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 the structure of the both models.
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 -category to the differential graded (DG) category (The DG-category is an -category where are trivial.) canonically extended from the derived category of coherent sheaves. Furthermore, the Fukaya category is also equivalent as an -category to a nontrivial -category extended from the DG-category by using homotopy operators [10–18]. In this extension, is explicitly constructed in the B-model [13], whereas have not been explicitly constructed yet.
In this paper, we extend the DG-category in a different way, based on the topological string amplitudes. are newly defined and explicitly constructed in the B-model. The -category that consists of these is shown to be equivalent to the Fukaya category (Thus, the -category defined in this paper is equivalent as an -category to the -category defined by the homotopy operators because they are equivalent to the Fukaya category as categories.). In this construction, we find a moduli space necessary to define that satisfy the 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 curve which is spanned by and . There are two ways to generalize the derived category of coherent sheaves on to a differential graded (DG) category. One way is based on ech cohomology [3–9] and the other is based on Dolbeault cohomology [2, 10–16]. They are equivalent by ech-Dolbeault isomorphism. We adopt Dolbeault cohomology. Coherent sheaves on are classified and constructed based on line bundles on [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 , (), where represent connections (gauge fields) over the D-branes. The morphisms representing open strings between and are , where represent the degrees of the grading (, ).
When , and the elements of are the basis of global sections of :where are theta functions with characteristics.
When , and the elements of are the basis of harmonic (0,1)-forms with values in the dual bundle :where is the complex conjugate of , , , , and .
We are concerned with the case for simplicity. One can easily introduce the connections. We simplify the expressions asIn this case, the strings are represented by
Topological string amplitudes on are defined as [9, 15, 20]where and . is the holomorphic (1,0)-form, , and . Here we explain what is. In order to define topological string amplitudes of more than one 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 string states is given as follows. We define by , where is a world sheet differential and is a BRST operator. Then, the deformation of the theory is to insert , where is a world sheet boundary.
In our case, we define this deformation by an isomorphism : byThat is, represent string states, when not only but also . This isomorphism will be justified later by mirror symmetry of the structure and . By using this isomorphism, (5) is written asOn the other hand, (5) should also be written by using in -category [15, 20] likeTherefore, should be defined like We will define completely that possesses structure in the next section.
Here we discuss consistency of the integration over with the periodicity. Whereas the theta functions are invariant under , they are transformed under asThen, the integrand is transformed asBecause this should be invariant for periodicity, needs to be and then .
3. and Structure
In order to define , we multiply theta functions with characteristics. They are defined by series as follows.
Definition 1 (theta functions with characteristics). where , .
A product formula is given by the following.
Theorem 2. where , , and .
While this formula was proved as an addition formula when in [21], it can also be proved as series when as follows.
Proof. where we have set , (), and (, ). On the other hand,If we set (), (15) coincides with (14).
In a special case, and , we obtainThis product is expanded by the theta functions with complex coefficients. The coefficients are independent of . We simplify this formula. Because on the last line, we can add and obtainBecause on the last line, we can add . By defining , we obtain
Lemma 3. Especially, when we obtain
From now on, we abbreviate to because does not vary. By using this formula, we obtain the following.
Theorem 4. where
Proof. We start with case. Explicitly,Then, Lemma 3 implies Theorem 4 is right when . For , explicitly,Then, Lemma 3 impliesTherefore,
Next, we study structure. We define an extended theta function whose configuration space is a universal cover of the configuration space of the theta function . That is, whereas . The product is defined as follows.
Definition 5 (product of ).
This definition leads to the following.
Lemma 6.
Proof. Clear.
The coefficients in this formula coincide with those of Lemma 3. Furthermore, the definition leads to the following.
Lemma 7 (local product in the configuration space).
Proof.
Next, we define a propagator in the configuration space as follows.
Definition 8 (propagator in the configuration space).
As a result, Feynman diagrams appear in the configuration space (Figure 1):
When external states propagate from to , we can parametrize . If two external states and propagate to the same point and interact locally, Lemma 7 leads tobecause . That is, , , and are preserved. Then, we define a canonical form of , including internal states, as , where and . As a result, represent a kind of preserved numbers of string states. represents how long the state propagates in an expression .
We define the direction of the propagator as the same as the direction of and ( and should have the same direction.). Because incoming states propagate from to , Coincidence of the signs of and implies that or . That is, incoming states with cannot propagate. Similarly, outgoing states with 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 space of the Feynman diagrams that satisfy the above conditions, for incoming states , , , , where , , and outgoing states , where . We also consider the zero- and one-dimensional subspaces of the moduli space: and , respectively. Then, we obtain the following theorem.
