TTropical Lagrangians in toric del-Pezzo surfaces
Jeff HicksAugust 18, 2020
Abstract
We look at how one can construct from the data of a dimer model a Lagrangiansubmanifold in ( C ∗ ) n whose valuation projection approximates a tropical hypersurface.Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropicalLagrangian submanifold, forming a Lagrangian mutation seed. Using this we findtropical Lagrangian tori L T in the complement of a smooth anticanonical divisor ofa toric del-Pezzo whose wall-crossing transformations match those of monotone SYZfibers.An example is worked out for the mirror pair ( CP \ E, W ) , X . We find asymplectomorphism of CP \ E interchanging L T and a SYZ fiber. Evidence is providedthat this symplectomorphism is mirror to fiberwise Fourier-Mukai transform on X . Contents a r X i v : . [ m a t h . S G ] A ug Lagrangian tori in toric del-Pezzos 29 CP . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 A -Model on CP \ E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Tropical geometry plays an important role in mirror symmetry, a duality proposed in[COGP91] between symplectic geometry on a space X , and complex geometry on a mirrorspace ˇ X . A proposed mechanism for constructing pairs of mirror geometries comes fromSYZ mirror symmetry ([SYZ96]) where X and ˇ X have dual Lagrangian torus fibrations overa common affine manifold Q . X ˇ XQ val ˇval From this viewpoint, mirror symmetry is recovered by degenerating the symplectic geometryof X and complex geometry of ˇ X to tropical geometry on the base Q . In the complexsetting, this degeneration was studied by [KS01; Mik05], where a correspondence betweenthe valuations of complex curves (called the amoeba) and tropical curves was established.More recently, tropical-Lagrangian correspondences have been constructed in the parallelworks of [Mik18; Mat18; Hic19; MR19]. These papers show that for a given tropical curve V ⊂ Q there exists a Lagrangian submanifold L ( V ) ⊂ X with val( X ) approximating V .A more precise relation of these two geometries is the homological mirror symmetryconjecture of [Kon94]. This predicts that Lagrangian submanifolds of X and complex sub-manifolds of ˇ X should be compared as objects via a mirror functor between the categoriesFuk( X ) and D b Coh( ˇ X ). An expectation is that homological and SYZ mirror symmetryinteract by relating Lagrangian torus fibers of val : X → Q to skyscraper sheaves of pointson ˇ X , and sections of the Lagrangian torus fibration to line bundles of ˇ X .This intuition was used in [Abo09] which proved that the Fukaya-Seidel categoryFuk(( C ∗ ) n , W Σ ) is equivalent to D b Coh( ˇ X Σ ), the derived category of coherent sheaves on amirror toric manifold. This was achieved by using tropical geometry to construct Lagrangiansections of val : ( C ∗ ) n → R n , and to show that these were mirror to line bundles on ˇ X Σ . In[Hic19], it was shown that the tropical-Lagrangian and tropical-complex correspondencesare compatible with this mirror functor, in the sense that when a tropical hypersurface V is approximated by ˇval( D ) for a divisor D , the Lagrangian L ( V ) is mirror to the sheaf O D .This extends the relation between homological and SYZ mirror symmetry to sheaves beyondline bundles and skyscrapers of points. 2 .2 Wall-Crossings and Lagrangian Mutation Lagrangian submanifolds have a moduli as objects of the Fukaya category. For example, themoduli space of Lagrangian torus fibers of the SYZ fibration is expected to be the mirrorspace ˇ X . A hands-on approach to understanding this moduli space is to build complexcoordinate charts. These coordinate functions are constructed by taking an appropriatelyweighted count of holomorphic disks with boundary on L . A first example are the producttori in C , whose holomorphic disks have areas dependent on the radii chosen to constructthe tori.The presence of bubbling of holomorphic disks in families of Lagrangians leads to a diffi-culty in this theory where a discontinuity appears in the disk counts used to construct thesecoordinates. In [Aur07] these discontinuities are explained in terms of a wall-crossing correc-tion which describes how such bubblings can be appropriately incorporated into coordinateson the space. For example, the monotone Chekanov and product tori in C are related by aLagrangian isotopy which exhibits one of these wall-crossing corrections.This technique inspired [Via14] to produce examples of non-Hamiltonian isotopic mono-tone Lagrangians in toric del-Pezzos. These Lagrangians are constructed by Lagrangianisotopies where wall-crossing occurs; thus the Lagrangians have related (but not equal) holo-morphic disk counts. This distinguishes the Hamiltonian isotopy classes of these Lagrangiansubmanifolds. A framework for this story was developed by [PT17] , which showed thatLagrangians constructed via Lagrangian mutation (a kind of Lagrangian surgery presentedin [Hau15]) had disk counts which were related by a wall-crossing transformation. The ex-amples considered in [Via14] were shown to be constructed via this Lagrangian mutationprocess. The goal of this paper is to extend the constructions of [Hic19] to Lagrangian fibrations X → Q which are almost toric (and so may admit some fibers with singularities). In doingso, we shed some light on questions laid out in [Mat18, section 6.3] regarding the homologicalmirror symmetry interpretation of monotone tropical Lagrangian tori in toric del-Pezzos.This paper first provides an alternate description of the tropical Lagrangian submanifoldsfrom [Hic19] using the combinatorics of dimers (classically, an embedded bipartite graph G ⊂ T ). To a dimer we construct a Lagrangian in ( C ∗ ) n whose valuation projection liesnear a tropical curve (definition 3.1.4). The argument projection arg : ( C ∗ ) n → T n of thisLagrangian is related to the dimer initially chosen. We can find a set of Lagrangian mutationsbased on the combinatorics of the dimer graph. Lemma (Dimer-Mutation Correspondance, Restatement of 3.3.3) . Let L be a Lagrangiandescribed by the dimer G ⊂ T . Suppose a face f of G has boundary satisfying the zeroweight condition (definition 3.3.1). Then we can construct another Lagrangian by mutation, µ D f L , whose argument projection can be explicitly described by another dimer. This motivates the construction of tropical Lagrangian submanifolds inside of toric del-Pezzos. In the case where dim C ( X ) = 2, the singular fibers of a toric fibration X → Q can3e chosen to be of a particularly nice form. We then call Q an almost-toric base diagram,which has the structure of a tropical manifold. We show that tropical curves V ⊂ Q meetingthe discriminant locus of Q admissibly admit Lagrangian lifts L ( V ) ⊂ Q . We then use thisto construct some tropical Lagrangian tori in toric del-Pezzos. As Lagrangian tori, these areinteresting because in the complement of an anticanonical divisor they are not isotopic tothose constructed in [Via14]. Theorem (Restatment of 4.1.3,5.1.1) . Let X be a toric del-Pezzo. Let E ⊂ X be a smoothanticanonical divisor chosen so that there is an SYZ fibration X \ E → Q obtained frompushing in the corners of the Delzant polytope. There exists a tropical Lagrangian torus L T ⊂ X \ E which is not isotopic to F q , the fiber of the moment map. Furthermore both L T and F q bound matching configurations of Lagrangian antisurgery disks, giving them matchingLagrangian mutations. The observation that there exists a correspondence between the antisurgery disks withboundary on L T and F q suggests that, although they represent different objects in theFukaya category, their moduli spaces match up. Furthermore, we show that a variationof this construction works more generally whenever one has a certain kind of Lagrangianmutation seed.We look to mirror symmetry for why mutation configurations (like those considered byVianna in toric Fanos) give tropical Lagrangian tori, and restrict to the example of X = CP .The mirror to CP \ E is known to be X , an extremal rational elliptic surface. There is anautomorphism of D b Coh( X ) (a fiberwise Fourier-Mukai transform) which interchangesthe moduli of points with the moduli of degree 0 line bundles supported on the elliptic fibers.Provided that a generation result for the Fukaya category of CP \ E is known, we can statewhat L T is as an object of the Fukaya category. Theorem (Restatement of 5.3.2,5.3.8) . There exists a symplectomorphism g : ( CP \ E ) → ( CP \ E ) interchanging L T to F q . With assumption 5.3.6, L T is mirror to a line bundlesupported on an elliptic fiber of X . We now outline the rest of this paper, focusing on the construction of the Lagrangian L T ,its surgery disks, and the symplectomorphism g : CP \ E → CP \ E . Section 2 starts withsome necessary background and notation. In section 2.1, we look at Lagrangian surgery,antisurgery, and mutations. These are the tools which we use to build tropical Lagrangiansubmanifolds and to describe the Lagrangian mutation phenomenon which becomes the focusof inquiry. ?? reviews tropical geometry on affine manifolds, with an emphasis on dimensiontwo.Section 3 extends the results of [Hic19] to construct tropical Lagrangian submanifoldsfrom the data of a dimer. This involves giving a definition for a dual-dimer (definition 3.0.2)in higher dimensions as a collection of polytopes { ∆ ◦ v } , { ∆ • w } in the torus T n whose verticeshave a matching condition imposed on them (see fig. 1a). We show that such a collection of4 a) A dual dimer. The threewhite hexagons correspond toantisurgery disks for muta-tion. × val( L ) val( L ) (b) Two tropical curves re-lated by a nodal trade.These curves have isotopicLagrangian lifts. ×× × L T (c) The Lagrangian L T ⊂ CP \ E . The center vertexcorresponds to fig. 1a. Figure 1polytopes corresponds to a tropical hypersurface in R n . In section 3.1 we construct from thiscollection of polytopes a Lagrangian whose valuation projection lies nearby the correspondingtropical hypersurface, and whose argument projection matches the dual dimer. Section 3.2is a slight detour from the main focus of the paper to provide a combinatorial model for theFloer-theoretic support of a tropical Lagrangian in terms of the Kasteleyn operator (similarto the computation in [TWZ18] for microlocal sheaf theory).The Lagrangian mutation story is introduced in section 3.3, where we show that each faceof the dimer builds a Lagrangian antisurgery disk on the corresponding Lagrangian. Thesefaces arise as sections of the argument projection over the complement of the polytopes inthe dual dimer. We additionally show that Lagrangian mutation across these disks can beunderstood as a modification of the underlying combinatorial dimer. See fig. 1a.In section 4, we generalize beyond tropical Lagrangians in ( C ∗ ) n to tropical Lagrangiansin almost toric fibrations. We show that a tropical curve in an almost toric base diagram hasa Lagrangian lift by constructing a local model for the Lagrangian lift near the discriminantlocus. In dimension 2, we prove that deformations of tropical curves lift to Lagrangianisotopies of their Lagrangian lifts. Lemma (Nodal Trade for Tropical Lagrangians) . The local models for Lagrangian subman-ifolds in fig. 1b are Lagrangian isotopic.
