Some Examples of Family Floer Mirror
SSome Examples of Family Floer Mirror
Man-Wai Cheung, Yu-Shen LinJanuary 19, 2021
Abstract
In this article, we give explicit calculations for the family Floer mirrors of some non-compact Calabi-Yau surfaces. We compare it with the mirror construction of Gross-Hacking-Keel for suitably chosen logCalabi-Yau pairs and the rank two cluster varieties of finite type. In particular, the analytifications ofthe later two give partial compactifications of the family Floer mirrors that we computed.
The Strominger-Yau-Zaslow (SYZ) conjecture predicts that Calabi-Yau manifolds have the structures ofspecial Lagrangian fibration and the mirror can be constructed via dual special Lagrangian fibrations. More-over, the metric receives the instanton corrections from holomorphic discs with boundaries on the specialLagrangian torus fibres. The conjecture not only gives a geometric way to construct the mirror, it alsogives the intuitive reasoning for mirror symmetry, for instance see [11, 30]. The SYZ philosophy becomes thehelpful tool of studying mirror symmetry and many of its implications are proved. However, the difficultyof the analysis involving the singular special Lagrangian fibration makes the progress toward the originalconjecture relatively slow (see [8, 9, 31] for the recent progress).To understand the instanton correction rigorously in the mathematical context, Fukaya [14] proposed howto understand the relation between the instanton correction from holomorphic curves/discs and the mirrorcomplex structure via the Floer theoretic approach. Kontsevcih-Soibelman [27] and Gross-Siebert [23] latersystematically formulated how the to construct the mirror in various settings from algebraic approaches.These approaches opened up an window to understand mirror symmetry intrinsically.In the algebro-geometric approach, Gross-Siebert first constructed an affine manifold with singularitiesfrom the toric degeneration. Then there is a systematic way of constructing the so-called scattering diagrams,which capture the information of the instanton corrections, on the the affine manifold. The data of thescattering diagrams encode how to glue the expected local models into the mirror Calabi-Yau manifolds. Onthe other hand, family Floer homology proposed by Fukaya [15] lays out the foundation to realize mirrorsymmetry via an intrinsic way from symplectic geometry point of view. Given a Lagrangian fibration, theFukaya’s trick introduced later in Section 4.1 provides pseudo-isotopies between the A ∞ structures of fibresafter compensation of symplectic flux. In particular, the pseudo-isotopies induce canonical isomorphisms ofthe corresponding Maurer-Cartan spaces. The family Floer mirror is then be the gluing of the Maurer-Cartanspaces via these isomorphisms. Not only the family Floer mirror are constructed [1, 39, 40, 41], Abouzaidproved the family Floer functor induces homological mirror symmetry [2,3]. It is natural to ask if the mirrorsconstructed via Gross-Siebert program and the family Floer homology approach coincide or not.The following is an expected dictionary connecting the two approaches:1 a r X i v : . [ m a t h . S G ] J a n amily Floer SYZ GHK mirror construction large complex structure limit toric degenerationbase of SYZ fibrationwith complex affine structure dual intersection complexof the toric degenerationloci of SYZ fibres bounding holomorphic discs rays in scattering diagramhomology of boundary of a holomorphic disc direction of the rayexp of generating functionof open Gromov-Witten invariantsof Maslov index zero wall functions attached to the raycoefficients of superpotential =open Gromov-Witten of Maslov index 2 discs coefficients of theta functions =counting of broken linesisomorphisms of Maurer-Cartan spacesinduced by pseudo isotopies wall crossing transformationLemma 4.1 in the article consistency of scattering diagramfamily Floer mirror GHK/GHKS mirrorTable 1: Dictionary between the symplectic and algebraic approaches of mirror construction.However, it is hard to have a good control of all possible discs in a Calabi-Yau manifold due to thewall-crossing phenomenon. Thus, it is generally hard to write down the family Floer mirror explicitly.In the examples family Floer mirror computed in the literature, there exists torus symmetries and onecan write down all the possible holomorphic discs explicitly. In particular, the loci of Lagrangian fibresbounding Maslov index zero discs to not intersect and thus exclude the presence of more complicated bubblingphenomenon.In this paper, we engineer some 2-dimensional examples that the family Floer mirrors are explicit andrealize most of the above dictionary step by step. We first prove that the complex affine structures of thebases of special Lagrangian fibrations coincide with the affine manifolds with singularities constructed inGross-Hacking-Keel [19] from some log Calabi-Yau surfaces. See the similar results in [29] for the case of P , general del Pezzo surfaces relative smooth anti-canoncial divisors [28] and rational elliptic surfaces [9]and the case of Fermat hypersurfaces [31]. When the 2-dimensional Calabi-Yau admits a special Lagrangianfibration, it is well-known that the special Lagrangian torus fibres bounding holomorphic discs supportsalong affine lines with respect to the complex affine coordinates. Using the Fukaya’s trick, the secondauthor identified a version of open Gromov-Witten invariants with tropical discs counting [35, 32], whichlays out a foundation to the connection between family Floer mirror and Gross-Siebert/Gross-Hacking-Keelmirror. The examples are engineer such that all the wall functions are polynomials. Therefore, there is noconvergence issue in the gluing procedure and the complication reduces to minimal. In particular, the familyFloer mirror has a model over complex numbers. On the other hand, one can compare it with the processof Gross-Hacking-Keel: we can construct a log Calabi-Yau pair ( Y, D ) such that the induced affine manifoldwith singularity coincides with the complex affine structure of the base of special Lagrangian fibration. Thenwe identify the loci of special Lagrangian fibres bounding holomorphic discs with the rays of the canonicalscattering diagram and the corresponding wall-crossing transformations in Gross-Hacking-Keel [19]. Thetechnical part is to prove that the family Floer mirror has a partial compactification to be the gluing ofrigid analytic tori. Comparing with the calculation of Gross-Hacking-Keel, we the know that the familyFloer mirror has a partial compactification to be the anaytification of the mirror from (
Y, D ) constructedin Gross-Hacking-Keel. The miror construction of Gross-Hacking-Keel is a family, which can be viewed asthe complexified K¨ahler moduli of Y . We further determine the distinguished point that correspond to thefamily Floer mirror. The following is a summary of Theorem 5.15, Theorem 6.6 and Theorem 7.4 Theorem 1.1.
The analytification of X -cluster variety of type A ( B and G ) or the Gross-Hacking-Keelmirror of suitable log Calabi-Yau pair ( Y, D ) is a partial compactification of the family Floer mirror of X II ( X III and X IV respectively). .1 Structure The structure of the paper is arranged as follows: In Section 2, we review the definition of cluster varietiesand the mirror construction in Gross-Hacking-Keel [19] and Gross-Hacking-Keel-Siebert [22]. In Section 3,we will formulate the surfaces that we are going to compute the family Floer mirror of those. They arecoming from the HyperK¨ahler rotation of the rational elliptic surfaces with singularities.In Section 4, we review the family Floer mirror construction and the relation between the open Gromov-Witten invariants. In Section 5, we will compute the family Floer mirror of a non-compact Calabi-Yau surface X II explicitly in full details. Then we compare it with the analytification of the A -cluster variety. We willalso compare it with the Gross-Hacking-Keel mirror for a del Pezzo surface of degree five. In particular, thefamily Floer mirror of X II can compactified to a del Pezzo surface of degree five via algebra structure ofthe theta functions. In Section 6 and Section 7, we will sketch the calculation for the family Floer mirror of X III and X IV , pointing out the differences from the case of X II . Acknowledgement
The author would like to thank Mark Gross and Shing-Tung Yau for the constant support and encouragement.The authors would also like to thank H¨ulya Arg¨uz, Dori Bejleri, Paul Hacking, Hansol Hong, Chi-Yun Hsu,Laura Friedrickson for helpful discussion. The first author is supported by NSF grant DMS-1854512. Thesecond author is supported by Simons Collaboration Grant
For log Calabi-Yau surfaces, Gross-Hacking-Keel [19] showed that the information from holomorphic discswith boundaries on SYZ fibres near the infinity boundary divisor are enough to construct the mirror. Heuris-tically, one scales the neighborhood of the infinity or equivalently collides all the singular fibres of the SYZfibration into one. As the SYZ fibres moved to infinity, the Maslov index zero holomorphic discs close up toholomorphic curves with exactly one intersection with the boundary divisor, known as the A -curves. The A -curves tropicalize to rays and the counting of A -curves will determine the wall functions of the canonicalscattering diagram. In this section, we will review the mirror construction of Gross-Hacking-Keel [19] andGross-Hacking-Keel-Siebert [22].Consider the pair ( Y, D ), where Y is a smooth projective rational surface, and D = D + · · · + D n is ananti-canonical cycle of rational curves. Then X := Y \ D is a log Calabi-Yau surface . The tropicalizationof ( Y, D ) would be a pair ( B GHK , Σ), where B GHK is homeomorphic to R and has the structure of integralaffine manifold with singularity at the origin, and Σ is a decomposition of B GHK into cones. The constructionof ( B GHK , Σ) starts by associating each node p i,i +1 := D i ∩ D i +1 with a rank two lattice M i,i +1 with basis v i , v i +1 and the cone σ i,i +1 ⊂ M i,i +1 ⊗ Z R generated by v i and v i +1 . Then σ i,i +1 are glued to σ i − ,i alongthe rays ρ i := R ≥ v i to obtain a piecewise-linear manifold B GHK and a decompositionΣ = { σ i,i +1 | i = 1 , . . . , n } ∪ { ρ i | i = 1 , . . . , n } ∪ { } ⊆ R . Define U i = Int( σ i − ,i ∪ σ i,i +1 ) . The integral affine structure on B GHK , = B GHK \ { } is defined by the charts ψ : U i → M R ,ψ i ( v i − ) = (1 , , ψ i ( v i ) = (0 , , ψ i ( v i +1 ) = ( − , − D i ) , with ψ i linear on σ i − ,i and σ i,i +1 . It may worth noting here that at the end of the gluing process, ρ n +1 may not agree with ρ . It would induce a nontrivial affine structure on B GHK , when we identify ρ n +1 with Note that X is denoted as U in [19, 22]. . We are going to demonstrate the affine structures explicitly in examples in this article. Note that if weconsider three successive rays ρ i − , ρ i , ρ i +1 , there is the relation ψ ( v i − ) + D i ψ ( v i ) + ψ ( v i +1 ) = 0 . (1)Consider a toric monoid P . A toric monoid P is a commutative monoid whose Grothendieck group P gp is a finitely generated free abelian group and P = P gp ∩ σ P , where σ P ⊆ P gp ⊗ Z R = P gp R is a convex rationalpolyhedral cone. We will assume that P comes with a homomorphism η : NE( Y ) → P of monoids. In laterdiscussion, we will in particular choose P = NE( Y ) and η to be the identity.Next we define a mutli-valued Σ-piecewise linear function as a continuous function ϕ : | Σ | → P gp R suchthat for each σ i,i +1 ∈ Σ max , ϕ i = ϕ | σ i,i +1 is given by an element ϕ σ i,i +1 ∈ Hom Z ( M, P gp ) = N ⊗ Z P gp . Foreach codimension cone cone ρ = R + v i ∈ Σ contained in two maximal cones σ i − ,i and σ i,i +1 , we have ϕ i +1 − ϕ i = n ρ ⊗ [ D i ] (2)where n ρ ∈ N is the unique primitive element annihilating ρ and positive on σ i,i +1 . Such data { ϕ i } gives alocal system P on B GHK , with the structure of P gp R -principal bundle π : P → B GHK , . To determine such alocal system, we first construct an affine manifold P by gluing U i × P gp R to U i +1 × P gp R along ( U i ∩ U i +1 ) × P gp R by ( x, p ) (cid:55)→ ( x, p + ϕ i +1 ( x ) − ϕ i ( x )) . The local sections x (cid:55)→ ( x, ϕ i ( x )) patch to give a piecewise linear section ϕ : B GHK , → P . Let Λ B denotethe sheaf of integral constant vector fields, and Λ B, R := Λ B ⊗ Z R . We can then define P := π ∗ Λ B, P ∼ = ϕ − Λ B, P on B GHK , . There is an exact sequence 0 → P gp → P r −→ Λ B → B GHK , , where r is the derivative of π . Then (2) is equivalent to ϕ i ( v i − ) + ϕ i ( v i +1 ) = [ D