Topological Space in Homological Mirror Symmetry

In the mirror symmetry including the T-duality, the observables coincide in the A- and B-model on different manifolds. Because the observables are determined by how the strings propagate on the manifolds, the observed geometry by the A- and B-model will coincide. In this paper, we prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the $A_{\infty}$ structure of the both models. Therefore, this homeomorphic topological space will be the observed geometry by the strings.


Introduction
In the mirror symmetry including the T-duality, the observables in the A-model on a manifold coincide with those in the B-model on a different manifold. Because the observables are determined by embedding the world sheets into the manifolds, this fact suggests coincidence of the observed geometries by the strings sweeping the two different manifolds.
Let us consider a direct way to find such a geometry. In order to quantize string theories on non-trivial manifolds, we study topological string theories. From the topological theories, we expect to obtain the topology of the observed geometry. Open strings can observe shorter length than closed strings. It is easy to study strings on complex one-dimensional manifolds.
Thus, by studying the homological mirror symmetry (HMS) [1] between one-dimensional manifolds, we search for a homeomorphism between topological spaces in the A-and Bmodels.
In the one-dimensional HMS, the A-model on a symplectic torus corresponds to the B-model on an elliptic curve [2]. The objects, representing D-branes, are the Lagrangian submanifolds in the A-model and the complexes of the coherent sheaves in the B-model. In one dimension, the Lagrangian submanifolds are denoted by straight lines and the coherent sheaves are denoted by line bundles. The morphisms, representing the open strings between the D-branes, are described by Abelian groups whose basis are given by intersecting points of the straight lines in the A-model, and the maps between the complexes of the line bundles in the B-model. From these objects and morphisms, we can canonically construct the Fukaya category in the A-model and the derived category of coherent sheaves in the B-model. It is proved in [3][4][5][6][7][8][9] that the Fukaya category is equivalent as an A ∞ -category to the differential graded (DG) category 1 canonically extended from the derived category of coherent sheaves. Furthermore, the Fukaya category is also equivalent as an A ∞ -category to a non-trivial A ∞ -category extended from the DG-category by using homotopy operators [10][11][12][13][14][15][16][17][18]. In this extension, m 3 is explicitly constructed in the B-model [13], whereas m d (d ≥ 4) have not been explicitly constructed yet.
In this paper, we extend the DG-category in a different way, based on the topological string amplitudes. m d are newly defined and explicitly constructed in the B-model. The A ∞ -category that consists of these m d is shown to be equivalent to the Fukaya category 2 .
In this construction, we find homeomorphic topological spaces necessary to define m d that satisfy the A ∞ relations in the A-and B-models.
We are concerned with the u = 0 case for simplicity. One can easily introduce the connections. We simplify the expressions as In this case, the strings are represented by Topological string amplitudes on E τ are defined as [9] Eτ where n i 1 −1,i 1 , · · · , n ic−1,ic > 0 and n i c+1 −1,i c+1 , · · · , n i d −1,i d , n < 0. Ω = dz is the holomorphic Here we explain what is ψ n [ m n ]. In order to define topological string amplitudes of more than one n < 0 states, we need to deform the theory by those states because more than one (0,1)-forms cannot enter the topological string amplitudes in one dimension. The derived category describes such a deformed theory. In the topological string theory, the deformation by n < 0 string states O (0) ∈ Ω 0,1 is given as follows. We define O (1) whered is a world sheet differential and Q is a BRST operator. Then, the deformation of the theory is to insert ψ := ∂Σ O (1) ∈ Ω 0,0 , where ∂Σ is a world sheet boundary.
In our case, we define this deformation by an isomorphism ψ: Ω 0,1 → Ω 0,0 by That is, θ n [ m n ](z, τ ) ∈ Ω 0,0 represent string states, when not only n > 0 but also n < 0. This isomorphism will be justified later by mirror symmetry of the A ∞ structure and m d . By using this isomorphism, (2.5) is written as On the other hand, (2.5) should also be written by using m d in A ∞ -category [15,20] like (2.9) We will define completely m d that possesses A ∞ structure in the next section.
Here we discuss consistency of the integration over E τ with the periodicity. Whereas the theta functions are invariant under z → z + 1, they are transformed under z → z + τ as Then, the integrand is transformed as Because this should be invariant for periodicity, n needs to be −(n 0,1 + · · · + n d−1,d ) and then n 0,1 + · · · + n d−1,d > 0.

