BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES, WALL-CROSSING AND PERIOD

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1 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES, WALL-CROSSING AND PERIOD HANSOL HONG, YU-SHEN LIN, AND JINGYU ZHAO Abstract. We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces. The computations makes use of wall-crossing formula, which matches with the tropical disc counting by Gross in the case of P 2. Moreover, we prove a big quantum period theorem for toric Fano surfaces which relates the descendant Gromov-Witten invariants with the oscillatory integrals of its Landau-Ginzburg mirror.. Introduction The mirror of a Fano symplectic manifold X is given by a Landau-Ginzburg model W : ˇX C, where ˇX is a smooth quasi-projective complex manifold and W is a holomorphic function on ˇX called the superpotential. For toric Fano manifolds, such Landau-Ginzburg mirrors were first written down by Hori-Vafa [3]. Then it is proved by Cho-Oh [7] that the Hori-Vafa superpotential W can be derived from the weighted counts of Maslov index two holomorphic discs with boundaries on Lagrangian torus fibers of the moment map. Explicitly, the Hori-Vafa potential is a Laurent polynomial defined on (C ) n which can be determined by the toric moment polytope. Later, this is generalized to toric semi-fano manifolds by the work of Chan-Lau-Leung-Tseng [6] and Fukaya-Oh-Ohta-Ono [7] for general toric manifolds in [2]. In particular, the closed string mirror symmetry conjecture has been proved in [7] for all toric Fano manifolds. It states that there is an isomorphism of Frobenius algebras between the small quantum cohomology of X and the Jacobian ring of the superpotential W, (QH (X), s ) = Jac(W ). In order to study the big quantum cohomology ring of X via mirror symmetry, Fukaya-Oh-Ohta-Ono further introduced bulk deformations of the superpotential W in [20] for toric manifolds. For each b H even (X), the bulk-deformed potential W b gives rise to a formal deformation of the Hori-Vafa potential W by counting holomorphic discs with boundaries on the moment map fibers and passing through torus invariant cycles D such that [D] = P D(b). It is also proved in [7] that for b in H 2 (X) that is a toric invariant class, there is a Kodaira-Spencer map ks which induces an isomorphism between the big quantum cohomology and Jacobian ring of the bulk-deformed superpotential ks: (QH (X), b ) = Jac(W b ). However, such an isomorphism is not known for non-toric invariant cocycles b and for b in H i (X) with i 4. In this paper, we study bulk deformations of the Hori-Vafa poentials of toric Fano surfaces by non-toric cycles in H 4 (X). Priori to our work, Gross [26] studied

2 2 HONG, LIN, AND ZHAO the bulk-deformed superpotential W b for X = P 2 via counting of tropical discs passing through k generic cycle constraints that represent b in H 4 (P 2 ). Inspired by this, we compute the kth order bulk-deformed potential W k which counts J- holomorphic discs with k-interior marked points with boundaries on the Lagrangian torus fibers L of the SYZ fibration π : X P given by the toric moment map. If we impose the condition that these k-interior marked points map to a given k-tuple (q,, q k ) of points in generic positions, then the loci of Lagrangian torus fibers that bounds such discs with suitable Maslov index will give -dimensional skeleton in the SYZ-base for X. As the virtual dimension of the moduli space M,k (L, β, J) of J-holomorphic discs passing through k generic point constraint in X and one generic point constraint in L is given by dim L + µ(β) 3 + 2k 4k =, when µ(β) = 2k one concludes that generic fibers of the SYZ fibration π do not bound such Maslov 2k discs passing through k generic marked points in X. We will call such a disc a generalized Maslov index 0 disc, since it behaves similarly to Maslov zero discs in usual wall crossing phenomenon in Floer theory [4, 5]. In this circumstances, the bulk-deformed potential W k is no longer a well-defined Laurent polynomial on (C ) 2, but it experiences discontinuities, or the wall-crossing phenomenons due to the existence of generalized Maslov zero discs with boundaries on certain torus fiber of the moment map. In fact, there are well-defined wall structures on the Log base R 2, which is the Legendre dual of the moment map fibration π. As the Lagrangian torus fibers vary across different chambers separated by walls in the Log base, the bulk-deformed potentials W k will undergone so-called cluster transformations. In the case of toric Fano surfaces, these cluster transformations are simply of the form (.) (.2) z z z 2 f(z, z 2 )z 2, up to a coordinate change. for some holomorphic function f and g on (C ) 2. To compute the bulk-deformed potential W k order by order, we first prove a tropicalholomorphic correspondence theorem which gives a detailed description of the locus of the Lagrangian torus fibers that bound generalized Maslov zero discs in the Log base (See section 5 for details) via tropical disc counting. Such description of tropical wall structures is given in Gross s work [26], our theorem below can be viewed as a generalization of Gross work to all toric Fano surfaces. Theorem.. For a toric Fano surface X, the chamber structure from generalized Maslov index 0 discs is homeomorphic to the one given by tropical disc counting. It should be remarked that there is a similar work by Nishinou [37], who used algebraic geometry to define a counting of tropical disc on toric varieties as certain log Gromov-Witten invariants and establish the tropical/holomorphic correspondence. While we provide a definition of holomorphic disc counting via Lagrangian Floer theory and establish the tropical/holomorphic correspondence under certain limit. We also provide a (counter)-example of tropical/holomorphic correspondence away from such limit in Section 3.3. Given the well-defined moduli space of such J-holomorphic discs with boundaries on Lagrangian torus fibers, we prove the first main theorem of the paper.

