Reconstruction of T P 2 via tropical Lagrangian multi-section
RRECONSTRUCTION OF T P VIA TROPICAL LAGRANGIANMULTI-SECTION
YAT-HIN SUEN
Abstract.
In this paper, we study the reconstruction problem of the holomorphic tangentbundle T P of the complex projective plane P . We introduce the notion of tropical Lagrangianmulti-section and cook up one tropicalizing the Chern connection associated the Fubini-Studymetric. Then we perform the reconstruction of T P from this tropical Lagrangian multi-section.Walling-crossing phenomenon will occur in the reconstruction process. Contents
1. Introduction 1Acknowledgment 32. SYZ mirror symmetry of P
33. A tropical Lagrangian multi-section associated to T P and reconstruction 53.1. Construction of the tropical Lagrangian multi-section 53.2. Reconstructing T P from the tropical Lagrangian multi-section 83.3. The wall-crossing factors 10Appendix A. Local model for caustics 11Appendix B. A cone complex of T P Introduction
Mirror symmetry is a duality between symplectic geometry and complex geometry. The famousSYZ conjecture [36] allows mathematicians to construct mirror pairs and explain homologicalmirror symmetry [30] geometrically via a fiberwise Fourier–Mukai-type transform, which we callthe
SYZ transform .The SYZ transform has been constructed and applied to understand mirror symmetry in thesemi-flat case [6, 34, 33, 17] and the toric case [1, 2, 19, 20, 21, 8, 10, 11, 9, 16, 14, 13, 18]. Butin all of these works the primary focus was on Lagrangian sections and the mirror holomorphicline bundles the SYZ program produces. For higher rank sheaves, Abouzaid [3] applied the ideaof family Floer cohomology to construct sheaves on the rigid analytic mirror of a given compactLagrangian torus fibration without singular fiber. Recently, applications of the SYZ transform forunbranched Lagrangian multi-sections have been study by Kwokwai Chan and the author in [15].In this paper, we study the holomorphic tangent bundle T P of P in terms of mirror symmetry.Via SYZ construction, the mirror of P is given by a Landau-Ginzburg model ( Y, W ) (see Section2 for a brief review). It carries a natural fibration p : Y → N R ∼ = R , which is dual (up to aLegendre transform) to the moment map fibration of P . Since T P is naturally an object in thederived category D b Coh( P ), it is natural to ask what is its mirror Lagrangian in Y . Being arank 2 bundle, the mirror Lagrangian of T P is expected to be a rank 2 Lagrangian multi-sectionof p : Y → N R with certain asymptotic conditions. As the base N R of the fibration p is simply Date : April 1, 2020. a r X i v : . [ m a t h . S G ] M a r Y.-H. SUEN connected, every unbranched covering map is trivial. But T P is certainly indecomposable. Hencewe are leaded to consider branched Lagrangian multi-section and the SYZ transform defined in[6, 15, 34] cannot be applied directly. To overcome this technicality, we introduce the notion of tropical Lagrangian multi-section and reconstruct T P from this tropical object. Definition 1.1 (=Definition 3.1) . Let B be an n -dimensional integral affine manifold withoutboundary. A rank r tropical Lagrangian multi-section is a triple L := ( L, π, ϕ ) , where a) L is a topological manifold. b) π : L → B a covering map of degree r with branch locus S ⊂ B being a union of locallyclosed submanifolds of codimension at least 2. c) ϕ = { ϕ U } is a multi-valued function on L such that on any two affine charts U, V ⊂ L \ π − ( S ) (with respective to the induced affine structure on L \ π − ( S ) via π ), ϕ U − ϕ V = (cid:104) m, x (cid:105) + b, for some m ∈ Z n and b ∈ R . Let Σ be the fan corresponds to P and v , v , v be its primitive generators. The tropicalLagrangian multi-section is obtained by “tropicalizing” the Chern connection associated to theFubini-Study metric. We will do this in Section 3.1 by considering a family of K¨ahler metrics on P , which gives rise to a family of Chern connections, parameterized by a small real number (cid:126) > (cid:126) → T N R . From the tropical connection, we cancook up six linear functions ϕ ± k , k = 0 , ,
2, for which ϕ ± k are defined on σ k , the cone generatedby the rays v i , v j , i, j (cid:54) = k . Let σ ± k be two copies of σ k and we consider ϕ + k (resp. ϕ − k ) as afunction defined on σ + k (resp. σ − k ). A piecewise linear function ϕ and an integral affine manifold L can be obtained by gluing ϕ ± k : σ ± k → R together in a continuous manner. By projecting σ ± k to σ k , we obtain a map π : L → N R . The triple L := ( L, π, ϕ ) forms a rank 2 tropical Lagrangianmulti-section. See Section 3.1 for the detailed construction.Let U k be the affine chart corresponds to the cone σ k . For each k , the linear functions ϕ ± k definea rank 2 equivariant bundle on U k by taking direct sum. The gluing of L tells us how to gluethese rank 2 bundles together so that the equivariant structures agree on U ij := U i ∩ U j . We canwrite down three transition functions τ sf , τ sf , τ sf for this naive gluing. However, this naive gluingis inconsistent. In order to obtain a consistent gluing, we modify the naive transition functionsby three invertible factors Θ , Θ , Θ . Put τ (cid:48) j := τ sfij Θ ij . Then these new transition functionssatisfy(1) (cid:0) τ (cid:48) J − (cid:1) τ (cid:48) ( Jτ (cid:48) ) = I. The J is the matrix (cid:18) −
11 0 (cid:19) and may wonder why it is present in the equation. This is due to a choice of a U (1)-local system L defined on L ∼ = N R \{ } with holonomy −
1. A more detailed explanation will be given in Section3.2.Equation (1) gives a rank 2 holomorphic vector bundle defined on P , which we call it the instanton-corrected mirror of L . Moreover, we have Theorem 1.2 (=Theorem 3.9) . The instanton-corrected mirror of the tropical Lagrangian multi-section L is isomorphic to the holomorphic tangent bundle T P of P . In the last section, Section 3.3, we will discuss the relationship between the factors { Θ ij } andthe family Floer theory of the (conjecturally exists) mirror Lagrangian of T P . The Fourier modes m k ∈ M of Θ ij , k (cid:54) = i, j , determine three cotangent directions of N R the origin. As the Lagrangian L is tropical, we cannot expect one can determine which fibers of p will bound holomorphic disk ECONSTRUCTION OF T P VIA TROPICAL LAGRANGIAN MULTI-SECTION 3 with the honest Lagrangian. Nevertheless, m k should be regarded as the normal directions of walls emitting from the branched point 0 ∈ N R .In Appendix A, we will give detailed review of Fukaya’s local model on caustic points and hisconstruction of mirror bundle via deformation theory [22], Section 6.4. This gives a symplecto-geometric explanation for the walling-crossing phenomenon in our reconstruction process.In Appendix B, we follow [35] to give a construction of another tropical Lagrangian multi-section L (cid:48) by using the natural equivariant structure on T P . One can also perform the reconstruction byusing L (cid:48) . But two reconstruction processes have some fundamental differences. For instance, thelocal system L is absent if we use L (cid:48) as the tropical model. We will give a heuristic explanation tothis difference in terms of special Lagrangian condition. Acknowledgment.