This lemma becomes a convenient tool for constructing isotopies of Lagrangian submani-folds, and is based on a method in used in [AS18] to compare Lagrangians inside of Lefschetzfibrations. Both the lifting and isotopies of tropical curves are achieved by modeling a node inthe almost toric base diagram with a Lefschetz fibration. Tropical Lagrangians are describedin these neighborhoods as Lagrangian surgeries of Lagrangian thimbles.In section 5 we apply the tropical lifting construction from the previous section to buildtropical Lagrangian tori in toric del-Pezzos disjoint from an anticanonical divisor (see fig. 1c).The vertex of the tropical Lagrangian is modeled on a dimer. A computation shows that themutation directions of this dimer match the ones known from [Via17; PT17]. This constructsthe tropical Lagrangian tori from the first theorem.Finally, in section 5.2 we present an in-depth example of homological mirror symmetry forthe example of CP \ E following [AKO06]. The main observation is that we may choose E
5o be a member of the Hesse pencil of elliptics, which has a large amount of symmetry. Usingthis observation, we take g : CP \ E → CP \ E to be a pencil automorphism which fixes E , but switches its meridional and longitudinal directions. The Lagrangian L T is comparedto a Lagrangian in a neighborhood of E using the mutation and nodal-trade operation fortropical Lagrangians. It is then observed that F q , a fiber of the SYZ fibration, also may beisotoped so that it too lives near E . In a neighborhood of E , we see that g interchangesthese two Lagrangians. We also present L T as a surgery of Lagrangian thimbles which areexpected to generate the Fukaya category of CP \ E , which characterizes the mirror objectto L T in D b Coh( X ). This project wouldn’t be possible without the support of my advisor Denis Auroux duringmy studies at UC Berkeley.While working on this project, I benefited from conversations with Ailsa Keating, MarkGross, Diego Matessi, Nick Sheridan who have been very generous with their time andadvice. I also thank Jake Solomon for providing useful feedback on the introduction to thispaper. Additionally, I would like to thank Paul Biran, who provided me with great amountof mathematical and professional advice during my visit at ETH Z¨urich.A portion of this work was completed at ETH Z¨urich. This work was partially sup-ported by NSF grants DMS-1406274 and DMS-1344991 and by a Simons Foundation grant(
Lagrangian surgery is a tool for modifying a Lagrangian along its self intersection locus.It was introduced by [Pol91] in the case where a Lagrangian is immersed with transverseself-intersections. In this setting, a neighborhood of the transverse intersection is replacedwith a Lagrangian neck. We will be using two similar notions of surgery. One extension isantisurgery along isotropic surgery disks [Hau15].
Theorem 2.1.1 ([Hau15]) . Suppose that D k is an isotropic disk with boundary contained in L and cleanly intersecting L along the boundary. Then there exists an immersed Lagrangian α D ( L ) ⊂ X called the Lagrangian antisurgery of L along D , which satisfies the followingproperties • α D ( L ) is topologically obtained by performing surgery along D k , • α D ( L ) agrees with L outside of a small neighborhood of D k , • If L was embedded and disjoint from the interior of D k , then α D ( L ) has a single self-intersection point. D n the resulting Lagrangian has a single self-intersection. There exists a choice of surgery neckso that the resolution of the self-intersection of α D n ( L ) by Lagrangian surgery is L . However,if we choose a Lagrangian surgery neck in the opposite direction of the disk D n to combineanti-surgery with surgery, we can obtain a new embedded Lagrangian. Definition 2.1.2.
Let L be an embedded Lagrangian submanifold, and D n a surgery disk. Let α D ( L ) be obtained from D n by antisurgery. The mutation of L along D n is the Lagrangian µ D ( L ) obtained from α D ( L ) by resolving the resulting single self-intersection point with theopposite choice of neck. It is expected that Lagrangians submanifolds which are related by mutation give differentcharts on the moduli space of Lagrangian submanifolds in the Fukaya category, and thatthese charts are related by a wall crossing formula [PT17]. A typical example of Lagrangiansrelated by mutation are the Chekanov and Clifford tori in C obtained by taking two differentresolutions of the Whitney sphere.The second variation of Lagrangian surgery that we use is surgery along a non-transverseintersection with a particular collared neighborhood. This surgery replaces two Lagrangianswith one in a neighborhood of their symmetric difference. Proposition 2.1.3.
Let L and L be two Lagrangians with boundary. Let U be an openneighborhood of L ∩ L . Suppose there exists a choice of collar neighborhood for the boundaryof U B (cid:15) ( ∂U ) ⊂ U = ∂U × (0 , r ) r and a function f : U → R with the following properties: • The function f is decreasing and convex in the r -variable • The function vanishes on the complement of B (cid:15) ( ∂U ) . • In a sufficiently small Weinstein neighborhood B ∗ c U , the Lagrangian L | B ∗ c U is the graphof the section df .Then there exists a Lagrangian L (cid:15)U L satisfying the following properties: • L (cid:15)U L ⊂ B (cid:15) (( L ∪ L ) \ ( L ∩ L )) . • There exists a Lagrangian cobordism (in the sense of [BC14]) K : ( L , L ) (cid:32) L (cid:15)U L The proof is analogous to the proof for the case when U is contractible presented in[Hic19, Proposition 3.1.1.]. The notation α D ( L ) is chosen as the character α results from applying antisurgery on the character c . .2 Affine and Tropical Geometry We summarize a description of these tropical manifolds from [Gro11].
Definition 2.2.1. An integral tropical affine manifold with singularities is a manifold withboundary Q containing an open subset Q such that • Q is an integral affine manifold with an atlas whose transition functions are in SL ( Z n ) (cid:110) R n . • ∆ := Q \ Q , the discriminant locus, is codimension 2 • ∂Q ⊂ Q can be locally modelled after a SL ( Z n ) (cid:110) R n coordinate change on R n − k × R k ≥ . We will be interested in tropical manifolds where the discriminant locus additionallycomes with some affine structure. A tropical manifold is a pair ( Q, P ), where P is a polyhe-dral decomposition of Q . For a full definition of the data of a tropical manifold ( Q, P ), werefer the reader to [Gro11, Definition 1.27], and provide a short summary here. The verticesof this polyhedral decomposition are decorated with fan structures which are required tosatisfy a compatibility condition so that the polyhedra may be glued with affine transitionsacross their faces. The compatibility need not extend to affine transitions in neighborhoodsof the codimension 2 facets of the polyhedra, giving rise to the discriminant locus, a union ofa subset of the codimension 2 faces. This determines the affine structure on Q completely.We call such a manifold an integral tropical manifold if all of the polyhedra are lattice poly-hedra. For most of the examples that we consider, Q will be real 2-dimensional, and thenotions of tropical manifold and tropical affine manifold agree with each other. The majority of our focus will be in dim( Q ) = 2, where there is a graphical notation fordescribing the affine geometry on Q and correspondingly the symplectic geometry of the4-dimensional symplectic manifold X [LS+10]. To describe the affine structure on Q , wedescribe the monodromy around the singular fibers. This can be done diagrammaticallywith the following additional data. Definition 2.2.2.
Let ( Q, P ) be a 2-dimensional tropical manifold. Let Q be the set ofsingular points. At each point q i ∈ Q we define the eigenray R i ⊂ Q to be the ray inthe base starting at q i pointing in the eigendirection of the monodromy around q i . A basediagram is a map from Q \ (cid:83) i R i to R with the standard affine structure, with eigenraysmarked with a dashed line at each singularity. We decorate the points q i with the marker × k ,where the monodromy around q i is a k -Dehn twist. The Lagrangian fibers F q of X → Q can be described by the points in the base diagram. • If a point q ∈ Q \ R has a standard affine neighborhood, then F q is a Lagrangian torus.8 × × Figure 2: The nodal trade applied three times to the toric diagram of CP . The toric divisorgiven by a nodal elliptic curve is transformed into a smooth symplectic torus. • If the point q ∈ Q \ R has an affine neighborhood modelled on R × R ≥ then fiber F q is an elliptic fiber of corank 1, corresponding to an isotropic circle in X . • If the point q ∈ Q \ R has an affine neighborhood modelled on R ≥ × R ≥ , the fiber F q is an elliptic fiber of corank 2, which is simply a point in X . • If a point q ∈ Q \ R belongs to the discriminant locus, then the fiber is a Whitneysphere (if k = 1) or a plumbing of Lagrangians spheres (if k > nodal slide, nodal trade, and cut transfer are operations which modify the affinestructure of a base diagram Q but correspond to symplectomorphisms of X → Q . Thenodal trade modifies a base diagram by replacing an elliptic corank 2 fiber with a nodal fiberin the neighborhood of an elliptic corank 1 fiber. This replaces a corner with a nodal fiberwhose eigenline points in the balancing direction to the corner. See fig. 2. In the setting where Q = R n , a tropical hypersurface is defined via the critical locus of atropical function φ : Q → R . However, in the general setting of tropical manifolds thereare sets which are locally described by the critical locus of tropical functions but cannotbe globally described by a tropical function due to monodromy around the singular fibers.Since the construction of tropical Lagrangians only requires the differential of the tropicalfunction, this is not problematic. Definition 2.2.3.
Let Q be a tropical manifold. The sheaf of tropical differentials on Q isthe sheaf Ω aff on the space Q . It is given by the sheafification of the quotient: Ω aff ( U ) = { φ : U → R } / R where φ : U → R is a piecewise linear polynomial satisfying the following conditions: • dφ ∈ T ∗ Z U whenever dφ is defined, • For every point q ∈ U there exists an integral affine neighborhood B (cid:15) ( q ) so that therestriction φ | B (cid:15) ( q ) is concave. he sheaf R here is the sheaf of constant functions. The sheaf of integral tropical differentials is the subsheaf of constant sections of T ∗ Z ( Q ) .Let i : Q (cid:44) → Q be the inclusion. We define the sheaf of tropical sections to be thequotient sheaf dTrop := i ∗ (Ω aff ) /i ∗ ( T ∗ Z Q ) . We will call the sections of this sheaf the tropical sections, and denote them φ ∈ dTrop( U ). Given a tropical section φ , we denote the locus of non-linearity as V ( φ ) ⊂ Q .Should φ have a representation in each chart by a smooth tropical polynomial, we say that φ is smooth. Remark 2.2.4.
A point of subtlety: the quotient defining the sheaf of tropical sections isperformed over Q , not Q . Importantly, while the presheaves i ∗ (Ω aff ) / pre i ∗ ( T ∗ Z ( Q \ ∆)) i ∗ (Ω aff / pre T ∗ Z ( Q \ ∆)) agree, their sheafifications do not. In particular, the sheaf of tropical differentials rememberthat in the neighborhood of the discriminant locus, the tropical section must actually arisefrom a representative tropical differential. When Q = R n , there is no difference between the global sections of dTrop and thedifferentials of global tropical polynomials.Given a triple ( Q, P , φ ), one can construct a dual triple ( ˇ Q, ˇ P , ˇ φ ) using a process calledthe discrete Legendre transform. Away from the boundary the base manifolds Q and ˇ Q agreeas topological spaces, however their affine structures differ at the singular points. At theboundary these spaces are modified so that the non-compact facets of Q are compactifiedin ˇ Q and vice-versa. The simplest example of this phenomenon is when ˇ Q = ∆ Σ ⊂ R isa compact polytope. The Legendre dual to ˇ Q is the plane Q = R , equipped with a fandecomposition whose non-compact regions correspond to the boundary vertices of ˇ P .Given a tropical manifold Q , we can produce a torus bundle X = T ∗ Q /T ∗ Z Q over Q .This space X comes with canonical symplectic and almost complex structure arising fromthe affine structure on Q . In good cases this compactifies to an almost toric fibration X over Q . Similarly, we may produce a associated manifold ˇ X over ˇ Q . The pair of spaces X and ˇ X are candidate mirror spaces. When Q is non-compact we expect that Q is equippedwith additional data in the form of a monomial admissibility condition or stops in orderto obtain a meaningful mirror symmetry statement. This admissibility condition should beconstructed by considering the open Gromov Witten invariants of ˇ F p . The computation ofthese invariants is beyond the scope of our exposition, and we’ll be content with constructingour admissibility conditions in an ad-hoc manner. In [Gro11], these are called piecewise linear affine multi-valued functions This is an abuse of notation, as there may not be a globally defined function whose differential describesthis section. However, this will make the remainder of our discussion consistent with the notation used toconstruct tropical Lagrangians. × × (a) ×× × (b) ×× × (c) Figure 3: Tropical subvarieties associated to some tropical sections on CP \ E . A running example that we will use is the symplectic manifold CP \ E . One can constructan almost toric fibration for CP \ E by starting with the toric base diagram for CP . Byapplying nodal trades at each corner, we obtain a toric fibration val : CP → Q CP , wherethe boundary of Q CP is an affine S (see fig. 2). The preimage of val − ( ∂Q CP ) = E ⊂ CP ,a symplectic submanifold isotopic to a smooth cubic. CP \ E is an almost toric fibrationover the interior of this set, Q CP \ E = Q CP \ ∂Q . The monodromy around the three singularfibers allows us to construct some more interesting tropical sections of Q . We give three suchexamples of these sections and their associated tropical subvarieties below. • Tropical sections which have critical locus close to the boundary of Q CP \ E . Figure 3agives an example of such a section. Even though the critical locus appears to have threecorners, the affine coordinate change across the branch cuts means that this criticallocus is actually an affine circle. • The example given in fig. 3b is an example of a tropical section which does not ariseas the differential of a globally defined tropical function. The critical locus terminatesat the nodal point, and points in the direction of the eigenray of the nodal point. • Tropical sections which meet the singular fibers coming from admissible tropical sec-tions as in fig. 3c. This gives us an example of a compact tropical curve in Q of genus1.The examples above are typical of the kind of phenomenon which may occur for tropicalcurves in affine tropical surfaces. Definition 2.2.5.
