅ E^{∗} between tangent bundle of B_{0} and the dual bundle of E, such that the dual connection ∇^{GM∗} induces a torsion free connection on T B_{0} by φ.

Proof. We first remark the following lemma.

Lemma 1.1. Any symplectic form ω as in (A) is ∼ equivalent to a symplectic form which is invariant of the T structure on M_{0}. T structure invariant symplectic form is unique in each ∼ equivalence class.

The proof is an immediate consequence of the existence of action angle coordinate (see for example [6]) and is left to the reader. From now on we consider symplectic form which is invariant of our T structure only. Using Gauss-Manin connection, we have an isomorphism, (splitting to horizontal and vertical directions),

T_{s_{0}(x)}M_{0} ≅ T_{x}B_{0} ⊕ T_{s_{0}(x)}F_{x} ≅ T_{x}B_{0} ⊕ E_{x}. (1)

Let ω be a T structure invariant nondegenerate 2 form on M_{0} such that it vanishes on the fibers F_{x} and on s_{0}(B_{0}). The restriction of ω to T_{x}B_{0}⊗E_{x} induces a homomorphism φ_{ω} : T_{x}B_{0} → E_{x}^{∗}. Since ω is nondegenerate and since it vanishes on the fibers F_{x} and on s_{0}(B_{0}), it follows that φ_{ω} is an isomorphism.

Using the T structure invariance of ω, it is easy to see that φ_{ω} = φ_{ω′} implies ω = ω′. Moreover, any isomorphism φ : T_{x}B_{0} → E_{x}^{∗} is φ_{ω} for some ω. Thus, to complete the proof of Proposition 1.1, it suffices to show the following lemma.

Lemma 1.2. dω = 0 if and only if φ_{ω}^{∗}∇^{GM∗} is torsion free.

Proof. The problem is local with respect to B_{0}. So we take a small open set U on B_{0} and study π_{M_{0}}^{−1}U. We use Gauss-Manin connection to split π_{M_{0}}^{−1}U = U ×T^{n}. Let x^{i} be a coordinate of U and y^{i} be an affine coordinate of T^{n}. We may regard the basis of E_{x} as \frac{∂}{∂y^{i}} i = 1, ⋯, n. Hence dy^{i} i = 1, ⋯, n may be regarded as a basis of E^{∗}.

Now we put φ_{ω} ( \frac{∂}{∂x^{i}} ) = ∑ h_{ij}(x)dy^{j}. Then ω = ∑ h_{ij}(x)dx^{i} ∧ dy^{j}. Since ω is T structure invariant it follows that h_{ij} depends only on x ∈ B_{0} and is independent of y^{j}. It is straight forward calculation to check :

dω = 0 ⇔ ∀i∀j∀k \frac{∂h_{ij}}{∂x^{k}} = \frac{∂h_{kj}}{∂x^{i}}  ⇔  T^{φ_{ω}^{∗}∇^{GM∗}} = 0.

Here T^{φ_{ω}^{∗}∇^{GM∗}} is the torsion tensor of φ_{ω}^{∗}∇^{GM∗}. The proof of Lemma 1.2 and hence of Proposition 1.1 is now complete.

Let us consider the situation of Proposition 1.1. Then, we have a flat torsion free affine connection φ_{ω}^{∗}∇^{GM∗} on the tangent bundle. Hence there exists an affine coordinate y^{j} on B_{0}. Namely \frac{∂}{∂y^{j}} are parallel. Then the function h_{ij} are constant. Hence by linear change of the affine coordinate y^{j} we may assume h_{ij} = δ_{ij}. It follows that ω = ∑ dx^{i}∧dy^{i}. Namely x^{i}, y^{j} i, j = 1, ⋯, n is the Darboux coordinate of our symplectic manifold M_{0}. We call x^{j} a symplectic coordinate of B_{0} and x^{j},y^{j} a symplectic coordinate of M in this case.

1.2 Complex structure

Proposition 1.2. There exists a bijection between two objects (A) and (B).

(A) A complex structure J on M_{0} which is invariant of T structure and which satisfies J(T_{x}B_{0}) ⊆ T_{s(x)}F_{x}, J(T_{x}F_{x}) ⊆ T_{s(x)}B_{0}. (Here we use the decomposition (1) to regard T_{x}B_{0} ⊂ T_{s_{0}(x)}M_{0}.)

(B) An isomorphism φ : T B_{0} ≅ E between the tangent bundle of B_{0} and E such that φ^{∗}∇^{GM} is a torsion free connection on T B_{0}.

Remark 1.2. The condition J(T_{x}B) ⊆ T_{s_{0}(x)}F_{x}, J(T_{s_{0}(x)}F_{x}) ⊆ T_{x}B we assumed in Proposition 1.2 (A) looks rather restrictive. To remove it we need to consider B field in the mirror side (the symplectic structure side). We do not discuss it in this article. See [64, 15, 3, 32, 35].

Proof. Let J be an almost complex structure which satisfies Proposition 1.2 (B) except integrability. The restriction of J to T_{x}B_{0} ⊂ T_{s_{0}(x)}M_{0} defines a homomorphism φ_{J} : T B_{0} → E. It is easy to see that φ_{J} is an isomorphism. On the other hand, any φ as in Proposition 1.2 (B) is obtained as φ_{J}. Therefore, to prove Proposition 1.2, it suffices to show the following.

Lemma 1.3. J is integrable if and only if φ_{J}^{∗}∇^{GM} is torsion free.

Proof. Let x^{i}, y^{j} be local coordinates of M_{0} as in the proof of Lemma 1.2. We put J ( \frac{∂}{∂x^{i}} ) = ∑ g_{i}^{j}(x) \frac{∂}{∂y^{j}} . It follows that φ_{J} ( \frac{∂}{∂x^{i}} ) = ∑ g_{i}^{j}(x) \frac{∂}{∂y^{j}} . We can check by a simple calculation that

\overline{∂}_{J} ∘ \overline{∂}_{J} = 0 ⇔ ∀i∀j∀k \frac{∂g_{i}^{j}}{∂x^{k}} = \frac{∂g_{k}^{j}}{∂x^{i}} ⇔ T^{φ_{J}^{∗} ∇^{GM}} = 0.

Here \overline{∂}_{J} is the Dolbaut operator associated to the almost complex structure J. Lemma 1.3 and Proposition 1.2 follow.

We take an affine local coordinate x^{i} on B_{0} with respect to the flat affine connection φ^{∗}∇^{GM}. We may choose it so that J ( \frac{∂}{∂x^{i}} ) = \frac{∂}{∂y^{i}} . Then z^{i} = x^{i} + \sqrt{−1}y^{i} is a complex coordinate of M_{0}. We say the coordinate x^{i} a complex coordinate (of B_{0}) in this case.

1.3 Kähler and Calabi-Yau structures

Let B_{0} = ∪U_{a} be an open covering such that each of U_{a} is contractible.

Proposition 1.3. There exists a bijection between the following two objects (A) and (B) below.

(A) A Kähler structure J, ω on M_{0} such that J satisfies Proposition 1.2 (A) and that ω satisfies Proposition 1.1 (A).

(B) A pair of an isomorphism φ : T B_{0} ≅ E^{∗} and a functions h_{a} : U_{a} → ℝ. We assume that φ^{∗}∇^{GM} is a torsion free connection on T B_{0}. We also assume that g_{ij} = \frac{∂^{2}h_{a}}{∂x^{i}∂x^{j}} is a Riemannian metric g_{a} on U_{a}. Here x^{i} is an affine coordinate with respect to the flat affine connection φ^{∗}∇^{GM} and we assume that g_{a} is independent of a such that x ∈ U_{a}.

Proof. Let J be as in Proposition 1.2 (A) and ω be as in Proposition 1.1 (A). We assume that (M_{0}, ω, J) is Kähler. Namely we assume that g(V, W) = ω(V, JW) is a Riemannian metric. We take a symplectic coordinate x^{i} and y^{j} and put J ( \frac{∂}{∂x^{i}} ) = ∑ g_{ij}(x) \frac{∂}{∂y^{j}} . Note ω = ∑ dx^{i} ∧ dy^{i}. Hence g ( \frac{∂}{∂x^{i}} , \frac{∂}{∂x^{j}} ) = ω ( \frac{∂}{∂x^{i}} , J \frac{∂}{∂x^{j}} ) = g_{ij}. Hence ∑ g_{ij}(x)dx^{i}dx^{j} is a Riemannian metric on B_{0}.

As we established in §1.1, the integrability of J is equivalent to \frac{∂g_{ij}}{∂x^{k}} = \frac{∂g_{kj}}{∂x^{i}} . This condition is equivalent to d(∑_{j} g_{ij}dx^{j}) = 0. Hence, locally, there exist smooth functions h^{i} such that ∑_{j} g_{ij}dx^{j} = dh_{i}. Since g_{ij} = g_{ji}, it follows that \frac{∂h_{i}}{∂x^{j}} = \frac{∂h_{j}}{∂x^{i}} . Therefore, there exists, locally, a smooth function h such that h_{i} = \frac{∂h}{∂x^{i}} . Thus we have :

g_{ij} = \frac{∂^{2}h}{∂x^{i}∂x^{j}} . (2)

We thus show that an element of (A) gives an element of (B). The proof of the converse is similar.

Remark 1.3. Affine flat manifolds with Riemannian metric satisfying (2) is called a Hessian manifold, which has been studied for a long time in differential geometry. (According to [71] it appeared in the first place in [62]. See also [1, 19].) Its relation to mirror symmetry is observed by [50].

Let us calculate the complex coordinate of B_{0} in terms of h and symplectic coordinate. We put

x̌^{i} = \frac{∂h}{∂x^{i}} . (3)

Since g_{ij} is nondegenerate, it follows from (3) that x̌^{i} is a local coordinate. We have

J ( \frac{∂}{∂y^{j}} ) = ∑ g^{ij}(x) \frac{∂}{∂x^{i}} = ∑ g^{ij}(x) \frac{∂^{2}h}{∂x^{i}∂x^{k}} \frac{∂}{∂x̌^{k}} = \frac{∂}{∂x̌^{j}} .

Namely x̌^{j} is a complex coordinate.

We remark that (3) is a classical Legendre transformation with respect to the convex function h. (See [6].) Thus we obtain :

Proposition 1.4. In the situation of Proposition 1.3 complex coordinate x̌^{j} is a Legendre transformation of symplectic coordinate x^{j}.

As for Calabi-Yau structure we have the following.

Lemma 1.4. In the situation of Proposition 1.3, the Kähler metric induced from J and ω is Ricci flat if and only if

det ( \frac{∂^{2}h}{∂x^{i}∂x^{j}} ) = 1.

We omit the proof. See [19, 64, 61, 47].

1.4 Mirror symmetry without quantum effect

The discussion of §§1.1, 1.2 imply a simple version of mirror symmetry as follows. Let M_{0},E,Λ be as above. Let E^{∗} be the dual bundle to E. It has a dual (flat) connection ∇^{GM∗}. We put Λ_{x}^{∗} = {γ ∈ E_{x}^{∗}|γ(Λ_{x}) ⊆ ℤ}, which is the dual lattice. Its union Λ^{∗} = ∪_{x}Λ_{x}^{∗} ⊆ E^{∗} is a ℤ^{n} bundle over B_{0}, and Λ^{∗} ⊗_{ℝ} ℝ = E^{∗}. We put M_{0}^{∨} = E^{∗}/Λ^{∗}, and call M_{0}^{∨} a mirror of M_{0}.

Proposition 1.5. The set of the symplectic structures ω on M_{0} satisfying Proposition 1.1 (A) corresponds one to one to the set of complex structures J on M_{0}^{∨} satisfying Proposition 1.2 (A).

Proof. They both correspond to an isomorphism φ : T B_{0} → E such that the pull back φ^{∗}∇^{GM∗} is torsion free, by Propositions 1.1 and 1.2.

We remark moreover that the symplectic coordinate for M_{0} will become a complex coordinate for M_{0}^{∨}. We may exchange the role of M_{0} and M_{0}^{∨} in Proposition 1.5. Then a complex structure on M_{0} corresponds to a symplectic structure on M_{0}^{∨}. Based on the discussion of §1.4 we may consider the situation where M_{0} is both symplectic and complex (that is Kähler). And also the situation when M_{0} is Calabi-Yau manifold. We omit it. (See [64, 61, 47].)

Proposition 1.5 is a mirror symmetry in the sense it exchanges the role of symplectic and complex structures. To study more interesting part of mirror symmetry, we need to consider the case when M is not globally a fiber bundle but has singular fibers. Then the complex structure J is not invariant of T structure (see §4). We also need to take into account “quantum effect”. Studying these points is the main purpose of this article. The purpose of this section is to provide a background to study “quantum effect”, which is the most interesting part of the story of mirror symmetry.

In this article, we start with a manifold M which admits a fibration π_{M} : M → B. Such a fibration is conjectured to exist in the case when the complex structure of M is in a neighborhood of large complex structure limit. (However we need to allow M to have a singular fiber unless it is a torus.) Several authors [47, 61] proposed and/or partially proved an existence of such singular fibration. To prove existence of torus fibration, the theory of collapsing Riemannian manifolds may be useful. See [47, 61].

Actually the following was proved.

Theorem 1.1. ([23, 24]) Let M_{i} be a sequence of Riemannian manifolds. We assume that the sectional curvature of M_{i} are bounded in absolute value by a number independent of i. We also assume that M_{i} converges to B, another (smooth) Riemannian manifold, with respect to the Gromov-Hausdorff convergence ([42]). Then, for sufficiently large i, there exists a fiber bundle M_{i} → B. The fiber is an infra nilmanifold and the structure group is reduced to the group of affine diffeomorphisms.

Here a nilmanifold is a quotient of nilpotent Lie group by its cocompact discrete subgroup. It has an equivariant affine structure. Infra nilmanifold is its quotient by an affine action of finite group. Remark 1.4. Yamaguchi [78] proved the first half of the conclusion under milder assumption that the sectional curvature is uniformly bounded from below. Remark 1.5. If ϵ is the diameter of the fiber and if the volume of M_{i} is of order ϵ^{k} where k = dim M_{i} − dim B in the situation of Theorem 1.1, then it follows from [16] Corollary A1.8 that the fibers are quotient of torus by a finite group.

In our situation, our Riemannian manifolds are of bounded Ricci curvature but not of bounded sectional curvature. So Theorem 1.1 does not directly apply. In the situation of bounded Ricci curvature Bando-Kasue-Nakajima [12] (see also [2]) proved the following.

Theorem 1.2. Let M_{i} be a sequence of Einstein-Kähler manifolds of complex dimension 2. We assume that the Ricci curvature of M_{i} are 0 or ±1. We also assume that the volume of M_{i} are greater that ϵ > 0 independent of i and the diameter of M_{i} are independent of i. Then the limit of M_{i} with respect to the Gromov-Hausdorff convergence is an orbifold M with A, D, E type singularities. Moreover M_{i} is diffeomorphic to the minimal resolution of M for large i.

Let (M, J_{i}) be a maximal degenerate family of Calabi-Yau manifolds. (See [65] for its definition.) Let us take the Ricci flat Kähler metric g_{i} on M whose existence was established by Yau’s proof of Calabi conjecture. We normalize the metric so that the diameter is 1. It follows from a famous result by Gromov ([42] Theorem 5.3) that there exists a subsequence which converges to some metric space. (However it is not know whether it converges without taking a subsequence.) In this situation, let us propose the following conjecture, which is a variant of one in [61, 47].

Conjecture 1.1. (M, g_{i}) converges to B in Gromov-Hausdorff convergence with the following property.

(1) B is homeomorphic to an n dimensional simplicial complex.

(2) There exists a codimension ≥ 2 subcomplex S(B) such that B\S(B) is a smooth Riemannian manifold.

(3) Let φ_{i} : (M, g_{i}) → B be ϵ_{i} Haudsorff approximations with lim_{i→∞} ϵ_{i} = 0, in the sense of [25] Definition 1.1. Then, for each δ, the sectional curvature of (M, g_{i}) on φ_{i}^{−1}(B\U_{δ}(S(B))) is uniformly bounded in its absolute value for large i. Here U_{δ}(S(B)) is the δ neighborhood of S(B).

2 A MODEL - PSEUDOHOLOMORPHIC CURVE - 12

The result of Gross-Wilson [47] implies Conjecture 1.1 in the case of elliptic K3 surface.

We remark that Conjecture 1.1, if proved, together with Theorem 1.1 will imply and existence of fiber bundle structure over B\S(B). Remark 1.5 may imply that the fibers are tori.

However, to estimate the size of the points where the sectional curvature goes to infinity, seems to be an extremely hard question. So the author feels that it is too much difficult to attach this problem using the technology of global Riemannian geometry established at this stage. We will explain a gauge theory analogue of Conjecture 1.1 in §5 (Conjecture 5.5), which seems more attachable.

2 A model - pseudoholomorphic curve -

2.1 Multi-valued function

Now we introduce the main new player in our story, a multi-valued function.

Let π_{M_{0}} : M_{0} → B_{0}, M_{0} = E/Λ be as in §1.

Assumption 2.1. (1) π_{M} : M → B is a map from a 2n dimensional manifold M to an n dimensional manifold B. We assume that dim(B\B_{0}) ≤ dim B − 2.

(2) We may identify M_{0} and B_{0} with dense open subsets of M and B, respectively.

(3) We assume that π_{M}^{−1}(B_{0}) = M_{0} and that the restriction of π_{M} to M_{0} coincides with π_{M_{0}}.

(4) There exists a symplectic structure on M which extends one on M_{0} satisfying Proposition 1.1 (A).

We put S(B) = B\B_{0} and call it singular locus. Remark 2.1. In Assumption 2.1 (1), we did not say so much about the singular locus S(B) and how the map π_{M} can be singular there. Actually the author does not know which condition we need to put on S(B) and on singular fiber in order to make the argument below works. In that sense, Assumption 2.1 is a working hypothesis rather than a definition. We add more assumptions on singular fiber in §2.4. Study of special Lagrangian fibration may give a way to find a good condition to be assumed. There are much progress in this direction especially by M. Gross. See [44, 46, 45, 41, 79, 43, 70, 53, 54] for example.

Under this assumption we will define a multivalued function on B_{0} as follows.

Definition 2.1. We put

B̃_{0} = \{(x, [φ]) | \begin{matrix}x ∈ B_{0}, φ : (D^{2}, ∂D^{2}) → (M, F_{x}), \\ [φ] ∈ π_{2}(M, F_{x}), [∂φ] ≠ 0 ∈ π_{1}(F_{x}). \end{matrix}\}

There exists a covering B̃_{0} → B_{0} : (x, [φ]) ↦ x.

Definition 2.2. We define a function f on B̃_{0} by f(x, [φ]) = ∫_{D^{2}} φ^{∗}ω. Here ω is the symplectic form of M.

Since f is a function on a covering space of B_{0} we may regard it as a multivalued function on B_{0} and write it f.

We need the following variant of f in the next section. Let B̃_{0}^{+} = {(x, [γ])|x ∈ B_{0}, [γ] ∈ π_{1}(F_{x})}. We define a map B̃_{0} → B̃_{0}^{+} by (x, [φ]) ↦ (x, [φ|_{∂D^{2}}]).

Lemma 2.1. There exists a closed one form θ on B̃_{0}^{+} whose pull back to B̃_{0} is df.

Proof. We fix x_{0}. For each (x, [γ]) ∈ B̃_{0}^{+} we take a path ℓ : [0, 1] → B_{0} joining x to x_{0}. We lift it to ℓ̃: [0, 1] → B̃_{0}^{+} such that ℓ̃(0) = (x, [γ]). We put ℓ̃(τ) = (ℓ(τ), [γ_{τ}]) where γ_{τ} : S^{1} → F_{ℓ(τ)} depends continuously on τ. We put u(τ, t) = γ_{τ} (t) and f^{+}(x, [γ]) = ∫_{[0,1]×S^{1}} u^{∗}ω. We remark that f^{+} is independent on the choice of x_{0} and the path ℓ up to constant. Hence θ = df^{+} is well defined. It is easy to see that the pull back of θ is df.

Hereafter we write df in place of θ by abuse of notation. (f then is well defined as a function on B̃_{0}^{+} locally up to constant difference.)

2.2 Free field

By free fields in A model, we mean a pseudoholomorphic cylinder. The purpose of this section is to describe pseudoholomorphic cylinder in terms of the gradient vector fields of our multivalued function f. For this purpose, we first need to choose an almost complex structure on M_{0}. We choose one which is T structure invariant and satisfies Proposition 1.2 (A). (We however do not assume J is integrable.)

ω and J induce a Riemannian metric g on B_{0} as follows. We identify T_{x}B_{0} with the subspace of T_{s_{0}(x)}M_{0} consisting of horizontal vectors. (Here we use Gauss-Manin connection.) Then we put g(V, W) = ω(V, JW), for V, W ∈ T_{x}B_{0} ⊆ T_{s_{0}(x)}M. Using the symplectic coordinate x^{i}, y^{j} we have g = ∑ g_{ij}(x)dx^{i}dx^{j}, where J ( \frac{∂}{∂x^{i}} ) = ∑ g_{ij}(x) \frac{∂}{∂y^{j}} . We are going to study J holomorphic map u : [a, b] × S^{1} → M. We use τ for the [a, b] coordinate and t for S^{1} = ℝ/ℤ coordinate. We use the complex structure J ( \frac{∂}{∂τ} ) = \frac{∂}{∂t} on [a, b] × S^{1}. Our main result in this section is the following Proposition 2.1.

Let ℓ : [a, b] → B_{0} be a smooth curve and ℓ̃: [a, b] → B̃_{0}^{+} be its lift. We put ℓ̃(τ) = (ℓ(τ), [γ_{τ}]), where γ_{τ} ∈ π_{1}(F_{ℓ(τ)}) and γ_{τ} : S^{1} → F_{ℓ(τ)}. We may choose γ_{τ} so that it is an affine geodesic and γ_{τ} (0) is contained in the zero section s_{0}(B_{0}). (Here we regard S^{1} = ℝ/ℤ.) We put

uℓ̃(τ, t) = γ_{τ} (t), [a, b] × S^{1} → M_{0}. (4)

The Riemannian metric g on B_{0} induces a Riemannian metric on B̃_{0}^{+}. Hence our closed one form df determines a vector field gradf on B̃_{0}^{+}.

Proposition 2.1. uℓ̃ is J-holomorphic if and only if \frac{dℓ̃}{dτ}(τ) = gradf.

Proof. The problem is local. We use symplectic coordinate x^{i}, y^{j}. We may take γ_{τ_{0}} = (γ_{1}, ⋯, γ_{n}) ∈ ℤ^{n} with respect to y^{j} coordinate of T^{n}. (We identify F_{ℓ(τ)} ≅ T^{n} ≅ ℝ^{n}/ℤ^{n}.) We put ℓ(τ) = (ℓ_{1}(τ), ⋯, ℓ_{n}(τ)). Then \frac{∂uℓ̃}{∂t} = ∑_{j=1}^{n} γ_{j} \frac{∂}{∂y^{j}} . Hence, uℓ̃ is J-holomorphic if and only if

∑_{j=1}^{n} γ_{j} \frac{∂}{∂y^{j}} = J (∑_{i} \frac{dℓ_{i}}{dτ} \frac{∂}{∂x^{i}} ) = ∑_{i,j} g_{ij}(x) \frac{dℓ_{i}}{dτ} \frac{∂}{∂y^{j}} . (5)

On the other hand, we find easily that f(x^{1}, ⋯, x^{n}) = ∑ x^{i}γ_{i} + const . Hence gradf(x, [γ]) = ∑_{i,j} γ_{i}g^{ij} \frac{∂}{∂x^{j}} . The proposition follows.

2.3 Interaction and/or Scattering

In this subsection, we discuss mainly the case of n = 2. The case n > 2 is discussed briefly at the end of this subsection. Interaction in our sense occurs when two gradient lines (which appeared in Proposition 2.1) intersect each other. Hence it is a local phenomenon in B_{0}. So we consider the case M_{0} = ℝ^{2} × T^{2}. Let x^{1}, x^{2} be coordinate of ℝ^{2} and y^{1}, y^{2} be coordinate of T^{2}. We take it so that ω = dx^{1} ∧ dy^{1} + dx^{2} ∧ dy^{2}, J ( \frac{∂}{∂x^{i}} ) = \frac{∂}{∂y^{i}}. We may choose such a coordinate without losing generality since the problem is local in B_{0}.

Let γ = (γ_{1}, γ_{2}) ∈ ℤ^{2} = π_{1}(T^{2}), μ = (μ_{1}, μ_{2}) ∈ ℤ^{2} = π_{1}(T^{2}), and w = (w_{1}, w_{2}) ∈ T^{2}. We put

u_{γ,w}(τ, t) = (τγ_{1}, τγ_{2}, tγ_{1} + w_{1}, tγ_{2} + w_{2}) ∈ ℝ^{2} × T^{2}. (6)

u_{γ,w} : ℝ × S^{1} is J-holomorphic by Proposition 2.1. We define u_{μ,w} and u_{γ+μ,w} in a similar way.

We consider Σ = ℙ^{1}\{0, 1, ∞}. It has three ends which we write End_{0}, End_{1}, End_{∞}. We conformally identify End_{0}, End_{1} with (−∞, 0] × S^{1} and End_{∞} with [0, ∞) × S^{1}. We fix sufficiently large positive number τ_{0} and consider the following boundary conditions :

Condition 2.1. (1) There exist t_{0}, C and ϵ, such that for each (τ, t) ∈ End_{0} ≅ (−∞, 0] × S^{1} we have :

∥u(τ, t) − u_{γ,ω_{0}}(τ + τ_{0}, t − t_{0})∥ < C exp(ϵτ).

We denote this condition by u ∼_{0;τ_{0}} u_{γ,ω_{0}}.

(2) There exists t_{1}, C and ϵ, such that for each (τ, t) ∈ End_{1} ≅ (−∞, 0] × S^{1} we have :

∥u(τ, t) − u_{μ,ω_{1}}(τ + τ_{0}, t − t_{1})∥ < C exp(ϵτ).

We denote this condition by u ∼_{1;T} u_{μ,ω_{1}}.

(3) There exists τ_{∞}, t_{∞}, C and ϵ, such that for each (τ, t) ∈ End_{∞} ≅ [0, ∞) × S^{1} we have :

∥u(τ, t) − u_{μ,w_{∞}}(τ − τ_{∞}, t − t_{∞})∥ < C exp(−ϵτ).

We denote this condition by u ∼_{∞} u_{γ+μ,w_{∞}}.

In case (3) is satisfied for some ω_{∞}, we write u ∼_{∞} u_{γ+μ,∗}.

Proposition 2.2. Let τ_{0} be a sufficiently large positive number and we fix ω_{0}, ω_{1} ∈ T^{2}. We assume that the greatest common divisor of four numbers γ_{1}, γ_{2}, μ_{1}, μ_{2} is 1. Then, the number of J-holomorphic maps u : Σ → ℝ^{2} × T^{2}, such that u ∼_{0;τ_{0}} u_{γ,ω_{0}}, u ∼_{1;τ_{0}} u_{μ,ω_{1}}, u ∼_{∞} u_{γ+μ,∗} is |γ_{1}μ_{2} − γ_{2}μ_{1}|.

Proof. We define an isomorphism ℑ : ℝ^{2} × T^{2} → ℂ_{×}^{2} by

ℑ(x_{1}, x_{2}, y_{1}, y_{2}) = (exp(2π(x_{1} + \sqrt{−1}y_{1})), exp(2π(x_{2} + \sqrt{−1}y_{2}))).

(Here ℂ_{×} = ℂ\{0}.) We define u_{α,β} by

ℑ ∘ u_{α,β}(z) = (αz^{γ_{1}}(z − 1)^{μ_{1}}, βz^{γ_{2}}(z − 1)^{μ_{2}}). (7)

Lemma 2.2. There exists w_{0}, w_{1}, w_{∞} (depending on α, β) such that u_{α,β} ∼_{0} u_{γ,w_{0}}, u_{α,β} ∼_{1} u_{μ,w_{1}}, and u_{α,β} ∼_{∞} u_{γ+μ,w_{∞}}.

Moreover any J-holomorphic curve u such that u ∼_{0} u_{γ,w_{0}}, to u ∼_{1} u_{μ,w_{1}}, and to u ∼_{∞} u_{γ+μ,w_{∞}} for some w_{0}, w_{1}, w_{∞}, is equal to u_{α,β} for some α, β.

Proof. The first half of the lemma is obvious. Let u be as in the second half of the lemma. We consider the composition ℑ ∘ u and regard it as ℙ^{1}\{0, 1, ∞} → ℂ_{×}^{2}. Our boundary condition implies that it extends to a holomorphic map : ℙ^{1} → ℙ^{2}. Moreover the intersection number of it with the divisors z_{1} = 0, z_{2} = 0, and ℙ^{2}\ℂ^{2} are the same as u_{α,β}. The lemma follows easily.

Let us write w_{0}(α, β), w_{1}(α, β) for w_{0}, w_{1} determined in Lemma 2.2. Our assumption on the greatest common divisor of four numbers γ_{1},

γ_{2}, μ_{1}, μ_{2} implies that if (α, β) ≠ (α′, β′) then u_{α,β} ≠ u_{α′,β′}

We remark that changing absolute value of α, β corresponds to the shift along τ direction in each end ≅ (−∞, 0] × S^{1} (or [0, ∞) × S^{1}). Since we fixed τ_{0} in Condition 2.1, there is a unique choice of the absolute value of α, β. Hence it suffices to study the argument of α, β. We remark w_{0}(uα, vβ) = ( \frac{Arg u}{2π} , \frac{Arg v}{2π} )+w_{0}(α, β), w_{1}(uα, vβ) = ( \frac{Arg u}{2π} , \frac{Arg v}{2π} )+ w_{1}(α, β). (Here we regard w_{0}(α, β), w_{1}(α, β) ∈ T^{2} = ℝ^{2}/ℤ^{2}.) We remark also that we have a freedom to choose t_{0}, t_{1} in Condition 2.1. This means that u is asymptotic to u_{γ,w} at 0 if and only if it is asymptotic to u_{γ,w+rγ} at 0, where r is an arbitrary real number. Therefore u_{α,β} satisfies Condition 2.1 if and only if ( \frac{Arg α}{2π} , \frac{Arg β}{2π} ) ∈ γ ∩ μ. (Here γ, μ in the right hand side is the closed geodesic representing the homotopy class γ, μ and containing the origin.)