Theorem 9 (correct Feynman diagram). If , , that is, , consists of only one element . Then, determines a correlation function , which satisfies
Proof. First, we classify the Feynman diagrams. There is no internal vertex because dim()=0. Then, the vertices are external only: , , , , . is at the end of the diagrams because of the nearest neighbour interaction (Figure 2). because the string needs to propagate. Because of the three point interactions, not at the end of the diagrams cannot propagate, namely, (Figure 3). Let us consider the outgoing vertex . When , a vertex propagate to (Figure 4). When , because no vertex propagates to , the last interaction must be as in (Figure 5). As a result, we obtain only (i) (Figure 6) when and (ii) (Figure 7) when . Therefore, we have shown that consists of only one element if .
Next, we calculate the correlation function determined by .
(i) When , from the diagram in Figure 6,From Lemma 6,(ii) When , from the diagram in Figure 7,From Lemma 6,
By using these Feynman diagrams, we define .
Definition 10 ().
should be determined so that satisfy the relation. We have given a general definition including more than one-dimensional case, whereas is unique in this one-dimensional situation. Although 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 of is 2-d.
Proof. Because in represent the degrees of the open string states, the states with and 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, , , and (). If we compare the degrees of the both sides of the formula in Definition 10, we obtain . Thus, the degree of is . Because , , (), and in the diagrams in Figure 7, . Thus, the degree of is .
Theorem 12 ( relation). where represent the degrees of the string states.
Proof. When take arbitrary values, the diagrams in Figure 8 are elements of , which is the one-dimensional subspace of , where is a closure of . Especially, are the diagrams in Figure 8 with , which are elements of , where is a boundary of . By choosing 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].
Here we compare in the B-model and A-model.
Theorem 13 (mirror symmetry). in the B-model in Definition 10 coincides with in the A-model by identifying with (This identification is a quasi-isomorphism [6, 9].).
Proof. The formula in Definition 10 can be regarded as in the A-model by replacing with . 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) is represented by a line with slope on the universal cover of the symplectic torus . The intersection of and is represented by in coordinates that is the universal cover of coordinates of . represents the string state with between the D-branes and . corresponds to in the B-model. The intersection of and is determined as . is a basis of the , which corresponds to in the B-model. The area of the convex surrounded by is determined as . is the coefficient of the , which is equal to the coefficient of the in the B-model as one can see in Theorems 4 and 9.
We are going to show that the Feynman diagrams, which determine the structure of the B-model, coincide with the tropical Morse trees, which determine the structure of the A-model. That is, both the structures coincide. The tropical Morse trees 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 to the universal cover () of () [6, 9]. (0)We attach 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 are preserved at vertices. We define an affine displacement vector on each edge.(1)The coordinates of the external incoming and outgoing vertices are and , respectively.(2)All the edges and vertices of the metric ribbon trees map to the edges and vertices in the universal cover, respectively.(3) at the external vertices.(4) on the edges. The directions of and the edges coincide.(5) are preserved at the vertices. The conditions 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 . (4) coincides with the condition that and directions of the propagators are the same as those of .
Because the Feynman diagrams in the B-model coincide with the tropical Morse trees in the A-model, , which is the moduli space of the tropical Morse trees. It is shown in [6, 9] that 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 and . If the decomposed tropical Morse trees are denoted as defined by , the decomposed pseudo holomorphic curves are denoted as defined by , where . Then, the map between these moduli spaces is given by defined by , that is, .) to the moduli space of the pseudo holomorphic maps from a disk with marked points to the symplectic torus in the A-model in one dimension. If a topology, for example Gromov topology [23], is defined on , a topology is uniquely induced on , by the bijective map (For example, if is equipped with the topology of the functional space, equivalently, compact-open topology, the same topology is induced to and . By patching them together, .). As a result, we obtain the following.
Theorem 14 (topological space in homological mirror symmetry). in the B-model is homeomorphic to in the A-model.
4. Conclusion and Discussion
In this paper, we have defined and explicitly constructed that form -category of the B-model on an elliptic curve. The way to define 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. are defined by the products of 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 to satisfy the relations. This algebra coincides with the algebra in the Fukaya category of the A-model on the corresponding symplectic torus. Thus, the -category formed by these is equivalent to the Fukaya category as an -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 define is naturally generalized in the case of the B-model on -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 by the products of morphisms with the configurations in the case that the moduli space of the Feynman diagrams is restricted to zero-dimensional. We conjecture that these satisfy the relations and form an -category, which is equivalent to the Fukaya category of the A-model on the corresponding -dimensional symplectic manifold, as an -category. We also conjecture that the moduli space of the Feynman diagrams in the B-model on the -dimensional complex manifolds is homeomorphic to the moduli space of the pseudo holomorphic curves in the A-model on the corresponding -dimensional symplectic manifold. These moduli spaces determine the structure of the both models.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that they have no conflicts of interest.