3 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 3 Theorem.2. (=Theorem 4.6) Let X be a toric Fano surface. Given generic points q,, q k in (C ) 2 X and a generic point u in R 2, we denote the Lagrangian torus fiber by L u := π (u). Then for any t, () there exists a neighborhood U t depending on t, which deformation retracts to the union of the tropical wall structure (described below Theorem 2.20). (2) the bulk-deformed superpotential W k (u) := W q,,q k t (u) := W H t (q ), H t (q k ) (Ht can be computed tropically if u is in the complement of U t, where H t : (C ) 2 (C ) 2, (x, y) ( x log t x x, y log t (u)) y y ) is the diffeomorphism of (C ) 2 introduced by Mikhalkin [36] and the superscripts of W above indicates the generic k point constraints. As a corollary, we obtain an inductive algorithm to compute the bulk-deformed potential W b for X as follows. With the help of Fukaya s pseudo-isotopies [5] for the bulk-deformed Fukaya A -algebra {m b k } k 0 defined in [20], we first proved that the expressions of the bulk-deformed potentials W k in adjacent chambers of the Log base differ by the wall-crossing map, which can be explicitly computed via Fukaya s pseudo-isotopies [5] associated to the bulk-deformed A -algebra {m b k } k 0 defined in [20] on the cohomology of the Lagrangian torus fiber. Then the wall-crossing formula in the bulk-deformed setting is essentially equivalent to the compatibility A - homomorphism induced by Fukaya s trick and the bulk-deformed A -structures on the Lagrangian Floer complex of torus fibers, namely, ˆm b ˆf b = ˆf b ˆm b. The details of this computation are provided in Section 2 below. Inductively, suppose we are given the expression of the bulk-deformed potential in each chamber of the Log base, we prove that the following fact to obtain W k for all k N. Proposition.3. If the expression of W k is given in each chamber, after adding the (k + )th point q k+ there is a one-to-one correspondence (.3) {terms in W k (q k+ )} {the walls of passing through q k+ }. Moreover, if W k (q k+ ) = β N βz β. The wall-crossing formula associated to the wall passing through p k+ that corresponds to the term N β z β under the bijection (.3) is given by z α z α ( + N β z β ) α,β, where N β denotes the weighted count of generalized Maslov two discs in class β in H 2 (X, L) for each Lagrangian torus fiber L. Given this algorithmic computation of W k for each k, one observes that although the non-toric bulk-deformed superpotential may depend on the choice of the moment torus fiber in different chambers of the Log base R 2, the corresponding LG periods, or oscillatory integrals e Wk(u)/ Ω are well-defined for Γ H n ( ˇX, Re(W b / ) 0) and Ω = dz z holomorphic volume form on (C ) 2. In the case when Γ = Γ dz2 z 2 is the standard (2πi) T 2 in (C ) 2, 2

4 4 HONG, LIN, AND ZHAO the independence of the chamber structure can be deduced from the following big quantum period theorem, which states that there is a direct relation between such oscillatory integrals of W k with closed Gromov-Witten invariants with descendants of the toric Fano surface X. Theorem.4. (=Theorem 6.) Let X be a toric Fano surface and let L be any Lagrangian torus fiber of the moment map π. The bulk-deformed potential W k associated to L satisfies for any k N (2πi) 2 e W k/ dx dx 2 T x 2 x 2 = + τ m 2 (α), α i,, α in X,m+n,β m t I I {,,k} m 2 β H 2(X;Z)/tor such that I = {i,, i n } and α ik = P D([p ik ]) and α = P D([q]) for q π (u), where,, X,d,β denotes the Gromov-Witten correlation function τ d γ,, τ dn γ n X,d,β = ψ d ev (γ ) ψn dn evn(γ n ) M 0,n(X,β) such that c (M)(β) = d = m + n. The formal parameter is usually referred as the descendant variable in Gromov-Witten theory. Remark.5. Let {T i } n i= be a basis of the cohomology of the toric Fano surfaces H (X) such that T 0 = P D[pt]. If one takes b = n i= t it i for formal parameters t i and denotes the corresponding bulk-deformed potential W b defined by Fukaya- Oh-Ohta-Ono [20], then the proof of the above theorem implies that the following relation + ψ m 2 T 0 m = (2πi) 2 e W b / dx dx 2, m 2 T x 2 x 2 where the notation ψ m 2 T 0 := k 0 k! ψm 2 T 0, b,, b }{{} X is the Gromovk Witten descendant correlation function. Such relations between descendant Gromov-Witten invariants and oscillatory integrals have been studied in closed-string mirror symmetry by Givental [24] and Barannikov [9] for X = P n under the name of quantum periods and semi-infinity variation of Hodge structures. Assuming closed-string mirror symmetry for Fano manifolds are proved, for instance for X = P n by Barannikov [0], the work of Gross [26] defined tropical bulk-deformed superpotential and studied the flat coordinates defined on the miniversal deformation space (or formal deformation space) of the Hori-Vafa potential for P 2 and match these oscillatory integrals with tropical descendant invariant that he defines (See Remark 6.2 below for details). More prominently, the recent work of Tonkonog [45] proved the small quantum period theorem for all monotone symplectic manifolds, which says that the -pointed descendant Gromov-Witten invariants match with the oscillatory integrals of its Laudu-Ginzburg mirror (which is not necessarily an affine variety). In another direction of interests, we study the oscillatory integrals of the bulk-deformed superpotential for toric Fano surfaces as it is conjecture in the work by Fukaya-Oh- Ohta-Ono [2] on mirror symmetry on compact toric manifolds that one proves an isomorphism of (formal) Frobenius manifolds whose tangent spaces are isomorphic (QH (X), b ) = Jac(W b ) as b varies in the germ of the origin in H (X). While on

5 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 5 the symplectic side, such formal Frobenius structure is equivalent to a cohomological field theory govern by Gromov-Witten correlation functions (c.f. [35, Chapter 3.4]), which are exactly the correlation functions that appear in our Theorem.4... An outline of the paper. In Section 2, we first review the construction by Fukaya-Oh-Ohta-Ono of the bulk deformations of the Fukaya s algebras associated to the Lagrangian torus fibers. For toric Fano surfaces, we first prove that the weak Maurer-Cartan elements associated to the bulk-deformed Fukaya s algebra is given by H (L). In particular, we proved the so-called Fukaya s trick in Proposition 2.6, which is crucial for the computations of the wall-crossing formula later. In Section 3, we illustrate the initial wall structures on the Log base for P 2. Here we demonstrate via a computation using [7] that the walls on the Log base are not given by straight lines in general, which complicates the determination of the wall structures in the holomorphic setting due to the ambiguities of the shape of the walls. In order to solve this problem, we appeal to the tropical degeneration of the standard complex structure on (C ) 2 defined by Mikhalkin in [36]. This is a diffeomorphism H t as defined above. For t, in Section 4 we prove the tropical-holomorphic correspondence theorem by making use of an anti-symplectic involution of (C ) 2, which then provides an explicit wall structures given by generalized Maslov zero discs in R 2 for all toric Fano surfaces. In Section 5, we appeal to the wall-crossing formula derived from Fukaya s trick in Section 2 and the wall structures obtain by the tropical-holomorphic correspondence in Section 4 to compute W k for all k N. Finally, in Section 6 we prove the big quantum period theorem by first defining the bulk-deformed Borman-Sheridan classes that are analogous to the Borman-Sheridan classes considered in Tonkonog s original proof [44] and we prove the Theorem.4 by a similar SFT neck stretching argument considered by Cieliebak-Mohnke [8] in section Acknowledgements. We thank Professors Shing-Tung Yau and Denis Auroux for their interests in this work. We are grateful to Cheol-Hyun Cho, Yoosik Kim, Siu-Cheong Lau for useful discussions. The third author would like to thank Mohammed Abouzaid, Kyler Siegel and Hsian-Hua Tseng for helpful discussions. The work of the first and third authours is substantially supported by the Simons Foundation grant (# , Simons Collaboration on Homological Mirror Symmetry). We also thank Harvard CMSA for its hospitality. 2. Preliminaries 2.. Basic Floer theory. In this section, we review Fukaya s A -algebra associated to each Lagrangian torus fiber L u := π (u) and pseudo-isotopies between such A -algebras defined in [5]. In particular, we will construct the A - homomorphism via pseudo-isotopies of A -algebras in the bulk-deformed setting The de Rham model for Fukaya s A -algebra. Let Λ 0 denote the Novikov ring over the real numbers R, Λ 0 := { a i T λi λ i 0, lim λ i = and a i R}. i i=

