# Moduli Space in Homological Mirror Symmetry

Authors: Matsuo Sato
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

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 theA∞ structure of the both models.


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## Body

## 1. Introduction

In the one-dimensional homological mirror symmetry (HMS) [1], 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, any real one-dimensional submanifold is the Lagrangian and the line bundles are coherent sheaves. The morphisms, representing the open strings between the D-branes, are described by Abelian groups whose bases are given by intersecting points of the real one-dimensional submanifolds 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 [3–9] that the Fukaya category is equivalent as an A∞-category to the differential graded (DG) category (The DG-category is an A∞-category where md(d≥3) are trivial.) canonically extended from the derived category of coherent sheaves. Furthermore, the Fukaya category is also equivalent as an A∞-category to a nontrivial A∞-category extended from the DG-category by using homotopy operators [10–18]. In this extension, m3 is explicitly constructed in the B-model [13], whereas md(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.md are newly defined and explicitly constructed in the B-model. The A∞-category that consists of these md is shown to be equivalent to the Fukaya category (Thus, the A∞-category defined in this paper is equivalent as an A∞-category to the A∞-category defined by the homotopy operators because they are equivalent to the Fukaya category as A∞ categories.). In this construction, we find a moduli space necessary to define md that satisfy the A∞ relations in the B-model. This moduli space is homeomorphic to the moduli space of the pseudo holomorphic curves in the A-model.


## 2. Topological String Amplitudes

The one-dimensional complex manifold in the B-model is an elliptic curveEτ which is spanned by 1 and τ∈C. There are two ways to generalize the derived category of coherent sheaves on Eτ to a differential graded (DG) category. One way is based on C˘ech cohomology [3–9] and the other is based on Dolbeault cohomology [2, 10–16]. They are equivalent by C˘ech-Dolbeault isomorphism. We adopt Dolbeault cohomology. Coherent sheaves on Eτ are classified and constructed based on line bundles on Eτ [19]. We study only the line bundles. The extension to general coherent sheaves is straight forward and the way is written in [2, 13]. The objects representing (d+1) D-branes are complexes of line bundles of degree ni, L(ni,ui) (i=0,…,d), where ui represent connections (gauge fields) over the D-branes. The morphisms representing