i ] − D i ϕ i ( v i ) , (4)which is the lifting of (1) to P . We will describe the symplectic meaning of P , P gp , and P in Section 5.2,particularly see (43).Next one would define the canonical scattering diagram D can on ( B GHK , Σ). We will first state thedefinition of scattering diagram as in [22] and then restrict to the finite case in this article. A ray in D can is a pair ( d , f d ) where • d ⊂ σ i,i +1 for some i , called the support of a ray, is a ray generated by av i + bv i +1 (cid:54) = 0, a, b ∈ Z ≥ ; • log f d = (cid:80) k ≥ kc k X − aki X − bki +1 ∈ k [ P ][[ X − ai X − bi +1 ]] with c k in the maximal ideal m ⊆ k [ P ] .The coefficient c k is the generating function of relative Gromov-Witten invariants, c k = (cid:88) β N β z β , where the summation is over all possible classes β ∈ H ( Y, Z ) with incidence relation β.D i = ak, β.D i +1 = bk and β.D j = 0, for j (cid:54) = i, i + 1. The coefficient N β is the counting of A -curves in such class β . We will referthe readers to [19, Section 3] for technical details of the definition of the relative Gromov-Witten invariants.Roughly speaking, a scattering diagram for the data ( B GHK , Σ) is a set D can = { ( d , f d ) } such that thereare only finitely many f d (cid:54) = 1. Note that scattering diagrams may give a refinement to the original fanstructure given by Σ. We will call the maximal cones of this refinement as chambers. At first glance, it looks like there is a sign change comparing with the wall functions of the scattering diagram from Floertheory in Definition 4.3. However, such discrepancy is explained in the discussion after Lemma 5.14 A I , where A I = k [ P ] /I and I ⊆ k [ P ] is anymonomial ideal with k [ P ] /I is Artinian. Now consider each ρ i as the support of a ray ( ρ i , f i ) in D can . Define R i,I := A I [ X i − , X ± i , X i +1 ] / ( X i − X i +1 − z [ D i ] X − D i i f i ) ,R i,i +1 ,I := A I [ X ± i , X ± i +1 ] ∼ = ( R i,I ) X i +1 , where z [ D i ] is the monomial in k [ P ] corresponding to the class of [ D i ]. Let U i,I := Spec R i,I and U i,i +1 ,I = Spec R i − ,i,I . Notice that if the fibre of U i,i +1 ,I → Spec A I over a point is a torus G m . Then the fibre of U i,I → Spec A I over a closed point is the partial compactifiaction of graph of the birational map U i − i,i,I (cid:57)(cid:57)(cid:75) U i,i +1 ,I ( X i − , X i ) (cid:55)→ ( X − i +1 z [ D i ] X − D i i f i , X i ) . (5)In particular, the the fibre of U i,I → Spec A I is the graph of (5) up to codimension two if V ( f i ) (cid:54) = ∅ . Onewould then like to glue U i,I and U i +1 ,I over the identified piece U i,i +1 ,I to obtain a scheme X ◦ I flat overSpec A I .To obtain a better behaved X ◦ I , one needs to consider an automorphism R i,i +1 ,I , called the path orderedproduct, associated to a path γ : [0 , → Int( σ i,i +1 ). Suppose γ crosses a given ray ( d = R ≥ ( av i + bv i +1 ) , f d ).The A I -algebra homomorphism θ γ, d : R i,i +1 ,I → R i,i +1 ,I is defined by X k i i X k i +1 i +1 (cid:55)→ X k i i X k i +1 i +1 f ± ( − bk i + ak i +1 ) d ,where the sign ± is positive if γ goes from σ i − ,i to σ i,i +1 when passing through d ; it is negative if γ goes inthe opposite direction and one can see this is the same as the wall crossing transformation stated in (12). If γ passes through more than one ray, one can define the path ordered product as composing each individualpath ordered product of each ray in the order according to the order of rays the γ passes. Choosing a path γ by starting very close to ρ i and ending near ρ i +1 in σ i,i +1 , then γ would pass all the rays in σ i,i +1 . Thendefine X ◦ I, D = (cid:96) i U i,I / ∼ with the gluing given by U i,I ← (cid:45) U i,i +1 ,I θ γ, D −−−→ U i,i +1 ,I (cid:44) → U i +1 ,I . The following observation is important later for the comparison between the Gross-Hacking-Keel mirror withthe family Floer mirror in the examples consider in this paper.
Remark 2.1.
When there are only finitely many rays d with nontrivial f d and I = m , one can replace ( Y, D ) be a minimal resolution such that all the A -curves are toric transverse. The procedure replaces Σ bythe refinement given by the original canonical scattering diagram and the integral affine manifold B GHK isremained the same. Then X ◦ I, D is gluing of tori, one corresponds to a chamber. The next step in [19, 22] is considering the broken lines to define consistency and to construct the thetafunctions. Since we will focus on the finite type in this paper, we can make use of path-ordered productdirectly without the use of broken lines. Instead, to define consistency, we can extend the definition of pathordered product to the path γ : [0 , → B ( Z ) with starting point q , and end point Q , where q and Q do notlie on any ray. Then the path ordered product θ γ, D can be defined similarly by composing θ γ, d ’s of the walls d passed by γ . Then the canonical scattering diagram D is consistent in the sense that the path orderedproduct θ γ, D only depends on the two end points q and Q .For a point q ∈ B ( Z ), let us assume q = av i − + bv i ∈ σ i − ,i and associate the monomial X ai − X bi to q . Consider now another point Q ∈ σ i,i +1 \ (cid:83) d ∈ D lim Supp d and a path γ from σ i − ,i to σ i,i +1 . Wewill define ϑ q,Q = (cid:16) z [ D i ] X − D i i f i X − i +1 (cid:17) a X bi . Note that the variables in R i − ,i,I are X i − , X i while thevariables in R i,i +1 ,I are X i , X i +1 . The change of variables are described the gluing from R i,I . We will dosimilarly if the γ goes in the opposite direction. Further if q, Q ∈ B ( Z ) not in adjacent chambers, we canconsider a path γ from q to Q and define ϑ q,Q by composing the changes of variables from the order of thechambers of how γ runs around D . This is well-defined since we have assumed our scattering diagram D is consistent. We will define ϑ ,Q = ϑ = 1. Thus the ϑ q,Q for various Q can be glued to give the globalfunction ϑ q ∈ Γ( X ◦ I, D , O X ◦ I, D ). Then, by [19, Theorem 2.28], X I, D := Spec Γ (cid:16) X ◦ I, D , O X ◦ I, D (cid:17) is a partialcompactification of X ◦ I, D . 5 .2 Cluster varieties Gross-Hacking-Keel-Kontsevich [21] constructed the cluster scattering diagrams and showed that the clustermonomials can be expressed as theta functions defined on the cluster scattering diagrams. The collectionsof the theta functions form the bases to the (middle) cluster algebras defined by Fomin-Zelevinsky [12].One can perform the similar construction as in the Gross-Hacking-Keel mirror construction by associatingeach chamber in the cluster scattering diagram with an algebraic torus. The path-ordered products (wallcrossings) give the birational maps between the tori. The A prin -cluster varieties are then defined as theschemes (up to codimension 2) obtained from gluing the tori associated to the chambers by the birationalmaps. The X -cluster varieties can be described as quotient of the A prin -varieties by torus action.Note that the underlying affine manifolds of the cluster scattering diagrams do not carry any monodromywhich are not exactly the same as canonical scattering diagrams. The cluster scattering diagrams can beseen as pushing the singularities of the affine structures of B of canonical scattering diagrams to infinity asexplained in [7]. We will illustrate how to choose branch cut and decompose the monodromy of B in Section5.3. Then we can translate from the canonical scattering diagrams to the cluster scattering diagrams. Theresulting schemes, no matter described by the canonical or the cluster scattering diagrams, are determined(up to codimension 2) by the algebras generated by the set of theta functions. We are going to see the casesin this articles are all associated to cluster algebras.For the dimension two case, the fix data are given by the bilinear form (cid:18) − (cid:19) and the scalars d , d ∈ N . Given fixed data, we can define the A (and X ) cluster varieties such that the rings of regularfunctions carry the A (and X ) cluster structures respectively. Gross-Hacking-Keel-Kontsevich [21] showedthat the middle A and X cluster algebras can be constructed from the theta functions of the correspondingschemes. Relations between the generators ϑ i in the cluster complex of the (middle) X cluster algebras canbe expressed as ϑ i − · ϑ i +1 = (cid:40) (1 + ϑ i ) d , if i is odd(1 + ϑ i ) d , if i is even , (6)where i ∈ Z . Conversely, given such relations between the variables, we can determine the algebras. Consider Y (cid:48) an extremal rational elliptic surface with singular configuration one of the following: II ∗ II , III ∗ III , IV ∗ IV , I ∗ I ∗ . We will denote Y (cid:48) = Y (cid:48)∗ , where ∗ = II, III or IV be the fibre over zero. Theserational elliptic surfaces can be constructed explicitly.We will first consider the case Y (cid:48) = Y (cid:48) II is the unique rational elliptic surface with singular configuration II ∗ II . The surface Y (cid:48) can be constructed as the minimal resolution of the surface { ty z = tx + atxz + uz } ⊆ P x,y,z ) × P u,t ) . (7)By the Tate algorithm [38], Y (cid:48) is an elliptic surface with a type II ∗ singular fibre over u = ∞ . Straight-forward calculation shows that Y (cid:48) has singular configuration II ∗ I if a (cid:54) = 0 and II ∗ II if a = 0. By theCalstelnuovo’s criterion of rationality, Y (cid:48) is rational and thus an rational elliptic surface. The other extremalrational elliptic surfaces can be constructed in a similar way with the corresponding affine equations below[36, p.545]: y = x + uy = x + t s y = x + at s x + bt s . It seems to the authors that the above examples are closely related to the geometry from SU(2) gauge theorystudied in [17]. We follow the definition of fixed data as in [20] Y (cid:48) has canonical bundle K Y (cid:48) = O Y (cid:48) ( − D (cid:48) ), where D (cid:48) denotesan elliptic fibre. Thus, there exits a meromorphic 2-form Ω (cid:48) with simple pole along a designate fibre whichis unique up to a C ∗ -scaling. In particular, the non-compact surface X (cid:48) = Y (cid:48) \ D (cid:48) can be viewed as a logCalabi-Yau surface. Indeed, Theorem 3.1. [25]
There exists a Ricci-flat metric ω (cid:48) on X (cid:48) for any choice of the fibre D (cid:48) . In particular, ω (cid:48) = Ω (cid:48) ∧ ¯Ω (cid:48) and X (cid:48) is hyperK¨ahler. Consider D (cid:48)∗ to be the infinity fibre in Y (cid:48)∗ and denote the hyperK¨ahler rotation of X (cid:48)∗ = Y (cid:48)∗ \ D (cid:48)∗ by X = X ∗ .Explicitly, X ∗ has the same underlying space as X (cid:48)∗ and equipped with K¨ahler form and holomorphic volumeform ω = ReΩ (cid:48) Ω = ImΩ (cid:48) + iω (cid:48) (8)on the underlying space of X (cid:48)∗ . Then the elliptic fibration X (cid:48)∗ → C implies the special Lagrangian fibration X ∗ → B , where B ∼ = R [24] (see the diagram below) topologically. We will refer the readers to [8, P.35]for more explicit calculation of hyperK¨ahler (HK) rotation. We will omit the subindex when there is noconfusion. Y (cid:48) X (cid:48) = Y (cid:48) \ D (cid:48) X P B ∼ = C B ∼ = R The fibrewise relative homology H ( X, L u ) glues to a local system of lattice over B . For any relativeclass γ ∈ H ( X, L u ), we denote the central charge Z γ ( u ) := (cid:90) γ Ω (cid:48) be a function from the local system Γ to C . Notice that B ∼ = C ∗ admits a complex structure structure and Z γ is locally a holomorphic function in u by Corollary 2.8 [33]. The central charge will help to locate thespecial Lagrangian torus fibre bounding holomorphic discs in Section 4.2. Let (
X, ω ) be a K¨ahler surface with holomorphic volume form Ω satisfying 2 ω = Ω ∧ ¯Ω. Assume that X admits a special Lagrangian fibration X → B possibly with singular fibres with respect to ( ω, Ω). We willuse L u to denote the fibre over u ∈ B . Let B be the complement of the discriminant locus. There are twonatural integral affine structures defined on B by Hitchin [26], one is called the symplectic affine structureand the other one is the complex affine structure. Given a reference point u ∈ B and a choice of thebasis ˇ e , ˇ e ∈ H ( L u , Z ), we will define the local affine coordinates around u . For any u ∈ B in a smallneighborhood of u , one choose a path φ contained in B connecting u, u . Let C k to be the S -fibrationover φ such that the fibres are in the homology class of parallel transport of ˇ e k . Then the local symplecticaffine coordinates can be defined by x k ( u ) = (cid:90) C k ω. (9)It is straight-forward to check that the transition functions fall in GL (2 , Z ) (cid:111) R , and thus the abovecoordinates give an integral affine structure. Remark 3.2.
