Proceedings of 7^{th} Gökova Geometry-Topology Conference, pp, 1 – 20

Galois symmetry on Floer cohomology

Kenji FUKAYA

1. Introduction

Floer cohomology of Lagrangian submanifold defined in [Fl, Oh, FOOO] is in general a module over a kind of formal power series ring, which is called Novikov ring [No]. In [FOOO, Fu2] we used the universal Novikov ring Λ_{ℂ} which consists of a formal sum

∑_{i} a_{i}T^{λ_{i}}  (1)

where a_{i} ∈ ℂ and λ_{i} ∈ ℝ such that λ_{i} < λ_{i+1}, lim_{i→∞} λ_{i} = +∞ and T is a formal parameter.

For a rational Lagrangian submanifold (we define it later in Definition 2.2), we take a smaller ring Λ_{ℂ}^{ℚ} which consists of elements (1) satisfying λ_{i} ∈ ℚ in addition.

Both of them are complete non Archimedean valuation field whose norm is defined by

∥∑_{i} a_{i}T^{λ_{i}} ∥ = exp (− min{λ_{i}|a_{i} ≠ 0}) .  (2)

The continuous Galois group of Λ_{ℂ}^{ℚ} over ℂ[[T]][T^{−1}], the Laurant power series ring, is ℤ̂, the profinite completion of ℤ. In fact, the topological (generator) ρ ∈ ℤ̂ acts by

∑_{i} a_{i}T^{λ_{i}} ↦ ∑_{i} a_{i}e^{2π\sqrt{−1}λ_{i}}T^{λ_{i}}  (3)

We denote the map (3) by x ↦ x^{ρ}. The rationality of λ_{i} implies that the ℤ action defined by (3) can be extended to a ℤ̂ action.

In the same way, ℤ acts as an automorphism on Λ_{ℂ}/ℂ[[T]][T^{−1}] by the formula (3). In this article, we discuss an action of ℤ̂ on Floer cohomologies which are compatible

with the Galois action on Novikov rings.

In section 2, we state our main result (Theorem 2.4) which establish a ℤ̂ action on the filtered A_{∞} category defined in [Fu2]. (See [Fu2] also for the definition of filtered A_{∞} category.) In section 3 we prove it. In section 4 we discuss briefly how the ℤ̂ action constructed in this article is related to mirror symmetry. Sections 3 and 4 are independent to each other.

Partially supported by Grant in-Aid for Science Research 13852001.

Kenji FUKAYA

The reader can find easily that there is influence of the paper [KS] to this article. Compare also [Fu3] §3.5, some of the argument of which is improved in this article.

2. Statement of main result

Let (M, ω) be a real 2n dimensional symplectic manifold. We assume c^{1}(M) = 0. (Here c^{1}(M) is the first Chern class of the tangent bundle of M. We remark that symplectic structure on M determines an almost complex structure on M uniquely up to homotopy. Hence the first Chern class of the tangent bundle of M is well defined.) We also assume that the cohomology class [ω] is contained in the image H^{2}(M; ℤ) → H^{2}(M; ℝ).

Definition 2.1. A pair (ξ, ∇^{ξ}) of a complex line bundle on M and a unitary connection on it is called a prequantum bundle if F_{∇^{ξ}} = 2π\sqrt{−1}ω as differential 2 forms. Here F_{∇^{ξ}} is the curvature of ∇_{ξ}.

By our assumption [ω] ∈ H^{2}(M; ℤ), prequantum bundle (ξ, ∇^{ξ}) exists. We choose and fix it throughout this article. Let L be a Lagrangian submanifold of M. Namely L is an n dimensional submanifold of M such that ω|_{L} = 0. By definition, the restriction of (ξ, ∇^{ξ}) to L is flat. Hence it determines a representation hol_{ξ} : π_{1}L → U(1).

Definition 2.2. L is called a rational Lagrangian submanifold if the image of hol_{ξ} : π_{1}L → U(1) is a finite group.

We remark that L is called a Bohr-Sommerfeld orbit, if hol_{ξ} : π_{1}L → U(1) is trivial. In [Fu2], the author defined a filtered A_{∞} category whose objects are system (L, 𝔏, s̃, b)

of Lagrangian submanifold L, a flat U(1) bundles L on it, together with some additional data (s̃, b). In this article, we consider the case when L is rational. Let us very briefly review the other data. See [Fu2] for detail.

We first assume that L is oriented. We also fix st ∈ H^{2}(M; ℤ_{2}). We can then define the notion that L is relatively spin in M as in [FOOO] or [Fu2] Definition 2.4. We assume that L is relatively spin and fix a relative spin structure.

We next define grading (see [Se, Fu2]). Let Lag(M) → M be a bundle on M whose fiber at x ∈ M is the set of all Lagrangian linear subspaces of T_{x}M. Note the fundamental group of Lagrangian Grassmanian (= the fiber of Lag(M) → M) is ℤ. It is proved in [Se] that, under our assumption c^{1}(M) = 0, there exists a ℤ cover L̃ag(M) of Lag(M) such that its restriction to each fiber of Lag(M) → M is the universal covering. Let x ∈ L. We put s(x) = T_{x}L ∈ Lag(M). We say that L is graded if there exists a lift s̃: L → L̃ag(M) of s, and call s̃ a grading of L.

Now an object of our A_{∞} category is (L, L, s̃, b). Here :

(1) L is a rational Lagrangian submanifold equipped with orientation and relative spin structure.

(2) L is a flat line bundle on L.

(3) L is assumed to be graded and s̃ is its grading.

To explain b, we need to review the main result of [FOOO]. In [FOOO], we developed an obstruction theory to the well-definedness of Floer cohomology of Lagrangian submanifold. We also found, even in the case the obstruction vanishes, Floer cohomology is not uniquely determined but depends on some extra parameter b. In the case when L is graded, b is represented by a chain

b = ∑ T^{λ_{i}}b_{i}

where b_{i} is a singular n − 1 chain which we regard as a 1 cochain. More precisely b is a solution of some formal equation, which we describe below. (See [FOOO] Theorem D more precisely.) We consider the cohomology group H^{k}(L; Λ_{ℂ}) or H^{k}(L; Λ_{ℂ}^{ℚ}). In [FOOO] Theorem D, we constructed a “map”

Q : H^{1}(L; Λ_{ℂ}) → H^{2}(L; Λ_{ℂ})  (4)

which is a kind of formal power series on T in the following sense.

Q = ∑_{i=1}^{∞} Q_{i} ⊗ T^{λ_{i}}  (5)

here

Q_{i} : H^{1}(L; ℂ) → H^{2}(L; ℂ)

is a formal power series, and λ_{i} ∈ ℝ, with lim_{i→∞} λ_{i} = +∞.

Q and Q_{i} depend on L,L,s̃ and λ_{i} depends on L,s̃. Hereafter we write Q_{L,L} and Q_{i,L,L} in place of Q, Q_{i}, in case we need to specify L, L.

We put “map” in the quote since Q(b) is not well defined in general by two reasons. One is that Q_{i} is a formal power series whose convergence (in the usual topology of ℂ vector space) is not known. The other is that the sum (5) is an infinite sum, whose convergence in the non Archimedean norm (which is induced by one on Λ_{ℂ}) is not yet proved. However Q(b) is well defined if b ≡ 0 mod Λ_{ℂ}^{+}, namely if b = ∑ b_{i}T^{λ_{i}} with all λ_{i} > 0.

Now we put

M(L, L; Λ_{ℂ}) = {b ∈ H^{1}(L; Λ_{ℂ}) | b ≡ 0 mod Λ_{ℂ}^{+}, Q(b) = 0}.  (6)

We also put

M(L, L; Λ_{ℂ}^{ℚ}) = M(L, L; Λ_{ℂ}) ∩ H^{1}(L; Λ_{ℂ}^{ℚ}).

