Disk counting and wall-crossing phenomenon via family Floer theory
FFamily Floer theory of toric manifolds and wall-crossing phenomenon H ANG Y UAN A BSTRACT : This paper is a follow-up research of [Yua20], giving examples and applications. Weapply the Floer-theoretic SYZ mirror construction over the Novikov field in [Yua20] to the momentmap fibration on a toric manifold and the Gross’s special Lagrangian fibration [Gro01a] on a toricCalabi-Yau manifold. In the first case, we explain a simple reinterpretation of [FOOO10a] by thefamily Floer theory. In the second case, we obtain a formal power series identity that involvesboth Maslov-zero and Maslov-two disk counting, which can particularly retrieve a computation ofone-pointed open Gromov-Witten invariants in [CLL12].
Contents
The main theorem in [Yua20] (see Theorem 1.2 below) states that we can associate to a smoothLagrangian fibration a rigid analytic space, equipped with a dual fibration, that describes the quantumcorrection and the wall crossing phenomenon. In this paper, we want to apply the construction to thefollowing two specific kinds of Lagrangian fibrations:(A) The moment map of the Hamiltonian torus action on a toric manifold;(B) The Gross’s special Lagrangian fibration [Gro01a, Gro01b] on a Calabi-Yau toric manifold.It turns out that the outcomes fit well with many previous works.
We first review the main result in [Yua20]. First, we recall that the
Novikov field
Λ = C (( T R ))consists of all the infinite series (cid:80) ∞ i = a i T λ i where a i ∈ C , T is a formal symbol, and { λ i } is a divergentstrictly-increasing sequence in R . Given a series x = (cid:80) i a i T λ i in Λ , if λ i = i , then wecall a i the energy-zero part of x ; also, for any λ ∈ R , the energy- λ part refers to the energy-zeropart of T − λ x . The Novikov field has a valuation map val : Λ → R ∪ {∞} defined by sending theabove series to the smallest λ i with a i (cid:54) = ∞ . It is equivalent to thenon-archimedean norm defined by | x | = exp( − val( x )). The Novikov ring is Λ = { x ∈ Λ | val( x ) ≥ } a r X i v : . [ m a t h . S G ] J a n Hang Yuan and its maximal ideal is Λ + = { x ∈ Λ | val( x ) > } . The multiplicative group U Λ is defined by thesubset { x ∈ Λ | val( x ) = } = { x ∈ Λ | | x | = } , resembling the subgroup U (1) ≡ S in C ∗ . Now,we consider the map trop : ( Λ ∗ ) n → R n defined by ( z i ) (cid:55)→ (val( z i )), called a non-archimedean torus fibration . Indeed, the trop is an analogof the natural torus fibration ( C ∗ ) n → R n , ( z i ) (cid:55)→ (log | z i | ), and every fiber of trop is isomorphic to U n Λ ≡ trop − (0) up to a translation z i (cid:55)→ T c i z i . We remark that the total space ( Λ ∗ ) n of trop is a rigidanalytic space.Let ( X , ω, J ) be a K¨ahler manifold of real dimension 2 n . Suppose there is a Lagrangian torusfibration π : X → B on some open domain X ⊂ X . We denote the Lagrangian fiber over q by L q : = π − ( q ). In the above two cases (A) and (B), it is easy to show that the Lagrangian fibration π satisfies the following assumption. Assumption 1.1
The Lagrangian torus fibration π : X → B is semipositive in the sense that there isno holomorphic stable disk of negative Maslov index bounding a Lagrangian fiber of π . Moreover, werequire all Lagrangian fibers of π are weakly unobstructed . Theorem 1.2 ( [Yua20])
Given ( X , π ) as above, there is a triple X ∨ = ( X ∨ , W ∨ , π ∨ ) consisting of- a rigid analytic space X ∨ over the Novikov field Λ ,- a global potential function W ∨ on X ∨ ,- a dual fibration map π ∨ : X ∨ → B over the same base B .unique up to isomorphism. Furthermore, if all the Lagrangian fibers are weakly unobstructed, the dualfiber over some point q ∈ B is set-theoretically equal to H ( L q ; U Λ ) ≡ U n Λ . Our convention is that we will call X ∨ , W ∨ , and π ∨ the mirror space, the mirror Landau-Ginzburg(super)potential, and the (SYZ) dual fibration map respectively. Recall that, as a set, we simply have X ∨ = (cid:70) q H ( L q ; U Λ ); the counts of Maslov-zero disks further lead to a rigid analytic space structureon this set, and the counts of Maslov-two disks give rise to the global potential function W ∨ .The rigid analytic space structure on X ∨ satisfies the following property: if the wall-crossingphenomenon does not happen in the sense that there is no non-trivial Maslov index zero holomorphicdisk bounding any Lagrangian fiber, then the mirror analytic space X ∨ is isomorphic to a part of thetotal space ( Λ ∗ ) n of the non-archimedean torus fibration trop . Specifically, we have: Proposition 1.3 (Corollary 2.3)
Suppose B ⊂ B is a contractible domain which admits an integralaffine chart ϕ : B (cid:44) −→ R n . If for every q ∈ B , the Lagrangian fiber L q bounds no non-constant Maslovindex zero holomorphic disk, then there is an isomorphism π ∨ | B ∼ = trop | ϕ ( B ) . In this paper, we apply Theorem 1.2 to the two kinds of Lagrangian fibrations in (A) and (B) above. The weak unobstructedness means that the A ∞ algebra associated to the Lagrangian fiber has a vanishingweak Maurer-Cartan equation. We give a precise definition in [Yua20]. It is proved by [FOOO10b] that the weakunobstructedness is unaffected if we take different choices like perturbation data. amily Floer theory of toric manifolds and wall-crossing phenomenon Figure 1: X = K P . Left: The base of a Gross’s fibration on X . Right: The base of a toric momentmap on X . (Both are taken from [CLL12]). Note that the first coordinates of the both pictures agree. We consider the case (A) first. Suppose X is a toric manifoldassociated to a polytope P ⊂ R n . Let ˆ π (cid:48) : X → P be the toric moment map, and we set π (cid:48) to be therestriction of ˆ π (cid:48) over the interior B = P ◦ of P . Theorem 1.4 (Theorem 4.4)
The mirror triple ( X ∨ , W ∨ , ( π (cid:48) ) ∨ ) for ( X , π (cid:48) ) as in Theorem 1.2 satisfiesthat ( π (cid:48) ) ∨ : X ∨ → P is isomorphic to the restriction of trop : ( Λ ∗ ) n → R n over P and that the potential W ∨ agrees with the one in [FOOO10a, Theorem 4.5 & 4.6]. The first half of the theorem is due to the absence of Maslov-zero holomorphic disks and Proposition1.3. In particular, the W ∨ can be described in a single chart and is thus basically the same as [FOOO10a],so the second half holds. In summary, [FOOO10a] provides an explicit example of Theorem 1.2. We next consider the case (B). Suppose X is a toric Calabi-Yau manifold associated to a fan Σ . The set of rays is denoted by Σ (1) = { v , v , . . . , v d } . TheCalabi-Yau condition says there exists some dual lattice point m so that (cid:104) v i , m (cid:105) = i . We alsodenote by D i the irreducible toric divisor associated to v i .The Gross’s fibration π (Definition 5.1) is a slight variant of the toric moment map fibration π (cid:48) : X → P ⊂ R n defined in § 1.2.1. Specifically, the first ( n −
1) coordinates in R n of π and π (cid:48) agreewith each other. However, the last coordinate of π is defined by ρ = | w − (cid:15) | − (cid:15) , where (cid:15) > w is the holomorphic function such that the principal divisor is ( w ) = − K X .Then, the π maps onto R n − × [ − (cid:15) , + ∞ ) = : B , and its discriminant locus Γ is the union of ∂ B andthe image of the corner points of the toric polytope P under the projection map R n → R n − × { } .Now, we still denote by π : X → B the smooth part of the Gross’s fibration, where the B is the B minus the discriminant locus Γ and the X is the preimage. Note that the SYZ mirror symmetry ofthe Gross’s fibrations over the complex field C are well studied in [Aur07, CLL12, AAK16], but in thispaper, we are going to study the SYZ mirror symmetry over the Novikov field Λ instead.The zero set H = { x ∈ B | ρ ( x ) = } of the last coordinate is called a wall in the sense that aLagrangian fiber L q bounds a non-trivial Maslov index zero holomorphic disk if and only if q ∈ H .The wall H separates the base B into two chambers B + : = { ρ > } and B − = { ρ < } and alsoseparates the mirror space into the two chambers ( π ∨ ) − ( B ± ) ∼ = trop − ( B ± ) in view of Proposition1.3. Note that we can decompose the wall H = (cid:70) di = H i into its connected components H i ’s that are Hang Yuan
Figure 2: X = K P . Pictures of the holomorphic disks bounding a fiber of the Gross’s fibration π over B + (Left) or a fiber of the moment map π (cid:48) (Right).labeled exactly by Σ (1). Indeed, if we denote by P i ⊂ ∂ P the facet of P labeled by v i , then the interiorof P i is homeomorphic to H i via the projection map R n → R n − . Note that the base B is covered bythe contractible open sets U i = B + ∪ H i ∪ B − for 0 ≤ i ≤ d . Example 1.5
Consider the case X = K P . The fan Σ is generated by v = (0 , v = (1 , v = ( − , π (cid:48) is a polytope P as in the right side of Figure 1.Also, the base B of the Gross’s fibration π is R × [ − (cid:15) , + ∞ ) minus two points r , r in R × { } asillustrated in the left side of Figure 1, where the singular values r , r correspond to the two corner pointsof the polytope P . The wall is H = ( R − { r , r } ) × { } ⊂ R , and its components are H = ( r , r ), H = ( −∞ , r ), and H = ( r , + ∞ ).Given q + ∈ B + , the Lagrangian fiber L q + is of Clifford type. There is a Lagrangian isotopy L within ( C ∗ ) n between the fiber L q + and a toric fiber of the π (cid:48) , thus, we have π ( X , L q + ) ∼ = π ( X , ( C ∗ ) n ) ∼ = Z (cid:104) β , . . . , β d (cid:105) ∼ = Z d + , where the β i ’s are the classes determined by the properties β i · D j = δ ij . Also, each β i can be represented by a Maslov index two holomorphic disk which isisotopic via L to the standard holomorphic disk [CO06] bounding the Lagrangian toric fiber; c.f.Figure 2. Given q − ∈ B − , the Lagrangian fiber L q − is of Chekanov type. One can show that there isonly one class in π ( X , L q − ) which can be represented by a holomorphic disk. The class has Maslovindex two and we denote it by ˆ β . If X = C and we adopt Auroux’s visualization [Aur07] of thefibration, the class ˆ β can be illustrated as in Figure 3. Besides, a path in B between q − and q + thatpasses through H i induces an isomorphism π ( X , L q − ) ∼ = π ( X , L q + ), and one can show that it sends ˆ β to β i ; see Figure 4.For any Lagrangian fiber L q , the counts of holomorphic disks in a Maslov index two topologicalclass β ∈ π ( X , L q ) define the so-called (one-pointed genus-zero) open Gromov-Witten invariant n β . In general, it may depend on various choices, and we will give a precise study later in § 3.1. Onecan show that if q ∈ B + , all the nonzero invariants are n β j and n β j + α for some class α of holomorphicsphere bubbles; if q ∈ B − , only n ˆ β is nonzero. Moreover, it is known that the open Gromov-Witteninvariants keep constant along a Lagrangian isotopy which has no Maslov index zero holomorphic diskinvolved (Theorem 3.5). In special, by applying this fact to the Lagrangian isotopy among the fibersover either B + or B − , we see that the numbers n β j and n β j + α about the fiber L q are independent of thebase point q ∈ B + , and the n ˆ β is also independent of the base point in B − . Recall that if q ∈ B + , thereis such a Lagrangian isotopy between L q and a toric fiber within ( C ∗ ) n , hence, this fact also impliesthat n β i = X , π ) produces a mirror triple ( X ∨ , W ∨ , π ∨ ). Note that we can use amily Floer theory of toric manifolds and wall-crossing phenomenon Figure 3: X = C and w = z z . Picture of ˆ β . Compare [Aur07, Figure 2]Figure 4: X = K P . A path between q ± induces π ( X , L q − ) ∼ = π ( X , L q + ).Proposition 1.3 to deduce that the dual fibration π ∨ restricted over B + or B − is isomorphic to a part ofthe fibration trop . But, unlike the case (A) before, the two chambers ( π ∨ ) − ( B ± ) ∼ = trop − ( B ± ) areglued along ( π ∨ ) − ( H ) in a non-trivial way by virtue of the wall crossing phenomenon [Aur07].More importantly, the mirror superpotential W ∨ has quite different expressions on the two chambers( π ∨ ) − ( B ± ). Note that the W ∨ locally encodes all the open Gromov-Witten invariants. Hence, by thelocal-to-global procedure in Theorem 1.2, we can understand how the open Gromov-Witten invariantsevolve along a Lagrangian isotopy among the fibers, no matter if the Maslov-zero disks involve or not.Fix 0 ≤ i ≤ d , and consider the Lagrangian isotopy over a path σ across the wall component H i ⊂ H . May assume the path σ is sufficiently small, and the various π ( X , L q ) for q ∈ σ can beidentified with each other without any monodromy issue. In special, we can think β i = ˆ β as discussedabove. The classes ˆ β, γ (cid:96) : = β (cid:96) − ˆ β ( (cid:96) (cid:54) = i ) also form a basis of π ( X , L q ), and the subgroup of all theMaslov index zero topological classes are generated by all the γ (cid:96) ’s ( (cid:96) (cid:54) = i ).Briefly, since the transition maps of X ∨ match the different expressions of W ∨ , one can show thefollowing main result: Theorem 1.6
For any ≤ i ≤ d , there exists F ( i ) ( Y , . . . , Y n ) ∈ Λ [[ Y ± , . . . , Y ± n ]] ∼ = Λ [[ π ( L q )]] ,which only depends on the counts of Maslov index zero holomorphic disks and whose coefficients areall contained in Λ + , such that the following formal power series identity holds: (1) n ˆ β = exp (cid:0) F ( i ) (cid:1) n β i + (cid:88) α ∈ H eff2 ( X ) \{ } T ω ∩ α n β i + α + (cid:88) (cid:96) (cid:54) = i T ω ∩ γ (cid:96) Y ∂γ (cid:96) (cid:88) α ∈ H eff2 ( X ) T ω ∩ α n β (cid:96) + α Moreover, we have ω ∩ γ (cid:96) > . Therefore, by taking the energy-zero part of the both sides, we obtain n ˆ β = n β i Hang Yuan
As mentioned above, the fact n β i = Corollary 1.7 ( [CLL12, Proposition 4.32]) n ˆ β = . In [CLL12], one proves n ˆ β = X = C n and by explicitlyexamining the existence and uniqueness of the holomorphic disk in C n . In contrast, by the generalresults of Theorem 1.2, we can prove a stronger formal power series identity (1) in Theorem 1.6; weparticularly retrieve n ˆ β = Acknowledgements.
I am heavily indebted to my advisor Kenji Fukaya for many enlighteningdiscussions and conversations. I would like to thank Yuhan Sun for his knowledge of toric geometry.I would also like to thank Mohammed Abouzaid, Jiahao Hu, Santai Qu, and Yi Wang for helpfuldiscussions. I am grateful to Siu Cheong Lau and Yu-Shen Lin for the invitation to Boston UniversityGeometry and Physics Seminar in Fall 2020.