Therefore the number of solution is equal to the intersection number |γ ·μ| which is |γ_{1}μ_{2} −γ_{2}μ_{1}|. The proof of Proposition 2.2 is complete.

Remark 2.2. It is easy to see from the proof above that our solution u is transversal if we work out the Fredholm theory of our J holomorphic curve equation. Remark 2.3. From the proof above we find that our solution in Proposition 2.2 is asymptotic to u_{γ+μ,w} where w runs on γ ∩ μ.

Remark 2.4. Let k, m be natural numbers whose greatest common divisor is 1. We may consider the holomorphic cylinders u_{kγ} and u_{mμ} as initial data in Proposition 2.2. We then obtain solutions which is asymptotic to u_{kγ+mμ,w}. So starting from a pair of “closed strings”, interaction produces an infinite number of “closed strings”. This remark will become important when we compare Propositions 2.2 to corresponding results in B-model. (See §4.) Remark 2.5. We remark that in Proposition 2.2 we consider only the case when two strings interact. We can find a similar solution related to the interaction of several closed strings. Namely we may generalize (7) to

ℑ ∘ u(z) = (α ∏_{i=1}^{k}(z − a_{i})^{γ_{i}}, β ∏_{i=1}^{k}(z − b_{i})^{μ_{i}}) . (8)

here a_{i}, b_{i} are points on ℂ disjoint to each other. The holomorphic map (8) is parametrized by the configuration space on ℂP^{1} and describes interaction of k closed strings.

Now we proceed to the case when n > 2. We will study J holomorphic curve u : Σ → ℝ^{n} ×T^{n}. Let x^{i} be coordinates of ℝ^{n} and y^{i} be coordinates of T^{n}. We use complex structure J ( \frac{∂}{∂x^{i}} ) = \frac{∂}{∂y^{i}} and a symplectic structure ω = ∑ dx^{i} ∧ dy^{i}. Let γ = (γ_{1}, ⋯, γ_{n}) ∈ ℤ^{n} ≅ π_{1}T^{n}, μ = (μ_{1}, ⋯, μ_{n}) ∈ ℤ^{n} = π_{1}T^{n}, w = (w_{1}, ⋯, w_{n}) ∈ T^{n} = ℝ^{n}/ℤ^{n}. We put

u_{γ,w}(τ, t) = (τγ, tγ + w) ∈ ℝ^{n} × T^{n}.

u_{γ,w} : ℝ × S^{1} → ℝ^{n} × T^{n} is J holomorphic by Proposition 2.1.

We can define asymptotic boundary conditions, u ∼_{0;τ_{0}} u_{γ,w_{0}}, u ∼_{1;τ_{0}} u_{μ,w_{1}}, u ∼_{∞} u_{γ+μ,w_{∞}}, u ∼_{∞} u_{γ+μ,∗} in the same way as Condition 2.1, where γ, μ ∈ ℤ^{2}, ω_{0}, ω_{1}, ω_{∞} ∈ T^{n}, and τ_{0} is a sufficiently large positive number.

Let γ ∈ ℤ^{n}. It determines an S^{1} action on T^{n} by

t · [t_{1}, ⋯, t_{n}] = [t_{1} + tγ_{1}, ⋯, t_{n} + tγ_{n}]. (9)

We write the quotient space of this action by T^{n}/S_{γ}^{1}. (Note T^{n}/S_{γ}^{1} ≅ T^{n}). We also write S_{γ}^{1} to show that this S^{1} acts by (9). For a cycle A_{γ} in T^{n}/S_{γ}^{1}, we write its pull back to T^{n} by S_{γ}^{1}A_{γ}.

Now we set the boundary condition as follows. Let A_{γ}, A_{μ} be cycles in T^{n}/S_{γ}^{1}, T^{n}/S_{μ}^{1}. We assume that S_{γ}^{1}A_{γ} is transversal to S_{μ}^{1}A_{μ} and S_{γ}^{1}A_{γ} ∩ S_{μ}^{1}A_{μ} is transversal to orbits of S_{γ+μ}^{1}. We consider J holomorphic map u : ℙ^{1}\{0, 1, ∞} → ℝ^{n} × T^{n} such that

u ∼_{0;τ_{0}} u_{γ,w_{0}} for some w_{0} ∈ S_{γ}^{1}A_{γ},
u ∼_{1;τ_{0}} u_{μ,w_{1}} for some w_{1} ∈ S_{μ}^{1}A_{μ},
u ∼_{∞} u_{γ+μ,∗}.

(10)

We denote the moduli space of such u by M(γ, μ, A_{γ}, A_{μ}).

Proposition 2.3. M(γ, μ, A_{γ}, A_{μ}) is diffeomorphic to S_{γ}^{1}A_{γ} ∩ S_{μ}^{1}A_{μ}.

Let u ∈ M(γ, μ, A_{γ}, A_{μ}). We assume γ + μ is prime in ℤ^{n}. The set of ω_{∞} such that u ∼_{∞} u_{γ+μ,ω_{∞}} is an orbit of S_{γ+μ}^{1}. Hence we have an evaluation map ev_{∞} : M(γ, μ, A_{γ}, A_{μ}) → T^{n}/S_{γ+μ}^{1}.

Proposition 2.4. ev_{∞∗}[M(γ, μ, A_{γ}, A_{μ})] = π_{∗}[S_{γ}^{1}A_{γ}∩S_{μ}^{1}A_{μ}] ∈ H(T^{n}/S_{γ+μ}^{1}), where π : T^{n} → T^{n}/S_{γ+μ}^{1}.

The proof of Propositions 2.3 and 2.4 are straightforward analogue to one of Proposition 2.2 and is omitted.

2.4 Singular locus.

As we mentioned already, the author is yet unable to state correct condition to be assumed for the singular fiber. However we state a conjecture (in case 2n = 4) which is expected to hold under an appropriate condition on singular fiber. Also we can check it in the case of simplest singular fiber.

Let x_{0} ∈ S(B). We choose a sufficiently small open neighborhood U of x_{0}. Let x ∈ U\S(B). The fiber π_{M}^{−1}(x) is a torus T^{n} which we write F_{x}.

Definition 2.3. An element (x, [u]) ∈ B̃_{0} is said to be a vanishing cycle if we can take a representative u of [u] ∈ π_{2}(M, F_{x}) such that u(D^{2}) ⊂ π_{M}^{−1}U.

We say [u] ∈ π_{2}(M, F_{x}), [∂u] ∈ π_{1}(F_{x}) vanishing cycle also.

Example 2.1. Let π : M → B be an elliptic fibration in the sense of Kodaira [57]. (We do not need to assume M to be compact.) The fibers of π are complex submanifold. We assume M is hyperkähler. Then by hyperkähler twist (see [51]) we find another Kähler structure such that the fibers of π are Lagrangian submanifolds with respect to ω. (It is actually a special Lagrangian submanifold.) Thus any singular fiber in Kodaira’s list appears as a singular fiber of our fibration satisfying Assumption 2.1.

The simplest among them is type I singular fiber. It is an immersed S^{2} which intersects with itself transversally. The inverse image π^{−1}(U) is homotopy equivalent to the image of S^{2}. Let F_{x} = T^{2} be a nearby fiber. Then the vanishing cycles are kℓ ∈ ℤ^{2}, where ℓ is a cycle which “shrinks” to the singular point of the singular fiber.

The open neighborhood U in the base space B is identified with a disk whose origin is the singular locus. D^{2}\{0} has a flat affine structure induced by the Gauss-Manin connection. This affine manifold is described as follows. Let us consider a connected manifold Ω together with a map π : Ω → D^{2}\{0} such that π is a diffeomorphism on the domain {(x, y) ∈ D^{2} | y > (1 + ϵ)x} ∪ {(x, y) ∈ D^{2} | y < −ϵx} ∪ {(x, y) ∈ D^{2}|x < 0} and is two to one on {(x, y) ∈ D^{2} | (1 + ϵ)x > y > −ϵx}. We consider diffeomorphism (x, y) ↦ (x, y + x). from {(x, y) ∈ D^{2} | ϵx > y > −ϵx} to {(x, y) ∈ D^{2} | (1+ϵ)x > y > (1−ϵ)x}. We use it to glue two different sheets of Ω and get a flat affine manifold. It is isomorphic to U\S(B) ≅ B\{0} as an affine manifold. To see this it suffices to recall the monodromy of type I singular fiber is (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}).

Figure 2.2

We can also calculate easily our multivalued function corresponding to the vanishing cycles and find f([x, y], kℓ) = ckx. Note the right hand side is independent of the gluing map (x, y) ↦ (x, y + x), and hence defines a function on B_{0}. Here c > 0 depends on the normalization of our Kähler form.

Let us take n fold covering D^{2}\{0} → D^{2}\{0} and pull back our M, and then obtain a type I_{n} singular fiber. Our function f will become f([x, y], kℓ) = ck Re(x + \sqrt{−1}y)^{n}.

In §2.2, 2.3 we found a relation of the gradient line of our multivalued function to holomorphic annuli. Our conjecture in this section is a relation of the pseudoholomorphic disk representing vanishing cycle to the gradient line of f which starts at singular locus.

We take and fix a Riemannian metric g on B. It induces a Riemannian metric on B̃_{0}

Definition 2.4. We define unstable manifold Unst(U; f; g) of f as the set of (x, [u]) ∈ B̃_{0} such that (1) x ∈ U. (2) [u] is a vanishing cycle. (3) We have :

(11)  lim_{τ→−∞} π exp(τ gradf)(x) ∈ S(B)

Here gradf is a gradient vector field of f with respect to the metric g, and exp(τ gradf_{γ}) is the one parameter group of transformations generated by it. π : B̃_{0} → B_{0} is the projection.

We next define a moduli space of pseudoholomorphic disks which bound the fiber F_{x}. In this subsection, we only need the case when the disk is contained in π^{−1}(U) a small neighborhood of singular fiber. But, since we study more general case in the next subsection, we define it in general.

Definition 2.5. We consider a pair (x, u) of u : D^{2} → M and x ∈ B_{0} such that (1) u is J holomorphic. (2) u(∂D^{2}) ⊆ F_{x}. We take the quotient of the set of all such (x, u) by an obvious action of P SL(2; ℝ) ≅ Aut(D^{2}, j_{D^{2}}). (Here j_{D^{2}} is the standard complex structure of 2 disk D^{2}.) We denote the quotient space by M(M; J; D^{2}).

We define a map π : M(M; J; D^{2}) → B̃_{0} by (x, u) ↦ (x, [u]). (Here [u] is the homotopy class of u in π_{2}(M, F_{x}).

We here need the following results :

Theorem 2.1. There exists η : π_{0}(B̃_{0}) → ℤ and a space CM(M; J; D^{2}) containing M(M; J; D^{2}) such that :

(1) CM(M; J; D^{2}) has a Kuranishi structure with corners of dimension η(β) + 2n − 3.

(2) π : M(M; J; D^{2}) → B̃_{0} extends to π : CM(M; J; D^{2}) → B̃_{0}.

(3) Let E < ∞ and K ⊂ B_{0} be a compact set. We consider B̃_{0}(K, E) = f^{−1}((−∞, E])∩π^{−1}(K) ⊂ B̃_{0}. Then the inverse image π^{−1}(B̃_{0}(K, E)) ⊂ CM(M; J; D^{2}) is compact.

CM(M; J; D^{2}) is the moduli space of stable maps from a Riemann surface with boundary. (See [37] §3 for its definition, which is a minor modification of one in [60].)

The homomorphism η(β) is called the Maslov index. See [7] for its definition.

We do not define the notion Kuranishi structure with corners. (See [38].) The reader who are not familiar with Kuranishi structure may read the statement (1) as follows : “after generic perturbation, CM(M; J; D^{2}) will become a smooth manifold with corner and of dimension η(β) + 2n − 3”. Note that Theorem 2.1(2) is a consequence of Gromov compactness. Namely the moduli space of pseudoholomorphic disk with bounded energy and compact family of boundary value is compact.

The proof of Theorem 2.1 is almost the same as a similar result (Proposition 17.1 in the 2000 December preprint version of [37]). We only need to remark the following point. In the definition of our moduli space we allow x to move in B_{0}. In [37], we considered the case when x (or F_{x}) is fixed. In that case the dimension is η(β) + n − 3. The freedom to move x increases the dimension by n.

Assumption 2.2. We assume η : B̃_{0} → ℤ is zero.

Remark 2.6. It is known that the Maslov index is zero for special Lagrangian submanifold in Calabi-Yau manifold. Hence Assumption 2.2 holds for special Lagrangian fibration. Remark 2.7. In case when M is simply connected, we can easily show the following. If Maslov index of vanishing cycles are all zero then Assumption 2.2 holds. Hence Assumption 2.2 may be regarded as a condition on singular fibers.

By Assumption 2.2, the dimension of CM(M; J; D^{2}) is 2n − 3. Let us assume n = 2. Then CM(M; J; D^{2}) is one dimensional. Hence it is a curve in B̃_{0}, after appropriate perturbation. We remark that the dimension of a space with Kuranishi structure is, by definition, its virtual dimension (index of linearized operator). Hence we need to take a perturbation to obtain a space whose actual dimension is equal to the virtual dimension. Existence of such perturbation is proved in general in [38]. Hereafter we write M(M; J; D^{2}) for its perturbation representing its virtual fundamental chain constructed in [38] by abuse of notation.

Now we restrict ourselves to a neighborhood U of a singular fiber F_{x_{0}}. We consider Ũ_{vanish} = {(x, [u]) ∈ B̃_{0}|x ∈ U, [u] is a vanishing cycle.} We put M_{vanish}(U) = π^{−1}Ũ_{vanish} ∩ M(M; J; D^{2}). Now we can state our conjecture. In this conjecture we assume [u] ∈ π_{2}(M; F_{x}) is primitive. (Namely it is not of the form k[u′] with k > 1.) We need this assumption since the author does not know the correct way to count multiple covered disks. (In physics literature it is conjectured that d^{−2} is the contribution of d fold covered disks. However no mathematical argument to justify this number d^{−2} is known at the time of writing this article.)

Conjecture 2.1. If n = 2, then M_{vanish}(U) is isotopic to Unst(U; f; g) if we take appropriate choice of g, J and perturbation. In case the fiber is a multiple fiber with multiplicity d we need to take d copies of Unst(U; f; g).

Theorem 2.2. Conjecture 2.1 is true in case of type I_{n} singular fiber in Example 2.1.

Proof. As we discussed in Example 2.1, the submanifold Unst(U; f; g) is an axis x ≥ 0, y = 0 for γ = ℓ and x ≤ 0, y = 0 for γ = −ℓ. It is proved in [37] Chapter 7 that the same holds for M_{vanish}(U).

The case of type I_{n} singular fiber can be proved by taking n fold cover of the type I case.

Remark 2.8. In case n ≥ 3 we need to be more careful to state the conjecture, since then dim M_{vanish}(U) ≥ n. We need to cut it by using cycles in T^{n}, the fiber. Compare [35] §5, where the author explains how to use M_{vanish}(U) to construct an object of derived category of the category of coherent sheaves in the mirror manifold. The idea to cut moduli space by cycles is naturally related to Propositions 2.3, 2.4. However at the time of writing this article the author does not find yet the best way to generalize Conjecture 2.1 to higher dimension. So he leaves it to the future research.

We next show another evidence of Conjecture 2.1. But it is not strictly an example of Conjecture 2.1, since M is singular there.

Example 2.2. Let us consider T^{4} = ℂ^{2}/ℤ^{4} the quotient of quaternion by its lattice. It is a hyperkähler manifold. We divide it by {±1} to obtain M, which is an orbifold. (If we blow up M, it will become a K3 surface. But we do not blow it up here.) There is a projection π : M → T^{2}/{±1} = S^{2}. The general fibers are T^{2} and there are 4 singular fibers which are S^{2}. The fibers are complex submanifolds with respect to the Kähler structure. We take hyperkhäler twist and the fibers will become special Lagrangian submanifolds. Let us take a neighborhood of a singular locus p ∈ S^{2}. We may identify it with ℝ^{2}/{±1} where p = 0. All of the elements of π_{1}(T^{2}) = ℤ^{2} are vanishing cycles. We can calculate easily that f((x_{1}, x_{2}); (k, ℓ)) = c(kx_{1} + ℓx_{2}). Here (x_{1}, x_{2}) is the coordinate of ℝ^{2}. The multivalued function c(kx + ℓy) induces a multivalued function on ℝ^{2}/{±1}. The unstable manifolds are union of all lines of rational slope.

We can find pseudoholomorphic disks corresponding to each slope as follows. Our M in a neighborhood of our singular fiber is obtained by dividing ℝ^{2} × ℝ^{2} by G. Where 1 → ℤ^{2} → G → {±1} → 1. Here ℤ^{2} acts as translation of the second factor of ℝ^{2} × ℝ^{2} and {−1} acts by (x_{1}, x_{2}, y_{1}, y_{2}) ↦ (−x_{1}, −x_{2}, −y_{1}, −y_{2}). We consider φ(u, v) = (ku + δ_{1}, ℓv + δ_{2}, ku, ℓv) : ℂ → ℝ^{2} × ℝ^{2} where δ_{i} = 0 or 1/2. φ is a pseudoholomorphic map. It induces a map ℂ/G → M. It is easy to see that if we restrict {(u, v)||v| < r} then φ defines a pseudoholomorphic disk which bounds F_{(kr,ℓr)}. By changing the choices of δ_{i} we have 2 pseudoholomorphic disks. (There are 4 choices for δ_{i}. But each two of them define the same map.) Thus an analogue of Conjecture 2.1 in case when M = T^{2}/{±1} holds.

To have more evidence of Conjecture 2.1, we introduce its analogue by including Lagrangian submanifolds and studying its caustics, in the next section.

2.5 Feynman diagram description of pseudoholomorphic disks - closed string version -

Combining the pictures developed in §§2.1 ∼ 2.4, we obtain a description of the moduli space M(M; J; D^{2}). In this subsection, we only consider the case when n = 2, since we can formulate Conjecture 2.1 precisely only in that case. However the author believes that a similar idea works in higher dimension.

We consider π_{M} : M → B^{2} which satisfies Assumptions 2.1, 2.2. We consider the following data, Γ,l,ψ̃,m. We will explain later how it corresponds to J holomorphic disks.

Definition 2.6. A finite oriented graph Γ consists of the following date :

(1) A finite set Vertex(Γ), the set of vertices. (2) A finite set Edge(Γ), the set of edges. (3) Maps ∂_{source} : Edge(Γ) → Vertex(Γ), ∂_{target} : Edge(Γ) → Vertex(Γ).

We take copies of intervals [0, 1]_{e} corresponding to each element of e ∈ Edge(Γ) and copies of points v corresponding to each element v ∈ Vertex(Γ). We identify {0} ∈ [0, 1]_{e} with ∂_{source}(e) and {1} ∈ [0, 1]_{e} with ∂_{target}(e). We thus obtain a one dimensional complex, which we write |Γ|. We put |e| = [0, 1]_{e} ⊂ |γ|. For a vertex v we let |v| ∈ |γ| a point corresponding to it.

We say that Γ is a tree if |Γ| is connected and simply connected.

We consider oriented trees Γ which satisfy the following conditions.

Condition 2.2. (1) Vertex(Γ) is decomposed into a disjoint union of Vertex_{int}(Γ) and Vertex_{ext}(Γ). (2) If v ∈ Vertex_{int}(Γ) then ♯∂_{target}^{−1}(v) = 2, ♯∂_{source}^{−1}(v) = 1. (3) There exists v_{last} ∈ Vertex_{ext}(Γ), the last vertex, such that ♯∂_{target}^{−1}(v) = 1, ♯∂_{source}^{−1}(v) = 0. (4) If v ∈ Vertex_{ext}(Γ)\{v_{last}} then ♯∂_{target}^{−1}(v) = 0, ♯∂_{source}^{−1}(v) = 1.

Figure 2.2

We say an element of Vertex_{int}(Γ) an interior vertex and an element of Vertex_{ext}(Γ) an exterior vertex. An edge e is called an exterior edge if {∂_{target}(e), ∂_{source}(e)}∩Vertex_{ext}(Γ) ≠ ∅. Otherwise it is called an interior edge. Hereafter we put Vertex_{sta}(Γ) = Vertex_{ext}(Γ)\{v_{last}}

We next explain the other data. Let Γ be an oriented tree satisfying Condition 2.2. A metric on Γ is a map l : Edge(Γ) → ℝ_{+} ∪ {∞}. We assume one of the following conditions.

Condition 2.3. (a) l(e) = ∞ if and only if e ∈ Vertex_{sta}(Γ). (b) l(e) = ∞ if and only if e ∈ Vertex_{ext}(Γ).

Let l satisfies Condition 2.3 (a). Let us consider |Γ| minus the union of |v| for v ∈ Vertex_{sta}(Γ). Then we define a metric on it by identifying the interior edge |e| with the arc of length l(e) and identify |e| minus |v| for v = ∂_{source}(e) ∈ Vertex_{sta}(Γ) with (−∞, 0]. We denote it by |(Γ, l)|.

In case l satisfies Condition 2.3 (b), we identify |e| ≅ [0, ∞) if ∂_{target}e = v_{last} and define |(Γ, l)| in a similar way. (Compare [36].)

The third data ψ̃ is a continuous map from |(Γ, l)|\{vertices} to B̃_{0} such that the composition π ∘ψ̃ extends to a continuous map |(Γ, l)| → B. The last data m is a multiplicity. Namely m : Edge(Γ) → ℤ_{+}.

Now our assumption for ψ̃ is as follows. We fix a Riemannian metric g on B. It induces a Riemannian metric on B̃_{0}. Hence we have a gradient vector field gradf of f on B̃_{0}.

Condition 2.4. The restriction of ψ̃ to Int|e| ≅ (−m(e), 0) satisfies the differential equation

\frac{dψ}{dτ} = gradf_{β(e)}.

We assume moreover the following conditions 2.5. Let e be an edge. We identify it with [0, l(e)]. We put

ψ̃(e; target) = lim_{t→l(e)} ψ̃(t), ψ̃(e; source) = lim_{t→0} ψ̃(t).

We define β by :

ψ̃(e; target) = (|∂_{target}(e)|, β_{target}(e)),
ψ̃(e; source) = (|∂_{source}(e)|, β_{source}(e)).

Condition 2.5. (1) If ∂_{source}(e) ∈ Vertex_{sta}(Γ) then lim_{t→−∞} ψ(t) converges to a point of the singular locus S(B). And β_{source}(e) is a vanishing cycle. (2) Let v ∈ Vertex_{int}(Γ). Let e_{1}(v),e_{2}(v),e_{0}(v) be three edges such that ∂_{source}(e_{1}(v)) = ∂_{source}(e_{2}(v)) = ∂_{target}(e_{0}(v)) = v. Then m(e_{1}(v))/m(e_{0}(v)), m(e_{2}(v))/m(e_{0}(v)) are integers and we have

m(e_{1}(v))β_{target}(e_{1}(v)) + m(e_{2}(v))β_{target}(e_{2}(v)) = m(e_{0}(v))β_{source}(e_{0}(v))

in π_{2}(M, F_{ψ(|v|)}). (3) If ∂_{target}(e) = v_{last} then m(e) = 1.

Definition 2.7. The moduli space M_{Morse}(M; g) is a set of all (Γ, l, ψ̃, m) satisfying Conditions 2.3(b), 2.4, 2.5.

We define a map ev_{last} : M_{Morse}(M; g) → B̃_{0} by ev_{last}(Γ, l, ψ̃, m) = ψ̃(e; target) where e is the edge such that ∂_{target}e = v_{last}.

We can prove transversally properties of M_{Morse}(M; g) in the same way as [36]. Our conjecture is a relation of M_{Morse}(M; g) to M(M; J; D^{2}) introduced in §2.4. To compare M_{Morse}(M; g) with M(M; J; D^{2}) we need to modify it by putting multiplicity appeared in Proposition 2.2. Here we only consider the case m ≡ 1. The case m ≠ 1 is more complicated and the author does not yet obtain an answer in the case m ≠ 1 at the time of writing this article.

Definition 2.8. Let m ≡ 1. We define multiplicity multi(Γ, l, ψ̃, m) of (Γ, l, ψ̃, m) ∈ M_{Morse}(M; g) by

multi(Γ, l, ψ̃, m) = ∏_{v∈Vertex_{int}(Γ)} |β_{target}(e_{1}(v)) ∩ β_{target}(e_{2}(v))|

where e_{1}(v), e_{2}(v) are as in Condition 2.5.

We take multi(Γ, l, ψ̃, m) copies of each (Γ, l, ψ̃, m) and obtain a moduli space M_{Morse}(M; g)

To state our conjecture we need to specify an almost complex structure. Namely we consider a family of almost complex structures J_{ϵ} defined by:

J_{ϵ} ( \frac{∂}{∂x^{i}} ) = ϵ^{−1} ∑_{j} g_{ij}(x^{1}, ⋯, x^{n}) \frac{∂}{∂y^{j}} (12)

outside a small neighborhood of π_{M}^{−1}(S(B)). Here x^{i}, y^{j} are symplectic coordinate as in §1.1 and g_{ij} is a Riemannian metric on B independent of the number ϵ.

Conjecture 2.2. For small ϵ, the moduli space M_{Morse}(M; g) is isotopic to M(M; J; D^{2}) as submanifolds in B̃_{0}.

In the same way as [36], we can reduce Conjecture 2.1 to Conjecture 2.2, Propositions 2.1, 2.2 and a gluing argument which now becomes standard in geometric analysis, as we explain below. It proves Conjecture 2.1 in the case when m ≡ 1 and all the singular fibers are of type I_{n} for some n.

Let (Γ, l) satisfies Condition 2.2 and Definition 2.7. We construct a family of Riemann surfaces D(Γ, l; ϵ; θ) as follows. For each edge e of Γ we take [−l(e)/ϵ, 0] × S^{1}. For each vertex v ∈ Vertex_{int}(Γ) we take a pair of pants Σ_{v}. We cut Σ_{v} by a union of three arcs as in Figure 2.3 below into three pieces. We identify them with [−l(e)/ϵ, 0]×S^{1} corresponding to the three edges e ∈ ∂_{target}^{−1}(v) ∪ ∂_{source}^{−1}(v). We thus obtain a Riemann surface with ♯(Vertex_{ext}(Γ)\{v_{last}}) punctures and one boundary. We conformally fill the punctures and obtain a Riemann surface which is isomorphic to a disk. We remark that there is a freedom to glue a pair of pants with [−l(e)/ϵ, 0] × S^{1}. We denote the resulting disk by D(Γ, l; ϵ; θ) where θ parametrizes the way to glue. (θ ∈ (S^{1})^{number of edges}.)

Figure 2.3

Now given ψ̃ and m, we obtain a map D(Γ, l; ϵ; θ) → M as follows. (We fix a choice of θ while we construct a map.) Let e be an edge. The restriction of ψ̃ to |e| defines a map uψ̃ : (−l(e)/ϵ, 0) × S^{1} → M by (6). It is J_{ϵ} holomorphic by Proposition 2.1

3 INCLUDING LAGRANGIAN SUBMANIFOLDS 24

We next consider Σ_{v}. Let e_{1}, e_{2}, e_{0} be as in Condition 2.5 (2). We apply Proposition 2.2 to γ = β_{target}(e_{1}(v)), μ = β_{target}(e_{2}(v)) to obtain u_{v} : Σ_{v} → ℝ^{2} × T^{2}. We identify a neighborhood of π^{−1}(ψ(|v|)) with ℝ^{2} × T^{2}. Hence we have u_{v} : Σ_{v} → M. We remark that Condition 2.5 (2) implies β(e_{0}) = β(e_{1}) + β(e_{2}) = γ + μ. We use it, Proposition 2.2 and the definitions of u_{v},u_{e} to see that u_{v} and u_{e} can be glued modulo small difference. We can adjust this difference by using partition of unity. (We have a choice of θ such that this gluing works.) The number of approximate solutions is the multiplicity multi(Γ, l, ψ̃, m) by Proposition 2.2.

We finally apply Conjecture 2.1 (which we proved in case of type I_{n} singular fibers), to construct J_{ϵ} holomorphic map in a neighborhood of the punctures corresponding to elements of Vertex_{sta}(Γ).

Thus we obtain an approximate solution of the J_{ϵ} holomorphic curve equation for D(Γ, l; ϵ; θ) → M. The final step is a standard implicit function theorem similar to [36].

We need careful analysis to show that we obtain all the pseudoholomorphic curves in this way. The author did not work it out yet, at the time writing this article. Remark 2.9. Actually we did not use at all the fact that Γ is a tree in the argument above. In case when Γ is not a tree, D(Γ, l; ϵ; θ) is not necessary a disk but is a Riemann surface with boundary and of higher genus. We thus can describe the holomorphic map from Riemann surface of higher genus with (or without) boundary using a similar moduli space to M_{Morse}(M; g). (Compare [27] and Remark 4.4.)

3 Including Lagrangian submanifolds

As we mentioned in §2.4, to handle singular fiber in arbitrary dimension is harder. To have more examples to play applying the ideas presented in the last section, we include Lagrangian submanifolds into the story. Then, we are attacking homological mirror symmetry conjecture by Kontsevich [59, 58]. Let π_{M} : M → B satisfy Assumption 2.1. Let L ⊂ M be a Lagrangian submanifold.