The author is grateful to Byung-Hee An, Kwokwai Chan, Ziming NikolasMa and Yong-Geun Oh for useful discussions. A special thanks goes to Katherine Lo for herencouragement during this work. The work of Y.-H. Suen. was supported by IBS-R003-D1.2.
SYZ mirror symmetry of P We begin with reviewing some elementary facts about the complex projective plane P .Let N ∼ = Z be a lattice of rank 2 and set N R := N ⊗ Z R , M := Hom Z ( N, Z ) , M R := M ⊗ Z R . Let Σ be the fan with primitive generators v := (1 , v := ( − , , v := (0 , − , The associate toric variety X Σ is the complex projective plane P . The dense torus of P can beidentified with T N R /N . Let ˇ p : T N R /N → N R be the natural projection. Denote the coordinateson N R by ξ i and the fiber coordinates of ˇ p by ˇ y i . Complex coordinates on T N R /N are given by w i := e z i := e ξ i + √− y i . There is an 1-1 correspondence between supporting functions on | Σ | = N R and ( C × ) -equivariantline bundles on P . Explicitly, the equivariant line bundle O ( a D + a D + a D ) corresponds tothe supporting function ϕ : N R → R , defined by setting ϕ ( v i ) := a i .Let σ := R ≥ (cid:104) v , v (cid:105) σ := R ≥ (cid:104) v , v (cid:105) , σ := R ≥ (cid:104) v , v (cid:105) . and U k ∼ = C be the affine chart corresponds to the cone σ k , for k = 0 , ,
2. We can trivialize T P on U k by τ k : T P | U k (cid:51) ([ ζ : ζ : ζ ] , v ) (cid:55)→ ( w ik , w jk , v ik , v jk ) ∈ C × C , for i, j, k = 0 , , i < j . Here, w ik := ζ i ζ k and v =: v ik ∂∂w ik + v jk ∂∂w jk . The transition functions τ ij := τ i ◦ τ − j are given by τ = (cid:32) − w ) − w ( w ) w (cid:33) , τ = (cid:32) w − w ( w ) − w ) (cid:33) , τ = (cid:32) − w ( w ) w − w ) (cid:33) . Remark 2.1.
The coordinates w i on T N R /N and the inhomogeneous coordinates w i are relatedby w i = w i . A wall is a codimension 1 submanifold W ⊂ N R for which there is a non-trivial holomorphic disk bounded bythe Lagrangian and those fibers over W . We use the convention that if a piecewise linear function f is given by f ( v i ) = a i , then the corresponding linebundle is given by O (cid:16)(cid:80) i =0 a i D i (cid:17) . Y.-H. SUEN
It is well-known that P carries a K¨ahler-Einstein metric, called Fubini-Study metric. It is theHermitian metric associated to the (1 , ω F S := 2 √− ∂ ¯ ∂φ ( ξ ) , where φ ( ξ ) := 12 log(1 + e ξ + e ξ ) . Let ∇ F S be the Chern connection associated to the the Fubini-Study metric. With respective tothe holomorphic frame { ∂∂w , ∂∂w } , it can be written as ∇ F S | U = d −
11 + | w | (cid:20)(cid:18) | w | w ¯ w | w | (cid:19) dz + (cid:18) | w | ¯ w w | w | (cid:19) dz (cid:21) where | w | := | w | + | w | . Also note that the induced connection on det( T P ) ∼ = O (3) is given by ∇ = d − (cid:18) | w | | w | dz + | w | | w | dz (cid:19) . Now, we jump to SYZ mirror symmetry of P . The mirror of P is given by the Landau-Ginzburgmodel ( Y, W ), where Y := T ∗ N R /MW ( z , z ) := z + z + qz z , and q > z j are given by z i := e x i + √− y i , where x i is the affine coordinates on ˚ P and y i are the fiber coordinates. One can equip Y with thestandard symplectic structure ω Y := dξ ∧ dy + dξ ∧ dy , and the holomorphic volume form Ω Y := dz z ∧ dz z . Let p : Y → N R be the natural projection, which is clearly dual to ˇ p : T N R /N → N R . Thehomological mirror symmetry conjecture predicts that Lagrangian branes in Y should be mirror tocoherent shaves in P . In [8], the author consider Lagrangian sections of p : Y → N R with certaindecay conditions at infinity and define its SYZ mirror line bundle on P . Roughly speaking, givensuch a Lagrangian section L , one can associate a line bundle ˇ L → P together with a connection ∇ ˇ L , so that with respective to a local unitary frame ˇ1, one has ∇ ˇ L := d + √− (cid:0) f ( ξ ) d ˇ y + f ( ξ ) d ˇ y (cid:1) . where ( f , f ) are local defining equations of L . In fact, every such connection is compatible withthe metric e − F , where F is a local potential function of the Lagrangian section L . In terms ofthe local holomorphic frame e − F ˇ1, the connection becomes d − (cid:0) f ( ξ ) dz + f ( ξ ) dz (cid:1) . Using the potential function φ , for each k ∈ Z , we can define the Lagrangian section L k := { ( ξ, k · dφ ( ξ )) ∈ Y : ξ ∈ N R } . The mirror bundle of L k is the line bundle O ( k ) together with the connection ∇ k = d − (cid:18) k | w | | w | dz + k | w | | w | dz (cid:19) . Note that k · φ is a smoothing of the supporting function corresponds to O ( kD ). Explicitly,lim t →∞ k t (1 + t ξ + t ξ ) = max { , kξ , kξ } . ECONSTRUCTION OF T P VIA TROPICAL LAGRANGIAN MULTI-SECTION 5
Thus the differential of the supporting function should be regarded as a singular Lagrangian sectionof p : T ∗ N R /N → N R . We apply this idea to tropicalize ∇ F S .3.