open strings between L(ni-1,ui-1) and L(ni,ui) are H0,p(Eτ,L(ni-1,i,ui-1,i)), where p represent the degrees of the grading (p=0,1, ni-1,i≔ni-ni-1ui-1,i≔ui-ui-1).Whenn>0, H0,1(Eτ,L(n,u))=0 and the elements of H0,0(Eτ,L(n,u)) are the basis of global sections of L(n,u):(1)θmnunz,nτ∈H0,0Eτ,Ln,um=0,…,n-1,where θab(z,τ) are theta functions with characteristics.Whenn<0, H0,0(Eτ,L(n,u))=0 and the elements of H0,1(Eτ,L(n,u)) are the basis of harmonic (0,1)-forms with values in the dual bundle L(n,u):(2)θmn-u-nz,-nτ¯exp⁡2πtnz22+2u2z2dz¯∈H0,1Eτ,Ln,u,where θ¯ is the complex conjugate of θ, τ=A+it, z=z1+τz2, u=u1+τu2, and A,t,z1,z2,u1,u2∈R.We are concerned with theu=0 case for simplicity. One can easily introduce the connections. We simplify the expressions as(3)θnpz,τ≔θp0nz,nτ.In this case, the strings are represented by(4)θnmnz,τ∈H0,0Eτ,Lnn>0,θn∗mnz,z¯,τ,τ¯dz¯≔θ-nmnz,τ¯exp⁡2πtnz22dz¯∈H0,1Eτ,Lnn<0.Topological string amplitudes onEτ are defined as [9, 15, 20](5)∫EτΩ∧θn∗mndz¯θni1-1,i1mi1-1,i1ni1-1,i1⋯θnic-1,icmic-1,icnic-1,icψnic+1-1,ic+1mic+1-1,ic+1nic+1-1,ic+1⋯ψnid-1,idmid-1,idnid-1,id,where ni1-1,i1,…,nic-1,ic>0 and nic+1-1,ic+1,…,nid-1,id,n<0. Ω=dz is the holomorphic (1,0)-form, θn∗[m/n]dz¯∈H0,1(Eτ,L(n)), and θnik-1,ik[mik-1,ik/nik-1,ik]∈H0,0(Eτ,L(nik-1,ik))  (k=1,…,c). Here we explain what ψn[m/n] is. 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)-form 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)∈Ω0,1 is given as follows. We define O(1) by d~O(0)={Q,O(1)}, where d~ is a world sheet differential and Q is a BRST operator. Then, the deformation of the theory is to insert ψ≔∫∂Σ‍O(1)∈Ω0,0, where ∂Σ is a world sheet boundary.In our case, we define this deformation by an isomorphismψ: Ω0,1→Ω0,0 by(6)O0=θn∗mnz,z¯,τ,τ¯dz¯↦ψ=θnmnz,τ.That is, θn[m/n](z,τ)∈Ω0,0 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 md. By using this isomorphism, (5) is written as(7)∫EτΩ∧θn∗mndz¯θn0,1m0,1n0,1θn1,2m1,2n1,2⋯θnd-1,dmd-1,dnd-1,d.On the other hand, (5) should also be written by using md in A∞-category [15, 20] like(8)~∫EτΩ∧θn∗mndz¯mdm0,1n0,1,…,md-1,dnd-1,d.Therefore, md should be defined like(9)mdm0,1n0,1,…,md-1,dnd-1,d~θn0,1m0,1n0,1⋯θnd-1,dmd-1,dnd-1,d. We will define completely md that possesses A∞ structure in the next section.Here we discuss consistency of the integration overEτ with the periodicity. Whereas the theta functions are invariant under z↦z+1, they are transformed under z↦z+τ as(10)θnmnz,τ↦e-πin2z+τθnmnz,τθn∗mnz,z¯,τ,τ¯dz¯↦e-πin2z+τθn∗mnz,z¯,τ,τ¯dz¯.Then, the integrand is transformed as(11)θn∗mndz¯θn0,1m0,1n0,1⋯θnd-1,dmd-1,dnd-1,d↦e-πi2z+τn+n0,1+⋯+nd-1,dθn∗mndz¯θn0,1m0,1n0,1⋯θnd-1,dmd-1,dnd-1,d.Because this should be invariant for periodicity, n needs to be -(n0,1+⋯+nd-1,d) and then n0,1+⋯+nd-1,d>0.


