Symmetry, Integrability and Geometry: Methods and Applications Examples of Matrix Factorizations from SYZ ⋆

We find matrix factorization corresponding to an anti-diagonal in ${\mathbb C}P^1 \times {\mathbb C}P^1$, and circle fibers in weighted projective lines using the idea of Chan and Leung of Strominger-Yau-Zaslow transformations. For the tear drop orbifolds, we apply this idea to find matrix factorizations for two types of potential, the usual Hori-Vafa potential or the bulk deformed (orbi)-potential. We also show that the direct sum of anti-diagonal with its shift, is equivalent to the direct sum of central torus fibers with holonomy $(1,-1)$ and $(-1,1)$ in the Fukaya category of ${\mathbb C}P^1 \times {\mathbb C}P^1$, which was predicted by Kapustin and Li from B-model calculations.


Introduction
The Strominger-Yau-Zaslow (SYZ for short) [21] conjecture provides a geometric way to understand mirror symmetry phenomenons. Recently, Chan and Leung [4] have shown that in CP 1 the Lagrangian Floer chain complex between the equator and a generic Lagrangian torus fiber corresponds by SYZ to the matrix factorization of the Landau-Ginzburg (LG for short) superpotential W . The general idea is as follows. To find a matrix factorization corresponding to a Lagrangian submanifold, say L 0 , they consider a family of Floer chain complex of the pair (L u , L 0 ) for all torus fibers L u (with all possible holonomies), and use the information of holomorphic strips (which varies as L u changes) and apply SYZ transformation to construct the matrix factorization for L 0 (see Section 2 for more details).
Their observation is very insightful to understand the homological mirror symmetry between Lagrangian Floer theory of toric manifolds and matrix factorization of LG superpotential W , but the procedure is known to work only for a CP 1 (or a product of CP 1 with Lagrangian submanifold given by product of equators). They also found the corresponding matrix factorization for CP 2 , but the description of Floer chain complex is not complete.
In this paper, we provide more evidence on this correspondence following their ideas. The first new example is the case of the anti-diagonal Lagrangian submanifold in the symplectic manifold CP 1 × CP 1 . In fact, Kapustin and Li already conjectured in [15] that the anti-diagonal should correspond to a specific matrix factorization of LG superpotential W = x + q x + y + q y , and we verify this conjecture using this procedure.
Proposition 1.1. For CP 1 × CP 1 , the anti-diagonal Lagrangian submanifold corresponds to the following matrix factorization by SYZ transformation (in the sense of [4]) (x + y) 1 + q xy = x + q x + y + q y .
For this, we deform generic Lagrangian torus fibers into specific forms via Hamiltonian isotopy, and analyze the Floer cohomology of the anti-diagonal, with "deformed" generic torus fibers, and apply Chan and Leung's SYZ transformation to find the corresponding matrix factorization.
From B-model calculations, Kapustin and Li further conjectured in [15] that in the Fukaya category, the direct sum of anti-diagonal A and its shift A[1], is isomorphic to the direct sum of two fibers T 1,−1 and T −1,1 of holonomy (1, −1) and (−1, 1) respectively. We also verify this conjecture by computing the Floer cohomology and products between these objects, and finding a homomorphism which induces this isomorphism.
Our second type of examples are weighted projective lines, which are toric orbifolds. There is an interesting new phenomenon due to bulk deformation by twisted sectors of toric orbifolds. First of all, these weighted CP 1 's have Landau-Ginzburg mirror superpotential W : C * → C, and the first author and Poddar has recently developed a Lagrangian Floer cohomology theory for toric orbifolds, and superpotential W can be defined from the data of smooth holomorphic discs in toric orbifolds. We consider the Floer chain complex of a central fiber of the weighted CP 1 and a generic torus fiber, and from this we can find the corresponding matrix factorization of W . Proposition 1.3. For a weighted CP 1 with Z/mZ-singularity on the left and Z/nZ on the right, the central fiber corresponds to the following matrix factorization by SYZ transformation: (See Sections 6 and 7.) Then, we can turn on bulk deformation b by twisted sectors to obtain a bulk deformed mirror superpotential W b . This potential has additional terms from the data of orbifold holomorphic discs in toric orbifolds. Once bulk deformation b is chosen (so that there is a torus fiber L whose Floer cohomology is non-vanishing), then we consider the Floer chain complex of L with a generic torus fiber to find a matrix factorization of bulk deformed LG superpotential W b . In this case, we find the corresponding matrix factorization of W b by additionally considering orbifold holomorphic strips.