6 6 HONG, LIN, AND ZHAO There is a non-archimedean valuation val: Λ 0 R, val( i a i T λi ) = inf{λ i a i 0} and val(0) =. The maximal ideal of Λ 0 is defined as Λ + := val ((0, )). Definition 2.. Suppose that G is a discrete monoid with a homomorphism ω : G R such that ω ([0, a)) <, a R. Let C be a Z/2-graded vector space over Λ 0. A G-gapped filtered A -structure on C is a sequence of homorphisms {m k,β } k 0,β G of degree m k,β : B k C[] := C[] C[] C[], }{{} k times where (C[n]) d = C d+n is a degree shift, and these operations induce a coderivation of degree ˆm := m k,β T ω(β) : BC[] = B k C[] C[] k 0 β G k=0 satisfying ˆm ˆm = 0 Definition 2.2. An A -homomorphism between two G-gapped filterd A -algebras (C, {m k,β }) and (C, {m k,β }) are defined by a sequence of homomorphism f k,β : B k C[] C [] of degree 0 such that the induced map ˆf = f k,β : BC[] BC [] k 0 β G satisfies ˆm ˆf = ˆf ˆm. For a fixed Lagrangian fiber L of toric moment map π : X P, we denote H as the graded vector space generated by smooth singular cycles of even dimensions of X over R and Ω(L) as the de Rham cochain complex of L. For each b H, there is a G-gapped filtered A -algebra structure on Ω(L) Λ 0 constructed via bulk deformations by Fukaya-Oh-Ohta-Ono in [20] as follows. For an ω-tamed almost complex structure J and a fixed class β H 2 (X, L; Z), we denote by M k+,l (L, β, J) the moduli space of stable J-holomorphic discs with k + boundary marked points and l interior marked points representing the class β. An element of M k+,l (L, β, J) is of the form (Σ, ψ, {z + i i =,, l}, {z j j = 0,,, k}), where Σ is a connected tree of discs, and ψ : (Σ, Σ) (X, L)) is a J-holomorphic curve with z + i ψ(int(σ)) and z j ψ( Σ) and ψ has only finite automorphisms. We impose the condition that the ordering of the boundary marked points z j agrees with the chosen orientation of Σ. It is shown in [Section 7.][7] that it has a Kuranishi structure with boundaries and corners of dimension n+2l+k+ 3+µ(β), where µ(β) is the Maslov index of β. There are interior evaluation maps evi int : M k+,l (L, β, J) X, (Σ, ψ, {z + i }, {z j}) ψ(z + i ) for i =,, l.

7 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 7 One fixes a basis {f i } N i= of H and each singular smooth cycle b H can be written as b = t i f i. i Since the ordering of the interior marked points z + i is not a priori fixed, we denote by Maps(l, N) the set of all maps p: {,, l} N = {,, N} and define the fiber product as l M k+,l (L, β, J; p) := M k+,l (L, β, J) (ev int f p(i).,,evl int ) X l i= Its virtual dimension is n + 2l + k µ(β) i (2n dim(f p(i))). There are evaluation maps from this fiber product to L k+ ev = (ev 0, ev,, ev k ): M k+,l (L, β, J; p) L k+ It is proved in [5, Corollary 3., Theorem 5.] that there is a consistent choice of Kuranishi structures on M k+,l (L, β, J; p) for each k, l 0 such that the evaluation map ev 0 is weakly submersive. We choose a Riemannian metric on the Lagrangian torus fiber L and represent elements in H (L, R)) by harmonic forms with respect to the fixed metric on L. Let h,, h k in Ω(L), since ev 0 is shown to be weakly submersive, one can define an operator by q l,k,β : B l (H[2]) B k (Ω(L)[]) Ω(L)[] q l,k,β (f p ; h,, h k ) = l! (ev 0)! (ev,, ev k ) (h h k ). Suppose that degrees of the differential forms h i satisfy (2.) ((deg(h j ) ) µ(β) + (2n dim(f p(i) )) + 2 = d for d N, j i the operation q l,k,β defines a degree d differential form in Ω(L). There is a symmetric group action of S l of order l! on B l (H[2]) given by σ (x x l ) = ( ) x σ() x σ(l), where = i<j;σ(i)>σ(j) deg x i deg x j. We denote by E l (H[2]) the subspace of S l - invariant elements in B l (H[2]) and denote the restriction of the operator q l,k,β to the symmetric tensors Sf p = f p(σ()) f p(σ(l)) l! σ S l by the same notation q l,k,β : E l (H[2]) B k (Ω(L)[]) Ω(L)[]. Then one obtains a family of operators q l,k := β T ω(β) q l,k,β : E l (H[2]) B k (Ω(L) Λ 0 []) Ω(L) Λ 0 []. We remark that this operation q l,k is well-defined of degree one as the Maslov index µ(β) is even for any Lagrangian torus fiber L. For each b H, we can define an operation m b k : B k(ω(l)[]) Ω(L)[] by m b k(h,, h k ) = m b β,k(h,, h k ) = q l,k,β (b,, b, h,, h k ). β β l 0