## 3.md and A∞ Structure

In order to definemd, we multiply theta functions with characteristics. They are defined by series as follows.Definition 1 (theta functions with characteristics).
⁡(12)θabz,τ≔∑m∈Zexp⁡πim+a2τ+2πim+az+b,where a∈R/Z, b∈C.A product formula is given by the following.Theorem 2.
⁡(13)θa1n10z1,n1τθa2n20z2,n2τ=∑d∈Z/n1+n2Zθn2a1-n1a2+n1n2dn1n2n1+n20n2z1-n1z2,n1n2n1+n2τθa1+a2+n1dn1+n20z1+z2,n1+n2τ,where n1,n2∈Z, a1,a2∈R, and z1,z2,τ∈C.While this formula was proved as an addition formula whenn1,n2∈N in [21], it can also be proved as series when n1,n2∈Z as follows.Proof.
⁡(14)l.h.s.=∑m1,m2∈Zexp⁡πin1τm1+a1n12+2πim1+a1n1z1+πin2τm2+a2n22+2πim2+a2n2z2=∑l,m∈Z,d∈Z/n1+n2Zexp⁡πia22n2n1+n2τl2+2a1+a2τ+z1+z2l+2n1τld+n1τd2+2a1τ+2z1d+n1τn1+n22m2+2n1n1+n2τlm+2a1τ+z1n1+n2m+2n1n1+n2τmd+a12n1τ+a22n2τ+2a1n1z1+2a2n2z2,where we have set m1=l+p, m2=l (l,p∈Z), and p=(n1+n2)m+d(m∈Z, d∈Z/(n1+n2)Z). On the other hand,(15)r.h.s.=∑m,n∈Z,d∈Z/n1+n2Zexp⁡πin1n2n1+n2τm+n2a1-n1a2+n1n2dn1n2n1+n22+2πim+n2a1-n1a2+n1n2dn1n2n1+n2n2z1-n1z2+πin1+n2τn+a1+a2+n1dn1+n22+2πin+a1+a2+n1dn1+n2z1+z2.If we set n=l+n1m (l∈Z), (15) coincides with (14).In a special case,z1=n1z+u1 and z2=n2z+u2(u1,u2∈C), we obtain(16)θa1n1u1n1z,n1τθa2n2u2n2z,n2τ=∑d∈Z/n1+n2Zθn2a1-n1a2+n1n2dn1n2n1+n2n2u1-n1u20,n1n2n1+n2τθa1+a2+n1dn1+n2u1+u2n1+n2z,n1+n2τ.This product is expanded by the theta functions with complex coefficients. The coefficients are independent of z. We simplify this formula. Because a1+a2+n1d/n1+n2∈R/Z on the last line, we can add -(n1+n2)/(n1+n2)d and obtain(17)θa1n1u1n1z,n1τθa2n2u2n2z,n2τ=∑d∈Z/n1+n2Zθn2a1-n1a2-n2dn1n2n1+n2n2u1-n1u2·0,n1n2n1+n2τθa1+a2-n2dn1+n2u1+u2·n1+n2z,n1+n2τ=∑m∈Z,d∈Z/n1+n2Zexp⁡πin1n2n1+n2τm+n2a1-n1a2-n2dn1n2n1+n22+2πim+n2a1-n1a2-n2dn1n2n1+n2n2u1-n1u2·θa1+a2-n2dn1+n2u1+u2n1+n2z,n1+n2τ.Because a1+(a2-n2d)/n1+n2∈R/Z on the last line, we can add -n2(n1+n2)m/(n1+n2)d. By defining α≔(n1+n2)m+d(α∈Z), we obtainLemma 3.
⁡(18)θa1n1u1n1z,n1τθa2n2u2n2z,n2τ=∑α∈Zexp⁡πiτn2a1-n1a2-n2α2n1n2n1+n2+2πin2a1-n1a2-n2αn1n2n1+n2n2u1-n1u2θa1+a2-n2αn1+n2u1+u2n1+n2z,n1+n2τ.Especially, when u=0 we obtain(19)θn1a1n1z,τθn2a2n2z,τ=∑α∈Zexp⁡πiτn2a1-n1a2-n2α2n1n2n1+n2θn1+n2a1+a2-n2αn1+n2z,τ.From now on, we abbreviateθn[p](z,τ) to θn[p] because (z,τ) does not vary. By using this formula, we obtain the following.Theorem 4.
⁡(20)θn0,1a0,1n0,1θn1,2a1,2n1,2⋯θnd-1,dad-1,dnd-1,d=∑α1,2,…,αd-1,d∈Zexp⁡2πiτSda~0,1n0,1,…,a~d-1,dnd-1,dθn0,1+⋯+nd-1,da~0,1+⋯+a~d-1,dn0,1+⋯+nd-1,d,where(21)a~i-1,i≔ai-1,i-ni-1,iαi-1,ii=1,…,dα0,1≡0Sdp0,1,…,pd-1,d≔12∑i=1dni-1,ipi-1,ipi-1,i-1-n0,dp0,dp0,d-1n0,d≔n0,1+⋯+nd-1,dp0,d≔1n0,d∑i=1dni-1,ipi-1,i.Proof.