Preliminaries
We recall Strominger-Yau-Zaslow conjecture briefly. The classical form of mirror symmetry considers mirror pairs of Calabi-Yau 3-folds X andX, and the symplectic geometry (Gromov-Witten invariants) of X corresponds to complex geometry (periods) ofX. The SYZ conjecture is, roughly speaking, a geometric tool to find the mirror manifold, as a dual torus fibration. We state the conjecture in the following form from [14] (see also [3]): Conjecture 2.1 ( [21]). If two Calabi-Yau n-folds X andX are mirror to each other, then there exist special Lagrangian fibrations f : X → B andf :X → B, whose generic fibers are tori. Furthermore, these fibrations are dual, namely X b = H 1 (X b , R/Z) anď Toric Fano manifolds X, which are torus fibrations over the moment polytopes, has a mirror given by a Landau-Ginzburg model (X, W ). Torus fibers become singular over the facets of the moment polytope, and the singularity of the fibration is measured by the Landau-Ginzburg superpotential W , which can be constructed from the Maslov index two holomorphic discs in X with boundary on torus fibers [9,13]. The homological mirror symmetry due to Kontsevich (in this setting) asserts that the derived Fukaya category DFuk(X) of a toric Fano manifold X is equivalent, as a triangulated category, to the category of matrix factorizations M F (X, W ) of the mirror Landau-Ginzburg model (X, W ). The latter category is equivalent to the category of singularites D Sg (X, W ) (see Orlov [19]).
A matrix factorization of a Landau-Ginzburg model (X, W ) is a square matrix M of even dimensions with entries in the coordinate ring C[X] and of the form for some λ ∈ C. It is well-known that M is a non-trivial element of M F (X, W ) only if λ is a critical value of W (see [19]). The idea of Chan and Leung will be explained in more detail in the next section, but we first explain the Lagrangian Floer theory behind this correspondence. Let L 0 , L 1 be a Lagrangian submanifold in a symplectic manifold (X, ω). Let J be a compatible almost complex structure. One considers J-holomorphic discs u : (D 2 , ∂D 2 ) → (X, L) with Lagrangian boundary conditions, and denote by M k (L, β) be the moduli space of such J-holomorphic discs of homotopy class β ∈ π 2 (X, L) with k boundary marked points. We denote by µ(β) the Maslov index of β. The dimension of the moduli space is given by n + µ(β) + k − 3.
We further assume that L 0 , L 1 are positive in the sense that any non-constant J-holomorphic discs have positive Maslov index. In particular, this implies that the (virtual) dimension of M 1 (L, β) is always at least n if β = 0 and non-empty. And dim(M 1 (L, β)) = n exactly when µ(β) = 2.
We define the Novikov ring The Lagrangian Floer chain complex CF (L 0 , L 1 ) is generated by intersection points L 0 ∩ L 1 with coefficients Λ, and its differential is defined by where the sum is over all q ∈ L 0 ∩ L 1 , and n α (p, q) is the count of isolated J-holomorphic strips with boundary on L 0 , L 1 (modulo translation action) of homotopy class α, and ω(α) is the area of such J-holomorphic strip. Such isolated strips have Maslov-Viterbo index one. We refer readers to [13,17] for details. In general, m 2 1 = 0 and hence the Lagrangian Floer cohomology cannot be defined in general. With the above positivity assumption, we have the following Floer complex equation where W L i is defined as follows: From the evaluation map ev 0,β : M 1 (L, β) → L at the marked point, if µ(β) = 2, the image of ev 0,β is a multiple of fundamental class [L] as it is of dimension n, and β is of minimal Maslov index. Then we define The Floer complex equation m 2 1 = (W L 1 − W L 0 )Id, is obtained by analyzing the moduli space of holomorphic strips of Maslov-Viterbo index two (see Fig. 1). Some sequences of J-holomorphic strips of Maslov-Viterbo index two, can degenerate into broken J-holomorphic strips, each of which has index one, which contributes to m 2 1 . Some sequence of J-holomorphic strips of index two can also degenerate into a constant strip together with a bubble holomorphic disc attached to either upper or lower boundary of the strip. Discs attached to upper (resp. lower) boundary contributes to W L 1 (resp. W L 0 ) and it gives Id map since the J-holomorphic strip is constant.
In fact, one needs to considers Lagrangian submanifolds L 0 , L 1 , equipped with flat line bundles L 0 → L 0 , L 1 → L 1 , and the above setting can be extended to this setting. In such a case, we put an additional contribution of holonomy hol L i (∂β) for each β in (2.1).
Chan and Leung's idea is to compare the Floer complex equation and that of matrix factorization M 2 = (W − λ)Id (via their Fourier transform). For this, we take L 0 to be a fixed torus fiber (corresponding to the critical value λ) and vary L 1 as generic torus fibers with holonomy to obtain W as a function on the mirror manifold. 3 Chan-Leung's construction for CP 1 We recall the result of Chan-Leung [4] in the case of X = CP 1 for readers' convenience. Recall thatX = C * , and the Landau-Ginzburg superpotential is W = z + q z where q = T t when [0, t] is the moment polytope of X. (W can obtained from the disc potential e x T u + e −x T t−u by substituting z = e x T u ). By removing north (N) and south (S) pole of X, we regard X \ {N, S} as a circle fibration over (0, t), and denote by u the coordinate in (0, t), by y that of the fiber circle. Then the standard symplectic form ω equals du ∧ dy on X \ {N, S}.