Let V ⊂ Q be a tropical curve in an affine tropical surface.We say that V avoids the critical locus if V is disjoint from ∆ and ∂Q .We say that the interior of V avoids the critical locus if V is disjoint from ∂Q ,and ateach node q ∈ Q \ ∆ , there is a neighborhood B (cid:15) ( q ) so that the restriction of V ∩ B (cid:15) ( q ) is aray parallel to the eigenray of q . a) (b) Figure 4: An example of a dimer and associated bipartite graph.
We now introduce a combinatorial framework generalizing some of the ideas discussed in[Mat18, Section 5.2], and the previous work of [TWZ18; UY13; STWZ15; FHKV+08].
Definition 3.0.1. A dimer is an embedded bipartite graph G on T so that V ( G ) = V ◦ (cid:116) V • .A zigzag configuration for a dimer G is a set of transverse cycles Σ ⊂ C ( T ) satisfying thefollowing conditions: • Each connected component in T \ Σ contains at most one vertex of G . • Each edge of the dimer is transverse to every cycle. Each edge passes through exactlyone intersection point between 2 cycles. • The oriented normals of the cycles point outward on the V ◦ dimer faces, and inwardon the V • dimer faces. We will now restrict to the setting of dual dimers , where Σ is a collection of affine cycles. It is the case that for every [Σ] ⊂ H ( T ) we can find a dimer whose zigzag collection is [Σ],however, it is not necessarily the case that we can find an affine dimer with this property[Gul08],[For19, Section 4]. A dimer picks out an oriented two chain whose boundary is Σ.This is similar to the data used in [STWZ15]. More generally, we will consider pairs of thefollowing form: Definition 3.0.2. A dual n -dimer is two collections of n -polytopes { ∆ ◦ v } , { ∆ • w } ⊂ R n which satisfy the following properties. • Each vertex set { ∆ • / ◦ v } is a set of distinct points on the torus in the sense that whenever w , w ∈ { ∆ ◦ v } and w ≡ w mod Z n ,then w = w . • We require that these two vertex sets match after quotienting by the lattice, { ∆ ◦ v } / Z n = { ∆ • w } / Z n . Somewhat counterintuitively, an dual dimer is one whose zigzags are straight circles. Let p ∈ ∆ ◦ v be a vertex, and let p ∈ ∆ • v be the corresponding vertex so that p ≡ p mod Z n . Let { e , . . . , e k } be the edges of ∆ ◦ v containing the vertex p . We require thatthe edges of ∆ • v containing p point in the opposite directions {− e , . . . , − e k } . If the interiors of the ∆ ◦ v and ∆ • w are disjoint mod Z n , we say that the dual dimer config-uration has no self-intersections.From this data, we obtain a bipartite graph G ⊂ T n , whose vertices are indexed by { ∆ ◦ v } ∪ { ∆ • w } , and whose edges are determined by which polytopes in the dual dimer share acommon vertex. We will usually index the polytopes by the vertices v • / ◦ ∈ V ◦ (cid:116) V • = V ( G ). The edgesof the bipartite graph are in bijection with { ∆ ◦ v } = { ∆ • w } . The graph G need not beembedded. If the polytopes { ∆ • / ◦ v } are disjoint, then G can be chosen to be embedded.A dual dimer prescribes the data of a n -chain in T n . Our requirement that G is bipartiteguarantees that this n -chain is oriented.We now briefly explore some of the combinatorics of these dual dimers to produce thedata of a tropical hypersurface in R n . Claim 3.0.3.
The edges of an dual dimer all have rational slope.Proof.
Let e be an edge of ∆ ◦ v , with ends on vertices p − , p + ∈ { ∆ • / ◦ v } . From our definitionof a dual dimer, there exists an edge e − in some ∆ • w which also has end on p − and is parallelto e . By concatenating e − and e + , we obtain a line segment. By repeating this process, weobtain an affine representative of a cycle in H ( T n , Z ) associated to each edge e . Claim 3.0.4.
Let { ∆ ◦ v } , { ∆ • w } be a dual n -dimer. Let α be a facet of some ∆ • v . Consider T α ⊂ T n , the affine ( n − subtorus spanned by α . The set of ( n − polytopes ∆ • / ◦ β givenby the facets of our original set of polytopes which satisfy { ∆ • β | β is a facet of ∆ • , β ⊂ T α }{ ∆ ◦ β | β is a facet of ∆ ◦ , β ⊂ T α } is the data of an ( n − dimer on T α . By induction, we get the same result for all faces.
Corollary 3.0.5.
Let α be a k -face of some ∆ • v . Consider T α ⊂ T n , the affine sub-torusspanned by α . The set of k polytopes given by the k -faces satisfying { ∆ • β | β is a k -face of ∆ • , β ⊂ T α }{ ∆ ◦ β | β is a k -face of ∆ ◦ , β ⊂ T α } is a dual k -dimer of T α . k dimensional dual dimers gives the data of a k -chain in T α . We denotethese k -chains of T n , { U β | β is a k -facet } ⊂ C k ( T n , Z ) . This can also be thought of an equivalence relation on the set of k -faces of the dual-dimer,where two faces are equivalent if they define the same dual k -dimer chain. A cone is thereal positive span of a finite set of vectors. Given a cone V ⊂ R n , a subspace U ⊂ R n , the U -relative dual cone of V is V ∨| U := { u ∈ U | (cid:104) u, V (cid:105) ≥ } . To each k -chain U β we can associate a cone in R n . Definition 3.0.6.
Let U β be a chain given by a facet β ⊂ ∆ • / ◦ v . Assume that we havetranslated ∆ • / ◦ v so that the origin is an interior point of the face β . Let R β be the affinesubspace generated by β . Let ( R β ) ⊥ be the corresponding perpendicular subspace. We definethe dual cone to the facet U β to beU β := (cid:26) ( R ≥ · ∆ • v ) ∨| ( R β ) ⊥ If β belongs to a • polytope − ( R ≥ · ∆ ◦ v ) ∨| ( R β ) ⊥ If β belongs to a ◦ polytope Suppose that α and β are facets in the same dual k -dimer so that U α = U β . Let α ⊂ ∆ • v ,and suppose that β ⊂ ∆ • w . After translating ∆ • v and ∆ • w so that 0 ∈ α and 0 ∈ β , we getan agreement of the cones R ≥ · ∆ • v = R ≥ · ∆ • w . Similarly, if γ ⊂ ∆ ◦ u and U γ = U α , then R ≥ · ∆ • v = − R ≥ · ∆ ◦ u . It follows that: Claim 3.0.7.
If U α ⊆ U β , then U α ⊇ U β This also shows that the definition of the cone is really only dependent on the data of the k -chain represented by the choice of facet α , in that U α = U β whenever U α = U β . Considerthe polyhedral complex containing the subset U β . This complex satisfies the zero tensioncondition , and therefore describes a tropical subvariety of R n . From the data of a dimer, we now construct a Lagrangian inside of X = ( C ∗ ) n . The construc-tion of these Lagrangians are similar to the construction of tropical Lagrangians in [Hic19,sections 3.1, 3.2]. Let Q = R n , and T ∗ Z Q be the lattice in the cotangent bundle generated by dx , . . . , dx n . We give X the symplectic structure via identification with T ∗ Q/T ∗ Z Q , and letval : X → Q be the valuation projection. The fibers of this projection are Lagrangian tori.We denote by arg : X → ( T ∗ ) Q/ ( T ∗ Z ) Q = T n the argument projection to a torus fiber. Definition 3.1.1.
Let ∆ v ⊂ R n be a polytope. The convex dual tropical function φ ◦ v : Q → R is the convex piecewise linear function with Newton polytope ∆ ◦ v . We choose the functionwhich is maximally degenerate in the sense that each domain of linearity contains the origin.Similarly, define φ • v to be the concave dual tropical function, φ • v = − φ ◦ v . φ ∆ : R n → R whose Newton polytope is∆, and whose tropical locus contains a single 0-dimensional strata. Given a dual dimer { ∆ ◦ v } , { ∆ • w } , let { φ ◦ v } , { φ • w } be the associated dual tropical functions.Following [Hic19], let ˜ φ • / ◦ : R n → R be smoothings of the convex functions by a kernelof small radius. Definition 3.1.2 ([Abo09]) . The tropical Lagrangian section σ φ : Q → X associated to φ is the composition T ∗ Q XQ /T ∗ Z Qd ˜ φ . For convenient of notation, when { ∆ ◦ v } , { ∆ • w } , we set σ • / ◦ k := σ φ • / ◦ k . We glue together thetropical Lagrangian submanifolds σ • / ◦ k along their overlapping regions. Claim 3.1.3.
Let { ∆ ◦ v } , { ∆ • w } be a dual dimer configuration without self-intersections. Thereis a decomposition of the intersections of the σ ◦ v and σ • w into convex subsets of R n , (cid:91) v,w ∈ G σ • v ∩ σ ◦ w = (cid:91) e ∈ G U e . Furthermore, the sections σ • / ◦ k have intersections with collared boundaries in the sense ofproposition 2.1.3 at each of the U e . The structure of a collared boundary on the intersections U e follows from the convex-ity/concavity of the primitive functions φ • / ◦ k . Definition 3.1.4.
Let { ∆ ◦ v } , { ∆ • w } be a dual dimer. The dimer Lagrangian is the Lagrangianconnect sum L ( φ • w , φ ◦ v ) := σ ◦ v U e | e ∈ G σ • w . The set U β is very close to the set U e , where β is the common vertex of the two dimerpolytopes corresponding to the edge e . As a result, the valuation of a dimer Lagrangian isclose to the tropical hypersurface associated to the dimer. Example 3.1.5.