In this section, we are going to sketch the construction of X = ( X ∨ , π ∨ , W ∨ ) in Theorem 1.2. We start with a brief review of the homological algebra usedin [Yua20]. A label group is a triple ( G , E , µ ) where G is an abelian group together with grouphomomorphisms E : G → R and µ : G → Z . In practice, we consider a symplectic manifold X anda compact oriented relatively-spin Lagrangian submanifold L ⊂ X , and we work with(2) G = G ( X , L ) : = im( π ( X , L ) → H ( X , L ))The homomorphisms E and µ are the energy/area and the Maslov index respectively.By a G -gapped A ∞ algebra on a vector space C , we mean an operator system m = ( m k ,β ), wherethe m k ,β : C ⊗ k → C is a k -multilinear operator labeled by β ∈ G , satisfying:• deg m k ,β = − k − µ ( β )• A ∞ associativity relation : (cid:80) m k ,β ◦ (id m ⊗ m k ,β ⊗ id m ) = ( x ) = ( − deg x + x is the signed identity map and id m is the tensor product id ⊗ · · · ⊗ id of m copies.• gappedness condition : m , = m ∗ ,β = E ( β ) ≤ , β (cid:54) =
0, and for any E > β ∈ G such that m ∗ ,β (cid:54) = E ( β ) ≤ E .For simplicity, we will often omit saying G -gapped. In general, m : = (cid:80) T E ( β ) m ,β does not satisfy m ◦ m =
0, but in the energy-zero level, we always have m , ◦ m , = A ∞ algebra ( C , m ) induces an underlying cochain complex ( C , m , ). A G -gapped A ∞ amily Floer theory of toric manifolds and wall-crossing phenomenon algebra ( C , m ) is called minimal if m , =
0. We can also define a G -gapped A ∞ homomorphism f = ( f k ,β ) : ( C , m ) → ( C (cid:48) , m (cid:48) ) in the obvious way, and it similarly induces an underlying cochain map f , : ( C , m , ) → ( C (cid:48) , m (cid:48) , ). Moreover, by a G -gapped A ∞ homotopy equivalence we mean an A ∞ homomorphism so that f , is a quasi-isomorphism of the underlying cochain complexes. Indeed, byWhitehead theorem, any A ∞ homotopy equivalence admits a homotopy inverse g in the sense that g ◦ f and f ◦ g are homotopic to the identity homomorphism id as A ∞ homomorphisms. Technically, weneed to include extra unitality and divisor axiom conditions into the homotopy theory of A ∞ algebras;see [Yua20] for more details.Furthermore, we need a notion called the pseudo-isotopy, which is roughly a family of A ∞ algebrastogether with certain information on ‘derivatives’. We first introduce C [0 , : = Ω ∗ ([0 , ⊗ C ∞ ([0 , , C ) C Keep in mind that when C = Ω ∗ ( L ) we have C [0 , = Ω ∗ ([0 , × L ). A ( G -gapped) pseudo-isotopy consists of an operator system M = ( M k ,β ) on C [0 , such that we can write M k ,β = ⊗ m sk ,β + ds ⊗ c sk ,β and the following conditions hold• For every s ∈ [0 , G -gapped A ∞ algebra ( C , m s ) so that m s , = : m , is independentof s . (In practice, C = Ω ∗ ( L ), and m , agrees with the exterior derivative d on L .)• We have c s , = dds , and c sk , = k ≥ • The following equation holds: dds m sk ,β + (cid:88) β + β = β ( (cid:96),β ) (cid:54) = (1 , ( − ∗ c s (cid:96),β ◦ (id • ⊗ m s β ⊗ id • ) − (cid:88) β + β = β ( (cid:96),β ) (cid:54) = (1 , m s β ◦ (id • ⊗ c s (cid:96),β ⊗ id • )An equivalent definition is that a pseudo-isotopy is a special sort of A ∞ algebra defined on C [0 , such that it is Ω ∗ ([0 , M , = ⊗ m , + ds ⊗ dds for an s -independent operator m , . In practice, the m , agrees withthe exterior derivative d on Ω ∗ ( L ), and it follows that the M , agrees with the exterior derivative on Ω ∗ ([0 , × L ).We say M is a pseudo-isotopy between m (cid:48) and m (cid:48)(cid:48) if m (cid:48) = m and m (cid:48)(cid:48) = m . Using a treeconstruction, a pseudo-isotopy between m (cid:48) and m (cid:48)(cid:48) naturally induces an A ∞ homotopy equivalence C from m (cid:48) to m (cid:48)(cid:48) . We refer to [Fuk10] or [Yua20] for more details of this construction.When c sk ,β = s and ( k , β ) (cid:54) = (1 , M a trivial pseudo-isotopy . It is as if all the‘derivatives’ vanish, and one can immediately check that the m s becomes s -independent.A metric g on L gives rise to the so-called g -harmonic contraction :(3) con ( g ) : H ∗ ( L ) i ( g ) (cid:47) (cid:47) Ω ∗ ( L ) π ( g ) (cid:111) (cid:111) G ( g ) (cid:7) (cid:7) It is the data of a Hodge decomposition for the given metric g . Fix an A ∞ algebra ˇ m on C = Ω ∗ ( L ).Applying the homological perturbation with respect to the con ( g ) = ( i ( g ) , π ( g ) , G ( g )), we can producea canonical model m g of ˇ m which is an A ∞ algebra defined on H ∗ ( L ). See Figure 5.Given a pseudo-isotopy ˇ M between ˇ m − and ˇ m + , we can choose a smooth family of metrics The condition c sk , = k ≥ Hang Yuan
Figure 5: Homological perturbation using the g -harmonic contraction. g = ( g s ), giving rise to a family version of harmonic contraction:(4) con ( g ) : H ∗ ( L ) [0 , i ( g ) (cid:47) (cid:47) Ω ∗ ( L ) [0 , π ( g ) (cid:111) (cid:111) G ( g ) (cid:7) (cid:7) For instance, one can write i ( g ) = ⊗ i ( g s ) + ds ⊗ h ( g s ) where h ( g s ) is roughly the ‘derivative’ of theoperator i ( g s ). See [Yua20] for more details.Finally, applying the homological perturbation to the chain-level pseudo-isotopy ˇ M with respect tothe con ( g ), we obtain a cohomology-level pseudo-isotopy M between m g − and m g + . ∞ algebras associated to Lagrangian submanifolds . Consider the (compactified) moduli space(5) M k + ,β ( J , L )of equivalence classes of ( k + J -holomorphic stable maps of genus zero with oneboundary component in the class β ∈ G ( X , L ). We often call a stable map in this moduli a stabledisk for simplicity. There exists a stable map topology on the set (5) which is Hausdorff and compactby [FOOO10c, Theorem 7.1.43]. Its interior part M ◦ k + ,β ( J , L ) is the subset of stable disks whosesource is a smooth disk (without any sphere or disk bubbles). Note that the virtual dimension of themoduli space (5) is equal to n + k − + µ ( β ).Once and for all, we fix a metric g and an ω -tame almost complex structure J . Given q ∈ B , wetake the moduli spaces in (5) with the Lagrangian torus fiber L q : = π − ( q ) in place of L there. Then,we obtain a G ( X , L q )-gapped A ∞ algebra(6) ( Ω ∗ ( L q ) , ˇ m J , q ) amily Floer theory of toric manifolds and wall-crossing phenomenon on the de Rham complex. If we perform the homological perturbation with respect to the g -harmoniccontraction con ( g ), then we get a minimal A ∞ algebra on the de Rham cohomology, denoted by(7) ( H ∗ ( L q ) , m g , J , q )For an arbitrary Lagrangian submanifold L , we will use the notations ˇ m J , L and m g , J , L respectively.Next, we consider the so-called Maurer-Cartan formal power series P : = P q : = P g , J , q defined by P = (cid:88) β T E ( β ) Y ∂β m g , J , q ,β This can be viewed as a finite collection of formal power series in the ring Λ [[ π ( L q )]] that can beidentified with Λ [[ Y ± , . . . , Y ± n ]] by specifying a basis of H ∗ ( L q ). Denote by the constant-onefunction; it gives rise to the generator of H ( L q ). By Assumption 1.1, the term m g , J , q ,β ∈ H − µ ( β ) ( L q ) isnon-zero only if µ ( β ) ≥
0. Hence, only those β ’s with µ ( β ) = µ ( β ) = P . Now, we can make the following decomposition(8) P = W · + Q where W : = W g , J , q = (cid:88) µ ( β ) = T E ( β ) Y ∂β m g , J , q ,β / Q : = Q g , J , q = (cid:88) µ ( β ) = T E ( β ) Y ∂β m g , J , q ,β Remark that the number m g , J , q ,β / is also known as the (one-pointed) open Gromov-Witten invariant .But it may depends on choices like J in general, and we will further study it in § 3. For a basis { θ , . . . , θ n } of H ( L q ), we obtain a basis θ pq : = θ p ∧ θ q , ≤ p < q ≤ n of H ( L q ) and write Q = (cid:80) p < q Q pq · θ pq . Before setting up the mirror local charts, we need to review some non-archimedeangeometry. The
Novikov field Λ consists of all series (cid:80) i ≥ a i T E i where T is a formal symbol, a i ∈ C and { E i } forms a divergent strictly-increasing sequence of real numbers. It comes with a valuationmap val : Λ → R ∪ {∞} defined by setting 0 to ∞ and sending (cid:80) i ≥ a i T E i ( a i (cid:54) =
0) to the smallest E . This valuation map is equivalent to a norm on Λ defined by | a | = exp( − val( a )), and there isan induced adic-topology on Λ for which we can talk about the convergence. Next, we consider theso-called non-archimedean torus fibration : trop : ( Λ ∗ ) n → R n , ( z i ) → (val( z i ))which is an analog of Log : ( C ∗ ) n → R n , ( z i ) (cid:55)→ (log | z i | )Given a rational polyhedron ∆ ⊂ R n , one can show trop − ( ∆ ) is an affinoid domain (an analogueof affine scheme in the theory of rigid analytic spaces), and we call it a polytopal domain . Note that trop − ( ∆ ) = Sp Λ (cid:104) ∆ (cid:105) is given by the spectrum of maximal ideals of the polyhedral affinoid algebra (9) Λ (cid:104) ∆ (cid:105) : = (cid:110) (cid:88) −∞ <ν ,...,ν n < ∞ a ν ··· ν n Y ν · · · Y ν n n (cid:12)(cid:12)(cid:12) val( a ν ··· ν n ) + ( ν , . . . , ν n ) · u → ∞ ∀ u ∈ ∆ (cid:111) Alternatively, the algebra Λ (cid:104) ∆ (cid:105) consists of all Laurent formal power series f ∈ Λ [[ Y ± , . . . , Y ± n ]] sothat for every point ( y , . . . , y n ) ∈ trop − ( ∆ ), the f ( y , . . . , y n ) forms a convergent series with respectto the aforementioned adic-topology.Our assertion is that the mirror fibration ( X ∨ , π ∨ ) over a sufficiently small domain ∆ ⊂ B is Hang Yuan isomorphic to trop − ( ∆ ). In short, the dual fibration is locally modeled on trop . Because the B isan integral affine manifold, we can similarly define a rational polyhedron ∆ in B . Given a rationalpolyhedron ∆ in B , we can find an integral affine chart ϕ q centered at some point q ∈ B . Then, ϕ q ( ∆ ) is a rational polyhedron in R n , and one can use the polytopal domain associated to Λ (cid:104) ϕ q ( ∆ ) (cid:105) .More intrinsically, we consider the algebra(10) Λ (cid:104) ∆ , q (cid:105) : = (cid:110) (cid:88) α ∈ π ( L q ) a α Y α (cid:12)(cid:12)(cid:12) val( a α ) + (cid:104) α, q (cid:48) − q (cid:105) → ∞ ∀ q (cid:48) ∈ ∆ (cid:111) in the formal power series ring Λ [[ π ( L q )]]. Here the Y is a formal symbol, and we can view q (cid:48) − q as avector in T q B ∼ = H ( L q ). Abusing the terminologies, we also call Λ (cid:104) ∆ , q (cid:105) a polyhedral affinoid algebra and call Sp Λ (cid:104) ∆ , q (cid:105) a polytopal domain . Similarly, one can define a map trop q : Sp Λ (cid:104) ∆ , q (cid:105) → ∆ which is isomorphic to the trop over ϕ q ( ∆ ).By Groman-Solomon’s reverse isoperimetric inequalities [GS14], one can prove that for a sufficientlysmall rational polyhedron ∆ , the series W and Q pq , ≤ p < q ≤ n in (8) are all contained in thepolyhedral affinoid algebra Λ (cid:104) ∆ , q (cid:105) . For q ∈ B , we denote by a : = a q : = a g , J , q the ideal in Λ (cid:104) ∆ , q (cid:105) generated by Q pq . We call it the ideal of weak Maurer-Cartan equations . Now, a local chart of themirror space X ∨ is defined by V ( a ) = Sp (cid:16) Λ (cid:104) ∆ , q (cid:105) / a (cid:17) . The remaining series W will gives a local pieceof the global potential W ∨ , and the dual map π ∨ is locally identified with the map trop q . However,by Assumption 1.1, the ideals a q of weak Maurer-Cartan equations are all zero. Hence, a local chartof the mirror X ∨ is simply given by a polytopal domain trop − q ( ∆ ). Recall that the ( Λ ∗ ) n is a rigidanalytic space glued by various polytopal domains. In contrast, the mirror ( X ∨ , π ∨ ) is only locallygiven by these polytopal domains, but there are additional twists in the gluing morphisms contributedby counting Maslov index zero stable disks.Suppose we have two local charts V = Sp( Λ (cid:104) ∆ , q (cid:105) ) and ˜ V = Sp( Λ (cid:104) ˜ ∆ , ˜ q (cid:105) ) defined as above. Therelated Lagrangian fibers are L : = L q and ˜ L : = L ˜ q . Strictly speaking, a transition map is a map betweenthe two subdomains Sp( Λ (cid:104) ∆ ∩ ˜ ∆ , q (cid:105) ) and Sp( Λ (cid:104) ∆ ∩ ˜ ∆ , ˜ q (cid:105) ), but we would rather write ∆ or ˜ ∆ insteadof ∆ ∩ ˜ ∆ for simplicity. Note that the category of affinoid spaces is equivalent to the opposite categoryof affinoid algebras. A transition map between the two polytopal domains is actually the same thing asa homomorphism between the two polyhedral affinoid algebra, φ : Λ (cid:104) ˜ ∆ , ˜ q (cid:105) → Λ (cid:104) ∆ , q (cid:105) . The two mainingredients to define ϕ are the Fukaya’s trick and the A ∞ homotopy equivalences. The local charts use the two A ∞ algebras m g , J , q and m g , J , ˜ q associated to theLagrangian fibers L = L q and ˜ L = L ˜ q . By Fukaya’s trick, we can unify the underlying Lagrangian,transferring an isotopy of Lagrangian fibers to a movement of almost complex structure, and the latteris studied in [Fuk10, FOOO10b].We first choose a small diffeomorphism F ∈ Diff ( X ) so that F ( L ) = ˜ L . We write ˜ β : = F ∗ β forany β ∈ G ( X , L ). We also put F ∗ J = dF ◦ J ◦ dF − and F ∗ g = ( F − ) ∗ g . Then, the Fukaya’s tricksays that there is a ‘push-forward’ A ∞ algebra m F ∗ ( g , J ) , ˜ q subject to the following relation(11) m F ∗ ( g , J ) , ˜ qk , ˜ β = F − ∗ ◦ m g , J , qk ,β ◦ ( F ∗ , . . . , F ∗ )Note that F ∗ : H ∗ ( ˜ L ) → H ∗ ( L ) only depends on the homotopy class of F . Intuitively, whenever u isa J -holomorphic disk bounding L , the composition map F ◦ u is F ∗ J -holomorphic disk bounding ˜ L ,and vice versa. This observation leads to a natural identification of the moduli spaces:(12) M k ,β ( J , L ) ∼ = M k , ˜ β ( F ∗ J , ˜ L ) u ↔ F ◦ u amily Floer theory of toric manifolds and wall-crossing phenomenon This identification first gives rise to a chain-level version of (11). Besides, the harmonic contraction(3) satisfies an analogous relation con ( F ∗ g ) = F − ∗ ◦ con ( g ) ◦ F ∗ . In the homological perturbationalgorithm, the two factors together can finally deduce (11). See [Yua20] for more details.The main difference between the two A ∞ algebras in (11) is the energy change. To be specific,one may first take a canonical 1-form λ so that ω = d λ on a Weinstein neighborhood. Note thatone can replace λ by any λ + df for some function f . By Stokes’ formula, we have E ( ˜ β ) − E ( β ) = ∂ ˜ β ∩ ( λ | ˜ L ) − ∂β ∩ ( λ | L ) in general. However, in our situation, we may choose an action-angle coordinates( x , . . . , x n ; α , . . . , α n ) for the Weinstein neighborhood. Note that the ( x i ) gives the local coordinate in B ; without loss of generality, we may assume the two base points q and ˜ q are given by the zero point(0 , . . . ,