A tropical Lagrangian multi-section associated to T P and reconstruction In this section, we introduce the notion of tropical Lagrangian multi-section and construct oneby tropicalizing the Chern connection associated to the Fubini-Study metric. We then perform thereconstruction of T P from the tropical Lagrangian multi-section. The wall-crossing phenomenonwill be discussed in the last subsection.We now introduce the following Definition 3.1.
Let B be an n -dimensional integral affine manifold without boundary. A rank r tropical Lagrangian multi-section is a triple L := ( L, π, ϕ ) , where a) L is a topological manifold. b) π : L → B a covering map of degree r with branch locus S ⊂ B and ramification locus S (cid:48) ⊂ L being a union of locally closed submanifolds of codimension at least 2. c) ϕ = { ϕ U } is a collection of local continuous functions on L such that on any two affinecharts U, V ⊂ L \ S (cid:48) (with respective to the induced affine structure on L \ S (cid:48) via π ), ϕ U − ϕ V = (cid:104) m, x (cid:105) + b, for some m ∈ Z n and b ∈ R . Definition 3.1 is a straightforward generalization of the notion of polarization in the famousGross-Siebert program [23, 25, 26]. We also remark that the domain L can be disconnected ingeneral. But in this paper, L is connected and the multi-valued function ϕ is a single-valuedcontinuous function.3.1. Construction of the tropical Lagrangian multi-section.
In order to obtain a tropicalLagrangian for T P , we need to “tropicalize” the Chern connection associated to the Fubini-Study.To do this, we construct a family { ( X (cid:126) , ω (cid:126) ) } (cid:126) > of K¨ahler manifolds. Let P := { ( x , x ) ∈ M R : x , x ≥ , x + x ≤ } be a moment polytope of P and ˚ P be its interior. Let g (cid:126) := g P + (cid:126) − ψ. where g P ( x ) := 12 ( x log( x ) + x log( x ) + (1 − x − x ) log(1 − x − x )) ,ψ ( x ) := x + x + x x − x − x . The Legendre dual coordinates are denoted by ξ i (cid:126) := ∂g (cid:126) ∂x i . They are related to the original coordinates via ξ i (cid:126) = ξ i + (cid:126) − ∂ψ∂x i . Put w i (cid:126) := e z i (cid:126) := e ξ i (cid:126) + √− y i , i = 1 , . A straightforward calculation shows that
Hess ( g P + (cid:126) − ψ ) > , det( Hess ( g P + (cid:126) − ψ )) = 1 α (cid:126) ( x ) x x (1 − x − x ) , Y.-H. SUEN for some smooth function α (cid:126) : P → R so that if we choose (cid:126) > α (cid:126) ( x ) > x ∈ ˚ P . These are precisely the compatibility conditions stated in [4, 5], which guaranteethe complex coordinates w i (cid:126) can be extended to P . Thus, we get a family of complex manifolds { X (cid:126) } (cid:126) > . The polytope P also gives an ample line bundle L (cid:126) → X (cid:126) and hence a family ofisomorphisms ι (cid:126) : X (cid:126) → P . On the torus T N R /N ⊂ X (cid:126) , the isomorphism ι (cid:126) is simply given by( w (cid:126) , w (cid:126) ) (cid:55)→ [1 : w (cid:126) : w (cid:126) ] . We define ω (cid:126) := ι ∗ (cid:126) ω F S . Remark 3.2.