8 8 HONG, LIN, AND ZHAO It is proved in [20, Lemma 2.2] that this defines a G-gapped filtered A -algebra structure on Ω(L) Λ The canonical model and its Maurer-Cartan elements. To define the canonical model on H (L; Λ 0 ), we choose a Riemannian metric on L and represent H(L; R) as the subspace of harmonic forms in Ω(L). By applying a version of the homological perturbation Lemma for filtered A -algebra, Fukaya [5, Section 0] constructed a G-gapped filtered A -structure m b,can k : B k (H (L; Λ 0 )[]) H (L; Λ 0 )[], which is quasi-isomorphic to the de Rham model (Ω(L) Λ 0, {m b k } k= ). Given the canonical model of L, one can define the space of weak Maurer-Cartan elements of positive valuation as (2.2) MC b +(L) := {b H odd (L; Λ + ) k=0 m b,can k (b,, b) = λp D[L], λ Λ + }. Here the infinite sum k=0 mb,can k (b,, b) converges in the non-archimedean topology defined by the norm = e val( ) for elements b H odd (L; Λ + ) with positive valuation. We say that two such weak Maurer-Cartan elements b 0, b are gauge equivalent if there exist b(t) of degree and c(t) of degree 0 for t [0, ] satisfying the following conditions (2.3) b(0) = b 0 and b() = b, d dt b(t) + m b,can k (b(t),, b(t), c(t), b(t),, b(t)) = 0, t (0, ). k The moduli space of weak Maurer-Cartan elements of positive valuations in Ω(L) Λ 0 is defined as MC b +(L) := MC b +(L)/, where is the gauge equivalence relation. In the specific case that we consider, one has the following Lemma. Lemma 2.3. Let L be a Lagrangian torus fiber of the moment map π : X P for a toric Fano surface and b = t 0 f 0 := t 0 P D[pt] H. The weak Maurer-Cartan space associated to the G-gapped filtered A -algebra (H (L; Λ 0 ), {m b,can k } k= ) is given by MC b +(L) = H (L; Λ + ). Proof. The lemma follows from a simple dimension argument. One needs to show that m b,can k,β (b,, b) k,β is a multiple of the unit class in H 0 (L; Λ) for any b H (L; Λ + ). Because of the compatibility of the forgetful maps, the term m b,can k,β (b,, b) can be non-zero only when the moduli space M 0,l (L, β, J; p) has the virtual dimension dim(l) + 2l 3 + µ(β) l deg(b) = µ(β) 2l 0 where l is the number of interior marked points and we have used the fact that dim(l) = 2 and deg(b) = 4 here. Therefore, we have µ(β) 2l +. This implies

9 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 9 that µ(β) 2l 2 since µ(β) is even. On the other hand, the degree of the output of m b,can (b,, b) is given by k,β 2 (dim(l) + 2l + 2k µ(β) l deg(b) k deg(b)) = 2 (µ(β) 2l) 0 since µ(β) 2l 2. Therefore the only possible degree of the output is zero. Since we are working on the canonical model, such a degree zero output must be a multiple of the unit class. To see that any two elements b 0, b MC b +(L) are not gauge equivalent, we assume by contradiction that there are b(t) of degree and c(t) of degree 0 satisfying equation (2.3). In the canonical model (H (L; Λ 0 ), {m b,can k } k ) the only element of degree zero is e = P D[L], so one has c(t) = f(t)e for some smooth function f(t). As the canonical model is shown to be unital [20, Lemma 6.4], we have that m b,can k (b(t),, b(t), e, b(t),, b(t)) = 0 for k 2, m b,can 2 (e, b(t)) = m b,can 2 (b(t), e) = b(t). Then equation (2.3) becomes d dt b(t) = 0 and hence b(t) = b 0 = b is the constant class, which completes the proof The A -functor between Fukaya A -algebras. For two different Lagrangian torus fibers of the moment map π : X P, denoted as L u := π (u) and L u := π (u ), the G-gapped A -algebra structures on H (L u ; Λ 0 ) and H (L u ; Λ 0 ) defines degree-one coderivations ˆm b (u) := m b,can : B(H (L u ; Λ 0 )[]) B(H (L u ; Λ 0 )[]), ˆm b (u ) := i=0 m b,can : B(H (L u ; Λ 0 )[]) B(H (L u ; Λ 0 )[]) i=0 satisfying ˆm b (u) ˆm b (u) = 0 and ˆm b (u ) ˆm b (u ) = 0. There is a notion of pseudoisotopies between G-gapped filtered A -algebras introduced in [6, Definition 8.5]. Definition 2.4. A pseudo-isotopy of G-gapped filtered A -algebras consists of the data (A, {m t k,β } k=, {ct k,β } k= ), where () The operations {m t k,β } k= and{ct β;k } k= ) are smooth in t; (2) For each fixed t, the operations {m t k,β } k= defines a G-gapped filtered A - algebras; (3) The following equation is satisfied for x i BA[] d dt mt k,β(x,, x k ) = k +k 2=k β +β 2=β k +k 2=k β +β 2=β k k 2+ i= k k 2+ i= m t β ;k (x,, c t β 2;k 2 (x i, ),, x k ) ( ) c t β ;k (x,, m t β 2;k 2 (x i, ),, x k ), where = deg(x ) deg(x i ) +. (4) When β = 0, we have that m t β;k is independent of t and ct β;k = 0 for all k..

10 0 HONG, LIN, AND ZHAO In our geometric situation, for any two interior points u, u Int(P ) of the moment polytope, one chooses a smooth path φ: [0, ] Int(P ) such that φ(0) = u and φ() = u and a family of diffeomorphisms φ t : L φ(0) = L u L φ(t) for each t [0, ]. One can construct a pseudo-isotopy between A -algebras (Ω(L u ) Λ 0, m b k (u)) and (Ω(L u ) Λ 0 ), m b k (u )) by considering the the parametrized moduli space (2.4) M k+,l (L, β, J ) = {t} M k+,l (L, β, J t ; p), t [0,] where J t is a family of ω-tamed almost complex structures such that J 0 = J u and J = φ J u, and M k+,l (L, β, J t ; p) is the fiber product M k+,l (L, β, J t ; p) := M k+,l (L, β, J t ) (ev int There are evaluation maps ev = (ev,, ev k ): M k+,l (L, β, J ) L k ev 0,t : M k+,l (L, β, J ) L [0, ], l,,evl int ) X l f p(i) i= {t} M k+,l (L, β, J t ; p) (ψ(z 0 ), t). It is shown in [5, Lemma.3] that the Kuranishi structures on M k+,l (L, β, J ) can be chosen such that the evaluation map ev 0,t is weakly submersive. We define the operations q l,k,β and q0 l,k,β by the formula (2.5) (ev 0,t )! (ev,, ev k )(h h k ) = q l,k,β(f p(i) ; h,, h k ) + q 0 l,k,β(f p(i) ; h,, h k )dt for differential forms h i in Ω(L u ). The operations {m b,t k,β } k= and {cb,t k,β } k= defining the pseudo-isotopy are given by m b,t k,β (h,, h k ) = β c b,t k,β (h,, h k ) = β q l,k,β(b,, b, h,, h k )T ω(β). l 0 q 0 l,k,β(b,, b, h,, h k )T ω(β), l 0 which are of degree µ(β) mod 2 = and µ(β) mod 2 = 0 respectively. By construction, the triple (Ω (L u ) ˆ Λ 0, {m b,t k,β }, {cb,t k,β }) gives rise to a pseudoisotopy. This gives rise to a pseudo-isotopy on the corresponding canonical model (H (L u ; Λ 0 ), {m b,t,can k,β }, {c b,t,can k,β }) by Theorem 8.4 in [5] and [46, Section 2.0]. From such a pseudo-isotopy, one can construct an A -homomorphism ˆf b := k 0 β H 2(X,L) f b,can k,β : B(H (L u ; Λ 0 )[]) H (L u ; Λ 0 )[] satisfying ˆf b ˆm b (u) = ˆm b (u ) ˆf b. The explicit definition of the operation f b,can k,β : B k (H (L u ; Λ 0 )[]) H (L u ; Λ 0 )[] of degree µ(β) = 0 mod 2 are given in [5, Section ] and [46, Section 2.7]. The A -structure {m b,can k } k= and the A -homomorphism {f b,can k } k= that we defined on the canonical model of Fukaya algebra associated to any Lagrangian torus fiber satisfy the following property.