An equator with trivial holonomy (fiber at t/2) has non-trivial Floer cohomology, and it corresponds to the critical point √ q of W . By homological mirror symmetry, this should correspond to a skyscraper sheaf at the critical point, and by Orlov's result, we have a matrix factorization corresponding to it. The critical value of W is 2 √ q and the corresponding factorization of Chan-Leung's idea is to recover this matrix factorization from the geometry of torus(S 1 ) fibration. Let L 0 be the central fiber with trivial holonomy. We deform L 0 to τ : [0, 3] → X as follows: (in (u, y) coordinate) Namely, τ first goes along the zero section from center to the left pole, comes back to the center and at last turns around L 0 . Let this deformation be denoted by L. Note that L still splits X into two equal halves. Since L is too singular, we slightly perturb L to L so that it is still area bisecting, and hence Hamiltonian isotopic to central fiber. It is helpful to think of L as a limit of L (see Fig. 2).
For each u ∈ (0, t), we have a corresponding fiber L u . Then L u and L meet at two points a and b, and there occur four holomorphic strips between them. Let [w] be a homotopy class of a holomorphic strip w between a and b, namely [w] ∈ π 2 (X; L, L u ; a, b). Let ∂ − [w] be the boundary of w on L u . Taking the limit of L , ∂ − [w] is identified as an element of π 1 (L u ). Now we define a function Ψ a,b L : (0, t/2) × π 1 (L u ) → R as follows: where the sign is due to the orientation of w, and n([w]) is the number of holomorphic discs representing [w]. Note that the area of a Maslov index 2 holomorphic disc whose boundary is a toric fiber L u is just given as u or (t − u) (up to times 2π). If we identify Z π 1 (L u ), then we have a complete list of (3.1): The correspondence of the function values and holomorphic strips is given as follows. D i are drawn in the above (Fig. 3) Observe that the areas of discs are computed in the limit L.
With these functions we make a matrix-valued function Ψ L by Finally, Fourier transform of (3.2) following [5] can be obtained. Each entry of (3.2) is a function of the form f = f v exp( u, v ), and for such a function f we define Fourier transform of f asf Since hol(v) = exp(iyv), if we adopt a complex coordinate z = exp(u + iy), then After the Fourier transform, we have Consider the anti-diagonal which is a Lagrangian submanifold of CP 1 × CP 1 , where both factors of CP 1 have the same standard symplectic form. Let µ : CP 1 ×CP 1 → R 2 be the moment map whose image is a square P = [0, l] 2 and L (a,b) be the moment fiber over (a, b) ∈ P . Then, L (a,b) is a torus isomorphic to L 1 × L 2 where L 1 has radius a and L 2 has b. Note that if a = b, L (a,b) does not intersect A. If a = b, they intersect along a circle.
In the example of CP 1 , the central fiber was deformed, and its Floer chain complex with a generic torus fiber was considered, whereas for our case of anti-diagonal, we deform a generic torus fiber while keeping the anti-diagonal A fixed. Namely, for each torus fiber, we deform the second component L 2 using the same methods as we did for CP 1 in the previous section and get L 2 as in Fig. 4. In fact, consider the "real" circle in CP 1 corresponding to the fixed points of complex conjugation of CP 1 , and we may choose the deformation L 2 so that it is symmetric with respect to this real circle. (In the Fig. 4, the real circle is the vertical circle which bisects the spike of L 2 .) Then, L (a,b) := L 1 × L 2 will meet the anti-diagonal A in at most two points. If 0 ≤ a ≤ b ≤ l, they intersect precisely at two points and we can explicitly find out these two points. Let α and β be two intersection points of L 1 and L 2 as in the picture below. Then, it is easy to check that Note that L 2 will be preserved under the conjugation action on CP 1 and (α, β) and (β, α) are two intersection points of L 1 and L 2 sinceᾱ = β in this case Now, we have to find holomorphic strips from (α, β) to (β, α) and vice versa. The following proposition classifies all those strips in terms of holomorphic strips in the CP 1 (which one might think of as the first or the second factor of CP 1 × CP 1 ). We introduce the following notation. We say that a holomorphic strip u : R × [0, 1] → M has a Lagrangian boundary condition (L a , L b ) if the image of (R, 0) maps to L a and that of (R, 1) maps to L b . Proposition 4.1. There is a one to one correspondence between holomorphic strips with boundary A, L (a,b) in CP 1 × CP 1 and holomorphic strips with boundary (L 2 , L 1 ) in CP 1 . Moreover, corresponding strips have the same symplectic area.