We call 1 cochain representing an element of M(L, L; Λ_{ℂ}^{ℚ}), a bounding cochain. The fourth data b consisting the objects of our category is a bounding cochian.

We thus described an object (L, L, s̃, b) of our filtered A_{∞} category.

Remark 2.1. In [FOOO] our Novikov ring has one additional formal parameter e of degree 2. In the case of graded Lagrangian submanifold, we do not need it. In general, bounding cochain has various degrees. In our case of graded Lagrangian submanifold, bounding cochain is of degree 1.

Remark 2.2. We introduced b as an element of cohomology group satisfying equation Q(b) = 0. In order to do so, we first need to work on chain level, introduce gauge equivalence and then identify gauge equivalence class to an element of the first cohomology. This is performed in [FOOO]. (In its revised version we will present it more algebraically. Compare also [Fu3].) We omit it here.

Actually the chain complex CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) over Λ_{ℂ}^{ℚ} in Proposition 2.1 defining Floer cohomology does depend on the representative of b in each gauge equivalence class. However it is well defined up to chain homotopy. So we choose a representative in each gauge equivalence class, and fix it.

The set of morphisms of our A_{∞} category is Floer cohomology. In [FOOO], we defined Floer cohomology

HF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}).  (7)

The Floer cohomology (7) is a module over Λ_{ℂ}, (here b_{i} ∈ M(L, L; Λ_{ℂ})).

Proposition 2.1. If L_{i} is rational and if b_{i} ∈ M(L, L; Λ_{ℂ}^{ℚ}) then Floer cohomology is defined over Λ_{ℂ}^{ℚ}. Namely we have a cochain complex CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) over Λ_{ℂ}^{ℚ} which defines HF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}).

Off course, the Floer cohomology in Proposition 2.1 will become (7) when tensored with Λ_{ℂ}.

We prove Proposition 2.1 in §3.

In the case of rational Lagrangian submanifolds, the operations 𝔪_{k} (that is the (higher) composition homomorphisms of our A_{∞} category) is defined also over Λ_{ℂ}^{ℚ}. More precisely we can prove the following Proposition 2.2. We take a countable set LA of rational Lagrangian submanifold equipped with relative spin structure, and will consider only the Lagrangian submanifolds contained in this set. We assume that each pair of elements of LA is transversal.

Proposition 2.2. There exists a filtered A_{∞} category LAG whose object is (L, L, s̃, b) where L ∈ LA, L is a rational flat bundle on it, s̃ is a grading of L and b ∈ M(L, L; Λ_{ℂ}^{ℚ}). The set of morphisms between two objects is CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) in Proposition 2.1.

The A_{∞} operations 𝔪_{k} are defined in a similar way as [FOOO, Fu2]. We prove Proposition 2.2 in §3. We remark that Proposition 2.2 implies that the operations 𝔪_{k} are multilinear over Λ_{ℂ}^{ℚ}.

The main result of this article asserts that the Galois group ℤ̂ acts as an automorphism on LAG. To state it precisely we first describe the action of ℤ̂ on the set of objects. We put

L^{ρ} = L ⊗ ξ|_{L}.  (8)

The rationality of L implies that there exists m such that

L^{ρ^{m}} = L.  (9)

(9) implies that our action of ρ induces a ℤ̂ action.

The Galois action does not move L, s̃.

We next discuss ℤ̂ action on the set of bounding cochains b. If the defining equation (4) of our moduli space M(L, L; Λ_{ℂ}^{ℚ}) was defined over ℂ[[T]][T^{−1}] then ℤ̂ would act on M(L, L; Λ_{ℂ}^{ℚ}) and (4) would be defined over ℂ[[T]][T^{−1}] if λ_{i} in (5) were integers. However this is not the case.

Proposition 2.3. The following diagram commutes.

H^{1}(L; Λ_{ℂ}^{ℚ}) ⟶^{Q_{L,L}} H^{2}(L; Λ_{ℂ}^{ℚ})
1⊗ρ↓  ^{1⊗ρ}↓
H^{2}(L; Λ_{ℂ}^{ℚ}) ⟶^{Q_{L,L^{ρ}}} H^{2}(L; Λ_{ℂ}^{ℚ})

Here the virtical arrow is 1 ⊗ ρ : H^{k}(L; Λ_{ℂ}^{ℚ}) → H^{k}(L; Λ_{ℂ}^{ℚ}), where H^{k}(L; Λ_{ℂ}^{ℚ}) = H^{k}(L; ℂ) ⊗ Λ_{ℂ}^{ℚ}.

Proposition 2.3 implies that there exists a map

ρ : M(L, L; Λ_{ℂ}^{ℚ}) → M(L, L^{ρ}; Λ_{ℂ}^{ℚ}).

We write ρ(b) = b^{ρ}.

Now we state the main result of this article.

Theorem 2.4. There exists a ℤ̂ action on LAG, such that it is compatible with ℤ̂ action on Λ_{ℂ}^{ℚ} and its action on objects is given by

ρ(L, L, s̃, b) = (L, L^{ρ}, s̃, b^{ρ}).

In place of defining the notion “ℤ̂ action on LAG compatible with ℤ̂ action on Λ_{ℂ}^{ℚ}” in general, we explain it in the case of Theorem 2.4.

We already defined a ℤ̂ action on the set of objects. We recall that the set of morphisms of LAG is Floer cohomology (more precisely a chain complex defining Floer cohomology). Existence of ℤ̂ action on the set of morphisms means that there exists a (canonical) isomorphism :

\begin{matrix}ρ : HF ((L_{1}, L_{1}, s̃_{1}, b_{1}), & (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) \\ ≅ HF ((L_{1}, L_{1}^{ρ}, s̃_{1}, b_{1}^{ρ}), (L_{2}, L_{2}^{ρ}, s̃_{2}, b_{2}^{ρ}); Λ_{ℂ}^{ℚ}). \end{matrix}  (10)

Here ρ is linear over ℂ[[T]][T^{−1}] and satisfies

ρ(xv) = x^{ρ}ρ(v)  (11)

for x ∈ Λ_{ℂ}^{ℚ}, v ∈ HF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}).

Note that in case when L_{1} and L_{2} are different to each other (and hence is transversal to each other) we will define (see §3)

CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) = ⨁_{p∈L_{1}∩L_{2}} Λ_{ℂ}^{ℚ}.

Hence the isomorphism

CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) ≅ CF ((L_{1}, L_{1}^{ρ}, s̃_{1}, b_{1}^{ρ}), (L_{2}, L_{2}^{ρ}, s̃_{2}, b_{2}^{ρ}); Λ_{ℂ}^{ℚ}) (12)

satisfying (11) obviously exists. Thus the essential point to check is that (12) commutes with the boundary operator 𝔪_{1}. We will prove it in §3.

As we mentioned above, 𝔪_{1} commutes with isomorphism (11). The other part of the statement of Theorem 2.4 is that (12) also commutes with higher compositions 𝔪_{k}. The operator 𝔪_{k} is a Λ_{ℂ}^{ℚ} multi linear map

\begin{matrix}𝔪_{k} : ∏_{ℓ=0}^{k−1} CF ((L_{ℓ}, L_{ℓ}, s̃_{ℓ}, b_{ℓ}), & (L_{ℓ+1}, L_{ℓ+1}, s̃_{ℓ+1}, b_{ℓ+1}); Λ_{ℂ}^{ℚ}) \\ → CF ((L_{0}, L_{0}, s̃_{0}, b_{0}), (L_{k}, L_{k}, s̃_{k}, b_{k}); Λ_{ℂ}^{ℚ}). \end{matrix}  (13)

Theorem 2.4 implies that

𝔪_{k} ∘ (ρ ⊗ ⋯ ⊗ ρ) = ρ ∘ 𝔪_{k}.  (14)

We thus described the statement of our main theorem. The proof is given in the next section. Actually the proof is not so difficult. We only need to check the consistency of the action. The main idea of this article is contained in the statement of the theorem and the definitions in this section.