From the construction, primitive classes ˇ e ∈ H ( L u , Z ) are one-to-one correspond to theprimitive integral vectors in T u B . Indeed, each v ∈ T u B has a corresponding functional (cid:82) − ι v Im Ω on H ( L u , Z ) and thus correspond to a primitive element in H ( L u , Z ) via its natural symplectic pairing andPoincare duality. Since there is monodromy, it is a multi-value function on B
7f there is a global Lagrangian section, then the transition functions fall in GL (2 , Z ). One can replace ω in (9) by ImΩ, then one gets the complex integral affine coordinates ˇ x k ( u ).We will use both integral affine structures later: the complex affine structures will be used to locate thefibres bounding holomorphic discs (see Section 4.2) while the symplectic affine structures will be used todefine the family Floer mirrors (see Section 4.3). In this section, we will talk about the background for the explicit calculation of the family Floer mirror inSection 5. We will review the construction of family Floer mirror of Tu [39] in Section 4.3. Recall thatgiven a Lagrangian torus fibration X → B with fibre L u over u ∈ B . Then Fukaya-Oh-Ohta-Ono [16]constructed an A ∞ on de Rham cohomologies of the fibres. Assume that the fibres are unobstructed, thenthe exponential of the corresponding Maurer-Cartan spaces are the analgue of the dual tori for the originalLagrangian fibration. Then the family Floer mirror are gluing of these exponential of Maurer-Cartan spaces.The gluing morphisms, known as the ”quantum correction” to the mirror complex structure, are induced bythe wall-crossing of the Maurer-Cartan spaces. Such wall-crossing phenomenons receive contributed from theholomoprhic discs of Maslov index zero with boundaries on SYZ fibres. We review the relation of the openGromov-Witten invariants with the gluing morphisms in Section 4.1. To further have better understandingof the gluing morphisms, in Section 4.2 we studied the location of all possible holomorphic discs of Maslovindex zero for the geometry discussed in Section 3, taking advantage of the special Lagrangina boundaryconditions. We will first review the so-called Fukaya’s trick, which is a procedure to compare the variation of the A ∞ structures of a Lagrangian and those of its nearby deformations.Let X be a symplectic manifold with special Lagrangian fibration X → B . Recall the definition ofNovikov field, Λ := (cid:40)(cid:88) i ∈ N c i T λ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ i ∈ R , lim i →∞ λ i = ∞ , c i ∈ C (cid:41) . Denote its maximal ideal by Λ + and Λ ∗ = Λ \{ } . There is a natural discrete valuationval : Λ ∗ −→ R (cid:88) i ∈ N c i T λ i (cid:55)→ λ i , where i is the smallest i with λ i (cid:54) = 0. One can extend the domain of val to Λ by setting val(0) = ∞ .Let B be the complement of the discriminant locus of the special Lagrangian fibration and L u be thefibre over u ∈ B . Given a relative class γ ∈ H ( X, L u ), we use M γ (( X, J ) , L u ) to denote the moduli spaceof stable J -holomorphic discs in relative class γ with respect to the (almost) complex structure J . We mayomit the J if there is no confusion. Fukaya-Oh-Ohta-Ono [16] constructed a filtered unital A ∞ structure { m k } k ≥ on H ∗ ( L u , Λ) by considering the boundary relations of M γ (( X, L ) , L u ), for all γ ∈ H ( X, L u ). Wewill assume that there exists only Maslov index zero discs in X . Due to the dimension reason, the modulispace M γ (( X, J ) , L u ) has virtual dimension negative one. In particular, the Maurer-Cartan space associateto the A ∞ structure is simply H ( L u , Λ + ).Now we explain the so-called Fukaya’s trick. Given p ∈ B and a path φ contained in a small neighborhoodof p such that φ (0) = u − , φ (1) = u + . One can choose a 1-parameter family of paths φ s ( t ) such that φ s ( t ) isa path from φ ( t ) to p and φ s ( t ) is contained in a small enough neighborhood of p . It is illustrated as follow:8 φ ( t ) φ (0)= u − φ (1)= u + Figure 1: Fukaya’s trickThen there exists a 2-parameter family of fibrewise preserving diffeomorphisms f s,t such that1. f s, = id .2. f s,t sends L φ s ( t ) to L p .3. f s,t is an identity outside a compact subset of B .Then J t = ( f ,t ) ∗ J is a 1-parameter family of almost complex structures tamed with respect to ω since φ is contained in a small enough neighborhood of p . There is a canonical isomorphism of moduli spaces ofholomorphic discs M k,β (( X, J ) , L φ ( t ) ) ∼ = M k, ( f ,t ) ∗ β (cid:0) ( X, ( f ,t ) ∗ J ) , L p (cid:1) (10)which carries over to the identification of the Kuranishi structures. However, the two sides of (10) give the A ∞ structures on are not the same under the parallel transport H ∗ ( L φ ( t ) , Λ) ∼ = H ∗ ( L p , Λ) because of thedifference of the corresponding symplectic area (or known as the flux) (cid:90) f ,t ∗ β ω − (cid:90) β ω = (cid:42) n (cid:88) k =1 (cid:0) x k ( φ ( t )) − x k ( p ) (cid:1) e k , ∂β (cid:43) , where e i ∈ H ( L p , Z ) is an integral basis.From the 1-parameter family of almost complex structures J t , one can construct a pseudo-isotopy ofunital A ∞ structures on H ∗ ( L p , Λ), connecting the A ∞ structures on H ∗ ( L p , Λ)from u ± . This induces apseudo-isotopy of the A ∞ structures from H ∗ ( L p , Λ) to itself. In particular, this induces an isomorphism onthe corresponding Maurer-Cartan spaces, which isomorphic to H ( L p , Λ + ) due to the dimension reason,Φ : H ( L p , Λ + ) → H ( L p , Λ + ) , (11)a priori is not identity if L φ ( t ) bounds holomorphic discs of Maslov index zero for some t ∈ [0 ,
1] [16]. Thefollow lemma states that Φ only depends on the homotopy class of the path φ . Lemma 4.1. [39, Theorem 2.7] Φ ≡ (mod Λ + ) and Φ only depends on the homotopy class of φ assumingno appearance of negative Maslov index discs in the homotopy. In particular, if φ is a contractible loop, thenthe corresponding Φ = 1 (before modulo Λ + ). The explicit form of Φ can be computed in the case of hyperK¨ahler surfaces with assumptions and onecan see that Φ acts like wall crossing in the Gross-Siebert program in the theorem below.
Theorem 4.2. (Theorem 6.15 [35] ) Assuming that there is only one primitive relative class γ such that L φ ( t ) bound holomorphic discs of class γ . Suppose that Arg Z γ ( u − ) < Arg Z γ ( u + ) (Check Remark 5.4 for thediscussion of the signs). Then the transformation Φ is given by K γ : z ∂γ (cid:48) (cid:55)→ z ∂γ (cid:48) f γ ( u ) (cid:104) γ (cid:48) ,γ (cid:105) , (12) for some power series f γ ( u ) ∈ T ω ( γ ) z ∂γ R [[ T ω ( γ ) z ∂γ ]] . Here (cid:104) γ (cid:48) , γ (cid:105) denotes the intersection pairing of thecorresponding boundary classes in the torus fibre. f γ ( u ) have enumerative meanings: counting of the Maslov index zero discs boundedby the 1-parameter family of Lagrangians [32] or counting of rational curves with certain tangency conditions[18]. This motivates the following definition. Definition 4.3.
Given u ∈ B and γ ∈ H ( X, L u ) primitive such that M kγ ( X, L u ) (cid:54) = ∅ for some k . Theopen Gromov-Witten invariants ˜Ω( γ ; u ) are defined via the formula log f γ ( u ) = (cid:88) d ∈ N d ˜Ω( dγ ; u )( T ω ( γ ) z ∂γ ) d , Then BPS rays are defined to be the support of loci with non-trivial open Gromov-Witten invariants ofthe same homology classes (up to parallel transport).
Definition 4.4.
Given u ∈ B and a relative class γ ∈ H ( X, L u ) , then the associated BPS ray is defined(locally) to be l γ := { u (cid:48) ∈ B | ˜Ω( γ ; u (cid:48) ) (cid:54) = 0 and Z γ ( u (cid:48) ) ∈ R + } . Remark 4.5.
Since special Lagrangians have vanishing Maslov classes, all the special Lagrangian torusfibres only bound Maslov index zero discs and the assumption of the Lemma 4.1 holds. In general, it is hardto control all the bubbling of pseudo-holomorphic discs (of Maslov index zero) and complicated to compute Φ . However, when the Lagrangian fibration is further special, the loci of special Lagrangian fibres boundingholomoprhic discs of a fixed relative class falls in an affine line with respect to the complex affine structure.Indeed, if u t be a path in B such that each L u t bounds a holomorphic disc in class γ (we identify the relativeclasses via parallel transport along the path u t ) for every t , then (cid:82) γ Im Ω = 0 along u t . In particular, l γ is affine line with respect to the complex affine structure. Notice that l γ is naturally oriented such that thesymplectic area of γ is increasing along l γ . From the expected dictionary in the introduction, these affinelines correspond to the rays in the scattering diagrams and the Lemma 4.1 translates to the consistency ofscattering diagrams. To compute the open Gromov-Witten invariants on X , we first recall the following fact: Given a rationalelliptic surfaces Y (cid:48) and a fibre D (cid:48) , there exists a 1-parameter deformation ( Y t , D t ) such that D (cid:48) t ∼ = D (cid:48) and Y (cid:48) t rational elliptic surfaces with only type I singular fibres except D (cid:48) t . The following theorem explains howto compute the local open Gromov-Witten invariants near a general singular fibre other than those of type I . We will denote X t to be the hyperK¨ahler rotation of Y (cid:48) t \ D (cid:48) t with relation similar to (8). Then X t → B t be a 1-parameter family of hyperK¨ahler surfaces with special Lagrangian fibration and X = X . We willidentify B t ∼ = B = B topologically. Theorem 4.6. [34, Theorem 4.3]
Given any u ∈ B , γ ∈ H ( X, L u ) , then there exists t and a neighborhood U ⊆ B of u such that1. If ˜Ω( γ ; u ) = 0 , then ˜Ω( γ ; u (cid:48) ) = 0 for u (cid:48) ∈ U .2. If ˜Ω( γ ; u ) (cid:54) = 0 , then l tγ ∩ U (cid:54) = ∅ and ˜Ω t ( γ ; u (cid:48) ) = ˜Ω( γ ; u ) , for u (cid:48) ∈ l tγ ∩ U and t with | t | < t .Here ˜Ω t ( γ ; u ) denotes the open Gromov-Witten invariant of X t . For instance, in the case for the singular configuration of Y (cid:48) is II ∗ II , then the BPS rays of X t wouldlook like the following picture with the notation defined in Section 5:10 γ + γ γ − γ − γ Figure 2: BPS rays on B t for the case discussed in Section 5 In this section, we will restrict to the the case Y (cid:48) has exactly two singular fibres at 0 , ∞ and the monodromyof the singular fibre is of finite order. The examples listed in Section 3 are exactly those possible Y (cid:48) . Wewill show that the BPS rays divide the base into chambers which are one-to-one correspondence to thetorus charts of the family Floer mirror later. In particular, the following observation simplifies the explicitcomputation of family Floer mirror. Lemma 4.7.
Let γ be one of the relative classes in Theorem 5.1. Then l γ i does not intersect each other. Inparticular, B is divided into chambers by l γ i .Proof. Let v ∈ T B and recall that one has vZ γ = (cid:82) ∂γ ι ˜ v Ω, where ˜ v is a lifting of v , by direct computation.Together with Ω is holomorphic symplectic, Z γ has no critical point in B . Let l γ be a BPS ray, then bydefinition the holomorphic function Z γ has phase 0 along l γ . Now take v to be the tangent of l γ at u ∈ l γ pointing away from the origin. Therefore, vZ γ ( u ) (cid:54) = 0. Otherwise, u is a critical point of Z γ and contracdictsto the fact that Ω is holomorphic symplectic. In other words, the function | Z γ | is strictly increasing along l γ . Next we claim that l γ can not wrap inside a compact set. Otherwise, there exists a sequence of points u i ∈ l γ converging to some point u ∞ ∈ B . Since the monodromy is finite, there are only finitely possibly relativeclasses among γ u i with respect to the trivialization of the local system H ( X, L u ) in a small neighborhood of u ∞ . After passing to a subsequence, one has lim i →∞ Z γ ( u i ) = Z γ ( u ∞ ). If u ∞ ∈ B , then l γ can be extendedover u ∞ and leads to a contradiction. Therefore, l γ connects 0 and ∞ . Then from the asymptotic geometrynear infinity, one has | Z γ | (cid:37) ∞ along l γ .Notice that the above argument holds for l θγ , where l θγ is the loci where Z γ has phase θ ∈ S . Thisimplies that | Z γ ( u ) | as u → ∞ . Recall that Z γ ( u ) is a multi-valued holomorphic function on B ∼ = C ∗ .Since π ( B ) ∼ = Z and the monodromy is of order k , we have Z γ ( u k ) is a well-defined holomorphic function C ∗ → C ∗ . By straight-forward calculation one has lim u → Z γ ( u k ) = 0 and thus u = 0 is a removablesingularity. The previous discussion implies that ∞ is a pole and the holomorphic function Z γ ( u k ) extendsto P → P and fixing 0 , ∞ . Thus, we reach that Z γ ( u k ) = cu, (13)for some constant c ∈ C ∗ and the lemma follows. Remark 4.8.