m d and A ∞ Structure
In order to define m d , we multiply theta functions with characteristics. They are defined by series as Definition 1 (theta functions with characteristics).
where a ∈ R/Z, b ∈ C.
A product formula is given by where n 1 , n 2 ∈ Z, a 1 , a 2 ∈ R, and z 1 , z 2 , τ ∈ C.
While this formula was proved as an addition formula when n 1 , n 2 ∈ N in [21], it can also be proved as series when n 1 , n 2 ∈ Z as follows. Proof.
In a special case: z 1 = n 1 z + u 1 and z 2 = n 2 z + u 2 (u 1 , u 2 ∈ C), we obtain θ a 1 n 1 This product is expanded by the theta functions with complex coefficients. The coefficients are independent of z. We simplify this formula. Because a 1 +a 2 +n 1 d we can add − (n 1 +n 2 ) (n 1 +n 2 ) d and obtain Because a 1 +(a 2 −n 2 d) n 1 +n 2 ∈ R/Z on the last line, we can add − n 2 (n 1 +n 2 )m (n 1 +n 2 ) d. By defining α := (n 1 + n 2 )m + d (α ∈ Z), we obtain Especially, when u = 0 we obtain From now on, we abbreviate θ n [p](z, τ ) to θ n [p] because (z, τ ) does not vary. By using this formula, we obtain Theorem 2.
The coefficients in this formula coincide with those of Lemma 1. Furthermore, the definition leads to Lemma 3 (local product in the configuration space). Proof.

Strings between (i-1)-th and i-th D-branes can interact only with strings between (i-2)-th
and (i-1)-th D-branes and strings between i-th and (i+1)-th D-branes. Therefore, we need to demand that the ordering of incoming states is non-commutative in the Feynman diagrams.
Then, we obtain the following theorem.

Definition 4 (m d ).
m d (p 0,1 , p 1,2 , · · · , p d−1,d ) |δ| ∈ Z should be determined so that m d satisfy the A ∞ relation. We have given a general definition including more than one-dimensional case, whereasδ is unique in this one-  the states with n i−1,i > 0 and n i−1,i < 0 have degrees 0 and 1, respectively. In the proof of Theorem 3, the Feynman diagrams are classified into two kinds (Fig. 6, 7). In the diagrams in Fig. 6, n 0,d < 0, n 0,1 > 0, and n i−1,i < 0 (i = 2, · · · d). If we compare the degrees of the both sides of the formula in Definition 4, we obtain 0 + (d − 1) + (the degree of m d ) = 1.
We are going to show that the Feynman diagrams, which determine the A ∞ structure of the B-model, coincide with the tropical Morse trees, which determine the A ∞ structure of the A-model. That is, both the A ∞ structures coincide. The tropical Morse treesφ are continuous maps that satisfy the following conditions, 4 from the metric ribbon trees S to the universal cover B (∋ỹ) of R/Z (∋ y) [9].
(0) We attach n i−1,i to the i-th external incoming edge and attach the sum of all numbers labelling the edges coming into a vertex to the edge coming out of the vertex. That is, the numbers n are preserved at vertices. We define an affine displacement vector v on each edge.
(1) The coordinates of the external incoming and outgoing vertices areỹ =p i−1,i and y =p 0,d , respectively.

Conclusion and Discussion
In this paper, we have defined and explicitly constructed m d that form The way to define m d is naturally generalized in the case of the B-model on 2n-dimensional complex manifolds: We derive Feynman rules in the configuration space of the morphisms from the products of the two morphisms in the DG-category of the B-model. Then we define m d by the products of d morphisms with the configurations in the case that the moduli space of the Feynman diagrams is restricted to zero-dimensional. We conjecture that these m d satisfy the A ∞ relations and form an A ∞ -category, which is equivalent to the Fukaya category of the A-model on the corresponding 2n-dimensional symplectic manifold, as an A ∞ -category. We also conjecture that the moduli space of the Feynman diagrams in the Bmodel on the 2n-dimensional complex manifolds is homeomorphic to the moduli space of the pseudo holomorphic curves in the A-model on the corresponding 2n-dimensional symplectic manifold. These moduli spaces determine the A ∞ structure of the both models. Therefore, the moduli space of the pseudo holomorphic curves will be the identical observed geometry by the strings living on the corresponding different manifolds.