11 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES Lemma 2.5 (Divisor axioms). For each fixed k 0, b, x,, x k H (L; Λ + ) and b H, the following equations hold ( ) (2.6) nm n 0+ +n k=n mb,can k+n,β (b n0, x, b n2,, b nk, x k, b nk = n! β, b b,can k (x,, x k ); (2.7) n f b,can ( ) nf 0+ +n k=n k+n,β, x (b n0, b n2,, b nk, x k, b nk = n! β, b b,can k (x,, x k ). Proof. The case when b = 0 is proved by Fukaya in [5, Lemma 3.2] and [46, Lemma 4.4]. The proof of the Lemma for general b is similar. Let n = (n 0, n,, n k ) be a tuple of non-negative integers such that i n i = n. There is a forgetful map forget n, : M k+n,l (L, β, J) M k,l (L, β, J) for each l, which forgets the marked points labeled by,, n, n + 2,, n + n 2 +, on the boundary. It is shown in Corollary 5. in [5] that there is a Kuranishi structure on the moduli spaces M k+,l (L, β, J) that are compatible with forgetful map forget n, and permutation group action on the interior marked points. One chooses a system of multisections s, s on M k+n,l (L, β, J) and M k,l (L, β, J) and denote their zero-sets of s, s by M k+n,l (L, β, J) s and M k,l (L, β, J) s. One can assume that the evaluation maps at the interior evi int are weakly submersive for i =,, l by increasing the dimensions of the obstruction bundles in each Kuranishi neighborhood of p M k,l (L, β, J) if necessary. So there is an induced forgetful maps on the fiber products forget s n, : M k+n,l (L, β, J; p) s M k,l (L, β, J; p) s. The preimage of any point p M,l (L, β, J; p) s is isomorphic the standard n- simplex n. This implies that (2.8) ev n (b b) = ( ) n, β b b Ω (L), (forget n s ) (p) n! where ev n : M k+n,l (L, β, J) L n is the evaluation map at the boundary marked points n and n! is the volume of (forgets n ) (p) = n. We remark that since b was assumed to have even degree and b Ω (L) ˆ Λ 0 [] is of degree two, there is no extra sign in equation (2.8) for all n. This complets the proof of (2.6) for the de Rham model. By Lemma 3.2 in [5], equation (2.6) holds in the canonical model as well. For equation (2.7), one first applies the above arguments to the moduli spaces M k+,l (L, β, J ) defining the pseudo-isotopy, and concludes that the operators } and {cb,t k,β } satisfy the divisor axiom (2.6) as {mb k,β }. Using formulae (2.6) and (2.7) in [46], we see that the induced operation {m b,t,can k,β } and {c b,t,can k,β } satisfy (2.6) in the canonical model H (L u ; Λ 0 ). Equation (2.7) then follows from a similar argument as in Lemma 4.4 in [46]. {m b,t k,β Given the A -homomorphism ˆf b, there is an induced map of degree zero on the corresponding weak Maurer-Cartan spaces of positive valuations (F b,can ) : MC + (L u ) = H (L u ; Λ + ) MC + (L u ) = H (L u ; Λ + ), (F b,can ) (b) := ˆf b,can (e b ) = f b,can 0 () + f b,can (b) + f b 2 (b b) +, where e b := + b + b b + b b b +. We prove next that this map (F b,can ) depends only on the homotopy class of the path φ relative to the end points. In

12 2 HONG, LIN, AND ZHAO the subsequent discussions, We will refer the following important consequence of Fukaya s pseudo-isotopies associate to φ as the Fukaya s trick. Proposition 2.6. [Fukaya s trick] Given two Lagrangian torus fibers L u and L u of the moment map π : X P if a toric Fano surface X with u, u Int(P )\W X. If φ is homotopic relative to end points to φ, then (F b,can φ ) = (F b,can φ ) : H (L u ; Λ + ) H (L u ; Λ + ). Equivalently, if φ is a contractible loop in a small open neighborhood of u Int(P ) such that φ(0) = φ() = u, then (F b,can φ ) = id: H (L u ; Λ + ) H (L u ; Λ + ). Proof. For (), given two pseudo-isotopies on the G-gapped filtered A -structures on H (L u ; Λ), there is a notion of pseudo-isotopy between pseudo-isotopies defined in [5, Definition 4.]. One important property of such pseudo-isotopies, shown in [5, Section 4], is that the A -functors that they define induce the same map on the weak Maurer-Cartan moduli spaces (2.9) (F b,can φ ) = (F b,can φ ) : MC b +(L u ) MC b +(L u ) By Lemma 2.3, it suffices to construct a pseudo-isotopy between the pseudo-isotopies defined by two different paths φ and φ. First, we choose a homotopy φ s relative to the end points between the two paths φ and φ, and choose a smooth map J from [0, ] 2 to the space of ω-compatible almost complex structures J (M, ω) such that J ([t, 0]) = J t, J ([t, ]) = J t, where J t and J t are the almost complex structures that define the pseudo-isotopy in (2.4) respectively. It is proved in [5, Section 4] that the moduli space M k+,l (L, β, J ) = {(t, s)} M k+,l (L, β, J t,s ; p), (t,s) [0,] 2 defines a pseudo-isotopy of pseudo-isotopies by a construction which is analogous to equation (2.5). To prove (2), we can assume that φ: [0, ] Int(P )\W X by virtue of (). For b H (L u ; Λ + ), one computes (F b,can φ ) (b) := ˆf b,can By Lemma 2.5, it suffices to compute ˆf b,can φ (e b ) = β We claim that φ (e b ) = f b,can 0 () + f b,can (b) + f b,can 2 (b b) + ( = f b,can,β (b) + f b,can 2,β (b b) + ). β H 2(X,L u) ( f b,can,β (b) + β, b f b,can,β (b) + b,can β, b f,β (b) + ), 2! f b,can (,0) b,can ˆf φ b,can (b) = b, f (b) = 0 if β 0, (,β) which implies that (e b ) = e b, or equivalently (F b,can φ ) = id as desired. To prove the claim, one first observes that only degree zero operations contribute to ). We compute the operations{c b,t } in the de Rham model (F b,can φ k,β c b,t (,0) (b) = q0 0,,0(b) + q 0,,0(b, b) + q 0 2,,0(b, b, b) +.