We start withd=2 case. Explicitly,(22)S2a~0,1n0,1,a~1,2n1,2=n1,2a~0,1-n0,1a~1,222n0,1n1,2n0,1+n1,2.Then, Lemma 3 implies Theorem 4 is right when d=2. For 2≤k≤d-1, explicitly,(23)Sk+1a~0,1n0,1,…,a~k,k+1nk,k+1-Ska~0,1n0,1,…,a~k-1,knk-1,k=nk,k+1a~0,1+⋯+a~k-1,k-n0,1+⋯+nk-1,ka~k,k+122n0,1+⋯+nk-1,knk,k+1n0,1+⋯+nk-1,k+nk,k+1.Then, Lemma 3 implies(24)θn0,1+⋯+nk-1,ka~0,1+⋯+a~k-1,kn0,1+⋯+nk-1,kθnk,k+1ak,k+1nk,k+1=∑αk+1∈Zexp⁡2πiτSk+1a~0,1n0,1,…,a~k,k+1nk,k+1-Ska~0,1n0,1,…,a~k-1,knk-1,kθn0,1+⋯+nk,k+1a~0,1+⋯+a~k,k+1n0,1+⋯+nk,k+1.Therefore,(25)θn0,1a0,1n0,1θn1,2a1,2n1,2⋯θnd-1,dad-1,dnd-1,d=∑α1,2∈Zexp⁡2πiτS2a~0,1n0,1,a~1,2n1,2θn0,1+n1,2a~0,1+a~1,2n0,1+n1,2θn2,3a2,3n2,3⋯θnd-1,dad-1,dnd-1,d=∑α1,2∈Zexp⁡2πiτS2a~0,1n0,1,a~1,2n1,2∑α2,3∈Zexp⁡2πiτS3a~0,1n0,1,a~1,2n1,2,a~2,3n2,3-S2a~0,1n0,1,a~1,2n1,2θn0,1+n1,2+n2,3a~0,1+a~1,2+a~2,3n0,1+n1,2+n2,3θn3,4a3,4n3,4⋯θnd-1,dad-1,dnd-1,d=∑α1,2,α2,3∈Zexp⁡2πiτS3a~0,1n0,1,a~1,2n1,2,a~2,3n2,3θn0,1+n1,2+n2,3a~0,1+a~1,2+a~2,3n0,1+n1,2+n2,3θn3,4a3,4n3,4⋯θnd-1,dad-1,dnd-1,d=⋯=∑α1,2,…,αd-1,d∈Zexp⁡2πiτSda~0,1n0,1,…,a~d-1,dnd-1,dθn0,1+⋯+nd-1,da~0,1+⋯+a~d-1,dn0,1+⋯+nd-1,d.Next, we studyA∞ structure. We define an extended theta function θ~n[p~] whose configuration space is a universal cover R(∋p~) of the configuration space R/Z(∋p) of the theta function θn[p]. That is, θ~n[a/n+1]≠θ~n[a/n] whereas θn[a/n+1]=θn[a/n]. The product is defined as follows.Definition 5 (product ofθ~).
⁡(26)θ~n0,1a0,1n0,1z,τθ~n1,2a1,2n1,2z,τ≔exp⁡πiτn1,2a0,1-n0,1a1,22n0,1n1,2n0,1+n1,2θ~n0,1+n1,2a0,1+a1,2n0,1+n1,2z,τ.This definition leads to the following.Lemma 6.
⁡(27)∑α∈Zθ~n0,1a0,1n0,1z,τθ~n1,2a1,2n1,2-αz,τ=∑α∈Zexp⁡πiτn1,2a0,1-n0,1a1,2-n1,2α2n0,1n1,2n0,1+n1,2θ~n0,1+n1,2a0,1+a1,2-n1,2αn0,1+n1,2z,τ.Proof.
Clear.The coefficients in this formula coincide with those of Lemma3. Furthermore, the definition leads to the following.Lemma 7 (local product in the configuration space).
⁡(28)θ~n0,1p~z,τθ~n1,2p~z,τ=θ~n0,1+n1,2p~z,τ.Proof.
⁡(29)l.h.s.=exp⁡πiτn1,2n0,1p~-n0,1n1,2p~2n0,1n1,2n0,1+n1,2θ~n0,1+n1,2n0,1p~+n1,2p~n0,1+n1,2z,τ=r.h.s..Next, we define a propagatorG~n[p~0,p~](z,τ) in the configuration space as follows.Definition 8 (propagator in the configuration space).
⁡(30)G~np~0,p~z,τθ~np~z,τ≔θ~np~0z,τ.As a result, Feynman diagrams appear in the configuration space (Figure1):(31)θ~n0,1a0,1n0,1θ~n1,2a1,2n1,2=G~n0,1a0,1n0,1,p~G~n1,2a1,2n1,2,p~θ~n0,1p~θ~n1,2p~=G~n0,1a0,1n0,1,p~G~n1,2a1,2n1,2,p~θ~n0,1+n1,2p~.=G~n0,1a0,1n0,1,p~G~n1,2a1,2n1,2,p~G~n0,1+n1,2p~,p~′θ~n0,1+n1,2p~′.Figure 1