Definition 3.1. A point x ∈ B is said to be in the caustic of L if F_{x} is not transversal to L. We denote the caustics by S(L).

We remark that the restriction of π_{M} to L is an immersion over M\S(L). S(L) plays a similar role to S(B). Caustics is studied extensively in symplectic geometry especially by Arnold and his school [8, 9].

3.1 Multi-valued function

We define a multivalued function f_{L} on B\S(L), as follows. We first define a covering space B̃_{L} over B\S(L).

We fix some notations. We consider (A; B_{1}, ⋯, B_{n}; p_{1}, ⋯, p_{m}) such that B_{i} ⊂ A and p_{i} ∈ A. A map φ : (A; B_{1}, ⋯, B_{n}; p_{1}, ⋯, p_{m}) → (A′; B′_{1}, ⋯, B′_{n}; p′_{1}, ⋯, p′_{m}) is by definition a continuous map φ : A → A′ such that φ(B_{i}) ⊆ B′_{i}, φ(p_{i}) = p′_{i}. Let

Π((A; B_{1}, ⋯, B_{n}; p_{1}, ⋯, p_{m}), (A′; B′_{1}, ⋯, B′_{n}; p′_{1}, ⋯, p′_{m}))

be the set of homotopy classes of such maps. We put

∂_{+}D^{2} = {z ∈ ∂D^{2}| Im z > 0}, ∂_{−}D^{2} = {z ∈ ∂D^{2}| Im z < 0}.

Let L_{1}, L_{2} ⊂ M and p, q ∈ L_{1} ∩ L_{2}. We put

π_{2}(M; L_{1}, L_{2}; p, q) = Π((D^{2}; ∂_{+}D^{2}, ∂_{−}D^{2}; −1, +1), (M; L_{1}, L_{2}; p, q)).

Let p, q ∈ X. We put

π_{1}(X; p, q) = Π(([0, 1]; 0, 1), (X; p, q)).

There exists an obvious products

\begin{aligned}π_{2}(M; L_{1}, L_{2}; p, q) × π_{2}(M; L_{1}, L_{2}; q, r) & → π_{2}(M; L_{1}, L_{2}; p, r) \\ π_{1}(X; p, q) × π_{1}(X; q, r) & → π_{1}(X; p, r) \end{aligned}

which are associative. In other words π_{2}(M; L_{1}, L_{2}; p, q) and π_{1}(X; p, q) are groupoids.

Moreover there exist boundary homomorphisms

\begin{aligned}∂_{+} : π_{2}(M; L_{1}, L_{2}; p, q) & → π_{1}(L_{1}; p, q), \\ ∂_{−} : π_{2}(M; L_{1}, L_{2}; p, q) & → π_{1}(L_{2}; p, q) \end{aligned}

which are groupoid homomorphisms.

Definition 3.2. We denote by B̃_{L} the set of all (x; p, q; [u]) where x ∈ B\S(L), p, q ∈ F_{x} ∩ L, and [u] ∈ π_{2}(M; L, F_{x}; p, q).

There exists an obvious covering map πB̃_{L} : B̃_{L} → B\S(L). (πB̃_{L} is a covering space over each connected component of B\S(L). The order of the inverse image may depend on the component.)

We define another covering space B̃_{L}^{+} as follows. B̃_{L}^{+} is the set of all (x; p, q; [γ]), such that p, q ∈ F_{x} ∩ L, [γ] ∈ π_{1}(F_{x}; p, q). The map πB̃_{L}^{+} : B̃_{L}^{+} → B\S(L) is defined in an obvious way. It is a covering space over each connected component of B\S(L). We also have a map ∂_{+} : B̃_{L} → B̃_{L}^{+}.

We now define

f_{L}(x; p, q; [u]) = ∫_{D^{2}} u^{∗}ω. (13)

We remark that the right hand side of (13) is independent of the representative u since L and F_{x} are Lagrangian submanifolds.

Lemma 3.1. There exists a closed one form θ on B̃_{L}^{+} which pulls back to df_{L} on B̃_{L}.

The proof is the same as the proof of Lemma 2.1 and is omitted. From now on we write df_{L} in place of θ by abuse of notation. We can define a multivalued function by using several Lagrangian submanifolds, in a similar way. We do not explain this generalization which is rather straight forward.

Example 3.1. Let h(y^{1}, ⋯, y^{n}) : ℝ^{n} → ℝ be a function. We assume dh(0) = 0. We put

L_{h} = \{( \frac{∂h}{∂y^{1}} , ⋯, \frac{∂h}{∂y^{n}} , y^{1}, ⋯, y^{n})| (y^{1}, ⋯, y^{n}) ∈ ℝ^{n}\}.

Then L_{h} is a Lagrangian submanifold of ℝ^{2n}. We divide the y direction by ℤ^{n} and obtain M = ℝ^{n} × T^{n}. We consider the projection π_{M} : M → ℝ^{n} defined by π(x^{1}, ⋯, x^{n}, y^{1}, ⋯, y^{n}) = (x^{1}, ⋯, x^{n}).

We remark that x ∈ S(L_{h}) if and only if x = dh(y) for some critical point y of h.

Suppose π^{−1}(x) intersect with L_{h} at two different points p = (x, y) and q = (x, y′). (Here dh(y) = dh(y′) = x.) We identify D^{2} with [0, 1]^{2} and define u by

u(s_{1}, s_{2}) = ((1 − s_{1})x + s_{1}dh(s_{2}y + (1 − s_{2})y′), s_{2}y + (1 − s_{2})y′).

Figure 3.1

Clearly (x; p, q, [u]) is an element of B̃_{L_{h}}. We can easily calculate

f_{L}(x; p, q; [u]) = \frac{1}{2} (h(y′) − h(y)).

We will use in §5.6 a similar but a bit different multivalued function, which we define here. Let L_{1}, L_{2} be two Lagrangian submanifolds of M. We assume that they intersect transversally each other. Let B_{00} = B_{0}\(S(L_{1}) ∪ S(L_{2})). We put χ = exp(2π\sqrt{−1}/3). Let ∂_{i}D^{2} be a part of ∂D^{2} between χ^{i} and χ^{i+1}. (i = 0, 1, 2.) We define :

Definition 3.3. B̃_{L_{1},L_{2}} is the set of all = (x; [u]; p, q; x_{0}) such that : (1) x ∈ B_{00}, (2) p ∈ F_{x} ∩ L_{1}, (3) q ∈ F_{x} ∩ L_{2}, x_{0} ∈ L_{1} ∩ L_{2}, (4) [u] ∈ Π((D^{2}; ∂_{0}, ∂_{1}, ∂_{2}; χ^{0}, χ^{1}, χ^{2}), (M; F_{x}, L_{1}, L_{2}; p, x_{0}, q)).

Figure 3.2

There exists an obvious immersion B̃_{L_{1},L_{2}} → B_{00}. We define

f_{L_{1},L_{2}}(x; [u]; p, q; x_{0}) = ∫_{D^{2}} u^{∗}ω. (14)

We define also

\begin{matrix}B̃_{L_{1},L_{2}}^{+} = \{(x; [γ]; p, q)| & x ∈ B_{00}, p ∈ F_{x} ∩ L_{1}, \\ q ∈ F_{x} ∩ L_{2}, [γ] ∈ π_{1}(F_{x}; p, q)\}. \end{matrix} (15)

There exists an obvious immersion B̃_{L_{1},L_{2}} → B̃_{L_{1},L_{2}}^{+}. In the same way as Lemma 3.1, we can find a closed one form θ on B̃_{L_{1},L_{2}}^{+} which pulls back to df_{L_{1},L_{2}}. We write θ = df_{L_{1},L_{2}} by abuse of notation. (Then f_{L_{1},L_{2}} is well defined locally up to constant difference on B̃_{L_{1},L_{2}}^{+}.)

3.2 Free field.

We discuss an open string analogue of the discussion of §2.2. Actually, as we mentioned in introduction, this case is classical and due to Floer [22].

In this section, we consider the case when S(B) = S(L) = ∅. (We do not assume B is complete. So, in other words, we are working on the complement of S(B) ∪ S(L).)

We fix a Riemannian metric g on B. It defines a family of almost complex structures J_{ϵ} by formula (12) in §2.4. We are going to study J_{ϵ} holomorphic map u(τ, t) : (a, b)×[0, 1] → M such that u(τ, 0), u(τ, 1) ∈ L.

Let ℓ(τ) : [a, b] → B be a path with bounded C^{1} norm. We take its lift

ℓ̃ = (ℓ(τ); p(τ), q(τ); γ_{τ}) : [a, b] → B̃_{L}^{+}. We define uϵ ; ℓ̃: (a/ϵ, b/ϵ) × [0, 1] →

M as follows. We remark that we have a canonical flat affine structure on the fibers F_{x}. Namely we take y^{i} as an affine coordinate. (Here y^{i} is a coordinate such that ω = ∑ dx^{i} ∧ dy^{i}. See the end of §1.1.) In other words y^{i} is an angle coordinate.

We take the representative γ_{τ} : [0, 1] → F_{ℓ(τ)} as the unique affine geodesic joining p(τ) with q(τ) in the homotopy class ∈ π_{1}(F_{ℓ(τ)}; p(τ), q(τ)). We now put

uϵ ; ℓ̃(τ, t) = γ_{ϵτ} (t). (16)

The boundary condition u(τ, 0), u(τ, 1) ∈ L is obviously satisfied.

We remark that J_{ϵ} and our symplectic structure ω determine a Riemannian metric g_{ϵ} on M. It then is easy to see

∥ \frac{∂uϵ ; ℓ̃}{∂τ}∥_{g_{ϵ}} ∼ Cϵ, ∥ \frac{∂uϵ ; ℓ̃}{∂t}∥_{g_{ϵ}} ∼ Cϵ. (17)

We also remark that g induces a Riemannian metric on B̃_{L}^{+}. Hence we have a gradient vector field gradf_{L} on B̃_{L}^{+}.

Proposition 3.1. If we assume

\frac{dℓ̃}{dτ} = gradf_{L}, (18)

then we have

∥ \frac{∂uϵ ; ℓ̃}{∂τ} + J_{ϵ} ( \frac{∂uϵ ; ℓ̃}{∂t})∥_{g_{ϵ}} ≤ Cϵ^{2}. (19)

In view of (17), Formula (19) means that uϵ ; ℓ̃ is an approximate solution of the J_{ϵ} holomorphic curve equation. The proof of Proposition 3.1 is a straight forward calculation and is omitted. Remark 3.1. We remark that Proposition 2.1 says that the gradient line of the multivalued function f corresponds to the exact solution of the pseudoholomorphic curve equation. On the other hand, Proposition 3.1 only gives an approximate solution. The difference is that we put here an extra data L which can be quite arbitrary. So it is in general impossible to get an explicit solution of our J_{ϵ} holomorphic curve equation together with L as a boundary value.

3.3 Open string vs Open string scattering - review -

This section is a review of [36] with only a minor modification. We study interaction of two open strings. The problem is local in B so we may consider the following situation. B = ℝ^{n}, M = ℝ^{n} × T^{n}. Moreover we may consider only a Lagrangian submanifold L which is a union of “horizontal” ones. Namely we take α_{i} ∈ T^{n}, and put L_{α_{i}} = ℝ^{n} × {α_{i}}. Then we study the case L = L_{α_{1}} ∪ L_{α_{2}} ∪ L_{α_{3}}.

Let x ∈ ℝ^{n}. We put p_{i}(x) = (x, α_{i}) ∈ F_{x}. The element of π_{1}(F_{x}; p_{i}(x), p_{j}(x)) corresponds to the element γ_{ij} ∈ ℝ^{n} such that γ_{ij} ≡ α_{j} − α_{i} mod ℤ^{n}. We put

ℓ̃_{ij}(τ) = (τγ_{ij}; p_{i}(τγ_{ij}), p_{j}(τγ_{ij}); γ_{ij}) ∈ B̃_{L}^{+}. (20)

The curve ℓ̃_{ij} : ℝ → B̃_{L}^{+} is a gradient line of df_{L}. In this case, we can easily show that uϵ ; ℓ̃_{ij} defined by (16) is J_{ϵ} holomorphic. (Here we take standard Euclidean metric on ℝ^{n} to define J_{ϵ}.) Since the discussion of this subsection is same for all ϵ, we omit ϵ hereafter in this subsection.

Proposition 3.2 below asserts that we have a holomorphic disk which is asymptotic to uℓ̃_{12}, uℓ̃_{23}, uℓ̃_{13} at their three ends.

Let us introduce some notations to state it. We put χ = e^{2π\sqrt{−1}/3} Let Σ be a Riemann surface with boundary conformal to D^{2}\{1, χ, χ^{2}}. Σ has three ends corresponding to 1, χ, χ^{2}. We write them End_{1}, End_{χ}, End_{χ^{2}}. We take and fix a conformal isomorphism End_{χ} ≅ End_{χ^{2}} ≅ (−∞, 0] × [0, 1], End_{1} ≅ [0, ∞) × [0, 1]. We use the symbol τ for the first factor and t for the second factor.

Proposition 3.2. ([36]) There exists a J_{ϵ} holomorphic map u : Σ → ℝ^{n} × T^{n} such that u(∂Σ) ⊂ L and that

(21a)  |u(τ, t) − uℓ̃_{12}(τ + τ_{1}, t)| < Ce^{τ} on End_{χ^{2}},
(21b)  |u(τ, t) − uℓ̃_{23}(τ + τ_{2}, t)| < Ce^{τ} on End_{χ},
(21c)  |u(τ, t) − uℓ̃_{23}(τ − τ_{3}, t)| < Ce^{−τ} on End_{1}.

Proof. We may identify Σ = {z ∈ ℂ | Im z ≥ 1, z ≠ ±1}, such that ends End_{χ}, End_{χ^{2}}, End_{1} correspond to −1,+1 ∞ respectively. We next identify ℝ^{n} × T^{n} ≅ ℂ_{∗}^{n}. We may identify L_{1}, L_{2}, L_{3} as L_{1} = ℝ_{+}^{n}, L_{2} = {(z_{1}, ⋯, z_{n})| arg z_{i} = πα_{i}}, L_{3} = {(z_{1}, ⋯, z_{n})| arg z_{i} = πβ_{i}}. We now put

u(z) = ((z + 1)^{α_{1}}(z − 1)^{β_{1}−α_{1}}, ⋯, (z + 1)^{α_{n}}(z − 1)^{β_{n}−α_{n}}).

Here we choose branch of z^{α} such that z^{α} ∈ ℝ_{+} if z ∈ ℝ_{+}. It is easy to check (21). (Note u({z ∈ ℝ|z > 1}) ⊆ L_{1}, u({z ∈ ℝ|1 > z > −1}) ⊆ L_{2}, u({z ∈ ℝ|z < −1}) ⊆ L_{3}.)

Remark 3.2. We remark that there is one difference between our case and the case of [36]. Namely here we are studying torus bundle and in [36] the case when fibers are vector space is studied. We specify γ_{ij} to handle this point. Then the two cases are almost the same as far as the argument of this section concerns except the following point. Let us consider the case α_{1} = α_{3}. In this case we obtain a solution u which corresponds to the interactions of many open strings. Let us describe it.

Let us consider Σ conformal to D^{2} minus k+1 points on the boundary. Σ has k + 1 ends End_{i}, i = 0, ⋯, k. We identify End_{i} ≅ (−∞, 0] × [0, 1] for i ≠ 0 and End_{i} ≅ [0, ∞) × [0, 1].

Figure 3.3

We put γ′_{12} = (k−1)(γ_{12} +γ_{21})/2+γ_{12} if k is odd and γ′_{11} = k(γ_{12} +γ_{21})/2 if k is even. We then put

ℓ′_{12,k} = (τγ′_{12}; p_{1}(τγ′_{12}), p_{2}(τγ′_{12}); γ′_{12}), if k is odd,
ℓ′_{11,k} = (τγ′_{11}; p_{1}(τγ′_{11}), p_{2}(τγ′_{11}); γ′_{11}), if k is even.

Then there exists a J holomorphic u : Σ → M such that u(∂Σ) ⊂ L = L_{1} ∪ L_{2} and

(22a)  |u(τ, t) − uϵ ; ℓ̃_{12}(τ − τ_{i}, t)| < Ce^{τ} on End_{i} with i odd,
(22b)  |u(τ, t) − uϵ ; ℓ̃_{21}(τ − τ_{i}, t)| < Ce^{τ} on End_{i} with i even,
(22c)  |u(τ, t) − uϵ ; ℓ̃′_{12}(τ − τ_{0}, t)| < Ce^{−τ} on End_{0} if k is odd,
(22d)  |u(τ, t) − uϵ ; ℓ̃′_{11}(τ − τ_{0}, t)| < Ce^{−τ} on End_{0} if k is even.

Such u is actually constructed also in [36] where the solution which gives interaction of arbitrary many open strings are constructed.

Interaction described by this solution u will become important when we discuss its B model analogue in §4.6.

We also point out that the gluing procedure given by our u here is the same one as studied by Hutchings and Li [52] in the case of closed orbit of ordinary differential equation which appears in Novikov homology [67].

3.4 Open string vs Closed string scattering.

We next discuss a mixture of the two kinds of scattering. Let us consider the same situation as the last section. (M = ℝ^{n} × T^{n} etc.) We suppose we have two horizontal Lagrangian submanifolds L_{i} = ℝ^{n} ×{α_{i}}, and put L = L_{1} ∪ L_{2}.

Let γ_{12} ∈ ℝ^{n} such that γ_{12} ≡ α_{2} − α_{1} mod ℤ^{n}. We have a J holomorphic curve uℓ̃_{12} defined by (16).

We next take γ ∈ ℤ^{n} = π_{1}(F_{x}) and w ∈ T^{n}. Then we put

u_{γ,w}(τ, t) = (τγ, tγ + w) ∈ ℝ^{n} × T^{n}.

u_{γ,w} : ℝ × S^{1} → ℝ^{n} × T^{n} is J holomorphic by Proposition 2.1. We put γ′_{12} = γ_{12} + γ and use it in place of γ_{12} to obtain uℓ̃′_{12} by (16).

Now main result of this subsection is about the J holomorphic curve which is asymptotic to uℓ̃_{12}, u_{γ,w}, uℓ̃′_{12} in its three ends. Let us define some notations to state it.

For s ∈ (−1, 1), let Σ_{s} be a Riemann surface with boundary which is conformal to D^{2}\\{ s\sqrt{−1}, −1, +1 \}. Σ_{s} has three ends End_{s\sqrt{−1}}, End_{±1} corresponding to the points removed. We fix a conformal diffeomorphisms End_{s\sqrt{−1}} ≅ (−∞, 0]×S^{1}, End_{−1} ≅ (−∞, 0]×[0, 1], End_{+1} ≅ [0, ∞)×[0, 1]. Let A be a codimension one cycle of T^{n}/S_{γ}^{1}. (Here we are using the notation of §2.3.) We take a representative γ_{12} ∈ π_{1}(T^{n}; α_{1}, α_{2}) which is an affine geodesic. We assume that AS_{γ}^{1} ∩ {α_{1}, α_{2}} = ∅ and γ_{12} intersects transversally with AS_{γ}^{1}.

Proposition 3.3. We consider the set of all pairs (u, s) of s ∈ (−1, 1) and J holomorphic maps u : Σ_{s} → ℝ^{n} × T^{n} with the following boundary conditions. There exists τ_{0}, τ_{±1} and w ∈ AS_{γ}^{1} such that :

(23a)  |u(τ, t) − u_{γ,w}(τ + τ_{0}, t)| < Ce^{+Cτ} on End_{s\sqrt{−1}},
(23b)  |u(τ, t) − uℓ̃_{12}(τ + τ_{−1}, t)| < Ce^{+Cτ} on End_{−1},
(23c)  |u(τ, t) − uℓ̃′_{12}(τ − τ_{+1}, t)| < Ce^{−Cτ} on End_{+1}.

Then, the number of such solutions is exactly equal to the intersection number |AS_{γ}^{1} ∩ γ_{12}|.

Proof. We prove only the case n = 2. The general case is similar. We identify Σ_{s} ≅ \{z ∈ ℂ| Im z ≥ 0, z ≠ s′ + \sqrt{−1}, 0\}. Then s′ moves on ℝ while s moves on (−1, 1). Here 0, s′ + \sqrt{−1}, ∞ corresponds the ends End_{−1}, End_{s\sqrt{−1}}, End_{+1} respectively. We also identify ℝ^{2} × T^{2} ≅ ℂ_{∗}^{2}. It suffices to consider the case when L_{1} = ℝ_{+}^{2} ⊂ ℂ_{∗}^{2}. L_{2} = {(z_{1}, z_{2}) ∈ ℂ_{∗}^{2}| Arg z_{1} = πα_{1}, Arg z_{2} = πα_{2}} for i = 1, 2. We may also take γ = (γ_{1}, γ_{2}) and A is one point. Hence AS_{γ}^{1} = {a + tγ|t ∈ ℝ} ⊂ ℝ^{2}/ℤ^{2}, a = (a_{1}, a_{2}).

An intersection point in AS_{γ}^{1} ∩ γ_{12} then corresponds to a pair of ρ ∈ [0, 1], θ ∈ [0, 1] such that

(ρα_{1}, ρα_{2}) = 2π(a_{1} + θγ_{1}, a_{2} + θγ_{2}). (24)

It follows that, for each point of AS_{γ}^{1} ∩ γ_{12}, we can take s′ such that

Arg(s′ + \sqrt{−1})^{α_{1}} = 2π(a_{1} + θ_{1}γ_{1}), Arg(s′ + \sqrt{−1})^{α_{2}} = 2π(a_{2} + θ_{2}γ_{2}).

Here we take the branch of z^{α_{i}} so that it is real on ℝ_{+}. (Note α_{i} ∈ (0, 2π) and γ_{i} are integers.) We now put

\begin{aligned}u(z) = & (z^{α_{1}}(z − s′ − \sqrt{−1})^{γ_{1}}(z − s′ + \sqrt{−1})^{γ_{1}}, \\ & z^{α_{2}}(z − s′ − \sqrt{−1})^{γ_{2}}(z − s′ + \sqrt{−1})^{γ_{2}}) . \end{aligned} (25)

It is easy to check (23). It is also easy to show that all solution is obtained in this way.

We can combine constructions of §§2.3, 3.3, 3.4 to obtain a holomorphic curve in ℝ^{n}×T^{n} which describes interactions between arbitrary many open and closed strings.

3.5 Caustics, elementary catastrophe, and bifurcation of pseudoholomorphic disks.

We now explain a conjecture which is an open string version of Conjecture 2.1. We are going to study a neighborhood of the point x ∈ M_{0} where L is tangent to the fiber. So we consider ℝ^{n} bundle instead of T^{n} bundle. We define an analogue of U(f_{γ}; g). Let x_{0} ∈ S(L). We take its neighborhood U in B. Let x ∈ U\S(L) and (x; p, q; [u]) ∈ B̃_{L}.

Definition 3.4. We say that [u] ∈ π_{2}(M; L, F_{x}; p, q) is a vanishing disk if we may choose its representative which is contained in π^{−1}(U). We say [∂_{+}u] ∈ π_{1}(F_{x}; p, q) is a vanishing arc if [u] is a vanishing disk.

We remark that an analogy of Assumption 2.2 is not appropriate in our situation, as we can see from examples we will show later. So we define Unst(U) as a subset of B̃_{L} as follows. We use a pull back of a Riemannian metric on B to define a Riemannian metric on B̃_{L}. Then a gradient vector field gradf_{L} is induced on B̃_{L}. Let ext(τ gradf_{L}) be the one parameter group of transformations generated by it.

Definition 3.5. We define Unst(U) as the set of all (x; p, q; [u]) ∈ B̃_{L}, such that :

(1) x ∈ U. u is a vanishing disk.

(2) π (ext(−τgradf_{L})(x; p, q; [u])) ∈ U for all τ > 0.

(3) lim_{τ→∞} π (ext(−τgradf_{L})(x; p, q; [u])) ∈ S(L).

We define moduli space of pseudoholomorphic disks as follows

Definition 3.6. We let M̃((M, J_{ϵ}); L) be the set of all (x; p, q; u) such that (x; p, q; [u]) ∈ B̃_{L}, where [u] is a homotopy class of u, and that u is J_{ϵ} holomorphic. It has a natural action of ℝ = Aut(D^{2}; ±1). We denote by M((M, J_{ϵ}); L) the quotient space by this action.

If U is a small neighborhood of x_{0} ∈ S(L), we define M_{vanish}(U; (M, J_{ϵ}); L) as the set of all [x; p, q; u] ∈ M((M, J_{ϵ}); L) such that x ∈ U and u(π(D^{2})) ⊂ U. (In other words, u is a vanishing disk.)

There is a natural map M((M, J_{ϵ}); L) → B̃_{L}.

Conjecture 3.1. M_{vanish}(U; (M, J_{ϵ}); L) is isotopic to Unst(U) as a submanifold of B̃_{L} after appropriate perturbation.

Let us give some examples and arguments which justify the conjecture.

Example 3.2. The simplest caustic is fold. Namely we take n = 1 and let L is given by x = 3y^{2} where x = const is the fiber. This is the case when we put h(y) = y^{3} in Example 3.1. Hence f_{L}(x) = x^{1/3}, which is defined only on the domain x > 0. In this case Unst(U) is the part x > 0 (and corresponds to the unique branch of the covering B̃_{L} → {x|x > 0}). It is easy to see that M_{vanish}(U; (M, J_{ϵ}); L) is the same.

Example 3.3. The next case to study is a cusp. We take h(y_{1}, y_{2}) = y_{1}^{4} + y_{1}y_{2}^{2} in Example 3.1. Then L is defined by (x_{1}, x_{2}) = Ψ(y_{1}, y_{2}) = (4y_{1}^{3} + y_{2}^{2}, 2y_{1}y_{2}). The caustics S(L) is given by Ψ({(y_{1}, y_{2}) | 6y_{1}^{3} = y_{2}^{2}}) and is given by ℓ(t) = (7t^{8}, 2\sqrt{6}t^{5}). Let us decompose ℝ^{2}\S(L) = Ω_{0} ∪ Ω_{1} where Ω_{0} contains the domain x_{1} < 0. If x ∈ Ω_{0} then F_{x} intersects with L at one point. If x ∈ Ω_{1} then F_{x} intersects with L at three points. Let p_{i}(x) i = 1, 2, 3 be those three points. We choose them so that |p_{1}(x)−p_{2}(x)| → 0 as x converges to S(L)_{−} = S(L) ∩ {x_{2} < 0} and |p_{3}(x) − p_{2}(x)| → 0 as x converges to S(L)_{+} = S(L) ∩ {x_{2} > 0}. There is one vanishing arc for each of S(L)_{−} and S(L)_{+}, that are the ones joining p_{1}(x) to p_{2}(x) and p_{3}(x) to p_{2}(x), respectively. The gradient lines of the corresponding branch of our function is obtained by Example 3.2 above. The gradient line starting S(L)_{−} dies when it meets with S(L)_{+} (since there will no longer be a corresponding branch of f_{L}.) There are two other branches of f_{L} which correspond to the vanishing arcs joining p_{1} to p_{3} and p_{3} to p_{1}. The gradient line of these branches starting origin do not exit.

Example 3.4. Let us take M = ℂ^{2}, and let (x, y) be the complex coordinate of it. We regard M as a hyperkähler manifold and take hyperkähler twist so that (x, y) ↦ x will have a Lagrangian submanifold as fibers. We put x = x_{1} + \sqrt{−1}x_{2}, y = y_{1} + \sqrt{−1}y_{2}. (Note that the symplectic form is dx_{1} ∧ dy_{1} + dx_{2} ∧ dy_{2}.) Let L is given by x = y^{2}. It is a complex submanifold with respect to the original complex structure. Hence after hyperkähler twist, L will become a (special) Lagrangian submanifold. The intersection L ∩ F_{x} consists of two points (x, ±x^{1/2}). Hence there are exactly two vanishing arcs, one from (x, −x^{1/2}) to (x, x^{1/2}), the other from (x, x^{1/2}) to (x, −x^{1/2}). (These two branch interchange when we go around origin once.)

We put x = re^{\sqrt{−1}θ} and parametrize L as (r, θ) ↦ (re^{2\sqrt{−1}θ}, \sqrt{r}e^{\sqrt{−1}θ}). Then (s, t) ↦ (sre^{2\sqrt{−1}t}, t\sqrt{sr}e^{\sqrt{−1}r}), (s, t) ∈ [0, 1] × [−1, 1] is a parameterization of a vanishing disk. We use it to calculate our multivalued function f_{L} and obtain

f_{L}(x) = ± \frac{4}{3} Re x^{3/2}.

Therefore there exists exactly three gradient lines starting from origin.

Figure 3.4

Let us explain an argument which suggests Figure 3.4 also appears as the moduli space of J holomorphic discs. This argument may work in more general situation. So we discuss the case of Example 3.1 in general. Namely our Lagrangian submanifold L is obtained as a graph of a one form dh where h is a function of y_{1}, ⋯, y_{n}. (We remark that in [36] we consider the case when our Lagrangian submanifold is a graph of one form which is an exterior derivative of a function of x. In that case, the Lagrangian submanifold is transversal to the fibers F_{x}. Here we are considering the opposite extreme namely the case when the Lagrangian submanifold is tangent to the fiber at origin.) We put

h_{x_{1},··· ,x_{n}}(y_{1}, ⋯, y_{n}) = h(y_{1}, ⋯, y_{n}) − ∑ x_{i}y_{i}.