The same holds for pairs (L (a,b) , A) and (L 1 , L 2 ).
and with asymptotic conditions From the boundary conditions of u, we can conclude that u 1 and u 2 agrees on one of boundary components, i.e. u 1 (s, 0) = u 2 (s, 0). Thus, if we define u : R × [−1, 1] → CP 1 by (see Fig. 5) where we use the complex coordinate as z = s + it, then u asymptotes to α(=β) and β(=ᾱ) at ∞ and −∞ respectively as we take complex conjugate of u 2 . Note that by the construction, L 2 is preserved by complex conjugation and hence, one of boundary components of the strip is still mapped to L 2 by u 2 .
Finally, since CP 1 × CP 1 has the product symplectic structure induced from one on each factor, u and u should have the same symplectic area.
Although the moduli spaces can be identified, the orientation of the moduli spaces work slightly differently (see [13] for details on the definition of canonical orientations). For example, for a holomorphic strip with boundary (L 2 , L 1 ), changing the orientation of L 2 and L 1 to the opposite orientation reverses the canonical orientation of the holomorphic strip. But for a holomorphic strip with boundary A, L (a,b) , if we change the orientation of L 2 and L 1 at the same time, the orientation of the product L (a,b) remains the same, and so does the canonical orientation. Hence, even though the holomorphic strips for the calculation of the Floer cohomology HF (L 2 , L 1 ) in CP 1 cancels in pairs (to produce a non-vanishing Floer cohomology of the equator), but the corresponding pairs of holomorphic strip with boundary A, L (a,b) do not cancel because they have the same sign from this consideration. This is why all the terms in the matrix factorization below (4.1) has positive signs.
By the above proposition, to find holomorphic strips for the anti-diagonal, and a deformed generic torus fiber, it suffices to find holomorphic strips bounding L 1 and L 2 which converge to α and β at ±∞. These are the same holomorphic strips, discussed in the previous section for CP 1 . Namely, the shape of the strip remain the same except that now we have deformed L 2 whose position is at b, not the central fiber of CP 1 .
Before we proceed, we recall the disc potential for CP 1 × CP 1 , whose terms correspond to holomorphic discs of index two, intersecting each toric divisor and having boundary lying in fibers of the moment map (see [11]): where (α, β) represents a induced holonomy on the boundary of holomorphic discs, which may be identified with an element of H 1 of the torus. We denote x = e α T u 1 and y = e −β T l−u 2 to obtain the potential (where q = T l ): Remark 4.2. We use the coordinate y = e −β T l−u 2 instead of y = e β T u 2 so that the upperhemisphere of the second factor CP 1 bounded by L 2 has the area y (Fig. 4). This is to get a symmetric form of factorization of W .
In Fig. 4, strips from α to β are strips of area 1 and q xy , and those from β to α are strips of area x and y. Therefore, the resulting factorization of W is These four strips contribute to m 1 with the same sign as we discussed above. This proves Proposition 1.1.

Remark 4.3.
We can also compute the matrix factorization corresponding to the central moment fiber. It turns out to be an exterior tensor product of matrix factorization of the central fiber of each factor CP 1 which is given in [4]. One can check that the following matrix factors (W − λ)I 4 , where λ = 4 √ q: For the tensor product of matrix factorization, see [2].
5 Lagrangian Floer cohomology in CP 1 × CP 1 In this section, we verify the conjecture that in the derived Fukaya category, the following two objects are the same: One is the direct sum of anti-diagonal A and its shift A[1]. The other is the direct sum of Lagrangian torus fiber at the center of the moment map image with holonomy (1, −1), denoted as T 1,−1 or (−1, 1), denoted as T −1,1 . We denote by T 0 the central fiber of CP 1 × CP 1 (without considering holonomies). We refer readers to Seidel's book [20] on the definition of derived Fukaya category. We just recall that in our case, we work with Z/2-grading and by definition we have HF * (L[1], L ) = HF * +1 (L, L ).

Floer cohomology
First we compute HF (T 1,−1 , A) and HF (T −1,1 , A). Note that T 0 ∩ A is a clean intersection, which is a circle S 1 . Instead of working with the Bott-Morse version of the Floer cohomology, we move T 0 by Hamiltonian isotopy so that it intersects A transversely at two points. The Hamiltonian isotopy we choose are rotations in each factor of CP 1 so that the equator of the circle is moved to the great circle passing through North and South pole. More precisely, if we identify CP 1 as C ∪ {∞} and the equator with the unit circle in C, then after isotopy, we obtain a Lagrangian submanifold L 0 obtained as a product of real line in the first component and imaginary line in the second component.