13 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 3 Here q 0 0,,0(b) = b is contributed by constant J t -holomorphic strips such that (ev 0 ) [M 0,,0 (L, β, J )] = b. For index reasons, these constant maps are all regular. There are no J t -holomorphic discs in class β = 0 that pass through extra interior marked points constraints, as the virtual index is negative in this case. This implies that q 0 k,,0 (b) = 0 for k and hence cb,t (,0)(b) = b in the de Rham model. For β 0, the operation c b,t (l,,β)(b) is of degree µ(β) + 2l 4l. So only generalized Maslov zero maps can contribute to c b,t (,β) = l cb,t (l,,β). However, we assume that the loop φ: [0, ] Int(P ) is a constant map to u Int(P )\W X and the Lagrangian torus fiber L u does not bound any generalized Maslov zero maps, so one concludes that c b,t (,β)(b) = 0 if β 0. Applying the formulae (2.6) and (2.7) in [46] again, we obtain that in the canonical model m b,t,can k,β (b) = 0 and c b,t,can (,0) (b) = b and c b,t,can (,β) (b) = 0 for β 0. This implies that f b,can (,0) (b) = b and (b) = 0 if β 0 as desired by appealing to formula (2.5) in [46]. f b,can (,β) 2.3. Counting of tropical discs in Toric Fano Surfaces. In this subsection, we recall the definitions of the counting of tropical curves and tropical discs. Our exposition follows that of Gross s in [26] for general toric Fano surfaces. Definition 2.7. A tropical curve in R 2 is a 3-tuple (h, T, w), where () T is a tree and contains possibly unbounded edges. The set of vertices is denoted by T [0] and the set of edges are denoted by T []. (2) Every vertex is trivalent. (3) h : T R 2 such that h(e) is an embedding of affine line segment/ray if e is bounded/unbounded. (4) w : T [] N be the weight on edges such that the balancing condition holds: for every vertex with adjacent edges e, e 2, e 3 then w(e )v(e ) + w(e 2 )v(e 2 ) + w(e 3 )v(e 3 ) = 0, where v(e i ) are the primitive vector tangent to h(e i ) and pointed away from v. Algebraically, one can understand a tropical curve as the corner locus of a Laurent polynomial in the tropical semi-ring. Geometrically, tropical curves are the Gromov-Hausdorff limit of holomorphic curves at certain adiabatic limit: Mikhalkin [36] introduced the following -parameter family of diffeomorphisms H t : (C ) 2 (C ) 2 (x, y) ( x log t x x, y log t y y ). This induces a -parameter family of almost complex structures J t. When t, the volume of the moment map torus approaches zero if one fixes te suitable torus invariant Kähler form. This is exactly the SYZ interpretation of the large complex structure limit [30][34]. One says that a Riemann surface V t is J t -holomorphic if and only if V t = H t (V ) for some holomorphic curve V (C ) 2 with respect to the standard complex structure J = J e. Let Log denote the moment map (up to a Legendre transformation) Log : (C ) 2 R 2 (x, y) (log x, log y ).

14 4 HONG, LIN, AND ZHAO One of the rudimental results in tropical geometry proved by Mikhalkin [36] states that there is a bijection between tropical curve in the adiabatic limit and a sequence of J t -holomorphic curves for t. Proposition 2.8. [36] Let V be a Gromov-Hausdorff limit of a sequence of J t - holomorphic curve. Then Log(V ) is a tropical curves. Conversely, a tropical curve can be realized as the image of a Gromov-Hausdorff limit of a sequence of J t -holomorphic curve under Log. Similarly, one can study the open analogue of tropical curves, which will be referred as tropical discs. Let N = Z 2 be a lattice and M := Hom(N, Z) be the dual lattice. Denote N R = N R and M R = M R. Let N R be an integral polytope and P be the associate toric variety. Then a tropical disc can be defined as follows. Definition 2.9. A tropical discs of is a 3-tuple (h, T, w), where () T is a rooted tree with root x and contains possibly unbounded edges. The set of vertices is denoted by T [0] and the set of edges are denoted by T []. (2) Every vertex other than x is trivalent. (3) h : T M R such that h(e) is an embedding of affine line segment/ray if e is bounded/unbounded. (4) w : T [] N be the weights on edges such that the balancing condition holds: for every vertex v x with adjacent edges e, e 2, e 3 then w(e )v(e ) + w(e 2 )v(e 2 ) + w(e 3 )v(e 3 ) = 0, where v(e i ) are the primitive vector tangent to h(e i ) and pointed away from v. (5) u = h(x) R 2 is called the end of the tropical disc. (6) If e is an unbounded edge, then v(e), n = a defines an boundary of for some a R. (7) relative homology [h] H 2 (X, L u ). Two tropical discs (h i, T i, w i ) are called () isomorphic if there exists a diffeomorphism f : T T 2 such that h 2 f = h and w 2 (f(e)) = w (e) for each e T []. (2) of the same type if there exists a diffeomorphism f : T T 2 such that h 2 (f(e)) is parallel to h (e) and w 2 (f(e)) = w (e) for each e T []. For simplicity of the notation, we will write h instead of (h, T, w) for a tropical disc in the following discussions. To each tropical curve, one can also associate the notion of Maslov indices as in the holomorphic case. Definition 2.0. Given a tropical disc (h, T, w), its Maslov index MI(h) is defined to be the twice of the sum of weights of unbounded edges. The weighted counted count N β of Maslov index two disc can be then replaced by the notion of Mikhalkin weight associated to each vertices. This gives a combinatorial realization of counting the limit of J t -holomorphic curve in the large complex structure limit. Definition 2.. Given a tropical disc (h, T, w) with end at u R 2 and only trivalent vertices except the root. Let v T [0] be a trivalent vertex with adjacent