Consider the dual dimer model drawn in fig. 5c. The six triangles drawn a) Assembling several dif-ferent dimer Lagrangians tobuild a tropical curve. × (b) Dimer giving an immersedLagrangian sphere. × (c) An embedded Lagrangiantorus lift of a non-smoothtropical curve. Figure 5: Three related Lagrangians are associated to the following six tropical functions. φ ◦ =( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) φ ◦ =( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) φ ◦ =( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) φ • = − ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) φ • = − ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) φ • = − ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) ⊕ ( x / (cid:12) x / ) All six functions give the same nonlinearity stratification to Q , V ( φ • i ) = V ( φ ◦ i ) = V ( x ⊕ x ⊕ ( x (cid:12) x ) − ) . There are nine Lagrangian surgeries that we need to perform in order to build L ( φ • w , φ ◦ v ) .The valuation projection of the Lagrangian submanifold approximates the tropical curve withthree legs. These dimer Lagrangians serve as a generalization of tropical Lagrangians constructedin [Hic19], where L ( φ ) = L ( φ v , . Example 3.1.6.
One can also assemble lifts of more complicated tropical curves by gluingseveral dimer Lagrangians together. For example, the genus 1 tropical curve drawn in fig. 5acan be built from taking three vertices. At each vertex we place a dimer whose cycles arenormal to the edges of the vertices.
Example 3.1.7.
It is not necessary for the dimer model to consist of disjoint faces. Infig. 5b we see a configuration with two triangular faces which overlap at a hexagon. TheLagrangian associated to this dimer is immersed, but has the same legs as the example infig. 5c.
We now restrict ourselves to the setting ( C ∗ ) = T ∗ F and describe a combinatorial approx-imation of CF • ( L ( φ • w , φ ◦ v ) , F ) , the Floer theory of our tropical Lagrangian against fibers of the SYZ fibration. Definition 3.2.1.
Let { ∆ ◦ v } , { ∆ • w } be a dual dimer configuration with affine bipartite graph G . Let ∇ be a C ∗ local system on T n . The Kasteleyn complex with weighting ∇ is the 2-termchain complex C • ( G, ∇ ) which as a graded vector space is C (cid:104) v ◦ i (cid:105) ⊕ C (cid:104) w • j (cid:105) [1] . The differential d ∇ is determined by the structure coefficients (cid:104) d Σ ∇ ( v ◦ ) , v • (cid:105) = (cid:88) e ∈ E ( G ) e = v ◦ v • ∇ e T w ( e ) . The support of { ∆ ◦ v } , { ∆ • w } is the set of local systems Supp( { ∆ ◦ v } , { ∆ • w } ) := {∇ | H ( G, ∇ ) (cid:54) = 0 } . In dimension 2, G is exactly a dimer, and the support is the zero locus of the polynomial Z G ( ∇ ) := det( d ∇ ) . The terminology comes from literature on dimers [KOS06]. By letting the local system ∇ determine a weight for each edge of the dimer, the terms of the determinant correspondsto the product of weights of a maximal disjoint set of edges (called its Boltzmann weight).A maximal disjoint set of edges in a dimer is called a dimer configuration , and the sum ofBoltzmann weights over all configurations gives the partition function Z G ( ∇ ) of the dimer.We now explain the relation between the Kasteleyn complex C • ( G, ∇ ) and the Lagrangianintersection Floer complex CF ( L ( φ • w , φ ◦ v ) , ( F , ∇ )). These complexes are isomorphic as vec-tor spaces, as the intersection points of F and L ( φ • w , φ ◦ v ) are in bijection with the ver-tices of the dimer. The Lagrangian L ( φ • w , φ ◦ v ) is built from taking a surgery of the pieces σ v • / ◦ . An expectation from [Fuk10] is that holomorphic strips contributing to the differen-tial µ : CF ( L p L , L ) are in correspondence with holomorphic triangles contributing to µ : CF ( L , L ) ⊗ CF ( L , L ). In our construction of L ( φ • w , φ ◦ v ) we smoothed regions largerthan intersection points between the sections σ • / ◦ v , however we expect a similar result tohold. These intersections are in correspondence with the edges of the dimer G , and so wepredict that the differential on CF ( L ( φ • w , φ ◦ v ) , ( F , ∇ )) should be given by weighted countof edges in the dimer. The local system ∇ on F determines the weight of the holomorphicstrips corresponding to each edge. 17 v v w w w Figure 6: The labelling of faces for the dimer model
Conjecture 3.2.2.
The isomorphism of vector spaces CF • ( L ( φ • w , φ ◦ v ) , F ) → C • ( G, ∇ ) is a chain homomorphism. If this conjecture holds, we have a new tool for computing the support of the Lagrangian L ( φ • w , φ ◦ v ), which will be determined by the zero locus of Z G ( ∇ ). Example 3.2.3.
A first example to look at is the Kasteleyn complex of example 3.1.5.We give the polygons of the dimer the labels from example 3.1.5. We can rewrite Z G ( ∇ ) as a polynomial by picking coordinates on the space of connections. Let z and z be theholonomies of a local system ∇ along the longitudinal and meridional directions of the torus.The differential on the complex C • ( G, ∇ ) in the prescribed coordinates is d Σ ∇ = z ( z z ) − z ( z z ) − z z z z ( z z ) − . The determinant of d Σ ∇ is Z G ( z , z ) = 3 − ( z + z + z z ) . This polynomial is a reoccurring character in the mirror symmetry story of CP ; forexample, it is the superpotential ˇ W Σ determining the mirror Landau-Ginzburg model. Thiscomputation motivates section 5. In previous examples, we exhibited different dimer models with the same associated tropicalcurve. We now describe how the different Lagrangian lifts of these dimers are related to eachother in dimension 2.
Definition 3.3.1.
Let L ( φ • w , φ ◦ v ) be a dimer Lagrangian. Let G be the associated graph. Give G the structure of a directed graph with edges going from ◦ to • . To each edge e , let γ e : [0 , → L ( φ • w , φ ◦ v )18 e a lift of the edge e to the dimer Lagrangian. We define the weight of an edge e to be theintegral w e := (cid:90) γ e η, where η = p · dq is the tautological one form on the cotangent bundle. We say that a cycle c ⊂ E ( G ) has zero weight if (cid:88) e ∈ c w e = 0 . Lemma 3.3.2.
Let { ∆ ◦ v } , { ∆ • w } be a dimer model with graph G . For each face f ∈ F ( G ) ,let c = ∂ f be the boundary cycle of the face. Suppose that c has zero weight. Let γ c : S → L ( φ • w , φ ◦ v ) be a lift of the cycle to the dimer Lagrangian, in the sense that arg( γ c ) = c. There exists a Lagrangian disk D f with ∂D f = c ⊂ L ( φ • w , φ ◦ v ) .Proof. Let V f ⊂ T be the subset of the Lagrangian torus T ⊂ T ∗ T corresponding to theface f . The zero weighting condition tells us that (cid:90) γ c η = 0 , and so there is no obstruction to finding a closed one form over V f whose value on theboundary matches ( γ c ) q . The Lagrangian disk D f is defined by the graph of this one form.The Lagrangian antisurgery α D f L ( φ • w , φ ◦ v ) is an immersed Lagrangian, which we nowdescribe with a dual dimer model. Let ∂f := { v • , v ◦ , . . . , v • k , v ◦ k } be the sequence of verticesof G corresponding to the boundary of f . Recall that Σ is the set of cycles in T given by theboundary polygons of the dual dimer model. Let Im(Σ) ⊂ T be the image of these cycles.After taking an isotopy of c , we may assume that arg( c ) ⊂ Im(Σ). We can also require thatarg( c ) is a homeomorphism onto its image.We now take a parameterization h : S × [ − , → L ( { ∆ ◦ v } , { ∆ • w } )for a neighborhood of γ c ⊂ L ( φ • w , φ ◦ v ), with h ( θ,
0) = γ c ( θ ). The boundary components ofthe collar h : S × [ − ,
1] give two cycles in L ( { ∆ ◦ v } , { ∆ • w } ), which we will label γ • c := γ c ( θ, − γ ◦ c := γ c ( θ, γ has argument contained within Σ, but we require the map h ( θ, t ) : S × [ − , → L ( { ∆ ◦ v } , { ∆ • w } ) have argument Im(arg ◦ γ c • ) ⊂ Σ ∪ { ∆ • i } Im(arg ◦ γ c ◦ ) ⊂ Σ ∪ { ∆ ◦ v } which “alternates” between bleeding into the ∆ ◦ v and ∆ • w polytopes. We now state thisalternating condition. We require at each θ exactly one of the three following cases occur: • That the ◦ component bleeds out of Σ into the interior of the dimer so arg ◦ h ( θ, (cid:54)∈ Σ • That the • component bleeds out of Σ into the interior of the dimer so arg ◦ h ( θ, − (cid:54)∈ Σ • Neither boundary component bleeds out of Σ, but the collar h passes through thevertex connected to polytopes in our dimer model so arg ◦ h ( t, θ ) maps to a vertex ofthe ∆ • / ◦ i .After performing the Lagrangian surgery, the band parameterized by h will be replacedwith two disks D • f and D ◦ f . The boundaries of D • / ◦ f are the cycles γ c • / ◦ .The disk D • f glues the polygons ∆ • v i which lie along the cycle γ c • to each other. Similarly,the disk D ◦ f connects the ∆ ◦ w i together. In summary, the polygons in the cycle c are replacedwith two larger polygons in the antisurgery:∆ f • :=Hull v • i ∈ ∂f (∆ • v )∆ f ◦ :=Hull v ◦ i ∈ ∂f (∆ ◦ v ) . Lemma 3.3.3.
Consider a dimer model { ∆ ◦ v } , { ∆ • w } . Let f be a face of G . Suppose that theboundary of f has zero weight. The antisurgery α D f L ( φ • w , φ ◦ v ) is again described by a higherdimer model, whose polygons are given by the collections { ∆ • v | for all v • (cid:54)∈ ∂f } ∪ { ∆ f • }{ ∆ ◦ w | for all w ◦ (cid:54)∈ ∂f } ∪ { ∆ f ◦ } The graph for this dimer is immersed. For example, the dimers in figs. 5a to 5c describesthe antisurgery and subsequent mutation of a dimer Lagrangian.
Besides using antisurgery to modify Lagrangian submanifolds, we may use the presence ofantisurgery disks for L ( φ • w , φ ◦ v ) to construct a Lagrangian seed in the sense of [PT17].20 a) (b) (c) Figure 7: An immersed dimer on the torus, along with the corresponding immersed dimeron the double cover. The last figure draws the resolution of this immersed dimer into anon-immersed dimer.
Definition 3.3.4 ([PT17]) . A Lagrangian seed ( L, { D i } ) is a monotone Lagrangian torus L ⊂ X along with a collection of antisurgery disks { D i } for L with disjoint interiors, andan affine structure on L making ∂D i affine cycles. Should the ∂D i ⊂ L be the edges of anaffine zigzag diagram, we say that this seed gives a dimer configuration on L . Whenever we have an mutation seed giving a dimer configuration on L , we can build adual Lagrangian using the same surgery techniques used to construct tropical Lagrangians.We start by taking a Weinstein neighborhood B ∗ (cid:15) L of L . Let { ∆ ◦ v } , { ∆ • w } be the dimer modelon L induced by the Lagrangian seed structure. Using definition 3.1.4, we can construct L ( { φ ◦ v } , { φ • w } ) in the neighborhood B ∗ (cid:15) L . The boundary of L ( { φ ◦ v } , { φ • w } ) is contained in the (cid:15) -cotangent sphere S ∗ (cid:15) L and consists of the (cid:15) -conormals Legendrians N ∗ (cid:15) ( ∂D i ). After taking aHamiltonian isotopy, the disks { D i } can be made to intersect S ∗ (cid:15) L along N ∗ (cid:15) ( ∂D i ). By gluingthe dimer Lagrangian to these antisurgery disks, we compactify L ( { φ ◦ v } , { φ • w } ) ⊂ B ∗ (cid:15) L to aLagrangian L ∗ ⊂ X . Definition 3.3.5.