It is easy to see that (x, y) ∈ L ∩ F_{x} if and only if y is a critical point of h_{x}. We now define another moduli space as follows :

Definition 3.7. We define an moduli space M_{Morse,vanish}(M; L) as the set of all (x; p_{1}(x), p_{2}(x); [ℓ])’s such that p_{i} = (x, y_{i}) ∈ L ∩ F_{x} and ℓ : ℝ → ℝ^{n} such that dℓ/dτ = gradh_{x} and lim_{τ→−∞} ℓ(τ) = y_{1}, lim_{τ→+∞} ℓ(τ) = y_{2}. Here [ℓ] is the ∼ equivalence class, and we define ∼ by ℓ(τ) ∼ ℓ(τ +τ_{0}).

There is an obvious map M_{Morse,vanish}(M; L) → B̃_{L}. Now we can divide Conjecture 3.1 into two pieces.

Conjecture 3.2. M_{vanish}(U; (M, J_{ϵ}); L) is isotopic to M_{Morse,vanish}(M; L) as a submanifold of B̃_{L} after appropriate perturbation.

Conjecture 3.3. M_{Morse,vanish}(M; L) is isotopic to Unst(U) as a submanifolds of B̃_{L}.

Remark 3.3. In Conjecture 3.1, we consider the general case and here we consider the case of Example 3.1. However we can divide B into two directions, one is reduced to Example 3.1 and the other direction is transversal to L. In that way, to study Conjecture 3.1, it suffices to consider the case of Example 3.1.

Conjecture 3.2 asserts that the moduli space of gradient lines is isotopic to the moduli space of J_{ϵ} holomorphic disks. This kinds of results are established in various setting, but not yet proved in the particular setting of Conjecture 3.2. The first result of this kind is due to Floer [22], that is the case of free field we discussed in §3.2. The author together with Y.G. Oh generalized it in [36] to the case when interaction exists.

Floer used in [22] another result (Proposition 5.1 in [22]) which is closely related to the case h(y) = y^{3} of Conjecture 3.2, to prove invariance of Floer homology under Hamiltonian diffeomorphism. The argument in [22] §5 is rather sketchy. The argument of [22] §19 is one using center manifold in the case appeared there, though the name center manifold is not used in [22]. The idea to use center manifold is used later by Morga-Mrowka-Ruberman in a related context of gauge theory. However their argument is also rather sketchy. To give more detail to the argument in [22] §5 should be a way to solve Conjecture 3.2. In a recent paper [63] Y.J. Lee justified a similar degenerate gluing in the situation of Floer homology of periodic Hamiltonian system. The author is sorry that he used this kind of degenerate gluing in his paper [30] without writing a detail of the proof.

Conjecture 3.3 is one on finite dimensional Morse theory. However it is not trivial one. The author can not prove it in general at the time of writing this article. Instead, let us check it in the case of Examples 3.3 and 3.4.

We first consider the case of Examples 3.3. We have h_{x_{1},x_{2}}(y_{1}, y_{2}) = y_{1}^{4} + y_{1}y_{2}^{2} − x_{1}y_{1} − x_{2}y_{2}. If x ∈ Ω_{1}, then h_{x_{1},x_{2}}(y_{1}, y_{2}) has three critical points y^{1}(x), y^{2}(x), y^{3}(x) ∈ ℝ^{2} such that p_{i}(x) = (x, y^{i}(x)). By direct calculation we find that y^{2}(x) is a saddle point and y^{1}(x), y^{3}(x) are local minimum. There exists a gradient lines of h_{x} from y^{2}(x) to y^{1}(x) and from y^{2}(x) to y^{3}(x). However there is no gradient line from y^{1}(x) to y^{3}(x) or from y^{3}(x) to y^{1}(x), for any x.

This is consistent with the description of M_{vanish}(U; (M, J_{ϵ}); L) we gave before. (We remark that M_{vanish}(U; (M, J_{ϵ}); L) is related to the gradient line of f_{L} which is different from h_{x_{1},x_{2}}.) Thus Conjecture 3.3 holds in this case.

We next consider the case of Example 3.4. We put h(y_{1}, y_{2}) = Re y^{3}/3. It is easy to check that the graph of dh is our Lagrangian submanifold L. For x ≠ 0, L intersects with the fiber F_{x} at two points p_{±}(x) = (x, ±\sqrt{x}) These two points are critical points h_{x}(y) = h(y)−x_{1}y_{1}−x_{2}y_{2}. We remark that they both are saddle points and have the same Morse index (=1). Hence the virtual dimension of the moduli space of gradient lines joining them is −1. It means that for generic x there exists no such gradient line but there may exist such a gradient line when x is contained in a codimension one subset of B.

We can study gradient vector field of it drawn below and calculate M_{Morse,vanish}(M; L) explicitly and finds that it consists exactly the same as the space Unst(U) before. Thus Conjecture 3.3 holds in this case also.

Figure 3.5

More generally, let us consider the case when L = {(y^{n}, y)|y ∈ ℂ}. Here we identify ℝ^{4} = ℂ^{2} with complex coordinate (x, y). We apply hyperkähler twist, then L will become a special Lagrangian submanifold. It is easy to see that L is obtained from h(y_{1}, y_{2}) = \frac{1}{n+1} Re y^{n+1} as in Example 3.1. In this case, we can check f_{L}(x) = \frac{n}{n+1} Re(exp(2π\sqrt{−1}k/n) − exp(2π\sqrt{−1}k′/n)x^{(n+1)/n}). One might check Conjecture 3.3 in this case also in a similar way by direct calculation. However the calculation looks rather cumbersome and the author did not complete it at the time of writing this article. Remark 3.4. The bifurcation of pseudoholomorphic disk as shown in Figure 3.4 is presented by the author in his talk at the conference “Integrable Systems in Differential Geometry” which was held in July 2000 at University of Tokyo. After his talk, Yousuke Ohyama pointed him out that Figure 3.4 looks quite similar to Stokes line which appears in exact WKB calculus. (See [75, 4].) Since lines in Figure 3.4 is related to asymptotic analysis as we will explain in the next two sections, it is natural to expect that it is related to WKB calculus. However the author does not yet know the precise relation of it to Stokes line. The difficulty for him to do so is that in our case the fibers are totally real while in exact WKB calculus the fibers are complex submanifolds.

The author lectured an outline of the article at Tokyo metropolitan University in 2002 January. After his lecture, Akishi Kato pointed him out that the story of string junction (see [39]) is similar to one discussed in this subsection. The essential difference is that in the story of string junction, fibers are complex tori and a Lagrangian submanifold of the total space is reduced to a graph of the base space. In our case, fibers are Lagrangian submanifolds and a pseudoholomorphic curve of the total space is reduced to a graph in the base space. The story of string junction may be more directly related to exact WKB calculus. In case dimension is 4 = 2 × 2 these two cases interchanges by hyperkähler twist. Remark 3.5. The Lagrangian submanifold in Example 3.4 is not generic in the sense of Lagrangian singularity. (In case n = 2 only cusp and fold appears as a generic singularity.) However it is generic among special Lagrangian submanifolds. It seems interesting to classify generic Lagrangian singularity for special Lagrangian submanifold in higher dimension, especially in the case n = 3.

During the conference “D-branes and Mirror Symmetry” held at RIMS in 2002 February, D. Joyce pointed out to the author that in case dimension is 6 = 3 × 2, fold also appears as a Lagrangian singularity of a special Lagrangian submanifold. He also pointed out that to have a good restriction of Lagrangian singularity, one needs to consider the limiting behavior of the special Lagrangian submanifold as the diameter of the fiber goes to zero.

We finally give another evidence of Conjecture 3.1. Namely we show its analogue in the case of an immersed Lagrangian submanifold.

Example 3.5. ([36, 32]) Let q_{1}, q_{2} : ℝ^{n} → ℝ be quadratic forms. Then the graph of its exterior derivatives are (linear) Lagrangian submanifolds in T^{∗}ℝ^{n} = ℝ^{2n}. We denote them by L_{1}, L_{2}. They intersect each other at 0 ∈ ℝ^{2n}. We assume that q_{2}(x) − q_{1}(x) is nondegenerate. Then L_{1} and L_{2} intersect transversally. We put L = L_{1} ∪ L_{2}. So L is an immersed Lagrangian submanifold. The analogy of the caustics S(L) in our situation is the image of the singular point 0. Namely S(L) = {0}.

Let x ∈ ℝ^{n}. p_{i}(x) = (x, dq_{i}(x)) ∈ L_{i} is a unique point in L_{i} ∩ T_{x}^{∗}(ℝ^{n}). Let u_{x} : D^{2} → T^{∗}ℝ^{n} be the map onto the affine triangle whose vertices are 0, p_{1}, p_{2}. The analogy of f_{L} in our case is

f(x) = ∫_{D^{2}} u_{x}^{∗}ω = q_{2}(x) − q_{1}(x).

Hence the analogy of Unst(U) is the unstable manifold of the gradient vector field of q_{2}(x) − q_{1}(x). It is then easy to see that the unstable manifold is the linear subspace spanned by the negative eigenvectors of q_{2}(x) − q_{1}(x).

We next consider the analogue of M_{vanish}(U; (M, J_{ϵ}); L) in our situation. Let x^{i} be the standard coordinate of ℝ^{n}. We take coordinate x^{i},y^{i} of ℝ^{2n} so that ω = ∑ dx^{i} ∧ dy^{i}. We define almost complex structure J_{ϵ} by J_{ϵ}(∂/∂x^{i}) = \frac{1}{ϵ} ∂/∂y^{i}. Let us put χ = exp(2π\sqrt{−1}/3). Let us define ∂_{i}D^{2} i = 1, 2, 0 so that ∪_{i=0}^{3}∂_{i}D^{2} = ∂D^{2} and ∂_{i}D^{2} ∩ ∂_{i+1}D^{2} = χ^{i+1} for i = 0, 1, 2. (∂_{3}D^{2} = ∂_{0}D^{2} by convention.)

We denote by M_{vanish}(ℝ^{n}; (ℝ^{2n}, J_{ϵ}); L) the set of all (x, u) where x ∈ ℝ^{n}, u : (D^{2}; ∂_{0}D^{2}, ∂_{1}D^{2}, ∂_{2}D^{2}; 1, χ, χ^{2}) → (T^{∗}ℝ^{n}; T_{x}^{∗}ℝ^{n}, L_{1}, L_{2}; 0, p_{1}, p_{2}) and u is J_{ϵ} holomorphic. We define ev : M_{vanish}(ℝ^{n}; (ℝ^{2n}, J_{ϵ}); L) → ℝ^{n} by (x, u) ↦ x.

It is proved in [36] that the image of ev : M_{vanish}(ℝ^{n}; (ℝ^{2n}, J_{ϵ}); L) → ℝ^{n} converges to the unstable manifold of q_{2}(x) − q_{1}(x) as ϵ goes to zero. Thus analogy of Conjecture 3.1 holds. (Note that the result of [36] implies a similar result also in case when q_{i} are not necessary quadratic.)

3.6 Feynman diagram description of pseudoholomorphic disks - open string version -

Now combining the discussion so far we can describe the J_{ϵ} holomorphic disk which bounds L (assuming Conjecture 3.1).

In this section, we describe the case when S(B) = ∅, that is the case when M is a torus. In the general case, we need both closed string vs open string and open string vs open string scattering. We will not discuss the general case since there are some points which are not yet clear to the author. (We will mention it at the end of this section.)

There are two cases we can study this way. One is the case of pseudoholomorphic disk u : D^{2} → M such that u(∂_{−}D) ⊂ L and u(∂_{+}D) ⊂ F_{x}, the other is the case of pseudoholomorphic disk u : D^{2} → M with u(∂D) ⊂ L. Remark 3.6. In the case of closed string, we did not discuss the case of J_{ϵ} holomorphic sphere. (That is a closed string analogue of u : (D^{2}, ∂) → (M, L).) The reason we did not do so is that, in case n = 2, (that is the only case we discussed there), the virtual dimension of the pseudoholomorphic sphere in Calabi-Yau 2 fold (K3 surface or torus) is −1. Hence in generic situation there is no such a sphere. If we consider the case n = 3 or if we consider a family of Calabi-Yau 2 fold then it is possible to analyze the J_{ϵ} holomorphic sphere in a similar way. (We need to clarify what happens in the neighborhood of singular fiber to do so.)

We use the notations of §2.5. Let Γ be a finite oriented graph.

Definition 3.8. A ribbon structure of Γ is a choice of cyclic order of the set ∂_{target}^{−1}(v) ∪ ∂_{source}^{−1}(v) for each vertex v.

Finite oriented graph equipped with a ribbon structure is called a finite oriented ribbon graph.

If there exists an embedding of |Γ| into an oriented surface, then it induces a ribbon structure. In case |Γ| is a tree then ribbon structure determines an embedding of it to ℝ^{2} up to isotopy.

We now consider the moduli space of data Γ,l,ψ̃ similar to ones in §2.7. Let us describe it precisely.

Condition 3.1. Γ is a finite oriented ribbon graph which satisfies Conditions 2.2 (1),(3),(4) and the following condition.

(2′) If v ∈ Vertex_{int}(Γ) then ♯∂_{target}^{−1}(v) ≥ 2, ♯∂_{source}^{−1}(v) = 1.

We assume either Condition 2.3 (a) or (b).

We next describe the data ψ̃. It is a continuous map ψ : |Γ| → B together with a lift ψ̃ : |Γ|\{vertices} → B̃_{L} which satisfies the following Condition 3.2. To state the condition we need some notations. For τ ∈ |Γ|\{vertices}, we put ψ̃(τ) = (ψ(τ); p(τ), q(τ); β(τ)), where β(τ) is an element of π_{2}(M; F_{ψ(τ)}; p(τ), q(τ)). Let v be an interior vertex of Γ. We number ∂_{target}^{−1}(v)∪∂_{source}^{−1}(v) as follows. There exists a unique edge e such that ∂_{source}(e) = v. We denote it by e_{0}(v). Now using cyclic order induced by the ribbon structure we order the edges in ∂_{target}^{−1}(v) and denote them by e_{1}(v), ⋯, e_{v(m)}(v).

Condition 3.2. (1) If we restrict ψ̃ to a edge |e| and if we identify |e| ≅ [0, l(e)] then

\frac{dψ}{dτ} = gradf_{L}. (26)

where we use τ as a coordinate of [0, l(e)] and the branch of our multivalued function f_{L} is determined by β(τ).

(2) Let τ_{i,j} ∈ |e_{i}(v)| such that lim_{j→∞} τ_{i,j} = |v| then

lim_{j→∞} p(τ_{i+1,j}) = lim_{j→∞} q(τ_{i,j}) ∈ F_{ψ(|v|)},  i = 1, ⋯, m(v) − 1,
lim_{j→∞} q(τ_{m(v),j}) = lim_{j→∞} q(τ_{0,j}) ∈ F_{ψ(|v|)},
lim_{j→∞} p(τ_{1,j}) = lim_{j→∞} p(τ_{0,j}) ∈ F_{ψ(|v|)}.

We also have

∑_{i=1}^{m(v)} lim_{j→∞} β(τ_{i,j}) = lim_{j→∞} β(τ_{0,j}).

Note we use the obvious sum π_{2}(M, F_{x}; p_{1}, p_{2}) × π_{2}(M, F_{x}; p_{2}, p_{3}) → π_{2}(M, F_{x}; p_{1}, p_{3}) in the left hand side. They both are elements of the homotopy set : π_{2}(M; F_{ψ(|v|)}; lim_{j→∞} p(τ_{0,j}), lim_{j→∞} q(τ_{0,j})).

Figure 3.6

(3) Let v ∈ Vertex_{sta}(Γ) and ∂_{source}(e) = v. we identify |e| ≅ (−∞, 0] and use τ as a coordinate. Then β(τ) is a vanishing disk and lim_{τ→−∞} ψ(τ) ∈ S(L).

(4) The case Condition 2.3 (b) is satisfied : Let e_{last} be the edge such that ∂_{target}(e_{last}) = v_{last}. We identify |e_{last}| ≅ [0, ∞) and use τ as a coordinate. Then −∂_{+}β(τ) is a vanishing disk and lim_{τ→∞} ψ(τ) ∈ S(L).

The case Condition 2.3 (a) is satisfied : We identify |e| ≅ [0, |l(e_{last})|] (Note l(e_{last}) < ∞ in this case.) We define

ẽv(Γ, l, ψ̃) = ψ̃(v_{last}), ev(Γ, l, ψ̃) = ψ(v_{last}).

Let α ∈ π_{2}(M, L). We consider also the following Condition 3.3 in case Condition 2.3 (b) is satisfied. We use the notation of Condition 3.2 (4). We choose a disk u bounding −∂_{+}β(t) and its image is in π^{−1}(U) where U is a small neighborhood of lim_{t→−∞} ψ(t) ∈ S(L). (Such u exists by (4)).

Condition 3.3. α = [β(t) ∪ u].

Definition 3.9. We denote by M_{Morse}(M, L; α) the set of all (Γ, l, ψ̃) satisfying Conditions 2.3(b), 3.1, 3.2, 3.3.

Let denote by M_{Morse}(M, L) the set of all (Γ, l, ψ̃) satisfying Conditions 2.3(a), 3.1, 3.2. We define ẽv : M_{Morse}(M, L) → B̃_{L} and ev : M_{Morse}(M, L) → B as in Condition 3.2(4).

4 B MODEL - ASYMPTOTIC ANALYSIS - 38

Now we can state an analogue of Conjecture 2.1 Conjecture 3.4. We assume S(B) = ∅. Then the moduli space of J_{ϵ} holomorphic disks u : D^{2} → M such that u(∂D^{2}) ⊂ L and [u] = α is cobordant to the moduli space M_{Morse}(M, L; α).

We consider the moduli space of (u, x) of pairs of point x ∈ B and a pseudoholomorphic map u : (D^{2}; ∂_{−}D^{2}, ∂_{+}D^{2}) → (M; L, F_{x}). We consider a map (u, x) ↦ x. Then it is bordant to the moduli space M_{Morse}(M, L) together with ev.

In a way similar to the discussion after Conjecture 2.1 one can reduce Conjecture 3.4 to the gluing argument in [36], results of §3.3 and Conjectures in §3.5. Remark 3.7. We do not need to restrict ourselves to the case of pseudoholomorphic disks in Conjecture 3.4. Namely we may generalize it to the case of Riemann surface of higher genus with boundary. In the case when M = T^{∗}N where N is a closed 3 manifold, it is related to Witten’s paper [77]. After [77] was distributed (in 1992), the author pointed out in [29] (which was written and submitted for publication in 1993) its relation to Morse theory. This point of view is further amplified by [27], [36], [28]. In those papers, the case when fibers are vector spaces is considered. Recently Kontsevich-Soibelman [61] showed that the same idea can be applied to the case when fibers are torus. (They however consider the case S(L) = B(L) = ∅.) Remark 3.8. After reading this and the last sections, it is easy to guess that there is a version which involves both open and closed string, using open string vs closed string scattering discussed in §3.4. It is expected to describe the moduli space of pseudoholomorphic disks (D^{2}, ∂D^{2}) → (M, L) in the case when S(B) ≠ ∅. The reason we did not state a conjecture in this general case is as follows. In §3.4 the open string vs closed string scattering is described by the intersection numbers of the bounding arcs u(∂_{+}u) and v(∂D^{2}). Here u : (D^{2}; ∂_{+}D^{2}, ∂_{−}D^{2}; −1, +1) → (M; L, F_{x}; p, q) is an open string and v : (D^{2}, ∂D^{2}) → (M, F_{x}) is a closed string. The trouble is that the intersection number u(∂_{+}u) ∩ v(∂D^{2}) is not an invariant of the homology (or homotopy) class [v(∂D^{2})] ∈ H_{1}(F_{x}). Actually if we move v continuously and when it path through one of the points p, q then the intersection number changes. So we need to study more to find a correct formulation of the conjecture. We leave it to the future research.

4 B model - asymptotic analysis -

Let us start with closed string version. Let M_{0} → B\S(B) be as Assumption 2.1. We consider its mirror M_{0}^{∨} constructed in §1. The symplectic structure ω on M_{0} induces a complex structure J_{0} on M_{0}^{∨}. We can extend ω to the compactification M. So we put it as a part of Assumption 2.1. On the other hand, the complex structure J_{0} does not extend to M. In physics literature, there are various arguments which suggest that the contribution of pseudoholomorphic disks bounding the fibers F_{x} modifies the complex structure J_{0} so that it can be extended to the singular fibers.

There are explicit formulae which describe modified complex structure locally in a neighborhood of some singular fibers. In Ooguri-Vafa [68] such a formula is given in the case of type I singular fiber. Recently it was used by [47] to study collapsing of of K3 surface in a neighborhood of large complex structure limit. More precisely, in [47], Gross-Willson used the metric constructed by [68] in each neighborhood of singular fiber (type I singular fiber), glue it with flat Calabi-Yau metric (which is a Kähler metric with respect to J_{0}) by partition of unity, and use functional analysis to show that there exists a Calabi-Yau metric which is close to the metric in a neighborhood of singular fiber and to a flat metric outside a neighborhood of singular fiber. The approximate solution they start with is fairly close to the actual solution. Namely the difference between the approximate solution and the actual solution is of order e^{−C/ϵ} where ϵ is the diameter of the fiber. It is known in physics literature that “instanton effect” (which is the contribution of pseudoholomorphic disks in the situation we are discussing now) is small of exponential order. However for various purposes we need more precise information of the metric or of complex structure.

One of the important consequences of mirror symmetry is coincidence of product structure of sheaf cohomology (Yukawa coupling) with the Gromov-Witten invariant of its mirror. When one tries to prove it using (singular) torus fibration picture we are discussing here, we need to study the “instanton effect” to the complex structure. So we need to study this exponentially small part of the complex structure. In asymptotic analysis, it is known that tunnel effect (see §4.2 for its definition in a situation which is closely related to one we are interested in) is also a term of exponentially small order. The main purpose of this section is to explain that Yukawa coupling is regarded as a kind of the tunnel effect (see §4.2) and show a relation of it to the Morse theory picture of pseudoholomorphic disks described in §2,3. However our discussion is incomplete by various reasons which are explained in §4.3. Since the case of instanton effect to J_{0} is harder, we study a simpler case of vector bundle, in the next section, which is related to homological mirror symmetry. In that case we can present the story in more detail.

4.1 Free Field - Fourier expansion of complex structure -

This section is influenced much by Arinkin-Polishchuk’s paper [5] which discuss relation of path space π_{1}(F_{x}; F_{x} ∩L) with Fourier expansion. Here we consider “closed string case” but the basic idea is the same. The open string version (which is more directly an analogue of Arinkin-Polishchuk’s paper), will be discussed in the next section.

Let us consider M_{0} → B\S(B) be as Assumption 2.1. We consider its mirror M_{0}^{∨} which has a T structure invariant complex structure J_{0} induced by the symplectic structure of M_{0}, by Proposition 1.5. Namely if we take a symplectic coordinate x^{i},y^{i} of M_{0} (namely ω = ∑ dx^{i} ∧ dy^{i}) then the complex coordinate of the mirror is z_{ϵ}^{i} = x^{i} + \sqrt{−1}ϵy^{i∗}. Here y^{i∗} is a dual coordinate in the following sense. There exists an open set U of B such that π_{M_{0}}^{−1}(U) = U × T^{n}. Then x^{i} is a coordinate of U and y^{i} is a coordinate of T^{n} = V/Γ such that V ≅ ℝ^{n}, Γ = ℤ^{n}. The corresponding open set of M_{0}^{∨} is U × T^{n∗} where T^{n∗} = V^{∗}/Γ^{∨}. Here V is a dual vector space to V and Γ^{∨} is a dual lattice of Γ. Then y^{i∗} is the dual basis of y^{i}.

We denote by \overline{∂}_{0,ϵ} the Dolbaut differential associated to J_{0,ϵ}. (Here J_{0,ϵ} ( \frac{∂}{∂x^{i}} ) = \frac{1}{ϵ} ( \frac{∂}{∂y^{i}} ).) We consider another almost complex structure on

M_{0} and let \overline{∂} is its Dolbaut differential. We put

\overline{∂} = \overline{∂}_{0,ϵ} + 𝔅 ∘ ∂_{0,ϵ} (27)

where 𝔅 ∈ Γ(M_{0}; T M_{0}⊗Λ^{0,1}). According to Kodaira-Spencer and Newlander-Nirenberg, the condition for \overline{∂} to be integrable is

\overline{∂}_{0,ϵ}𝔅 + \frac{1}{2} [𝔅, 𝔅] = 0. (28)

We are going to study equation (28) using fiberwise Fourier expansion.

Lemma 4.1. There exists a canonical isomorphism

Γ(B\S(B); T B ⊗ Λ^{k}(B)) ≅ Γ(M_{0}^{∨}; M_{0}^{∨} ⊗ Λ^{0,k}(M_{0}^{∨}))^{T} . (29)

Here the right hand side is the set of T structure invariant sections.

Proof. We use local complex coordinate z_{ϵ}^{i} = x^{i} + \sqrt{−1}ϵy^{i∗} as above. T structure invariant section locally can be written as

s = ∑ s_{J}^{i}(x^{1}, ⋯, x^{n}) \frac{∂}{∂z_{ϵ}^{i}} ⊗ d\overline{z}_{ϵ}^{J},

where J = (j_{1}, ⋯, j_{k}) is a multi index. Here the T structure invariance of s is equivalent to the fact that s_{J}^{i} depends only on x^{k} and is independent of y^{k∗}. We put

\overline{s} = ∑ s_{J}^{i}(x^{1}, ⋯, x^{n}) \frac{∂}{∂x^{i}} ⊗ dx^{J}.

It is easy to see that s ↦ \overline{s} gives the required isomorphism.

Hereafter we denotes the isomorphism (29) by ℑ_{ϵ}.

Let x ∈ B\S(B). We remark that F_{x} = π_{M_{0}^{∨}}^{−1} is the dual torus of F_{x}^{∨} =

π_{M_{0}}^{−1}. Hence by Pontrjyagin duality we have a canonical isomorphism

π_{1}(F_{x}) = Hom(F_{x}^{∨}, U(1)).

For γ ∈ π_{1}(F_{x}), we denote by exp γ^{∗} the corresponding element of Hom(F_{x}^{∨}, U(1)). We then have immediately the following :

Lemma 4.2. Let U be an contractible open neighborhood of x in B\S(B). Then any element s of Γ(π_{M_{0}^{∨}}^{−1}(U); M_{0}^{∨} ⊗ Λ^{0,k}(M_{0}^{∨})) can be uniquely expanded as

s = ∑_{γ∈π_{1}(F_{x}^{∨})} (exp γ^{∗}) s_{γ} (30)

such that s_{γ} is invariant of T structure.

Now let 𝔅 be as in (27). We expand 𝔅 as

𝔅 = ∑_{γ∈π_{1}(F_{x}^{∨})} exp γ^{∗} ℑ_{ϵ}(𝔅_{γ}) (31)

locally. We assume 𝔅_{0} = 0. Namely the T structure invariant part of 𝔅 is zero. (We recall that the T structure invariant part can be included to J_{0,ϵ}. Hence we may assume it is zero without loosing generality.)

In this section we discuss Free field. Namely we study an equation

\overline{∂}_{0,ϵ}𝔅 = 0, (32)

which is a linearization of (20).

Proposition 4.1. 𝔅 satisfies (32) if and only if

d(e^{2πf_{γ} /ϵ}𝔅_{γ}) = 0. (33)

Here f_{γ} is a branch of the multivalued function f, defined in §2.1, corresponding to [u] ∈ π_{2}(N; F_{x}) such that [∂u] = [γ] and d is the exterior derivative of T B valued one form 𝔅_{γ}. We define exterior derivative of T B valued forms by using flat affine structure of B\S(B).

Proof. The problem is local so we may use local coordinate x^{i}, y^{i∗}. Let (γ_{1}, ⋯, γ_{n}) ∈ ℤ^{n} ≅ π_{1}(F_{x}). As we see before f_{γ} = ∑ x^{i}γ_{i} + c where c is a constant. On the other hand, exp γ^{∗} = exp (2π\sqrt{−1} ∑ γ_{i}y^{i∗}). Hence

\overline{∂}_{0,ϵ}𝔅 = \frac{1}{2} ∑ ( \frac{∂}{∂x^{i}} + \frac{2π}{ϵ} γ_{i}) exp γ^{∗}d\overline{z}_{ϵ}^{i} ∧ ℑ_{ϵ}(𝔅_{γ}).

The proposition follows.

Thus linearization (32) of (20) is equivalent to a system of differential equations (33). Equation (33) is a multivalued version of the deformed De Rham complex studied by a celebrated paper of Witten [76], which we review in the next subsection.

4.2 Super symmetry and Morse theory, revisited.

In this section we review [76, 49] and propose some conjectures which generalize it. The generalization we discuss in this section is based on Morse homotopy [29, 36] (see also [14]) and is closely related to [77], §8 of [27] and also to [61].

Let X be a Riemannian manifold and f be a Morse function on it. (In our situation of multivalued Morse theory, X will correspond to B and f to our multivalued function.) We consider the De Rham complex (Γ(X; Λ(X)), d). Let ∗ : Γ(X; Λ^{k}(X)) → Γ(X; Λ^{n−k}(X)) be the Hodge star operator (where n = dim X). Witten considered the differential operator

d_{f,ϵ} = exp(−f/ϵ) ∘ d ∘ exp(f/ϵ) (34)

and

Δ_{f,ϵ} = d_{f,ϵ} ∘ d_{f,ϵ}^{∗} + d_{f,ϵ}^{∗} ∘ d_{f,ϵ}.

Here d_{f,ϵ}^{∗} is the L^{2} dual to d_{f,ϵ} and is

d_{f,ϵ}^{∗} = − exp(−f/ϵ) ∘ ∗ ∘ d ∘ ∗ ∘ exp(f/ϵ).