15 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 5 edges e, e 2, e 3. Then the (Mikhalkin) weight Mult v at v is defined by w(e )v(e ) w(e 2 )v(e 2 ) 2 T Z R 2 = Z. The weight Mult(h) of the tropical disc (h, T, w) is defined to be Mult(h) = Mult v. v T [0] \{x} For each u in the log base R 2, we denote the Lagrangian torus fiber by L u = π (u). Since the moment map Lagrangian fibration is topologically trivial, we can choose an basis e, e 2 H (L u, Z). Given [ β] H (L u, Z), set z [ β] = z [ β],e z [ β],e2 2, where, is the natural pairing between H (L u, Z) and H (L u, Z). Definition 2.2. For any u R 2, the tropical superpotential (also known as the Hori-Vafa mirror) is defined to be (2.0) W (u) := Mult(h)z [h], h where the summation is over all tropical discs of Maslov index two with stop at u. Cho-Oh [7] proved that (2.0) indeed coincide with the superpotential of Maslov index two holomorphic discs in Floer theory. M. Gross [26] further considered the bulked deformation of the tropical superpotential, which is a tropical analogue of the bulk-deformed potential that we defined in Section 2. Definition 2.3. Let p,, p k R 2 be k generic points. For a tropical disc (h, T, w) with end u generic, we define the following: () p h := {i {,, k} p i Im(h)}. (2) The generalized Maslov index MI (h) of a tropical disc h is defined to be MI (h) = MI(h) 2 p h. To simplify the wall-crossing phenomenon of the tropical discs, M. Gross [26] restricted the coefficients ring to R k = C[t,, t k ]/(t 2 i = 0). The use of this coefficient ring is crucial in our computation of the bulk-deformed potential, as it only record the first-order information in the wall-crossing map defined in Definition 2.4 below. We now define the bulk-deformed potential W k (u) tropically. Definition 2.4. Let p,, p k, u R 2 be in generic position. The bulk-deformed tropical superpotential W k (u) is defined by W trop k (u) = h Mult(h)z [h] t h Q[z, z 2 ] R k, where t h = i p h t i and the summation is over all generalized Maslov index two discs with stop at u. By the work of Nishinou [37] on tropical disc counting, we can conclude that the above expression is well-defined and it defines a Laurent polynomial in R k [z ±, z± 2 ] for each k.

16 6 HONG, LIN, AND ZHAO Proposition 2.5. ([37] Proposition 3.5) There are only finitely many tropical discs with end on u of generalized Maslov index 2 with respect to a given configuration p,, p k. Therefore, W trop k (u) is a Laurent polynomial. Moreover, all the generalized Maslov index two discs are trivalent. To study the wall-crossing phenomenon of the tropical bulk-deformed potential W trop k (u), we define the so-called slab function appear in the Gross-Siebert program [28]. These functions are the analogue of the wall-crossing maps in Floer theory. Definition 2.6. Let (h, T, w) be a tropical disc ending at u R 2 and of generalized Maslov index 2. Let e be the edge adjacent to the root. We call the extension of f(e) in the direction of u to infinity a tropical wall. For each tropical wall d, we associate a slab function and an symplectomorphism f d = + Mult(h)z [h] t h Q[z, z 2 ] R k, K d : Q[z, z 2 ] Q[z, z 2 ] z i z i f ei, [h] d, which preserves the standard holomorphic symplectic 2-form dz z dz2 z 2. The tropical wall structure W trop associated to p,, p k is the union of all tropical walls. Then W trop k (u) in Definition 2.4 is defined when u is in the complement of the wall structure. Given two tropical walls d, d 2, it can be verified that K d K d2 = K d2 K d if and only if d, d 2 are parallel. To study the analogue of the Fukaya s trick that we proved in Proposition 2.6, we will consider a generic tropical loop on the log base R 2 defined as follows. Definition 2.7. We say a loop φ in R 2 is generic if () it avoids all the intersections of the tropical walls, and (2) every intersection of φ with the tropical wall d is transversal. Given such a generic loop φ, it induces an symplectomorphism F φ := Kd ɛd, d:d φ φ where the product order is with respect to the path and ɛ d = sgn φ (t d ), [h]. The following illustrates the wall-crossing phenomenon of the tropical discs. Lemma 2.8. [27] Assume that there are two tropical walls d, d 2 intersect at p transversally and no other walls passing through p by imposing the point constraints configuration p,, p k generic. Let φ be a small loop around p, then F φ = id if and only if there exists exactly one tropical wall d emanating from p with f d = f d f d2. Remark 2.9. Here the coefficients in R k is essential to greatly simplify the calculation. By induction, one can conclude the following theorem. Theorem ([26] Proposition 4.7, Theorem 4.5) This is the simplified version of the unique factorization theorem, see [33][27]

17 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 7 () Given any generic loop φ in R 2, then F φ = id. (2) Given u, u R 2 in adjacent chambers bounded by a tropical wall d, then W trop k (u) = K d W trop k (u ). Given this Theorem, we can inductively compute the tropical bulk-deformed potential W trop k (u) for any given configuration of k generic points and any Lagrangian torus fiber L u Algorithm of Constructing Tropical Wall Structures. 2 We now provides the descriptions of the algorithm computing the bulk-deformed potential W trop k tropically. Let p,, p k be k generic points p,, p k in (C ) 2. By induction, suppose that one can construct the wall structure for any generic k points and compute the tropical bulk-deformed potential W trop k (u) in every chamber for any arbitrary generic k points configuration. To construct the tropical wall structure for p,, p k+, one first construct tropical wall structure for p,, ˆp i,, p k+, where ˆp i indicates that we omit the ith point constraint. The one computes the tropical bulk-deformed potential in each of the chamber for the constraint p,, ˆp i,, p k+. In particular, the expression of W trop k (u) at the point p i can be computed. Every term n β z β in W trop k (p i ) implies that there are n β tropical discs (counted with weight) ending on p i. Therefore, extending the edges with ends of those tropical discs over p i gives the corresponding family of tropical discs of generalized Maslov index zero. As a conclusion, we show that the tropical walls emanating from p i are in one to one corresponding to the monomials in W trop k (p k ). Conversely, any tropical discs of generalized Maslov index zero with p i on the edge with the end arise this way. Apply the procedure to each p i and for each intersection of the walls we add one more wall by Lemma 2.8. There can only be finitely many of walls added thanks to the artificial constraints t 2 i = 0 in the coefficient ring R k. Now given a tropical disc (h, T, w) of generalized Maslov index zero. Let v is the other vertex of the edge e adjacent to the end. Deleting the edge e of T provide another two tropical discs (h, T, w ), (h 2, T 2, w 2 ) with ends on h(v), where T T 2 = T \e and h i = h Ti, w i = w Ti. Since generalized Maslov index tropical discs are rigid (up to enlogation of the ends) and MI (h ) + MI (h 2 ) = MI (h) = 0. We have both h, h 2 are both of generalized Maslov index zero. Then by induction on the number of edge, the above algorithm produces all possible tropical discs of generalized Maslov index zero. 3. Analysis on initial walls for P 2 In this section, we provides the initial wall structure for the holomorphic bulkdeformed potential W k (u). We illustrate some of the difficulties lying ahead of determining the holomorphic wall structure on the log base without a tropicalholomorphic correspondence Theorem. in the case of X = P 2. We begin with computing the first and the second-order bulk-deformed potentials for X = P 2. 3 For the first-order potential, all the relevant walls are induced by Maslov index 2 discs which are known to be regular by Cho-Oh [7], and hence 2 This section is known to the experts but the authors cannot find it in the literature. So we include it for self-containness. 3 The argument in this section is also valid for all toric Fano surfaces, but we focus on P 2 in this section to make the exposition more explicit.