Let ( L, { D i } ) be a Lagrangian seed giving a dimer configuration on L .We call the Lagrangian L ∗ ⊂ X the dual Lagrangian to ( L, { D i } ) . One way to interpret this construction is that a Lagrangian seed has a small symplecticneighborhood which may be given an almost toric fibration. The dual Lagrangian L ∗ is acompact tropical Lagrangian built inside of this almost toric fibration.By lemma 3.3.3 the Lagrangian L ∗ possesses a set of antisurgery disks given by the facesof the dimer graph on L . Should the antisurgery disks D f with boundary on L ∗ form amutation configuration, we call ( L ∗ , { D f } ) the dual Lagrangian seed. Remark 3.3.6.
The geometric portion of this construction does not require L or L ∗ tobe tori, although statements about mutations of Lagrangians from [PT17] and relations tomirror symmetry use that L is a torus. Much of the machinery we have constructed for building Lagrangians lifts of tropical hy-persurfaces in the fibration ( C ∗ ) n → R n carries over to building tropical Lagrangian hyper-surfaces for almost toric fibrations X → Q with the dimension of the base dim Q = 2. In21ection 4.1 we look at the local model of a node in a almost toric base diagram and showthat lifts of tropical curves can be constructed for tropical curves with edges meeting thesingular strata of Q along the eigendirection. Section 4.2 continues using local models fromthe node based on Lefschetz fibrations to show that isotopy of tropical curves “through” anode of the base extend to isotopies of the Lagrangian lifts. Recall that X → Q is an almost toric fibration, X → Q = Q \ Q is an honest toricfibration in the complement of the discriminant locus ∆. By abuse of notation, when we aregiven a tropical section φ ∈ dTrop( U ) where U ⊂ Q , we will write σ φ : U → X | U to meanthe Lagrangian section defined over the bundle X | U → U given by some choice of smoothingparameter (see [Hic19]). It is immediate that we can use the existing surgery lemma to buildtropical Lagrangians away from the critical locus. Claim 4.1.1.
Let val : X → Q be an almost toric Lagrangian fibration. Suppose that V ( φ ) ⊂ Q is a tropical curve which is disjoint from the critical locus. Then there exists aLagrangian submanifold L ( φ ) ⊂ X whose valuation projection lies in a small neighborhoodof V . Furthermore, if Q has no boundary, there exists a tropical section φ so that L ( φ ) = σ σ − φ . In the case where dim( Q ) = 2, we can find a Lagrangian lift when the interior of V avoidsthe critical locus. This is built on the following local model. Claim 4.1.2.
Let X = C \ { z z = 1 } be the symplectic manifold with symplectic fibration W : C \ { z z = 1 } → C \ { } ( z , z ) (cid:55)→ z z and let val : X → Q be the almost toric Lagrangian fibration described in [Aur07, Section5.1]. Then Q has a single node q × of multiplicity 1, and there exists a tropical Lagrangianlift of the eigenray of q × .Proof. The claim follows from considering the construction of the almost toric fibrationarising from the Lefschetz fibration W . The rotation ( z , z ) (cid:55)→ ( e iθ z , e − iθ z ) is a globalHamiltonian S symmetry which preserves the fibers of the fibration. Let µ : ( C ∗ ) → R be the moment map of this Hamiltonian action, which also descends to a moment map µ : W − ( z ) → R . This map gives an SYZ fibration on the fibers of the Lefschetz fibration.The base of the Lefschetz fibration C \ { } comes with a standard SYZ fibration by circles1 + re πiθ . The symplectic parallel transport map given by the Lefschetz fibration preservesthe SYZ fibration on W − ( z ); as a result, one can build an SYZ fibration for the total space { C \ z z = 1 } by taking the circles val − W − ( z ) ( s ) and parallel transporting them along circles1 + re iθ of the second fibration to obtain Lagrangian tori F r,s = { ( z , z ) | | z z − | = r, µ ( z , z ) = s } . C = z z W = z Q × z r = | z | Figure 8: Lagrangian tori constructed from a Lefschetz fibration giving an almost toricfibration. The colored fibers correspond to cycles (cid:96) being parallel transported around acircle in the base.The nodal degeneration occurs from parallel transport of vanishing cycle through the path1 + e iθ . This corresponds to the single almost toric fiber of this fibration, a Whitney sphere,which occurs in the base when q × = (1 , . Q comes with an affine structure by identifyingthe cotangent fiber at q with H ∗ ( F q , R ), and taking the lattice to be the integral homologyclasses. The monodromy of this fibration around the Whitney sphere acts by a Dehn twiston the vanishing cycle (i.e. for s = 0) of F q . As a result, the coordinate s is a global affinecoordinate on Q near q x , but r is not. The eigenray is s = 0. The Lagrangian tori F q with q in the eigenray of q × are those tori which are built from parallel transport of the vanishingcycle. See fig. 8 for the correspondence between Lagrangians in the Lefschetz fibration andalmost toric fibration.We now consider the Lagrangian thimble τ drawn from the critical point ( z , z ) =(0 , F q with q on the eigenray of q × . Therefore, thisLagrangian thimble has valuation projection travelling in the eigenray direction of q × , provingthe claim. Corollary 4.1.3.
Let val : X → Q be an almost toric Lagrangian fibration over an integraltropical surface Q . Let V be a smooth tropical variety whose interior avoids the discriminantlocus ∆ . Then there exists a tropical Lagrangian lift L ⊂ X of V .Proof. First, construct the lift of V to a Lagrangian ˚ L on X \ X . It remains to compactify˚ L to a Lagrangian submanifold of X . At each point q i ∈ X , we take a neighborhood B i of q i and model it on the standard neighborhood from claim 4.1.2. The portion of ˚ L withvaluation over B i is a Lagrangian cylinder given by the periodized conormal to the eigenrayof q i . Similarly, the thimble τ i restricted to this valuation is a Lagrangian cylinder givenby the periodized conormal to the eigenray of q i . Therefore, we may compactify ˚ L to aLagrangian L ⊂ X by gluing the thimbles τ i to L at each nodal point such that q i ∈ V .This allows us to build tropical Lagrangian lifts of the tropical curves described in figs. 3ato 3c. We may generalize the examples of compact Lagrangian tori in CP to more toricsymplectic manifolds with dim C ( X ) = 2. Let X Σ be a toric surface, and let val : X Σ → Q dz Σ be the standard moment map projection. The moment polytope Q dz is an example of an23 × ×× (a) ×× ×× (b) Figure 9: Some more examples of tropical sectionsalmost toric base diagram. Consider the almost toric fibration val : X Σ → Q Σ obtained byapplying a nodal trade to each corner of the moment polytope. The boundary of Q is nowan affine circle, corresponding to a symplectic torus E ⊂ X Σ . Example 4.1.4.
The neighborhood of ∂Q Σ is topologically ∂Q Σ × [0 , (cid:15) ) t . For fixed realconstant < r < (cid:15) , we construct the tropical function r ⊕ t , which only has dependenceon collar direction t . This extends to a tropical function over Q Σ , whose critical locus isan affine circle pushed off from the boundary ∂Q Σ . The critical locus is a tropical curvewhich avoids the discriminant locus, so there is an associated Lagrangian torus L ∂Q Σ r ⊂ X Σ corresponding to this tropical curve.This Lagrangian torus can also be constructed without using the machinery of Lagrangiansurgery. Let γ ⊂ E be a curve. There is a neighborhood D of E ⊂ X Σ which is a disk bundle D → E . There is a standard procedure to take γ and lift it to a Lagrangian ∂D γ , the unionof real boundaries of this disk bundle along the curve γ . See fig. 9a. As one increases the parameter r , the Lagrangian L ∂Q Σ r approaches the critical locus ∆ Σ .One can continue this family of Lagrangian submanifolds past the critical locus. Example 4.1.5.
In the above example, each nodal point q i corresponds to a corner of theDelzant polytope Q dz Σ . The index i is cyclically ordered by the boundary of the Delzant poly-tope. Let Σ i be the fan generated by vectors v − i , v + i given by the edges of the corner corre-sponding to q i . Let v λi be the eigenray of q i . Then Σ i ∪ { v λi } is a balanced fan. At each nodalpoint q i , consider the tropical pair of pants with legs in the directions Σ i ∪ { v λi } .The legs of adjacent pairs of pants (from the cyclic ordering) match so that v − i = − v + i +1 .This means that if the pairs of pants are properly placed (say so that the distance from thevertex of the pair of pants along the eigenray direction to the boundary ∂Q Σ are all equal)these assemble into a tropical curve.This is a tropical curve whose interior is disjoint from the critical locus, and thus lifts toa tropical Lagrangian with the topology of a torus in X Σ . See fig. 9b The tropical curves from figs. 9a and 9b are related via an isotopy of tropical curves. We nowintroduce some notations for Lagrangians in Lefschetz fibrations which will allow us to show24igure 10: The three different building blocks for a Lagrangian glove: parallel transport,thimbles, and trace of a surgery.that this isotopy of tropical curves can be lifted to their corresponding tropical Lagrangians.The local model for the nodal fiber in an almost toric base diagram is built from a Lefschetzfibration. The goal of this section is to build some geometric intuition for interchanging thesetwo different perspectives. We now describe three Lagrangian submanifolds which will serveas building blocks in Lefschetz fibrations, similar to those considered in [BC17]. See fig. 10.The first piece is suspension of Hamiltonian isotopy. Given a path e : [0 , → C avoid-ing the critical values of W : X → C , and Hamiltonian isotopic Lagrangians (cid:96) and (cid:96) in W − ( e (0)) and W − ( e (1)), we can create a Lagrangian L (cid:96)e which is the suspension of Hamil-tonian isotopy along the path e . We assume that this Hamiltonian isotopy is small enoughso that the trace of the isotopy similarly avoids the critical fibers. This Lagrangian has twoboundary components, one above e (0) and one above e (1). In practice, we will simply specifythe Lagrangian (cid:96) and assume that the Hamiltonian isotopies are negligible.The second building block that we consider are the Lagrangian thimbles , which are thereal downward flow spaces of critical points in the fibration. These can also be characterizedby taking a path e : [0 , → C with e (0) a critical value of W : X → C , and letting (cid:96) be avanishing cycle for a critical point in W − ( e (1 / L (cid:96)e , has single boundary component above e (1).The third building block we will use comes from Lagrangian cobordisms. In any smallcontractible neighborhood U ⊂ C of C which does not contain a critical value of W : X → C ,we can use symplectic parallel transport to trivialize the fibration so it is W − ( p ) × D forsome p ∈ U . We then consider cycles (cid:96) , (cid:96) , (cid:96) ⊂ W − ( p ) so that (cid:96) (cid:96) = (cid:96) with neck size (cid:15) . There is the trace cobordism of the Lagrangian surgery between these three cycles whichproduces a Lagrangian cobordism in the space W − ( p ) × C . Given paths e , e , e ⊂ D indexed in clockwise order, with e i (1) = p , we let L (cid:96) i e i be the trace cobordism of the surgerybetween the (cid:96) i with support living in a neighborhood of the edges e i . This Lagrangian hasthree boundary components, which live above e i (0).These pieces glue together to assemble smooth Lagrangian submanifolds of X wheneverthe ends of the pieces (determined by their intersection with the fiber) agree with each other. Definition 4.2.1.