Three-point interaction.When external statesθ~n[m/n] propagate from m/n to p~, we can parametrize p~=m+v/n. If two external states θ~n0,1[m0,1/n0,1] and θ~n1,2[m1,2/n1,2] propagate to the same point p~=m0,1+v0,1/n0,1=m1,2+v1,2/n1,2 and interact locally, Lemma 7 leads to(32)θ~n0,1m0,1+v0,1n0,1θ~n1,2m1,2+v1,2n1,2=θ~n0,1+n1,2m0,1+m1,2+v0,1+v1,2n0,1+n1,2,because p~=m0,1+m1,2+v0,1+v1,2/n0,1+n1,2. That is, n, m, and v are preserved. Then, we define a canonical form of θ~, including internal states, as θ~n[m+v/n], where n,m∈Z and v∈R. As a result, m represent a kind of preserved numbers of string states. Δp~=Δv/n=v′-v/n represents how long the state propagates in an expression G~n[m+v/n,m+v′/n].We define the direction of the propagatorG~n[m+v/n,m+v′/n] as the same as the direction of v and v′ (v and v′ should have the same direction.). Because incoming states propagate from m/n to m+v/n, Δp~=m+v/n-m/n=v/n. Coincidence of the signs of Δp~ and v implies that n>0 or Δp~=0. That is, incoming states with n<0 cannot propagate. Similarly, outgoing states with n>0 cannot propagate.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 noncommutative in the Feynman diagrams.We consider the moduli spaceF(p~0,d;p~0,1,p~1,2,…,p~d-1,d) of the Feynman diagrams that satisfy the above conditions, for incoming states θ~n0,1[p~0,1], θ~n1,2[p~1,2], ⋯, θ~nd-1,d[p~d-1,d], where p~i-1,i=m~i-1,i/ni-1,i=mi-1,i/ni-1,i-αi-1,i(αi-1,i∈Z, i=1,…,d), and outgoing states θ~n0,d[p~0,d], where p~0,d=m~0,d/n0,d=∑i=1d‍m~i-1,i/n0,d. We also consider the zero- and one-dimensional subspaces of the moduli space: F0(p~0,d;p~0,1,p~1,2,…,p~d-1,d) and F1(p~0,d;p~0,1,p~1,2,…,p~d-1,d), respectively. Then, we obtain the following theorem.Theorem 9 (correct Feynman diagram).
IfF0(p~0,d;p~0,1,p~1,2,…,p~d-1,d)≠∅, F0(p~0,d;p~0,1,p~1,2,…,p~d-1,d)={δ~}, that is, F0(p~0,d;p~0,1,p~1,2,…,p~d-1,d), consists of only one element δ~. Then, δ~∈F0(p~0,d;p~0,1,p~1,2,…,p~d-1,d) determines a correlation function p~0,d;p~0,1,p~1,2,…,p~d-1,dδ~, which satisfies(33)∑α1,2,…,αd-1,d∈Zp~0,d;p~0,1,p~1,2,…,p~d-1,dδ~θn0,dp~0,d=θn0,1p0,1θn1,2p1,2⋯θnd-1,dpd-1,d.Proof.
First, we classify the Feynman diagrams. There is no internal vertex because dim(F0)=0. Then, the vertices are external only: p~0,d, p~0,1, p~1,2, …, p~d-1,d. p~0,1 is at the end of the diagrams because of the nearest neighbour interaction (Figure 2). n0,1>0 because the string needs to propagate. Because of the three point interactions, p~i-1,i not at the end of the diagrams cannot propagate, namely, ni-1,i<0 (Figure 3). Let us consider the outgoing vertex p~0,d. When n0,d<0, a vertex propagate to p~0,d (Figure 4). When n0,d>0, because no vertex propagates to p~0,d, the last interaction must be as in (Figure 5). As a result, we obtain only (i) (Figure 6) when n0,d<0 and (ii) (Figure 7) when n0,d>0. Therefore, we have shown that F0(p~0,d;p~0,1,p~1,2,…,p~d-1,d) consists of only one element δ~ if F0(p~0,d;p~0,1,p~1,2,…,p~d-1,d)≠∅.
Next, we calculate the correlation function determined byδ~.