3. Proofs

We start with the proof of Proposition 2.3. During the proof we review briefly the definition of Q_{L,L}. Let β ∈ π_{2}(M, L). We put

E(β) = ∫_{D^{2}} φ^{∗}ω  (15)

where φ : D^{2} → M is a map representing the homotopy class β. We also put

H(β; L) = hol_{L}(φ(S^{1})) ∈ U(1)  (16)

where the right hand side is the holonomy of the flat bundle L along the curve φ(S^{1}). The key observation in the proof of Proposition 2.3 is the following Lemmata 3.1, 3.2.

Lemma 3.1. If L is rational then E(β) ∈ ℚ.

Proof. By the definition of prequatum bundle we have

2π\sqrt{−1}E(β) = ∫_{D^{2}} F_{∇_{ξ}}.  (17)

Hence

exp(2π\sqrt{−1}E(β)) = hol_{ξ}(∂β),  (18)

where the right hand side is the holonomy of the prequatum bundle along the curve φ(S^{1}). By the rationality of L there exists m such that hol_{ξ}(∂β)^{m} = 1. Hence E(β) ∈ ℚ as required.

Lemma 3.2. exp(2π\sqrt{−1}E(β))H(β; L) = H(β; L^{σ}).

Proof. Immediate from (18) and the definition.

We next need a moduli space M_{k+1}(L; β). We fix a compatible almost complex structure J on M. M_{k}(L; β) is the moduli space of pseudoholomorphic maps φ : (D^{2}, S^{1}) → (M, L) of homotopy class β, together with k marked points on the boundary ∂D^{2} = S^{1}. Namely :

Definition 3.1. M̃_{k+1}(L; β) is the set of (φ; z⃗). Here φ : (D^{2}, S^{1}) → (M, L) is J holomorphic and of homotopy class β, and z⃗ = (z_{1}, ⋯, z_{k+1}) ∈ (∂D^{2})^{k+1}, such that z_{i} are mutually distinct and respects the cyclic order on ∂D^{2}.

Aut(D^{2}, J_{D^{2}}) ≅ P SL(2; ℝ) acts on M̃_{k+1}(L; β) by

u · (φ; z_{1}, ⋯, z_{k+1}) = (φ ∘ u^{−1}; u(z_{1}), ⋯, u(z_{k+1})).

We denote the quotient space by M_{k+1}(L; β).

M_{k+1}(L; β) can be compactified by including stable maps. (See [FOOO] §3). The compactification CM_{k+1}(L; β) is a space with Kuranishi structure with corners of dimension n + k − 2. We remark that the dimension is independent of β. This is because we assumed that c^{1}(M) = 0 and L is graded. (We refer to [FO] for the definition of Kuranishi structure with corners and to [FOOO] for the proof of this statement.) Moreover since we fixed a relative spin structure on L, it follows that CM_{k+1}(L; β) is oriented in the sense of Kuranishi structure. (See [FOOO] Chapter 6.) We remark that there exists an evaluation map

ev : CM_{k+1}(L; β) → L^{k+1}.  (19)

Namely

ev(φ, z⃗) = (φ(z_{0}), ⋯, φ(z_{k})).  (20)

We put ev = (ev_{0}, ⋯, ev_{k}).

Using the machinery developed in [FO], we can use CM_{k+1}(L; β) to define a ℚ cycle on L^{k+1}. We will use CM_{k+1}(L; β) to define Q. (To justify the construction below we need to work out carefully the transversality issue. We omit it since it is discussed in detail in [FOOO].)

By Gromov compactness, there exist β_{1}, β_{2}, ⋯ ∈ π_{2}(M, L), with the following properties : E(β_{i}) ≤ E(β_{i+1}) : for each C there exists only a finite number of indices i with E(β_{i}) ≤ C : if β ∉ {β_{1}, β_{2}, ⋯ } then CM_{k+1}(L; β) is empty. (See [FOOO] Proposition 5.8 for detail.)

Let \overline{S}^{k}(L; ℂ) be the set of all distribution valued k form on L which is represented by an n − k singular chain. (See [FOOO] §A1 for the detail of this argument.) We choose a countably generated subcomplex of it satisfying appropriate transversality conditions. We denote it by C^{k}(L; ℂ). The module C^{k}(L; Λ_{ℂ}^{ℚ}) is the completion of C^{k}(L; ℂ) ⊗_{ℂ} Λ_{ℂ}^{ℚ} with respect to the norm (2). The map Q is constructed by using

𝔪_{k,β_{i}} : (C^{1}(L; ℂ))^{⊗k} → C^{2}(L; ℂ).  (21)

Here (21) is defined by

𝔪_{k,β_{i}}(P_{1}, ⋯, P_{k}) = ±ev_{0,∗} (CM_{k+1}(L; β)_{ev_{1},··· ,ev_{k}} ×_{L^{k}} (P_{1} × ⋯ P_{k})) .

We omit the discussion on sign which is given in detail in [FOOO] Chapter 6. Using 𝔪_{k,β_{i}} we define a formal map

Q′_{L,L} : (C^{1}(L; Λ_{ℂ}^{ℚ}))^{⊗k} → C^{2}(L; Λ_{ℂ}^{ℚ})

by

Q′_{L,L} = ∑_{i} Q′_{L;β_{i}} ⊗ H(β_{i}; L)T^{E(β_{i})}.  (22)

Here

Q′_{L,L;β_{i}} : C^{1}(L; ℂ) → C^{2}(L; ℂ)

is a formal map defined by

Q′_{L,L;β_{i}}(b) = ∑_{k} 𝔪_{k,β_{i}}(b, ⋯, b).

In case b ≡ 0 mod Λ_{ℂ}^{ℚ}, Q′_{L,L}(b) converges in the topology induced by the norm of Λ_{ℂ}^{ℚ}. We also remark that Lemma 3.1 implies that E(β_{i}) is a rational number. Hence the right hand side of (22) is contained in C^{2}(L; Λ_{ℂ}^{ℚ}).

Lemma 3.3. The following diagram commutes.

C^{1}(L; Λ_{ℂ}^{ℚ}) ⟶^{Q′_{L,L}} C^{2}(L; Λ_{ℂ}^{ℚ})
1⊗ρ↓  ^{1⊗ρ}↓
C^{1}(L; Λ_{ℂ}^{ℚ}) ⟶^{Q′_{L,L^{ρ}}} C^{2}(L; Λ_{ℂ}^{ℚ})

Proof. We remark that Q′_{L;β_{i}} is defined over ℚ and is independent of L. Hence by (22) and Lemma 3.2, we have

ρ(Q′_{L,L}(b)) = ∑_{i} Q′_{L;β_{i}}(b^{ρ}) ⊗ ρ(H(β_{i}; L)T^{E(β_{i})})

= ∑_{i} Q′_{L;β_{i}}(b^{ρ}) ⊗ exp(2π\sqrt{−1}E(β_{i}))H(β_{i}; L)T^{E(β_{i})}

= ∑_{i} Q′_{L;β_{i}}(b^{ρ}) ⊗ H(β_{i}; L^{ρ})T^{E(β_{i})} = Q′_{L,L^{ρ}}(b^{ρ}).

The proof of the lemma is complete.

The formal map Q_{L,L} is obtained from Q′_{L,L} by homological algebra. (See [FOOO] §A6.) Inspecting the construction there it is easy to see that Lemma 3.3 implies Proposition 2.3. The proof of Proposition 2.3 is now complete.

We next prove Proposition 2.1. We only consider the case L_{1} ≠ L_{2}. The case L_{1} = L_{2} is easier. We remark that then L_{1} is transversal to L_{2}. We begin with the following lemma.