0) and c = ( c , . . . , c n ) in the local coordinate. Then, we may choose λ = (cid:80) i ( x i − c i ) d α i , as d λ = (cid:80) i dx i ∧ d α i = ω Hence, λ | ˜ L = λ | L = − (cid:80) i c i d α i . So, we obtain a concise formula asfollows:(13) E ( ˜ β ) − E ( β ) = (cid:104) ∂β, ˜ q − q (cid:105) where ˜ q − q denotes the class in H ( L ; R ) ∼ = T q B identified with the point c (c.f. [Abo17] or [Fuk10]). To compare m g , J , q and m g , J , ˜ q , it suffices to compare m F ∗ ( g , J ) , ˜ q and m g , J , ˜ q by the Fukaya’s trick (11). Choose a path J = ( J t ) t ∈ [0 , of ω -tame almost complex structures from J to F ∗ J , and then we consider the following parameterized moduli space(14) M k + ,β ( J , L ) : = (cid:71) t M k + ,β ( J t , L )producing a chain-level pseudo-isotopy ˇ M between ˇ m J , ˜ q and ˇ m F ∗ J , ˜ q . Recall that we also need tochoose a path g = ( g t ) t ∈ [0 , of metrics from g to F ∗ g and consider the contraction con ( g ) as in (4).Applying the homological perturbation about con ( g ) to the chain-level pseudo-isotopy ˇ M yieldsa pseudo-isotopy M in the cohomology-level between m g , J , ˜ q and m F ∗ ( g , J ) , ˜ q . It is known that anypseudo-isotopy naturally determines an A ∞ homotopy equivalence. Then, we denote by(15) C F : m g , J , ˜ q → m F ∗ ( g , J ) , ˜ q the A ∞ homotopy equivalence determined by the pseudo-isotopy M . Recall that the C F consists of acollection of multi-linear maps C Fk ,β labeled by both k ∈ N and β ∈ G ( X , L ) so that the A ∞ equationshold.Now, the ring homomorphism for the transition map is defined as follows:(16) φ : Λ [[ π ( L )]] → Λ [[ π ( ˜ L )]] , Y α (cid:55)→ T (cid:104) α, ˜ q − q (cid:105) · Y F ∗ α · exp (cid:68) F ∗ α, (cid:88) γ (cid:54) = C F ,γ T E ( γ ) Y ∂γ (cid:69) Note that deg C F ,γ = − µ ( γ ); thus, by Assumption 1.1, we see that C F ,γ (cid:54) = µ ( γ ) =
0. Thisis saying that only Maslov index zero holomorphic disks have contributions to the exponential parts in(16). In particular, if there is no Maslov-zero disks, we will see that the homomorphism φ is given by Y k (cid:55)→ T c k Y k , and the same kind of homomorphisms define the transition maps between the polytopaldomains in ( Λ ∗ ) n . This point also confirms Auroux’s observation [Aur07] about the wall crossingphenomenon. Furthermore, one can show the above homomorphism restricts to φ : Λ (cid:104) ˜ ∆ , ˜ q (cid:105) → Λ (cid:104) ∆ , q (cid:105) Recall the decomposition P = W · + Q in (8), and we similarly have ˜ P = ˜ W · + ˜ Q . To define the transition map, it is crucial to establish the following wall crossing formula : Hang Yuan for any η ∈ H ∗ ( L ), there exists a collection of formal power series R η so that(17) φ ( (cid:104) η, P (cid:105) ) = (cid:104) F ∗ η, (cid:105) · ˜ W + (cid:80) R η pq · ˜ Q pq The basic ideas for a proof should be as follows: First, regarding appropriate chosen bases, one canrestrict (17) on U n Λ ∼ = H ( ˜ L ; U Λ ) without losing information (because a Laurent formal power series f ∈ Λ [[ Y ± , . . . , Y ± n ]] vanishes identically if and only if the restriction of f on U n Λ vanishes). Next,since every y ∈ U Λ can be written in the form y = e x for some x ∈ Λ , one can use the divisoraxiom to transform (17) to the A ∞ equation of C F . In fact, fix y = ( y , . . . , y n ) ∈ U n Λ , and we choose b = ( x , . . . , x n ) ∈ Λ n so that y i = e x i . Utilizing the divisor axiom and Fukaya’s trick, one can compute φ ( (cid:104) η, P (cid:105) ) | y = (cid:104) F ∗ η, m F ∗ ( g , J ) , ˜ q ( C F ( b , . . . , b ) . . . C F ( b , . . . , b )) (cid:105) . Then, using the A ∞ relation, it is equalto (cid:104) F ∗ η, C F ( b , . . . , b , m g , J , ˜ q ( b , . . . , b ) , b , . . . , b ) (cid:105) . Finally, by the divisor axiom, we can pull out the b ’sand obtain (17). See [Yua20] for more details. By choosing various η , one can use the wall crossingformula to show that φ ( a ) ⊂ ˜ a and φ ( W ) ∈ ˜ W + ˜ a . In special, the φ induces a quotient homomorphism ϕ = [ φ ] : Λ (cid:104) ˜ ∆ , ˜ q (cid:105) / ˜ a → Λ (cid:104) ∆ , q (cid:105) / a . But we note that the ideals a and ˜ a are zero in our examples.Now, according to the family Floer program construction [Yua20], the transition map (18) ψ : = ϕ ∗ : V = Sp( Λ (cid:104) ∆ , q (cid:105) ) → ˜ V = Sp( Λ (cid:104) ˜ ∆ , ˜ q (cid:105) )is defined by taking the spectrum of maximal ideals. We explain why the transition map need to be defined in this way [Yua20].On the one hand, we can show that the transition map ψ (or the homomorphism ϕ ) is independentof the various choices, such as F , J , g In reality, one can first show that an A ∞ homomorphism, say C F (cid:48) , obtained by making different choices, is ud-homotopic to the original C F . If φ (cid:48) is the ringhomomorphism obtained from C F (cid:48) , then according to the defining formula (16), we observe that φ (cid:48) ( Y α ) = φ ( Y α ) · exp (cid:68) α, (cid:88) T E ( β ) ( C F (cid:48) ,β − C F ,β ) Y ∂β (cid:69) and it suffices to study the error termS : = (cid:88) T E ( β ) ( C F (cid:48) ,β − C F ,β ) Y ∂β in the exponent. Indeed, one can exactly use the ud-homotopy between C F and C F (cid:48) to measure the errorterm, and it turns out that the error term S is contained in the ideal a of weak Maurer-Cartan equations.Therefore, the two homomorphisms and the two transition maps must agree with each other. By asimilar argument, the cocycle conditions among these transition maps can also be proved. Ultimately,we can construct a mirror triple X ∨ = ( X ∨ , W ∨ , π ∨ ). See [Yua20] for more details. A key step in the mirror reconstruction of X ∨ = ( X ∨ , W ∨ , π ∨ ) is that we prove the transition map isindependent of various choices as explained in § 2.1.7. Accordingly, one can usually take some specificchoice to extract information as follows.Let L = L q and ˜ L = L ˜ q be two adjacent Lagrangian fibers as before. Suppose σ : [0 , → B isa sufficiently small path in the base manifold so that σ (0) = ˜ q and σ (1) = q . The path σ defines aLagrangian isotopy L = ( L t ) where(19) L t : = L σ ( t ) = π − ( σ ( t )) 0 ≤ t ≤ The prefix ‘ud’ here means ‘unitality with divisor axiom’, and note that we have developed a refined homotopytheory of A ∞ algebras in [Yua20] with them. amily Floer theory of toric manifolds and wall-crossing phenomenon May assume σ is sufficiently small. Then, we can find a smooth family F t ∈ Diff ( X ) so that F t ( L t ) = L = ˜ L t ∈ [0 , π ( X , L t ) for various t ∈ [0 ,
1] can be identified with each other,and only the homotopy classes of F t matter. Thus, any class β t in some π ( X , L t ) naturally inducesa family ( β t ) of topological classes in (cid:83) t ∈ [0 , π ( X , L t ). For simplicity, we may use the same notation β to represent every β t . The Maslov index is preserved along the isotopy L , but the energy is varied: E ( β t ) − E ( β t ) = (cid:104) ∂β, σ ( t ) − σ ( t ) (cid:105) due to (13).Define J t : = F t ∗ J for t ∈ [0 , F t is close to the identity map, all the J t are still ω -tame.Hence, using the Fukaya’s trick (12) yields the identification of moduli spaces as follows:(20) M k ,β ( J , L t ) ∼ = M k ,β ( J t , ˜ L )Because we have proved the choice-independence, any path of ω -tame almost complex structuresbetween J and F ∗ J will lead to the same transition map. Therefore, we can use the following specificchoice:(21) J [0 , = ( J t ) t ∈ [0 , = ( F t ∗ J ) t ∈ [0 , Proposition 2.1
Let ψ be a transition map between the two local charts associated to two adjacentLagrangian fibers L = L q and ˜ L = L ˜ q as in (18). If there is a path σ connecting q and ˜ q , over whichthe Lagrangian fibers do not bound any Maslov index zero J -holomorphic disks, then the underlyingring homomorphism φ of ψ is given by (22) Y α (cid:55)→ T (cid:104) α, ˜ q − q (cid:105) Y F ∗ α Proof.
By the choice-independence, we can use the specific family in (21) in the parameterized modulispace M k ,β ( J [0 , , ˜ L ) = (cid:83) t ∈ [0 , M k ,β ( J t , ˜ L ) to define the transition map. As in § 2.1.5, these modulispaces define a chain-level pseudo-isotopy ˇ M and a cohomology-level pseudo-isotopy M by thehomological perturbation (§ 2.1.1).It follows from (20) that M k ,β ( J [0 , , ˜ L ) ≡ (cid:71) t ∈ [0 , M k ,β ( J , L t )which is an empty set whenever µ ( β ) =
0. Hence, we conclude that ˇ M β = β . Besides, note that the M is obtained by a tree summation (c.f. Figure 5); if µ ( β ) =
0, asummand in the definition of M β is determined by several ˇ M β k with (cid:80) k β k = β . By Assumption 1.1,we have µ ( β k ) ≥ µ ( β k ) =
0. Then, the above arguments implies ˇ M β k =
0, and we alsohave M β = β .The A ∞ homotopy equivalence C F associated to the M is also defined by a tree summation.Specifically, we can write M = ⊗ m s + ds ⊗ c s , and a summand in the definition of C F β is determinedby a finite collection of families c s β k with (cid:80) k β k = β . By degree reasons, we may assume µ ( β ) = µ ( β k ) =
0. So, as above, we conclude that M β k =
0, which implies c s β k =
0. Insummary, we conclude that whenever µ ( β ) =
0, we have C F β =
0, which by the defining formula (16)completes the proof.In (22), we note that α ∈ π ( L ) and F ∗ α ∈ π ( ˜ L ) can be both identified with a lattice point( α , . . . , α n ) ∈ Z n , regarding some F -related bases. So, the monomial Y α or Y F ∗ α can be bothregarded as Y α · · · Y α n n . If we further make the identification ˜ q − q = ( c , . . . , c n ) ∈ R n , then we can Hang Yuan think of (22) as(23) ( Y , . . . , Y n ) (cid:55)→ ( T c Y , . . . , T c n Y n )The following example may give the reader the ideas underlying the non-archimedean SYZ constructionin Theorem 1.2. In short, if there is no Maslov index zero holomorphic disk contributed, then the (local)structure of the mirror agrees with the one of the non-archimedean torus ( Λ ∗ ) n . Example 2.2
The total space ( Λ ∗ ) n of the non-archimedean torus trop : ( Λ ∗ ) n → R n is a rigid analyticspace. It admits an admissible covering( Λ ∗ ) n = (cid:83) ∞ r = trop − ( ∆ r ) ≡ (cid:83) ∞ r = Sp Λ (cid:104) ∆ r (cid:105) of polytopal domains, where ∆ r = [ − r , r ] n . Such an admissible covering is not unique in that onecan replace the sequence ( ∆ r ) by any other increasing sequence ( ∆ (cid:48) r ) of rational polyhedrons sothat (cid:83) r ≥ ∆ (cid:48) r = R n . If we take a polyhedron covering ( ∆ i ) of R n , then ( Λ ∗ ) n is the union of thecorresponding polytopal domains. Let c = ( c , . . . , c n ) ∈ R n be a vector and ∆ ⊂ R n be a rationalpolyhedron. By (9) and by definition, one can easily show that the transformation Y i (cid:55)→ T c i Y i as in (23)gives rise to an algebra isomorphism Λ (cid:104) ∆ (cid:105) → Λ (cid:104) ∆ − c (cid:105) , hence it induces an isomorphism betweenthe two polytopal domains Sp Λ (cid:104) ∆ (cid:105) ≡ trop − ( ∆ ) and Sp Λ (cid:104) ∆ − c (cid:105) ≡ trop − ( ∆ − c ), namely, it givesthe transition map for the rigid analytic space ( Λ ∗ ) n . Corollary 2.3
Suppose B ⊂ B is contractible and admits an integral affine chart x : B (cid:44) −→ R n andsatisfies that for every ˜ q ∈ B the Lagrangian fiber L ˜ q bounds no non-constant J -holomorphic stabledisk whose Maslov index is zero. Then, there is an isomorphism: (24) π ∨ | B ∼ = trop | x ( B ) In particular, if we set q = x − (0) , then for any rational polyhedron ∆ ⊂ B , we have ( π ∨ ) − ( ∆ ) ∼ = Sp Λ (cid:104) ∆ , q (cid:105) ∼ = Sp Λ (cid:104) x ( ∆ ) (cid:105) ∼ = trop − ( x ( ∆ )) . Proof.
First, by condition, the ideals of weak MC equations which determine the local charts are allzero. Hence, the local charts reduce to the polytopal domains Sp Λ (cid:104) ∆ , q (cid:105) ∼ = trop − ( ∆ ); moreover, thering homomorphisms in (22) define the transition maps due to Proposition 2.1. The both sides of (24)share the same local charts and the same transition maps; thus, they are isomorphic to each other.The non-archimedean SYZ mirror is locally defined by using the formal power series ring Λ [[ π ( L q )]].Since there is no canonical isomorphism π ( L q ) ∼ = Z , we need to be careful about the monodromy of π ( L q ), when the base point q moves along a non-contractible loop in B . Recall that the superpotential function W ∨ on the mirror space is locally given by the degree-zeropart of the Maurer-Cartan formal power series, namely, given by the counts of Maslov index two stabledisks. The counts are also known as the one-pointed open Gromov-Witten invariant. Recall that G ( X , L )is the image of the Hurewicz map π ( X , L ) → H ( X , L ). amily Floer theory of toric manifolds and wall-crossing phenomenon Definition 3.1
A class β ∈ G ( X , L ) is called separable (with respect to J ), if for some m ≥
2, thereexist β k (cid:54) = k = , . . . , m , each of which can be represented by a J -holomorphic stable disk, suchthat (cid:80) ≤ k ≤ m β k = β . Otherwise, we call β non-separable . Proposition 3.2
Let β ∈ G ( X , L ) be such that µ ( β ) = . Suppose Assumption 1.1 holds. If theredoes not exist any Maslov-zero J -holomorphic stable disk bounding L , then β is non-separable. Proof.