Let Y t be a small deformation of Y such that Y t has a fibre isomorphic to D and all othersingular fibres are of type I , then Lemma 4.7 still holds. Remark 4.9.
Let Y be the del Pezzo surface of degree five and D be an anti-canonical divisor consists ofa wheel of five rational curves. Set X = Y \ D . It is known that X is the moduli space of flat connectionson punctured sphere. There exists a hyperK¨hlaer metric on it such that suitable hyperK¨hler rotation becomessome meromorphic Hitchin moduli space, which is X (cid:48) , the complement of the II ∗ fibre of the rational ellipticsurface Y (cid:48) with singular configuration II ∗ II . It is not clear if the holomorphic volume form Ω (cid:48) on X (cid:48) extendsas a meromorphic form with a simple pole along the II ∗ fibre. However, the Hitchin metric is exponentiallyasymptotic to the semi-flat metric at infinity [13] , the proof of Lemma 4.7 also applies to this case. .3 Construction of the Family Floer Mirror We will briefly recall the construction of family Floer mirror constructed by Tu [39] in this section. We willrefer the details of the analytic geometry to [10].
Definition 4.10.
Let U (cid:48) ⊆ B be an open set and ψ : U (cid:48) → R n be the affine coordinate. Then U = ψ − ( P ) ⊆ U (cid:48) for some rational convex polytope P ⊆ R n is called a rational domain. The Tate algebra T U associated to a rational domain U consists of the power series of the form (cid:88) k ∈ Z n a k z k · · · z k n n , where k = ( k , · · · , k n ) with the following conditions:1. a k ∈ Λ the Novikov field and2. (convergence in T -adic topology) lim k →∞ val ( a k ) + (cid:104) k , x (cid:105) → ∞ , (14) as k → ∞ , for each x = ( x , · · · , x n ) ∈ U . Take a contractible open cover { U i } i ∈ I of B . For each φ i : U i → R n , we take the maximum spectrumof the associated Tate algebra U i := Spec( T U i ) which is called an affinoid domain. For each pair i, j with U i ∩ U j (cid:54) = ∅ , there is a natural gluing data Ψ ij : U i → U j , which now we will explain below: Let ( x i , · · · , x in ) be the local symplectic affine coordinates on U i . Thecorresponding functions in T U i are denoted by ( z i , · · · , z in ), where val( z ik ) = x ik . Choose p ∈ U i ∩ U j and f u i ,p fibrewise preserving diffeomorphism sending L u i to L p and is identity outside U i . The difference ofsymplectic affine coordinates is (cid:90) f ui,p ∗ β ω − (cid:90) β ω = (cid:42) n (cid:88) k =1 (cid:0) x k ( p ) − x k ( u i ) (cid:1) e ik , ∂β (cid:43) . Denote T U i,p the Tate algebra satisfying the convergence in T -adic topology (14) on the rational domain φ i ( U i ) − φ i ( p ) and U i,p as its spectrum. Then there is the transition map S u i ,p : U i → U i,p z ik (cid:55)→ T x k ( u i ) − x k ( p ) z ik . (15)Then define the gluing data Ψ ij be the compositionΨ ij : U ij S ui,p −−−→ U ij,p Φ ij −−→ U ji,p S − uj,p −−−→ U ji , (16)where Φ ij is defined in (11). The gluing data Ψ ij satisfies (Section 4.9 [39])1. Independent of the choice of reference point p ∈ U i ∩ U j .2. Ψ ij = Ψ ji and Ψ ij Ψ jk = Ψ ik .3. For p ∈ U i ∩ U j ∩ U k , we have Ψ ij ( U ij ∩ U ik ) ⊆ U ji ∩ U jk .Then the family Floer mirror ˇ X is defined to by the gluing of the affinoid domainsˇ X := (cid:91) i ∈ I U i / ∼ , (17)where ∼ is defined by (16). The natural projection map T U → U from the valuation glue together and givesthe family Floer mirror a projection map Trop : ˇ X → B . The following example is straight forward from the construction.12 xample 4.11.
Recall that the rigid analytic torus ( G anm ) admits a valuation map Trop : ( G anm ) → R .Let π : X → U be a Lagrangian fibration such that for any path φ connecting u i , u j ∈ U , the corresponding F φ = id . Assume that the symplectic affine coordinates give an embedding U → R and we will simplyidentify U with its image. Then the gluing Ψ ij : U ij → U ji z ik (cid:55)→ T x k ( u i ) − x k ( u j ) z jk , is simply translation from equation (16) . Thus the corresponding family mirror ˇ X is simply Trop − ( U ) → U .In particular, when U ∼ = R , then the family Floer mirror is simply the rigid analytic torus ( G anm ) n . It worthnoticing that if U ⊆ R is a proper subset, then Trop − ( U ) is not a dense subset of ( G anm ) . X II In this section, we will have a detailed computation of the family Floer mirror of X = X II from the extremalrational elliptic surface Y (cid:48) II with singular configuration II ∗ II and II ∗ at infinity. We sketch the proof below:We will first identify the locus of special Lagrangian fibres bounding holomorphic discs to be simply fiverays l γ i connecting 0 , ∞ . Then we compute their corresponding wall-crossing transformations which areanalytification of some birational maps. Thus, the family Floer mirror ˇ X can be glue from five charts. Thenwe will prove that the embedding of each of the five charts into ˇ X can be extended to an embedding of theanalytic torus G anm into ˇ X . In other words, ˇ X is gluing five analytic torus. On the other hand, consider thedel Pezzo surface Y of degree 5 and D be the cycle of five rational curves. Let B GHK be the affine manifoldwith the singularity constructed in Section 2.1 after choosing suitable branch cut. We identify the complexaffine structure on B with the one on B GHK , the rays and the corresponding wall-crossing transformations.Then from [19, Example 3.7], we know that ˇ X is the analytification of the del Pezzo surface of degree five.Furthermore, we would choose the branch cuts on B in a different way. This would induce another realizationof ˇ X as gluing to five tori but with different gluing morphisms, which we will later identify ˇ X as the X -clustervariety of type A .First we apply Theorem 4.6 to the 1-parameter family of hyperK¨ahler rotation of the rational ellipticsurfaces described in Section 3, one get Theorem 5.1. [34, Theorem 4.11]
Choose a branch cut from the singularity to infinity and a basis { γ (cid:48) , γ (cid:48) } of H ( X, L u ) ∼ = Z such that (cid:104) γ (cid:48) , γ (cid:48) (cid:105) = 1 and the counter-clockwise monodromy M around the singularity is γ (cid:48) (cid:55)→ − γ (cid:48) γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) (18) with respect to the basis. Set γ = − γ (cid:48) , γ = γ (cid:48) , γ = γ (cid:48) + γ (cid:48) , γ = γ (cid:48) , γ = − γ (cid:48) . Then1. f γ ( u ) (cid:54) = 1 if and only if u ∈ l γ i and γ = γ i for some i = 1 , · · · , .2. In such cases, f γ i = 1 + T ω ( γ i ) z ∂γ i . We will put the branch cut between l γ and l γ . Therefore, locally, we have the following picture:13 γ l γ l γ l γ l γ Figure 3: BPS rays near the singularity.Straight-forward calculation shows that γ i +2 = − γ i + γ i +1 , (19)which is the analogue of (1).Next we claim that above five families of discs are the only contributing to the construction of the familyFloer mirror. Corollary 5.2. If ˜Ω( γ ; u ) (cid:54) = 0 , then u ∈ l γ i , γ = γ i for some i and ˜Ω( γ ; u ) = ( − d − d , where d is thedivisibility of γ .Proof. This is a direct consequence of the split attractor flow mechanism of the open Gromov-Witten invari-ants ˜Ω( γ ; u ) (see [35, Theorem 6.32]). We will sketch the proof here for self-containednes. Let l γ be a rayemanating from u such ω γ is decreasing along l γ . From Gromov compactness theorem, the loci where ˜Ω( γ )jumps are discrete. Assume that ˜Ω( γ ) is invariant along l γ , then the holomorphic disc representing γ canfall into a tubular constant C by [8, Proposition 5.3]. Then by Lemma 5.1, γ = γ i for some i . Otherwise,assume u is the first point where ˜Ω( γ ) jumps. Apply Lemma 4.1 to a small loop around u , there exists γ α , α ∈ A such that ˜Ω( γ α ; u ) (cid:54) = 0 and γ = (cid:80) α ∈ A γ α . In particular, ω ( γ α ) < ω ( γ ). One may replace ( γ, u ) by( γ α , u ) and run the procedure. Again by Gromov compactness theorem, after finitely many splittings, allthe relative classes are among { γ i } i =1 , ··· . To sum up, there exists a rooted tree T and a continuous map f such that the root maps to u , each edge is mapped to an affine line segment and all the 1-valent vertex aremapped to 0. Since l γ i s do not intersect by Lemma 4.7, the lemma follows. Lemma 5.3.
The composition of the wall-crossing transformations cancel out the monodromy. Explicitly, K γ K γ K γ K γ K γ ( z γ ) = z M − γ . Proof.