More generally, given a projective toric manifold ( X, ω ) with toric symplectic struc-ture ω and an ample line bundle L , Guillemin [28] proved that [ ω ] = [ ι ∗ ω F S ] = c ( L ) , where ι : X (cid:44) → P N is the holomorphic embedding and N + 1 = (cid:93) ( P ∩ M ) . Now, if we have a family { ( X (cid:126) , L (cid:126) ) } (cid:126) > of polarized toric manifolds with holomorphic embeddings ι (cid:126) : X (cid:126) (cid:44) → P N , then [ ω ] = [ ι ∗ (cid:126) ω F S ] = c ( L (cid:126) ) and the K¨ahler metric (cid:126) · ω ( J (cid:126) − , − ) degenerates to the Hessian metric Hess ( ψ ) on the polytope P as (cid:126) → (see [7] ). This fits into the context of SYZ conjecture [27, 31] that certain “good”representative ( ω in this case) in c ( L (cid:126) ) , after rescaling, degenerates to a metric on the base asthe family approaching the larger complex structure limit point. Put g i := ∂g P ∂x i and ψ i := ∂ψ∂x i . Note that ψ =2 x + x − ,ψ = x + 2 x − ,ψ − ψ = x − x . Fix a fiber F ξ := ˇ p − ( ξ ) ⊂ T N R /N ⊂ P . With respective to the coordinates w i (cid:126) , we computethe limit (cid:126) → ∇ F S | F ξ . We decompose P intothree pieces P := P ∩ { ψ ≤ , ψ ≤ } ,P := P ∩ { ψ ≥ , x ≥ x } ,P := P ∩ { ψ ≥ , x ≥ x } . For x ∈ ˚ P , we have lim (cid:126) → | w (cid:126) | | w (cid:126) | = lim (cid:126) → e g e (cid:126) − ψ e g e (cid:126) − ψ + e g e (cid:126) − ψ = 0 , lim (cid:126) → | w (cid:126) | | w (cid:126) | = lim (cid:126) → e g e (cid:126) − ψ e g e (cid:126) − ψ + e g e (cid:126) − ψ = 0 , lim (cid:126) → | w (cid:126) w (cid:126) | | w (cid:126) | = lim (cid:126) → e g + g e (cid:126) − ( ψ + ψ ) e g e (cid:126) − ψ + e g e (cid:126) − ψ = 0as ψ , ψ <
0. For x ∈ ˚ P , we havelim (cid:126) → | w (cid:126) | | w (cid:126) | = lim (cid:126) → e g e − (cid:126) − ψ + e g + e g e (cid:126) − ( ψ − ψ ) = 1 , lim (cid:126) → | w (cid:126) | | w (cid:126) | = lim (cid:126) → e g e − (cid:126) − ψ + e g e (cid:126) − ( ψ − ψ ) + e g = 0 , lim (cid:126) → | w (cid:126) w (cid:126) | | w (cid:126) | = lim (cid:126) → e g + g e g − g e (cid:126) − ( ψ − ψ ) + e g − g e (cid:126) − ( ψ − ψ ) = 0 . ECONSTRUCTION OF T P VIA TROPICAL LAGRANGIAN MULTI-SECTION 7
Similarly, we have, for x ∈ ˚ P , lim (cid:126) → | w (cid:126) | | w (cid:126) | =0 , lim (cid:126) → | w (cid:126) | | w (cid:126) | =1 , lim (cid:126) → | w (cid:126) w (cid:126) | | w (cid:126) | =0 . The potential function φ of the Fubini-Study metric defines an isomorphism dφ : N R → ˚ P , whichis known as the Legendre transform . Since dφ maps ˚ σ i to ˚ P i , we have a singular connection ∇ tropF S on the total space T N R , define by ∇ tropF S := d if ξ ∈ ˚ σ ,d − √− (cid:18) (cid:19) d ˇ y if ξ ∈ ˚ σ ,d − √− (cid:18) (cid:19) d ˇ y if ξ ∈ ˚ σ . (2)Now we construct a tropical Lagrangian multi-section from (2). For each cone σ k , we let σ ± k betwo copies of σ k . Define six linear functions ϕ +0 : σ +0 (cid:51) ( ξ , ξ ) (cid:55)→ ,ϕ − : σ − (cid:51) ( ξ , ξ ) (cid:55)→ ,ϕ +1 : σ +1 (cid:51) ( ξ , ξ ) (cid:55)→ ξ ,ϕ − : σ − (cid:51) ( ξ , ξ ) (cid:55)→ ξ ,ϕ +2 : σ − (cid:51) ( ξ , ξ ) (cid:55)→ ξ .ϕ − : σ +2 (cid:51) ( ξ , ξ ) (cid:55)→ ξ We obtain a topological space L by gluing σ ± with σ ∓ and σ ∓ along v and v , respectively, andglue σ ± with σ ∓ along v . The topological space L is homeomorphic to N R and the projectionmap π : L → | Σ | ∼ = N R given by mapping σ ± k → σ k can be identified with the square map z (cid:55)→ z on C . Moreover, { ϕ ± k } glue to a continuous piecewise linear function ϕ on L . See Figure 1. If wetake the “trace” of ϕ , we obtain the piecewise linear function corresponds to the toric line bundle O (3 D ), which is of course isomorphic to det( T P ) as a holomorphic line bundle. Figure 1.
The tropical Lagrangian L .In summary, we obtain Y.-H. SUEN
Proposition 3.3.
The data L := ( L, π, ϕ ) defines a tropical Lagrangian multi-section. Remark 3.4.
One should think of the manifold L and the differential of the function ϕ as thetropical limit of certain rank 2 Lagrangian multi-section of p : Y → N R that is mirror to T P . Ifwe formally apply the SYZ transform to the Lagrangian multi-section dϕ : L → Y , we obtain (2).See [15] for precise definition of SYZ transform of unbranched Lagrangian multi-sections. Remark 3.5. In [35] , Payne used equivariant structure of T P to construct another tropical La-grangian multi-section (he called it a cone complex instead). We will discuss the difference betweenthese two constructions in Appendix B. Reconstructing T P from the tropical Lagrangian multi-section. The functions ϕ ± k , k = 0 , ,
2, correspond to six equivariant line bundles. ϕ +0 ↔ O| U ,ϕ − ↔ O| U ,ϕ +1 ↔ O (2 D ) | U ,ϕ − ↔ O ( D ) | U ,ϕ +2 ↔ O ( D ) | U .ϕ − ↔ O (2 D ) | U . Hence give rise to three equivariant rank 2 bundles on each affine charts: O| U ⊕O| U , O (2 D ) | U ⊕O ( D ) | U , O (2 D ) | U ⊕O ( D ) | U . The ( C × ) -action on each piece is given by minus of the slope of the supporting functions, explicitly,( λ , λ ) · ( w , w , v , v ) =( λ w , λ w , v , v ) , ( λ , λ ) · ( w , w , v , v ) =( λ − w , λ − λ w , λ − v , λ − v ) , ( λ , λ ) · ( w , w , v , v ) =( λ − w , λ λ − w , λ − v , λ − v ) . According to the gluing of (
L, ϕ ), we can try to glue these rank 2 bundles together equivariantly.Then we obtain the following set of naive transition functions τ sf := (cid:32) a ( w ) b w (cid:33) , τ sf := (cid:32) a ( w ) b w (cid:33) , τ sf := (cid:32) b w a ( w ) (cid:33) , for some constants a i , b i ∈ C × . Since tr ( ϕ ) is the piecewise linear function defining O (3), we shouldimpose the condition (cid:89) i =0 a i b i = − . However, it is clear that they can’t composed to the identity, which means they cannot be glued toan equivariant bundle on P ! This is due to the non-trivial monodromy present around the originof N R . Remark 3.6.