Obviously (Γ(X; Λ(X)), d_{f,ϵ}) is a cochain complex and its cohomology is isomorphic to the cohomology of (Γ(X; Λ(X)), d), the De Rham cohomology. We remark that operator (33) is similar to the operator (34).

Witten observed the following Theorem 4.1 which is proved rigorously, for example, in [69]. Let Cr(f) be the set of all critical points of f. Let Cr_{k}(f) ⊆ Cr(f) be the subset of critical points of Morse index n − k. Hereafter we denote the Morse index of p by n − η(p). For δ > 0 we put

H_{δ}^{k}(X; Δ_{f,ϵ}) = {u ∈ Γ(X; Λ^{k}(X)) | Δ_{f,ϵ}u = λu, λ < δ}.

Theorem 4.1. For sufficiently small ϵ, δ, the dimension of the vector space H_{δ}^{k}(X; Δ_{f,ϵ}) is equal to the order of Cr_{k}(f). Moreover there exists u_{p} ∈ H_{δ}^{k}(X; Δ_{f,ϵ}) for each p ∈ Cr_{k}(f) such that

|u_{p}| < C exp(−c/ϵ)∥u_{p}∥_{L^{2}}

outside a small neighborhood of p.

Hereafter we normalize so that ∥u_{p}∥ = 1.

For our purpose we need more detailed information. Namely we need to study tunnel amplitude and also interactions (that is k point function with k ≥ 3). We start with tunnel amplitude. Let p ∈ Cr_{k}(f), q ∈ Cr_{k+1}(f). We are going to study ⟨d_{f,ϵ}u_{p}, u_{q}⟩_{L^{2}}. An observation by Witten, which is proved rigorously by B.Helffer and J.Sjöstrand in [49], is that this number is described by the number of gradient lines joining p to q. To state it precisely we need some notations.

We assume that gradf is Morse-Smale. Namely its stable manifolds and unstable manifolds intersect transversally. Let us consider the set of all ℓ : ℝ → X such that dℓ/dτ = grad_{ℓ(τ)}f and lim_{τ→−∞} ℓ(τ) = p, lim_{τ→+∞} ℓ(τ) = q. We say ℓ ∼ ℓ′ if there exists τ_{0} such that ℓ′(τ) = ℓ(τ + τ_{0}). We denote by M(X; f; p, q) the set of all ∼ equivalence classes of such ℓ. Since we assumed gradf to be Morse-Smale it follows that M(X; f; p, q) is a smooth manifold of dimension μ(q) − μ(p) − 1.

To fix orientation on M(X; f; p, q) we need to fix the following extra data. (See [38] the beginning of page 1037 for detail.) We fix an orientation of T_{p}W^{u}(f; p) where W^{u}(f; p) = {x ∈ X lim_{τ→−∞} exp(τgradf)(x) = p} is the unstable manifold of the gradient flow. (Here exp(τgradf) is the one parameter group of transformations associated to gradf.) We also fix an orientation of T_{q}W^{u}(f; q) As explained in [38] it induces an orientation of M(X; f; p, q). As we remarked in [38] the orientation induced on M(X; f; p, q) is independent of the orientation of X. (Actually we do not need to assume that X is orientable.)

We also remark after normalization there are two choices u_{p}. (Namely we may replace u_{p} by −u_{p}. To fix this choice we need to take an orientation of W^{u}(f; p). (See [69] page 166. The eigenform e_{0} there depends on the oriented orthonormal basis dx^{i}.)

Now we have :

Theorem 4.2. (Helffer-Sjöstrand )

⟨d_{f,ϵ}u_{p}, u_{q}⟩_{L^{2}} ∼ e^{(f(p)−f(q))/ϵ}[M(X; f; p, q)].

Here [M(X; f; p, q)] ∈ ℤ is the order counted with sign and ∼ means that the difference C_{ϵ} between left and right hand side is estimated as

lim_{ϵ→0} e^{(f(q)−f(p))/ϵ}|C_{ϵ}| = 0.

Theorem 4.1 means that u_{p} is concentrated on a neighborhood of p. Theorem 4.2 means that u_{p} propagates along the gradient line of f.

For our purpose, we need even more detailed information which is related to the product structure. Namely we need to find a relation between a number similar to ⟨d_{f,ϵ}u_{p}, u_{q}⟩ and Morse homotopy ([29, 36]). For this purpose we need to take several Morse functions as follows. Let f_{i} i = 0, 1, 2 be functions. We assume f_{i} − f_{j} i ≠ j are Morse functions. We take p_{1} ∈ Cr(f_{1} − f_{0}), p_{2} ∈ Cr(f_{2} − f_{1}) and q ∈ Cr(f_{2} − f_{0}). We assume μ(p_{1}) + μ(p_{2}) = μ(q) where n − μ is the Morse index. We want to study asymptotic behavior of ⟨u_{p_{1}} ∧ u_{p_{2}}, u_{q}⟩. We let M(X; f_{1}, f_{2}, f_{3}; p_{1}, p_{2}, q) be the set of all x ∈ X such that

lim_{τ→−∞} exp(τ(gradf_{1} − gradf_{0}))(x) = p_{1},
lim_{τ→−∞} exp(τ(gradf_{2} − gradf_{1}))(x) = p_{2},
lim_{τ→+∞} exp(τ(gradf_{2} − gradf_{0}))(x) = q.

M(X; f_{0}, f_{1}, f_{2}; p_{1}, p_{2}, q) is an intersection of two unstable manifolds and a stable manifold. For generic f_{1}, f_{2}, f_{3}, it is a smooth manifold of dimension μ(p_{1}) + μ(p_{2}) − μ(q) = 0. So we assume that M(X; f_{0}, f_{1}, f_{2}; p_{1}, p_{2}, q) consists of finitely many points. The orientations of T_{p_{1}}W^{u}(f_{1} − f_{0}; p_{1}), T_{p_{2}}W^{u}(f_{2} − f_{1}; p_{2}), T_{q}W^{u}(f_{2} − f_{0}; q) induce an orientation of the moduli space M(X; f_{0}, f_{1}, f_{2}; p_{1}, p_{2}, q).

Conjecture 4.1. Using notations similar to Theorem 4.1 we have :

\begin{matrix}⟨u_{p_{1}} ∧ u_{p_{2}}, u_{q}⟩ \\ ∼ e^{−(f_{1}(p_{1})−f_{0}(p_{1})+f_{2}(p_{2})−f_{1}(p_{2})+f_{0}(q)−f_{2}(q))/ϵ}[M(X; f_{0}, f_{1}, f_{2}; p_{1}, p_{2}, q)]. \end{matrix}

An idea behind the Conjecture 4.1 is as follows. As we mentioned before, u_{p_{i}} are concentrated in a neighborhood of p_{i}. Hence u_{p_{1}} ∧ u_{p_{2}} is zero up to exponentially small term. To study those small terms, we need to see how they propagate. Theorem 4.1 suggests that they propagate along the gradient lines of f_{i+1} − f_{i}. Hence u_{p_{1}} ∧ u_{p_{2}} will be nonzero at the point where two gradient lines, one of f_{1} − f_{0} the other of f_{2} − f_{1}, intersect each other. From the intersection points, it propagates along the gradient line of f_{2} − f_{1}. Hence the left hand side may be described by the moduli space M(X; f_{0}, f_{1}, f_{2}; p_{1}, p_{2}, q).

Conjecture 4.1 is on three points function. We need further generalize it to 4 points functions or more. Let us first discuss the generalization of the left hand side. (Compare [61] §6.4 and [37] §A6.)

Let Γ be a ribbon tree with k + 1 exterior vertices. We assume Condition 2.2 (1),(2),(3),(4). There exists an embedding |Γ| ⊂ D^{2} which is compatible with ribbon structure and such that the exterior vertices are on ∂D^{2}. We let v_{last} = v_{0} (the last exterior vertex) and number the other exterior vertices v_{1}, ⋯, v_{k} so that v_{0}, v_{1}, ⋯, v_{k} respects the cyclic order of ∂D^{2}. The domain D^{2}\|Γ| has exactly k + 1 connected components D_{i}, i = 0, ⋯, k. We number them so that v_{i} ∈ \overline{D}_{i} ∩\overline{D}_{i−1} for i = 1, ⋯, k and v_{0} ∈ \overline{D}_{0}∩\overline{D}_{k}. For each edge e of Γ there exists exactly two of D_{i}’s D_{right(e)} and D_{left(e)}, such that |e| ⊂ \overline{D}_{right(e)} ∩ \overline{D}_{left(e)} and that left(e) > right(e). We take smooth functions f_{i}, i = 0, ⋯, k such that f_{i} − f_{j} are Morse functions.

We put f_{e} = f_{left(e)} − f_{right(e)} and consider the operators Δ_{f_{e},ϵ}, d_{f_{e},ϵ} in §4.1. Let Π_{f_{e},ϵ,δ}^{k} : L^{2}(X; Λ^{k}(X)) → H_{δ}^{k}(X; Δ_{f_{e},ϵ}) be the orthonormal projection. The following lemma can be proved in the same way as usual Harmonic analysis.

Lemma 4.3. There exists a pseudo differential operator

Q_{f_{e},ϵ,δ}^{k} : Γ(X; Λ^{k}(X)) → Γ(X; Λ^{k}(X))

of degree −2 such that

Δ_{f_{e},ϵ} ∘ Q_{f_{e},ϵ,δ}^{k} − Q_{f_{e},ϵ,δ}^{k} ∘ Δ_{f_{e},ϵ,δ}^{k} = 1 − Π_{f_{e},ϵ,δ}^{k}.

Definition 4.1. We define the propagator

G_{f_{e},ϵ,δ}^{k} : Γ(X; Λ^{k}(X)) → Γ(X; Λ^{k+1}(X))

by

G_{f_{e},ϵ,δ}^{k} = d_{f_{e},ϵ}^{∗} ∘ Q_{f_{e},ϵ,δ}^{k}.

We can easily prove that

d_{f_{e},ϵ} ∘ G_{f_{e},ϵ,δ}^{k} + G_{f_{e},ϵ,δ}^{k−1} ∘ d_{f_{e},ϵ} = 1 − Π_{f_{e},ϵ,δ}^{k}. (35)

Remark 4.1. We consider two chain maps id : (H_{δ}^{∗}(X; Δ_{f,ϵ}); d_{f,ϵ}) → (Γ(X; Λ(X)); d_{f_{e},ϵ}) and Π_{f_{e},ϵ,δ} : (Γ(X; Λ(X)); d_{f_{e},ϵ}) → (H_{δ}^{∗}(X; Δ_{f_{e},ϵ}); d_{f_{e},ϵ}). The composition Π_{f_{e},ϵ,δ} ∘ id is obviously an identity. On the other hand,

(35) implies that G_{f_{e},ϵ,δ} is a chain homotopy from identity to id ∘Π_{f_{e},ϵ,δ}. Now using propagator G_{f_{e},ϵ,δ} we define operators

𝔪_{Γ} : H_{δ}^{∗}(X; Δ_{f_{1}−f_{0},ϵ}) ⊗ ⋯ ⊗ H_{δ}^{∗}(X; Δ_{f_{k}−f_{k−1},ϵ}) → H_{δ}^{∗}(X; Δ_{f_{k}−f_{0},ϵ})
φ_{Γ} : H_{δ}^{∗}(X; Δ_{f_{1}−f_{0},ϵ}) ⊗ ⋯ ⊗ H_{δ}^{∗}(X; Δ_{f_{k}−f_{k−1},ϵ}) → Γ(X; Λ(X))

as follows. The definition is by induction on k, the number of exterior vertices plus 1.

In the case k = 0, there exists only one Γ say Γ_{0}. We put

𝔪_{Γ_{0}} = d_{f_{e},ϵ}, φ_{Γ_{0}} = id .

We assume that we have constructed 𝔪_{Γ} up to k − 1. Let v_{last} be the last vertex of Γ. There exists a unique edge e_{last} which has v_{last} as an vertex. We remove |e_{last}| from |Γ|. Then |Γ|\|e_{last}| is a union of two ribbon trees Γ_{1},Γ_{2}. We now put

(36)  φ_{Γ} = G_{f_{e_{last}} ,ϵ,δ} ∘ ∧ ∘ (φ_{Γ_{1}} ⊗ φ_{Γ_{2}})
(37)  𝔪_{Γ} = Π_{f_{e_{last}} ,ϵ,δ} ∘ ∧ ∘ (φ_{Γ_{1}} ⊗ φ_{Γ_{2}}).

The operator 𝔪_{Γ} defines an A_{∞} structure and φ_{Γ} defines a homotopy equivalence from it to the De Rham complex of X. More precisely we have the following Theorem 4.3.

We put 𝔪_{k} = ∑_{Γ} 𝔪_{Γ}, φ_{k} = ∑_{Γ} φ_{Γ}, where the sum is taken over all ribbon trees Γ which satisfies Condition 2.2 (1),(2),(3),(4) and has k + 1 vertices.

Theorem 4.3. 𝔪_{k} satisfies A_{∞} formulae. Namely for any elements x_{i} ∈ H_{δ}^{∗}(X; Δ_{f_{i}−f_{i−1},ϵ}), we have :

∑_{k+ℓ=n+1} ∑_{i=1}^{n+1−ℓ}(−1)^{∗}𝔪_{k}(x_{1}, ⋯, x_{i−1}, 𝔪_{ℓ}(x_{i}, ⋯, x_{i+ℓ−1}), x_{i+ℓ}, ⋯, x_{n}) = 0. (38)

Here the sign is defined by : ∗ = deg x_{1} + ⋯ + deg x_{i−1} + i − 1.

The homomorphisms φ_{k} : H_{δ}^{∗}(X; Δ_{f_{1}−f_{0},ϵ})⊗⋯⊗H_{δ}^{∗}(X; Δ_{f_{k}−f_{k−1},ϵ}) → Γ(X; Λ(X)) define an A_{∞} functor.

We do not prove Theorem 4.3 since we do not use it. We omit the definition of A_{∞} functor. See [56, 40, 34, 33, 37] for more detail on homological algebra of A_{∞} algebra and of A_{∞} category. Remark 4.2. In the case when f_{j} − f_{i} are perfect Morse functions (that is the case when H_{δ}^{∗}(X; Δ_{f_{k}−f_{k−1},ϵ}) is isomorphic to the De Rham cohomology) Theorem 4.3 implies that there exists an A_{∞} structure on cohomology group which is homotopy equivalent to the De Rham complex (and an A_{∞} algebra). (Actually we need to say A_{∞} category instead of A_{∞} algebra, since 𝔪_{k} depends on the choice of f_{i}.) This statement was first proved by Kadeishvili in [55]. See also [48, 66, 61, 33]. We can prove Theorem 4.3 in the same way. The relation of this construction to Feyman diagram is found in [61] and in [37].

Now let us consider p_{i} ∈ Cr(f_{i} − f_{i−1}), q ∈ Cr(f_{k} − f_{0}). We assume ∑ μ(p_{i}) − μ(q) = k − 2. It is easy to see that the degree of the operator 𝔪_{k} is 2 − k. So the number

⟨𝔪_{Γ}(u_{p_{1}} ⊗ ⋯ ⊗ u_{p_{k}}), u_{q}⟩

is well defined. It is a generalization of the left hand side of Conjecture 4.1.

We next describe the generalization of the right hand side of Conjecture 4.1. It is quite similar to one we discussed in §3.6. Let Γ, f_{i}, p_{i}, q be as above. We consider the pair (ℓ, ψ) where ℓ : Edge → ℝ_{+} ∪ {∞}, and ψ : |Γ|\{|v_{0}|, ⋯, |v_{k}|} → X is a continuous map which is smooth on each edge. We require the following condition.

Condition 4.1. (1) ℓ(e) is ∞ if e is exterior. ℓ(e) is finite if e is interior.

(2) For each interior edge e, we identify |e| ≅ [0, ℓ(e)]. If ∂_{source}(e) ∈ Vertex_{ext}(Γ) we identify |e| ≅ (−∞, 0]. It ∂_{target}(e) = v_{last}, we identify |e| ≅ [0, ∞). Now we have

\frac{dψ}{dτ} = gradf_{e}

on each edge e. (3) If ∂_{source}(e) = v_{i} ∈ Vertex_{ext}(Γ) and if τ_{i} ∈ (−∞, 0] ≅ |e| with lim_{i} τ_{i} = −∞, then we have lim_{i} ψ(τ_{i}) = p_{i}. (4) If ∂_{target}(e) = v_{last}, τ_{i} ∈ [0, +∞, 0) ≅ |e| with lim_{i} τ_{i} = ∞, then we have lim_{i} ψ(τ_{i}) = q.

Definition 4.2. M(X; Γ; f_{0}, ⋯, f_{k}; p_{1}, ⋯, p_{k}, q) is the set of all (ℓ, ψ) satisfying Condition 4.1.

The following lemma is proved in [36].

Lemma 4.4. For generic f_{i} the moduli space M(X; Γ; f_{0}, ⋯, f_{k}; p_{1}, ⋯, p_{k}, q) is a smooth manifold of dimension ∑ μ(p_{i}) − μ(q) − k + 2.

Thus by our assumption, M(X; Γ; f_{0}, ⋯, f_{k}; p_{1}, ⋯, p_{k}, q) consists of finitely many points. We can now state our conjecture. We put

λ = f_{k}(q) − f_{1}(q) − ∑_{i}(f_{i}(p_{i}) − f_{i−1}(p_{i})).

Conjecture 4.2. Orientations of unstable manifolds determine an orientation of M(X; Γ; f_{0}, ⋯, f_{k}; p_{1}, ⋯, p_{k}, q), so that the following asymptotic expansion holds.

⟨𝔪_{Γ}(u_{p_{1}} ⊗ ⋯ ⊗ u_{p_{k}}), u_{q}⟩ ∼ e^{−λ/ϵ}[M(X; Γ; f_{0}, ⋯, f_{k}; p_{1}, ⋯, p_{k}, q)].

To explain the meaning of Conjecture 4.2 we recall the following Theorem 4.4 proved in [26, 36]. We define the Morse complex MC(f_{i} − f_{i−1}) by

\begin{aligned}MC(f_{i} − f_{i−1})^{k} & ≅ ⨁_{p∈Cr_{k}(f_{i}−f_{i−1})} ℝ[p] \\ d[p] & = ∑_{q∈Cr_{k+1}(f_{i}−f_{i−1})}[M(X; f; p, q)][q]. \end{aligned}

We define A_{∞} operation by using the moduli space M(X; Γ; f_{0}, ⋯, f_{k}; p_{1}, ⋯, p_{k}, q) as follows.

\begin{aligned}𝔪_{Γ}([p_{1}], ⋯, [p_{k}]) & = ∑[M(X; Γ; f_{0}, ⋯, f_{k}; p_{1}, ⋯, p_{k}, q)][q], \\ 𝔪_{k} & = ∑ 𝔪_{Γ} k ≥ 2, \\ 𝔪_{1} & = d. \end{aligned}

Theorem 4.4. ([36]) 𝔪_{k} satisfies the A_{∞} formula (38).

Now we consider an isomorphism I : MC(f_{i}−f_{i−1}) ≅ H_{δ}(X; Δ_{f_{1}−f_{0},ϵ}) by

I([p]) = e^{f_{i}(p)−f_{i−1}(p)}u_{p}.

Then Conjecture 4.2 and Theorem 4.2 will imply that I is an isomorphism of the A_{∞} category. On the other hand, Theorem 4.3 implies that (H_{δ}(X; Δ_{f_{1}−f_{0},ϵ}), 𝔪_{k}) is homotopy equivalent to the De Rham complex.

We remark that, in [61], it is proved that MC(f_{i} − f_{i−1}) is homotopy equivalent to the De Rham complex. However Conjecture 4.2 itself is not discussed there. One may try to generalize the argument of Helffer-Sjöstrand [49] to prove Conjectures 4.1, 4.2 though the author did not succeed it yet at the time of writing this article.

4.3 Quantum effect to complex structure - a plan -

We now pursuit the analogy of the results and conjectures in §4.2 to the study of (28). Proposition 4.1 shows that the operator \overline{∂}_{0,ϵ} reduces to the operator d_{f,ϵ} in each Fourier mode. (Here f is our multivalued function.)

Let us next study the solution of (28) itself (not its linearization). In this section we state a conjecture of the asymptotic behavior of 𝔅 solving (28).

To state our conjecture, we need to assume that dim_{ℂ} M = 2, since otherwise closed string closed string scattering is more complicated to describe. In other words, we consider the case when M is a K3 surface.

Let us consider the covering space B̃_{0} of B_{0} = B\S(B) introduced in §2.1. Let 𝔅̃ ∈ Γ(B̃_{0}^{+}, T B̃_{0} ⊗Λ^{1}B̃_{0}). We can associate 𝔅 ∈ Γ(M_{0}^{∨}, T_{ℂ}M_{0}^{∨} ⊗ Λ^{0,1}M_{0}^{∨}) as follows.

Let p ∈ B_{0}. For each [u] ∈ π_{2}(M_{0}, F_{x}) we have 𝔅̃(p, [u]) which can be identified as an element of of the fiber of T B ⊗ Λ^{1}B at p. We take it as 𝔅_{[u]} at p. Then we put

𝔅 = ∑_{[u]∈π_{2}(M_{0},F_{x})} e^{−2πf_{u}/ϵ}[∂u]^{∗} ⊗ ℑ_{ϵ}(𝔅_{[u]}), (39)

as in (31). Here f_{u} is the branch of our multivalued function f specified by [u] ∈ π_{2}(M_{0}, F_{x}), and [∂u] ∈ π_{1}(F_{x}). (Actually the author does not know whether the right hand converges. So to be precise, we need to regard the right hand side as an asymptotic expansion. See Conjecture 4.3.)

Thus 𝔅_{[u]} in (39) and 𝔅_{γ} in (31) is related by the formula

𝔅_{γ} = ∑_{[u]∈π_{2}(M_{0},F_{x}),∂[u]=γ} e^{−2πf_{u}/ϵ}𝔅_{[u]}.

\overline{∂}𝔅 = 0 then becomes d𝔅_{[u]} = 0.

Now we take ev_{last} : M_{Morse}(M; g) → B̃_{0} as in Definitions 2.7,2.8. (We remark that the multiplicity of the moduli space M_{Morse}(M; g) was left as an open problem in §2.5 in the case of multiple covered disk. So the definition of M_{Morse}(M; g) was not rigorous in that case.)

It is easy to check that the dimension of M_{Morse}(M; g) is one in our case. We regard it as a distribution valued 1 form on it. (1 is the codimension of M_{Morse}(M; g) in B̃_{0}. Note dim_{ℂ} M^{∨} = 2.) To add T B̃_{0}^{+} ≅ π^{∗}T B factor, we proceed as follows. We consider the canonical isomorphism

H_{1}(F_{x}) ≅ H^{1}(F_{x}) ≅ T_{s_{0}(x)}^{∗}F_{x} ≅ T_{x}(B_{0}). (40)

Here the first isomorphism is the Poincaré duality. (Note we are considering the case dim F_{x} = 2.) There exists a second isomorphism since F_{x} is a torus equipped with an affine structure. The third isomorphism is induced by the symplectic form on M_{0}. We remark that all the isomorphisms in (40) are canonical. By (40) element [γ] ∈ H_{1}(F_{x}) corresponds to an element of T_{x}(B_{0}) which we write d_{[γ]}.

Let U_{x} be a small open neighborhood of x in B_{0}. We consider an open subset π_{M_{0}^{∨}}^{−1}(U_{x}) of M_{0}^{∨}. Our vector field d_{[γ]} determines a T structure

invariant section of Λ^{0,1}M_{0}^{∨} on π_{M_{0}^{∨}}^{−1}(U_{x}) as before. Namely using complex

coordinate x^{i},y^{i∗}, we send ∂/∂x^{i} to ∂/∂\overline{z}_{ϵ}^{i}, where z^{i} = x^{i} + \sqrt{−1}ϵy^{i∗}. We write it as ∂_{[γ],ϵ}.

Now let x ∈ B_{0}. For each [u] ∈ π_{2}(M, F_{x}) with nonzero [∂u] ∈ π_{1}(F_{x}), we have a neighborhood U_{(x,[u])} of (x, [u]) ∈ B̃_{0}. We restrict distribution valued one form ev_{last ∗} M_{Morse}(M; g) to U_{(x,[u])} and write it as ev_{last ∗} M_{Morse}(M; g)|_{U_{(x,[γ])}}. It induces a section of Λ^{0,1}M_{0}^{∨} as before. (Namely we send dx^{i} to d\overline{z}_{ϵ}^{i}). We write it by the same symbol ev_{last ∗} M_{Morse}(M; g)|_{U_{(x,[u])}} by abuse of notation.

Now on a neighborhood U_{x} of x in B_{0}, we “define” 𝔅 by

𝔅 “ = ” ∑_{γ∈π_{1}(F_{x})} ∑_{[u]∈π_{2}(M,F_{x}),∂[u]=γ} e^{−2πf_{u}/ϵ} exp γ^{∗}
∂_{[γ],ϵ} ⊗ ev_{last ∗} M_{Morse}(M; g)|_{U_{(x,[u])}}
“ ∈ ” W^{−∞}(π_{M_{0}^{∨}}^{−1}(U_{x}), T_{ℂ}M_{0}^{∨} ⊗ Λ^{0,1}M_{0}^{∨}).

(41)

Here W^{−∞} denotes the space of distribution valued sections. We put the equality in (41) in the quote because of the convergence problem mentioned before. To go around the convergence problem we take (41) as an asymptotic expansion. (It is proposed in [61] to use rigid analytic geometry to go around convergence problem in a related context. To use asymptotic expansion is similar to their proposal.) To state our conjecture, we also need to take smoothing of distribution valued forms. Let us discuss those points below.

We consider a family of smoothing of the distribution valued one form ev_{last ∗} M_{Morse}(M; g) and denote it by

𝔐_{δ} (ev_{last ∗} M_{Morse}(M; g)) ∈ Γ(B̃_{0}, T B̃_{0} ⊗ Λ^{1}B̃_{0}).

Here we say that 𝔐_{δ}(T) is a smoothing of a distribution T if the support of 𝔐_{δ}(T) is in the δ neighborhood of T, if lim_{δ→0} 𝔐_{δ}(T) = T as distribution, and if 𝔐_{δ}(T) is smooth.

We next take a large positive number λ. We consider the set B̃_{0,λ} of all (x, [u]) such that f(x, [u]) < λ. The restrictions to B̃_{0,λ} of the distribution valued section ev_{last ∗} M_{Morse}(M; g) and of its smoothing 𝔐_{δ} (ev_{last ∗} M_{Morse}(M; g)), have compact support, by a Morse theory analogue of Gromov compactness, (which is easy to prove). We denote the restriction of 𝔐_{δ} (ev_{last ∗} M_{Morse}(M; g)) to B̃_{0,λ} by the symbol 𝔐_{δ,λ} (ev_{last ∗} M_{Morse}(M; g)).

Now, instead of (41), we consider

𝔅(ϵ, δ, λ) = ∑_{γ∈π_{1}(F_{x})} ∑_{[u]∈π_{2}(M,F_{x}),∂[u]=γ} e^{−2πf_{u}/ϵ} exp γ^{∗}
∂_{[γ],ϵ} ⊗ 𝔐_{δ,λ} (ev_{last ∗} M_{Morse}(M; g)) |_{U_{(x,[u])}}
∈ Γ(π_{M_{0}^{∨}}^{−1}(U_{x}), T_{ℂ}M_{0}^{∨} ⊗ Λ^{0,1}M_{0}^{∨}).

(42)

We remark that (42) is a rigorously defined smooth section if we specify the choice of the smoothing 𝔐_{δ} and the multiplicity we put for multiple covered disks. We also remark that (42) is patched together and gives a section on M_{0}^{∨}.

Now we are ready to state the conjecture on the asymptotic behavior of the complex structure of the mirror.

Conjecture 4.3. Let M be a K3 surface with Lagrangian torus fibration and M^{∨} be its mirror. Then, for sufficiently small ϵ, δ and sufficiently large λ, there exists a section 𝔅̃(ϵ, δ, λ) ∈ Γ(M_{0}^{∨}, T_{ℂ}M_{0}^{∨} ⊗ Λ^{0,1}M_{0}^{∨}), the choice of the smoothing 𝔐_{δ} and of the multiplicity of multiple covered disks. They have the following properties.

(1) 𝔅̃(ϵ, δ, λ) is a solution of

\overline{∂}_{0,ϵ}𝔅̃(ϵ, δ, λ) + \frac{1}{2} [𝔅̃(ϵ, δ, λ), 𝔅̃(ϵ, δ, λ)] = 0.

In other words, \overline{∂}_{0,ϵ} + 𝔅̃(ϵ, δ, λ) ∘ ∂_{0,ϵ} is an (integrable) complex structure on M_{0}^{∨}.

(2) The complex structure \overline{∂}_{0,ϵ} + 𝔅̃(ϵ, δ, λ) ∘ ∂_{0,ϵ} extends to M^{∨}.

(3) There exists C(λ) independent of ϵ such that

∥𝔅̃(ϵ, δ, λ) − 𝔅(ϵ, δ, λ)∥ < C(λ)e^{−λ/ϵ},

where 𝔅(ϵ, δ, λ) is as in (42).

Compared to the discussion of §2, the following two points are missing in the discussion of this section.

(1) The nonlinear term of (28) is the mixture of the bracket [ , ] and wedge product. On the other hand, we consider usual wedge product of the differential form in Conjecture 4.2. Hence we need to discuss interaction or scattering of the equation (28) which is different from one in the last section. We conjecture that it coincides with closed string closed string scattering we described in §2.3.