We argue by contradiction. Suppose one has a decomposition β = (cid:80) mk = β k as in Definition3.1 for some m ≥
2. As (cid:80) µ ( β k ) = µ ( β ) =
2, it follows from Assumption 1.1 that µ ( β k ) ≥ k . Therefore, there exists some k with µ ( β k ) = k (cid:48) (cid:54) = k we have µ ( β k (cid:48) ) = k (cid:48) exists since m ≥
2. But, by condition we cannot allow such a Maslov-zero class β k (cid:48) ; this is acontradiction.Fix β ∈ G ( X , L ) with µ ( β ) =
2. We recall the minimal A ∞ algebra m g , J , L as in (7) which isdefined by applying the homological perturbation with respect to the g -harmonic contraction. Noticethat the degree of m g , J , L ,β is 2 − µ ( β ) =
0, so it is a class in H ( L ) ∼ = R · , where we denote by theconstant-one function. Now, we can introduce the following definition. Definition 3.3
We call(25) n β : = n β, J : = n β, g , J = m g , J , L ,β / the one-pointed genus-zero open Gromov-Witten invariant or the open GW invariant of the class β .By the singular chain model [FOOO10b], the value n J ,β can be regarded as the intersection numberof the ‘ n -chain’ ev : M ,β ( J , L ) → L with the point class [ pt ] in L . Theorem 3.4 If β is non-separable, then the number n β only depends on J and β . The non-separability of β relies on J ; thus, we cannot eliminate the J -dependence. Proof.
Recall that the cohomology-level A ∞ algebra m g , J , L comes from a chain-level one ˇ m J , L . Bewarethat it also depends on a choice of ‘virtual fundamental chain’ Ξ , which roughly consists of a systemof Kuranishi structures and CF-perturbations on all moduli spaces M k + ,β ( J , L ) [FOOO17, § 22]. Forclarity, we temporarily make the Ξ explicit in the notations, namely, we write the A ∞ algebras as ˇ m J , L , Ξ and m g , J , L , Ξ .Take different g (cid:48) and Ξ (cid:48) . We can first construct a pseudo-isotopy ˇ M between ˇ m J , Ξ , L and ˇ m J , Ξ (cid:48) , L in the chain-level . Then, the remaining parts of the proof can be completed by pure homologicalalgebra arguments as follows. Choosing a path of metrics between g and g (cid:48) , we can construct fromthe ˇ M another pseudo-isotopy M in the cohomology-level between m g , J , Ξ , L and m g (cid:48) , J , Ξ (cid:48) , L , usingthe parameterized harmonic contraction (4). Now, the A ∞ associativity (with labels) of the pseudo-isotopy M implies that (cid:80) β + β = β M ,β (cid:0) M ,β (cid:1) =
0. Recall that we must have M , = M , = ds ⊗ dds hold by the definition of pseudo-isotopy. Due to the non-separability of β , one canapply the similar argument in the proof of Proposition 2.1 to prove that M ,β = M ,β = Concretely, the trivial [0 , , t × M k + ,β ( J , L ) admit the extended per-turbation data ΞΞΞ , which restricts to Ξ and Ξ (cid:48) in the two ends t = t =
1, using [FOOO17, Proposition22.6 (4)] (see also [FOOO17, Property 22.22]). Then, we can use them to obtain the pseudo-isotopy ˇ M by [FOOO17, Theorem 21.35 (6)]. Hang Yuan whenever β (cid:54) = (cid:54) = β . Hence, the A ∞ relation actually reduces to M , ◦ M ,β =
0. Note that wecan write M ,β = ⊗ m s ,β + ds ⊗ c s ,β and M , = ⊗ m , + ds ⊗ dds . Consequently, we deduce that dds m s ,β = m ,β = m ,β , which completes the proof.Now, we go back to our setting of mirror reconstruction. Assume ( X , ω ) is fixed and π : X → B is a Lagrangian torus fibration on an open domain X ⊂ M . Let X ∨ = ( X ∨ , W ∨ , π ∨ ) be the mirrorgiven by Theorem 1.2. If there exists no Maslov index zero J -holomorphic stable disk bounding anyLagrangian torus fiber L q , then we claim that the super-potential W ∨ = W ∨ J only depends on J . Inreality, by Proposition 3.2, we know β is non-separable. Recall that the local model of W ∨ is locallygiven by the formal power series W = W g , J , q associated to the A ∞ algebra m g , J , q as defined in (8). Bydefinition, we have W = (cid:80) µ ( β ) = T E ( β ) Y ∂β n β, J , and every n β, J does not depend on the virtual dataand the metric g due to Theorem 3.4. More generally, we study how the open GW invariants evolve along a smooth Lagrangian isotopy L = ( L t ) t ∈ [0 , . We assume every L t is oriented, compact and relatively-spin. Note that the relativehomotopy groups π ( X , L t ) for various t ∈ [0 ,
1] can be identified with each other; now, suppose β = β t ∈ π ( X , L t ) is a family of topological classes that are identified with each other in the abovesense. Assume µ ( β ) = Theorem 3.5
If every L t does not bound any Maslov index zero J -holomorphic stable disk, then theopen GW invariants n β t , J keep constant along the Lagrangian isotopy L = ( L t ) . Namely, we have n β t , J = n β t , J for any ≤ t , t ≤ . Proof.
Without loss of generality, we may assume the Lagrangian isotopy L = ( L t ) is small enough sothat all L t are contained in a fixed Weinstein neighborhood. Now, we take a smooth family F t ∈ Diff ( X )such that F t ( L t ) = L for all t and we put J t : = F t ∗ J . We set F = F , and so F ( L ) = L . We alsotake a path J = ( J t ) between J and F ∗ J .The parameterized moduli space M k ,γ ( J , L ) ≡ (cid:71) t M k ,γ ( J t , L ) ∼ = (cid:71) t M k ,γ ( J , L t )must be empty for any nonzero Maslov index zero class γ = γ t ∈ π ( X , L t ). Recall that, given a metric g , we can take a family of metrics g = ( g t ) between F ∗ g and g , from which we obtain a cohomology-level pseudo-isotopy M between m g , J , ˜ L and m F ∗ ( g , J ) , ˜ L . They satisfy the relation of Fukaya’s trick (11).As in the proof of Proposition 2.1, one can show that M k ,γ = k ∈ N and all such γ (cid:54) = A ∞ relation (cid:80) γ + γ = β M ,γ (cid:0) M ,γ (cid:1) =
0. Since µ ( γ i ) ≥ µ ( γ ) + µ ( γ ) = µ ( β ) =
2, one of µ ( γ ) and µ ( γ ) has to be zero. Hence, we must have either γ = γ = M k ,γ i (cid:54) =
0, which contradicts to what we just obtain). It follows that the A ∞ relation reduces to M , (cid:0) M ,β (cid:1) =
0. Therefore, as in the proof of Definition-Theorem 3.4, we canshow that m g , J , ˜ q ,β = m F ∗ ( g , J ) , ˜ q ,β , or equivalently, m g , J , ˜ q ,β = ( F − ) ∗ m g , J , q ,β using the Fukaya’s trick, whichexactly implies that n β , J = n β , J . amily Floer theory of toric manifolds and wall-crossing phenomenon We want to reinterpret [FOOO10a] in the framework of family Floer theory, i.e. Theorem 1.2. Thisis quite straightforward, and we also need the notations and results for later uses in § 5.
Let N ∼ = Z n be a lattice of rank n , and let M = Hom Z ( N , Z ) be its dual. Weset N R = N ⊗ R and M R = M ⊗ R . Denote by T N = N ⊗ U (1) ∼ = T n and T C N = N ⊗ C ∗ ∼ = ( C ∗ ) n the compact real torus and the complex torus respectively. There is a pairing (cid:104)· , ·(cid:105) : N ⊗ M → Z whichcan be naturally extended over Q or R as well. Unless we further specify, by a cone σ in N R we alwaysmean a strongly convex rational polyhedral cone in the sense that σ ∩ ( − σ ) = { } . A fan Σ in N R is afinite collection of cones in N R so that every face of a cone in Σ is also in it, and the intersection of anytwo cones in Σ is a face of either cone. Besides, a cone is called smooth if its minimal generators formpart of some Z -basis of N , and it is called simplicial if its minimal generators are linearly independentover R . Note that a cone itself can be also viewed as a fan. A fan is called smooth (resp. simplicial )if all of its cones are smooth (resp. simplicial). Let Σ ( k ) be the set of all k -dimensional cones in Σ ;thus, Σ (1) is the set of rays in Σ , and Σ ( n ) is the set of maximal cones.We denote by X = X Σ the toric variety associated to the fan Σ ⊂ N R . We set Σ (1) = { v , . . . , v d } .Note that X Σ is compact if and only if the fan Σ is complete, i.e. its support | Σ | = N R . Also, X Σ issmooth if and only if the fan Σ is smooth.In practice, we want to realize Σ as the normal fan Σ ( P ) of some full-dimensional polyhedron P ⊂ M R . But, notice that P is not unique yet since the normal fan of P is only determined by the shapeof P . For instance, the kP and P + m for some k ∈ N and m ∈ M R also have the same normal fan with P . A polyhedron P in M R is called a Delzant polyhedron if it is a compact convex polyhedron so thatthe slope vectors of all the edges meeting at any vertex form a Z basis of M . Specifically, if P ⊂ M R is a Delzant polyhedron, one can find constants c i ∈ R and v i ∈ N so that the P is equal to the set ofall m ∈ M R so that (cid:104) v i , m (cid:105) ≥ c i holds for all i = , , . . . , d .From now on, we fix Σ = Σ ( P ) to be the normal fan of P , and we define X : = X Σ = X Σ ( P ) tobe the toric variety associated to P . Note that each inner normal vector v i of P corresponds to an irreducible toric divisor D i . By Delzant’s theorem [Del88], as P is a Delzant polyhedron, we know X is a compact smooth toric manifold . Since Σ is smooth (or just simplicial), we can associate to aaa = (cid:80) i a i [ D i ] ∈ A n − ( X ) ⊗ R ( a i ∈ R ≥ ) a collection { m σ | σ ∈ Σ } in M R so that the identity (cid:104) v i , m σ (cid:105) = − a i holds whenever v i ∈ σ . Note that when σ ∈ Σ ( n ), the datum m σ is unique. We call aaaconvex if (cid:104) v i , m σ (cid:105) ≥ − a i holds for all arbitrary σ ∈ Σ , and it is called strictly convex if (cid:104) v i , m σ (cid:105) > a i whenever v i / ∈ σ ∈ Σ ( n ). We consider the cone cpl( Σ ) consisting of aaa = (cid:80) di = a i D i ∈ A n − ( X ) ⊗ R such that all a i ≥ aaa is convex. Its interior cpl ◦ ( Σ ) is the subset of strictly convex aaa ’s, and it isidentified with the K¨ahler cone of X [CK00, § 3.3.1] A subset
P ⊂ Σ (1) is called a primitive collection if P is not the set of raysof any cone σ in Σ but every proper subset of P is contained in some cone σ (cid:48) in Σ . We associateto it the affine subspace A ( P ) of C Σ (1) defined by setting x i = i with v i ∈ P . Define Z ( Σ ) = (cid:83) P A ( P ), where the union runs over all primitive collections in Σ (1). Let A n − ( X ) be theChow group of Weil divisors modulo linear equivalence. Consider the group ˜ K = Hom Z ( A n − ( X ) , C ∗ ) Hang Yuan which acts freely on U ( Σ ) : = C Σ (1) − Z ( Σ ); so, X Σ ∼ = U ( Σ ) / ˜ K . On the other hand, let K be the kernelof the homomorphism p : Z d → N ∼ = Z n defined by ( a , . . . , a d ) (cid:55)→ (cid:80) di = a i v i . Then, the maximalcompact subgroup K of K is the kernel of the induced map T d → T n between torus. So, there areshort exact sequences as follows(26) 0 → K i −→ Z d p −→ N → , → K i −→ T d p −→ T n → , → ˜ K → ( C ∗ ) d p −→ ( C ∗ ) n → T n , K , and ˜ K are identified with T N ≡ N ⊗ U (1), K ⊗ U (1), and K ⊗ C ∗ respectively. Inparticular, we note that ˜ K ∼ = ( C ∗ ) d − n . Denote by k the Lie algebra of K , and we get another exactsequence 0 → k i −→ R d p −→ N R →
0, and its dual sequence is then given by 0 → M R p ∗ −→ ( R d ) ∗ i ∗ −→ k ∗ →
0. Now, we consider the natural moment map µ : C d → ( R d ) ∗ given by µ ( z , . . . , z d ) = ( | z | , . . . , | z d | ) for the standard Hamiltonian action of T d on C d . Then, the moment map of the K -action on C d is µ : = i ∗ ◦ µ : C d → ( R d ) ∗ → k ∗ By [CK00, § 3.3.4], we have K ∼ = Hom Z ( A n − ( X ) , U (1)) and k ∗ ∼ = A n − ( X ) ⊗ R . For any aaa ∈ cpl ◦ ( Σ ),the level set µ − ( aaa ) ⊂ C d − Z ( Σ ) is a compact submanifold on which K acts freely. Also, thereis a diffeomorphism from µ − ( aaa ) / K onto X , which by symplectic reduction [MW74] gives rise to acanonical symplectic form ω aaa on X Σ . The T d -action on C d leaves the map µ invariant and henceinduces an T d -action on the level set µ − ( aaa ). Then, there is an induced action of the quotient torus T N ≡ N R / N ∼ = T n ∼ = T d / K on X ∼ = µ − ( aaa ) / K which turns out to be a Hamiltonian T n -action. Theform ω aaa depends on aaa and we need to choose aaa appropriately in order to recover the original polyhedron P . To achieve this, we first denote by e k the standard basis of C d , and we take λλλ = i ∗ ( − λ , . . . , − λ d ) = − (cid:80) dk = λ k · i ∗ e k ∈ k ∗ Then, we consider Z : = µ − ( λλλ ) = (cid:110) ( z , . . . , z d ) ∈ C d | (cid:88) k (cid:0) | z k | + λ k ) · i ∗ e k = (cid:111) The canonical symplectic form ω : = ω λλλ on X ∼ = Z / K is compatible with the complex structure J on X ; that is, X is K¨ahler [Gui94, Theorem 2.5]. We also note that that X = U ( Σ ) / ˜ K . The Lagrangianfibration we begin with is provided by the Delzant’s theorem [Del88] (see also [Aud12, TheoremVII.2.1] and [Gui94, Theorem 2.3]): Theorem 4.1
Let P ⊂ M R is a Delzant polyhedron, and X be the compact toric manifold associatedto the normal fan of P . Suppose ω is the symplectic form given by the reduction X ∼ = µ − ( λλλ ) / K and (27) π : X → ( R n ) ∗ ∼ = M R is the moment map associated to the induced T n -action. Then, ω is K¨ahler and T n -invariant; moreover,we have π ( X ) = P . In reality, Guillemin further discovers the following explicit formula for ω . We begin with somenotations. Recall that the v i ’s are inner normal vectors of P . Consider the linear functionals defined on M R as follows:(28) (cid:96) i ( q ) = (cid:104) v i , q (cid:105) − c i i = , , . . . , d (cid:96) ∞ ( q ) = (cid:104) (cid:80) di = v i , q (cid:105) amily Floer theory of toric manifolds and wall-crossing phenomenon Remark that P = { q ∈ M R | (cid:96) i ( q ) ≥ i = , , . . . , d } and that ∂ P = (cid:83) i P i for the facet P i of P defined by the zero locus (cid:96) i =
0, then the irreducible toric divisor D i associated to the inner normalvector v i can be described by D i = π − ( T i ). Theorem 4.2 ( [Gui94, Theorem 4.5])
In the situation of Theorem 4.1, the symplectic form ω on theinterior P ◦ of P is given by: (29) ω = √− π ∂ ¯ ∂ (cid:16) π ∗ (cid:0) m (cid:88) i = (log (cid:96) i ) + (cid:96) ∞ (cid:1)(cid:17) Once and for all, we fix a Delzant polyhedron P , and fix the ( X , π ) as above. n -orbits . For any point q ∈ P ◦ , the fiber L q : = π − ( q ) is a Lagrangian T n -orbit ;c.f. [CO06]. Note that we may regard T n as T N ∼ = N R / N ∼ = R n / Z n . Let e , . . . , e n be a Z -bases of N and let e (cid:48) , . . . , e (cid:48) n be the dual basis of M . Let S i ∼ = S be the subgroup generated by the i -th generatorelement in T N ∼ = ( S ) n . For any q ∈ P ◦ the orbits S i ( q ) of these subgroups S i in the T n -orbit L q giverise to a basis of π ( L q ) ≡ H ( L q ; Z ). Accordingly, we have natural isomorphisms(30) N ∼ = H ( L q ; Z ) M ∼ = H ( L q ; Z )for all q ∈ P ◦ . We remark that since P ◦ is contractible, there is no monodromy issue. Recallthat the label group G ( X , L q ) = im( π ( X , L q ) → H ( X , L q ) as in (2), but in this situation we havenatural isomorphisms H ( X , π − ( P ◦ )) ∼ = H ( X , L q ) ∼ = π ( X , L q ) ∼ = G ( X , L q ). Moreover, it has a basisconsisting of(31) β i ≡ β i , q ∈ G ( X , L q ) i = , , . . . , d which is topologically defined by taking a small disk transversal to D i ; see e.g. [FOOO10a, (6.7)]. Inparticular, we know the topological intersection numbers:(32) β i · D j = δ ij and we also know(33) ∂β i ∼ = v i regarding the natural isomorphism H ( L q ; Z ) ∼ = N in (30); c.f. [FOOO10a, (6.8)].By [CO06, Theorem 5.1], we have the following Maslov index formula: µ ( β ) = (cid:80) di = β · D i forany class β ∈ G ( X , L q ) ∼ = Z d that can be represented by a holomorphic disk. Moreover, all holomorphicdisks bounding a Lagrangian toric orbit can be classified by [CO06, Theorem 5.2] as follows: Denote by p : ( C ∗ ) d → ( C ∗ ) n the natural projection, and by (26) its fiber is identified with ˜ K ∼ = ( C ∗ ) d − n . Also, themap p extends to the quotient ˆ p : C d − Z ( Σ ) → X ≡ (cid:0) C d − Z ( Σ ) (cid:1) / ˜ K . Note that L q ⊂ ( C ∗ ) n ⊂ X . Thereal torus K ⊂ ˜ K acts on ( C ∗ ) d and preserves the fibers of p . Fix ccc = ( c , . . . , c d ) ∈ p − ( L q ) ⊂ ( C ∗ ) d and take the Lagrangian orbit ˜ L = T d · ccc . Any holomorphic disk w : ( D , ∂ D ) → ( X , L q ) can be liftedto a holomorphic disk ˜ w = ( z , . . . , z d ) : ( D , ∂ D ) → ( C d − Z ( Σ ) , ˜ L ) so that z j ( z ) = c j (cid:81) n j k = z − α j , k − ¯ α j , k z for n j ∈ N . Moreover, if ˜ w (cid:48) is another lift, then ˜ w (cid:48) = t · ˜ w where t is a constant element in the real torus K . Hence, w = ˆ p ◦ ˜ w = ˆ p ◦ ˜ w (cid:48) .Following [CO06, § 7], we study holomorphic disks of Maslov index two. Every β i in (31) can berepresented by a holomorphic disk(34) u i : ( D , ∂ D ) → ( X , L q ) Hang Yuan that is defined by the lift holomorphic disk ˜ u i = ( c , . . . , c i − , c i z , c i + , . . . , c d ). It is clear that µ ( β i ) =
2. There is a natural S -action on the image of u i given by S ( v i ) (cid:44) −→ T N ≡ T n . In thelevel of Lie algebra, this corresponds to v i ∈ N R ∼ = Lie ( T N ). If we use the polar coordinate ( r , θ ) on D , then u i ( r , θ ) is equivariant in θ ∈ S and the image π ◦ u i under the moment map π : X → M R is a one-dimensional line which meets the i -th facet P i of the polytope P . The areas or energiesof these β i = β i , q actually depend on the base point q ∈ P ◦ , and they can be computed explicitlyby [CO06, Theorem 8.1]:(35) E ( β i , q ) = (cid:96) i ( q )Be aware that we adopt a slightly different convention that the symplectic form ω in (29) is actually theone in [CO06] further divided by 2 π . The holomorphic disks u i in (34) are Fredholm regular by [CO06, Theorem 6.1] in the sense thatthe interior part M ◦ ,β i ( J , L ) of the moduli space in (5) is a smooth manifold for each i . Here by theinterior part we mean the subset of those stable maps without any disk or sphere bubbles. Denote by H eff2 ( X ; Z ) the effective cone , the cone in H ( X ; Z ) generated by holomorphic spheres in X . Theorem 4.3 ( [FOOO10a, Theorem 11.1]) (a) If M ,β ( J , L q ) (cid:54) = ∅ , then there exists ( k , . . . , k d ) ∈ Z d ≥ \ { (0 , . . . , } and α ∈ H eff2 ( X ; Z ) sothat β = (cid:80) di = k i β i + α .(b) We have M ,β i ( J , L q ) = M ◦ ,β i ( J , L q ) which is a smooth manifold so that the evaluation map ev : M ,β i ( J , L q ) → L q is an orientation-preserving diffeomorphism.(c) For an arbitrary class β , the interior of the moduli space M ◦ ,β ( J , L ) is a smooth manifold, and ev : M ◦ ,β ( J , L q ) → L q is a submersion. The moment map π in (27) gives a Lagrangian torus fibration over the interior P ◦ of the momentpolytope P = π ( X ) ⊂ R n ∼ = M R , still denoted by π . Given q ∈ P ◦ , we write P ◦ q : = P ◦ − q ⊂ R n forsimplicity. By the identification (30), we can just work with Λ [[ N ]] instead of Λ [[ π ( L q )]] for various q in the mirror reconstruction. The cap product π ( L q ) ⊗ H ( L q ) → R can be identified with the pairing N ⊗ M R → R . Recall (25) that the open GW invariant of a Maslov index two class β is denoted by n β = n β, J , and it is well known that n β i = ≤ i ≤ d . Theorem 4.4
Suppose that there is no nontrivial holomorphic sphere with negative Maslov index .Fix q ∈ P ◦ . The mirror associated to ( X , π ) by Theorem 1.2 is given by X q : = ( X ∨ q , W ∨ q , π ∨ q ) of ( X , ω, π ) , where X ∨ q ≡ trop − ( P ◦ q ) , π ∨ q = trop : X ∨ q → P ◦ q ⊂ R n and (36) W ∨ q = d (cid:88) i = (cid:16) + (cid:88) α ∈ H eff2 ( X ; Z ) \{ } T ω ∩ α c β i + α (cid:17) · T (cid:96) i ( q ) Y v i where the open GW invariant n β i + α ∈ R for the class β i = β i , q ∈ π ( X , L q ) is independent of the basepoint q . In particular, when X is Fano, the potential function is given by (37) W ∨ q = d (cid:88) i = T (cid:96) i ( q ) Y v i compare [AAK16, Assumption 2.2] or [FOOO10a, p138] amily Floer theory of toric manifolds and wall-crossing phenomenon In all situations, for q (cid:54) = q , we have a natural isomorphism X ∨ q ∼ = X ∨ q given by (38) Y v (cid:55)→ T (cid:104) v , q − q (cid:105) Y v Proof.
By assumption and Theorem 4.3 (a), for every q ∈ P ◦ , the Lagrangian torus fiber L q boundsno stable disk whose Maslov index is zero. By Corollary 2.3, we can obtain X ∨ q and π ∨ q as desired. Itremains to show the formula of W ∨ q . Denote by ˇ m = ˇ m J , q the A ∞ algebra associated to L q and denoteby m = m g , J , q its canonical model. Due to Theorem 4.3 (a), ˇ m β (cid:54) = β = (cid:80) di = k i β i + α for some α ∈ H eff2 ( X ; Z ). By homological perturbation, we also know that m β (cid:54) = β is in theform β = (cid:80) di = k i β i + α .Since µ ( β i ) = c ( α ) ≥
0, and deg m ,β = − µ ( β ), it is enough to only consider those β in theform β = β i + α for some i and α ∈ H eff2 ( X ) with c ( α ) =
0. Note that β = β i + α is non-separablesatisfying that µ ( β i + α ) = µ ( β i ) = ∂ ( β i + α ) = ∂β i ≡ v i . By Theorem 3.4, we have a well-definednumber n β i + α . In addition, we can also apply Theorem 3.5 to deduce that the number n β i + α ≡ n β i , q + α is independent of q ∈ B . Also, it is well-known that n β i =
1. Put things together, and we see that W ∨ q = (cid:80) µ ( β ) = T E ( β ) Y ∂β n β = (cid:80) di = Y ∂β i (cid:80) α T E ( β i + α ) n β i + α agrees with (36). As for the Fano case, we only need to consider the case β = β i , and thus it isstraightforward to obtain (37). Finally, let q and q be two distinct points in P ◦ . By (28), we have (cid:96) i ( q ) − (cid:96) i ( q ) = (cid:104) v i , q − q (cid:105) . So it is routine to check W ∨ q is transformed to W ∨ q .For later use, we remark that the above proof already shows the following result: Proposition 4.5 If n β (cid:54) = , then there exists α ∈ H eff2 ( X ) and ≤ j ≤ d − such that β = β j + α .Moreover, we have n β j = . As before, let N and M be lattices of rank n , dual to each other, and we put N R = N ⊗ R and M R = M ⊗ R . We denote by (cid:104) , (cid:105) the pairing N ⊗ M → Z . Let Σ ⊂ N R be astrongly-convex smooth fan, and its primitive generators of rays are denoted by v , v , . . . , v d ∈ N .Further, we require that there exists some m ∈ M so that (cid:104) v i , m (cid:105) = i = , , . . . , d and (cid:104) v , m (cid:105) ≥ v ∈ | Σ | ∩ N − { } .In this case, the (noncompact) toric variety X = X Σ defined by the fan Σ is Calabi-Yau. We oftenidentify a ray with its primitive generator in N . Recall that each ray v i corresponds to an irreducibletoric divisor D i ⊂ X . Furthermore, every m ∈ M gives a character χ m : T N → C ∗ . It can be viewedas a meromorphic function on X , and then its principal divisor is given by(39) ( χ m ) = d (cid:88) i = (cid:104) v i , m (cid:105) D i Without loss of generality, we fix, once and for all, a Z -basis { e (cid:48) = m , e (cid:48) , . . . , e (cid:48) n − } of the lattice M and the dual basis of N : { e = v , e , . . . , e n − } Hang Yuan
Consider the codimension-one sublattice(40) ˜ N = { n ∈ N | (cid:104) n , m (cid:105) = } ≡ Z { e , . . . , e n − } and then its dual lattice ˜ M = Hom Z ( ˜ N , Z ) can be naturally identified with M / Z m ∼ = { m ∈ M |(cid:104) v , m (cid:105) = } ∼ = Z { e (cid:48) , . . . , e (cid:48) n − } . The projection maps are denoted by pr : M → ˜ M and pr : M R → ˜ M R ≡ ˜ M ⊗ R . In practice, we always use the following identification M R ∼ = ˜ M R ⊕ R m ∼ = R { e (cid:48) , . . . , e (cid:48) n − } ⊕ R e (cid:48) ∼ = R n − ⊕ R N R ∼ = ˜ N R ⊕ R v ∼ = R { e , . . . , e n − } ⊕ R e ∼ = R n − ⊕ R where the e (cid:48) or e corresponds to the last coordinate rather than the first one.Consider a polyhedral complex P of the convex hull of Σ (1) = { v , . . . , v d } in the hyperplane R n − ×{ } ≡ { v ∈ N R | (cid:104) v , m (cid:105) = } such that the set P (0) of vertices is precisely Σ (1). Hence, | Σ | = R ≥ · P ⊂ N R . Following [AAK16, § 3.1], we choose a set of constant real numbers { c , c , . . . , c d } which is adapted to P in the sense that there is a convex piecewise linear function ρ on the convexhull of Σ (1) whose maximal domains of linearity are exactly the cells of P . Now, we consider thetropical polynomial ϕ : ˜ M R ∼ = R n − → R defined by ϕ ( ξ ) = max {−(cid:104) v i , ξ (cid:105) − c k | k = , , . . . , d } It is a convex function, and its epigraph P = { ( η, ξ ) ∈ M R | η ≥ ϕ ( ξ ) } = { m ∈ M R | (cid:104) v i , m (cid:105) ≥ c i ∀ ≤ i ≤ d } is a convex polytope that defines the fan structure Σ . The projection map pr : M R → ˜ M R induces ahomeomorphism from ∂ P to ˜ M R ; see e.g. [CLL12, Proposition 4.6]. Moreover, the vectors v i ’s can beregarded as the inner normal vectors of P whose end points lie in the hyperplane R n − × { } . Define P I = { m ∈ P | (cid:104) v i , m (cid:105) = c i ∀ i ∈ I } for I ⊂ { , , . . . , d } , and then we have v i = ( −∇ ϕ, | pr( P i ) . Notice that by convex analysis, given p ∈ pr( P i ) and p (cid:48) ∈ pr( P (cid:96) ), we have(41) (cid:104) v (cid:96) − v i , p (cid:48) − p (cid:105) < Π ⊂ ˜ M R the tropical hypersurface defined by ϕ , i.e. the set of points wherethe maximum is achieved at least twice. For instance, when X = K P , the set Π consists of two points;see Figure 1. In general, we have Π = (cid:91) | I |≥ pr( P I )If for any 0 ≤ i ≤ d , we denote by H i the interior of pr( P i ), then we have(42) ˜ M R \ Π = d (cid:91) i = H i = : H Observe that Π is the dual cell complex of P , and the connected components H i of H = ˜ M R \ Π canbe naturally labeled by a vertex in P (0) ≡ σ (1) = { v , v , . . . , v d } , determined by the term of ϕ thatachieves the maximum. Recall that we denote by T N = N ⊗ U (1) ∼ = T n and T C N ∼ = ( C ∗ ) n the real andcomplex torus respectively. Choose a toric K¨ahler form ω associated to the polytope P , and we denote amily Floer theory of toric manifolds and wall-crossing phenomenon (a) The fan Σ for X . The set P (0) of vertices of thepolyhedral complex is the endpoints of v , v , v . (b) A polytope P which describes X . Figure 6: Example: X = K P (Both figures are taken from [CLL12])the moment map by µ X : X → P ⊂ M R Also, note that T ˜ N ∼ = T n − , and we denote the moment map for the T ˜ N -action on X by˜ µ X : X → ˜ M R Observe that ˜ µ X = pr ◦ µ X .Note that the projection pr : M R → ˜ M R restricts to a homeomorphism from ∂ P to ˜ M R , and itfurther restricts to a linear homeomorphism from the facet P i to the closure ¯ H i of the wall component H i . For any 0 ≤ i ≤ d , the irreducible toric divisor D i is also a toric manifold on which the ˜ µ X inducesa moment map µ D i : = ˜ µ X | D i : D i → pr( P i ) ≡ ¯ H i for a Hamiltonian T n − -action. Then, the irreducibletoric divisors in the toric manifold D i are of the form D i ∩ D k for some 0 ≤ k ≤ d . For clarity, weset I ( i ) = { k | D i ∩ D k (cid:54) = ∅ } . The fiber of µ D i over a point in the interior H i of ¯ H i is a Lagrangian T n − -orbit. Suppose L (cid:48) ∼ = T n − is such a Lagrangian fiber. Then, as in (31) and (34), the classes β ik ∈ H ( D i , L (cid:48) ) ∼ = π ( D i , L (cid:48) ) for k ∈ I ( i ) form a basis and satisfy that β ik · D (cid:96) = δ k (cid:96) for any (cid:96) ∈ I ( i ).Also, every β ik can be represented by a holomorphic disk u ik : ( D , ∂ D ) → ( D i , L (cid:48) ) so that(43) u ik · D k = u ik · D k (cid:48) = k (cid:48) (cid:54) = k , ≤ k (cid:48) ≤ d5.1.3 Gross’s fibration . The Gross’s fibration is a special Lagrangian fibration on a toric Calabi-Yauvariety. In this paper, we will mainly use the notations and conventions in [CLL12].
Definition 5.1
Define w ( x ) = χ m ( x ) to be the meromorphic function corresponding to m ∈ M . Fixa constant (cid:15) >
0, and we define ρ ( x ) = | w ( x ) − (cid:15) | − (cid:15) . Let B = ˜ M R × [ − (cid:15) , ∞ ). In this paper,the Gross’s fibration refers to: ˆ π : X → B defined by setting ˆ π ( x ) = ( ˜ µ X ( x ) , ρ ( x )). The discriminantlocus , namely the set of critical values, of ˆ π is given by Γ : = ∂ B ∪ ( Π × { } )(see e.g. [CLL12, Proposition 4.9]). Then, we set B = B − Γ and X = π − ( B ), and the restrictionof ˆ π over B yields a smooth Lagrangian torus fibration: π : X → B Abusing the terminology, we will also call π the Gross’s fibration.In Example 1.5, X = K P and the discriminant locus Γ is the union of ∂ B = R × {− (cid:15) } ∪ { r , r } (see Figure 1). Hang Yuan
Proposition 5.2 ( [Gro01a] or [CLL12, Proposition 4.3 & 4.7])
Let ζ i be the meromorphic functioncorresponding to e (cid:48) k ∈ M for ≤ k ≤ n − . Then d ζ ∧ d ζ ∧ · · · d ζ n − extends to a nowhere-zeroholomorphic n -form on X . Moreover, π : X → B is a special Lagrangian torus fibration with respectto the holomorphic volume form ( w − (cid:15) ) − d ζ ∧ d ζ ∧ · · · d ζ n − . Motivated by Proposition 5.2, we consider the following boundary divisor:(44) E : = π − ( ∂ B ) = { x ∈ X | w ( x ) = (cid:15) } We also consider the following subsets of B : B + = ˜ M R × (0 , + ∞ ) B − = ˜ M R × ( − (cid:15), H = (cid:0) ˜ M R \ Π (cid:1) × { } ∂ B = ˜ M R × {− (cid:15) } We will call H the wall , since only the Lagrangian fibers over a point in H can bound a nontrivialMaslov zero holomorphic disk by Lemma 5.7 below. Further, we introduce the following contractibledomains(45) U i : = B + ∪ H i ∪ B − for 1 ≤ i ≤ d . For simplicity, we often identify H i with H i × { } and H with H × { } . Then, weobserve that H = (cid:70) i H i B = B + ∪ H ∪ B − U i ∩ U j = B + (cid:116) B − The zero locus w − (0) = (cid:83) i D i of w is contained in the zero locus ρ − (0) = π − ( R n − × { } ) = π − ( H ) of ρ = | w − (cid:15) | − (cid:15) . It follows that the map ρ is zero on the irreducible toric divisor D i .Since µ X ( D i ) = P i ⊂ ∂ P and ˜ µ X ( D i ) = pr( P i ) = ¯ H i , we see that π ( D i ) = ( ˜ µ X , ρ )( D i ) is contained in¯ H i × { } . Now, we consider the contractible set U i introduced in (45) for a fixed 0 ≤ i ≤ d . Because U i ∩ ( ¯ H (cid:96) × { } ) = ∅ for (cid:96) (cid:54) = i , we have the following useful observation:(46) π − ( U i ) ∩ D (cid:96) = ∅ whenver (cid:96) (cid:54) = i Lemma 5.3 ( [Aur07, Lemma 3.1])
Let Y be a K¨ahler manifold of dimension n , and let σ be anowhere-zero meromorphic n -form on Y with pole divisor D . If L ⊂ Y \ D is a compact orientedspecial Lagrangian submanifold with respect to σ , then for any β ∈ π ( Y , L ) which can be representedby a holomorphic disk, we have µ ( β ) = β · D . Recall that we denote the fiber of π over q ∈ B by L q = π − ( q ). Corollary 5.4 If β ∈ π ( X , L q ) can be represented by a stable map, then we have µ ( β ) ≥ . Proof.
Since X is Calabi-Yau, we can discard sphere bubbles without affecting the Maslov index.Then we just need to apply Proposition 5.2 and Lemma 5.3. n − -action . In the toric case (§ 4), the Lagrangian fibers are just the orbits of Hamiltonian T n -actions. In contrast, the Lagrangian fibers of the Gross’s fibration has a weaker symmetry that amily Floer theory of toric manifolds and wall-crossing phenomenon comes from the action of sub-torus T ˜ N ∼ = T n − . First, the Gross fiber over q = ( q , q ) ∈ B ⊂ B = ˜ M R × [ − (cid:15) , ∞ )can be explicitly described as follows: L q = { x ∈ X | ˜ µ X ( x ) = q , | w ( x ) − (cid:15) | = (cid:15) + q } Since both ˜ µ X and w are T ˜ N -invariant, we see that the L q is also invariant under the T ˜ N -action. Further,we can decompose L q into a S -family of T ˜ N -orbits as follows:(47) L q = (cid:71) θ ∈ [0 , π ) S q ( θ )where(48) S q ( θ ) : = { x ∈ X | ˜ µ X ( x ) = q , w ( x ) = (cid:15) + e i θ (cid:112) (cid:15) + q } is an orbit of the T ˜ N -action. The following statement is clear from the definition. Proposition 5.5
Fix c ∈ C . The intersection L q ∩ w − ( c ) is either the empty set or some S q ( θ ) where c = (cid:15) + e i θ (cid:112) (cid:15) + q . In other words, the holomorphic function w : X → C restricts to a T n − fibration on L q over the circle { z ∈ C | | z − (cid:15) | = (cid:112) (cid:15) + q } . Corollary 5.6
When c (cid:54) = , since w − ( c ) is topologically ( C ∗ ) n − we have π ( w − ( c ) , L q ∩ w − ( c )) ∼ = π (( C ∗ ) n − , T n − ) = . In particular, there is no non-trivial holomorphic disk in w − ( c ) bounding L q ∩ w − ( c ) . When c =
0, the zero level set w − (0) is simply the union (cid:83) i D i of all irreducible toric divisors. Lemma 5.7
For q = ( q , q ) ∈ B , the Gross fiber L q = π − ( q ) bounds a non-constant Maslov indexzero holomorphic disk if and only if q = , i.e. q ∈ H . Moreover, the image of a Maslov index zeroholomorphic disk that bounds L q for q ∈ H is always contained in the divisor w − (0) = (cid:83) i D i . Proof.
This is basically proved in [CLL12, Lemma 4.27]; see also [Aur07, Lemma 5.4]. We reviewthe proof here. Suppose u : ( D , ∂ D ) → ( X , L q ) is a non-constant Maslov-zero holomorphic disk. ByLemma 5.3 and Proposition 5.2, we have u · E = E = w − ( (cid:15) ). So, the holomorphicfunction w ◦ u − (cid:15) is never zero. Notice also that | w ◦ u − (cid:15) | ≡ (cid:15) + q keeps constant on ∂ D . Byapplying the maximal principle to both | w ◦ u − (cid:15) | and | w ◦ u − (cid:15) | − , w ◦ u must be constant with value c in the circle | z − (cid:15) | = (cid:15) + q in C . If c (cid:54) =
0, then the u has to be constant due to Corollary 5.6.Hence, c = w ◦ u ≡
0, which implies q = u is completely contained in thezero level set w − (0). Conversely, if q =
0, then one can easily find a nontrivial holomorphic diskcontained in w − (0) as in § 5.1.2. On the one hand, let q = ( q , q ) ∈ B , and we are interested in thefollowing smooth Lagrangian isotopy(49) L ( t ) : = L q ( t ) : = { x ∈ X | ˜ µ X ( x ) = q ; | w ( x ) − t | = (cid:15) + q } t ∈ [0 , (cid:15) ]Note that L q ( (cid:15) ) = L q = π − ( q ). We further introduce the following notation(50) L q = L q (0) Hang Yuan and observe that L q is a Lagrangian T N -orbit. Concerning the decomposition (47) of L q , we alsointroduce the induced isotopy for each S q ( θ ) as follows: S q ( t , θ ) : = { x ∈ X | ¯ µ X ( x ) = q ; w ( x ) = t + e i θ (cid:112) (cid:15) + q } Note also that S q ( (cid:15), θ ) = S q ( θ ). Proposition 5.8
When q ∈ B + , all the Lagrangians L q ( t ) in the isotopy do not admit non-trivialMaslov-zero holomorphic disks. Moreover, when q ∈ B + , every L q ( t ) is contained in T C N ∼ = ( C ∗ ) n . Proof.
The first half is proved in [CLL12, Lemma 4.29]. Indeed, we just need to apply the argumentin Lemma 5.7 to another Gross’s fibration (Definition 5.1) with (cid:15) (cid:48) = t , ρ (cid:48) = | w − t | − t , and q (cid:48) = q + (cid:15) − t . For the second statement, note that T C N is the complement set of w − (0) = (cid:83) i D i in X . Therefore, it suffices to show every L q ( t ) cannot intersect any toric divisor D i . Otherwise, if x ∈ L q ( t ) ∩ D i , then since w vanishes on D i and q >
0, we have (cid:15) ≥ t = | w ( x ) − t | = (cid:15) + q > (cid:15) ,a contradiction.On the other hand, we are also interested in the Lagrangian isotopies among the fibers of π . Givena path σ : [0 , → B , we denote the Lagrangian isotopy among the fibers over σ by L σ , which givesrise to an isomorphism P σ : π ( X , L σ (0) ) → π ( X , L σ (1) )on the second relative homotopy groups and also an isomorphism P σ : π ( L σ (0) ) → π ( L σ (1) )on the fundamental groups such that the following diagram commutes:(51) π ( X , L σ (0) ) P σ (cid:47) (cid:47) ∂ (cid:15) (cid:15) π ( X , L σ (1) ) ∂ (cid:15) (cid:15) π ( L σ (0) ) P σ (cid:47) (cid:47) π ( L σ (1) ) π ( T C N ) and π ( X , T C N ) . Note that T C N ⊂ X is an open dense complex torus. Asbefore, we denote by β i the class of a small closed disk transversal to the toric divisor D i = µ − X ( T i ).Then, we have(52) π ( X , T C N ) = Z (cid:104) β , β , . . . , β d (cid:105) It is well-known that π ( T C N ) is naturally isomorphic to N in that every v ∈ N determines a one-parameter subgroup of T C N . For clarity, we always stick to the following specific coordinate. For thepreviously chosen bases e i ∈ N and e (cid:48) i ∈ M , we have an isomorphism(53) π ( T C N ) ∼ = Z n by declaring the class [ γ ] for a loop γ ⊂ T C N corresponds to the n -tuple(54) (cid:0) deg( χ e (cid:48) ◦ γ ) , . . . , deg( χ e (cid:48) n − ◦ γ ) , deg( w ◦ γ ) (cid:1) ∈ Z n amily Floer theory of toric manifolds and wall-crossing phenomenon Recall that w = χ e (cid:48) . A useful observation is that(55) deg( w ◦ γ ) = (cid:40) deg( w ◦ γ − (cid:15) ) when γ ⊂ π − ( B + )0 when γ ⊂ π − ( B − ∪ ∂ B )Since | w − (cid:15) | > (cid:15) holds over B + , the winding number (degree) of w ◦ γ : S → C ∗ is the same as thatof w ◦ γ − (cid:15) , which explains the first half of (55) above. Similarly, as | w − (cid:15) | < (cid:15) over B − ∪ ∂ B , weknow the loop w ◦ γ in C ∗ is contractible.Next, we consider the boundary homomorphism(56) ∂ : π ( X , T C N ) ≡ Z d + → π ( T C N ) ≡ Z n Due to (39), the principal divisor ( χ e (cid:48) k ) = (cid:80) di = v ki D i for any k , where for simplicity we write(57) v ki : = (cid:104) v i , e (cid:48) k (cid:105) ∈ Z ≤ k ≤ n − v i = (cid:104) v i , m (cid:105) = ≤ i ≤ d . Notice that v i = v i e + · · · v n − i e n − ∈ N .Now, by definition, the class ∂β i only goes around the toric divisor D i once. So, we have deg( χ e (cid:48) k ◦ ∂β i ) = v ki and deg( w ◦ ∂β i ) = v i =
1. In summary, respecting the identification π ( T C N ) = Z n in the above(53), the following holds:(58) ∂β i = ( v i , v i , . . . , v n − i , ∈ Z n We can further take an identification Z n ∼ = N through ( m , . . . , m n − , m ) (cid:55)→ m e + m e + · · · + m n − e n − . Then, we exactly obtain that ∂β i = v i just like (33). For simplicity, we often abuse thenotation and write v i = ( v i , . . . , v n − i , v = e = (0 , . . . , , π ( L q ) for q ∈ B ± . When q ∈ B + or q ∈ B − , we have an inclusion map(59) j q : L q (cid:44) −→ T C N ⊂ X It induces a group homomorphism j q ∗ : π ( L q ) → π ( T C N ) ≡ Z n where we use the identification (53).Therefore, the j q ∗ is also defined by the formula (54), that is, the k -th coordinate of j q ∗ [ γ ] is deg( χ e (cid:48) k ◦ γ )for 1 ≤ k ≤ n − n -th coordinate is deg( w ◦ γ ). Besides, regarding (47) and (48), the inclusionmap s θ q : S q ( θ ) (cid:44) −→ L q of the sub-torus induces a group embedding(60) s θ q ∗ : π ( S q ( θ )) (cid:44) −→ π ( L q ) Lemma 5.9
For any q ∈ B + or q ∈ B − , the image of the composition π ( S q ( θ )) s θ q ∗ (cid:44) −→ π ( L q ) j q ∗ −→ π ( T C N ) ≡ Z n is im( j q ∗ ◦ s θ q ∗ ) = Z n − × { } . Proof.
Take a set { γ , . . . , γ n − } of loops in S q ( θ ). Then, they give rise to a basis of the imagegroup im s θ q ∗ ⊂ π ( L q ). Note also that changing the angle θ does not affect the homotopy class[ γ i ] ∈ π ( L q ). Recall that ˜ N = Z { e , . . . , e n − } , and we may assume that for each 1 ≤ i ≤ n −
1, theloop γ i is the S -orbit of the action of S · e i ⊂ T ˜ N ≡ T n − . Now, take an arbitrary loop γ in S q ( θ ).Then, by (48), we know w ◦ γ is constant and so deg w ◦ γ =
0. By the formula (54), we concludeim( j q ∗ ◦ s θ q ∗ ) = Z n − × { } . Hang Yuan
Proposition 5.10
When q ∈ B + , we have an isomorphism j q ∗ : π ( L q ) → Z n . When q ∈ B − , the j q ∗ is an embedding such that im j q ∗ = Z n − × { } ; in particular, the rank of the kernel of j q ∗ is one.Moreover, given a path σ in B + (resp. in B − ) between q (cid:48) = σ (0) and q (cid:48)(cid:48) = σ (1) , the isomorphism P σ : π ( L q (cid:48) ) → π ( L q (cid:48)(cid:48) ) induced by the Lagrangian isotopy L σ is compatible with the j q (cid:48) ∗ and j q (cid:48)(cid:48) ∗ in the sense that the following diagram commutes: π ( L q (cid:48) ) j q (cid:48)∗ (cid:47) (cid:47) P σ (cid:15) (cid:15) Z n π ( L q (cid:48)(cid:48) ) j q (cid:48)(cid:48)∗ (cid:47) (cid:47) Z n Proof.
By Lemma 5.9, it suffices to understand the degree of a loop γ (cid:48) ⊂ L q that meets every sub-torus S q ( θ ) exactly once in order to completely describe the j q ∗ . Indeed, by (48), it is easy to see that w ◦ γ (cid:48) forms the circle | z − (cid:15) | = (cid:112) (cid:15) + q in C ∗ . So, we know deg( w ◦ γ (cid:48) ) is equal to ± q > q <
0. Moreover, note that the maps χ e (cid:48) k and w have no zeros or poles in π − ( B + ) or π − ( B − ). So, the deg( χ e (cid:48) k ◦ γ (cid:48) ) and deg( w ◦ γ (cid:48) ) keep unchanged along the Lagrangian isotopy. Corollary 5.11
For appropriate coordinates T C N ∼ = ( C ∗ ) n , the fiber L q for q ∈ B + is homotopyequivalent to the standard torus T n in ( C ∗ ) n . Not only the previous inclusion map j q : L q → T C N but also thefollowing various inclusion maps of (pointed) space or space pairs are of our interest:(61) λ q : ( T C N , L q ) → ( X , L q ) (cid:96) q : ( X , L q ) → ( X , T C N ) k q : T C N → ( T C N , L q )Similar to ∂ : π ( X , T C N ) → π ( T C N ) in (56), we also have the following boundary homomorphisms ∂ : π ( X , L r ) → π ( L r ) ∂ : π ( T C N , L r ) → π ( L r ) ∂ : π ( X , T C N ) → π ( T C N )These boundary maps satisfy several obvious commutative diagrams.Regarding the pair ( T C N , L q ), there is a long exact sequence(62) 0 (cid:47) (cid:47) π ( T C N , L q ) ∂ (cid:47) (cid:47) π ( L q ) j q ∗ (cid:47) (cid:47) Z n k q ∗ (cid:47) (cid:47) π ( T C N , L q ) (cid:47) (cid:47) π ( T C N ) = π ( L q ) =
0, and we also have used the identification π ( T C N ) ∼ = Z n in (53). Regarding the triple L q ⊂ T C N ⊂ X , we also have the following long exactsequence:(63) 0 (cid:47) (cid:47) π ( T C N , L q ) λ q ∗ (cid:47) (cid:47) π ( X , L q ) (cid:96) q ∗ (cid:47) (cid:47) Z (cid:104) β , . . . , β d (cid:105) k q ∗ ◦ ∂ (cid:47) (cid:47) π ( T C N , L q )where we have used π ( X , T C N ) = Z (cid:104) β , . . . , β d (cid:105) in (52). In reality, by [May99, p.86], the leftmostarrow is at first given by the composition π ( X , T C N ) ∂ −→ π ( T C N ) k q ∗ −−→ π ( T C N , L q ) and then we can usethe fact π ( T C N ) = amily Floer theory of toric manifolds and wall-crossing phenomenon Proposition 5.12
When q ∈ B + , the induced map (cid:96) q ∗ : π ( X , L q ) → π ( X , T C N ) ≡ Z (cid:104) β , . . . , β d (cid:105) isan isomorphism. Proof.
By Corollary 5.11, we first know π k ( T C N , L q ) ∼ = π k (( C ∗ ) n , T n ) = k = , q ∈ B + .Then, using the long exact sequence (63) completes the proof.Abusing the notations, when q ∈ B + , we set(64) β i : = β i , q : = ( (cid:96) q ∗ ) − ( β i )Thus, regarding the isomorphism j q ∗ , it follows from (58) that(65) j q ∗ ( ∂β i , q ) = ∂(cid:96) q ∗ ( β i , q ) = ∂β i = ( v i , v i , . . . , v n − i , q ∈ B − . Definition-Proposition 5.13
When q ∈ B − , there exists a generator, denoted by ˆ β = ˆ β q , in therank-one group π ( T C N , L q ) such that ˆ β · E = Proof.
Denote D (cid:48) = { z ∈ C | | z − (cid:15) | ≤ (cid:112) (cid:15) + q } , and the holomorphic function w restricts to a T n − -fibration on L q over the circle ∂ D (cid:48) (Proposition 5.5). Then, we take a section γ : ∂ D (cid:48) → L q of w | L q over ∂ D (cid:48) which by definition intersects every S q ( θ ) exactly once. By Lemma 5.9 and Proposition5.10, we have j q ∗ ( γ ) =
0, that is, γ is contractible in T C N . Hence, we can fill it into a disk s : D (cid:48) → T C N such that ∂ s = γ . By deforming the disk s within T C N = ( C ∗ ) n , we may assume w ◦ s = id. Since E = w − ( (cid:15) ) and (cid:15) ∈ D (cid:48) , the topological intersection number [ s ] · E is either + −
1. Finally, wedefine ˆ β to be either [ s ] or − [ s ] such that ˆ β · E = π ( L q ) over U i . We consider the sub-lattice˜ M ( i ) : = { m ∈ M | (cid:104) v i , m (cid:105) = } The motivation is that if m ∈ ˜ M ( i ) , then by (46), the principle divisor ( χ m ) = (cid:80) (cid:96) (cid:54) = i (cid:104) v (cid:96) , m (cid:105) D (cid:96) hasno zeros or poles in π − ( U i ), obtaining a non-vanishing holomorphic function χ m : π − ( U i ) → C ∗ .Clearly, there is an isomorphism ˜ M (0) → ˜ M ( i ) defined by m (cid:55)→ m − (cid:104) v i , m (cid:105) e (cid:48) so that the Z -basis { e (cid:48) , . . . , e (cid:48) n − } of ˜ M ≡ ˜ M (0) induces a natural Z -basis of ¯ M ( i ) given by f ( i ) k : = e (cid:48) k − v ki · e (cid:48) for k = , . . . , n −
1. We denote the corresponding meromorphic functions by(66) ζ ( i ) k : = χ f ( i ) k = χ e (cid:48) k · w − v ki With the addition of w − (cid:15) , we find out n non-varnishing holomorphic functions on π − ( U i ). Accord-ingly, we obtain the following coordinate map(67) h ( i ) = h ( i ) q : π ( L q ) → Z n for q ∈ U i , defined as follows: the image of [ γ ] for a loop γ ⊂ L q is given by the following n -tuple:(68) (cid:0) deg( ζ ( i )1 ◦ γ ) , . . . , deg( ζ ( i ) n − ◦ γ ) , deg( w ◦ γ − (cid:15) ) (cid:1) ∈ Z n Compared to the previous coordinate (54), one difference is that the base point q is allowed to be in thelarger domain U i here. If q ∈ B + or q ∈ B − , applying (66) infers that(69) deg( ζ ( i ) k ◦ γ ) = deg( χ e (cid:48) k ◦ γ ) − v ki · deg( w ◦ γ )which will be used to study the monodromy of π ( L q ). Similar to Proposition 5.10, we have: Hang Yuan
Proposition 5.14
The map h ( i ) = h ( i ) q is invariant under a Lagrangian isotopy among fibers over U i .In other words, given a path σ in U i between q (cid:48) = σ (0) and q (cid:48)(cid:48) = σ (1) , the induced isomorphism P σ makes the following diagram commutes: π ( L q (cid:48) ) h ( i ) q (cid:48) (cid:47) (cid:47) P σ (cid:15) (cid:15) Z n π ( L q (cid:48)(cid:48) ) h ( i ) q (cid:48)(cid:48) (cid:47) (cid:47) Z n Proof.
Just notice that the zeros and poles of either ζ ( i ) k or w − (cid:15) are all contained in the complement of π − ( U i ). Hence, moving a loop γ along such a Lagrangian isotopy does not change the degrees. π ( L q ) . To study the monodromy, we want to compare various coordinates of π ( L q ). Assume q ∈ B + first. By (57, 55, 69), the group homomorphism(70) A ( i ) : = h ( i ) q ◦ ( j q ∗ ) − : Z n → Z n can be described by a matrix as follows:(71) A ( i ) = · · · · · · − v i · · · · · · − v i . . . ... ...1 0 − v n − i − v n − i n × n Explicitly, this means the homomorphism A ( i ) sends the tuple ( m , . . . , m n ) ∈ Z n to ( m − v i m n , . . . , m n − − v n − i m n , m n ) ∈ Z n . In particular, we observe that A ( i ) ( Z n − × { } ) = Z n − × { } . Note also that thematrix is independent of the choice of the base point q ∈ B + .Next, assume q ∈ B − . Although the j q ∗ is no longer an isomorphism, the h ( i ) q is still an isomorphism.If γ is a loop in L q , then using (69) and (55) yields that deg( ζ ( j ) k ◦ γ ) = deg( ζ ( i ) k ◦ γ ) + ( v ki − v kj ) deg( w ◦ γ ) = deg( ζ ( i ) k ◦ γ ) for all i , j . Hence, we have h ( i ) q = h ( j ) q for q ∈ B − , and so h ( j ) q ◦ ( h ( i ) q ) − ≡ id : Z n → Z n π ( X , L q ) . For clarity, we use q + and q − to represent some base points in B + and B − respectively. Also, we often abbreviate L + = L q + and L − = L q − .Given a path σ in the base B between q + and q − , recall that we denote the Lagrangian isotopyamong the fibers over σ by L σ which gives rise to an isomorphism P σ : π ( X , L − ) → π ( X , L + ). Bymonodromy issue we mean the isomorphism depends on the homotopy class of the path σ . Note thatthe homotopy class of the path σ is determined by the wall component that the path passes through. Lemma 5.15
For ≤ i ≤ d , let σ i be a path in B that passes through the wall component H i . Then P σ i ( ˆ β ) = β i Proof.
Assume P σ i ( ˆ β ) = k β + · · · + k d β d , where k i ∈ Z . By (46), π − ( U i ) ∩ D (cid:96) = ∅ for any (cid:96) (cid:54) = i . Therefore, since the Lagrangian isotopy L σ i is totally contained in π − ( U i ), the intersection amily Floer theory of toric manifolds and wall-crossing phenomenon number of a disk class with D (cid:96) is preserved along L σ i . In particular, we have ˆ β · D (cid:96) = P σ i ( ˆ β ) · D (cid:96) for any (cid:96) (cid:54) = i . Clearly, ˆ β · D (cid:96) =
0. As β k · D (cid:96) = δ k (cid:96) , we conclude that k (cid:96) = (cid:96) (cid:54) = i . Hence, P σ i ( ˆ β ) = k i β i for some k i ∈ Z . Moreover, the Lagrangian isotopy preserves the Maslov index, whichimplies that 2 k i = µ ( k i β i ) = µ ( ˆ β ) =
2. Thus, k i =
1. The proof is now complete.Recall that for any path σ between q + and q − , we use the same notation P σ to denote the inducedisomorphism π ( L − ) → π ( L + ) on the fundamental groups. By (51), we have P σ ◦ ∂ = ∂ ◦ P σ .Because the path σ i is contained in the domain U i , we can use the coordinate maps h ( i ) q + and h ( i ) q − . ByProposition 5.14, we may denote both of them by h ( i ) for simplicity, and we have h ( i ) ◦ P σ i = h ( i ) .Then, using Lemma 5.15 deduces that h ( i ) ∂ ˆ β = h ( i ) ∂β i . First of all, applying Theorem 1.2 to the above pair ( X , π ) produces a mirror triple( X ∨ , W ∨ , π ∨ ). By Lemma 5.7, the Gross’s fibration π restricted over B + or B − does not encounterany Maslov-zero disk. Then, by Corollary 2.3, we know X ∨± : = ( π ∨ ) − ( B ± ) ∼ = trop − ( B ± )The restrictions of W ∨ over the two chambers X ∨± have quite different expressions. Therefore, the wayhow transition maps glue various local pieces of W ∨ will tell us how open GW invariants evolve alonga Lagrangian isotopy along the fibers. The main purpose of this section is to make an application ofcomputing open GW invariants from this viewpoint. ∨ . Given a Maslov index two class β ∈ π ( X , L q ), we recall that its openGW invariant n β = n β, J is given by the counts of holomorphic disks in the class β (Definition 3.3).For q ∈ B + , we will often identify β with (cid:96) q ∗ β due to Proposition 5.12. Proposition 5.16
Suppose q ∈ B + . If n β (cid:54) = , then there exists α ∈ H eff2 ( X ) and ≤ j ≤ d − suchthat β = β j + α . Moreover, we have n β j = . Proof.
This is basically identical to [CLL12, Proposition 4.30]. Consider the Lagrangian isotopy L q ( · )in (49), and we denote by P the induced isomorphism π ( X , L q ) → π ( X , L q ). By Proposition 5.8, itis legitimate to apply Theorem 3.5 to the isotopy, which implies that n β = n P ( β ) . In conclusion, thestatement reduces to the previous Proposition 4.5 in the toric case.The case q ∈ B − is more delicate. We need to first study the behaviors of holomorphic curves. Lemma 5.17
Suppose q ∈ B − . Any non-trivial holomorphic disk u : ( D , ∂ D ) → ( X , L q ) is containedin the set S − : = π − ( B − ∪ ∂ B ) = { x ∈ X | | w ( x ) − (cid:15) | < (cid:15) } Moreover, there exists some k > so that [ u ] = k ˆ β . In particular, its Maslov index is µ ( u ) = k . Proof.
Write q = ( q , q ) and D (cid:48) = { z ∈ C | | z − (cid:15) | ≤ (cid:112) (cid:15) + q } as before. We observe that w ◦ u : ( D , ∂ D ) → ( D (cid:48) , ∂ D (cid:48) ) is a holomorphic map such that w ◦ u ( ∂ D ) ⊂ ∂ D (cid:48) . By maximum principle, | w ◦ u − (cid:15) | ≤ (cid:112) (cid:15) + q < (cid:15) on D . Thus, the image of u is always contained in S − , which impliesthat u cannot meet any toric divisor D i , as w − (0) = (cid:83) i D i . It follows that the class [ u ] lives in thesubgroup π ( T C N , L q ) ∼ = Z · ˆ β (Definition-Proposition 5.13), say [ u ] = k ˆ β . It suffices to show k > Hang Yuan
Since the u is non-trivial, the w ◦ u cannot be a constant function due to Corollary 5.6. So, we have w ◦ u ( D ) = D (cid:48) . Accordingly, the u has at least one intersection point with the divisor E = w − ( (cid:15) ).Moreover, since u is holomorphic, any intersection is positive. So, 0 < [ u ] · E = k ˆ β · E = k . Finally,due to Lemma 5.3, we know µ ( u ) = k . Lemma 5.18
Any non-trivial holomorphic sphere h : P → X is contained in w − (0) = (cid:83) i D i . Inparticular, the sphere h does not intersect with S − . Proof.