(i) Whenn0,d<0, from the diagram in Figure 6,(34)∑α1,2,…,αd-1,d∈Zp~0,d;p~0,1,p~1,2,…,p~d-1,dδ~θ~n0,1+⋯+nd-1,dp~0,d=∑α1,2,…,αd-1,d∈ZG~n0,1m~0,1n0,1,m~1,2n1,2G~n0,1+n1,2m~1,2n1,2,m~2,3n2,3⋯G~n0,1+n1,2+⋯+nd-2,d-1m~d-2,d-1nd-2,d-1,m~d-1,dnd-1,dG~n0,1+n1,2+⋯+nd-1,dm~d-1,dnd-1,d,m~0,dn0,dθ~n0,1+n1,2+⋯+nd-1,dm~0,dn0,d=∑α1,2,…,αd-1,d∈ZG~n0,1m~0,1n0,1,m~1,2n1,2G~n0,1+n1,2m~1,2n1,2,m~2,3n2,3⋯G~n0,1+n1,2+⋯+nd-2,d-1m~d-2,d-1nd-2,d-1,m~d-1,dnd-1,dθ~n0,1+n1,2+⋯+nd-1,dm~d-1,dnd-1,d=∑α1,2,…,αd-1,d∈ZG~n0,1m~0,1n0,1,m~1,2n1,2G~n0,1+n1,2m~1,2n1,2,m~2,3n2,3⋯G~n0,1+n1,2+⋯+nd-2,d-1m~d-2,d-1nd-2,d-1,m~d-1,dnd-1,dθ~n0,1+n1,2+⋯+nd-2,d-1m~d-1,dnd-1,dθ~nd-1,dm~d-1,dnd-1,d⋯=∑α1,2,…,αd-1,d∈Zθ~n0,1m~0,1n0,1θ~n1,2m~1,2n1,2⋯θ~nd-1,dm~d-1,dnd-1,d.From Lemma 6,(35)∑α1,2,…,αd-1,d∈Zp~0,d;p~0,1,p~1,2,…,p~d-1,dδ~θn0,1+⋯+nd-1,dp~0,d=θn0,1p0,1θn1,2p1,2⋯θnd-1,dpd-1,d.
(ii) Whenn0,d>0, from the diagram in Figure 7,(36)∑α1,2,…,αd-1,d∈Zp~0,d;p~0,1,p~1,2,…,p~d-1,dδ~θ~n0,1+⋯+nd-1,dp~0,d=∑α1,2,…,αd-1,d∈ZG~n0,1m~0,1n0,1,m~1,2n1,2G~n0,1+n1,2m~1,2n1,2,m~2,3n2,3⋯G~n0,1+n1,2+⋯+nd-2,d-1m~d-2,d-1nd-2,d-1,m~0,dn0,dθ~n0,1+n1,2+⋯+nd-2,d-1m~0,dn0,dG~nd-1,dm~d-1,dnd-1,d,m~0,dn0,dθ~nd-1,dm~0,dn0,d=∑α1,2,…,αd-1,d∈Zθ~n0,1m~0,1n0,1θ~n1,2m~1,2n1,2⋯θ~nd-2,d-1m~d-2,d-1nd-2,d-1θ~nd-1,dm~d-1,dnd-1,d.From Lemma 6,(37)∑α1,2,…,αd-1,d∈Zp~0,d;p~0,1,p~1,2,…,p~d-1,dδ~θn0,1+⋯+nd-1,dp~0,d=θn0,1p0,1θn1,2p1,2⋯θnd-1,dpd-1,d.Figure 2
p ~ 0,1.Figure 3
The external vertices that cannot propagate.Figure 4
End point whenn0,d<0.Figure 5
End point whenn0,d>0.Figure 6
Whenn0,d<0.Figure 7
Whenn0,d>0.By using these Feynman diagrams, we definemd.Definition 10 (md).
⁡(38)mdp0,1,p1,2,…,pd-1,d≔∑α1,2,…,αd-1,d∈Z∑δ~∈F0p~0,d;p~0,1,p~1,2,…,p~d-1,d-1δ~p~0,d;p~0,1,p~1,2,…,p~d-1,dδ~θn0,dp~0,d| δ ~ | ∈ Z should be determined so that md satisfy the A∞ relation. We have given a general definition including more than one-dimensional case, whereas δ~ is unique in this one-dimensional situation. Although md are essentially products of the theta functions as one can see in Theorem 9, the naturally defined directions of the propagators and noncommutativity of the external vertices restrict the states of the external vertices. As a result, we obtain the following theorems.Theorem 11.
The degree ofmd is 2-d.Proof.
Becausep in H0,p(Eτ,L(ni-1,i,ui-1,i)) represent the degrees of the open string states, the states with ni-1,i>0 and ni-1,i<0 have degrees 0 and 1, respectively. In the proof of Theorem 9, the Feynman diagrams are classified into two kinds (Figures 6 and 7). In the diagrams in Figure 6, n0,d<0, n0,1>0, and ni-1,i<0 (i=2,⋯d). If we compare the degrees of the both sides of the formula in Definition 10, we obtain 0+(d-1)+(thedegreeofmd)=1. Thus, the degree of md is 2-d. Because n0,d>0, n0,1>0, ni-1,i<0 (i=2,⋯d-1), and nd-1,d>0 in the diagrams in Figure 7, 0+(d-2)+0+(thedegreeofmd)=0. Thus, the degree of md is 2-d.Theorem 12 (A∞ relation).