(2) In the case of our multivalued function f, its critical point set is not on B\S(B). In other words, S(B) plays a role of Cr(f) in the last section. So we need to study the behavior of 𝔅 in a neighborhood of singular locus and compare it to the unstable manifold of f (which we discussed in §2.4), in order to study Conjecture 4.3. The behavior of 𝔅 in a neighborhood of singular fiber should be determined by (2) of Conjecture 4.3. In the case when the singular locus is type I, Ooguri-Vafa [68] found a complex structure (in fact they found a hyperkähler metric) of the neighborhood of the singular locus. They found that it is related to the counting of holomorphic disks. (They count the multiplicity of dth multiple covered disk as d^{−2}.)

The author can not solve these points at the time of writing this article. So, in the next section, we study a slightly simpler case, that is the case of holomorphic structure of mirror bundle and provides more detail related to the above points, in its open string version. Remark 4.3. Our conjecture 4.3 is on complex structure. The symplectic structure of M^{∨} is not included in the story. One may try to go further and to formulate a conjecture on the asymptotic behavior of Calabi-Yau metric in a similar way. The trouble to do so is that the Einstein-Kähler equation is more complicated than the equation \overline{∂}∘\overline{∂} = 0. Namely we need one more equation to solve. The equation \overline{∂} ∘\overline{∂} = 0 is not elliptic hence its solution is far from being unique. We need gauge fixing to have an elliptic equation. Requiring Einstein-Kähler condition is one (the ideal) way to do so. The author does not know yet appropriate Feynman rule for the Einstein-Kähler equation. He does not know either how it is related to the holomorphic sphere in the mirror. Remark 4.4. In [35], the author discussed the quantum effect to the complex structure in a different way. Namely we study the family of Floer homologies and the quantum effect to the complex structure of the D-Brane moduli space (the moduli space of the pair of Lagrangian submanifold and a flat U(1) bundle on it) is described so that the family of Floer homology will become a holomorphic family. Then the quantum effect is related to the family version of the obstruction theory of well definedness of Floer homology in [37].

The two points of view (one of [35] and one of this article) actually coincide to each other. As we discussed in [35], the family of Floer homology will not give a holomorphic family with respect to the complex structure J_{0,ϵ} because of the wall crossing of Floer homology. On the other hand, Conjecture 2.2 implies that wall crossing occurs exactly on ev_{last ∗} M_{Morse}(M; g). Remark 4.5. It seems likely that the discussion of this section can be generalized to the case of higher loop. (Namely we may include the graph which is not a tree.) Then, if everything works ideally, we might prove the coincidence of the Quantum Kodaira Spencer theory [13] with the Gromov-Witten invariant of higher genus.

In the case when M is the cotangent bundle and the open string version, this possibility was suggested already in [27, 36]. Namely it was conjectured that higher genus Morse homotopy coincides with the invariant of Chern-Simons Perturbation theory [10, 11].

5 ASYMPTOTIC ANALYSIS ON MIRROR BUNDLE 51

If we continue our heuristic argument, we may give an idea which might imply the coincidence of Gromov-Witten potential of M with Yukawa coupling of M^{∨}. We do not try to do so in this article since the author is unable to realize this idea yet. We present its open string analogue in the next section. In that case we can state at least a conjecture in a precise way.

5 Asymptotic analysis on mirror bundle

5.1 Semiflat homological mirror symmetry - review -

In this subsection, we study semiflat mirror symmetry in the case when there is a Lagrangian submanifold. The construction of this subsection is not new. It first appeared in 1998 July in [32] and was used by [5]. Several other authors found the same construction independently. See for example [3, 15, 61, 64]. The content of this subsection may be regarded as a real analogue of Fourier-Mukai transformation. (In other words, classical part of homological mirror symmetry is a Fourier-Mukai transformation.)

Let π_{M_{0}} : M_{0} → B_{0} be as in §1. Let M satisfy Assumption 2.1. We consider a Lagrangian submanifold L of M. Let S(L) ⊂ B be its caustics. We put B_{00} = B_{0}\S(L). We assume the following for simplicity.

Assumption 5.1. (1) dim B − dim S(L) ≥ 2. (2) The intersection between F_{x} then L at each intersection is positive for x ∈ B_{00}. (Namely contribute +1 to the intersection number.)

We remark that Assumption 5.1 (1) does not hold for generic Lagrangian submanifold L since the generic Lagrangian singularity is fold which is of codimension 1. However Assumption 5.1 holds for generic special Lagrangian submanifold if dim_{ℂ} M = 2. We remark that Assumption 5.1 (2) is not generic either. It holds if M is hyperhähler and if L is obtained from a complex submanifold by hyperkähler twist. We may remove Assumption 5.1. We need some more discussions to do so, which we do not mention in this article. As for as the argument of this subsection concerns, we do not need to use Assumption 5.1.

We put M_{00}^{∨} = π_{M^{∨}}^{−1}(B_{00}) ⊂ M_{0}^{∨}. We put the complex structure J_{0,ϵ} on it. We put

(43) B̃_{L}^{0} = {(x, p) | x ∈ B_{0}, p ∈ F_{x} ∩ L, F_{x} is transversal to L at p}.

Note B̃_{L}^{0} is different from B̃_{L} defined in Definition 3.2. We put B̃_{L}^{00} = B̃_{L}^{0} ∩ π^{−1}(B_{00}). By Assumption 5.1, the order of F_{x} ∩ L is independent of x ∈ B_{00}, hence (x, p) ↦ x defines a covering space πB̃_{L}^{00} : B̃_{L}^{00} → B_{00}. We put r = ♯(F_{x} ∩ L), which will be the rank of the mirror bundle.

Before going further let us mention the plan of this section. We defined a family of complex structures J_{ϵ} on M_{0} that is J_{ϵ} ( \frac{∂}{∂x_{i}} ) = \frac{1}{ϵ} \frac{∂}{∂y_{i}^{∗}}. In this subsection, we are going to construct a holomorphic vector bundle (E(L)_{0}, \overline{∂}_{0,ϵ}). The holomorphic structure \overline{∂}_{0,ϵ} we define in this subsection is a “classical” one. Namely we do not put instanton effect which is caused by pseudoholomorphic disks. Actually we need to include instanton effect to obtain mirror bundle. (We will discuss it in §5.2 ∼ §5.5. Also we need a twist by a local system ρ : π_{1}(B̃_{L}^{0}) → {±1}. ρ is expected to be related to orientation problem of the moduli space of pseudo holomorphic disks. (See [37] §25.6 for family version of orientation problem of pseudoholomorphic disks.) We call it orientation twist. (See §5.5 and 5.6 for its definition.) It modifies E(L)_{0} to E(L)_{0}^{ρ}. Holomorphic structure \overline{∂}_{0,ϵ} induces one on E(L)_{0}^{ρ} which we denote by the same symbol. The mirror bundle is (E(L)_{0}^{ρ}, \overline{∂}_{0,ϵ} + 𝔅), where 𝔅 is a section of End(E(L)_{0}^{ρ}) ⊗ Λ^{0,1} satisfying the equation \overline{∂}_{0,ϵ}𝔅 + 𝔅 ∧ 𝔅 = 0.

Let us start the construction of mirror bundle E(L)_{0}^{ρ}. We recall that there exists a flat SL(n, ℤ) bundle E → M_{0} and its lattice Λ ⊂ E such that M_{0} = E/Λ. The mirror manifold is the dual fibration M_{0}^{∨} = E^{∗}/Λ^{∨} (See §1). We first define a bundle Ẽ(L)_{0}^{ρ} on E^{∗}.

Let ρ : π_{1}(B̃_{L}^{0}) → {±1} be a homomorphism which we will define later. Let ℂ_{ρ} → B̃_{L}^{0} be the flat U(1) bundle associated to ρ. We define

Ẽ(L)_{0}^{ρ} = π_{E^{∗}}^{∗}(πB̃_{L}^{00}!(ℂ_{ρ})). (44)

We next lift the Λ^{∨} action on E^{∗} to Ẽ(L)_{0}^{ρ}. Let (x, y^{∗}) ∈ E_{x}^{∗} and (x, y) ∈ B̃_{L}^{0}. (Here x ∈ B_{00}). Let v ∈ (ℂ_{ρ})B̃_{L}^{0} (the fiber of the flat bundle ℂ_{ρ} at B̃′_{L}). v determines an element of the fiber of Ẽ(L)^{ρ} at (x, y^{∗}) which we write [x, y^{∗}; y; v]. Let (x, α) ∈ Λ_{x}^{∨}. We remark that ⟨α, y^{∗}⟩ is well defined as an element of ℝ/ℤ. Hence exp(2π\sqrt{−1}⟨α, y^{∗}⟩) ∈ U(1) is well defined.

Definition 5.1. We put :

α · [x, y^{∗}; y; v] = exp(2π\sqrt{−1}⟨α, y^{∗}⟩)[x, y^{∗} + α; y; v]. (45)

We define E(L)_{0}^{ρ} → M_{0}^{∨} to be the quotient bundle of Ẽ(L)_{0}^{ρ} by this Λ^{∨} action.

Hereafter we omit y^{∗} from the notation [x, y^{∗}; y; v] and will write [x; y; v].

We next define a holomorphic structure on E(L)_{0}^{ρ}. For this purpose we first define a local holomorphic frame of Ẽ(L)_{0}^{ρ}. Let s_{0} : B_{0} → M_{0} be the zero section. We put

B̃_{L_{0},L} = {(x, p, [ℓ]) | (x, p) ∈ B̃_{L}^{00}, [ℓ] ∈ π_{1}(F_{x}; s_{0}(x), p)}. (46)

In other words, [ℓ] is the homotopy class of an arc in F_{x} joining s_{0}(x) ∈ F_{x} ∩ L_{0} with p. The map (x, p, [ℓ]) ↦ x defines a covering map πB̃_{L_{0},L} :

B̃_{L_{0},L} → B_{00}. There exists also another covering map π_{B̃_{L_{0},L},B̃_{L}^{00}} :

B̃_{L_{0},L} → B̃_{L}^{00}.

Now let U be a contractible open subset of B_{00}. We choose a section π−1 ; B̃_{L}^{00}(U) → π−1 ; B̃_{L_{0},L}(U) to π_{B̃_{L_{0},L},B̃_{L}^{00}}. Such section exists since U is con-

tractible. We denote it by (x, p) ↦ (x, p, [ℓ_{U}(x, p)]). In other words, we choose a homotopy class of a path ℓ_{U}(x, p) joining s_{0}(x) to p ∈ F_{x} ∩ L.

We remark that π−1 ; B̃_{L}^{00}(U) is a disjoint union of r copies of U. In

other words, there exist (continuous) sections p_{i} : U → π−1 ; B̃_{L}^{00}(U), i = 1, ⋯, r such that the union of their images is π−1 ; B̃′_{L}(U). We put ℓ_{i}(x) =

ℓ_{U}(x, p_{i}(x)). We also fix a trivialization of ℂ_{ρ} on π−1 ; B̃_{L}^{00}(U).

Now let (x, y^{∗}) ∈ E_{x}, x ∈ U. The basis of the fiber of Ẽ(L)_{0}^{ρ} is [x; p_{i}(x); 1]. (Here 1 is a canonical basis of the fiber of Ẽ(L)_{0}^{ρ} at (x, p_{i}(x)) induced by its trivialization.) We lift ℓ_{i}(x) to a path ℓ̃_{i}(x)(t) in E_{x} such that ℓ̃_{i}(x)(0) = 0. We put p̃_{i}(x) = ℓ̃_{i}(x)(1) ∈ E_{x}. (Note p̃_{i}(x) ≡ p_{i}(x) mod Λ_{x}.) We put

H_{i}(x, y^{∗}) = ⟨y^{∗}, p̃_{i}(x)⟩ ∈ ℝ. (47)

We next fix a base point x_{U} in U. Let x ∈ U. We joint x_{U} with x by a path m in U. We may assume that

φ_{i,x}(s, t) = ℓ_{i}(m(s))(t) ∈ F_{m(s)}

is a smooth map φ_{i,x} : [0, 1]^{2} → M_{0}. (We remark that the image of φ_{i,x} is a disk which bounds F_{x_{U}} ∪ F_{x} ∪ L ∪ s_{0}(U) as in Figure 5.1.

Figure 5.1

We put

Q_{i}(x) = ∫_{[0,1]^{2}} φ_{i,x}^{∗}ω. (48)

We now define local frame s_{i,ϵ} by

s_{i,ϵ}(x, y^{∗}) = e^{−2π( \frac{1}{ϵ} Q_{i}(x)+\sqrt{−1}H_{i}(x,y^{∗})))}[x; p_{i}(x); 1].

Here [x; p_{i}(x); 1] i = 1, ⋯, r is a basis of the fiber of Ẽ(L)_{0}^{ρ} at (x, y^{∗}) we introduced before.

Using a symplectic coordinate x^{j} we have H_{i} = ∑_{j=1}^{n} \frac{∂Q_{i}}{∂x^{j}} y^{j∗}.

Lemma 5.1. There exists a unique holomorphic structure of the bundle E(L)_{0}^{ρ} on the complex manifold (M_{0}^{∨}, J_{ϵ}) such that s_{i,ϵ} is holomorphic with respect to the induced holomorphic structure on Ẽ(L)_{0}^{ρ}.

Proof. We need to show that the holomorphic structure of Ẽ(L)_{0}^{ρ} defined by s_{i,ϵ} can be patched on the intersection of different U and also to show that it is invariant of the Λ action.

To prove the first point we only need to show that the holomorphic structure on Ẽ(L)_{0}^{ρ} is independence of ℓ_{i}(x) and of the base point p_{U}. The later is immediate since by changing p_{U} the function Q_{i} changes by a constant and H_{i} does not change. To show the independence of ℓ_{i}(x) we consider alternative choice of ℓ_{i} say ℓ′_{i}. We write p̃′_{i}(x), Q′_{i}, H′_{i} the ones obtained from ℓ′_{i}. We remark that p_{i}(x) − p′_{i}(x) = q(x) ∈ Λ_{x}. Since we are working in a small open set, we may take symplectic coordinate x^{j}, y^{j}. We may then regard q(x) ∈ ℝ^{n} (by using y coordinate.) Now, it is easy to see from the definition that

Q_{i}(x) − Q′_{i}(x) = ⟨x − x_{U}, q(x)⟩

here the right hand side is the standard inner product of ℝ^{n}. On the other hand, we have

H_{i}(x, y^{∗}) − H′_{i}(x, y^{∗}) = ⟨y^{∗}, q(x)⟩.

We remark that x^{j} + \sqrt{−1}ϵy^{j∗} is complex coordinate of (M_{0}^{∨}, J_{ϵ}). Hence \frac{1}{ϵ} (Q_{i}(x) − Q′_{i}(x)) + \sqrt{−1}(H_{i}(x, y^{∗}) − H′_{i}(x, y^{∗})) is a holomorphic function of (x, y^{∗}).

The proof of Λ invariance is easy and is omitted. The proof of Lemma 5.1 is now complete.

We thus defined a holomorphic structure \overline{∂}_{0,ϵ} on Ẽ(L)_{0}^{ρ}. In later sections, we use a section of End(E(L)_{0}^{ρ}) ⊗ Λ^{0,1} to modify the holomorphic structure \overline{∂}_{0,ϵ}. So let us describe a local holomorphic frame of End(E(L)_{0}^{ρ}).

Actually we need more general case later that is the case of Hom(E(L_{2})_{0}^{ρ_{2}}, E(L_{1})_{0}^{ρ_{1}}) where L_{1}, L_{2} are Lagrangian submanifolds. So we discuss this case here. We put B_{00} = B_{0}\S(L_{1})∪S(L_{2}) and consider the fiber product B̃_{L_{1}}^{00} ×_{B_{00}} B̃_{L_{2}}^{00}. Its elements is a triple (x; p_{1,i}(x), p_{2,j}(x)) where p_{a,i} : U → B̃_{L_{a}}^{00}, i = 1, ⋯, r_{a}, a = 1, 2 are sections as before. (r_{a} = ♯(F_{a} ∩ L_{a}) = rank E(L_{a})_{0}^{ρ_{a}}.) By using the trivialization of ℂ_{ρ_{a}} we introduced before, a basis of a fiber of Hom(E(L_{2})_{0}^{ρ_{2}}, E(L_{1})_{0}^{ρ_{1}}) at (x, y^{∗}) is [x; p_{1,i}(x), p_{2,j}(x); 1] where i = 1, ⋯, r_{1}, j = 1, ⋯, r_{2}. Then for each i, j we have a local holomorphic frame s_{i,j;ϵ} of Hom(E(L_{2})_{0}^{ρ_{2}}, E(L_{1})_{0}^{ρ_{1}}) defined by

s_{i,j;ϵ}(x, y^{∗}) = e^{−2π( \frac{1}{ϵ} Q_{i,j} (x)+\sqrt{−1}H_{i,j} (x,y^{∗})))}[x; p_{1,i}(x), p_{2,j}(x); 1], (49)

Here Q_{i,j}, H_{i,j} are defined as follows. Let us consider Q_{1,i},H_{1,i} which is defined by using Lagrangian submanifold L_{1} as before. We define Q_{2,j},H_{2,j} using L_{2} in a similar way. We now put Q_{i,j} = Q_{2,i} − Q_{1,j}, H_{i,j} = H_{2,i} − H_{1,j}. Actually Q_{i,j} is a branch of the function f_{L_{1},L_{2}} defined at the end of §3.1. In particular, in case L_{1} = L_{2} = L, we have

Q_{i,j}(x) = f_{L}(x; p_{i}(x), p_{j}(x); [γ]) + const. (50)

where the right hand side is as in (13), (x; p_{i}(x), p_{j}(x); [γ]) ∈ B̃_{L}^{+} and γ is chosen as follows. We recall that we took lifts p̃_{i}(x), p̃_{j}(x) ∈ E_{x}. We put γ̃(t) = (1 − t)p̃_{i}(x) + tp̃_{j}(x). It is a path in E_{x} and induces a path γ in F_{x}. (We write [γ] ∈ π_{1}(F_{x}; p_{i}(x), p_{j}(x)) in (50) since f_{L} depends only on (x; p_{i}(x), p_{j}(x); [γ]) and is independent of [u] with [∂_{+}u] = [γ] up to constant difference.) The formula (50) is immediate from definition.

5.2 Free field - the case of vector bundle -

We consider sections 𝔅 ∈ End(E(L)_{0}^{ρ}) ⊗ Λ^{0,k}, s ∈ E(L)_{0}^{ρ} ⊗ Λ^{0,k} and study linear equations

\overline{∂}_{0,ϵ}𝔅 = 0, \overline{∂}_{0,ϵ}𝔰 = 0. (51)

We recall that the condition that \overline{∂}_{0,ϵ} +𝔅 defines a holomorphic structure on E(L)_{0}^{ρ} is

\overline{∂}_{0,ϵ}𝔅 + 𝔅 ∧ 𝔅 = 0 (52)

(k = 1) and 𝔰 is a holomorphic section of \overline{∂}_{0,ϵ} + 𝔅 is

\overline{∂}_{0,ϵ}𝔰 + 𝔅 ∧ 𝔰 = 0. (53)

Hence the equations (51) are their linearizations. We are studying (51) by using Fourier expansion along fiber. (As we mentioned before this idea is similar to one in [5] and is also quite natural from the point of view of T -duality.) To describe Fourier expansion, we proceed as follows. Let x ∈ B_{00}. We consider (x, y), (x, y_{1}), (x, y_{2}) ∈ F_{x} ∩L and let (x, 0) = s_{0}(x). We then consider the relative homotopy groups π_{1}(F_{x}; (x, y_{1}), (x, y_{2})) and π_{1}(F_{x}; (x, 0), (x, y)). For elements [ℓ] ∈ π_{1}(F_{x}; (x, y_{1}), (x, y_{2})), [γ] ∈ π_{1}(F_{x}; (x, 0), (x, y)), and (x, y^{∗}) ∈ F_{x}^{∨}, we define

⟨[ℓ], y^{∗}⟩ ∈ ℝ/ℤ, ⟨[γ], y^{∗}⟩ ∈ ℝ/ℤ

as follows. We lift the path ℓ to ℓ̃: [0, 1] → E_{x}. Then ℓ̃(1) − ℓ̃(0) ∈ E_{x} is independent of the choice of the lift. So if we lift y^{∗} to an element ỹ^{∗} of E_{x}^{∗} then we have ⟨ℓ̃(1) − ℓ̃(0), ỹ^{∗}⟩ ∈ ℝ. This number is independent of the lift ỹ^{∗} modulo ℤ. Hence we obtain ⟨[ℓ], y^{∗}⟩ ∈ ℝ/ℤ. The definition of ⟨[γ], y^{∗}⟩ ∈ ℝ/ℤ is similar.

We next identify π_{M^{∨}}^{∗}Λ^{k}B ≅ Λ^{0,k}M_{0}^{∨} by sending dx^{i} to d\overline{z}_{ϵ}^{i}. Here x^{i} are symplectic coordinate of M_{0} → B_{0} (and hence is a complex coordinate of M_{0}^{∨} → B_{0}) and z_{ϵ}^{i} = x^{i} + ϵ\sqrt{−1}y^{i∗}. We write this isomorphism as ℑ_{ϵ} : π_{M^{∨}}^{∗}Λ^{k}B ≅ Λ^{0,k}M_{0}^{∨}.

Now we expand

𝔅(x, y^{∗}) = ∑_{p_{1},p_{2},ℓ} e^{−2π\sqrt{−1}⟨[ℓ],y^{∗}⟩}[x; p_{1}, p_{2}; 1] ⊗ ℑ_{ϵ}(𝔅_{p_{1},p_{2},ℓ}(x)). (54)

Here we remark that [x; p_{1}, p_{2}; 1] is a basis of the fiber of End(E(L)_{0}^{ρ}) at each (x, y^{∗}). 𝔅_{p_{1},p_{2},ℓ} are 1 forms on B. (Then ℑ_{ϵ}(𝔅_{p_{1},p_{2},ℓ}) is a T structure invariant (0, 1) form on M_{0}.)

In a similar way we have

𝔰(x, y^{∗}) = ∑_{p,γ} e^{−2π\sqrt{−1}⟨[γ],y^{∗}⟩}[x; p; 1] ⊗ ℑ_{ϵ}(𝔰_{p,γ}(x)). (55)

Now the following lemma is obtained in a way similar to Proposition 4.1 and the definition of the holomorphic structure in the last subsection. To state it we recall some notations. Let [γ] ∈ π_{1}(F_{x}; p_{1}, p_{2}). Then (x; p_{1}, p_{2}; [γ]) is an element of B̃_{L}^{+}. Hence f_{L}(x; p_{1}, p_{2}; [ℓ]) is well defined up to constant by Lemma 3.1.

Lemma 5.2. The first equation of (51) is equivalent to

d(e^{\frac{2π}{ϵ} f_{L}(x;p_{1},p_{2};[γ])}𝔅_{p_{1},p_{2},ℓ}(x)) = 0. (56)

Let us denote by L_{st} the image of s_{0} : B → M, the zero section. (It will correspond to the structure sheaf of the mirror manifold. See [32].)

We remark that

f_{L_{st},L}(x, p_{i}(x); [γ]) = Q_{i}(x) + const (57)

where Q_{i} is as in the last subsection, and [γ] is chosen as follows. We define a lift p̃_{i}(x) ∈ E_{x} in the last subsection. Then γ(t) ≡ tp̃_{i}(x) mod Λ_{x}.

Lemma 5.3. The second equation of (51) is equivalent to

d(e^{\frac{2π}{ϵ} f_{L_{st},L}(x;p;[γ])}𝔰_{p,γ}(x)) = 0. (58)

Let us now explain how the solutions of (56), (58) are related to the Morse theory of our multivalued functions f_{L}, f_{L_{st},L}.

Here we discuss the case of 𝔅. We first remark that we may regard 𝔅_{p_{1},p_{2},ℓ} as a differential form on B̃_{L}^{+} rather than one on B. In other words we consider a map

ℑ̃_{ϵ} : Γ_{c}(B̃_{L}^{+}; Λ^{k}) → Γ(M_{0}^{∨}; End(E(L)_{0}^{ρ}) ⊗ Λ^{0,k})

as follows. (Here Γ_{c} denotes the space of sections of compact support.) Let 𝔄 ∈ Γ_{c}(B̃_{L}^{+}; Λ^{k}) and x ∈ B_{0}. There exists finitely many (p, q; [γ]) in π−1 ; B̃_{L}^{+}(x) such that 𝔄(x; p_{1}, p_{2}; [γ]) is nonzero. We identify Λ_{x;p,q;[γ]}^{k}B̃_{L}^{+} ≅

Λ_{x}^{k}B. We then put

\begin{aligned}ℑ̃_{ϵ}(𝔄)(x, y^{∗}) = ∑_{p_{1},p_{2},[γ]} & e^{−2π\sqrt{−1}⟨[ℓ],y^{∗}⟩} \\ & [x; p_{1}, p_{2}; 1] ⊗ ℑ_{ϵ}(𝔄(x; p_{1}, p_{2}; [γ])). \end{aligned} (59)

(59) is a generalization of (54). We have an obvious analogue of Lemma 5.2

We recall that the function f_{L} is well defined on B̃_{L}^{+} up to constant difference. Hence the gradient vector field gradf_{L} on B̃_{L}^{+} is well defined. Here we used the Riemannian metric on B̃_{L}^{+} induced by one on B_{0}. We remark that this Riemannian metric is not complete since at the caustics the map B̃_{L}^{+} → B_{00} is not a covering map. Let ℭB̃_{L}^{+} be the completion of B̃_{L}^{+}. Note however the vector field gradf_{L} is complete, since caustics is a critical point of f_{L}.

Let τ ↦ e^{τgradf_{L}} be the one parameter group of transformations on B̃_{L}^{+} generated by gradf_{L}. For 𝔄 ∈ Γ_{c}(B̃_{L}^{+}; Λ^{k}) we put

ψ_{τ} 𝔄 = (e^{−τgradf_{L}})^{∗} 𝔄,
G_{τ_{1},τ_{2}}𝔄 = ∫_{τ_{1}}^{τ_{2}} i_{gradf_{L}}(ψ_{τ} 𝔄)dτ

Lemma 5.4. We have

d ∘ G_{τ_{1},τ_{2}} + G_{τ_{1},τ_{2}} ∘ d = ψ_{τ_{2}} − ψ_{τ_{1}}.

Proof. Easy calculation using d ∘ i_{X} + i_{X} ∘ d = L_{X}, L_{gradf_{L}}(ψ_{τ} 𝔄) = \frac{d}{dτ} (ψ_{τ} 𝔄).

We set

U(f_{L}) = {(x; p_{1}, p_{2}; [γ]) ∈ B̃_{L}^{+}| lim_{τ→∞} e^{τgradf_{L}}(x; p_{1}, p_{2}; [γ])
is bounded in ℭB̃_{L}^{+}}.

Let x = (x; p_{1}, p_{2}; [γ]) ∈ B̃_{L}^{+}. We put ℓ(τ) = e^{τgradf_{L}}(x) and

f_{L,τ_{0}}(x) = ∫_{0}^{τ_{0}} ℓ^{∗}df_{L}(e^{τgradf_{L}}(x))dτ,
f′_{L,τ_{0}}(x) = f_{L,τ_{0}}(e^{−τ_{0}gradf_{L}}x).

(60)

We remark that f_{L} is well defined only up to constant. But the right hand side is well defined. Hence we define the left hand side by (60). The following lemma is easy to see.

Lemma 5.5. If x ∉ U(f_{L}) then lim_{τ→+∞} f_{L,τ} (x) = +∞.

We next put

𝔊_{τ_{1},τ_{2}}𝔄 = ∫_{τ_{1}}^{τ_{2}} ℑ̃_{ϵ} (e^{−2πf′_{L,τ}} i_{gradf_{L}}(ψ_{τ} 𝔄))dτ, (61)

and

Ψ_{τ} 𝔄 = ℑ̃_{ϵ}(e^{−2πf′_{L,τ}} ψ_{τ} 𝔄). (62)

Note Ψ_{0}𝔄 = ℑ̃_{ϵ}(𝔄). Combining Lemmata 5.2, 5.4, we have

\overline{∂} ∘ 𝔊_{τ_{1},τ_{2}} + 𝔊_{τ_{1},τ_{2}} ∘ d = Ψ_{τ_{2}} − Ψ_{τ_{1}}. (63)

Lemma 5.5 implies that

lim_{τ→+∞} Ψ_{τ} 𝔄 = 0

if the support of 𝔄 is disjoint with U(f_{L}). This implies that lim_{τ→+∞} 𝔊_{τ,0} plays a role of propagator. Namely we have the following :

Proposition 5.1. If d𝔄 = 0 and the support of 𝔄 is disjoint with U(L) then lim_{τ→+∞} 𝔊_{τ,0}(𝔄) converges and we have

−\overline{∂} ( lim_{τ→+∞} 𝔊_{τ,0}(𝔄)) = ℑ̃_{ϵ}(𝔄).

Moreover we may take the generator of the \overline{∂} cohomology so that its support is near U(f_{L}). This is related to the Witten’s argument we discussed in §4.2.

Proposition 5.1 implies that we may take a solution of (51) so that its Fourier component propagates along the gradient vector field of f_{L}. We can prove a similar proposition for 𝔰.

5.3 Interaction

In §5.2, we studied the linearized equation (51). As we discussed in §5.2 we are studying the solution which propagates along the gradient vector field of f_{L}. The effect of the nonlinear term appears when two of such gradient lines intersect each other (in B). The problem is thus local in B. Hence we consider the following case. M = ℝ^{n} × T^{n} and L is a disjoint union of Lagrangian submanifolds obtained by ℝ^{n}×{p_{i}}, i = 1, ⋯, k. We take a symplectic coordinate x^{i}, y^{i} of M. (We take the metric ∑ dx^{i} ⊗dx^{i} on B = ℝ^{n}. The mirror manifold then is M^{∨} = ℝ^{n} ×T^{n} whose coordinate is x^{i} and y^{i∗} with complex structure J_{ϵ}( \frac{∂}{∂x^{i}}) = \frac{1}{ϵ} \frac{∂}{∂y^{i∗}}. The mirror bundle E_{0} is a trivial bundle with frame [x; p_{i}]. A holomorphic frame is

e_{i;ϵ}(x, y^{∗}) = e^{−2π( \frac{1}{ϵ} ⟨p_{i},x⟩+\sqrt{−1}⟨p_{i},y^{∗}⟩)}[x; p_{i}]. (64)

Here p_{i}, x, y^{∗} ∈ ℝ^{n} and ⟨ · , · ⟩ is the standard inner product of ℝ^{n}.