18 8 HONG, LIN, AND ZHAO we can see the wall structure more explicitly. For the second order, we can still compute the loci of corresponding walls, but we will see that they are not as simple as the first order ones, which motivates us to look for another method for higher orders in the next section. Our setting in this section is as follows. We fix a generic point q in X = P 2, and bulk-deform the Lagrangian Floer complex of toric fibers L by b = t q where the parameter t is taken from Λ[[t ]]/t 2. The condition t 2 = 0 automatically discard the contribution by a holomorphic disc with more than one interior marking, and the resulting bulk-deformed Floer theory can be interpreted as the first-order deformation. 3.. Wall-crossing for first-order bulk-deformed potentials. The first-order bulk-deformed potential W,u for a torus fiber L u is given by the coefficient of the unit class appearing in m b k,β (b,, b), where b = t q as above, and b = x e + x 2 e 2 H (L u ) for a toric fiber L u in P 2. Due to the relation t 2 = 0, W has only two components W,u (z, z 2 ) = W 0,u (z, z 2 ) + t F u (z, z 2 ) where z i = e xi and W 0,u (z, z 2 ) = T u z + T u2 z 2 + T u u 2 z z 2 is the Hori-Vafa potential studied in [7], and F counts holomorphic discs with one interior insertion constrained by q. Set β 0, β, β 2 to be generators of π 2 (X, L u ) represented by the standard Maslov 2 discs that are responsible for the terms z, z 2, z z 2 in W 0,u. Then β can be written as a linear combination of β i s. Then by degree reason, F u is contributed by Maslov 4 discs which are in the classes β i + β j with i j. Here, we do not have a contribution from classes of the type 2β i by genericity of L u. Figure. First order walls drawn on the moment polytopes

19 BULK-DEFORMED POTENTIALS FOR TORIC FANO SURFACES 9 Therefore we are interested in the behavior of the moduli space M k, (L u, β; q ) = M k, (L u, β) ev int {q } for β = β i + β j when L u varies among toric fibers. (Unless specified, the moduli consists of holomorphic discs with respect to the standard complex structure on X.) We will see that this moduli space jumps when we travel across one of rays depicted in Figure 4. Such a phenomenon is called the wall crossing. These rays are indeed the images boundaries of Maslov 2 discs that pass through q i under the moment map. As drawn in the figure, there is an obvious change in countings of tropical discs which pass through p := µ(q ) for different chambers A, B and C. See the later sections for more details on tropical disc counting. In what follows, we shall provide precise Floer theoretical analysis on this. Essentially wall crossing occurs since the map between Maurer-Cartan spaces of a torus fiber in one chamber and that in another chamber is nontrivial. As explained in 2.2, this map is contributed by the following moduli space of discs (3.) M k, (L u, β i ; q ) := M k, (L u, β i ) ev int {q } (or, more precisely one parameter family of such). It is easy to see that the virtual dimension of (3.) is, and hence for generic L u, this moduli is empty. For this reason, we sometimes call an element of this moduli space a generalized maslov index 0 disc. By the classification of Maslov 2 discs in [7], this moduli space becomes nonempty precisely when L u lies over the three rays l 0, l, l 2 in Figure 2, and the Fukaya s trick gives an identity unless two Lagrangians are in different chambers separated by one of these rays. Thus we have the following. Lemma 3.. M k,0 (L u, β i ; q ) φ if and only if L u lies over the rays l i in the moment polytope. Moreover, the wall crossing maps are given as follows. Proposition 3.2. When we travel in clockwise direction around p, the wall crossing maps for three rays l 0, l and l 2 are given as follows. (i.e., A B, C A and B C in Figure 2.) (3.2) (z, z 2 ) ( z ( + t z z 2 T ɛ0 ), z 2 ( + ) t z z 2 T ɛ0 ) for l 0 (z, z 2 ) (z, z 2 ( + t z T ɛ )) for l (z, z 2 ) ( z ( + t z 2 T ɛ2 ), z 2 ) for l 2 If we introduce new variable z 0 = z z 2, then the first wall crossing (for l 0 ) also simplifies into a standard form of the cluster transformation. Proof. We will only prove the one for l, and the rest are similar. As in Figure 2, choose two Lagrangian torus fibers L u and L u on both sides of l, and a smooth isotopy ϕ s (0 s q) between them along the path drawn in the Figure. Recall that the moduli space (3.3) s M k, (L u, β, (ϕ s ) J std ; q ) (= s M k, (L u, β, (ϕ s ) J std ) ev int {q }) 4 It is more natural to draw these on the log-base. Especially the tropical discs drawn this way are not precise, but we give this naive picture for intuition.

20 20 HONG, LIN, AND ZHAO induces a map {f k } : CF (L u, L u ) CF (L u, L u ), and in particular the map between the Maurer-Cartan spaces is given by f : H (L u ) H (L u ) x f 0 () + f (b) + f 2 (b, b) +. Observe that the only nonempty slice of (3.3) is when φ s (L u ) precisely sits over l. Since the corresponding disc has class β, the divisor axiom (Lemma 2.5) tells us that f (x e + x 2 e 2 ) = x e + (x 2 + t e x T ɛ0 )e 2 which is equivalent to the desired one in exponential coordinates. Notice that since t 2 = 0. e x2+tex T ɛ 0 = e x2 ( + t e x T ɛ0 ) = z 2 ( + t z T ɛ0 ) 3.2. Computation of W. Recall that we set W,u = W 0,u +t F u where F u counts Maslov 4 discs passing through q and bounding L u. Due to wall crossing, F u varies from chamber to chamber. We choose generic points u A, u B and u C in the chambers A, B and C in Figure 2, respectively, and compute F ua, F ub and F uc using wall crossing maps. Figure 2 Since W should be compatible with wall-crossing maps by the algebraic relations between A -algebra homomorphisms and algebra structures up to modification of T -th power from the flux, we see that F ua does not depend on torus fibers L ua as long as u A stays in A again up to a power of T, and similarly for F ub and F uc By the Classification Theorem of Maslov two discs in [7, Section 4], only possible contributions to F are from the classes β 0 + β, β + β 2 and β 2 + β 0 (recall that we do not have 2β i here due to genericity of L u )), which will induce, z z 2, z 2 z respectively. Let us first investigate the term z z 2. If there exists a holomorphic disc U : (D 2, D 2 ) (X, L u ) of the class β + β 2 passing through q, then its entire image should lie in X \{z 0 = 0} by the classification of such discs (otherwise it would have β 0 component in its class). Therefore one can compose z : X \{z 0 = 0} R with U to get a well-defined function on D 2. Since this function achieves the value q 2