Although { τ sfij } does not form a vector vector bundle on P , they do form a rank2 bundle E (cid:48) on the singular space D ∪ D ∪ D because each divisor D k is covered by the twocharts U i ∩ D k , U j ∩ D k for i, j, k = 0 , , begin distinct and there are no triple intersections, sothe cocycle condition is vacuous. Since the torus ( C × ) ⊂ P corresponds to the point ∈ N R ,which is exactly the branched point of π : L → N R , from the SYZ transform perspective, E (cid:48) shouldbe regarded as the mirror bundle of L := ( L \ π − (0) , π | L \ π − (0) , ϕ | L \ π − (0) ) . ECONSTRUCTION OF T P VIA TROPICAL LAGRANGIAN MULTI-SECTION 9
In order to obtain a consistent gluing and reconstruct T P , we have to modify each τ sfij by aninvertible factor. We choose the correction factors to beΘ := I + (cid:32) − a b a w w (cid:33) ∈ Aut ( O| U ⊕ O| U ) , Θ := I + (cid:32) − b − b − a − w w (cid:33) ∈ Aut ( O (2 D ) | U ⊕ O ( D ) | U ) , Θ := I + (cid:32) a − b − b − w w (cid:33) ∈ Aut ( O (2 D ) | U ⊕ O ( D ) | U ) . For each j = 0 , ,
2, the factor Θ ij is written in terms of the frame on U j and is only defined on U ij . Define τ (cid:48) ij := τ sfij Θ ij . But still, they do not compose to the identity. Let J := (cid:18) −
11 0 (cid:19) . Then we have
Proposition 3.7.
With the condition (cid:81) i a i b i = − , we have (3) (cid:0) τ (cid:48) J − (cid:1) τ (cid:48) ( Jτ (cid:48) ) = I. The present of J can be understood as a U (1)-local system L with holonomy − L ,where L is identified with N R \{ } under the identification L ∼ = N R . The construction of L is asfollows: we cover N R \{ } by two charts V , V , obtained by deleting the rays R ≥ (cid:104) v (cid:105) , R ≥ (cid:104) v (cid:105) ,respectively. Then we can cover L by π − ( V ) =: V +1 (cid:113) V − , π − ( V ) =: V +2 (cid:113) V − . We define L by declaring its transition functions on each overlapping region:1 on V +1 ∩ V +2 , − V +1 ∩ V − , − V − ∩ V +2 , V − ∩ V − . Pushing forward L to N R \{ } via π , we obtain a rank 2 local system on N R \{ } with transitionfunctions given by I on π ( V +1 ∩ V +2 ) and J on π ( V +1 ∩ V − ). This rank 2 local system contributeswhen one goes into the cones for which J is present. This explains the appearance of J and J − in Equation (3). One compares this reconstruction process with the construction given in [22],Section 6.4. or Appendix A, by using deformation theory. Remark 3.8.
We can also put − on V ± ∩ V ± and on V ± ∩ V ∓ . Equation (3) will not beaffected by this change as J − = − J . Proposition 3.7 gives a rank 2 holomorphic vector bundle E on P , which we called the instanton-corrected mirror of the tropical Lagrangian multi-section L . Furthermore, we have Theorem 3.9.
The instanton-corrected mirror of the tropical Lagrangian multi-section L is iso-morphic to the holomorphic tangent bundle T P of P . Proof.
We define f : T P → E by f | U := f := (cid:18) a b a (cid:19) on U ,f | U := f := (cid:18) a − b − a (cid:19) on U ,f | U := f := (cid:18) − b − a b (cid:19) on U . Using (cid:81) i a i b i = −
1, one can check that (cid:0) τ (cid:48) J − (cid:1) f = f τ , τ (cid:48) f = f τ , ( Jτ (cid:48) ) f = f τ . Hence f defines an isomorphism. (cid:3) Remark 3.10.
As pointed out by Fukaya [22] , the local system L (which he called an orientationtwist) is related to the orientation of certain moduli space of holomorphic disks. Another worthmentioning since a generic caustic point of a 2-dimensional special Lagrangian can always bemodeled by ( C , L ) , where L := { ( x , ¯ x ) ∈ C : x ∈ C } . In our reconstruction of T P , we tropicalize the connection ∇ F S , which is Hermitian-Yang-Millsconnection. Since Hermitian-Yang-Mills condition is supposed to be mirror to special Lagrangiancondition [34, 37] in dimension 2, the tropical Lagrangian multi-section L , which is obtained fromthe tropicalization of ∇ F S , should be a tropical limit of a rank 2 special Lagrangian multi-sectionof p : Y → N R . That’s why the local system L and the sign difference appear naturally in ourreconstruction process. Another way to see that L is the local model is by smoothing the piecewiselinear function ϕ . We first identify the projection π : L → B with the square map x (cid:55)→ x on C and approximate ϕ : L → R by
12 ( x − x ) near . This function is nothing but the potential function of L . The wall-crossing factors.
In Section 3.2, we have introduced three invertible factorsΘ := I + (cid:32) − a b a w w (cid:33) ∈ Aut ( O| U ⊕ O| U ) , Θ := I + (cid:32) − b − b − a − w w (cid:33) ∈ Aut ( O (2 D ) | U ⊕ O ( D ) | U ) , Θ := I + (cid:32) a − b − b − w w (cid:33) ∈ Aut ( O (2 D ) | U ⊕ O ( D ) | U ) , to modify the naive transition functions τ sfij . In this section, we give a heuristic explanation abouthow Θ ij are related to holomorphic disks bounded by the (conjecturally exists) mirror Lagrangianof T P . In terms of the coordinates w i = w i , each factor Θ ij determines a Fourier mode m k ∈ M ,for k (cid:54) = i, j , where m := (0 , − , m := (1 , , m := ( − , . They define three covectors of N R at 0 and hence three lines l k , k = 0 , , N R by taking kernels of m k ’s. Let n k ∈ N be the primitive integral tangent vector of l k so that if we identify N R , M R with R and the pairing (cid:104)− , −(cid:105) with the standard inner product on R , { m k , n k } forms an orientablebasis with respective to the standard volume form dx ∧ dy on R . Then we have n := (1 , , n = (0 , , n := ( − , − . The Fourier mode of e ( m,z ) is m ∈ M ECONSTRUCTION OF T P VIA TROPICAL LAGRANGIAN MULTI-SECTION 11
See Figure 2.