We first explain the simplest case to illustrate the problem and then proceed to more general case. We first consider the case k = 3, n = 2. Let w_{i,j} ∈ ℝ_{>0}^{2}. We assume w_{i,j} ≡ p_{j} − p_{i} mod ℤ^{2}. Let Π_{i,j} be the projection

Π_{i,j}(x) = x − \frac{⟨w_{i,j}, x⟩}{⟨w_{i,j}, w_{i,j}⟩} w_{i,j} (65)

to the plain orthogonal to w_{i,j}. We identify Π_{i,j}(ℝ^{2}) ≅ ℝ. Let 𝔟_{δ} be a 1 form on ℝ such that the support of 𝔟_{δ} is on [−δ, δ] and that ∫ 𝔟_{δ} = 1. (In other words 𝔟_{δ} is a smooth approximation of delta current.) We consider

\begin{aligned}𝔅_{i,j;ϵ;δ} & = e^{−2π(c_{0}+ \frac{1}{ϵ} ⟨w_{i,j} ,x⟩+\sqrt{−1}⟨w_{i,j} ,y^{∗}⟩)} \\ & ℑ_{ϵ}(Π_{i,j}^{∗}(𝔟_{δ})) ⊗ [x; p_{j}, p_{i}] ∈ Γ(ℝ^{2} × T^{2}; End(E_{0}) ⊗ Λ^{0,1}). \end{aligned}

Here c_{0} is a positive constant. Here and hereafter we put [x; p_{j}, p_{i}] = [x; p_{j}] ⊗ [x; p_{i}]^{∗}.

It follows from the construction of the last subsection that \overline{∂}_{0,ϵ}𝔅_{i,j;ϵ;δ} = 0. Moreover since 𝔅_{i,j;ϵ;δ} ∧ 𝔅_{i,j;ϵ;δ} = 0 (for i ≠ j) it follows that \overline{∂}_{0,ϵ} + 𝔅_{i,j;ϵ;δ} is another holomorphic structure of E_{0}. However if we consider 𝔅_{1,2;ϵ;δ} + 𝔅_{2,3;ϵ;δ}, it no longer satisfies the Maurer-Cartan equation, since 𝔅_{2,3;ϵ;δ} ∧ 𝔅_{1,2;ϵ;δ} is nonzero in a neighborhood of origin. We thus need to study the following scattering problem. We assume that w_{1,2} and w_{2,3} are linearly independent.

Problem 5.1. Find an element 𝔅 ∈ Γ(ℝ^{2} ×T^{2}; End(E_{0})⊗Λ^{0,1}) with the following properties.

(1) \overline{∂}_{0,ϵ}𝔅 + 𝔅 ∧ 𝔅 = 0.

(2) There exists C > 0 such that if x^{1} + x^{2} < −C then 𝔅 = 𝔅_{1,2;ϵ;δ} + 𝔅_{2,3;ϵ;δ}.

Figure 5.2

To solve this problem it suffices to find 𝔅′_{1,3} = B_{1,3}[x; p_{3}, p_{1}] such that

\overline{∂}𝔅′_{1,3} = 𝔅_{2,3;ϵ;δ} ∧ 𝔅_{1,2;ϵ;δ}

and that the support of 𝔅′_{1,3} is away from x^{1} +x^{2} < −C. If we put 𝔅′_{1,3} = B_{1,3}[x; p_{3}, p_{1}] then 𝔅′_{1,3} ∧𝔅_{1,2;ϵ;δ}, 𝔅′_{1,3} ∧𝔅_{2,3;ϵ;δ}, 𝔅_{1,2;ϵ;δ} ∧𝔅′_{1,3}, 𝔅_{2,3;ϵ;δ} ∧ 𝔅′_{1,3} are all zero. This means that once two particles (corresponding to 1, 2 and 2, 3 components) interact, new particle (corresponding to 1, 3 component) are born. But they never interact again. So it suffices to find 𝔅′_{1,3} in order to solve Problem 5.1. Such 𝔅′_{1,3} can be obtained by applying Proposition 5.1. Let us describe the answer below.

To find 𝔅′_{1,3} we consider the two form Π_{2,3}^{∗}(𝔟_{δ})∧Π_{1,2}^{∗}(𝔟_{δ}) = ξ(x^{1}, x^{2})dx^{1}∧ dx^{2}, and define

ρ(x) = ∫_{t>0} ξ(x − tw_{1,3})dt.

Then

d(ρ(x)i_{w_{1,3}}(dx^{1} ∧ dx^{2})) = −Π_{2,3}^{∗}(𝔟_{δ}) ∧ Π_{1,2}^{∗}(𝔟_{δ}).

(Here we regard w_{1,3} as a vector field on ℝ^{2}.) Now we put

B_{1,3} = e^{−2π(2c_{0}+ \frac{1}{ϵ} ⟨w_{1,3},x⟩+\sqrt{−1}⟨w_{1,3},y^{∗}⟩)}ρ(x)ℑ_{ϵ}(i_{w_{1,3}}(dx^{1} ∧ dx^{2})).

Thus we have :

Proposition 5.2. 𝔅 = 𝔅_{1,2;ϵ;δ} + 𝔅_{2,3;ϵ;δ} + B_{1,3}[x; p_{3}, p_{1}] satisfies (1),(2) of Problem 5.1.

The next case to study is the case when n = 2 and k = 2 and we start with 𝔅_{1,2} and 𝔅_{2,1}. (We recall w_{1,2}, w_{2,1} are both in ℝ_{>0}^{2}.)

We define 𝔅_{1,2;ϵ;δ} and 𝔅_{2,1;ϵ;δ} in the same way. Now our problem is :

Problem 5.2. Find an element 𝔅 ∈ Γ(ℝ^{2} ×T^{2}; End(E_{0})⊗Λ^{0,1}) with the following properties.

(1) \overline{∂}_{0,ϵ}𝔅 + 𝔅 ∧ 𝔅 = 0.

(2) There exists C > 0 such that if x^{1} + x^{2} < −C then 𝔅 = 𝔅_{1,2;ϵ;δ} + 𝔅_{2,1;ϵ;δ}.

This problem looks quite similar to Problem 5.1 but its answer is much more complicated. The reason is as follows. We first find 𝔅′_{1,1} and 𝔅′_{2,2} such that

\begin{aligned}\overline{∂}_{0,ϵ}𝔅′_{1,1} & = −𝔅_{2,1;ϵ;δ} ∧ 𝔅_{1,2;ϵ;δ} \\ \overline{∂}_{0,ϵ}𝔅′_{2,2} & = −𝔅_{1,2;ϵ;δ} ∧ 𝔅_{2,1;ϵ;δ}. \end{aligned}

Contrary to the case of Proposition 5.1, 𝔅_{2,1;ϵ;δ} ∧ 𝔅′_{1,1} etc. are again nonzero. Hence we add more terms whose \overline{∂} boundaries are 𝔅_{2,1;ϵ;δ} ∧𝔅′_{1,1} etc. In this way we need to add infinitely many terms. Remark 5.1. This phenomenon is B model analogue of the phenomenon we discussed in Remark 3.2.

To describe those infinitely many terms, we use Feynman diagram as follows. We consider a graph Γ which satisfies Condition 2.2. Let l be a metric on it. We put one more additional data on it. Let I : |Γ| → D^{2} be an embedding which preserves ribbon structure and such that I^{−1}(∂D^{2}) consists of the exterior vertices. Let α : π_{0}(D^{2}\{I(|Γ|)}) → {0, 1} be a map with the following condition.

For each e ∈ Edge(Γ) let left(e) ∈ π_{0}(D^{2}\I(|Γ|)) [resp. right(e) ∈ π_{0}(D^{2}\I(|Γ|)) ] be the component which is in the left [resp. right] hand side of e. (Note e is oriented.) Let v_{1}, ⋯, v_{k} be the set of all exterior vertices minus v_{last}. We assume that v_{last}, v_{1}, ⋯, v_{k} respects the cyclic order. Let e_{i} be the unique edge such that ∂_{source}e_{i} = v_{i}. We assume that α(left(e_{i})) ≠ α(right(e_{i})) for each i. (We remark that α(left(e)) ≠ α(right(e)) may happen for interior edge e and for the last edge.)

We define a form B_{(Γ,l,α)} on ℝ^{2} by induction on the number of vertices of Γ.

To start the induction, we consider the case of Γ_{−1} and Γ_{0}. Here Γ_{−1} is the tree without interior vertex and Γ_{0} has three edges and one interior vertex. There are two choice of α for Γ_{−1}. The first one α_{1,2} is one such that α_{1,2}(right(e)) = 1, α_{1,2}(left(e)) = 2, where e is the unique edge of Γ_{−1}. We call the other choice of α as α_{2,1}. The metric l is trivial. We now put

B_{(Γ_{−1},·,α_{1,2})} = Π_{1,2}^{∗}(𝔟_{δ})[x; p_{2}, p_{1}].

We define B_{(Γ_{−1},·,α_{2,1})} in the same way by exchanging 1 and 2.

We next consider the case of Γ_{0}. There is only one edge e_{last} of Γ_{0} with finite length. (e_{last} is the edge which has v_{last} as a vertex.) Let l(e_{last}) be its length. There are two choices of α. One of them α_{1} is the case when α(D) = 1 for two of the domains D. The other, α_{2} is the case when α(D) = 2 for two of the domains D. (See Figure below.)

Figure 5.3

We then put

B_{(Γ_{0},l,α_{1})} = i_{w_{1,2}+w_{2,1}} exp(−l(e_{last})(w_{1,2} + w_{2,1}))^{∗}
B_{(Γ_{−1},·,α_{2,1})} ∧ B_{(Γ_{−1},·,α_{1,2})}.

Let us denote by Met(Γ) the set of all metrics l on Γ satisfying Condition 2.3 (a). Clearly Met(Γ_{0}) = (0, ∞). The following lemma can be proved in the same way as Lemma 5.4.

Lemma 5.6.

d (∫_{l=0}^{∞} B_{(Γ_{0},l,α_{1})}) = −B_{(Γ_{−1},·,α_{2,1})} ∧ B_{(Γ_{−1},·,α_{1,2})}.

Now we describe inductive step of the construction of B_{(Γ,l,α)}. We assume that B_{(Γ,l,α)} is constructed in the case the number of interior vertices of Γ is smaller than k. Let us consider Γ with k interior vertices. We take the last vertex v_{last} and the last edge e_{last}. We remove it from Γ. Then we get a union of two trees, which we call Γ_{1} and Γ_{2}. (We use the ribbon structure at the vertex ∂_{source}e_{last} to number Γ_{1}, Γ_{2} so that e_{last}, Γ_{1}, Γ_{2} respects the cyclic order.)

α for Γ induces ones α_{1},α_{2} for Γ_{1}, Γ_{2} respectively. There exists an obvious map π : Met(Γ) → Met(Γ_{1}) × Met(Γ_{2}), whose fiber is (0, ∞), parametrizing the length of e_{last}. For l_{i} ∈ Met(Γ_{i}) let l = l(l_{1}, l_{2}; s) be the element of Met(Γ) such that π(l) = (l_{1}, l_{2}) and l(e_{last}) = s.

To define B_{(Γ,l(l_{1},l_{2},s),α)} we define w(e) for each edge e. Let e,e′ be edges of Γ. We say e < e′ if any path joining e to v_{last} intersects with e′. < is a partial order. We define w(e) by induction on this order.

If e is minimal then e is an exterior vertex. We put a = α(right(e)), b = α(left(e)) and w(e) = w_{a,b}. (Note a ≠ b by assumption. Hence w(e) = w_{1,2} or v_{2,1}.) Suppose w(e) is defined for all edges smaller than e. Let ∂_{source}e = v. The three edges which has v as a vertex are denoted by e and e_{1}, e_{2}. Obviously e_{1}, e_{2} < e. Hence w(e_{i}) are defined. We put w(e) = w(e_{1}) + w(e_{2}). We thus defined w(e).

We now put :

B_{(Γ,l(l_{1},l_{2},s),α)} =i_{w(e_{last})} exp(−sw(e_{last}))^{∗}

(B_{(Γ_{1},l_{1},α_{1})} ∧ B_{(Γ_{2},l_{2},α_{2})}) . (66)

The following lemma is easily proved in a way similar to the proof of Lemma 5.6.

Lemma 5.7.

d (∫_{s=0}^{∞} B_{(Γ,l(l_{1},l_{2},s),α)}) = −B_{(Γ_{1},l_{1},α_{1})} ∧ B_{(Γ_{2},l_{2},α_{2})}.

Now we consider the infinite sum

B = ∑_{Γ} e^{−c_{0}k(Γ)} ∫_{l∈Met(Γ)} B_{(Γ,l,α)}

where k(Γ) + 1 is the number of exterior edges. (We put an obvious measure on Met(Γ).) We can easily show that B converges if we take c_{0} large.

Lemmata 5.6,5.7 imply the following :

Proposition 5.3.

dB + B ∧ B = 0.

We now modify B_{(Γ,l,α)} to obtain 𝔅_{(Γ,l,α)} so that a similar statement to Proposition 5.3 holds for \overline{∂} in place of d. In order to do so, we define a function f similar to f_{L}. For this purpose, we first define exp_{(Γ,l,α)} : ℝ^{2} × |(Γ, l)| → ℝ^{2}. (The definition below is the same as one used in [36] §16.) Here |(Γ, l)| is as in §2. We define exp_{(Γ,l,α)} on ℝ^{2} × |e| for edges e by opposite induction with respect to the order < on edges. The biggest edge with respect to < is e_{last}. We identify |e_{last}| = [−l(e_{last}), 0]. We put

exp_{(Γ,l,α)}(x, τ) = x + τw(e_{last}).

Let us assume that exp_{(Γ,l,α)} is defined on ℝ^{2} × |e| for e bigger than e_{0}. We put ∂_{target}e_{0} = v. By induction hypothesis exp_{(Γ,l,α)} is defined on ℝ^{2} × {|v|}. We identify |e_{0}| = [−l(e_{0}), 0]. (In case when l(e_{0}) = ∞ we identify |e_{0}| = (−∞, 0].) We now put

exp_{(Γ,l,α)}(x, τ) = x + τw(e_{0}).

Now let v_{1}, ⋯, v_{k}, v_{last} be all the exterior edges of Γ. Let e_{i} be the exterior edges such that ∂_{source}e_{i} = v_{i}. We put v′_{i} = ∂_{target}e_{i}. Let C_{1,f} (Γ) be the set of all edges of finite length. (Namely the set of edges other than e_{i}.) We define w(α, i) = w_{a,b} where a = α(right(e)), b = α(left(e)). We put

f_{(Γ,l,α)}(x) = ∑_{i}⟨w(α, i), exp_{(Γ,l,α)}(x, |v′_{i}|)⟩ + ∑_{e∈C_{1,f} (Γ)} l(e)∥w(e)∥^{2}.

We now define

𝔅_{(Γ,l,α)} = e^{−2π(kc_{0}+ \frac{1}{ϵ} f_{(Γ,l,α)}(x)+\sqrt{−1}⟨y^{∗},w(e_{last})⟩)}ℑ_{ϵ}(B_{(Γ,l,α)}),
𝔅 = ∑_{Γ,α} ∫_{l∈Met(Γ)} 𝔅_{(Γ,l,α)}.

𝔅 converges if c_{0} is large. The following theorem is a consequence of Proposition 5.3 and the argument of the last subsection.

Theorem 5.1. 𝔅 satisfies (1), (2) of Problem 5.2.

We remark again the procedure we construct a solution of Problem 5.2 exactly corresponds to Remark 3.2.

In this subsection, we considered only two cases. They are generic in (real) dimension 4. The other cases in dimension 4 can be treated in a similar way. The difference between 4 dimensional case and higher dimensional case is as follows. Here we can start with the gauge field propagating along union of two lines. In higher dimension we need to study the gauge field propagating on submanifolds (foliated by lines). The discussion however is basically the same and can be organized by a similar Feynman diagram. We leave the precise study of it to the future research.

5.4 Monodromy around caustics and orientation twist.

We next study the Maurer-Cartan equation \overline{∂}_{0,ϵ}𝔅 + 𝔅 ∧ 𝔅 = 0 in a neighborhood of the caustics. Hence again our discussion is local. For the sake of preciseness we only consider the following example in this subsection, (though the author believes that the story can be generalized to the caustics of arbitrary generic Lagrangian submanifold of arbitrary dimension).

Let us consider ℝ^{4} = ℂ^{2}. Let x^{1}, x^{2}, y^{1}, y^{2} be the coordinate such that ω = ∑ dx^{i} ∧ dy^{i} and J ( \frac{∂}{∂x^{i}} ) = \frac{∂}{∂y^{i}}. Let L be a special Lagrangian submanifold in ℝ^{4} which is tangent to the fiber x = 0. L is a complex submanifold with respect to the other complex structure I ( \frac{∂}{∂x^{1}} ) = \frac{∂}{∂x^{2}} ,

I ( \frac{∂}{∂y^{1}} ) = \frac{∂}{∂y^{2}} . We assume that the restriction of π : (x, y) ↦ x to L is a branched double covering and 0 is the branch point. Namely with respect to the complex structure I, the restriction of π to L is holomorphic which is conjugate to z ↦ z^{2} at 0.

The mirror manifold is ℝ^{4} with coordinate x^{1}, x^{2}, y^{1∗}, y^{2∗}. The complex structure is J_{ϵ} ( \frac{∂}{∂x^{i}} ) = \frac{1}{ϵ} \frac{∂}{∂y^{i∗}}.

The mirror bundle E_{0} without quantum correction is given as follows. For each x ≠ 0, F_{x} ∩ L consists of two points p_{1}(x) = (x, y_{1}(x)), p_{2}(x) = (x, y_{2}(x)). (We remark however that p_{1}(x) and p_{2}(x) exchange when we go around origin. So p_{i}(x) is not globally well defined.) The fiber of E_{0} at (x, y^{∗}) is of rank 2 and its basis consists of [x; p_{1}(x)] and [x; p_{2}(x)]. They however are not holomorphic basis and we need to put some weight function to obtain a holomorphic basis (as in §5.1). Let us apply the definition of §5.1 in this case. There are multivalued (2 valued) closed one form θ such that L is a graph is θ. We may find a two valued function f such θ = df and f(0) = 0. Here f is a function on double cover of ℝ^{2} and will change the sign if we go around origin once. We have f(x) = ± \frac{2}{3} Re(x^{1} + \sqrt{−1}x^{2})^{3/2}. (See Example 3.4.)

If p_{i}(x) = (x, df(x)) then we put

Q_{i}(x) = \frac{1}{2} ∫_{0}^{1} df = \frac{1}{2} f(x), H_{i}(x, y^{∗}) = ⟨df(x), y^{∗}⟩ = y^{1∗} \frac{∂f}{∂x^{1}} + y^{2∗} \frac{∂f}{∂x^{2}} .

e^{2π( \frac{1}{ϵ} Q_{i}(x)+\sqrt{−1}H_{i}(x,y^{∗}))}[x; p_{i}(x)] is the holomorphic local frame we defined in §5.1 in this case. (Note Q_{2} = −Q_{1}, H_{2} = −H_{1} in our case.)

The holomorphic local frame of End(E_{0,ϵ}) is

\begin{matrix}e^{4π( \frac{1}{ϵ} Q_{i}(x)+\sqrt{−1}H_{i}(x,y^{∗}))}[x; p_{j}(x), p_{i}(x)] & if i ≠ j, \\ [x; p_{i}(x), p_{i}(x)] & i = 1, 2. \end{matrix} (67)

We next define the orientation twist ρ : π_{1}(L\{(0, 0)}) → {±1}. Note π_{1}(L\{(0, 0)}) = ℤ. ρ in our case is the unique nontrivial homomorphism ℤ → {±1}. The reason we need it becomes clearer later in this subsection.

We thus obtain a holomorphic vector bundle E_{0}^{ρ}. To describe it precisely we proceed as follows. We cut ℝ^{2}\{0} by the half line {(x^{1}, x^{2})|x^{2} = 0, x^{1} < 0}. We denote it by ℝ_{cut}^{2}. The two branches f_{1}, f_{2} of f is well defined globally. We choose f_{i} so that f_{1} ≥ 0 for x = (x^{1}, 0) with x^{1} ≥ 0.

On ℝ_{cut}^{2} × ℝ^{2}, we define holomorphic vector bundle by taking the local holomorphic frame e^{2π( \frac{1}{ϵ} Q_{i}(x)+\sqrt{−1}H_{i}(x,y^{∗}))}[x; p_{i}(x)]. Let (x^{1}, 0) be the point on the cut. We consider (x^{1}, +0) [resp. (x^{1}, −0)] which are limit of (x^{1}, x^{2}) when x^{2} goes to zero from above [resp. below]. We now put

\begin{aligned}[(x^{1}, +0); p_{1}(x)] & = −[(x^{1}, −0); p_{2}(x)], \\ [(x^{1}, +0); p_{2}(x)] & = [(x^{1}, −0); p_{1}(x)]. \end{aligned} (68)

Gluing the trivial bundle by this isomorphism we obtain E_{0}^{ρ}.

Now an important observation is that neither E_{0} nor E_{0}^{ρ} can be extended to the fiber at origin {0}×ℝ^{2}. This is because there is a nontrivial monodromy of our bundle. This phenomenon is “gauge theory analogue” of the phenomenon we met in §4, that is the T structure invariant complex structure can not be extended to the singular fiber. The idea we explained in §4 (which in case of type I singular fiber is already established by [68, 47]) is that the instanton effect gives a “quantum correction” to the complex structure and then the complex structure with “quantum correction” can be extended to the singular fiber. The “quantum correction” in the case of complex structure of manifold itself is a family version of the obstruction class 𝔪_{0} of the well-definedness of Floer homology of Lagrangian submanifold (see [37, 35]). Namely it is obtained by using the moduli space of pseudoholomorphic disks which bounds the fiber F_{x} of the fibration M_{0} → B_{0}. In our case of holomorphic structure of vector bundle, “quantum correction” is obtained by counting pseudoholomorphic strip, which bounds union of the two Lagrangian submanifolds L and the fiber F_{x}. We discussed the moduli space of such pseudoholomorphic disks in §3 especially in subsection 3.5. Conjecture 3.1 there asserts that such moduli space is described by the gradient line of the function f_{L} which is nothing but the function f we used above. (In §3.5 we verified half of the conjecture in our case, namely Conjecture 3.2 was proved there in our case.) In the rest of this subsection, we use the unstable manifold of f to modify the complex structure of E_{0}^{ρ} so that it can be extended to {0}×ℝ^{2}. Remark 5.2. We remark that our situation, where holomorphic bundle on ℝ^{2}\{0} × T^{2} does not extend, is a bit unusual. Usually in complex geometry, we meet a situation where a coherent sheaf is singular on a complex subvariety. In our case {0}×T^{2} is totally real. Hence the bundle E_{0}^{ρ}, E_{0} can not be extended even as a coherent sheaf, without including quantum effect.

We first recall that the unstable manifold of our multivalued function f is a union of the three half lines

(0, ∞) → ℝ^{2}, s ↦ (s, 0), (s cos ±2π/3, s sin ±2π/3).

Let w_{0} = (1, 0), w_{±} = (s cos ±2π/3, s sin ±2π/3) be their tangent vectors. We define Π_{0}, Π_{±} : ℝ^{2} → ℝ using w_{0}, w_{±} in the same way as (65). We take sufficiently small δ and a one form 𝔟_{δ} whose support is in [−δ, δ] and ∫ 𝔟_{δ} = 1, as in §5.3. We now define

B_{0} = − Hev(⟨x, w_{0}⟩)Π_{0}^{∗}(𝔟_{δ})[x; p_{2}(x), p_{1}(x)],
B_{+} = Hev(⟨x, w_{+}⟩)Π_{+}^{∗}(𝔟_{δ})[x; p_{1}(x), p_{2}(x)],
B_{−} = Hev(⟨x, w_{−}⟩)Π_{−}^{∗}(𝔟_{δ})[x; p_{1}(x), p_{2}(x)].

(69)

Here Hev is the Heaviside function.

We regard B_{0}, B_{±} as smooth forms on ℝ^{2}\D^{2} with coefficient in the bundle End(π!ℂ_{ρ}). Here π : L → ℝ^{2} is the projection and ℂ_{ρ} is a flat line bundle on L\({0} × ℝ^{2}) with monodromy ρ. (We remark that E_{0}^{ρ} is a pull back of π!ℂ_{ρ}.) The support of them (restricted to ℝ^{2}\D^{2}) is disjoint. And they are closed forms. Hence they define a flat connection on π!ℂ_{ρ} which we write d_{0,ρ} + B. (We remark that the rank two bundle π!ℂ_{ρ} has a flat connection d_{0,ρ} induced by the flat connection on ℂ_{ρ}.) Now we have :

Lemma 5.8. The bundle π!ℂ_{ρ} on ℝ^{2}\D^{2} equipped with the flat connection d_{0,ρ} + B can be extended to a flat bundle on ℝ^{2}.

Proof. B_{0},B_{+},B_{−} gives the monodromy (\begin{matrix}1 & −1 \\ 0 & 1 \end{matrix}), (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}), (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) re-

spectively. We have

(\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) (\begin{matrix}1 & −1 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (70)

which cancels the monodromy (68).

Remark 5.3. The cancellation of the holonomy as in (70) looks rather accidental. However, if we use the fact that the gradient line here coincides with the wall of the wall crossing phenomenon of the Floer homology as is discussed in [37, 35], then one finds that it should occur in general.

Remark 5.4. The reason we need the orientation twist ρ and the minus sign in the definition of B_{0} (69) can be seen from the proof of Lemma 5.8 above. However this reason is rather computational. It seems likely that there is a better explanation based on the orientation problem of the pseudoholomorphic disks.

Now we put weight to B_{0}, B_{±} in a way similar to §5.2 and will prove a complex analogue of Lemma 5.8. Namely we put

𝔅_{0,ϵ} = −e^{4π( \frac{1}{ϵ} Q_{1}(x)+\sqrt{−1}H_{1}(x))} Hev(⟨x, w_{0}⟩)
ℑ_{ϵ}Π_{0}^{∗}(𝔟_{δ})[x; p_{2}(x), p_{1}(x)],
𝔅_{+,ϵ} = e^{4π( \frac{1}{ϵ} Q_{2}(x)+\sqrt{−1}H_{2}(x))} Hev(⟨x, w_{+}⟩)
ℑ_{ϵ}Π_{+}^{∗}(𝔟_{δ})[x; p_{1}(x), p_{2}(x)],
𝔅_{−,ϵ} = e^{4π( \frac{1}{ϵ} Q_{2}(x)+\sqrt{−1}H_{2}(x))} Hev(⟨x, w_{−}⟩)
ℑ_{ϵ}Π_{−}^{∗}(𝔟_{δ})[x; p_{1}(x), p_{2}(x)].

(71)

We then set 𝔅_{ϵ} = 𝔅_{0,ϵ} + 𝔅_{+,ϵ} + 𝔅_{−,ϵ}. We can easily show

\overline{∂}_{0,ϵ}𝔅_{ϵ} + 𝔅_{ϵ} ∧ 𝔅_{ϵ} = 0. (72)

Now we have :

Proposition 5.4. The bundle E_{0,ρ,ϵ} with holomorphic structure \overline{∂}_{0,ϵ} +𝔅_{ϵ} can be extended to a holomorphic vector bundle on ℝ^{2} × T^{2}.

Proof. Let we take sections s_{i}(x) (i = 1, 2) of π!ℂ_{ρ} such that (d+B)s_{i} = 0 and are linearly independent. (Lemma 5.8 implies the existence of such sections). We write them as

s_{i}(x) = s_{i,1}(x)[x; p_{1}(x)] + s_{i,2}(x)[x; p_{2}(x)].

Then we define e_{i}(x, y^{∗}) by

\begin{matrix}e_{i}(x, y^{∗}) = & e^{−2π( \frac{1}{ϵ} Q_{1}(x)+\sqrt{−1}H_{1}(x,y^{∗}))}s_{i,1}(x)[x; p_{1}(x)] \\ + e^{−2π( \frac{1}{ϵ} Q_{2}(x)+\sqrt{−1}H_{2}(x,y^{∗}))}s_{i,2}(x)[x; p_{2}(x)]. \end{matrix}

By a straightforward calculation, we can check

(\overline{∂}_{0,ϵ} + 𝔅_{ϵ})e_{i} = 0.

The proposition follows easily.

5.5 Feynman diagram description of holomorphic structure of vector bundles.

Using the result of subsections 5.2,5.3,5.4, we can define holomorphic structures of the mirror bundle (together with quantum effect) at least in the asymptotic sense. The result (which we describe in this subsection) is parallel to the description on the pseudoholomorphic disks we gave in §3.6.

We consider the following situation. Let us take L ⊆ T^{4} a complex submanifold (Riemann surface) with respect to the complex structure I. It then becomes a (special) Lagrangian submanifold of T^{4} (with respect to the symplectic structure ω which is a Kähler form of the complex structure J.) We consider Lagrangian fibration T^{4} → T^{2}. We restrict it to L. Then, with respect to the complex structure I, this map L → T^{2} is holomorphic. Hence it is a branched covering. We assume that it is locally conjugate to z ↦ z^{2} at caustics. (This holds for generic L.) Remark 5.5. The author thanks to N. Leung who showed this example L to the author and asked what is its mirror bundle, after the author’s talk at RIMS during the conference “Algebraic geometry and Integrable system” in 2000 June.