Let W : X → C be a symplectic fibration. A Lagrangian glove L ⊂ X is Lagrangian submanifold so that for each point z ∈ C , there exists a neighborhood U (cid:51) z sothat W − ( U ) ∩ L is one of the three building blocks given above. The reason that we look at Lagrangian gloves is that they can be specified by the followingpieces of data: • A planar graph G ⊂ C . This graph is allowed to have semi-infinite edges and loops. • A Lagrangian submanifold (cid:96) e ⊂ W − ( e (0)) labelling each edge e ∈ G .This data will correspond to a Lagrangian glove if it satisfies the following conditions: • The interior of each edge is disjoint from the critical values of W . • Outside of a compact set, the semi-infinite edges are parallel to the positive real axis. • All vertices of G have degree 1 or degree 3. • Every vertex of degree 1 must lie at a critical value. Furthermore, the incoming edge e to the vertex v is labelled with a vanishing cycle of the corresponding critical fiber. • Every vertex of degree 3 with incoming edges e , e , e must have corresponding La-grangian labels (cid:96) , (cid:96) and (cid:96) which satisfy the relation (cid:96) (cid:96) = (cid:96) for a surgery of necksize small enough that there exists a disk D ⊃ v containing the trace of this surgery.Such a collection of data gives us a Lagrangian L (cid:96) e G ⊂ X .We will diagram these Lagrangians by additionally picking a choice of branch cuts b i for C so that W : ( X \ W − ( b i )) → ( C \{ b i } ) is a trivial fibration. We can then consistently labelthe edges of the graph G ⊂ C with Lagrangians in (cid:96) e ∈ W − ( p ) for some fixed non-criticalvalue p . Graph isotopies which avoid the critical values correspond to isotopic Lagrangians;furthermore, as long as the label of an edge does not intersect the vanishing cycle of a criticalvalue, we are allowed to isotope an edge over a critical value.There is another type of isotopy which comes from interchanging Lagrangian cobordismswith Dehn twists [MW15; AS18], which we now describe. Let v be a trivalent vertex withedges e = vw , e = vw , e = vw . Suppose that the degree of w is one. Supposeadditionally that the Lagrangians (cid:96) and (cid:96) , the labels above e and e , intersect at a singlepoint so that the surgery performed is the standard one at a single transverse intersectionpoint. Let G be the graph obtained by replacing e , e , e with a new edge f , which hasvertices w , w , and is obtained travelling along e , out along e and around the critical value w , and returning along e and e (See fig. 11). Then the graph H = G ∪{ f , }\{ e i } equippedwith Lagrangian labelling data inherited from G (with the additional label (cid:96) f , = (cid:96) e ) is againa Lagrangian glove. We call the Lagrangian obtained via this exchanging operation τ w L (cid:96) e G .In summary: Proposition 4.2.2.
The following operations produce Lagrangian isotopic Lagrangiangloves. ve e e w w f , w w Figure 11: One can add or remove Lagrangian thimbles by exchanging them for Dehn twists. • Any isotopy of the graph G where the interior of the edges stay outside the complementof the critical values of W . • Any isotopy of the graph G where an edge passes through a critical value, but theLagrangian label of the edge is disjoint from the vanishing cycles of the critical fibers. • Exchanging the Lagrangian L (cid:96) e G with τ w L (cid:96) e G at some vertex w .Proof. The first two types of modifications are clear. For the third kind of modification, see[AS18, Lemma A.25].
We now will provide a construction of a Lagrangian pair of pants in the setting of ( C \{ z z =1 } ) from the perspective of the Lefschetz fibration considered in section 4.1: W : C \ { z z = 1 } → C ( z , z ) (cid:55)→ z z See fig. 8 for the correspondence between Lagrangian tori in the Lefschetz fibration andalmost toric fibration.In this setting we build a Lagrangian glove. We start with the Lagrangian (cid:96) = R ⊂ W − (1). For small (cid:15) <
1, we consider the loop γ (cid:15) = (cid:15)e iθ −
1. The parallel transport of (cid:96) along this loop builds a Lagrangian L (cid:96)γ (cid:15) . The Lagrangian L (cid:96)γ (cid:15) only pairs against tori F (cid:15),s , soits support in the almost toric fibration will be a line. See the blue Lagrangian as drawn infig. 12.By exchanging a Dehn twist for an additional vertex in the glove (proposition 4.2.2), wecan build a new Lagrangian τ L (cid:96)γ (cid:15) (drawn in red in fig. 12). This description provides us withanother construction of the Lagrangian pair of pants. These local models are compatiblewith the discussion from section 4. Let Q × be the integral tropical manifold which is thebase of X = C \ { z z = 1 } . Q × can be covered with two affine charts. Call the charts Q = { ( x , x ) } \ { ( x, x ) | x > } Q = { ( y , y ) } \ { ( y, y ) | y < } . C z z ‘ γ (cid:15) ◦ − (a) Two Lagrangian gloves for the Lefschetz fibration z z : C \ { z z = 1 } → C . × τ L ‘γ (cid:15) QL ‘γ (cid:15) x x y y (b) The valuation projection ofthese Lagrangian gloves. Figure 12: Comparing Lefschetz and tropical pictures at nodal fibers.The charts are glued with the change of coordinates( y , y ) = (cid:26) ( x , x ) x > x (2 x − x , x ) x < x We now consider two tropical curves inside of Q × . The first is an affine line, which is givenby the critical locus of a tropical polynomial defined over the Q chart φ ( x , x ) = 1 ⊕ x . The second tropical curve we consider is a pair of pants with a capping thimble (as describedin section 4,) given by the critical locus of a tropical polynomial defined over the Q chart, φ ( y , y ) = y ⊕ y ⊕ . From proposition 4.2.2, we get the following corollary:
Corollary 4.2.3 (Nodal Trade for Tropical Lagrangians) . Consider the tropical curves V ( φ ) and V ( φ ) inside of Q × . The Lagrangians L ( φ ) and L ( φ ) are Lagrangian isotopic in C \ { z z = 1 } . This corollary allows us to manipulate tropical Lagrangians by manipulating the tropicaldiagrams in the affine tropical manifold instead.
Example 4.2.4.
Consider the Lefschetz fibration with fiber C ∗ given by the smoothed A n sin-gularity as in fig. 13. We construct the Lagrangian glove where we parallel transport the realarc (cid:96) = R ⊂ C ∗ around the loop of the glove. The monodromy of the symplectic connectionfrom travelling around the large circle corresponds to n twists of the same vanishing cycles.By attaching n vanishing cycles to this arc, we get a Lagrangian glove. In the moment map ×× × × × Figure 13: The resolved A singularity, a Lagrangian glove, and its associated tropical curve. picture, all of the singularities lie on the same eigenray, and we get the tropical Lagrangianwhich is a n + 2 punctured sphere with n of the punctures filled in with thimbles. Though itappears that the n thimbles of the Lagrangian coincide with each other in the moment mappicture, they differ by some amount of phase in the fiber direction, which is easily seen inthe Lefschetz fibration. Corollary 4.2.5.
The Lagrangians from figs. 9a and 9b are Lagrangian isotopic.
We now introduce a monotone Lagrangian torus which exists in a toric del-Pezzo. We showthat in the setting of CP this Lagrangian L T is isotopic to F q , a fiber of the momentmap. Finally, we speculate on homological mirror symmetry for L T ⊂ CP \ E , where thisLagrangian is no longer isotopic to F q . We exhibit a symplectomorphism g : CP \ E → CP \ E expected to be mirror to fiberwise Fourier-Mukai transform on the mirror. Monotone Lagrangian tori and Lagrangian seeds in del-Pezzo surfaces have been studied in[Via17; PT17]. Let X Σ be a toric del-Pezzo. There exists a choice of symplectic structureon X Σ so that the monotone Lagrangian torus F Σ at the barycenter of the moment polytopehas a Lagrangian seed structure { D i, Σ } given by the Lagrangian thimbles extending fromthe corners of the moment polytope. The Lagrangian thimbles and corresponding dimersare drawn in figs. 14a to 14e. In these 5 examples, the dimer Lagrangian F ∗ Σ constructedfrom the data of ( F Σ , D i, Σ ) again has the topology of a torus. This can be checked from thecomputation of the Euler characteristic of the dual Lagrangian, χ ( L ∗ ) = | V ( G ) | − | E ( G ) | + | Σ | , where | Σ | is the number of antisurgery disks with boundary on L .29 × ××× ××× × (a) CP ×× ××× ××× × (b) CP × CP ×× ××× ××× × (c) Bl CP ×× ××× ××× × (d) Bl CP ×× ××× ××× × (e) Bl CP Figure 14:
Top : Lagrangian seeds in toric del Pezzo surfaces. The antisurgery disks aredrawn in red.
Middle : The corresponding dual dimer models associated the Lagrangianseeds. In the first example of CP , we additionally draw the classes of the cycles ∂D f i, Σ ⊂ F ∗ Σ . Bottom:
Cycle classes of the zigzag diagram, corresponding to mutation directions.One method of distinguishing Lagrangians is to compute their open Gromov-Wittenpotentials. In the case of toric Fanos, it was proven in [Ton18] that all Lagrangian tori havethe potentials given by one of those in [Via17]. A computation shows that the Lagrangians F Σ and F ∗ Σ have the same mutation configuration. Claim 5.1.1.
Let X Σ be a toric Fano, F Σ the standard monotone Clifford torus in X Σ , and F ∗ Σ be the dual torus constructed using the Lagrangian seed structure on F Σ . There is a setof coordinates for H ( F ∗ Σ ) and H ( F Σ ) so that the mutation directions determined by theirLagrangian seed structures are the same.Proof. This is done by an explicit computation of the homology classes of the disk boundariesin F ∗ Σ . Remark 5.1.2.
In the example fig. 14c, there are more faces of G than mutation directions.However, some of the disks represent the same homology classes. As a corollary, the wall and chamber structure on the moduli space of Lagrangians F Σ obtained by mutations may be replicated in a similar fashion on the moduli space of theLagrangians F ∗ Σ . Corollary 5.1.3.
In the setting of toric Fanos, the Landau-Ginzburg potential of F Σ is thesame as F ∗ Σ . In both figs. 14a and 14b we may mutate the diagram to give us a dimer model withtwo polygons, which is the balanced tropical Lagrangian for some tropical polynomial. Asa result, the Lagrangians figs. 14a and 14b are Lagrangian isotopic to tropical Lagrangiansconstructed in section 4. It is unclear how much of this story extends beyond the toric case.30 uestion 5.1.4.
Is there a relation between ( L, D i ) and ( L ∗ , D ∗ f ) that can be stated in thelanguage of mirror symmetry? We conclude our discussion with a collection of observations for mirror symmetry of CP \ E and the elliptic surface ˇ X . Here, ˇ X is the extremal elliptic surface in thenotation of [Mir89]. This elliptic surface W : ˇ X → CP has 3 singular fibers of type I , and one singular fiber of type I . We can present this elliptic surface [AGL16, Table Two]as the blowup of a pencil of cubics on CP ,( z z + z z + z z ) + t · ( z z z ) = 0 . From this pencil, we get a map ˇ π bl : ˇ X → CP , which has nine exceptional divisors. Threeof the exceptional divisors correspond to the base points of the pencil giving us three sectionsof the fibration ˇ W : ˇ X → CP . We’ve already looked at homological mirror symmetryfor tropical Lagrangians when we place the A -model on X \ ( I ∪ { D i } i =1 ) = ( C ∗ ) , andthe B model on CP . We now switch the model used to study each space, and instead studythe A -model on CP . Of principle interest will be the Lagrangian discussed in fig. 3c, whichwe will call L inner ⊂ CP . The Lagrangian discussed in fig. 3a will be called L outer ⊂ CP .In section 5.2, we use methods from section 4.2 to compare the Lagrangian L T to a fiber F q ⊂ CP of the moment map. Finally, we make a homological mirror symmetry statementfor L T and the fibers of the elliptic surface ˇ X in section 5.3. CP . We now apply the tools from Lefschetz fibrations to give us a better understanding of thetropical Lagrangians in CP from fig. 14a. Proposition 5.2.1.