Figure 2
Hence Equation (3) can be understood as the wall-crossing diagram as shown in Figure 2.Furthermore, in view of [29], the walls of the mirror Lagrangian should concentrate on a smallneighborhood of (cid:83) k =0 R ≥ (cid:104) n k (cid:105) as (cid:126) → tropical vertex group . They are responsible for correcting thesemi-flat complex structure by Maslov index 0 disks bounded by those SYZ fibers over a wall. Inour case, which is an open theory, the factors Θ ij ’s are responsible for correcting the “semi-flatbundle” { τ sfij } by non-trivial holomorphic disks bounded by SYZ fibers over walls and the mirrorLagrangian of T P .We end this section by stating the following Conjecture 3.11.
There exists a connected rank 2 special Lagrangian multi-section L of p : Y → N R so that for any (cid:15) > , there exists δ > such that if (cid:126) ∈ (0 , δ ) , the (cid:15) -tubular neighborhood U (cid:15) of (cid:91) k =0 R ≥ (cid:104) n k (cid:105) contains the walls of L . Furthermore, L is quasi-isomorphic to a cone between L and L ⊕ , where L is the zero section of p and the section L . As the SYZ transform of L and L is given by the structural sheaf O and the line bundle O (1),respectively, this conjecture is nothing but a symplecto-geometric analog of the Euler sequence for P . Appendix A. Local model for caustics
In this appendix, we give a review on Fukaya’s local model on caustic points [22], Section 6.4.Let B := C and X := C . Equip X with the standard symplectic structure ω = dx ∧ dy + dx ∧ dy and holomorphic volume form Ω = dz ∧ dz , where z i = x i + √− y i are the standard complex coordinates on X . After a hyperk¨ahler rotation,we have complex coordinates x = x + √− x , y = y − √− y . Fukaya considered the Lagrangian L := { ( x , ¯ x ) | x ∈ B } ⊂ X With respective to p : X → B , the projection onto the first coordinate, L is a special Lagrangianmulti-section of rank 2. Parameterizing L in terms of polar coordinates: L = { ( re √− θ , √ re −√− θ ) ∈ C : r ≥ , θ ∈ R } . Let u : [0 , × [ − , → X be given by u ( s, t ) := ( sre √− θ , t √ sre −√− θ ) . Define f L ( x ) := (cid:90) − (cid:90) u ∗ ω. An elementary calculation shows that f L ( x ) = 43 r cos (cid:18) θ (cid:19) . Proposition A.1.
There are precisely three gradient flow lines of f L starting from the origin.Proof. In polar coordinates, we have ∇ f L = 2 √ r cos (cid:18) θ (cid:19) ∂∂r − (cid:126) − √ r sin (cid:18) θ (cid:19) ∂∂θ . Then the gradient flow equation ( ˙ r, ˙ θ ) = ∇ f L ( r, θ )has solution given by r sin (cid:18) θ (cid:19) = C, where C is a real constant. If the gradient flow lines start from the origin, then we have C = 0.Hence the gradient flow lines are precisely those straight lines along the directions θ = 0 , θ = 2 π , θ = 4 π . (cid:3) The 3 gradient flow lines emitting from the origin of B , namely,(0 ,
12 ) (cid:51) t (cid:55)→ te √− θ , with θ = 0 , π , π , should correspond to three holomorphic disks bounded by L and those fibers of p : X → B supported on these rays.Let ˇ p : ˇ X → B be the dual fibration of p : X → B and define B > := B \{ x ∈ C : x ≤ } ,B > := B \{ x ∈ C : x ≥ } , ˇ X > :=ˇ p − ( B > ) , ˇ X < :=ˇ p − ( B < ) . Equip L := L \{ } with a local system L with holonomy −
1. The mirror bundle E of ( L \{ } , L )has local holomorphic frame ˇ e , ˇ e on ˇ X > . More precisely,ˇ e j = e − π (cid:126) f i ˇ1 i , i = 1 , , The monodromy action around the fiber { (0 , } × T is given byˇ e (cid:55)→ ˇ e , ˇ e (cid:55)→ − ˇ e . ECONSTRUCTION OF T P VIA TROPICAL LAGRANGIAN MULTI-SECTION 13
Let ¯ ∂ be the Dolbeault operator of E . We need extend this complex structure to the wholespace X . Let’s delete a small disk D around the origin of B . Let δ > b δ be a 1-fromon R , supported on [ − δ, δ ] and (cid:82) R b δ = 1. Define three elements in A , ( ˇ X > , End ( E )):ˇ B := − Hev ( (cid:104) x, v (cid:105) ) F (Π ∗ b δ )ˇ e ∗ ⊗ ˇ e , ˇ B := Hev ( (cid:104) x, v (cid:105) ) F (Π ∗ b δ )ˇ e ∗ ⊗ ˇ e , ˇ B := Hev ( (cid:104) x, v (cid:105) ) F (Π ∗ b δ )ˇ e ∗ ⊗ ˇ e , where Hev : R → R is the Heaviside function: Hev ( x ) = (cid:26) x ≥ , x < , Π j : R → R · v ⊥ j , j = 0 , ,
2, are the orthogonal projections:Π j ( x ) := x − (cid:104) x, v j (cid:105) v j , and F is the Fourier transform sending dx i to d ¯ z i . By choosing δ > δ depends on the radius of the disk D ), we may assume ˇ B , ˇ B , ˇ B have disjoint support onˇ p − ( B > \ D ). Let ˇ B := ˇ B + ˇ B + ˇ B ∈ A , (ˇ p − ( B > \ D ) , End( E )) . Since the support of ˇ B i ’s are all away from the ray { ( x , x ) | x ≤ } , ˇ B can be extended toˇ p − ( B \ D ). Clearly, ¯ ∂ ˇ B = 0 and [ ˇ B, ˇ B ] = 0 since ˇ B j ’s have disjoint support. Therefore, we have¯ ∂ ˇ B + 12 [ ˇ B, ˇ B ] = 0 , which means ¯ ∂ + ˇ B defines a holomorphic structure on the rank 2 complex vector bundle E | ˇ p − ( B > \ D ) . Proposition A.2.