Lemma 3.4. We assume that L_{1} and L_{2} are rational. If φ : S^{1} × [0, 1] → M is a smooth map such that S^{1} × {0} ⊂ L_{1}, S^{1} × {1} ⊂ L_{2}, then

∫_{S^{1}×[0,1]} φ^{∗}ω ∈ ℚ.

The proof is similar to the proof of Lemma 3.1 and is omitted.

Now we recall that in [FOOO, Fu4], we defined Floer’s chain complex by :

CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}) = ⨁_{p∈L_{1}∩L_{2}} Hom(L_{1,p}, L_{2,p}) ⊗_{ℂ} Λ_{ℂ}.  (23)

(The case when L_{i} is trivial is written in [FOOO]. The case when L_{i} is nontrivial is in [Fu4].)

We need to modify boundary operator 𝔪_{1} a bit so that it is defined over Λ_{ℂ}^{ℚ}. Let p, q ∈ L_{1} ∩ L_{2}. We assume that the degree (Maslov-Viterbo index) of q is 1+ the degree of p.

Remark 3.1. We do not discuss the definition of degree. We need grading s̃ for this purpose. See [Se, Fu2].

Hereafter we denote by 𝔪′_{1} the boundary operator defined in [FOOO, Fu4] in order to distinguish it from one we use in this article. A component of the boundary operator 𝔪′_{1} is

𝔪_{1}^{′p,q} : Hom(L_{1,p}, L_{2,p}) ⊗_{ℂ} Λ_{ℂ}^{ℚ} → Hom(L_{1,q}, L_{2,q}) ⊗_{ℂ} Λ_{ℂ}^{ℚ}.  (24)

It is defined by using moduli space of pseudoholomophic disks, which we discuss now. Let us put

∂_{1}D^{2} = {z ∈ ∂D^{2}| Im z < 0}, ∂_{2}D^{2} = {z ∈ ∂D^{2}| Im z > 0}.

Definition 3.2. We denote by π_{2}(M; L_{1}, L_{2}; p, q) the set of all homotopy classes of maps φ : D^{2} → M such that

(1) φ(1) = q, φ(−1) = p.

(2) φ(∂_{1}D^{2}) ⊂ L_{1}, φ(∂_{2}D^{2}) ⊂ L_{2}.

For β ∈ π_{2}(M; L_{1}, L_{2}; p, q), we put

E(β) = ∫_{D^{2}} φ^{∗}ω,

where φ is a map in the homotopy class of β. (Note that the right hand side depends only on the homotopy class of φ since L_{i} are Lagrangian submanifolds.) Lemma 3.4 implies that

E(β) − E(β′) ∈ ℚ  (25)

for any β, β′ ∈ π_{2}(M; L_{1}, L_{2}; p, q). We next define

H(β; L_{1}, L_{2}) ∈ Hom(Hom(L_{1,p}, L_{2,p}), Hom(L_{1,q}, L_{2,q})),

by

H(β; L_{1}, L_{2})(v) = Pal_{L_{2}}(φ(∂_{2}D^{2})) ∘ v ∘ (Pal_{L_{1}}(φ(∂_{1}D^{2})))^{−1}.  (26)

Here Pal_{L_{1}}(φ(∂_{1}D^{2})) : L_{1,p} → L_{1,q} is the parallel transport of the flat bundle L_{1} along the path φ(∂_{1}D^{2}).

We next define the moduli space we use. Let β ∈ π_{2}(M; L_{1}, L_{2}; p, q). We consider the set of maps φ : D^{2} → M which is J holomorphic and is of homotopy class β. We denote the set of all such φ by M̃(L_{1}, L_{2}; p, q; β). The group of authomorphisms of D^{2} which preserves ±1 is isomorphic to ℝ. It acts on M̃(L_{1}, L_{2}; p, q; β). We denote by M(L_{1}, L_{2}; p, q; β) the quotient space. Using stable map it has a compactification which we denote by CM(L_{1}, L_{2}; p, q; β). (See [FOOO] §3.) If the difference of the degrees of p and q is 1, then CM(L_{1}, L_{2}; p, q; β) has a Kuranishi structure of dimension 0 with corners. (See [FOOO] Chapter 5.) Hence we obtain a rational number ♯CM(L_{1}, L_{2}; p, q; β).

We now review the boundary operator 𝔪_{1}^{′p,q} defined in [FOOO, Fu4]. Its leading term 𝔪_{1;0}^{′p,q} is defined as follows.

𝔪_{1;0}^{′p,q} = ∑ ♯CM(L_{1}, L_{2}; p, q; β_{i})H(β_{i}; L_{1}, L_{2}) ⊗ T^{E(β_{i})}.  (27)

Here β_{i} are elements of π_{2}(M; L_{1}, L_{2}; p, q) such that CM(L_{1}, L_{2}; p, q; β_{i}) is nonempty. By using Gromov compactness we can prove that the right hand side converges in the norm of Λ_{ℂ}.

In general, we need correction terms to define 𝔪_{1}^{′p,q}, satisfying 𝔪′_{1} ∘ 𝔪′_{1} = 0. However in case b_{1} = b_{2} = 0, we have 𝔪_{1}^{′p,q} = 𝔪_{1;0}^{′p,q}. See [FOOO] the definition of correction terms.

For simplicity, we only consider the case b_{1} = b_{2} = 0 here. (In general Q(0) ≠ 0. Hence we do need to include correction terms.)

We remark that E(β_{i}) is not necessary a rational number even in the case when L_{1}, L_{2} are rational. So the homomorphism 𝔪_{1;0}^{′p,q} in (30) is not defined on Λ_{ℂ}^{ℚ}. We need to modify 𝔪_{1;0}^{′p,q} so that it is defined over Λ_{ℂ}^{ℚ}.

First we remark that there is an obvious map

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),

which we write (β, β′) ↦ β♯β′. Clearly

E(β♯β′) = E(β) + E(β′), H(β♯β′; L_{1}, L_{2}) = H(β′; L_{1}, L_{2}) ∘ H(β; L_{1}, L_{2}).  (28)

Next we fix p_{0} ∈ L_{1} ∩ L_{2}. For each p ∈ L_{1} ∩ L_{2} we fix β^{p} ∈ π_{2}(M; L_{1}, L_{2}; p_{0}, p).

Definition 3.3. Let p, q ∈ L_{1} ∩ L_{2}, β ∈ π_{2}(M; L_{1}, L_{2}; p, q). We define

\begin{aligned}E′(β) & = E(β) − E(β^{p}) + E(β^{q}), \\ H′(β; L_{1}, L_{2}) & = H(β^{q}; L_{1}, L_{2})^{−1} ∘ H(β; L_{1}, L_{2}) ∘ H(β^{p}; L_{1}, L_{2}). \end{aligned}

We remark that

H′(β; L_{1}, L_{2}) ∈ Hom(L_{1,p_{0}}, L_{2,p_{0}}), Hom(L_{1,p_{0}}, L_{2,p_{0}})) ≅ ℂ.

Moreover H′(β; L_{1}, L_{2}) ∈ U(1).

We remark that Lemma 3.4 implies that E′(β) ∈ ℚ.

Now we put

CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) = ⨁_{p∈L_{1}⊕L_{2}} Λ_{ℂ}^{ℚ}[p].  (29)

Remark 3.2. More precisely we put

CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) = ⨁_{p∈L_{1}⊕L_{2}} Λ_{ℂ}^{ℚ}[p] ⊗_{ℂ} Hom((L_{1})_{p_{0}}, (L_{2})_{p_{0}}).

where we fixed p_{0} ∈ L_{1} ∩ L_{2} as before. But, since Hom((L_{1})_{p_{0}}, (L_{2})_{p_{0}}) is independent of p, we omit this factor.