We will use the identification as in (29). Let us consider a = a γ (cid:48) + a γ (cid:48) ∈ H ( L p , Z ), where p ∈ B is a reference point, and a loop from l γ anticlockwise to itself:We will first compute the case without any singularities. This is very standard from [18]. We are onlyrepeating it as there may be confusion about signs. 1 + z γ (cid:48) z γ (cid:48) + γ (cid:48) z γ (cid:48) l γ l γ l γ l γ l γ δ emark 5.4. Before we go into the calculation, let us unfold the sign convention in Theorem 4.2. Todetermine the sign, we have the condition Arg Z γ ( u − ) < Arg Z γ ( u + ) . This means that the loop δ is going inanti-clockwise direction.In the calculation of the exponents, we consider γ (cid:55)→ (cid:104)· , γ (cid:105) . Note that (cid:104)· , ·(cid:105) is the intersection pairing butnot the usual inner product. Together with (cid:104) γ (cid:48) , γ (cid:48) (cid:105) = 1 , we have (cid:104)· , γ (cid:105) is the normal of l γ pointing in thesame direction as δ in the language of [18] . Let us consider the transformation K δ = K δ,l γ K δ,l γ K δ,l γ K δ,l γ K δ,l γ , where K δ,l γk = K γ k for k = 1 , , K δ,l γk +3 = K γ k for k = 1 , K δ,lγk (cid:55)−−−−→ for the wall crossing over the wall l γ k according to thecurve δ . z a K δ,lγ (cid:55)−−−−→ z a (1 + z γ (cid:48) ) a , K δ,lγ (cid:55)−−−−→ z a (1 + z γ (cid:48) + γ (cid:48) ) a − a (cid:16) z γ (cid:48) (1 + z γ (cid:48) + γ (cid:48) ) − (cid:17) a , = z a (1 + z γ (cid:48) + γ (cid:48) ) − a (1 + z γ (cid:48) + z γ (cid:48) + γ (cid:48) ) a , K δ,lγ (cid:55)−−−−→ z a (1 + z γ (cid:48) ) − a (cid:16) z γ (cid:48) + γ (cid:48) (1 + z γ (cid:48) ) − (cid:17) − a (cid:16) z γ (cid:48) (1 + z γ (cid:48) ) − (1 + z γ (cid:48) ) (cid:17) a , = z a (1 + z γ (cid:48) + z γ (cid:48) + γ (cid:48) ) − a (1 + z γ (cid:48) ) a , K δ,lγ (cid:55)−−−−→ z a (1 + z γ (cid:48) ) − a (cid:16) z γ (cid:48) (1 + z γ (cid:48) ) − (1 + z γ (cid:48) ) (cid:17) − a (1 + z γ (cid:48) ) a = z a (1 + z γ (cid:48) ) − a , K δ,lγ (cid:55)−−−−→ z a (1 + z γ (cid:48) ) a (1 + z γ (cid:48) ) − a , = z a . Thus we obtain the consistency as usual. Next we investigate the wall crossing transformation over themonodromy deduced by focus-focus singularities on l γ (cid:48) .1 + z − γ (cid:48) z γ (cid:48) β Let us consider the wall crossing K β = K β, K β, over the curve β , where K β, = K γ (cid:48) , and K β, = K − γ (cid:48) .The first wall crossing will lead us to K β, ( z a ) = z a (1 + z γ (cid:48) ) a . Then passing over the wall again by using β will get us K β ( z a ) = K β, ◦ K β, ( z a ) = z a (1 + z − γ (cid:48) ) − a (1 + z γ (cid:48) ) a = z a γ (cid:48) +( a + a ) γ (cid:48) . To have z a γ (cid:48) +( a + a ) γ (cid:48) goes back to z a , we have the monodromy M γ (cid:48) (cid:55)→ γ (cid:48) − γ (cid:48) , (20) γ (cid:48) (cid:55)→ γ (cid:48) . (21)Let us first consider the monodromy over the focus-focus singularities on l γ (cid:48) :15 + z γ (cid:48) z − γ (cid:48) α Consider the transformation according to the loop α . Let K α, = K γ (cid:48) , and K α, = K − γ (cid:48) . We have K α, ( z a ) = z a (1 + z γ (cid:48) ) − a . Then the whole loop α leads us to K α = K α, ◦ K α, ( z a ) = z a (1 + z − γ (cid:48) ) a (1 + z γ (cid:48) ) − a = z ( a − a ) γ (cid:48) + a γ (cid:48) . Then we obtain the monodromy M γ (cid:48) (cid:55)→ γ (cid:48) , (22) γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) . (23)Thus, we can compute the monodromy while singularity is at the origin. There are two ways checkingit. The first one is doing a similar calculation as in the beginning of the proof. Now we consider1 + z γ (cid:48) z γ (cid:48) + γ (cid:48) z γ (cid:48) z − γ (cid:48) z − γ (cid:48) l γ l γ l γ l γ l γ The first three wall crossings are the same and let us recap here: K δ,l γ K δ,l γ K δ,l γ ( z a ) = z a (1 + z γ (cid:48) + z γ (cid:48) + γ (cid:48) ) − a (1 + z γ (cid:48) ) a . Now to pass over l γ , we will have K ( z a (1 + z γ (cid:48) + z γ (cid:48) + γ (cid:48) ) − a (1 + z γ (cid:48) ) a ) = z a (1 + z − γ (cid:48) ) − a (cid:16) z γ (cid:48) (1 + z − γ (cid:48) ) − (1 + z γ (cid:48) ) (cid:17) − a (1 + z γ (cid:48) ) a = z a γ (cid:48) +( a + a ) γ (cid:48) (1 + z γ (cid:48) + γ (cid:48) ) − a . The monodromy M would then be γ (cid:48) (cid:55)→ − γ (cid:48) ; γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) . and gives us K M ( z a γ (cid:48) +( a + a ) γ (cid:48) (1 + z γ (cid:48) + γ (cid:48) ) − a ) = z ( a + a ) γ (cid:48) + a γ (cid:48) (1 + z γ (cid:48) ) − a . The last wall crossing would then be K δ,l γ (cid:16) ( z ( a + a ) γ (cid:48) + a γ (cid:48) (1 + z γ (cid:48) ) a (cid:17) = z ( a + a ) γ (cid:48) + a γ (cid:48) (1 + z − γ (cid:48) ) a (1 + z γ (cid:48) ) − a = z a . The second way is to use the following meta-lemma by direct computation16 laim 5.5. K − γ K γ ( z γ (cid:48) ) = z M − γ (cid:48) , where M is transformation γ (cid:48) (cid:55)→ γ (cid:48) + (cid:104) γ, γ (cid:48) (cid:105) γ . Note that if γ is primitive, then M is the Picard-Lefschetz transformation of a focus-focus singularitywith Lefschetz thimble γ . Recall that if (cid:104) γ (cid:48) , γ (cid:105) = 1, then the pentagon equation reads K γ K γ (cid:48) = K γ (cid:48) K γ + γ (cid:48) K γ . (24)Let M , M denote the transformation in the Claim 5.5 with respect to γ (cid:48) , γ (cid:48) respectively.With the branch cut as in Figure 3, one has K γ K γ K γ K γ K γ = (cid:18) K − γ K γ (cid:19)(cid:18) K − γ K γ K γ K γ K − − γ (cid:19)(cid:18) K − γ K γ (cid:19) . Notice that the middle of the right hand side is identity by the pentagon identity (24). From Lemma 5.5,we have K γ K γ K γ K γ K γ ( z γ ) = z M − M − γ = z ( M M ) − γ and the lemma follows from the fact that M = M M . Notice the the proof is motivated by deforming thetype II singular fibre into two I singular fibres as in Figure 4. However, the proof does NOT depend onthe actual geometric deformation. 1 + z γ (cid:48) z γ (cid:48) + γ (cid:48) z γ (cid:48) z − γ (cid:48) z − γ (cid:48) Figure 4: Geometric interpretation of Lemma 5.3.
Remark 5.6.
It worth noticing that the above calculation a priori may be different from the composition ofwall-crossing for the A cluster variety for two reasons. The first difference comes from the appearance ofthe monodromy at the origin while there is no such in the cluster scattering diagram. We will explain theidentification in Section 5.3. The second difference comes from the fact that the in the calculation for clustervariety there is a preferred choice of basis in each chamber while the calculation in Floer theory uses a fixedbasis (up to parallel transport). However, thanks to (19) , the two calculations thus coincide. X II From the construction of the family Floer mirror in the last section and Example 4.11, we learn thatthe construction starts with choosing a open cover { U i } of B and then look for the gluing between the U i := Spec( T U i ). 17 γ l γ l γ l γ l γ U Let U k be the chamber bounded by l γ k and l γ k +1 in B , i = 1 , · · · , U be the chamber bounded by l γ and l γ . Thus there are only 5 chambers. Recall that the dotted line represents a branch cut between l γ and l γ . With such branch cut and monodromy, we trivialized the local system H ( X, L u ) over thecomplement of the branch cut. It is easy to check that M γ i = γ i +1 .Denote the symplectic and complex affine coordinate (with respect to γ k , γ k +1 ) discussed in Section 3.1by x k = (cid:90) γ k ω, y k = (cid:90) γ k +1 ω ˇ x k = (cid:90) γ k ImΩ , ˇ y k = (cid:90) γ k +1 ImΩ . We will also denote x = (cid:90) γ (cid:48) ω, y = (cid:90) γ (cid:48) ω, ˇ x = (cid:90) γ (cid:48) ImΩ , ˇ y = (cid:90) γ (cid:48) ImΩ , which give another set of symplectic/complex affine coordinates.From the discussion of the hyperK¨ahler rotation in Section 3, we view B as a projective line afterhyperK¨ahler rotation. We have x k − i ˇ x k is a (multi-valued) holomorphic function with respect to the abovecomplex structure on B . Notice that x k > x k = 0 along l γ k . From Remark 4.5, after choosing asuitable complex coordinate u on B such that l γ i is the locus where Arg z = 0, one have x k − i ˇ x k = c k u a , k = 1 , . . . , , (25)for some constant a ∈ Z , c k ∈ C ∗ . With more analysis, we have the following lemma Lemma 5.7.
With suitable choice of coordinate u on B ∼ = C ∗ , we have x k − i ˇ x k = e πi ( k − u . (26) In particular, the angle between l γ k and l γ k +1 is π with respect to the conformal structure after hyperK¨ahlerrotation .Proof. We will first assume that u is normalized that x − i ˇ x = u a . Recall that Z γ k := x k − i ˇ x k . From themonodromy M γ k = γ k +1 , we have Z γ k +1 ( u ) = Z γ k ( ue πi ) = e πi a Z γ k ( u ) . Now It suffices to show that a = 5 or show that Z γ i ( u ) = O ( | u | ). This can be seen by direct computation.Indeed, locally one can write Ω = f ( u ) du ∧ dxy for some holomorphic function f ( u ) with f (0) (cid:54) = 0. Then Z γ ( u ) = (cid:90) γ Ω = (cid:90) u (cid:18) (cid:90) ζu u dx ( x + u ) (cid:19) du, Notice that there is no well-defined notion of angle with only an affine structure on B . ζ = −
1. Direct calculation shows that (cid:82) ζu u dx ( x + u ) / = O ( | u | − ) and the lemma follows. The lastpart of the lemma comes from the fact that Z γ k +1 ( u ) ∈ R + when u ∈ l γ k +1 .Next, we compare the affine structure from the SYZ fibration with the one from Gross-Hacking-Keel (seeSection 2.1). Lemma 5.8.
The complex affine structure on B coincides with the affine manifold B GHK with singularityconstructed from del Pezzo surface of degree five relative to a cycle of five rational curves in [19] .Proof.
From Lemma 5.7, one has l γ = { ˇ y = 0 , ˇ x > } l γ = { ˇ x = 0 , ˇ y > } . (27)Therefore, we may identify l γ , l γ with R > (1 , , R > (0 ,
1) respectively. Then ( − , , ( − , , (0 , −
1) arethe tangents of l γ , l γ , l γ respectively by Lemma 5.7 and the relation − Z γ i + Z γ i +1 = Z γ i +2 which is theanalogue of (1). Notice that monodromy around the singularity acting on the coordinatesˇ x (cid:55)→ ˇ x + ˇ y ˇ y (cid:55)→ − ˇ x. (28)Then the affine monodromy around the singularity, which is the inverse dual of (28), is given by (cid:0) −
11 1 (cid:1) . Inparticular, the monodromy glues U with U and thus identifies the base of the special Lagrangian fibration(with the complex affine structure) with B GHK . • l γ i = { ˇ y i = 0 } l γ i +1 l γ i +2 l γ i +3 U i y i = 0 x i = 0 U (cid:48) i V i Figure 5: Illustration for the notations in the beginning of Section 5.1.Notice that a priori l γ i is only an affine line with respect to the complex affine coordinates. To computethe family Floer mirror, we need to have a better control of the BPS rays in terms of the symplectic affinestructure. The following observation comes from (25) directly. Lemma 5.9.