⁡(39)∑1≤p≤d,0≤q≤d-p-1degp0,1+⋯+degpq-1,q-qmd-p+1p0,1,…,pq-1,q,mppq,q+1,…,pp+q-1,p+q,pp+q,p+q+1,…,pd-1,d=0,where deg(pi-1,i)  (i=1,…,d) represent the degrees of the string states.Proof.
⁡(40)md-p+1p0,1,…,pq-1,q,mppq,q+1,…,pp+q-1,p+q,pp+q,p+q+1,…,pd-1,d=∑αq+1,q+2,…,αp+q-1,p+q∈Z∑δ~1∈F0r~;p~q,q+1,…,p~p+q-1,p+q-1δ1~r~;p~q,q+1,…,p~p+q-1,p+qδ~1md-p+1p0,1,…,pq-1,q,r~,pp+q,p+q+1,…,pd-1,d=∑α0,1,…,αq-1,q,αq+1,q+2,…,αd-1,d∈Z∑δ~1∈F0r~;p~q,q+1,…,p~p+q-1,p+q∑δ~2∈F0p~0,d;p~0,1,…,p~q-1,q,r~,p~p+q,p+q+1,…,p~d-1,d-1δ1~+δ1~r~;p~q,q+1,…,p~p+q-1,p+qδ~1p~0,d;p~0,1,…,p~q-1,q,r~,p~p+q,p+q+1,…,p~d-1,dδ~2θn0,1+⋯+nd-1,dp~0,d.When r~ take arbitrary values, the diagrams in Figure 8 are elements of F¯1(p~0,d;p~0,1,p~1,2,…,p~d-1,d), which is the one-dimensional subspace of F¯(p~0,d;p~0,1,p~1,2,…,p~d-1,d), where F¯(p~0,d;p~0,1,p~1,2,…,p~d-1,d) is a closure of F(p~0,d;p~0,1,p~1,2,…,p~d-1,d). Especially, δ~1⊗δ~2 are the diagrams in Figure 8 with r~=nq,q+1p~q,q+1+⋯+nq+p-1,q+pp~q+p-1,q+p/nq,q+1+⋯+nq+p-1,q+p, which are elements of ∂F¯1(p~0,d;p~0,1,p~1,2,…,p~d-1,d), where ∂F¯1(p~0,d;p~0,1,p~1,2,…,p~d-1,d) is a boundary of F¯1(p~0,d;p~0,1,p~1,2,…,p~d-1,d). By choosing |δ~| appropriately, (40) becomes 0 because the contributions from the boundary of the one-dimensional space cancel with each other as in the same mechanism in the A-model [6, 9, 22].Figure 8
δ ~ 1 ⊗ δ ~ 2 ∈ ∂ F ¯ 1.Here we comparemd in the B-model and A-model.Theorem 13 (mirror symmetry).
m d in the B-model in Definition 10 coincides with md in the A-model by identifying θn[p~] with [p~] (This identification is a quasi-isomorphism [6, 9].).Proof.
The formula in Definition10 can be regarded as md in the A-model by replacing θn[p~] with [p~]. The following is the definition of the formula in Definition 10 in the A-model [6, 9]. In the A-model, the i-th D-brane (Lagrangian submanifold) L~i(i=1,…,d) is represented by a line with slope -ni on the universal cover of the symplectic torus TB. The intersection of L~i-1 and L~i is represented by y~=p~i-1,i in (y~,x~) coordinates that is the universal cover of (y,x) coordinates of TB(0≤y,x<1). [p~i-1,i] represents the string state with ni-1,i between the D-branes L~i-1 and L~i. [p~i-1,i] corresponds to θni-1,i[p~i-1,i] in the B-model. The intersection of Ld and L0 is determined as p~0,d=1/n0,d∑i=1d‍ni-1,ip~i-1,i. [p~0,d] is a basis of the md(p0,1,p1,2,…,pd-1,d), which corresponds to θn0,d[p~0,d] in the B-model. The area of the convex surrounded by L0,…,Ld is determined as Sd(p~0,1,…,p~d-1,d)=1/2(∑i=1d‍ni-1,ip~i-1,i(p~i-1,i-1)-n0,dp~0,d(p~0,d-1)). exp⁡(2πiτSd(p~0,1,…,p~d-1,d)) is the coefficient of the md(p0,1,p1,2,…,pd-1,d), which is equal to the coefficient of the md(p0,1,p1,2,…,pd-1,d) in the B-model as one can see in Theorems 4 and 9.