We define

𝔪_{1;0}^{p,q} = ∑_{β_{i}∈π_{2}(M;L_{1},L_{2};p,q)} ♯CM(L_{1}, L_{2}; p, q; β_{i})H′(β_{i}; L_{1}, L_{2})T^{E′(β_{i})} ∈ Λ_{ℂ}^{ℚ}, (30)

and then

𝔪_{1,0}([p]) = ∑_{q} 𝔪_{1;0}^{p,q}[q].

In the case when b_{1} = b_{2} = 0 we can prove 𝔪_{1,0} ∘ 𝔪_{1,0} = 0. In general we need to add correction terms which we do not discuss here. (See [FOOO].) We put 𝔪_{1} = 𝔪_{1,0} + ⋯ where ⋯ are correction terms we do not define. We thus defined the chain complex (CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}), 𝔪_{1}). We prove the following :

Lemma 3.5. The chain complex (CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}), 𝔪_{1}) ˆ⊗_{Λ_{ℂ}^{ℚ}}Λ_{ℂ} is isomorphic to the chain complex (CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}), 𝔪′_{1}).

Here ˆ⊗ is the completion of the algebraic tensor product.

Proof. We recall

CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}) = ⨁_{p∈L_{1}∩L_{2}} Hom(L_{1,p}, L_{2,p}) ⊗_{ℂ} Λ_{ℂ}.

We first define an isomorphism

\begin{matrix}I : CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), & (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) ˆ⊗_{Λ_{ℂ}^{ℚ}}Λ_{ℂ} \\ ≅ CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}). \end{matrix}  (31)

Let p ∈ L_{1} ∩ L_{2}. We take φ : D^{2} → M which is of homotopy class β^{p}. We then put :

I_{0}([p]) = Pal_{L_{2}}(φ(∂_{2}D^{2})) ∘ (Pal_{L_{1}}(φ(∂_{1}D^{2})))^{−1} ∈ Hom(L_{1,p}, L_{2,p})

where the notation is as in (26) and

I([p]) = I_{0}([p]) ⊗ T^{E(β^{p})}.

It is easy to see that I induces an isomorphism (31). It is easy to check that I is a chain map.

The proof of Proposition 2.1 is now complete.

The proof of Proposition 2.2 is similar to the proof of Proposition 2.1. Namely we need to modify operations 𝔪_{k} so that it is defined over Λ_{ℂ}^{ℚ}. We can do it in the same way as the proof of Proposition 2.1. We leave the detail to the reader.

Now we are in the position to prove Theorem 2.4. We already constructed an action on the set of objects. We next define the action on the set of morphisms. Namely we define :

ρ : CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) → CF ((L_{1}, L_{1}^{ρ}, s̃_{1}, b_{1}^{ρ}), (L_{2}, L_{2}^{ρ}, s̃_{2}, b_{2}^{ρ}); Λ_{ℂ}^{ℚ}).

Actually CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) is a free Λ_{ℂ}^{ℚ} module with basis [p], p ∈ L_{1} ∩ L_{2} and is independent of L_{i}, b_{i} by (29). We define

ρ(∑ x_{i}[p_{i}]) = x_{i}^{ρ}[p_{i}].

We will prove that ρ is compatible with 𝔪_{k}, k = 1, 2, · · ·. We prove it only in case k = 1. The other case is similar. Namely we prove the following :

Proposition 3.6. The following diagram commutes.

CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ}) ⟶^{𝔪_{1}} CF ((L_{1}, L_{1}, s̃_{1}, b_{1}), (L_{2}, L_{2}, s̃_{2}, b_{2}); Λ_{ℂ}^{ℚ})
ρ↓  ^{ρ}↓
CF ((L_{1}, L_{1}^{ρ}, s̃_{1}, b_{1}^{ρ}), (L_{2}, L_{2}^{ρ}, s̃_{2}, b_{2}^{ρ}); Λ_{ℂ}^{ℚ}) ⟶^{𝔪_{1}} CF ((L_{1}, L_{1}^{ρ}, s̃_{1}, b_{1}^{ρ}), (L_{2}, L_{2}^{ρ}, s̃_{2}, b_{2}^{ρ}); Λ_{ℂ}^{ℚ})

Proof. We need the following :

Lemma 3.7. Let φ : S^{1} × [0, 1] → M be a smooth map such that S^{1} × {0} ⊂ L_{1}, S^{1} × {1} ⊂ L_{2}, then we have

\begin{matrix}exp (2π\sqrt{−1} ∫_{S^{1}×[0,1]} φ^{∗}ω) & hol_{L_{1}}(φ(S^{1} × \{0\}))^{−1}hol_{L_{2}}(φ(S^{1} × \{1\})) \\ = hol_{L_{1}^{ρ}}(φ(S^{1} × \{0\}))^{−1}hol_{L_{2}^{ρ}}(φ(S^{1} × \{1\})). \end{matrix}

Proof. It is easy to see from definition of prequantum bundle that

exp (2π\sqrt{−1} ∫_{S^{1}×[0,1]} φ^{∗}ω) = hol_{ξ}(φ(S^{1} × \{0\}))^{−1}hol_{ξ}(φ(S^{1} × \{1\})).  (32)

Lemma 3.7 follows easily from (32) and the definitions.

Let β ∈ π_{2}(M; L_{1}, L_{2}; p, q). Then Lemma 3.7 and the definition implies

exp(E′(2π\sqrt{−1}β))H′(β; L_{1}, L_{2}) = H′(β; L_{1}^{ρ}, L_{2}^{ρ}).

Proposition 3.6 in the case when b_{i} = 0 then follows immediately. The proof of the general case is similar.

The proof of Theorem 2.4 is now complete.

4. Relation to Mirror symmetry

In this section, we describe what the ℤ̂ action of Theorem 2.4 corresponds to, in the mirror, the complex manifold. The argument of this section is rather sketchy since the construction here is not actually new. The author describes it in the way so that it will be the direct analogue of the construction of §2 in the complex category.

Before starting the construction we review several points which are widely believed (though not proved) among the workers of mirror symmetry. Let us consider a symplectic manifold (M, ω) such that c^{1}(M) = 0 and [ω] ∈ H^{2}(M; ℤ). Let us assume that there exists a mirror M^{∨} of M. The manifold M^{∨} is a complex manifold whose complex structure depends on ω. So we write it as (M^{∨}, J_{ω}). First it is believed that complex structure of mirror manifold depends on the complexified symplectic structure. Namely, for Ω = ω + \sqrt{−1}B where B is a closed 2 form and ω is a symplectic structure on M, the mirror M^{∨} has corresponding complex structure J which we denote by J_{Ω}. Ω is called a complexified symplectic structure.

Moreover it is believed that if [Ω] − [Ω′] ∈ \sqrt{−1}H^{2}(M; ℤ) then two complex manifolds (M^{∨}, J_{Ω}) and (M^{∨}, J_{Ω′}) are isomorphic to each other.

Now we consider a family of complexified symplectic structures (M, −\sqrt{−1}τω) where (M, ω) is as above and τ ∈ ℂ with Im τ > 0. We denotes its mirror by (M^{∨}, J_{τ}). (Namely we write J_{τ} in place of J_{−2π\sqrt{−1}ω}.) Since −\sqrt{−1}τω − \sqrt{−1}(τ + 1)ω ∈ \sqrt{−1}H^{2}(M; ℤ) we expect that there exists a biholomorphic map

ψ_{τ} : (M^{∨}, J_{τ}) ≅ (M^{∨}, J_{τ+1}).

We put

q = exp(2π\sqrt{−1}τ) ∈ D_{∗}^{2}.