The holomorphic structure ¯ ∂ + ˇ B is monodromy free around the fiber ˇ p − (0) .Hence ( E , ¯ ∂ + ˇ B ) extends to a holomorphic bundle E on ˇ X . Appendix B. A cone complex of T P In this appendix, we give another of construction of a tropical Lagrangian multi-section corre-sponds to T P by following [35].On each affine chart U k , k = 0 , ,
2, the tangent bundle splits equivariantly as T P | U k ∼ = O ( D i ) | U k ⊕ O ( D j ) | U k , for i, j, k distinct. One constructs a cone complex Σ T P as follows: Let σ ki , σ kj be two copiesof the cone σ k . They are responsible for O ( D i ) | U k and O ( D j ) | U k , respectively. We have thecorrespondence between toric line bundles and supporting functions: O ( D ) | U ↔ ϕ (cid:48) ( ξ ) := − ξ , O ( D ) | U ↔ ϕ (cid:48) ( ξ ) := − ξ , O ( D ) | U ↔ ϕ (cid:48) ( ξ ) := ξ , O ( D ) | U ↔ ϕ (cid:48) ( ξ ) := ξ − ξ , O ( D ) | U ↔ ϕ (cid:48) ( ξ ) := ξ , O ( D ) | U ↔ ϕ (cid:48) ( ξ ) := − ξ + ξ . Let Σ T P be obtained by gluing σ ij so that the function ϕ (cid:48) | σ ij := ϕ (cid:48) ij is continuous. Explicitly, σ ij is glued with σ ji and σ ik is glued with σ jk , both gluing processes are obtained by identifying theircommon ray v k for i, j, k distinct. The projection π (cid:48) : | Σ T P | → | Σ | is given by projecting σ ij to σ i .The space | Σ T P | is homeomorphic to N R and the projection π (cid:48) : | Σ T P | → | Σ | is a 2-fold branchedcovering, which can be identified with the map z (cid:55)→ z on C . Denote the underlying topologicalspace | Σ T P | by L (cid:48) . The triple L (cid:48) := ( L (cid:48) , π (cid:48) , ϕ (cid:48) ) clearly defines a tropical Lagrangian multi-section. One can also perform a naive gluing for O ( D i ) | U k ⊕ O ( D j ) | U k , k = 0 , ,
2, respecting theequivariant structures( λ , λ ) · ( w , w , v , v ) =( λ w , λ w , λ v , λ v ) , ( λ , λ ) · ( w , w , v , v ) =( λ − w , λ − λ w , λ − v , λ − λ v ) , ( λ , λ ) · ( w , w , v , v ) =( λ − w , λ λ − w , λ − v , λ λ − v ) , and cook up three wall-crossing factors Θ (cid:48) j by comparing with the transition functions τ ij of P .However, in this case, a simple calculation shows that we don’t need to equip L (cid:48) ∼ = N R \{ } withany local system! This suggests that, around the caustic point, any smoothing of the tropicalLagrangian multi-section L (cid:48) cannot be modeled (see Appendix A) by L := { ( x , ¯ x ) ∈ C : x ∈ C } . near 0. This is reasonable. Note ϕ (cid:48) is a strictly convex function, so a smoothing of ϕ (cid:48) is modeledby 12 ( x + x )near 0, while the potential function of L is given by12 ( x − x ) . This also suggests L (cid:48) is not a tropical limit of a special Lagrangian multi-sections (see Remark3.10). However, it doesn’t explain why L , L (cid:48) give the same mirror bundle. Let’s consider thedifference ϕ − ϕ (cid:48) = (cid:40) ξ on σ − ∪ σ +1 ∪ σ +2 ,ξ on σ +0 ∪ σ − ∪ σ − . Although it is not an affine function, “half” of it does define an affine function on N R because π ( σ ∓ ∪ σ ± ∪ σ ± ) = N R . This phenomenon reflects that the two Lagrangian multi-sections (again,conjecturally exist) L , L (cid:48) have the same asymptotic behaviour at infinity of Y and so they are relatedby a Hamiltonian diffeomorphism that respecting certain asymptotic conditions. In particular, theyshould give rise to isomorphic mirror bundles. References
1. M. Abouzaid,
Homogeneous coordinate rings and mirror symmetry for toric varieties , Geom. Topol. (2006),1097–1157 (electronic). MR 2240909 (2007h:14052)2. , Morse homology, tropical geometry, and homological mirror symmetry for toric varieties , Selecta Math.(N.S.) (2009), no. 2, 189–270. MR 2529936 (2011h:53123)3. Mohammed Abouzaid, Family Floer cohomology and mirror symmetry , Proceedings of the International Con-gress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 813–836. MR 37286394. Miguel Abreu,
K¨ahler geometry of toric varieties and extremal metrics , Internat. J. Math. (1998), no. 6,641–651. MR 16442915. , K¨ahler geometry of toric manifolds in symplectic coordinates , Symplectic and contact topology: inter-actions and perspectives (Toronto, ON/Montreal, QC, 2001), Fields Inst. Commun., vol. 35, Amer. Math. Soc.,Providence, RI, 2003, pp. 1–24. MR 19692656. D. Arinkin and A. Polishchuk,
Fukaya category and Fourier transform , Winter School on Mirror Symmetry,Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math., vol. 23,Amer. Math. Soc., Providence, RI, 2001, pp. 261–274. MR 18760737. T. Baier, C. Florentino, J. M. Mour˜ao, and J. P. Nunes,
Toric K¨ahler metrics seen from infinity, quantizationand compact tropical amoebas , J. Differential Geom. (2011), no. 3, 411–454. MR 28792478. K. Chan, Holomorphic line bundles on projective toric manifolds from Lagrangian sections of their mirrors bySYZ transformations , Int. Math. Res. Not. IMRN (2009), no. 24, 4686–4708. MR 2564372 (2011k:53125)9. ,