Let x^{1}, x^{2}, y^{1}, y^{2} be the (symplectic) coordinate of T^{4}. Let T^{4} be the mirror torus with complex structure J_{ϵ}. (z_{ϵ}^{i} = x^{i} +ϵ\sqrt{−1}y^{i∗} is its complex coordinate.)

Let S(L) ⊂ T^{2} be the caustics. We put B_{0} = T^{2}\S(L). We put

L_{0} = {p ∈ L | dπ : T_{p}L → T_{πp}T^{2} is zero.}

We define ρ : π_{1}(L_{0}) → {±1} so that holonomy around each point in L\L_{0} is nontrivial.

Using these data, we defined (in subsection 5.1) a holomorphic vector bundle E(L)_{0}^{ρ} on B_{0} × T^{2}. We denote its holomorphic structure by \overline{∂}_{0,ϵ}. The main result of this section is the following Theorem 5.2. We consider the following assumption. We use the notation of §3.6 (Definition 3.9).

Assumption 5.2. λ is a positive number. The moduli space M_{Morse}(M, L; α) is empty if α ∩ ω ≤ λ.

Theorem 5.2. We assume Assumption 5.2, Then, for each ϵ, there exists a section 𝔅_{ϵ;λ} ∈ Γ(B_{0} × T^{2}; End(E(L)_{0}^{ρ}) ⊗ Λ^{0,1}) with the following properties.

(1) There exists C_{λ,k} depending on λ but is independent of ϵ such that

∥(\overline{∂}_{0,ϵ} + 𝔅_{ϵ;λ})^{2}∥_{C^{k}} ≤ C_{λ,k}e^{−λ/ϵ}.

Here the left hand side is the C^{k} norm.

(2) The complex vector bundle E(L)_{0}^{ρ} together with holomorphic structure ∂_{0,ϵ} + 𝔅_{ϵ;λ} extends to one on T^{4}.

Remark 5.6. If Conjecture 3.4 is correct, Assumption 5.2 is equivalent to the following : there exists no pseudoholomorphic disk φ : (D^{2}, ∂D^{2}) → (T^{4}, L) such that ∫ φ^{∗}ω ≤ λ.

Hence Assumption 5.2 is related to the vanishing of the obstruction to define Floer homology of Lagrangian submanifolds, introduced in [37].

We remark that Assumption 5.2 is satisfied for generic L in our case when dimension is 4 and if Maslov index of L vanishes.

Proof. We prove more. Namely we give an explicit description of 𝔅_{ϵ;λ} based on Feynman diagram.

We first take sufficiently small c (which depends on λ and will be specified later) and identify c neighborhood B_{c}(q) of each point of the caustics q with a ball in ℝ^{2} of large radius. The restriction of the bundle E(L)_{0}^{ρ} to (B_{c}(q)\{q}) × T^{2} can be divided as a direct sum of the two bundles E(q)_{1} ⊕E(q)_{2}. E(q)_{1} is the rank two bundle whose basis is identified to the two points in F_{p} ∩ L which are boundary points of the vanishing arc. We rescale the form B_{ϵ} constructed in the last section and regard it as the section of End(E(q)_{1}) ⊗ Λ^{0,1}. We denote it by 𝔅_{ϵ}(q). Then ∂_{0,ϵ} + ∑_{q} 𝔅_{ϵ}(q) is a holomorphic structure defined in the union of (B_{c}(q)\{q}) × T^{2} and it extends to the union of B_{c}(q) × T^{2}.

We are going to extend it globally so that Theorem 5.2 (1) is satisfied. To do so, we need to define a moduli space which is a kind of smoothing

of one in §3.6. In §3.6 we considered the gradient lines which start exactly at caustics. We relax this condition a bit as follows.

Let ℓ_{i,q} : (−∞, 0] → B_{c}(q), i = 1, 2, 3 be the three gradient lines starting the caustics and corresponding to the vanishing cycles. There exist small transversals u_{i,q} : [−1, 1] → B_{c}(q) and a smooth section 𝔟 : [−1, 1] → u_{i,q}^{∗}Λ^{1} of compact support, such that 𝔅_{ϵ}(q) in a neighborhood of the image u_{i,q} is

e^{−4π( \frac{1}{ϵ} f_{L}+\sqrt{−1} ∑_{i} \frac{∂f_{L}}{∂x^{i}} y^{∗i})}ℑ_{ϵ}B (73)

where

\begin{aligned}B = ∫_{−1}^{1} dt ∫_{−δ}^{δ} ds & exp(−sgradf_{L})^{∗}𝔟(t) \\ & ⊗ [u_{i,q}(v); p_{b}(u_{i,q}(v)), p_{a}(u_{i,q}(v))]. \end{aligned} (74)

Here f_{L} is a branch of our multivalued function and p_{a}(x), p_{b}(x) are two intersection points of F_{x} ∩ L which are two boundary points of the vanishing arc.

The existence of such u_{i,q} and 𝔟 is immediate from the construction of the last subsection.

Now we consider ribbon graph Γ satisfying Condition 3.1. Let l be a metric on it. We assume here that all the edges are of finite length. Let ψ : |Γ| → B be a continuous map and ψ̃ : |Γ|\{vertices} → B̃_{L} be its lift. We assume Condition 3.2 (1),(2). In place of Condition 3.2 (3), we assume the following.

Condition 5.1. Let v ∈ Vertex_{sta}(Γ) and ∂_{source}(e) = v. We identify |e| = [0, l(e)]. Then ψ(0) = u_{i,q}(t) where u_{i,q} is as above and t ∈ [−1, 1]. We put ψ̃(s) = (ψ(s); p(s), q(s); β(s)) where β(s) ∈ π_{1}(F_{ψ(s)}; p(s), q(s)). Then β(0) is the vanishing arc at q.

Definition 5.2. The set of all (Γ, l, ψ, ψ̃) such that Conditions 3.2 and 3.4 (1)(2), 5.1 are satisfied and that l is finite, is denoted by M_{c}(L).

We define a function E on M_{c}(L) by

E(Γ, l, ψ, ψ̃) = ∫_{β(l(e_{last}))} ω. (75)

We set

ev(Γ, l, ψ, ψ̃) = ψ(v_{last}). (76)

We will define one form 𝔟(Γ, l, ψ, ψ̃) at ev(Γ, l, ψ, ψ̃) as follows. We define 𝔟(v) for each vertex v by the induction on the partial order <. Here we say v < v′ if every arc joining v with v_{last} contains v′. First for v ∈ Vertex_{sta}(Γ) we have ψ(0) = u_{i,q}(t). Then we put 𝔟(v) = 𝔟(t), where 𝔟(t) is as in (74). Let us assume that 𝔟(v) is defined for any v with v < v_{0}. Let us define 𝔟(v_{0}). We consider two edges e_{1}, e_{2} such that ∂_{target}e_{i} = v_{0}. Let v_{i} = ∂_{source}e_{i}. We now put

𝔟(v_{0}) = i_{gradf_{0}}(− exp(l(e_{2})gradf_{2})^{∗}𝔟(v_{2})

∧ exp(−l(e_{1})gradf_{1})^{∗}𝔟(v_{1})) (77)

Here f_{1}, f_{2}, f_{0} are three branches of f_{L} which correspond to the lift ψ̃ on e_{1},e_{2} and e_{0} respectively. Here e_{0} is the edge such that ∂_{source}e_{0} = v_{0}.

We then define

𝔟(Γ, l, ψ, ψ̃) = 𝔟(v_{last}).

We put

M_{c}(L; λ) = {(Γ, l, ψ, ψ̃) ∈ M_{c}(L)|E(Γ, l, ψ, ψ̃) ≤ λ}

The following is easy to prove :

Lemma 5.9. Under Assumption 5.2, we may choose the positive number c so small that M_{c}(L; λ) is compact.

For (Γ, l, ψ, ψ̃) ∈ M_{c}(L) we may fix a branch of f_{L} at ev(Γ, l, ψ, ψ̃). We use this branch to define

e^{−2π( \frac{1}{ϵ} f_{L}+\sqrt{−1} ∑_{i} \frac{∂f_{L}}{∂x^{i}} y^{∗i})} ∘ ev.

Hereafter we omit ev for simplicity. The choice of branch determines a basis of End(E_{0,ϵ}^{ρ}) also. We write it as e(Γ, l, ψ, ψ̃). We choose δ > 0 small such that Assumption 5.2 holds for λ + δ in place of λ. Let χ : ℝ → [0, 1] be a smooth function such that χ = 0 on [λ + δ, ∞) and χ = 1 on [0, λ]. Now we define :

Definition 5.3.

\begin{aligned}𝔅_{ϵ,λ} = ∫_{M_{c}(L;λ+δ)} & χ(E(Γ, l, ψ, ψ̃))e^{−2π( \frac{1}{ϵ} f_{L}+\sqrt{−1} ∑_{i} \frac{∂f_{L}}{∂x^{i}} y^{∗i})} \\ & ℑ_{ϵ}(𝔟(Γ, l, ψ, ψ̃)) ⊗ e(Γ, l, ψ, ψ̃). \end{aligned}

Theorem 5.2 now follows from the construction of the last three subsections.

Theorem 5.2 gives only an asymptotic solution. It seems likely that we can use implicit function theorem to find an actual solution nearby. Since the author did not work it out at the time of writing this article, he states it as a conjecture.

Conjecture 5.1. We assume Assumption 5.2 for all λ. Then, for each λ, there exists ϵ(λ, k) such that there exists 𝔅′_{ϵ,λ} for each ϵ < ϵ(λ, k) with the following properties.

(1) \overline{∂}_{0,ϵ}𝔅′_{ϵ,λ} + 𝔅′_{ϵ,λ} ∧ 𝔅′_{ϵ,λ} = 0.

(2) ∥𝔅_{ϵ;λ} − 𝔅′_{ϵ;λ}∥_{C^{k}} ≤ C_{λ,k}e^{−λ/ϵ}.

5.6 Homoloical mirror symmetry

In §5.5 we described a procedure to construct a mirror bundle from an unobstructed Lagrangian submanifold in the 4 dimensional case. This is a part of the homological mirror symmetry conjecture by Kontsevich [58, 59]. (See [32, 33] for precise formulation.) In this section, we explain how the other parts of the homological mirror symmetry conjecture may follow from a similar argument. We recall that the first part of the homological mirror symmetry conjecture is HF (L_{1}, L_{2}) ≅ Ext(L_{1}, L_{2}), where HF (L_{1}, L_{2}) is the Floer cohomology and Ext(L_{1}, L_{2}) is the extension. We remark that Ext(L_{1}, L_{2}) is isomorphic to the \overline{∂} cohomology group H\overline{∂}(M^{∨}; Hom(E(L_{1})_{ϵ}, E(L_{2})_{ϵ})).

Let us fix a hermitian metric on E(L_{a})_{ϵ} a = 1, 2 and let Δ_{\overline{∂},ϵ} be the Laplace operator acting on Γ(M^{∨}; Hom(E(L_{1})_{ϵ}, E(L_{2})_{ϵ}) ⊗ Λ^{0,k}). Now analogy of Witten’s statement Theorem 4.1 in §4.2 is the following. We assume that L_{1} is transversal to L_{2} for simplicity.

Conjecture 5.2. There exists δ > 0 such that if ϵ is sufficiently small then the rank of the vector space

\begin{matrix}ℌ_{ϵ}^{k}(E(L_{1})_{ϵ}, E(L_{2})_{ϵ}) \\ = \{u ∈ Γ(M^{∨}; Hom(E(L_{1})_{ϵ}, E(L_{2})_{ϵ}) ⊗ Λ^{0,k})|Δ_{\overline{∂},ϵ}u = λu, λ < e^{− \frac{δ}{ϵ}} \} \end{matrix}

is equal to the number of the points in the intersection L_{1} ∩ L_{2} whose Maslov index is k. Moreover element u_{q} corresponding to (q, p) = L_{1} ∩ L_{2} is concentrated in a neighborhood of q in the following sense. For any c there exists C such that

|u_{q}| < Ce^{−1/Cϵ}∥u_{q}∥_{L^{2}}

outside π_{M^{∨}}^{−1}B_{c}(q).

Then the next thing to study is the tunnel amplitude and interactions. Our plan is to reduce both of them to the Morse homotopy of multivalued functions. In the A model side (pseudoholomorphic disk) we discussed it already in §3. (We discussed there the case when there is one Lagrangian submanifold only. But to generalize it to our situation is fairly straight forward.) We start with B-model side. Our discussion here is very close to §4.2.

We first consider

\overline{∂} : ℌ_{ϵ}^{k}(E(L_{1})_{ϵ}, E(L_{2})_{ϵ}) → ℌ_{ϵ}^{k+1}(E(L_{1})_{ϵ}, E(L_{2})_{ϵ}) (78)

and study ⟨\overline{∂}(u_{p}), u_{q}⟩. In case there is no quantum effect to the holomorphic structure of E(L_{a})_{ϵ} its asymptotic behavior is described by the gradient line of f_{L_{1},L_{2}} in the same way as Theorem 4.2. Namely the asymptotic behavior of \overline{∂}_{0,ϵ}𝔰 = 0 can be analyzed by the Laplace operator deformed by f_{L_{1},L_{2}} in each Fourier mode. However, since there is a quantum effect on the holomorphic structure on E(L_{a})_{ϵ}, we need to study the equation

\overline{∂}_{0,ϵ}𝔰 + 𝔅^{2} ∧ 𝔰 + 𝔰 ∧ 𝔅^{1} = 0. (79)

Here \overline{∂}_{0,ϵ} + 𝔅^{a} is the holomorphic structure of E(L_{a})_{ϵ}.

Before stating a conjecture describing the asymptotic behavior of the solution of (79) we generalize the story. Namely we study more general operations than (78). Let us consider several Lagrangian submanifolds L_{a}, a = 0, ⋯, A. We assume that they are mutually transversal. Let

Π_{L_{a},L_{b}} : L^{2}(M^{∨}; Hom(E(L_{a})_{ϵ}, E(L_{b})_{ϵ})) → ℌ_{ϵ}^{k}(E(L_{a})_{ϵ}, E(L_{b})_{ϵ})

be the orthonormal projection. We denote by Δ_{L_{a},L_{b}} the \overline{∂} Laplace operator on Γ(M^{∨}; Hom(E(L_{a})_{ϵ}, E(L_{b})_{ϵ})). In a way similar to §4.2 we can construct a propagator G_{L_{a},L_{b}} such that

G_{L_{a},L_{b}} ∘ \overline{∂} + \overline{∂} ∘ G_{L_{a},L_{b}} = id −Π_{L_{a},L_{b}}.

Using it we will define

𝔪_{Γ,α} : ⨂_{j=1}^{r} ℌ_{ϵ}(E(L_{α(j−1)})_{ϵ}, E(L_{α(j)})_{ϵ}) → ℌ_{ϵ}(E(L_{α(0)})_{ϵ}, E(L_{α(r)})_{ϵ})

in a way similar to §4.2. Here Γ,α are as follows. Γ is a ribbon tree satisfying Condition 2.2. We embed it as |Γ| ⊂ D^{2} such that the inverse image of ∂D^{2} is the set of exterior vertices. α is a map α : π_{0}(D^{2}\|Γ|) → {1, ⋯, A}. In other words, we assign a Lagrangian submanifold to each connected component of D^{2}\|Γ|. Let r +1 be the order of π_{0}(D^{2}\|Γ|). We number the element of π_{0}(D^{2}\|Γ|) clockwise. We write them D_{0}, ⋯, D_{r} respectively and put α(a) = α(D_{a}).

Figure 5.4

We define 𝔪_{Γ,α} together with φ_{Γ,α} by induction on the number of vertices of Γ. In case Γ = Γ_{−1} has no interior vertex, 𝔪_{Γ_{−1},α} is the operator (78) and φ_{Γ_{−1},α} is the identity.

We assume that 𝔪_{Γ,α}, φ_{Γ,α} is defined for Γ the number of whose interior vertices is < v. Let Γ be the ribbon tree with v interior vertices. We remove the last edge from it in the same way as in §4.2. We then get two ribbon trees Γ_{1},Γ_{2}. α also determines α_{1}, α_{2}. Now we put

\{ \begin{matrix}𝔪_{Γ,α} & = Π_{L_{α(0),α(r)}} ∘ (𝔪_{Γ_{1},α_{1}} ∧ 𝔪_{Γ_{2},α_{2}}) \\ φ_{Γ,α} & = G_{L_{α(0),α(r)}} ∘ (𝔪_{Γ_{1},α_{1}} ∧ 𝔪_{Γ_{2},α_{2}}) . \end{matrix} (80)

We then define

𝔪_{k} = ∑_{Γ,α} 𝔪_{Γ,α}, φ_{k} = ∑_{Γ,α} φ_{Γ,α},

where the sum is taken over all Γ with k + 1 exterior vertices. In a way similar to the proof of Theorem 4.3 we can prove that 𝔪_{k} satisfies the A_{∞} relation and hence defines an A_{∞} category and φ_{k} defines an A_{∞} functor.

We now are going to describe the asymptotic behavior of the operator 𝔪_{Γ,α} by Morse homotopy of several multivalued functions f_{L_{a},L_{b}}. Because of the presence of quantum effect to the holomorphic structures of E(L_{a}) we need to use a bit more complicated Feynman diagram to do so.

Assumption 5.3. We consider a ribbon tree Γ_{+} which satisfies Condition 2.2 (1),(3),(4) and Condition 3.1(2′). We assume that the set of edges of Γ_{+} is splitted into a disjoint union Edge(Γ_{+}) = Edge(Γ)_{0} ∪ Edge(Γ)_{+}.

(1) We assume that after removing the edges in Edge(Γ)_{+} we still have a ribbon tree which we write Red(Γ_{+}) = Γ. We assume Γ satisfies Condition 2.2 (1),(2),(3),(4).

(2) Let |Γ_{+}| ⊂ D^{2} be the embedding as before. α : π_{0}(D^{2}\|Γ|) → {1, ⋯, A} is a map as before.

(3) We divide the difference |Γ_{+}|\|Γ| into the connected components Γ_{+}^{(i)}. We assume the closures of Γ_{+}^{(i)} are disjoint each other. We also assume that each of Γ_{+}^{(i)} satisfies Condition 2.2 (1),(3),(4) and Condition 3.1 (2′).

(4) Let v be the vertex on |Γ_{+}^{(i)}|∩|Γ|. (We remark that |Γ_{+}^{(i)}|∩|Γ| consists of one point.) We assume that v is contained in the interior of the edge of Γ. We also assume that v is the last vertex of Γ_{+}^{(i)}.

We draw the Figure of |Γ_{+}| below. In Figure 5.5 we draw the edge of Γ by a real line and the edge in Edge(Γ)_{+} by dotted line.

Figure 5.5

We next discuss some other data we need. l is a metric on Γ_{+}. We assume Condition 2.3 (b). It induces a metric on Γ and Γ_{+}^{(i)}.

ψ : |Γ_{+}| → B is a continuous map. ψ̃ is a lift of it at each edges such that :

Condition 5.2. Let e ∈ Edge(Γ)_{+}. |e| is contained in a component D(e) of D^{2}\|Γ|. Then the restriction of ψ̃ to |e| is a map to B̃_{L_{α(D(e))}}^{+}.

Let e ∈ Edge(Γ). We have two components left(e),[resp. right(e)] which is in the left [resp. right] hand side of e. Then the restriction of ψ̃ to |e| is a map to B̃_{L_{α(left(e))},L_{α(right(e))}}^{+}.

We assume the lift ψ̃ is a gradient line. We specify the branch as follows. We write ψ̃ = (ψ; p, q; β).

Assumption 5.4. (1) Let v be an exterior vertex of Γ_{+}. We choose τ_{i} ∈ |Γ_{+}| converging to v. Let e be the unique edge such that v ∈ ∂e. We first consider the case v ∈ Γ. Then

lim_{i→∞} ψ(τ_{i}) ∈ π_{M} (L_{α(left(e))} ∩ L_{α(right(e))}) .

We assume moreover that β(τ_{i}) (which is in π_{1}(F_{x}; F_{x}∩L_{α(left(e))}, F_{x}∩

L_{α(right(e))}) is a vanishing arc.

(2) We next consider the case when v ∈ Γ_{+}^{(i)}. We may assume |Γ_{+}^{(i)}| ⊂ D_{a}.

Then

lim_{i→∞} ψ(τ_{i}) ∈ S(L_{α(D_{a})}).

We also assume that β(τ_{i}) is a vanishing arc.

(3) Let v be an interior vertex of Γ_{+}. We first assume that v is an interior vertex of Γ. Let e_{1}, e_{2}, e_{0} be the three edges of Γ such that ∂_{target}e_{1} = ∂_{target}e_{2} = v = ∂_{source}e_{0}. We assume e_{1}, e_{2}, e_{0} respects the cyclic order of the ribbon structure at v. We choose τ_{i,j} on e_{i} converging to v as j → ∞. Then

lim_{j→∞} β(τ_{1,j}) + lim_{j→∞} β(τ_{2,j}) = lim_{j→∞} β(τ_{0,j}).

Here the left hand side is the product π_{1}(F_{x}; F_{x}∩L_{left(e_{1})}, F_{x}∩L_{right(e_{1})})× π_{1}(F_{x}; F_{x}∩L_{left(e_{2})}, F_{x}∩L_{right(e_{2})}) → π_{1}(F_{x}; F_{x}∩L_{left(e_{1})}, F_{x}∩L_{right(e_{2})}).

(4) We consider each of Γ_{+}^{(i)}. Then we assume that the restriction of ψ̃ there satisfies Condition 3.2 (2).

(5) Let v be the vertex in |Γ| ∩ |Γ_{+}^{(i)}|. Let us take τ_{j} ∈ |Γ_{+}^{(i)}| converging to v. Let v = τ_{0} ∈ |e| where e is an edge of Γ. (Compare Assumption 5.3 (4).) Suppose |Γ_{+}^{(i)}| ∈ D_{a}, a = right(e). We take points τ ± 1/j on |e| converging to v. We then have

lim_{j→∞} β(τ_{j}) + lim_{j→∞} β(τ − 1/j) = lim_{j→∞} β(τ + 1/j).

Here the sum in the left hand side is π_{1}(F_{x}; F_{x} ∩ L_{a}, F_{x} ∩ L_{a}) × π_{1}(F_{x}; F_{x} ∩L_{left(e)}, F_{x} ∩L_{right(e)}) → π_{1}(F_{x}; F_{x} ∩L_{left(e)}, F_{x} ∩L_{right(e)}). In case a = left(e) we have

lim_{j→∞} β(τ − 1/j) + lim_{j→∞} β(τ_{j}) = lim_{j→∞} β(τ + 1/j).

Figure 5.6

Now the main assumption is that ψ̃ is a gradient line. Namely

Assumption 5.5. On |e| with e ∈ Edge(Γ)_{0}, we have

\frac{∂ψ̃}{∂τ} = gradf_{L_{right(e)},L_{left(e)}}.

We also assume that the restriction of ψ̃ to each of Γ_{+}^{(i)} satisfies Condition 3.2 (1).

Definition 5.4. Let M_{Morse}(M; L_{0}, ⋯, L_{A}; Γ_{+}, α) be the set of all (l, ψ̃) satisfying Assumptions 5.3, 5.4, 5.5 and Condition 5.2.

We define an energy E of an element (l, ψ̃) of M_{Morse}(M; L_{0}, ⋯, L_{A}; Γ_{+}, α) as follows.

E(l, ψ̃) = ∑_{e∈Edge(Γ_{+})} ∫_{τ∈|e|} ∥ \frac{∂ψ}{∂τ} ∥ dτ. (81)

Let us fix (Γ, α). Let us number the components of D^{2}\|Γ| in clockwise way and denote the components by D_{0}, ⋯, D_{r}. Let v_{1}, ⋯, v_{r} be exterior vertices such that v_{last}, v_{1}, ⋯, v_{r} respects the cyclic order. We take edges e_{i} such that ∂_{source}e_{i} = v_{i}. Let us take Γ_{+} such that Red(Γ_{+}) = Γ. We assume that right(e_{1}) = D_{0}, left(e_{r}) = D_{r}. Let (l, ψ̃) ∈ M_{Morse}(M; L_{0}, ⋯, L_{A}; Γ_{+}, α). For each v_{i} ∈ Vertex(Γ), the map ψ̃ determines an element of L_{i−1} ∩ L_{i}. We denote it by ev_{i}(l, ψ̃).

For p_{i} ∈ L_{i−1} ∩ L_{i} we have an element u_{p_{i}} of ℌ_{ϵ}(E(L_{i−1})_{ϵ}, E(L_{i})_{ϵ}) (if we assume Conjecture 5.2.) Now we can state the following.

Conjecture 5.3.

⟨𝔪_{Γ,α}(u_{p_{1}}, ⋯, u_{p_{r}}), u_{p_{0}}⟩ ∼ ∑ ± ext(−E(l, ψ̃)) (82)

where the sum is taken over all Γ_{+} with Red(Γ_{+}) = Γ and (l, ψ̃) ∈ M_{Morse}(M; L_{0}, ⋯, L_{A}; Γ_{+}, α) such that ev_{i}(l, ψ̃) = p_{i}.

The sign ± is determined by the orientation of the moduli space M_{Morse}(M; L_{0}, ⋯, L_{A}; Γ_{+}, α).

It might be possible to prove Conjecture 5.3 by using a similar argument as ones given in this section. In fact the conditions we put on the Feynman diagram for M_{Morse}(M; L_{0}, ⋯, L_{A}; Γ_{+}, α) is the Feynman rule of the equation (79).

An argument similar to one in §3 may imply that the same number as in the right hand side of (82) is the matrix element of the product structure of the Floer homology. In fact this is correct only in the case when there is no holomorphic disks bounding L_{a}. In case such holomorphic disk exists we need to include the correction terms as in [34, 37]. It seems likely that one can modify the argument of this section to include those correction terms. But we do not discuss it here since the formula will become very complicated and since we assumed that there is no holomorphic disk bounding L in Theorem 5.2. So our conjecture is :

Conjecture 5.4. Suppose that there is no holomorphic disks bounding L_{a}. Then the A_{∞} structure on Floer homology defined in [34] is

⟨𝔪_{k}(p_{1}, ⋯, p_{r}), p_{0}⟩ = ∑_{Γ} ∑ ± ext(−E(l, ψ̃))

where the sum in the right hand side is taken in the same way as Conjecture 5.3.

Conjectures 5.3 and 5.4, if proved, will imply homological mirror symmetry conjecture. Actually the way it was proved in the case of affine Lagrangian submanifold in complex torus in [32] is similar to one we are discussing here.

REFERENCES 74

We finally state another conjecture which is related to the construction to the opposite direction. Namely the construction of Lagrangian submanifold out of holomorphic vector bundles. As we mentioned at the beginning of §1.1, this is a gauge theory analogue of the theory of collapsing Riemannian manifolds. Let π_{M^{∨}} : M^{∨} → B be as in §1. Namely π_{M^{∨}} is a singular fibration such that the general fibers are Lagrangian tori. We assume that there exists a family of complex structures J_{ϵ} on M^{∨} and a family of Calabi-Yau metrics g_{ϵ} on M^{∨} such that J_{ϵ} satisfies the assumption of §1.2 Proposition 1.2 and the diameter of the fibers with respect to g_{ϵ} are ∼ ϵ. We also suppose that there exists a family of holomorphic vector bundles E_{ϵ} on (M^{∨}, J_{ϵ}). We assume that they are stable. Then by [20, 74] there exists a family of Einstein-Hermitian connections ∇_{ϵ} on E_{ϵ}.

Conjecture 5.5. There exists a rectifiable subset S(E) ⊆ B of codimension ≥ 2 such that for each δ the curvature of the restriction of ∇_{ϵ} to M^{∨}\π_{M^{∨}}^{−1}(U_{δ}(S(E))) is uniformly bounded. Here U_{δ}(S(E)) is the δ neighborhood of S(E).

For x ∉ S(E). The restriction of ∇_{ϵ} to the fiber F_{x}^{∨} converges to a flat connection as ϵ → 0. Hence it determines an element of S_{k}(F_{x}). Here S_{k} is the k-th symmetric power and k is the rank of the bundle E. We denote this element by s(x). The set {(x, y)|y ∈ s(x)} is a submanifold of M the mirror manifold. Its closure is a Lagrangian submanifold. We write it L(E).

There exists a family of special Lagrangian submanifolds L_{ϵ} on the mirror manifold M_{ϵ} such that L_{ϵ} converges to L.

Remark 5.7. In the case when the dimension of M is 4, the Einstein-Hermitian connections ∇_{ϵ} consist a family of self dual connections. Hence we have a family of self dual connections on a family of 4 manifolds which collapses to a 2 manifold. This is the situation studied extensively in the study of Atiyah-Floer conjecture. So in this situation, by using the method of [21, 31], one may solve Conjecture 5.5. Remark 5.8. Let us consider the case when the dimension of M is 6. Then the situation is harder. However we still may prove the conjecture in the following way. We may use the result by Tian [73] to conclude that the set where the curvature of ∇_{ϵ} diverges is a complex subvariety of complex codimension 2. Namely it is a holomorphic curve. As we studied in section 2, the holomorphic curve (with finite energy) on M^{∨} is in a small neighborhood of π_{M^{∨}}^{−1}(Γ) where Γ is a graph in B. Thus we may take S(E) = Γ.

Thus we described a series of conjectures, which, if proved, will prove most of the consequences of mirror symmetry. Some of the conjectures may be proved near future. However to prove several of the conjectures are yet very hard and so there is yet a long way to go.

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