Any ray with a constant phase is affine with respect to the symplectic affine structure. Inparticular, l γ i is an affine line with respect to the symplectic affine structure.Proof. Any such ray can be parametrized by z = Ct for some complex number C . From (25), the symplecticcoordinates along the ray are given by x k = C (cid:48) k t πk , y k = C (cid:48)(cid:48) k t πk , for some C (cid:48) k , C (cid:48)(cid:48) k ∈ R and the lemmafollows. In other words, such ray is given by the affine line C (cid:48)(cid:48) k x k = C (cid:48) k y k with respect to the symplecticaffine coordinates ( x k , y k ). 19y using the symplectic affine coordinates, we can identify the U i with a subset of standard affine plane R i as affine manifolds, which we will abuse the notation and denote it by U i . Let Trop i : ( G anm ) i → R i be the standard valuation map. Here we put an subindex i for each analytic tori and later it would correspondto the five different tori. Let U (cid:48) i be the (slightly bigger) open neighborhood containing U i and recall that thefamily Floer mirror is defined to be Trop − i ( U (cid:48) i ) / ∼ . Note thatˇ X = (cid:83) i Trop − i ( U (cid:48) i ) / ∼ ( G anm ) Trop − i ( U i ) ⊇ ⊆ To distinguish the two inclusion, we will always view
Trop − i ( U i ) as a subset of ( G anm ) and consider α i : Trop − i ( U i ) → ˇ X. Notice that
Trop − i ( U i ) only occupies a small portion of ( G anm ) . Thus we need to extend α i to most part of( G anm ) i . For the simplicity of the notation, we will still denote those extension of α i be the same notation.Let V i , V i +1 be some small enough rational domains on B such that V i ⊆ U (cid:48) i , V i +1 ⊆ U (cid:48) i +1 and theFukaya’s trick applies. Let p ∈ V i ∩ V i +1 be the reference point and one has( G anm ) i ⊇ Trop − i ( V i ) ⊇ Trop − i ( V i ∩ V i +1 ) Φ i,i +1 −−−−→ Trop − i +1 ( V i ∩ V i +1 ) ⊆ Trop − i +1 ( V i +1 ) ⊆ ( G anm ) i +1 , where Φ i,i +1 = α − i +1 ◦ α i is given byΦ i,i +1 : z ∂γ (cid:55)→ z ∂γ (1 + T ω ( γ i +1 ) z ∂γ i ) (cid:104) γ,γ i +1 (cid:105) from Definition 4.3 and Theorem 5.1. From (19), we have (cid:104) γ i +1 , γ i (cid:105) = 1. Denote z γ i := T ω ( γ i ) z ∂γ i , thenΦ i,i +1 is simply the polynomial map z γ i (cid:55)→ z γ i (1 + z γ i +1 ) − z γ i +1 (cid:55)→ z γ i +1 . (29)Recall that there is a natural identification (Λ ∗ ) ∼ = ( G anm ) as sets such that the below diagram commutes.(Λ ∗ ) ( G anm ) R val Trop (30)Thus, we have val( z γ i ) = 2 x i . Since near l γ i +1 one has ω ( γ i +1 ) >
0, one hasval( z γ ) = val( z γ (1 + z γ i ) − ) . (31)Thus, the following commutative diagram holds, Trop − i ( V i ) ⊇ Trop − i ( V i ∩ V i +1 ) Trop − i +1 ( V i ∩ V i +1 ) ⊆ Trop − i +1 ( V i +1 ) R i ⊇ V i ∩ V i +1 V i ∩ V i +1 ⊆ R i +1 Trop i Φ i,i +1 Trop i +1 (32)We may view (Λ ∗ ) as the Λ-points of the scheme ( G m ) = SpecΛ[ z ± γ i , z ± γ i +1 ]. Then we have the commu-tative diagram from GAGA functor ( G anm ) ( G anm ) ( G m ) ( G m ) i,i +1 GAGA GAGA (33)20nder the identification (Λ ∗ ) ∼ = ( G anm ) , Φ i,i +1 is simply the restriction of the map ( G anm ) → ( G anm ) withthe same equation as in (29). Therefore, we have the same commutative diagram as in (32) with V i , V i +1 replaced by U + i , U i +1 for any open subset U + i ⊆ R such that ω ( γ i +1 ) > U + i , which we will choose itexplicitly later.To see the largest possible extension U + i and thus largest possible extension of the above diagram, wewould want to know explicitly where ω ( γ i +1 ) >
0. Viewing B ∼ = C , we may take U + i as the interior ofthe sector bounded by l γ i and the ray by rotating π counter-clockwisely from l γ i +1 and this is the largestpossible region (extending U i counter-clockwisely) such that ω ( γ i +1 ) > α i : Trop − i ( U i ) (cid:44) → ˇ X to α i : Trop − i ( U + i ) (cid:44) → ˇ X , i = 1 , · · · ,
5. In particular, we have α i +1 : Trop − i +1 ( U + i +1 ) (cid:44) → ˇ X .To further extend α i , the commutative diagram (32) no longer holds sinceval( z γ i (cid:0) z γ i +1 (cid:1) − ) = val( z γi ) − val(1 + z γ i +1 ) = val( z γ i ) − val( z γ i +1 ) (34)outside of U + i , which is no longer val( z γ i ) on the right hand side as in (31). Now for V i disjoint from U + i and V i +1 ⊆ U i +2 ⊆ U + i +1 , the diagram becomes Trop − i ( V i ) ⊇ Trop − i ( V i ∩ V i +1 ) \ { z γ i +1 = 0 } ( G an ) R i ⊇ V i ∩ V i +1 R i +1 , Trop i Φ i,i +1 Trop i +1 φ i,i +1 (35)where from (34), we have φ i,i +1 : x i (cid:55)→ x i − y i y i (cid:55)→ y i . (36)Notice that Φ i,i +1 is only defined when 1 + z γ i +1 (cid:54) = 0. Lemma 5.10. φ i,i +1 ( U i +2 \ U + i ) ⊆ U + i +1 . In particular, α i (cid:0) Trop − i ( U i +2 ) (cid:1) ⊆ α i +1 (cid:0) Trop − i +1 ( U + i +1 ) (cid:1) ⊆ ˇ X. Proof.
The left boundary of U + i is characterized by x i +1 = 0 , y i +1 > U + i +1 ischaracterized by x i +1 < , y i +1 = 0. Therefore, we may identify the region bounded by the above two affinelines with the third quadrant of R x i +1 ,y i +1 as affine manifolds. Notice that this is a subset of U + i +1 . Undersuch identification, we have U i +2 \ U + i is the region bounded by x i +1 + y i +1 = 0 and y i +1 -axis in the thirdquadrant by Lemma 5.9. In terms of ( x i +1 , y i +1 ), (36) becomes φ i,i +1 : x i +1 (cid:55)→ x i +1 y i +1 (cid:55)→ x i +1 + y i +1 , from the relation γ i + γ i +2 = γ i +1 . The lemma then follows from direct computation.To sum up, one can extend the original inclusion α i (cid:0) Trop − i ( U i ) (cid:1) ⊆ ˇ X in the counter-clockwise directionto α i (cid:0) Trop − i ( U i ∪ U i +1 ∪ U i +2 ) \ { z γ i +1 = 0 } (cid:1) ⊆ ˇ X. (37)Here we use U to denote the interior of the compactification of U . Lemma 5.11.
The inclusion (37) extends over { z γ i +1 = 0 } \ Trop − i (0) .Proof. Let W i be small neighborhood of (a component of ) ∂U + i such that { z γ i +1 = 0 } ⊆ Trop − i ( W i ).Notice that from Lemma 5.10, we have that Trop (cid:0) α i ( Trop − i ( W i )) (cid:1) ⊆ U i +2 . We will show that α i (cid:0) Trop − i ( W i ) (cid:1) ⊆ α i +1 (cid:0) Trop − i +1 ( U + i +1 ) (cid:1) ∪ α i +2 (cid:0) Trop − i +2 ( U i +2 ) (cid:1) ∪ α i +3 (cid:0) Trop − i +3 ( U i +2 ) (cid:1) . (38)21rom the earlier discussion, we have α i (cid:0) Trop − i ( W i ) \ { z γ i +1 = 0 } (cid:1) ⊆ α i +1 (cid:0) Trop − i +1 ( U + i +1 ) (cid:1) . From the earlier discussion, we haveΦ i +1 ,i +2 : Trop − i +1 ( U i +2 ) ∼ = Trop − i +2 ( U i +2 )Φ i +3 ,i +2 : Trop − i +3 ( U i +2 ) ∼ = Trop − i +2 ( U i +2 ) . (39)Recall that Φ i,j = α − j ◦ α i . It suffices to check that A = { z γ i +1 = 0 } ⊆ Φ i +2 ,i (cid:0) Trop − i +2 ( U i +2 ) (cid:1) ∪ Φ i +3 ,i (cid:0) Trop − i +3 ( U i +2 ) (cid:1) (40)as subsets of ( G anm ) i . Straight calculation shows thatΦ i,i +2 : Trop − i ( W i ) → Trop i +2 ( U i +2 ) z γ (cid:55)→ z γ (1 + z γ i +2 ) (cid:104) γ,γ i +2 (cid:105) (cid:18) z γ i +1 z γ i +2 (cid:19) (cid:104) γ,γ i +2 (cid:105) Since (cid:104) γ, γ i +2 (cid:105) > (cid:104) γ, γ i +1 (cid:105) > U i +2 . We have Φ i,i +2 is not defined only on B = { z γ i +2 = 0 } ∪ { z γ i +1 + z γ i +2 = 0 } . Therefore, we have α i can be extended over Trop − i ( W i ) \ B . Similarly, Φ i,i +3 is defined except C = { z γ i +3 = 0 } ∪ { z γ i +2 + z γ i +3 = 0 } ∪ { z γ i +1 + z γ i +2 + z γ i +3 = 0 } . Therefore, α i can be extended over Trop − i ( W i ) \ C . It is easy to check that A ∩ B ∩ C = { z γ i +1 = z γ i +2 = − } ⊆ Trop − (0). Since Φ i,j = α − j ◦ α i and thus the extension is compatible. Now the lemma is proved.For the same reason, one can extend the inclusion in the clockwise direction α i (cid:0) Trop − i ( U i ∪ U i − ∪ U i − ) (cid:1) ⊆ ˇ X. (41)Notice that l γ i +3 = l γ i − is the the boundary of both U i +2 and U i − . Then (37)(41) together imply theinclusion α i (cid:0) Trop − i ( R \ l γ i +3 ) (cid:1) ⊆ ˇ X. (42)Then Lemma 5.3 guarantees that the inclusion extends over the ray l γ k and we reach an extension α i : Trop − i ( R \ { } ) → ˇ X. Finally we claim that α i is an embedding restricting on Trop − ( U ) for small enough open subset U ⊆ R .On the other hand, α i is fibre-preserving with respect to Trop i : ( G anm ) → R and Trop : ˇ X → B and theinduced map on the base is piecewise-linear. Direct computation shows that induced map on the base isinjective. Therefore, α i is an embedding. Therefore ˇ X has a partial compactification (cid:83) i =1 ( G anm ) i / ∼ , withthe identification Φ i,j : ( G anm ) i → ( G anm ) j .In next section, we will show that the later has a compactification to the analytification of the del Pezzosurface of degree five by adding a cycle of five rational curves via ring of theta functions following [19]. Remark 5.12.
One would naturally expect that the family Floer mirror of the hyperK¨ahler rotation of X (cid:48) t still compactifies to the del Pezzo surface of degree five. In this case, one there is only two families ofholomorphic discs in each of the singularities and one can glue the local model in [27, Section 8] and geta partial compactification of the family Floer mirror. The author will compare it with the Gross-Siebertconstruction of the mirror in the future work. Remark 5.13.
Shen-Zaslow-Zhou uses the homological mirror symmetry for the A cluster variety with ancanonical equivariant Z action [37] 22 .2 Comparison with GHK Mirror of dP Let Y be the del Pezzo surface of degree five and D be the anti-conical divisor consists of wheel of fiverational curves. Here we will explain the comparison of the family Floer mirror of X II with the GHK mirrorof ( Y, D ). Recall that in Lemma 5.8, we identify the integral affine structures on B and B GHK . Moreover,the BPS rays naturally divide B into cones which is exactly the cone decomposition of B GHK . The canonicalscattering diagram in this case is computed in [19, Example 3.7] and all the A - curves are shown in Figure8. Lemma 5.14.