Homological mirror symmetry for A n -resolutions as a T -duality , J. Lond. Math. Soc. (2) (2013),no. 1, 204–222. MR 302271310. K. Chan and N. C. Leung, Mirror symmetry for toric Fano manifolds via SYZ transformations , Adv. Math. (2010), no. 3, 797–839. MR 2565550 (2011k:14047)
ECONSTRUCTION OF T P VIA TROPICAL LAGRANGIAN MULTI-SECTION 15
11. ,
Matrix factorizations from SYZ transformations , Advances in geometric analysis, Adv. Lect. Math.(ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 203–224. MR 307725812. K. Chan, N. C. Leung, and Z. M. Ma,
Scattering diagrams from asymptotic analysis on Maurer-Cartan equa-tions , preprint (2017), arXiv:1807.08145.13. K. Chan, D. Pomerleano, and K. Ueda,
Lagrangian sections on mirrors of toric Calabi-Yau 3-folds , preprint(2016), arXiv:1602.07075.14. ,
Lagrangian torus fibrations and homological mirror symmetry for the conifold , Comm. Math. Phys. (2016), no. 1, 135–178. MR 343922415. K. Chan and Y.-H. Suen,
SYZ transforms for immersed Lagrangian multi-sections , to be appear in Transactionof American Mathematical Society, arXiv:1712.02586.16. K. Chan and K. Ueda,
Dual torus fibrations and homological mirror symmetry for A n -singlarities , Commun.Number Theory Phys. (2013), no. 2, 361–396. MR 316486817. J. Chen, Lagrangian sections and holomorphic
U(1) -connections , Pacific J. Math. (2002), no. 1, 139–160.MR 189592918. B. Fang,
Central charges of T-dual branes for toric varieties , preprint (2016), arXiv:1611.05153.19. ,
Homological mirror symmetry is T -duality for P n , Commun. Number Theory Phys. (2008), no. 4,719–742. MR 2492197 (2010f:53154)20. B. Fang, C.-C. M. Liu, D. Treumann, and E. Zaslow, The coherent-constructible correspondence and homologicalmirror symmetry for toric varieties , Geometry and analysis. No. 2, Adv. Lect. Math. (ALM), vol. 18, Int. Press,Somerville, MA, 2011, pp. 3–37. MR 288243921. ,
T-duality and homological mirror symmetry for toric varieties , Adv. Math. (2012), no. 3, 1875–1911. MR 287116022. Kenji Fukaya,
Multivalued Morse theory, asymptotic analysis and mirror symmetry , Graphs and patterns inmathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI,2005, pp. 205–278. MR 213101723. Mark Gross,
Tropical geometry and mirror symmetry , CBMS Regional Conference Series in Mathematics, vol.114, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the AmericanMathematical Society, Providence, RI, 2011. MR 272211524. Mark Gross, Rahul Pandharipande, and Bernd Siebert,
The tropical vertex , Duke Math. J. (2010), no. 2,297–362. MR 266713525. Mark Gross and Bernd Siebert,
Affine manifolds, log structures, and mirror symmetry , Turkish J. Math. (2003), no. 1, 33–60. MR 197533126. , From real affine geometry to complex geometry , Ann. of Math. (2) (2011), no. 3, 1301–1428.MR 284648427. Mark Gross and P. M. H. Wilson,
Large complex structure limits of K surfaces , J. Differential Geom. (2000), no. 3, 475–546. MR 186373228. Victor Guillemin, Moment maps and combinatorial invariants of Hamiltonian T n -spaces , Progress in Mathe-matics, vol. 122, Birkh¨auser Boston, Inc., Boston, MA, 1994. MR 130133129. Hansol Hong, Yu-Shen Lin, and Jingyu Zhao, Bulk-deformed potentials for toric Fano surfaces, wall-crossingand period , preprint (2019), arXiv:1812.08845.30. M. Kontsevich,
Homological algebra of mirror symmetry , Proceedings of the International Congress of Mathe-maticians, Vol. 1, 2 (Z¨urich, 1994), Birkh¨auser, Basel, 1995, pp. 120–139. MR 140391831. Maxim Kontsevich and Yan Soibelman,
Homological mirror symmetry and torus fibrations , Symplectic geometryand mirror symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, pp. 203–263. MR 188233132. ,
Affine structures and non-Archimedean analytic spaces , The unity of mathematics, Progr. Math., vol.244, Birkh¨auser Boston, Boston, MA, 2006, pp. 321–385. MR 218181033. N. C. Leung,
Mirror symmetry without corrections , Comm. Anal. Geom. (2005), no. 2, 287–331. MR 215482134. N. C. Leung, S.-T. Yau, and E. Zaslow, From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukaitransform , Adv. Theor. Math. Phys. (2000), no. 6, 1319–1341. MR 189485835. Sam Payne, Toric vector bundles, branched covers of fans, and the resolution property , J. Algebraic Geom. (2009), no. 1, 1–36. MR 244827736. A. Strominger, S.-T. Yau, and E. Zaslow, Mirror symmetry is T -duality , Nuclear Phys. B (1996), no. 1-2,243–259. MR 142983137. Hikaru Yamamoto, Special Lagrangian and deformed Hermitian Yang-Mills on tropical manifold , Math. Z. (2018), no. 3-4, 1023–1040. MR 3856842
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic ofKorea
E-mail address ::