Here we write D_{∗}^{2} = D^{2} − {0}. Since J_{τ} depends only on q we write J_{q} instead of J_{τ}. Using ψ as a monodromy we have a holomorphic map

π : 𝔐^{∘} → D_{∗}^{2}

whose fiber at q is isomorphic to (M^{∨}, J_{q}). Thus when we start with a single symplectic manifold, its mirror is expected to be a family of complex manifold parametrized by D_{∗}^{2}. It is also believed that the family can be compactified to

π : 𝔐 → D^{2}  (33)

and the fiber of 0 (= π^{−1}(0)) is singular. Also it is expected that π^{−1}(0) is of maximal degeneration. (See [LTY] for its definition.)

Actually what I have been described in this section so far are conjectures which are not yet proved in the general case. So logically speaking, we start from here, and the above arguments are regarded as a motivation. More precisely, we start with the following situation : There exists a family of complex manifolds (33) such that the fiber π^{−1}(q) is smooth except q = 0, and π^{−1}(0) is of maximal degeneration.

Now we consider the category whose object is a family of sheaves on the fibers of (33). More precisely we proceed as follows. Let P_{m} : D^{2} → D^{2} be the map P_{m}(z) = z^{m}. We put

𝔐_{m} = 𝔐 _{π} ×_{P_{m}} D^{2}.

There exist projections π_{m} : 𝔐_{m} → D^{2} and P_{m,m′} : 𝔐_{mm′} → 𝔐_{m}.

We take a neighborhood U of 0 in D^{2} and consider a coherent sheaf 𝔉 on π^{−1}P_{m}^{−1}(U) ⊆ 𝔐_{m}. Let us consider the set of all such (𝔉, m, U) and define an equivalence relation on it as follows.

Let (𝔉, m, U),(𝔊, ℓ, V) be in this set. we say (𝔉, m, U) ∼ (𝔊, ℓ, V), if there exist a neighborhood W of 0 with W ⊂ U ∩ V and if there exists an isomorphism

Ψ : (P_{m,ℓ}^{∗}𝔉)|_{π^{−1}P_{mℓ}^{−1}(W −{0})} ≅ (P_{ℓ,m}^{∗}𝔊)|_{π^{−1}P_{mℓ}^{−1}(W −{0})}

such that q^{k}Ψ extends to a morphism Ψ : (P_{m,ℓ}^{∗}𝔉)|_{π^{−1}P_{mℓ}^{−1}(W)} → (P_{ℓ,m}^{∗}𝔊)|_{π^{−1}P_{mℓ}^{−1}(W)} and that q^{k}Ψ^{−1} extends to a morphism Ψ : (P_{ℓ,m}^{∗}𝔊)|_{π^{−1}P_{mℓ}^{−1}(W)} → (P_{m,ℓ}^{∗}𝔉)|_{π^{−1}P_{mℓ}^{−1}(W)}, for some k. It is easy to see that ∼ is an equivalence relation.

An object of our category SH(𝔐) is a ∼ equivalence class of such triples (𝔉, m, U). We denote this set by 𝔒𝔅(SH(𝔐)).

Now, for each pair of objects [𝔉, m, U], [𝔊, ℓ, V] ∈ 𝔒𝔅(SH(𝔐)), we are going to construct a chain complex of Λ_{ℂ}^{ℚ} modules as follows.

For each object of SH(𝔐), we fix its representative (𝔉, m, U) and its injective resolution 𝔉 → ℜ𝔉^{∗}.

Let [𝔉, m, U], [𝔊, ℓ, V] ∈ SH(𝔐) and W ⊆ U ∩ V with 0 ∈ W. We consider a complex

Hom (P_{ℓ,m}^{∗}ℜ𝔉^{∗}|_{π^{−1}P_{mℓ}^{−1}(W)}, P_{m,ℓ}^{∗}ℜ𝔊|_{π^{−1}P_{mℓ}^{−1}(W)})  (34)

of coherent sheaves on π^{−1}P_{mℓ}^{−1}(W). Let 𝔒_{P_{mℓ}^{−1}(W)} be the sheaf of holomorphic function on P_{mℓ}^{−1}(W) (⊂ D^{2}). We then obtain a complex 𝔒_{P_{mℓ}^{−1}(W)} module sheaves whose stark at q ∈ P_{mℓ}^{−1}(W) is

Γ (π_{ℓm}^{−1}(q); Hom (P_{ℓ,m}^{∗}ℜ𝔉^{∗}, P_{m,ℓ}^{∗}ℜ𝔊)) .

We denote this complex of sheaves by C_{W}^{∗} (ℜ𝔉, ℜ𝔊).

For any open neighborhood W′ of 0 contained in W, we consider the set of sections s of C_{W}^{∗} (ℜ𝔉, ℜ𝔊) on P_{ℓm}^{−1}(W′) − {0} such that q^{k}s extends to P_{ℓm}^{−1}(W′) for some k. We denote this set by Γ(P_{ℓm}^{−1}(W′_{∗}); C_{W}^{∗} (ℜ𝔉, ℜ𝔊)).

We put :

C_{0}^{∗}(ℜ𝔉, ℜ𝔊) = lim_{⟵} Γ(P_{ℓm}^{−1}(W′_{∗}); C_{W}^{∗} (ℜ𝔉, ℜ𝔊))  (35)

Here the right hand side is the projective limit as W′ → {0}.

We denote by ℂ⟨⟨q^{1/ℓm}⟩⟩[q^{−1}] the set of all germs at 0 ∈ π_{ℓm}^{−1}(W) ⊆ D^{2} of meromorphic functions. C_{0}^{∗}(ℜ𝔉, ℜ𝔊) is a complex of ℂ⟨⟨q^{1/ℓm}⟩⟩[q^{−1}] modules.

We may identify ℂ⟨⟨q^{1/ℓm}⟩⟩[q^{−1}] with a subring of Λ_{ℂ}^{ℚ} by sending q^{1/mm′} to T^{1/mm′}. Now we put

C^{∗}([𝔉, m, U], [𝔊, ℓ, V]) = C_{0}^{∗}(ℜ𝔉, ℜ𝔊) ˆ⊗_{ℂ⟨⟨q^{1/ℓm}⟩⟩[q^{−1}]} Λ_{ℂ}^{ℚ}.  (36)

Here ˆ⊗ is the completion of the algebraic tensor product.

It is straightforward to check that the right hand side is independent of the representative (𝔉, m, U), (𝔊, ℓ, V) and of resolution up to chain homotopy. (However since we are going to construct a filtered A_{∞} category, we need to work in the chain level. This is the reason we fixed the representative and the resolution. The resulting filtered A_{∞} category is independent of the choice of them up to homotopy equivalence of filtered A_{∞} category.)

Proposition 4.1. There exists a filtered A_{∞} category SH(𝔐) such that the set of its objects is 𝔒𝔅(SH(𝔐)), and the modules of morphisms are C^{∗}([𝔉, m, U], [𝔊, ℓ, V]).

Moreover there exists a ℤ̂ action on SH(𝔐) compatible with its action on Λ_{ℂ}^{ℚ}.

Proof. The boundary operator 𝔪_{1} is the boundary operator of the complex C(𝔉, 𝔊). 𝔪_{2} is induced by the composition of homomorphisms. We put 𝔪_{k} = 0 for k ≥ 3. It is easy to check the axiom of filtered A_{∞} category.

We next construct a ℤ̂ action on SH(𝔐). We remark that ℤ̂ is the algebraic fundamental group of D_{∗}^{2}. The ℤ̂ action on SH(𝔐) is induced from this fact. More precisely we proceed as follows.

We first construct an action on the set of objects. Let ρ ∈ ℤ̂ be the topological generator. We let is act as the generator of the deck transformation group of P_{1,m} :

𝔐_{m} → 𝔐. Let [𝔉, m, U] ∈ SH(𝔐). Then [ρ^{∗}𝔉, m, U] is also an object of SH(𝔐). So we put

ρ[𝔉, m, U] = [ρ^{∗}𝔉, m, U].

It is easy to see that this gives an action of ℤ̂ on the set of objects.