Locally on C × C we use (a, b) and (x, y) as coordinates of the first and the second factor, respectively. Let L 0 be the torus in CP 1 × CP 1 given by the following equations We denote L R (resp. L I ) the great circle in CP 1 corresponding to the real axis (resp. imaginary axis). We have L 0 = L R × L I . The anti-diagonal A (for which we will write L 1 from now on) can be expressed as Let us calculate the Floer cohomology of the pair (L 0 , L 1 ). They intersect at two points, (0, 0) and (∞, ∞). We denote p = (0, 0) and q = (∞, ∞). As explained in the previous section, given the holomorphic strips with boundary on (L 0 , L 1 ), we can glue the first and the conjugate of second component of the strip to obtain a holomorphic strip with boundary on (L I , L R ) (lower boundary on L I , and upper boundary on L R ).
There are four such strips as seen in Fig. 6 (two strips from p to q, the other two from q to p) and these four strips have the same symplectic area.
As explained in the previous section, each of these strips are counted with the same sign in CP 1 × CP 1 (different from the case of CP 1 ). Hence two strips from p to q do not cancel out but adds up. In fact m 2 1 = 0 also, since and the potential W T 0 for the central fiber T 0 with a trivial holonomy (1, 1) is non-trivial, which is a sum of 4 terms corresponding to 4 holomorphic discs with boundary either on L I or L R in CP 1 . But for T 1,−1 or T −1,1 (which induces the flat line bundles of the same holonomy on L 0 ), two strips from p to q cancel out due to holonomy contribution, and note that the corresponding potential W T 1,−1 = W T −1,1 = 0. Thus from this cancellation of holomorphic strips we have m 1 (p) = m 1 (q) = 0.
Hence the Floer cohomology HF (T 1,−1 , A) (or HF (T −1,1 , A)) is generated by p, q and hence isomorphic to the homology of S 1 with Novikov ring coefficient. The similar argument works for HF (A, T 1,−1 )(or HF (A, T −1,1 )), which is again generated by p, q.
The Floer cohomology HF (T 1,−1 , T 1,−1 ) is a Bott-Morse version of the Floer cohomology (see [13]) and can be computed as in [7] or [9], and is isomorphic to the singular cohomology of the torus H * (T 0 , Λ). The Floer cohomology HF (A, A) is also isomorphic to the singular cohomology H * (A, Λ), as it is monotone and minimal Maslov index is 4 (see [17]).

Products
From now on, we don't distinguish L 0 and T 0 since they are clearly isomorphic in the Fukaya category. We assume that L 0 is the central torus fiber in CP 1 × CP 1 which is equipped with a flat complex line bundle of holonomy (1, −1) (or (−1, 1)), but we will omit it from the notation for simplicity. And by L 1 , we denote the anti-diagonal Lagrangian submanifold (for which we used the notation A before).      . For convenience, we also call 0 as p.
Note that both L R and L I are preserved by the complex conjugation so that we can freely use this kind of process. Note also that after gluing, the marked point for evaluation lies in the interior of the strip.
From [16] such holomorphic strips can be decomposed into simple ones, and in this case, homotopy class of any holomorphic strip is given by the union of strips (in fact an even number of unions to come back to p). Since we are interested in the case that the dimension of the evaluation image is either 0 or two, the number of strips must be less than or equal to two. Since it starts and ends at p, the number is either 0 or 2.
First we consider the case of 0, or equivalently a constant triangle. In this case, we can use the following theorem of the first author in preparation The notion of an angle is defined in [1]. In our case, L a = L c = L 0 and L b = L 1 and hence ∠L c L a = 0 and also it is not hard to see from the definition of an angle that if L a and L b intersect transversely, Thus, n+∠L a L b +∠L b L c +∠L c L a is equal to 2n. Hence, we have p as an energy-zero component of m 2 (p, p).  Now, we consider the case that the holomorphic strip covers half of CP 1 . It is in fact easy to find such holomorphic strips, covering half of CP 1 , starting from p ending at p. The image of the strip is a disc with boundary either on real or imaginary circle in CP 1 , and one of the lower or upper boundary covers the circle once, and the other boundary covers part of the segment and comes back to p. (This strip of index two usually appears to explain the bubbling off in C with Lagrangian submanifolds R and unit circle S 1 .) These holomorphic strips of Maslov-Viterbo index two, lies inside a holomorphic disc in CP 1 (of Maslov index two) with boundary on L I or L R , and there are 4 such discs (see Fig. 7). Thus, there are 4 homotopy classes of Maslov-Viterbo index two holomorphic strips from p to p and we denote them as β 1 , . . . , β 4 .
Consider M 1 (L 0 , L 1 , β i , p, p) the moduli space of holomorphic strips of (Maslov-Viterbo) index two as described above starting and ending at p, with one marked point in the upper boundary of the strip for i = 1, . . . , 4. The boundary ∂M 1 (L 0 , L 1 , β i , p, p) is well understood, and exactly has two possible components, one is from the broken strip of from p to q and to p, and the other is the bubbling off of a Maslov index two disc attached to a constant strip at p. In the former case, the marked point is located in either component of the broken holomorphic strips, and in the latter case, one of the coordinate of the marked point is free to move along the bubbled disc.