There exists a homeomorphism X II ∼ = Y \ D .Proof. From the explicit equation in Section 3, a deformation of X II has two singular fibres of type I andthe vanishing cycles has intersection number 1. On the other hand, [4, Example 3.1.2] provides the localmodel of Lagrangian fibration near the blow-up of a point on the surface. Since Y can be realized as the blowup of two non-toric boundary point on del Pezzo surface of degree 7, One can topologically glue the pull-backof the moment map torus fibration with the local Lagrangian fibration to get a torus fibration on Y \ D withtwo nodal fibres such that the vanishing cycles has intersection 1. This gives the homeomorphism between X II and Y \ D topologically and the identification of the class of tori among H ( X II , Z ) ∼ = H ( Y \ D, Z ). Inparticular, we can use Y as an auxiliary topological compactification of X II .We will take P = NE( Y ) in the Gross-Hacking-Keel construction. We have P gp R ∼ = Pic( Y ) ∗ ∼ = H ( Y, Z ),where the first isomorphism comes from Poincare duality and Y is projective while the second isomorphismcomes from H , ( Y ) = H , ( Y ) = 0 . The rank two lattice H ( L u , Z ) glues to a local system of lattice over B and naturally identified with Λ B by Remark 3.2. Then we have the commutative diagram except themiddle map. Here H ( Y, Z ) denotes the constant sheaf with fibre H ( Y, Z ) over B .0 (cid:47) (cid:47) P gp R (cid:47) (cid:47) P r (cid:47) (cid:47) Λ B (cid:47) (cid:47) (cid:47) (cid:47) H ( Y, Z ) (cid:47) (cid:47) ∼ = (cid:79) (cid:79) (cid:83) u ∈ B H ( Y, L u ) ∂ (cid:47) (cid:47) Ψ (cid:79) (cid:79) (cid:83) u ∈ B H ( L u , Z ) ∼ = (cid:79) (cid:79) (cid:47) (cid:47) Y, D ) is modeled by ( C x i ,y i , { x i y i = 0 } ) near a node D i ∩ D i +1 . The torus fibre in Y \ D is isotopic to L = {| x i | = | y i | = 1 } . It is easy to see that L boundstwo family of holomorphic discs {| x i | ≤ , y i = const } and { x i = const, | y i | ≤
1. Denote β i ∈ H ( Y, L ) berelative class of the discs intersecting D i . Over the simply connected subset U i ⊆ B , both of the shortexact sequence in (43) splits (non-canonically) and we define the middle map by Ψ( β i ) = φ ρ i ( v i ). FromRemark 3.2, the right hand side square commutes and ∂β i (up to parallel transport) generate H ( L u , Z ).Therefore, the five lemma implies Ψ is an isomorphism over U i and the two short exact sequences in (43)over U i are identified. To see that the middle map is independent of i , one has the following observation:We may choose u to be in a neighborhood of D i , which is diffeomorphic to N D i /Y ∼ = O P ( D i ). Use therelation x i +1 = y − i , y i +1 = x i y − D i i , one has ∂β i − + D i ∂β i + ∂β i +1 = 0, which is the analogue of (1). Tosee lifting the relation (1) in (cid:83) u ∈ B H ( Y, L u ), notice that the 2-chains realizing β i − , β i +1 , D i β i with thesame boundary condition Q ∈ B glue to a 2-cycle from (1) up to a multiple of fibres, which is contractiblein Y . As the boundary condition u moves towards D i , the resulting 2-cycle is homotopic to D i (see Figure44). Therefore, this implies that β i − + D i β i + β i +1 = [ D i ] , (44)which is the analogue of (4). 23 D i +1 D i D i − D i +1 D i D i − convergesFigure 6: Illustration for (44).Therefore, the middle map is well-defined from (44) and the middle map is an isomorphism by the fivelemma. Notice that β i + γ i represents a 2-chain, which defines a 2-cycle up to a multiple of the fibre. Sincethe fibre is contractible in Y , thus we may view β i + γ i as a 2-cycle in H ( Y, Z ). Notice that [ E i ] is theunique class with intersections [ E i ] . [ D j ] = δ ij , we have z [ E i ] − φ ρi ( v i ) identified with z γ i (see Figure 7). D i γ i β i Figure 7: The class [ E i ] decomposes into sum of γ i and β i In particular, the transformation Φ i,i +1 coincides with (5). This will leads to the identification of ˇ X andthe GHK mirror of ( Y, D ) as gluing to tori. Notice that the Gross-Hacking-Keel mirror of (
Y, D ) comeswith a family over Spec C [NE( Y )]. We will have to determine which particular point in Spec C [NE( Y )] thefamily Floer mirror ˇ X corresponds to. Notice that the monodromy sends γ i to γ i +1 . This implies that ˇ X corresponds to the point such that the value of z [ E i ] all coincides. From the explicit relation of curve classes[ E i ], ˇ X corresponds to the point where z [ D i ] = z [ E i ] = 1.Indeed, one can see this via the identification ˇ X with in the subset of the analyticiation of del Pezzosurface of degree 5. We will see in the next section (Section 5.3) that this is the cluster variety of type A . Recall that the Gross-Hacking-Keel mirror is determined by the algebraic equations (45) from the thetafunctions [19, Equation (3.2)], ϑ i − ϑ i +1 = z [ D i ] ( ϑ i + z [ E i ] ) . Comparing with (6) (and later (45)), we see that the family Floer mirror ˇ X corresponds to the fibre with z [ D i ] = z [ E i ] = 1 . To conclude Section 5.1, Section 5.3 and Section 5.2, we have
Theorem 5.15.
The analytification of X -cluster variety of type A or the Gross-Hacking-Keel mirror of ( Y, D ) is a partial compactification of the family Floer mirror of X II . A -Cluster Variety In this section, we will prove that the family Floer mirror constructed in Section 5.1 is simply the X -clustervariety of type A . The X -cluster algebra of type A are defined in Section 2.1 with d = d = 1. The24 y E x H − E x H − E y H − E x − E y Figure 8: The canonical scattering diagram and the A -curves in del Pezzo surfaces of degree 5.following observation helps to link the scattering diagram in Theorem 5.1 and X the scattering diagram oftype A .The operation we are going to have can be viewed as a symplectic analogue of “pushing singularities toinfinity” in [19]. Recall that if one has a special Lagrangian fibration with a focus-focus singularity at u and Lefschetz thimble γ . Then there exist two affine rays l ± γ emanating from u on the base, parametrizingspecial Lagrangian fibres bounding holomorphic discs in classes ± γ . Then l ± γ divide a neighborhood of u into two chambers U ± , where U ± is characterized by (cid:82) ± γ ImΩ >
0. The corresponding wall-crossing across l ± γ from U − to U + is K ± γ and the monodromy around u is given by M in Claim 5.5. We make a branch cutfrom u to infinity and the parallel transport should changed by M when crossing the branch cut. Noticethat the three transformations K ± γ and M commute. If we choose the cut coincides with l − γ , then thetransformation crossing l − γ from U − to U + is K γ , coincides with the transformation crossing l γ from U − to U + . Similarly, if we choose the cut coincides with l γ , then the transformation crossing l γ from U + to U − is K − γ , coincides with the transformation crossing l − γ from U + to U − .To sum up, choosing the branch cut coinciding with l − γ makes the transformation across l ± γ from U − to U + both equal to K γ , as if the singularity u is moved to infinity along l − γ . Similarly, if we choose thebranch cut coincides with l γ , then the transformation from U − to U + is K − γ as if the singularity is movedto infinity along l γ .Now back to the scattering diagram in Theorem 5.1. We can express the underlying integral affinestructure on B in a different way by choosing different branch cuts. First we decompose M = M M , where M , M are the Picard-Lefschetz transformations with vanishing cycles γ (cid:48) , γ (cid:48) . Choose the branch cut to be l γ (and l γ ) with the corresponding identifications to be M (and M respectively) as in Figure 9. Thenfrom the previous discussion in this section and the same argument in Section 5.1, the family mirror is thusgluing of five tori with the gluing coincide with those of the A -cluster variety ˇ X C .25 γ l γ l γ l γ l γ M M Figure 9: The different choice of branch cuts for X II .Note that one can similarly define theta function in the analytic situation. Since we are working withfinite type, we can express theta functions in different torus charts by path ordered products. The functionsare well defined since the scattering diagram is consistent (see Lemma 5.3). Further note that, in the finitecase, we can replicate (6) to define multiplications between theta functions without broken lines . Standardand straight-forward calculation shows that ϑ v i − · ϑ v i +1 = 1 + ϑ v i , (45)where v i denotes the primitive generator of l γ i i ∈ { , . . . , } ordered cyclically. We can see it agrees withthe exchange relations as in Section 2.1. This gives a natural embedding of ˇ X C into P after suitablehomogenization of (45) thus compactified to a del Pezzo surface of degree five. X III
In this section, we will consider the case when Y (cid:48) = Y (cid:48) III be a rational elliptic surface with singular config-uration
III ∗ III , D (cid:48) is the type III ∗ fibre. We claim that the family Floer mirror of X = X III is then thedel Pezzo surface of degree 6. The argument is similar to that in Section 5.First of all, such Y (cid:48) has the explicit affine equation y = x + u. It is easy to see that the fibre over u = 0 is a singular fibre of type III , while the fibre at infinity is of type
III ∗ . There is a natural deformation Y (cid:48) t be the minimal resolution of the surface { z y = x + 4 t x z + uz } ⊆ P x : y : z ) × P s : u ) such that there are two singular fibres of type I , I with near u = 0, | t | (cid:28)
1. With vanishing thimbles γ (cid:48) and γ (cid:48) , γ (cid:48) . By Theorem 4.6, we have the analogue of Theorem 5.1. Theorem 6.1. [34, Theorem 4.12]
There exist γ (cid:48) , γ (cid:48) , γ (cid:48) ∈ H ( X, L u ) ∼ = Z such that (cid:104) γ (cid:48) , γ (cid:48) (cid:105) = (cid:104) γ (cid:48) , γ (cid:48) (cid:105) = 1 , (cid:104) γ (cid:48) , γ (cid:48) (cid:105) = 0 and Z γ (cid:48) = Z γ (cid:48) . Moreover, if we set γ = − γ (cid:48) , γ = γ (cid:48) , γ = γ (cid:48) + γ (cid:48) + γ (cid:48) , γ = γ (cid:48) + γ (cid:48) , γ = γ (cid:48) , γ = − γ (cid:48) . Then1. f γ ( u ) (cid:54) = 1 if and only if u ∈ l γ i and γ = γ i for some i ∈ { , · · · , } .2. In such cases, f γ i = (cid:40) T ω ( γ i ) z ∂γ i if i odd, (1 + T ω ( γ i ) z ∂γ i ) if i even. In general, the products of theta functions can be expressed as the linear combination of theta functions [19, 21], which thecoefficients can be computed via broken lines. . If we choose the branch cut between l γ and l γ , then the counter-clockwise mondoromy M across thebranch cut is given by γ (cid:48) (cid:55)→ − ( γ (cid:48) + γ (cid:48) + γ (cid:48) ) γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) . (46)Notice that from the condition Z γ (cid:48) = Z γ (cid:48) , we have l γ (cid:48) = l γ (cid:48) and l γ (cid:48) + γ (cid:48) = l γ (cid:48) + γ (cid:48) . Then we compute thecentral charges Z γ i , which is parallel to Lemma 5.7. Taking the branch cut between l γ and l γ , we wouldobtain the diagram as in Figure 10. γ = − γ (cid:48) γ = γ (cid:48) γ = γ (cid:48) + 2 γ (cid:48) γ = γ (cid:48) + γ (cid:48) γ = γ (cid:48) γ = − γ (cid:48) Figure 10: BPS rays near the singular fibre in X III . Lemma 6.2.
With suitable choice of coordinate u on B ∼ = C ∗ , we have Z γ k ( u ) = (cid:40) e πi ( k − u if k odd, − i e πi ( k − u if k even. (47) In particular, the angle between l γ k and l γ k +1 is π . See how the BPS rays position as demonstrated in Figure10.Proof. Straight-forward calculation shows that Z γ k ( u ) = O ( | u | ). Normalize the coordinate u such that Z γ ( u ) = u . Notice that M γ k = γ k +2 , the case for k being odd follows immediately. Similarly, when k iseven, Z γ k ( u ) = ce πi ( k − u , for some c ∈ C . With Z γ + Z γ = Z γ we gets c = − i .We will take U i be the sector bounded by l γ i and l γ i +1 . Let ˇ X to be the family Floer mirror constructedby Tu [39]. Again we denote the embedding α i : Trop − i ( U i ) → ˇ X . From Lemma 6.2, x i > l γ i and angle π ×
2. Thus, α i can be extended to Trop − i (cid:18)(cid:83) k = i +2 k = i − U k (cid:19) . Followingthe same line of Lemma 5.10 and Lemma 5.11, α i extends to Trop − (cid:18)(cid:83) k = i +3 k = i − U k (cid:19) . Finally, α i extends over l γ i +4 from the following analogue of Lemma 5.3. The proof is similar and we will omit the proof. Lemma 6.3.
The composition of the wall-crossing transformations cancel out the monodromy. Explicitly, K γ K γ K γ K γ K γ K γ ( z γ ) = z M − γ . Similar to the argument of Section 5.3, we may change the branch cut in Figure 10 into two, as in Figure11. The explicit gluing functions of B -cluster variety can be found in [5, p.54 Figure 4.1]. Then the familyFloer mirror ˇ X can be partially compactified to gluing of six tori (up to GAGA) with the gluing functionsame as the X cluster variety of type B . One can compute the product of the theta functions via broken27ines and obtain ϑ v ϑ v = 1 + ϑ v ,ϑ v ϑ v = (1 + ϑ v ) ,ϑ v ϑ v = 1 + ϑ v ,ϑ v ϑ v = (1 + ϑ v ) ,ϑ v ϑ v = 1 + ϑ v ,ϑ v ϑ v = (1 + ϑ v ) , (48)where v i denotes the primitive generator of l γ i for i ∈ { , . . . , } ordered cyclically. By [6], Cheung andMagee showed that the compactification of the cluster variety of type B is the del Pezzo surface of degree6. l γ l γ l γ l γ l γ l γ M M Figure 11: The choice of a different branch cut for X III .To compare with the mirror constructed by Gross-Hacking-Keel, we take the corresponding log Calabi-Yau pair (
Y, D ) with Y the del Pezzo surface of degree six. Since all del Pezzo surfaces of degree 6 areisomorphic, we will identify it with the blow up of P at three points, two non-toric points on y -axis andone non-toric point on x -axis. The anti-canonical divisor D is the proper transform of the x, y, z -axis of P .Denote H be the pull-back of the hyperplane class, E , (and E , E ) be the exceptional divisor of the blowup on x -axis (and y -axis). Lemma 6.4.