We choose resolution so that if 𝔉 → ℜ𝔉^{∗} is the resolution chosen for 𝔉 then ρ^{∗}𝔉 → ρ^{∗}ℜ𝔉^{∗} is the resolution chosen for ρ^{∗}𝔉.

Then it is easy to construct a homomorphism

ρ : C^{∗}([𝔉, m, U], [𝔊, ℓ, V]) → C^{∗}([ρ^{∗}𝔉, m, U], [ρ^{∗}𝔊, ℓ, V])

which satisfies (11) and which is a chain map. It is also easy to see that ρ commutes with compositions. The proof of Proposition 4.1 is now complete.

We thus described a filtered A_{∞} category on which ℤ̂ acts. Using it and one constructed in §2,3, we can formulate homological Mirror symmetry conjecture by M.Kontsevitch in the level of formal power series. (In [Fu1], for example, it was stated assuming the convergence of 𝔪_{k}.) Namely :

Conjecture 4.2. Let M be Calabi-Yau manifold M with Kähler form ω. We assume [ω] ∈ H^{2}(M; ℤ). Let us assume that there exists a mirror family 𝔐 → D_{∗}^{2} as above.

Then there exists an filtered A_{∞} functor Mir : LAG → SH(𝔐) which preserves ℤ̂ actions. Moreover induced homomorphisms Mir : HF ((L_{1}, L_{1}, b_{1}), (L_{2}, L_{2}, b_{2})) → H(C^{∗}(Mir(L_{1}, L_{1}, b_{1}), Mir(L_{2}, L_{2}, b_{2})), 𝔪_{1}) on cohomologies are isomorphisms.

We also remark that the ℤ̂ action described in this section is closely related to the mixed Hodge structure which plays an important role in Mirror symmetry.

We next exhibit our construction in the case of elliptic curve T^{2}. In this case, the isomorphism of categories was established in [PZ, Ko] for 𝔪_{1}, 𝔪_{2} and in [Fu1] for 𝔪_{k}, k ≥ 3. Let us calculate ℤ̂ action in this case.

The prequantum bundle ξ is (T^{2}×ℂ, π\sqrt{−1}(xdy−ydx))/ℤ^{2} where ℤ^{2} acts by ((x, y), v) ↦ ((x + 1, y), e^{π\sqrt{−1}y}v), ((x, y), v) ↦ ((x, y + 1), e^{−π\sqrt{−1}x}v).

We here apply the construction of [Fu1] to the case of elliptic curve. Let us consider Lagrangian submanifolds L_{pt}(a) = {(a, y)|y ∈ S^{1}}, L_{st}(b) = {(x, b)|x ∈ S^{1}}, L = {(x, x)|x ∈ S^{1}}.

It is easy to see that L_{pt}(a), L_{st}(b) are rational if a, b ∈ ℚ. Moreover L is rational and ξ|_{L}^{⊗2} is trivial. (ξ|_{L} is nontrivial, however, since we identify (x, y; v) with (x+1, y+1; −v).)

So the object corresponding to (L, ℂ) is fixed by ρ^{2}. (Here ℂ stands for the trivial line bundle.) The object corresponding to (L_{st}(0), ℂ) is fixed by ρ. We remark that (L_{st}(0), ℂ) will become a structure sheaf on the mirror and (L, ℂ) will become a polarization E, that is a line bundle of degree 1. For (x, y^{∗}) we consider the object L_{x,y^{∗}} = (L_{pt}(x), L(y^{∗})). Here L(y^{∗}) stands for the complex line bundle on S^{1} ≅ L_{pt}(x) with monodromy e^{2π\sqrt{−1}y^{∗}}. L_{x,y^{∗}} corresponds to the skyscraper sheaf at (x, y^{∗}) ∈ T_{τ}^{2} by mirror symmetry. Here we regard T_{τ}^{2} = ℝ^{2}/ℤ^{2} such that (x, y^{∗}) ↦ τx − y^{∗} is a complex isomorphism. (So if we take z = z(x, y^{∗}) = τx − y^{∗} as a complex coordinate, we have T_{τ}^{2} = ℂ/(ℤ ⊕ τℤ).)

Remark 4.1. Let us put τ = a + \sqrt{−1}b. Then complexified symplectic structure is (b − \sqrt{−1}a)ω. We defined a complex structure in [Fu1] so that (b − \sqrt{−1}a)x + \sqrt{−1}y^{∗} is a complex coordinate. It is easy to see that (b − \sqrt{−1}a)x + \sqrt{−1}y^{∗} = \sqrt{−1}z.

Since the holonomy of the flat bundle ξ|_{L_{pt}(x)} is

exp (∫_{0}^{1} π\sqrt{−1}xdy) exp (π\sqrt{−1}x) ,

where the first factor is the integration of the connection form and the second factor is induced by the action ((x, y), v) ↦ ((x, y + 1), e^{−π\sqrt{−1}x}v). It follows that

ρ(L_{x,y^{∗}}) = L_{x,y^{∗}+x}.  (37)

Let ℂ be the trivial line bundle on T_{τ}^{2}, that is the structure sheaf. Let 𝔉_{z} be the skyscraper sheaf supported at z ∈ T_{τ}^{2}. We consider the operation (Yoneda product)

𝔪_{2} : Ext^{0}(T_{τ}^{2}; ℂ, E) ⊗ Ext^{1}(T_{τ}^{2}; E, 𝔉_{z}) → Ext^{1}(T_{τ}^{2}; ℂ, 𝔉_{z}).  (38)

It is easy to see that Ext^{1}(T_{τ}^{2}; E, 𝔉_{z}) ≅ E_{z}^{∗} the dual to the fiber of E at z. On the other hand, Ext^{1}(T_{τ}^{2}; ℂ, 𝔉_{z}) is canonically isomorphic to ℂ. Moreover Ext^{0}(T_{τ}^{2}; ℂ, E) is one dimensional. We find that the map (38) is described as follows. Let s ∈ Ext^{0}(T_{τ}^{2}; ℂ, E) which is a global holomorphic section of E. Let v ∈ E_{z}^{∗} ≅ Ext^{1}(T_{τ}^{2}; E, 𝔉_{z}). Then

𝔪_{2}(s, v) = v(s(z)) ∈ ℂ ≅ Ext^{1}(T_{τ}^{2}; ℂ, 𝔉_{z}).

The mirror of (38) is

\begin{aligned}𝔪_{2} : HF^{0}((L_{st}(0), ℂ), (L, ℂ)) & ⊗ HF^{1}((L, ℂ), ((L_{pt}(x), L(y^{∗}))) \\ & → HF^{1}((L_{st}(0), ℂ), ((L_{pt}(x), L(y^{∗}))). \end{aligned}  (39)

Remark 4.2. Here the coefficient ring of the Floer cohomology HF above is ℂ and is not Novikov ring. In the case of elliptic curve (or more generally affine Lagrangian submanifolds of symplectic torus) the operator 𝔪_{k} converges when we put T = e^{−1}, as was shown in ([Fu1]). So Floer cohomology of ℂ coefficient can be defined in those cases.

HF^{0}((L_{st}(0), ℂ), (L, ℂ)) is canonically isomorphic to ℂ. We next remark that

HF^{1}((L, ℂ), ((L_{pt}(x), L(y^{∗}))) ≅ L(y^{∗})|_{(x,0)}

(see Remark 3.2) is canonically isomorphism to ℂ. Finally

HF^{1}((L_{st}(0), ℂ), ((L_{pt}(x), L(y^{∗}))) ≅ L(y^{∗})|_{(x,x)}.

We can identify it with ℂ by regarding L(y^{∗}) as the trivial line bundle with connection \sqrt{−1}y^{∗}dy. Here y is a coordinate of L_{pt}(x) = \{(x, y)|y ∈ ℝ\}/ℤ.