The holomorphic map w ◦ h : P → C must be constant, say w ◦ h = c . In other words, thesphere h is contained in the level set w − ( c ). If c (cid:54) =
0, then the level set w − ( c ) is topologically( C ∗ ) n − . But, we know π (( C ∗ ) n − ) =
0, which implies h must be trivial. So, we have c =
0, andthus the image of h is contained in w − (0) = (cid:83) i D i . Proposition 5.19 If q ∈ B − and β ∈ π ( X , L q ) with µ ( β ) = , then n β (cid:54) = only if β = ˆ β Proof.
Suppose u is a holomorphic stable disk in the class β that contributes to n β . Since X is Calabi-Yau, discarding sphere bubbles in u does not affect the Maslov index. So, by Lemma 5.17, there is atmost one non-trivial disk component, denoted by u : ( D , ∂ D ) → ( X , L q ), such that [ u ] = ˆ β , and all thesphere bubbles have total Maslov index zero. By Lemma 5.17 again, the image of u is contained in S − . However, by Lemma 5.18, any non-trivial holomorphic sphere does not meet S − . Thereby, the u cannot hold any sphere bubbles, and β = [ u ] = [ u ] = ˆ β . The proof is now complete.Actually, by [CLL12, Proposition 4.32], we can further show n ˆ β = X = C n and finding the holomorphic disk in the class ˆ β explicitly. However, as discussed in theintroduction, we are going to provide a new proof of n ˆ β = n ˆ β .Now, we denote by W ∨± the restriction of W ∨ on the chamber X ∨± . On the one hand, we have W ∨ + = d (cid:88) j = T E ( β j ) Y ∂β j (cid:16) n β j + (cid:88) (cid:54) = α ∈ H eff2 ( X ) T E ( α ) n β j + α (cid:17) using Proposition 5.16, and we particularly single out the part without sphere bubbles. Temporarily, westill keep writing n β j , although we already know n β j =
1. On the other hand, by Proposition 5.19, wehave W ∨− = T E ( ˆ β ) Y ∂ ˆ β n ˆ β Notice that the various topological classes β j ’s in the expression of W ∨ + actually depend on a base point q + ∈ B + , and the W ∨− has a similar issue. But, this point is not essential thanks to the Fukaya’s trick. From now on, we fix 0 ≤ i ≤ d . As in Lemma 5.15, we take the path σ = σ i in B that passes through the component H i so that σ (0) = q − ∈ B − and σ (1) = q + ∈ B + . Recallthat we denote by L σ the Lagrangian isotopy among the fibers over σ , and we denote by P σ theinduced isomorphism on either π ( L q ) or π ( X , L q ). May assume the path σ is sufficiently small. So,in particular, the two base points q ± are close to the wall component H i ⊂ H . By Fukaya’s trick, wecan adopt the following simplification of notations.First, the two Lagrangian fibers L − and L + can be identified with each other by a small diffeo-morphism. For simplicity, we will not always specify the isomorphisms P σ : π ( L − ) ∼ = π ( L + ) and amily Floer theory of toric manifolds and wall-crossing phenomenon π ( X , L − ) ∼ = π ( X , L + ). For example, we know P σ ( ˆ β ) = β i by Lemma 5.15, but we would ratherwrite:(72) ˆ β = β i Observe that all the β i for 0 ≤ i ≤ d form a basis of π ( X , L + ), producing a basis of π ( X , L − ) as wellthrough P σ . In addition, we consider a new basis of π ( X , L − ) ∼ = Z d + defined as follows:(73) ˆ β, γ (cid:96) : = β (cid:96) − ˆ β (0 ≤ (cid:96) ≤ d , (cid:96) (cid:54) = i )By Lemma 5.7, there is a unique Lagrangian, denoted by L , in the isotopy L σ that admits a Maslovindex zero holomorphic disk. We can further make the identifications π ( L − ) ∼ = π ( L ) ∼ = π ( L + ) ∼ = Z n and π ( X , L − ) ∼ = π ( X , L ) ∼ = π ( X , L + ) ∼ = Z d + .In the remaining part of paper, we will always adopt the above simplifications. It is clear that the topological class γ (cid:96) has Maslov index zero. Thesubgroup π (cid:48) ( X , L ) in π ( X , L ) consisting of Maslov index zero classes has codimension one and isspanned by all γ (cid:96) ( (cid:96) (cid:54) = i ). In particular, we have π (cid:48) ( X , L ) ∼ = Z d by associating a class the tuple kkk : = ( k , k , . . . , k i − , k i + , . . . , k d ) ∈ Z d to the class γ kkk : = (cid:80) (cid:96) (cid:54) = i , ≤ (cid:96) ≤ d k (cid:96) γ (cid:96) . For simplicity, we write kkk ≥ kkk ∈ Z d ≥ , i.e. all k (cid:96) ≥
0, and we write kkk > kkk ≥ kkk (cid:54) = Proposition 5.20
If the class γ kkk is represented by a non-trivial Maslov index zero holomorphic stabledisk, then kkk > and E ( γ kkk ) = ω ∩ γ > . Proof.
For m (cid:54) = i , we consider the intersection number γ kkk · D m = (cid:80) (cid:96) (cid:54) = i k (cid:96) ( β (cid:96) · D m − ˆ β · D m ) = k m .If there is a stable disk representing the class γ kkk , then the intersection number is non-negative, thus, k m ≥
0. Finally, if all k m =
0, then the stable disk must be trivial.Motivated by Proposition 5.20, we should conjecture that the converse is also true. Equivalently,this means every γ (cid:96) can be represented by a nontrivial Maslov index zero holomorphic disk. A directevidence is given in § 5.1.2 if D (cid:96) ∩ D i (cid:54) = ∅ , but the case when D (cid:96) ∩ D i = ∅ is not clear. Nevertheless,the following weaker statement is enough for our application: Lemma 5.21
In the above situation, we have E ( β (cid:96) ) > E ( ˆ β ) , namely E ( γ (cid:96) ) > , for any (cid:96) (cid:54) = i . Proof.
To begin with, we claim that the lemma holds if D (cid:96) ∩ D i (cid:54) = ∅ . In fact, we notice that theintersection L (cid:48) : = L ∩ D i = L ∩ w − (0) is exactly an orbit of the action of T ˜ N ∼ = T n − by Proposition5.5. So, as in § 5.1.2, we can find a holomorphic disk u contained in D i so that u · D (cid:96) = u · D (cid:96) (cid:48) = (cid:96) (cid:48) (cid:54) = i , (cid:96) . Also, since u · E =
0, one can check we much have [ u ] = γ (cid:96) = β (cid:96) − ˆ β . Since E ( u ) >
0, we conclude that E ( β (cid:96) ) > E ( ˆ β ).In general, we first note that the purpose is a strict inequality. Thus, by Fukaya’s trick, we maychoose to work with the Lagrangian fiber L = L + over a base point q = q + slightly above the wall.Specifically, we set q = ( q , δ ) ∈ ˜ M R × R for q ∈ H i and a sufficiently small positive real number δ >
0. Choose q (cid:48) ∈ H (cid:96) , and draw a straight line segment L in ˜ M R ∼ = R n − between q and q (cid:48) .Perturbing q (cid:48) slightly if necessary, we may assume L ∩ Π = ∅ . Successively, we denote all thecomponents of H = ∪ k H k that meets L by H (cid:96) = i , H (cid:96) , H (cid:96) , . . . , H (cid:96) m − and H (cid:96) m = (cid:96) . Then, we consider β (cid:96), q − ˆ β q = m − (cid:88) k = ( β (cid:96) k + , q − β (cid:96) k , q ) Hang Yuan where for clarity we specify the base point q as in (64). It suffices to prove E ( β (cid:96) k + , q ) > E ( β (cid:96) k , q ) for any0 ≤ k ≤ m −
1. To show this, we pick up a point q k in H (cid:96) k ∩ L and set q k = ( q k , δ ) ∈ B + . Notice that H (cid:96) k and H (cid:96) k + are adjacent along the line segment L , thus, D (cid:96) k ∩ D (cid:96) k + (cid:54) = ∅ . Then, using the claim atthe start of proof, we conclude that E ( β (cid:96) k + , q k ) > E ( β (cid:96) k , q k ). Consider the line segment L + = ( L , δ ) in B + between q and q (cid:48) : = ( q (cid:48) , δ ), and then the point q k lies in L + . Also, consider the parametrization q ( t ) = ((1 − t ) q + tq (cid:48) , δ ) of the line segment L + . Now, it suffices to show that the function f : [0 , → R , t (cid:55)→ E ( β (cid:96) k + , q ( t ) ) − E ( β (cid:96) k , q ( t ) )is decreasing in t ∈ [0 , E ( β (cid:96) k + , q ) − E ( β (cid:96) k , q ) = f (0) > E ( β (cid:96) k + , q k ) − E ( β (cid:96) k , q k ) >
0. Inreality, due to the energy formula (13), we first have E ( β q ( t ) ) − E ( β q ( t ) ) = (cid:104) ∂β, q ( t ) − q ( t ) (cid:105) = ( t − t ) ·(cid:104) ∂β, ( q (cid:48) − q ) (cid:105) . Then, in our situation, it follows that f (cid:48) ( t ) = (cid:104) ∂β (cid:96) k + − ∂β (cid:96) k , q (cid:48) − q (cid:105) = (cid:104) v (cid:96) k + − v (cid:96) k , q (cid:48) − q (cid:105) .By (41), we finally deduce that f (cid:48) ( t ) <
0, and we complete the proof.
By applying the Fukaya’s trick as above, we may assume the ring isomor-phism φ underlying the transition map is in the following form:(74) φ : Λ [[ π ( L − )]] → Λ [[ π ( L + )]] , Y α (cid:55)→ Y α · exp (cid:68) α, (cid:88) γ (cid:54) = ,µ ( γ ) = C γ T E ( γ ) Y ∂γ (cid:69) where the C γ lives in H ( L − ) ∼ = H ( L + ) which is determined by counting stable disks in the class γ ∈ π ( X , L − ) ∼ = π ( X , L + ). Note that compared to (16), we omit the coefficient T (cid:104) α, ˜ q − q (cid:105) , as we haveapplied Fukaya’s trick at here. Because the transition map matches W ∨− and W ∨ + (or one can directlyutilize the wall crossing formula (17)), we conclude that(75) φ ( W ∨− ) = W ∨ + Proof of Theorem 1.6.
Applying (74) to the left side of the identity (75) deduces that φ ( W ∨− ) = φ (cid:16) T E ( ˆ β ) Y ∂ ˆ β n ˆ β (cid:17) = T E ( ˆ β ) Y ∂ ˆ β n ˆ β exp (cid:68) ∂ ˆ β, (cid:88) µ ( γ ) = C γ T E ( γ ) Y ∂γ (cid:69) We set kkk = ( k , . . . , k i − , k i + , . . . , k d ) ∈ Z d . If C kkk : = C γ kkk (cid:54) =
0, then by Proposition 5.20, we musthave kkk > E kkk : = E ( γ kkk ) >
0. Now, the identity (75) deduces that T E ( ˆ β ) Y ∂ ˆ β n ˆ β exp (cid:32)(cid:88) kkk > (cid:104) ∂ ˆ β, C kkk (cid:105) T E kkk Y ∂γ kkk (cid:33) = d (cid:88) j = T E ( β j ) Y ∂β j (cid:0) n β j + (cid:88) (cid:54) = α ∈ H eff2 ( X ) T E ( α ) n β j + α (cid:1) Define F = − (cid:80) (cid:104) ∂ ˆ β, C kkk (cid:105) T E kkk Y ∂γ kkk , and by (72) the above identity is equivalent to n ˆ β = exp ( F ) · n β i + (cid:88) α (cid:54) = T E ( α ) n β i + α + (cid:88) (cid:96) (cid:54) = i T E ( γ (cid:96) ) Y ∂γ (cid:96) (cid:88) α T E ( α ) n β (cid:96) + α viewed as an identity in Λ [[ π ( L − )]] ∼ = Λ [[ π ( L + )]] ∼ = Λ [[ Y ± , . . . , Y ± n ]]. By Proposition 5.20, thecoefficients of F are all in Λ + . Also, we have E ( γ (cid:96) ) > E ( α ) > α ∈ H eff2 ( X ). Finally, by extracting the energy-zero parts of both sides, we obtain n ˆ β = n β i . References [AAK16] Mohammed Abouzaid, Denis Auroux, and Ludmil Katzarkov. Lagrangian fibrations on blowupsof toric varieties and mirror symmetry for hypersurfaces.
Publications math´ematiques de l’IH ´ES , amily Floer theory of toric manifolds and wall-crossing phenomenon Journal of the European MathematicalSociety , 19(7):2139–2217, 2017.[Aud12] Michele Audin.
The topology of torus actions on symplectic manifolds , volume 93. Birkh¨auser, 2012.[Aur07] Denis Auroux. Mirror symmetry and T-duality in the complement of an anticanonical divisor.
Journalof G¨okova Geometry Topology , 1:51–91, 2007.[CK00] David A Cox and Sheldon Katz.
Mirror symmetry and algebraic geometry . Mathematical surveys andMonographs, 2000.[CLL12] Kwokwai Chan, Siu-Cheong Lau, and Naichung Conan Leung. SYZ mirror symmetry for toricCalabi-Yau manifolds.
Journal of Differential Geometry , 90(2):177–250, 2012.[CO06] Cheol-Hyun Cho and Yong-Geun Oh. Floer cohomology and disc instantons of Lagrangian torus fibersin Fano toric manifolds.
Asian Journal of Mathematics , 10(4):773–814, 2006.[Del88] Thomas Delzant. Hamiltoniens p´eriodiques et images convexes de l’application moment.
Bulletin dela Soci´et´e Math´ematique de France , 116(3):315–339, 1988.[FOOO10a] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian Floer theory on compacttoric manifolds, I.
Duke Mathematical Journal , 151(1):23–175, 2010.[FOOO10b] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.
Lagrangian intersection Floertheory: anomaly and obstruction, Part I , volume 1. American Mathematical Soc., 2010.[FOOO10c] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.
Lagrangian intersection Floertheory: anomaly and obstruction, Part II , volume 2. American Mathematical Soc., 2010.[FOOO17] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Kuranishi structure, Pseudo-holomorphic curve, and Virtual fundamental chain: Part 2. arXiv preprint arXiv:1704.01848 , 2017.[Fuk10] Kenji Fukaya. Cyclic symmetry and adic convergence in Lagrangian Floer theory.
Kyoto Journal ofMathematics , 50(3):521–590, 2010.[Gro01a] Mark Gross. Examples of special lagrangian fibrations. In
Symplectic geometry and mirror symmetry ,pages 81–109. World Scientific, 2001.[Gro01b] Mark Gross. Topological mirror symmetry.
Inventiones mathematicae , 144(1):75–137, 2001.[GS14] Yoel Groman and Jake P Solomon. A reverse isoperimetric inequality for J-holomorphic curves.
Geometric and Functional Analysis , 24(5):1448–1515, 2014.[Gui94] Victor Guillemin. Kaehler structures on toric varieties.
Journal of differential geometry , 40(2):285–309,1994.[KS00] Maxim Kontsevich and Yan Soibelman. Homological mirror symmetry and torus fibrations. arXivpreprint math/0011041 , 2000.[KS06] Maxim Kontsevich and Yan Soibelman. Affine structures and non-archimedean analytic spaces. In
Theunity of mathematics , pages 321–385. Springer, 2006.[May99] J Peter May.
A concise course in algebraic topology . University of Chicago press, 1999.[MW74] Jerrold Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry.
Reports onmathematical physics , 5(1):121–130, 1974.[ST16] Jake P Solomon and Sara B Tukachinsky. Differential forms, Fukaya A ∞ algebras, and Gromov-Wittenaxioms. arXiv preprint arXiv:1608.01304 , 2016.[Yua20] Hang Yuan. Family Floer program and non-archimedean SYZ construction. arXiv preprint arXiv:2003.06106 , 2020., 2020.