We are going to show that the Feynman diagrams, which determine theA∞ 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 trees ϕ~ are continuous maps that satisfy the following conditions,(The condition (6) in [9] determines not the tropical Morse trees but the length of the metric ribbon trees from the tropical Morse trees.) from the metric ribbon trees S to the universal cover B (∋y~) of R/Z (∋y) [6, 9]. (0)
We attachni-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 arey~=p~i-1,i and y~=p~0,d, respectively.(2)
All the edges and vertices of the metric ribbon trees map to the edges and vertices in the universal cover, respectively.(3)
v=0 at the external vertices.(4)
Δv=nΔy~ on the edges. The directions of v and the edges coincide.(5)
v are preserved at the vertices.
The conditions(0),…,(5) coincide with the properties of the Feynman diagrams in the B-model: (1) is satisfied in the B-model by Theorem 9. (2) is automatically satisfied in the B-model. (0), (3), and (5) follow from the definition of the canonical form of the θ~. (4) coincides with the condition that Δv=nΔp~ and directions of the propagators are the same as those of v.Because the Feynman diagrams in the B-model coincide with the tropical Morse trees in the A-model,F(p0,d;p0,1,p1,2,…,pd-1,d)=Strop(p0,d;p0,1,p1,2,…,pd-1,d), which is the moduli space of the tropical Morse trees. It is shown in [6, 9] that Strop(p0,d;p0,1,p1,2,…,pd-1,d) is bijective (The bijection map is given as follows [6, 9]: if the tropical Morse trees are decomposed to the edges, the corresponding pseudo holomorphic curves are also decomposed and, thus, the moduli spaces are decomposed as Strop=⋃eSetrop and M/PSL(2,R)=⋃eMe/PSL(2,R). If the decomposed tropical Morse trees are denoted as Setrop∋ϕ~:Se→B defined by s↦ϕ~(s)  (0≤s≤1), the decomposed pseudo holomorphic curves are denoted as Me/PSL(2,R)∋Rϕ~:De→TB defined by (s,t)↦(ϕ~(s),-neϕ~(s)-td/dsϕ~(s))  (0≤t≤1), where d/dsϕ~(s)=v(s). Then, the map between these moduli spaces is given by F:Setrop→Me/PSL(2,R) defined by ϕ~↦Rϕ~, that is, ϕ~(s)↦(ϕ~(s),-neϕ~(s)-td/dsϕ~(s)).) to the moduli space M(p0,d;p0,1,p1,2,…,pd-1,d)/PSL(2,R) of the pseudo holomorphic maps from a disk with d+1 marked points to the symplectic torus TB in the A-model in one dimension. If a topology, for example Gromov topology [23], is defined on M(p0,d;p0,1,p1,2,…,pd-1,d)/PSL(2,R), a topology is uniquely induced on Strop(p0,d;p0,1,p1,2, …,pd-1,d) by the bijective map (For example, if Me/PSL(2,R) is equipped with the topology of the functional space, equivalently, compact-open topology, the same topology is induced to Setrop and Setrop≅Me/PSL(2,R). By patching them together, Strop≅M/PSL(2,R).). As a result, we obtain the following.Theorem 14 (topological space in homological mirror symmetry).
F ( p 0 , d ; p 0,1 , p 1,2 , … , p d - 1 , d ) in the B-model is homeomorphic to M(p0,d;p0,1,p1,2,…,pd-1,d)/PSL(2,R) in the A-model.


## 4. Conclusion and Discussion

In this paper, we have defined and explicitly constructedmd that form A∞-category of the B-model on an elliptic curve. The way to define md is summarized as follows. The morphisms in the DG-category of this model are represented by theta functions with characteristics. From the products of the two theta functions, we can derive Feynman rules in the space of the characteristics, in other words, the configuration space of the morphisms. md are defined by the products of d theta functions with the characteristics in the case that the moduli space of the Feynman diagrams is restricted to zero-dimensional. This restriction is necessary for md to satisfy the A∞ relations. This md algebra coincides with the md algebra in the Fukaya category of the A-model on the corresponding symplectic torus. Thus, the A∞-category formed by these md is equivalent to the Fukaya category as an A∞-category. We have also shown that the moduli space of the Feynman diagrams in the B-model is homeomorphic to the moduli space of the pseudo holomorphic curves in the A-model.The way to definemd 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 md 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 md 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 B-model 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.



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