The Lagrangian L inner drawn in fig. 14a is Lagrangian isotopic to themoment map fiber F p of CP . This relation is already somewhat expected. [Via14] provides an infinite collection ofmonotone Lagrangian tori which are constructed by mutating the product monotone torialong different mutation disks. It is conjectured that these are all of the monotone toriin CP . From corollary 5.1.3 we know that the Lagrangian L T has the same Lagrangianmutation seed structure as T prod,mon , so if this conjecture on the classification of Lagrangiantori in CP holds, these two tori must be Hamiltonian isotopic. Proof.
The outline is as follows: we first use the isotopy provided by corollary 4.2.5 between L inner and L outer . We then compare the Lagrangians L outer to a Lagrangian glove for a Lef-schetz fibration. This Lefschetz fibration is constructed from a pencil of elliptic curves chosenfor a large amount of symmetry. Finally, we compare F p to the Lagrangian constructed viaa Lefschetz fibration. The Lagrangians F p and L outer are matched via an automorphism ofthe pencil of elliptic curves. 31e first will talk about the geometry of the pencil and the automorphism we consider.The Hesse pencil of elliptic curves is the one parameter family described by( z + z + z ) + t · ( z z z ) = 0which has four degenerate I fibers at equidistant points p , p , p , p ∈ CP . Let E ⊂ CP be the member of the pencil whose projection to the parameter space CP is the midpoint p between p and p . The generic fiber of the projection W : CP \ E → C is a9-punctured torus. From each I fiber we have three vanishing cycles. After picking pathsfrom these degenerate fibers to a fixed point p ∈ C , we can match the vanishing cycles tothe cycles in E p as drawn in fig. 15. Remark 5.2.2.
A small digression, useful for geometric intuition but otherwise unrelated tothis discussion, concerning the apparent lack of symmetry in the vanishing cycles of X .One might expect that the configuration of vanishing cycles which appear in fig. 15 to beentirely symmetric. While the Hesse pencil has symmetry group which acts transitively onthe I fibers, to construct the vanishing cycles one must pick a base point p and a basis ofpaths from E p to the critical fibers of the Hesse configuration, which breaks this symmetry.Each path from a point p to one of the four critical values p i gives us 3 parallel vanishingcycles. The 4 critical fibers of the Hesse configuration lie at the corners of an inscribedtetrahedron on CP . By choosing p = p to be the center of a face spanned by three of thesecritical values, 3 paths (say, γ , γ , γ ) from p to the critical values are completely symmetric.From such a choice, we obtain vanishing cycles (cid:96) j , (cid:96) j , (cid:96) j , where j ∈ { , , } . The homologyclasses (and in fact, honest vanishing cycles) | (cid:96) j | = (cid:104) (cid:105) | (cid:96) j | = (cid:104) , (cid:105) | (cid:96) j | = (cid:104) , (cid:105) are indistinguishable after action of SL (2 , Z ) , reflecting the overall symmetry of both the X configuration and the symmetry of the paths. The action of SL (2 , Z ) which inter-changes these cycles also permutes the 9 points of E p which are the base points of thisfibration.However, the introduction of the last path from the fourth critical fiber to p breaksthis symmetry. At best, this path can be chosen so that there remains one symmetry, whichexchanges (cid:96) and (cid:96) . In this setup, the vanishing cycles (cid:96) i lies in the class (cid:104) , − (cid:105) . Corre-spondingly, the class (cid:104) , − (cid:105) distinguishes the class (cid:96) from the other classes by intersectionnumber. This pencil is sometimes called the anticanonical pencil of CP . The automorphismgroup of the Hesse pencil is called the Hessian Group [Jor77]. This group acts on CP by permuting the critical values by even permutations. Consider a pencil automorphism g : CP → CP which acts on the 4 critical values via the permutation ( p p )( p p ). Thepoint p is fixed under this action, therefore g ( E ) = E . While the fiber E is mappedto itself, the map is a non-trivial automorphism of the fiber, swapping the vanishing cycles32 ××× p p p p p • ◦ ◦ ◦◦◦ ◦ ◦◦◦◦ ◦ ◦◦◦ ◦ ◦◦◦◦ ◦ ◦◦◦ ◦ ◦◦◦◦ ◦ ◦◦◦ ◦ ◦◦◦ Figure 15: A basis for the vanishing cycles for X given in [Sei17].for p and p : g ( (cid:96) ) = (cid:96) g ( (cid:96) ) = (cid:96) . We can use the Lefschetz fibration to associate to each cycle (cid:96) in E a Lagrangian in CP by taking the Hamiltonian suspension cobordism of (cid:96) in a small circle p + (cid:15)e iθ aroundthe point p in the base of the Lefschetz fibration. Call the Lagrangian torus constructedthis way T (cid:15),(cid:96) . The automorphism of the pencil g : CP → CP interchanges the Lagrangians T (cid:15),(cid:96) and T (cid:15),(cid:96) The standard moment map val dz : CP → Q CP ,dz can be chosen so that one of the I fibers of the Hesse configuration projects to the boundary of the Delzant polygon Q CP . Wechoose the moment map so that val − dz ( ∂Q CP ,dz ) = E , the I fiber lying above the point p .When one performs a nodal trade exchanging the corners of the moment map for interiorcritical fibers, we obtain a new toric base diagram, Q CP . The boundary of the base of thealmost toric fibration val : CP → Q CP corresponds to a smooth symplectic torus. Wearrange that val − ( ∂Q CP ) = E ⊂ CP . By comparison to the standard moment map, one sees that the cycle (cid:96) ⊂ E projects to apoint in the boundary of the moment map, while the cycle (cid:96) ⊂ E projects to the wholeboundary cycle. This gives us an understanding of the valuation projections of Lagrangian T (cid:15),(cid:96) and T (cid:15),(cid:96) . T (cid:15),(cid:96) has valuation projection which roughly looks like a point, and T (cid:15),(cid:96) hasvaluation projection which is a cycle that travels close to the boundary of Q CP . As a result33 ××× p p p p p ◦◦◦ ◦◦◦ ◦◦◦ ‘ ‘ γE Q CP L inner F q ˇ W ( z ) ∈ C E = ˇ W − ( p ) × ×× Figure 16: Relating tropical Lagrangians to thimbleswe have Hamiltonian isotopies identifying the Lagrangians T (cid:15),(cid:96) ∼ F p T (cid:15),(cid:96) ∼ L outer . See fig. 16, where L outer is drawn in red, and F p is drawn in blue.We conclude g ( L outer ) ∼ F p . As the projective linear group is connected, the morphism g is symplectically isotopic to the identity, and since H ( CP ) is trivial, all symplectic isotopiesare Hamiltonian isotopies. Therefore the Lagrangians L outer and F p are Hamiltonian isotopic.By corollary 4.2.3, the Lagrangians L inner and F p are Lagrangian isotopic.This shows that L inner is obtained from a Lagrangian that we’ve seen before, but pre-sented from a very different perspective. By taking a Lagrangian isotopy, L outer can be movedto L inner . We obtain the following relationships between Lagrangian submanifolds. Here,the equalities are taken up to Hamiltonian isotopy, and the dashed lines are Lagrangianswhich we expect to be Hamiltonian isotopic. L T L inner L outer T prod,mon T chek,mon F p . mutation Lag. Isotopymutation Lag. Isotopy . These tori are isomorphic objects of the Fukaya category, but this is a consequence ofFuk( CP ) having so few objects. A -Model on CP \ E . We now study the map g : CP → CP given by the automorphism of the Hesse configuration.The category Fuk( CP ) does not contain many objects, so the automorphism of the Fukaya34ategory induced by g is not so interesting. By removing an anticanonical divisor E = E we obtain a much larger category. For example, the Lagrangians L outer and F q are no longerHamiltonian isotopic in CP \ E . Claim 5.3.1. L outer and F q are not isomorphic objects of Fuk( CP \ E ) Proof.
The symplectic manifold CP \ E contains a Lagrangian thimble τ which is con-structed from the singular fiber of the almost toric fibration and extends out towards theremoved curve E (see fig. 3b). This thimble τ intersects L outer at a single point, and there-fore CF • ( L outer , τ ) is nontrivial. However, τ is disjoint from the fiber F q , so CF • ( F q , τ ) istrivial. As a result, F q and L outer are not isomorphic objects of the Fukaya category. Since E was fixed by the symplectomorphism g : CP → CP , the restriction to thecomplement g : CP \ E → CP \ E is still defined. Corollary 5.3.2.
The automorphism of the Fukaya category induced by the symplectomor-phism g g ∗ : Fuk( CP \ E ) → Fuk( CP \ E ) acts nontrivially on objects. This section of the paper is a series of observations and conjectures outlining homologicalmirror symmetry with the A -model on CP \ E , and B -model on ˇ X which hope to shedlight on the following conjecture. Conjecture 5.3.3.