We can compare the orientations of the bubbled discs for each β i 's and they correspond to the potential W of L 0 , and with the holonomy (1, −1) or (−1, 1), all these terms cancel out. Similarly, the evaluation images of the first type of boundary from the broken strips also are mapped exactly twice, since given an index one strip, there are two adjacent strips to it. And as the signs cancelled in W , the signs of the images for the first type of boundary should be opposite too. Thus, this shows that actually the boundaries of M 1 (L 0 , L 1 , β i , p, p) for i = 1, . . . , 4 matches with opposite signs and the union gives a cycle in L 1 ∼ = CP 1 .
Hence this shows that m 2 (p, p) is a constant multiple of [L 1 ]. And it is enough to find the constant. Given such an index two strip, we consider the glued strip in CP 1 , and we evaluate at the marked point which is in the middle line of the glued strip. By varying the strip, it is not hard to see that the image of evaluation map covers "half" of the disc, or a spherical lune (Fig. 8), connecting p and q once. But there are 4 discs and these 4 lunes together cover the whole CP 1 . This shows that the constant is one, and we have m 2 (p, p) = ±[L 1 ]T l/2 .
(ii) One can show that these strips are Fredholm regular from the following explicit formulation. First we identify the holomorphic strip with the upper half-disc D + = {z ∈ C | |z| ≤ 1, Im (z) ≥ 0} with punctures at −1, +1 ∈ D 2 , which are identified with −∞, ∞ of the strip. Then, consider a holomorphic map u : D + → C with semi-circle of ∂D + mapping to the unit circle of C, and real line segment of ∂D + mapping to real line of C. All such maps of degree two (whose images covers D 2 once) are given by for a real number a, b ∈ (−1, 1), or for some α ∈ D 2 .
(To see this one starts with the generic form of a product of two Blaschke factors, and define an involution u(z) → u(z) and find its fixed elements.) Since the holomorphic discs in C with boundary on S 1 are always Fredholm regular, the fixed elements by involution are again Fredholm regular.
Proof of Lemma 5.2. All the arguments are the same as the proof of Lemma 5.1 except on the sign in front of [L 1 ]. Hence, we only need to compare orientations. Note that the moduli space of holomorphic strips from p to p of index two with boundary on (L R , L I ) in CP 1 , gives rise to the moduli space of holomorphic strips from q to q by rotating 180 with boundary on (L R , L I ) in CP 1 . If L R lies at the center of the disc which contains the strip, then this this process reverses the orientation of L R , but not the orientation of L I . (If L I lies at the center, orientation of L I is reversed, but that of L R is fixed.) As the rest of the ingredients for the orientation of the moduli space and evaluation map to the anti-diagonal remain the same, the resulting evaluation image has the opposite sign.
Proof of Lemma 5.3. The proof is somewhat similar to that of Lemma 5.1.
A holomorphic triangle contributing to m 2 in this case can be considered as a holomorphic strip u : and a marked point z 0 = [t 0 , 1] on the upper boundary of the strip, which is used as an evaluation to L 0 . Hence the first and second factor of holomorphic triangle can be again glued as in the previous section to give a holomorphic strip in CP 1 with boundary on (L I , L R ) of CP 1 , but with both ends converging to p.
Again, the same argument as in the previous lemma shows that constant strip do contribute to m 2 in this case, and also the Maslov-Viterbo index two strips are to be considered. The relevant moduli space of holomorphic strips have 4 connected components, and their boundaries cancel out. Hence the evaluation image defines a 2-dimensional cycle in L 0 , or a constant multiple of unit [L 0 ].
Thus it is enough to find the constant. For this we use the explicit form of the holomorphic strip (5.1), (5.2). We may find a holomorphic strip of index two, sending z 0 to (t 1 , t 2 ) ∈ L 0 .
After gluing of first and second component of the strip, we may find a holomorphic map from D + sending 0 to t 1 ∈ R ⊂ C and i to t 2 ∈ S 1 ⊂ C (up to automorphism of a strip, we may assume that (0, i) corresponds to z 0 ).
By inserting these numbers to (5.1), we obtain Or from (5.2), we obtain Thus a, b or α, α are (real or conjugate) pair of solutions of the quadratic equation x + t 1 = 0.
(one can check that the coefficient of x is real). If we choose t 1 < 1 to be almost as big as 1, and choose t 2 to be close to −1, then the coefficient of x is very close to 0 whereas t 1 is almost equal to 1. Thus the quadratic equation has a unique conjugate pair of complex solutions, both of which lies in the unit disc (since |α| 2 < 1). Thus, this shows that the constant c of m 2 (p, p) = c[L 0 ]T l/2 equals ±1. Hence, this proves the lemma.
The proof of Lemma 5.4 is exactly the same as that of Lemma 5.2 and omitted.
Proof of Lemma 5.5. We begin the proof of Lemma 5.5. The products HF (L 1 , L 0 ) × HF (L 0 , L 1 ) → HF (L 1 , L 1 ), given by m 2 (p, q) or m 2 (q, p) are zero since HF (L 1 , L 1 ) ∼ = H * (CP 1 , Λ) has no degree one classes. (This is because holomorphic strips connecting p and q have odd Maslov-Viterbo index, which is the dimension of the moduli space of holomorphic strips).

Floer cohomology between torus with dif ferent holonomies
First we consider the case of a cotangent bundle of a torus. Let L be a Lagrangian torus T n ⊂ T * T n Let L 1 and L 2 be two different flat line bundles on L. We prove that the Floer cohomology of the pairs HF (L, L 1 ), (L, L 2 )) vanishes if L 1 = L 2 .
Proposition 5.8. The Floer cohomology HF ((L, Proof . Since L is a torus, we identify L as R n /Z n , and define a Morse function f : T n → R by It is immediate to check that the critical points set is {(a 1 , a 2 , . . . , a n )| a i = 0 or 1/2 for i = 1, . . . , n}.
Denote the holonomy of L 0 (or L 0 ) along the i-th generator of T n by h 0 i (or h 1 i ). Let I = (1/6, 5/6), J = (4/6, 8/6). Define S defines an open cover of T n . The line bundle L 0 (and L 1 ) may be described by local charts on S as follows. We explain how to glue trivial lines bundles on the open sets of S φ : L 1 × · · · × L n × C → L 1 × · · · × L n × C sends (x 1 , . . . , x n , l) → (x 1 , . . . , x n , l ) where l = b 1 b 2 · · · b n l with It is easy to check that this defines the flat line bundle L 0 . Now we compute the boundary map in the Floer complex. First, we fix some sign convention about Morse complex. Recall the following rules, for a submanifold P ⊂ L and x ∈ P , determines the orientation of P 1 ∩ P 2 at x. Now, we denote W u (x), W s (x) to be the unstable and stable manifold of x for the given Morse function f on L. Then, we set Finally, we set the orientation of the moduli space M(x, y) of the trajectory moduli space as Now, we consider the function f given by (5.3). Unstable manifolds of f can be written as products of intervals [0, 1/2) or (1/2, 0], and intervals are canonically oriented. Hence we assign the product orientations on the unstable manifolds. Lemma 5.9. Let x = [a 1 , a 2 , . . . , a n ], y = [b 1 , . . . , b n ] where for a fixed i, a i = 0, b i = 1/2 and b j = a j for j = i. Then, the trajectory space M(x, y) has the canonical orientation (−1) A ∂ i where A is the number of j < i with a j = 0. Here ∂ i is the ith standard basis vector of R n .
Proof . First, from the orientation convention, we can identify N W u = T W s . Hence, It is easy to check that where A is the number of j < i with a j = 0, by comparing two unstable manifolds. Hence, from (5.4), we have Hence, combining with (5.5) and denoting T M(x, y) = (−1) B ∂ i , we have Hence, we have T L · (−1) A+B = T L, which proves the lemma.
The lemma implies that actual Morse boundary map is given as follows by comparing the coherent orientation with the flow orientation.
If L 0 = L 1 , we have h 0 i /h 1 i = 1, hence all boundary maps vanish and we obtain the singular cohomology of the torus T n . If L 0 = L 1 , we first assume that h 0 i = h 1 i for all i, and show that the complex has vanishing homology.
In fact, the above complex, with an assumption h 0 i = h 1 i for all i, is chain isomorphic to the same complex with the following new differential ∂ (a 1 , . . . , a n ) = for each a i =0 (−1) A i (1)(a 1 , . . . , a i−1 , a i + 1/2, a i+1 , . . . , a n ).
Here chain isomorphism can be defined as Ψ([a 1 , . . . , a n ]) = It is easy to check that Ψ∂ = ∂Ψ, and there is an obvious inverse map.
The new complex with ∂ may be considered as the reduced homology complex of the standard simplex ∆ n−1 , hence has a vanishing homology. The face corresponding to [a 1 , . . . , a n ] contains i-th vertex if and only if a i = 0. Now, consider the general case that h 0 i = h 1 i if and only if i ∈ {i 1 , . . . , i k } where k ≥ 1. The chain complex we obtain has non-trivial differential only for the terms containing (1 − h 0 i /h 1 i ) for i ∈ {i 1 , . . . , i k } and hence the chain complex decomposes into several chain sub-complexes with only non-trivial differentials within. And by using the result in the first case, we obtain the proposition.
So far, we have discussed the case in the cotangent bundle of the torus. For our case, it follows from the spectral sequence of [18]. Proof . T 0 (and hence T 1,−1 and T −1,1 ) is a monotone Lagrangian submanifold, and hence by [18], there is filtration of the Floer differential where N is the minimal Maslov number of T , which can be easily modified to the case of flat complex line bundles. By usual spectral sequence argument, we obtain the vanishing of the homology of m 1 differential, since the homology of m 1,0 vanishes in our case.

Teardrop orbifold
We show that the correspondence between the Floer complex equation and the matrix factorization continues to hold for a teardrop orbifold. Such correspondence can be divided into two levels. The first level is regarding smooth discs. Namely, we consider the Floer complex equation, only involving smooth holomorphic strips (and discs). Then, we obtain a smooth potential or the Hori-Vafa Landau-Ginzburg potential and the correspondence holds on this level. Here, by smooth holomorphic strips or discs, we mean a holomorphic maps from a smooth domain Riemann surface with boundary, and by definition of holomorphicity, they locally lift to uniformizing chart of the target orbifold point, and hence when their images contain an orbifold point, it meets the point with multiplicity (see [10]).
For the second level we consider bulk deformations by twisted sectors, and hence obtain the corresponding bulk potential or bulk Landau-Ginzburg potential which has additional terms corresponding to orbi-discs. We consider the Floer complex equation, involving smooth and orbifold holomorphic strips (and discs), which are maps from orbifold Riemann surfaces with boundary. Then the correspondence between the Floer complex equation and the matrix factorization continues to hold for bulk deformed cases.
Let X be the orbifold obtained from the following stacky fan: Then, X is an orbifold with one singular point with (Z/3Z)-singularity.

The case of Hori-Vafa potential
The Hori-Vafa Landau-Ginzburg potential can be constructed (see [10]) in this case by considering smooth holomorphic discs of Maslov index two: where z 3 is due to the fact that smooth holomorphic discs around the orbifold point has to wrap around it 3 times (see [11] for a general procedure of boundary deformation to construct such a potential from the moduli of holomorphic discs). We briefly review how to obtain the above expression of the potential (6.1) as above. Since index 2 discs correspond to the vectors in the stacky fan [10], we have the following description of index 2 holomorphic discs, e 3x T 3(u−(− 1 3 )) + e −x T 1−u = e 3x T 1+3u + e −x T 1−u , where the power of e represents the holonomy factors, and that of T represents the area of discs (see (3.1)). In particular, u is a position in the interior of the moment polytope. Note that we multiply 3 to (u + 1/3) to obtain the area of the smooth discs.
One get the expression of W as in (6.1), by substituting z = e x T 1+3u and q = T 1/3 . Then, the total area of the teardrop orbifold will give the term Denote L u by the torus fiber over u ∈ [−1/3, 1] where we identify P with the interval [−1/3, 1] ⊂ R. Let L be the balanced fiber L 0 (i.e. the moment fiber over u = 0).
Let α be a holonomy around L which is one of solutions of 3z 2 − 1 z 2 = 0 or equivalently, 3α 4 = 1. Here, the holonomy α is not unitary but, the first author proved in [8] that one can define a Floer cohomology with non-unitary line bundles. As in the picture, we can list up all strips which bound L u and L. Then, the similar technique to the one given in Section 3 will give the corresponding holomorphic functions. Note below that there are two more discs D 5 and D 6 which are not easily visible in Fig. 9. We will be able to find these in the development picture (see Fig. 10).  (1) strips from a to b: (i) The disc D 1 in the pictures leads to the term − z αq . (ii) In the limit, D 2 degenerates so that D 2 corresponds to 1.

Weighted projective lines
Finally, we study the case of weighted CP 1 s with general weights at ends. Let X be a weighted projective line with Z/mZ-singularity on the left and Z/nZ on the right, i.e. X is obtained by dividing C 2 \ {(0, 0)} by the following action of C * (m, n are assumed to be relatively prime): ρ ∈ C * : (z, w) → (ρ n z, ρ m w).
Recall that − 1 m , 1 n is the moment polytope of X. (Similarly, one can also work on the case of toric orbifold of dimension one, which corresponds to the interval as a polytope with integer labels m, n at each end point. In this setting, m, n do not need to be relatively prime. But we leave the details of this general case to the interested reader.) Thus, in this case, smooth holomorphic discs of Maslov index 2 can be described as follows: e mx T 1+mu + e −nx T 1−nu .
As before, we make a substitution for this equation by z = e x T 1 m +u and q = T 1/m , which is coherent to the computation for the teardrop. Therefore, the LG potential W is given as follows: In terms of the smooth Floer theory, the fiber at u = 0 has a non-vanishing Floer homology. This is because two smooth holomorphic discs (left, and right) has the same area at u = 0. As before, we consider the deformation L of the central fiber L 0 and consider intersection with a general fiber L u at u ∈ 0, 1 n ⊂ − 1 m , 1 n . The total area of X corresponds to and L splits the total area into q and q m/n (see Fig. 11).