We thus identify three Floer cohomology groups with ℂ. Then (39) is

Θ(τ; (x, y^{∗})) = ∑_{n} exp(π\sqrt{−1}τ(x + n)^{2} − 2π\sqrt{−1}(x + n)y^{∗})  (40)

(40) was first obtained by [Ko]. See also [PZ]. The above formulation is due to [Fu1].

We recall that ρ^{2}(L, ℂ) = (L, ℂ), ρ(L_{st}, ℂ) = (L_{st}, ℂ), ρ(L_{x,y^{∗}}) = L_{x,y^{∗}+x}.

By a direct calculation, it is easy to check

Θ(τ + 2; x, y^{∗} + 2x) = exp(−2π\sqrt{−1}x^{2})Θ(τ; x, y^{∗}).  (41)

We remark that the isomorphism

ξ|_{L_{pt}(x)} ⊗ (L_{pt}(x) × ℂ, 2π\sqrt{−1}y^{∗}dy) ≅ (L_{pt}(x) × ℂ, 2π\sqrt{−1}(y^{∗} + x)dy)

is ((x, y), v) ↦ (x, e^{−π\sqrt{−1}xy}v). Hence at (x, x) it is a multiplication by e^{−π\sqrt{−1}x^{2}}. Thus we have a commutative diagram :

HF^{1}((L, ℂ), ((L_{pt}(x), L(y^{∗}))) ⟶ ℂ
ρ^{2}↓  ^{exp(−2π\sqrt{−1}x^{2})}↓
HF^{1}((L, ℂ), ((L_{pt}(x), L(y^{∗} + 2ρ))) ⟶ ℂ

Here the horizontal arrow is the isomorphism explained above.

Thus (41) is equivalent to the ρ^{2} invariance of 𝔪_{2}, which is a part of the statement of Theorem 2.4.

In other words, we can proceed as follows. Let us discuss it in the mirror side. The canonical generator 1 of HF^{0}((L_{st}(0), ℂ), (L, ℂ)) will turn out to be a global section of E which we write s. The isomorphism HF^{1}((L, ℂ), ((L_{pt}(x), L(y^{∗}))) ≅ ℂ explained above will give a unitary frame of E. We multiply it by exp(π\sqrt{−1}τx^{2} − 2π\sqrt{−1}xy^{∗}) to get a holomorphic frame e_{z} : E_{z} → ℂ. Using it we obtain

\begin{aligned}𝔪_{2}(s, e_{z}) = e_{z}(s(z)) & = Θ(τ; x, y) exp(−π\sqrt{−1}τx^{2} + 2π\sqrt{−1}xy^{∗}) \\ & = ∑_{n} exp(π\sqrt{−1}n^{2}τ + 2π\sqrt{−1}nz). \end{aligned}

(z = τx − y^{∗}). We denote by ϑ(τ, z) the right hand side. We have

ϑ(τ, z) = ϑ(τ + 2, z),

which is a standard identity of theta function.

We remark that there is another (more interesting) identity

ϑ(−1/τ, z/τ) = e^{−π\sqrt{−1}/4}τ^{1/2}ϑ(τ, z)  (42)

of theta function. These two symmetries generate an index two subgroup of P SL(2; ℤ). It is harder to see (42) from symplectic geometry side of the story. In fact, the action T^{τ} ↦ T^{−1/τ} is not well-defined on Λ_{ℂ}^{ℚ}. So, one needs to find a smaller field (whose Galois group over ℂ(T) (the field of rational function on ℂP^{1}) might be a completion of P SL(2; ℤ)) on which Floer cohomology and operations 𝔪_{k} is defined.

In the case of elliptic curve, it seems possible to work out such a construction by using the fact that there exists such symmetry in the complex side and that mirror symmetry is proved in this case. However to carry it out in detail, one needs to work out what happens for 𝔪_{k}, k ≥ 3, in complex side of the story. It is nontrivial to do so, since 𝔪_{3} etc. are secondary operators so in which sense it is symmetric with respect to P SL(2; ℤ) is not so obvious.

Remark 4.3. In the case of Novikov homology of closed one form on finite dimensional manifold, the operator 𝔪_{1} is proved to be a rational function (that is a meromorphic function on ℂP^{1}) by [Pa]. It seems likely true for higher 𝔪_{k}. In other words, the “mirror of finite dimensional Novikov homology” gives a family parametrized by ℂP^{1}. In our case of infinite dimensional Floer theory, the story should be more involved even in the case of elliptic curve.

It might be possible to generalize the story of elliptic curve mentioned above to the case of higher dimensional torus and affine Lagrangian submanifolds, by using [Fu1] (at least as far as 𝔪_{2} concerns).

However beyond that case, to establish a symmetry begger that ℤ̂ symmetry on symplectic side of the story, is extremely difficult.

Let us consider, for example, the case of quintic M. We have H^{2}(M) = ℂ, so the moduli space of complexified symplectic structure of M is one dimensional. Let M^{∨} be the mirror quintic. The moduli space of its complex structure is a quotient of upper half plane by symmetries τ ↦ τ + 1, and, τ ↦ \frac{τ cos 2π/5−sin 2π/5}{τ sin 2π/5+cos 2π/5}, an element of P SL(2, ℝ), of order 5 which fixes \sqrt{−1}. (See for example [COGP].) The first symmetry corresponds to the ℤ̂ symmetry which we now understood. However the second symmetry in the symplectic side is mysterious and is very interesting.

One may also try to study a similar symmetry in the case of K3 surface. Namely the action of the group O(3, 19; ℤ) are supposed to exist on the category of Lagrangian submanifold. (In this case we need to move symplectic structure of K3 surface also to get full symmetry.)

The existence of such symmetries in the symplectic side seems to be one of the deepest consequences of Mirror symmetry.

Remark 4.4. We remark that there are several works comparing two symmetries by mirror symmetry. (See [ST, SK]). However in [ST, SK] the symmetry is induced by a symplectic diffeomorphism (generalized Dehn twist) in the symplectic side, and in the complex side, it is not induced by a biholomorphic map but is a Fourier-Mukai transform. In our situation, in the complex side the symmetry is induced by a biholomorphic map, and in the symplectic side it is not induced by a symplectic diffeomorphism.

We also remark that the ℤ̂ action on objects of LAB is easy to calculate. If we assume that the mirror is obtained by a dual torus fibration as in [SYZ], a point of the mirror manifold M^{∨} can be identified with an object of LAB of M. Therefore, Conjecture 4.2 gives a nice description of the monodromy ψ_{τ} : (M^{∨}, J_{τ}) ≅ (M^{∨}, J_{τ+1}). For example, the following will be proved if Conjecture 4.2 is true.

Conjecture 4.3. In the situation of Conjecture 4.2, the monodromy ψ_{τ} : (M^{∨}, J_{τ}) ≅ (M^{∨}, J_{τ+1}). can be taken as a “complete integrable system”.

Namely the following holds. There exists a Baire set M_{0}^{∨} of M^{∨} such that if x ∈ M_{0}^{∨} then the closer of the orbit {ψ_{τ}^{k}(x)|k ∈ ℤ} is an n dimensional Lagrangian torus. (n = dim_{ℂ} M^{∨}). Moreover for any x, the closure of {ψ_{τ}^{k}(x)|k ∈ ℤ} is a finite union of isotropic torus.

For example in the case of elliptic curve, the monodromy is (x, y^{∗}) ↦ (x, x+y^{∗}). Hence the closure of its orbit is S^{1} if x is not rational.

Solving Conjecture 4.3 might give a way to find a Lagrangian torus fibration of a Calabi-Yau manifold near maximal degeneration point.

References

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Department of Mathematics, Faculty of Sciences, Kyoto University, Kitashirakawa, Sakyoku, Kyoto, Japan

   E-mail address: fukaya@kusm.kyoto-u.ac.jp