The symplectomorphism g : CP → CP is mirror to fiberwise FourierMukai transform on the elliptic surface ˇ X which interchange the points of ˇ X with linebundles supported on the fibers of the elliptic fibration. CP \ E To that end, we study L T ⊂ CP \ E . An intermediate Blowup and Base Diagrams for X : We will begin with a de-scription of the elliptic surface X as an iterated blow up of CP along the base points ofan elliptic pencil following [AKO06]. Consider the pencil( z z + z z + z ) + tz z z = 0 . This elliptic fibration has 3 base points of degree 4, 4, and 1. Let ˇ W : ˇ X → CP beprojection to the parameter of the pencil. We can arrange for 6 of the blowups (3 on thetwo base points of degree 4) to be toric. We therefore obtain an intermediate step betweenˇ CP and ˇ X which is the toric symplectic manifold ˇ X Σ int . The toric diagram Q Σ int is theDelzant polytope with 9 edges. The remaining 3 blowups introduce nodal fibers in the toric In fact, the same argument shows that L outer and F q are not topologically isotopic. ( C ∗ ) , W CP )ˇ CP → ˇ Q CP ,dz (a) (( C ∗ ) , W Σ )ˇ X int Σ → ˇ Q int Σ (b) × ×× ( CP \ E, W E ) → Q CP \ E ˇ X → ˇ Q (c) Figure 17:
Top : obtaining ˇ X as a toric base diagram by first blowing up CP Bottom : Admissibility conditions for the A -model mirrors.base diagram ˇ Q for ˇ X which has 9 edges and 3 nodal fibers. The 9 edges of the toricbase correspond to the nine CP ’s making the I fiber of the fibration. The eigenray at eachcut in the diagram is parallel to the boundary curves. See fig. 17 for the base diagrams ofthese different blowups. B -model of X : Let ˇ π : ˇ X → ˇ X Σ int be the projection of the blowup. By [BO95]have a semiorthogonal decomposition of the category of the blowup as D b Coh( ˇ X ) = (cid:104) ˇ π − D b Coh( ˇ X Σ int ) , O D , O D , O D (cid:105) . For sheaves O H ∈ D b Coh( ˇ X Σ int ) with support on a hypersurface H , this semiorthogonaldecomposition states that there is a corresponding sheaf in X whose support is on thetotal transform of H . Should H avoid the points of the blow-up, the total transform willhave the same valuation projection as H . Should H contain the point of the blowup, thetotal transform includes the exceptional divisor of the blow-up. A fiber of the elliptic surface W − ( p ) is the proper transform of E p ⊂ X Σ int , a member of the pencil. On sheaves, wehave an exact sequence: (cid:32) (cid:77) i =1 O D i (cid:33) → π − ( O E p ) → O W − ( p ) . (1)We will now set up some background necessary to state a similar story for the A -model,summarized in assumption 5.3.6. Base for CP \ E : Running the machinery of [GS03] on ˇ Q , the SYZ base for ˇ X , willyield the SYZ base Q CP \ E for CP \ E . The base diagram Q CP \ E can also be constructed by36 Σ int C α y y = 0 arg − y y (0) X | C α (a) ×× × C α Q CP \ E × y y = (cid:15)y y = 0arg − y y − (cid:15) (0) Y | C α (b) Figure 18: Relating Lagrangians and Admissibility conditions between ( C ∗ ) and CP \ E with local Lefschetz models near corners.first constructing the mirror to the space ˇ X Σ int . As ˇ X Σ int is a toric variety, the mirror spaceis a Landau Ginzburg model ( X Σ int , W Σ int ) = (( C ∗ ) , W Σ int ), where the superpotential W Σ int yields a monomial admissibility condition (in the sense of [Han18]) ∆ Σ int on Q Σ int = R . Assumption 5.3.4 (Monomial Admissible Blow-up) . There is notion of monomial admis-sibility condition for CP \ E . This monomial admissibility condition is constructed from thedata of the monomial admissibility condition (( C ∗ ) , W Σ int ) . We now provide some motivation for this assumption. Recall, a monomial admissibilitycondition assigns to each monomial c α z α a closed set C α on which arg c α z α ( L | C α ) = 0. Fora set C α , denote by X | C α the portion of the SYZ valuation with valuation lying inside of C α . The restriction of admissible Lagrangians L | C α are contained within arg − c α z α (0). Theprojection arg − c α z α (0) → C α is an S subbundle of the SYZ fibration X | C α → C α .To obtain Q CP \ E from Q Σ int , we add in three cuts mirror to the three blowups. Thesethree cuts are added by replacing the regions C z z , C z z − and C z − z with affine charts C (cid:48) z z , C (cid:48) z z − and C (cid:48) z − z each containing a nodal fiber. The charts C α can be locally modelledon C \ { y y = 0 } with monomial admissibility condition ( y y ) − . We replace these withcharts containing a nodal fiber modeled on Y := C \ { y y = (cid:15) } and admissibility conditioncontrolled by the monomial ( y y − (cid:15) ) − . The valuation map Y | (cid:48) C α → C (cid:48) α is an almost toricfibration. We still have an S subbundle arg − y y − (cid:15) (0) ⊂ Y | (cid:48) C α of the SYZ fibration Y | (cid:48) C α → C (cid:48) α whenever (cid:15) is not negative real. This S -subbundle, and the monomial ( y y − (cid:15) ) − , shouldbe used to construct a monomial admissibility condition on CP \ E . See figs. 18 and 19In terms of the almost toric base diagrams, this compatibility can be stated as a matchingbetween the eigendirection of the introduced cuts and the ray of the fan corresponding tothe controlling monomial over the region including the cut.37 -model on CP \ E We now conjecture the existence of a mirror to the inverse-image func-tor on the B -model. Lagrangian submanifolds which lie in the S subbundle arg − c α z α (0) → C α should be in correspondence with Lagrangians which lie in the subbundle arg − y y − (cid:15) (0) ⊂ Y | C (cid:48) α . In particular monomial admissible Lagrangians of X give us monomial admissibleLagrangians of CP \ E . This allows us to transfer Lagrangians L in Fuk(( C ∗ ) , W Σ int ) toLagrangians π − ( L ) ∈ Fuk( CP \ E, W E ). Remark 5.3.5. π − ( L ) does not arise from a map between the spaces CP \ E and ( C ∗ ) .The symplectic manifold CP \ E is constructed from ( C ∗ ) by handle attachment. We keepthe notation π − so that it is consistent with the inverse image functor from our earlierdiscussion on the B -model. We observe that the thimbles of the newly introduced nodes (as in fig. 19) do not arise aslifts of Lagrangians in ( C ∗ ) . When constructing the Lagrangian thimble, there is a choiceof argument in the invariant direction of the node. We take the convention that in the localmodel Y | C (cid:48) α , the argument of the constructed thimble is positive and decreasing to zero alongthe thimble. With this choice of argument an application of the wrapping Hamiltonian willseparate the τ i and π − ( L ) so that π − ( L ) ∩ θ ( τ i ) = ∅ , and hom( π − ( L ) , τ i ) = 0. In summary: see figs. 18 and 19 Assumption 5.3.6 (Monomial Admissible Blow-up II) . There exists a Lagrangian corre-spondence between (( C ∗ ) , W Σ ) and ( CP , W E ) , giving us a functor π − : Fuk ∆ (( C ∗ ) , W Σ ) → Fuk ∆ ( CP \ E, W E ) . This functor gives us a semi-orthogonal decompositions of categories: (cid:104) π − Fuk ∆ (( C ∗ ) , W Σ ) , τ , τ , τ (cid:105) . We furthermore assume that this is mirror to the decomposition: (cid:104) ˇ π − D b Coh( X Σ int ) , O D , O D , O D (cid:105) . Remark 5.3.7.
While to our knowledge this has not been proven for the monomial admissi-bility condition, this statement is understood by experts in the symplectic Lefschetz fibrationadmissibility setting [HK; AKO06]. We give a translation of our statement into the Lefschetzviewpoint. Consider the pencil of elliptic curves p ( z , z , z ) + t · ( z z z ) . where p ( z z z ) = 0 is homogeneous degree 3 polynomial defining a generic elliptic curve E meeting z z z = 0 at 9 distinct points. Consider the elliptic fibration W E : X E → CP × × L T τ i ×× × π − ( L ( φ )) × τ L T π − ( L ( φ )) Figure 19: The Lagrangians in CP \ E relevant to our homological mirror symmetry state-ment. obtained by blowing up the 9 base points of this elliptic pencil, with exceptional divisors P , . . . P ⊂ X E . Let z ∞ ∈ CP be a critical value so that W − E ( z ∞ ) = I . Then ( C ∗ ) (cid:39) X E \ ( I ∪ P ∪ · · · ∪ P ) , and we may look at the restriction W E | ( C ∗ ) : ( C ∗ ) → CP \ { z ∞ } = C ( z , z ) (cid:55)→ p ( z , z , z z By construction, this is a rational function which expands into 9 monomial terms, and has 9critical points. The nine monomial terms correspond to the 9 directions in the fan drawn infig. 17b. The Fukaya-Seidel category constructed with W E | ( C ∗ ) → CP is mirror to X Σ int ,where the 9 thimbles drawn from these critical points are mirror to a collection of 9 linebundles generating D b Coh( ˇ X Σ int ) . These 9 thimbles correspond to 9 tropical Lagrangiansections σ φ : Q Σ int → ( C ∗ ) in the monomial admissible Fukaya category with fan fig. 17b.We now consider X = ( CP \ E ) = ( X E \ ( E ∪ P ∪ · · · ∪ P )) . The restriction W E | X : X → ( CP \ { } ) = C has 12 critical points, 9 of which may be identified with the critical points from the examplebefore. Conjecturally, this is mirror to X , where the thimbles from the three additionalcritical points are mirror to the exceptional divisors introduced in the blowup X → X Σ int .In the monomial admissible picture, the three additional thimbles are matched to the tropicalLagrangian thimbles introduced from the nodes appearing in the toric base diagram Q CP \ E drawn in fig. 17c L T ⊂ CP \ E . We now look at the Lagrangian three punctured torus L φ T ⊂ ( C ∗ ) = CP \ I described inexample 3.1.5. In order to make a homological mirror symmetry statement, we need to use39-side B-side( C ∗ ) , W Σ int X Σ int W Σ int L ( φ T ) Member of 9111-pencil (a) A-side B-side CP \ E, W E X Thimbles τ i Exceptional Divisors D i π − ( L ( φ T )) Total transform of member of 9111 Pencil L T Fiber of X → CP . (b) Table 1: A summary of the mirror correspondences that we use for this section.the non-Archimedean mirror ˇ X Λ9111 , however the intuition should be independent of the useof Novikov coefficients.Let φ T = x ⊕ x ⊕ ( x x ) − be the tropical polynomial whose critical locus passesthrough the rays of the nodes added in the modification of Q Σ int to Q . Theorem 5.3.8.
There exists a Lagrangian cobordism with ends ( L T , τ ∪ τ ∪ τ ) (cid:32) π − ( L φ T ) . Provided that assumption 5.3.6 holds and the cobordism is unobstructed, the Lagrangian L T is mirror to a divisor Chow-equivalent to a fiber of the elliptic fibration ˇ W : ˇ X → CP .Proof. We first construct the Lagrangian cobordism. At each of the 3 nodal points in thebase of the SYZ fibration Q CP \ E the Lagrangian L T meets τ i at a single intersection point.In our local model for the nodal neighborhood, this is the intersection of two Lagrangianthimbles. The surgery of those two thimbles is a smooth Lagrangian whose argument in theeigendirection of the node avoids the node. This was our local definition for π − ( L ( φ T )) ina neighborhood of the node.Recall that in this setting, we have an exact sequence of sheaves (cid:32) (cid:77) i =1 O D i (cid:33) → π − ( O E p ) → O W − ( p ) . (2)In the event that the cobordism constructed above is unobstructed, by [BC14] we havea similar exact triangle on the A -side, (cid:71) i =1 τ i → π − ( L ( φ T )) → L T . emark 5.3.9. It is reasonable to expect that the Lagrangian cobordism in question is un-obstructed, as the intersections between the τ i and L T are all in the same degree, thereforefor index reasons we can rule out the existence of holomorphic strips on L T ∪ τ i . In com-plex dimension 2, one can additionally choose an almost complex structure to rule out theexistence of Maslov 0 disks with boundary on L T ∪ τ i . These are similar to the conditionsused to prove unobstructedness of tropical Lagrangian hypersurfaces [Hic19]. Under the assumptions of [Hic19, A.3.2] on the existence of a restriction morphism forthe pearly model of Lagrangian Floer theory of Lagrangian cobordisms , the first and thirdterm in these exact triangles are mirror to each other. This identifies the mirror of the middleterm in the Chow group, proving the theorem.This mirror symmetry statement ties together several lines of reasoning. To each fiber F q ⊂ CP \ E equipped with local system ∇ , we can associate a value OGW ( F q , ∇ ) which isa weighted count of holomorphic disks with boundary F q in the compactification F q ⊂ CP .By viewing X as the moduli space of pairs ( F q , ∇ ), we obtain a function W OGW : X \ I → C . This function matches the restriction W | X \ I . In the previous discussion we conjec-tured that sheaves supported on W − OGW (0) are mirror to L T . Recall that L T can also beconstructed as the dual dimer Lagrangian (definition 3.3.5 )to the mutation configurationfor the monotone fiber F . In this example, these two constructions of L T suggest thatthe dual dimer Lagrangian for a mutation configuration is mirror to the fiber of the OpenGromov-Witten superpotential. References [Abo09] Mohammed Abouzaid. “Morse homology, tropical geometry, and homologi-cal mirror symmetry for toric varieties”.
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