There is an isomorphism of affine manifolds B GHK ∼ = B .Proof. From [19, Lemma 1.6], toric blow-ups corresponds to the refinement of cone decomposition but notchange the integral affine structure. We will find a successive toric blow-ups of ( ˜
Y , ˜ D ) → ( Y, D ) suchthat not only the corresponding integral affine structure with singularity coincides with B but also its conedecomposition coincide with the chamber structure bounded by the BPS rays. Such ˜ Y is the ordered blowup the intersection point of the x, z -axis, the proper transform of the z -axis and the exceptional divisor,the proper transform of y, x -axis. Then we take ˜ D to be the pull-back of the x, y, z -axis. If we take theproper transform of y -axis as ˜ D and number the boundary divisors in counter-clockwise order, then we have˜ D i = − i odd and ˜ D i = − i even.Use (6.2), we have l γ = { ˇ x > , ˇ y = 0 } l γ = { ˇ y > , ˇ x = 0 } . and we will identify l γ = R > (1 ,
0) and l γ = R > (0 ,
1) and the rest of the proof is similar to that in Lemma5.8.Same argument of Lemma 5.14, we have a homeomorphism between X III ∼ = Y \ D ∼ = ˜ Y \ ˜ D and ˜ Y provides a compactification of X III . For the later discussion, we will replace (
Y, D ) by ( ˜
Y , ˜ D ) for the restof the section (see Remark 2.1). Similarly, we have the identification of the short exact sequence (43).Next we need to compute the canonical scattering diagram for ( Y, D ). Let D i be the components of D with D i are exceptional curves when i even. 28 emma 6.5. Under the identification of integral affine structures with singularities B ∼ = B GHK , the canon-ical scattering diagram of Gross-Hacking-Keel coincides with the scattering diagram in Theorem 6.1 viaidentification z [ C i ] − φ ρi ( v i ) = z γ i (or z [ C ji ] − φ ρi ( v i ) = z γ i ) for i is odd (or even).Proof. We will first compute all the A -curves of ( Y, D ), which is standard and we just include it for self-completeness. Any irreducible curves, in particular the irreducible A curves in ( Y, D ) are either exceptionalcurves of blow-up from P or proper transform of a curve C ⊆ P . All the three exceptional curves are A -curves intersecting D i for i odd. If C is of degree one and its proper transform is an A -curve, then iteither1. passes through two of the blow up points and its proper transform intersect ˜ D i for i odd. There arethree such lines.2. passes through one blow up point and one intersection of toric 1-stratum. There are three such linesand intersect ˜ D i for i even.There are no higher degree curves with proper transform are A -curves and we draw the canonical scatteringdiagram and the corresponding A -curves in Figure 12.Since D ∈ | − K Y | is ample, there is no holomorphic curves contained in Y \ D . In particular, all thesimple A -curves are irreducible and all the possible A -curves are the multiple covers of the above ones. Thecontribution of multiple covers of degree d is ( − d − /d by [18, Proposition 6.1]. Then the lemma followsfrom the definition of the canonical scattering diagram [19, Definition 3.3]. Then the function attached tothe ray ρ i is f i = (cid:40) (1 + z [ C i ] − φ ρi ( v i ) ) , if i is odd, (cid:81) j =1 (1 + z [ C ji ] − φ ρi ( v i ) ) , if i is even, (49)where C i , C ji are the A -curve classes corresponding to l γ i in Figure 12. The assumption Z γ = Z γ impliesthat z E ] = z [ E ] . Notice that the monodromy of the only singular fibre shifts γ i to γ i +2 . This implies thatone would also need to identify z [ E ] = z [ H − E i ] = z [2 H − E − E − E ] z [ E i ] = z [ H − E i ] = z [ H − E − E i ] , i = 2 , . Equivalently, this corresponds to z [ D i ] = z [ C i ] = z [ C ji ] = 1 .E E i , i = 2 , H − E i , i = 2 , H − E H − E − E i , i = 2 , H − E − E − E l γ l γ l γ l γ l γ l γ Figure 12: The canonical scattering diagram and the A -curves in del Pezzo surfaces of degree 6.29he GHK mirror can be computed via the spectrum of the algebra generated by theta functions. Theproducts of the theta functions ϑ i − ϑ i +1 = z [ D i ] 2 (cid:89) j =1 (cid:16) ϑ i + z [ C ji ] (cid:17) for i even, ϑ i − ϑ i +1 = z [ D i ] (cid:16) ϑ i + z [ C i ] (cid:17) for i odd . Again compare it with the analogue relations (48) from X -cluster algebra of type B , we conclude thatthe family Floer mirror ˇ X corresponds to the particular fibre of the GHK mirror characterized by z [ D i ] = z [ C i ] = z [ C ji ] = 1 . To sum up, we conclude the section with the following theorem.
Theorem 6.6.
The family Floer mirror of X III has a partial compactification as the analytification of the B -cluster variety or the Gross-Hacking-Keel mirror of suitable pair ( Y, D ) . In particular, the family Floermirror of X III can be compactified as the analytification of a del Pezzo surface of degree . X IV In this section, we will consider the case when Y (cid:48) be a rational elliptic surface with singular configuration IV ∗ IV and D (cid:48) is the type IV ∗ fibre. We claim that the family Floer mirror of X is then the del Pezzosurface of degree 4. The argument is also similar to that in Section 5. Such rational elliptic surface Y (cid:48) hasWeiestrass model y = x + t s . (50) Theorem 7.1. [34, Theorem 4.14]
There exist γ (cid:48) , γ (cid:48) , γ (cid:48) , γ (cid:48) ∈ H ( X, L u ) ∼ = Z such that (cid:104) γ (cid:48) , γ (cid:48) i (cid:105) = 1 , (cid:104) γ (cid:48) i , γ (cid:48) j (cid:105) = 0 and Z γ (cid:48) i = Z γ (cid:48) j , for i, j ∈ { , , } . Moreover, if we set γ = − γ (cid:48) , γ = γ (cid:48) , γ = γ (cid:48) + γ (cid:48) + γ (cid:48) + γ (cid:48) , γ = γ (cid:48) + γ (cid:48) + γ (cid:48) ,γ = 2 γ (cid:48) + γ (cid:48) + γ (cid:48) + γ (cid:48) , γ = γ (cid:48) + γ (cid:48) , γ = γ (cid:48) , γ = − γ (cid:48) . Then1. f γ ( u ) (cid:54) = 1 if and only if u ∈ l γ i and γ = γ i for some i ∈ { , · · · , } .2. In such cases, f γ i = (cid:40) T ω ( γ i ) z ∂γ i if i odd, (1 + T ω ( γ i ) z ∂γ i ) if i even.3. If we choose the branch cut between l γ and l γ , then the counter-clockwise mondoromy M across thebranch cut is given by γ (cid:48) (cid:55)→ − ( γ (cid:48) + γ (cid:48) + γ (cid:48) + γ (cid:48) ) γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) (51) γ (cid:48) (cid:55)→ γ (cid:48) + γ (cid:48) . (52) Lemma 7.2.
With suitable choice of coordinate u on B ∼ = C ∗ , we have Z γ k ( u ) = (cid:40) e π i ( k − u if k odd, √ e π i ( k − e − πi u if k even. (53) In particular, the angle between l γ i and l γ i +1 is π . See how the BPS rays position as demonstrated in Figure13. = − γ (cid:48) γ = γ (cid:48) γ = γ (cid:48) + γ (cid:48) + γ (cid:48) + γ (cid:48) γ = γ (cid:48) + γ (cid:48) + γ (cid:48) γ = 2 γ (cid:48) + γ (cid:48) + γ (cid:48) + γ (cid:48) γ = γ (cid:48) + γ (cid:48) γ = γ (cid:48) γ = − γ (cid:48) Figure 13: BPS rays near the singular fibre in X IV . Note in Theorem 7.1, we have Z γ (cid:48) i = Z γ (cid:48) j , for i, j ∈{ , , } . Proof.
One can check that Z γ ( u ) = O ( | u | ) and let Z γ k ( u ) = c k u . Using the relations between γ i andstraight-forward calculation show that c = 1 , c = 1 √ e − πi , c = e − πi , c = − i √ u . Then use the relation M γ i = γ i +4 to determines the rest of c k . With the data above, the similar argument in Section 5.1 shows that the family Floer mirror of X IV isgluing of eight copies of Trop ( R \ { } ) ⊆ ( G anm ) , with the gluing functions in Theorem 7.1. Similar to theargument of Section 5.3, we may change the branch cut in Figure 13 into two, as in Figure 14. γ = − γ (cid:48) γ = γ (cid:48) γ = γ (cid:48) + γ (cid:48) + γ (cid:48) + γ (cid:48) γ = γ (cid:48) + γ (cid:48) + γ (cid:48) γ = 2 γ (cid:48) + γ (cid:48) + γ (cid:48) + γ (cid:48) γ = γ (cid:48) + γ (cid:48) γ = γ (cid:48) γ = − γ (cid:48) Figure 14: A choice of a different branch cut for X IV The scattering diagram of cluster type G can be found in [21, Figure 1.2]. One can show that the corre-sponding gluing functions of the X case are the same as those in Theorem 7.1 under suitable identification.Then the family Floer mirror of X IV can be partially compactified to gluing of eight tori (up to GAGA)with the gluing functions same as the X -cluster variety of type G .Next we will construct a log Calabi-Yau pair ( Y, D ) such that the corresponding Gross-Hack-Keel mirrorcorresponds to the family Floer mirror of X IV . We will take1. Y to be the blow up of P at 4 points, three of them are the non-toric points on y -axis and one non-toricpoint on x -axis.2. D is the proper transform of x, y, z -coordinate axis.Let ˜ Y be the successive toric blow up of ( Y, D ) at the intersection of x, z -axis, the proper transform of z -axisand the exceptional divisor, the two nodes on the last exceptional divisor and then the proper transform31f y, z -axis in order. Then take ˜ D to be the proper transform of D . Denote H to be the pull-back of thehyperplane class on P , E (and E , E , E ) to be the exceptional divisor of the blow up on the non-toricpoint on the x -axis (and y -axis).Similar to the argument Section 5.2 we have the following lemma. Lemma 7.3.
The complex affine structure on B together with l γ i is isomorphic to the integral affinemanifold B GHK of ( ˜ Y , ˜ D ) . Moreover, the BPS rays l γ i give the correoding cone decomposition on B GHK from ( ˜
Y , ˜ D ) , the wall function with restriction z [ D i ] = z [ E i ] = 1 and the identification d coincide with thefunctions in Theorem 7.1 We then can compute the canonical scattering diagram for (
Y, D ). Actually all the simple A -curvescontributing to the scattering diagram are toric transverse in ( ˜ Y , ˜ D ), which are depicted in Figure 15 below. E E i , i = 2 , , H − E i , i = 2 , , H − E H − E − E i H − ( (cid:80) i =1 E i )2 H − E − E i − E j ,3 H − E − E − E − E { i, j } = { , , } Figure 15: The canonical scattering diagram and the A -curves corresponding to X IV .We conclude the section with the following theorem. Theorem 7.4.
The family Floer mirror of X IV has a partial compactification as the analytification of the B -cluster variety or the Gross-Hacking-Keel mirror of a suitable pair ( Y, D ) . Here we consider the family Floer mirror of X without the geometry of its compactification. Following theidea of the Gross-Hacking-Keel as summarized in Section 2.1, one would need to use the theta functions,the tropicalization of the counting of Maslov index two discs, to construct a (partial) compactification ofthe original mirror. Assuming that X = X ∗ in the previous sections admit a compactification to a rationalsurface with an anti-canonical cycle at infinity. Moreover, assume that the there is certain compatibilitybetween the compactification and the asymptotic of the metric behaviour. Then one can follow the similarargument in the work of the second author [32] and prove that the counting of the Maslov index two discswith Lagrangian fibre boundary conditions can be computed by the weighted count of broken lines. However,the authors are unaware of such asymptotic estimates of the metrics in the literature.One can further construct the pair ( Y, D ) such that the corresponding monodromy is conjugate to themonodromy of the type IV ∗ , III ∗ , II ∗ , I ∗ . For instance, the case of I ∗ can be realized by a cubic surfacewith anti-canonical cycle consisting of three ( − X = Y \ D coincides with a particular fibre in the mirror family constructed by Gross-Hacking-Keel. Moreover, the families of Maslov index zero discs emanating from the singular fibres in X areone-to-one corresponding to the A -curves of the pair ( Y, D ). This may help to understand the Floer theoryof more singular Lagrangians. In this case, the wall functions are algebraic functions and the GAGA canstill apply. Although the walls are dense, it is likely the mirror can be covered by finitely many tori up to32ome codimension two locus. In general, the wall functions may not be algebraic a priori and GAGA maynot apply directly. The authors will leave it for the future work.
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Man-Wai CheungDepartment of Mathematics, One Oxford Street Cambridge, Harvard University, MA 02138,USA e-mail: [email protected]
Yu-Shen LinDepartment of Mathematics and Statistics, 111 Cummington Mall, Boston, Boston Univer-sity, MA 02215, USA e-mail:e-mail: