Tropical Lagrangian Hypersurfaces are Unobstructed

We produce for each tropical hypersurface V (φ) ⊂ Q = R a Lagrangian L(φ) ⊂ (C∗)n whose moment map projection is a tropical amoeba of V (φ). When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.

The recent parallel works of [27,29,26] provide methods for constructing tropical Lagrangian submanifolds L(φ) ⊂ X whose image under val : X → Q is nearby a tropical hypersurface V (φ). In this paper, we show how such a tropical Lagrangian submanifold can be constructed via Lagrangian surgery. With this method of construction we show that tropical Lagrangians L(φ) are unobstructed by bounding cochain and can be therefore considered as objects of the Fukaya category. Provided an appropriate version of the Fukaya category exists, we additionally prove a homological mirror symmetry for these Lagrangians, thereby extending the intuition above to holomorphic sheaves supported on hypersurfaces.

Summary of Results
In Section 2, we provide necessary background related to tropical geometry, Lagrangian cobordisms, and mirror symmetry for toric varieties. The review of tropical geometry is included to fix notation, and introduce the non-self-intersecting property of a tropical hypersurface. The section on Lagrangian cobordisms may be safely skipped by experts who are familiar with the results of [5,13]. Our notation for Lagrangian cobordism follows [20]. In Subsection 2.3, we review the monomial admissibility condition of [19], and state homological mirror symmetry for toric varieties.
Section 3 associates to each tropical polynomial φ : Q = R n → R a Lagrangian submanifold L(φ) ⊂ X = (C * ) n whose projection under the valuation map val : X → Q lies near the tropical variety V (φ) ⊂ Q. These tropical Lagrangian submanifolds are constructed using Lagrangian surgery. We prove that this construction is independent of choices made up to exact Lagrangian isotopy. In Section 4 we show that the constructed tropical Lagrangians are unobstructed by bounding cochain, and are monomial-admissible Lagrangians. Between these two sections, the main result of this paper can be summarized as (see Theorem 3.18 for details): Theorem 1.1. Let V (φ) be a tropical hypersurface of R n without self-intersections. For every > 0 there exists a tropical Lagrangian L(φ) ⊂ (C * ) n whose valuation projection is -close to V (φ) in the Hausdorff metric. Furthermore, this Lagrangian is Floer-theoretically unobstructed.
The proof is of Theorem 1 is given modulo hypothesis stated in Subsection 4.1, which relate to extending the definition of the pearly complex of [9] to the monomial-admissible setting. We additionally include an outline of how domain-dependent perturbations or abstract perturbation techniques can be employed to remove these hypothesis. In Section 5 we prove that the constructed Lagrangians L(φ) are mirror to structure sheaves of divisors in a mirror toric varietyX Σ (see Theorem 5.2 for details) Theorem 1.2. LetX Λ Σ be a toric variety, and let X = ((C * ) n , W Σ ) be its mirror Landau Ginzburg model. Let D be a base point-free divisor ofX Λ Σ with tropicalization given by the tropical variety V (φ). The corresponding tropical Lagrangian L(φ) is homologically mirror to a structure sheaf O D , where D and D are rationally equivalent.
The construction of a bounding cochain for the tropical Lagrangian uses some general statements about filtered A ∞ algebras and deforming cochains, a review of which is included in Appendix A.
This work is a portion of the author's PhD thesis, [21], which additionally explores applications and examples of tropical Lagrangians in the context of homological mirror symmetry. The tropical Lagrangians constructed in this paper are Hamiltonian isotopic to existing parallel constructions of tropical Lagrangians presented in [27,29,26]. These constructions have been recently employed in the works of [33,36], which look at how tropical geometry can be used to obstruct the existence of certain unobstructed Lagrangian cobordisms, and how the combinatorics of dimers is related to these tropical Lagrangians. I would like to thank my advisor Denis Auroux whose guidance and comments have helped me at every step of this project. I would also like to thank Andrew Hanlon for walking me through the construction of the monomial admissible Fukaya-Seidel category, and an anonymous reviewer whose comments and suggestions greatly improved the exposition of this article. This project has also benefited from useful conversations with Diego Matessi, Nick Sheridan, and Ivan Smith. Finally, I am especially grateful to Paul Biran and ETH Zürich for their hospitality while hosting me.
A portion of this work was completed at ETH Zürich. This work was partially supported by NSF grants DMS-1406274 and DMS-1344991; EPSRC Grant EP/N03189X/1; and by a Simons Foundation grant.
2 Background: Tropical Geometry and Lagrangian Cobordisms

Tropical Geometry
The tropical semiring (R, ⊕, ) is the set R ∪ {+∞} equipped with the following two binary operations These two operations are called tropical plus and tropical times respectively, and they obey the distributive law. Tropical polynomials of multiple variables describe piecewise linear concave functions φ : Q := R n → R of rational slope. It will frequently be useful for us to use the following characterization of tropical linear polynomials.
Proposition 2.1. The tropical polynomials are exactly the piecewise linear concave functions with dφ(x) ∈ T * Z R n at all points where φ is differentiable.
One can approximate tropical polynomials with regular polynomials via logarithms and the estimates for q goes to 0. We'll frequently describe tropical polynomials of two variables in terms of their tropical varieties by drawing a planar graph whose faces describe the domains of linearity of φ. This graph generalizes in higher dimensions to a stratification of Q which describes many of the combinatorial properties of a tropical polynomial.
Definition 2.2. Let φ : Q → R be a tropical polynomial. Each monomial term in φ can be labelled by its exponent v ∈ Z n . The linearity stratification of Q is the stratification where p ∈ Q k if and only if there is no k + 1 dimensional open subset of an affine subspace A ⊂ Q, with p ∈ A on which the restriction φ| A is a k + 1-affine map. Each stratum will be denoted U {vi} , where {v i } is the collection of monomial terms which achieve their minimum along the strata. We define the tropical variety of Q to be V (φ) = Q n−1 , which describes the locus of non-linearity of φ. 1 One interpretation to the approximation given in Equation 2 is that when f : (C * ) n → C is a Laurent polynomial, the tropical variety V (φ) provides a dominating term approximation of val(f −1 (0)).
There is an involution on smooth concave functions f with convex domains ∆ given by the Legendre transform. An analogous involution exists in the setting of tropical polynomials.
φ be the set of integer points v for which φ matches the monomial a v x v on an open subset. Define the Newton polytope ∆ φ ⊂ T * 0 Q to be the convex hull of ∆ Z φ . We define the Legendre transformφ(v) to be the minimal-fit concave piecewise linear function to the datǎ We will denote the linearity stratification induced byφ on the Newton polytope as We will denote 2 the stratum with vertices v i by U {vi} .
See Figure 1 for examples. The Newton polytope is a lattice polytope which is determined by the leading order behavior of the tropical polynomial φ. It will also be convenient for us to interpret this Newton polytope as the image of dφ under the projection π 0 : T * R n → T * 0 R n ∆ φ = Hull({π 0 • dφ(x) | φ is differentiable at x}).
As the Legendre transform contains the data of the coefficients of φ, the polynomials φ andφ determine each other completely. This relation is also reflected in the duality between the stratifications U {vi} and U {vi} .
Definition 2.4. We say that a non-maximal stratum U {vi} is smooth if U {vi} is a standard simplex. The self-intersection number of a strata U {vi} is defined as the number of interior lattice points of U {vi} .
When φ has no self-intersections, then ∆ φ ∩Z n = ∆ Z φ . When a tropical variety comes as the tropicalization of a family of complex curves, the self-intersection number gives the genus of the family of curves which degenerate in the family. In our constructions of tropical Lagrangians, this self-intersection number will give the number of self-intersection points of our tropical Lagrangian (see discussion following Definition 3.16).
Example 2.5. We now look at a few examples which demonstrate the difference between smooth, non-smooth, and self-intersecting tropical hypersurfaces. Consider the tropical polynomial This has 3 domains of linearity, where each of the monomial terms dominate. Notice that the dual stratum U x1,x2,(x1x2) −1 is not a primitive simplex, and so this vertex is non-smooth (see Figure 1a). As (0, 0) ∈ U x1,x2,(x1x2) −1 is an interior lattice point of the simplex, this stratum has 1-self intersection.
If we modify the coefficients of the monomials in the tropical polynomial to with c > 0, we get a smooth tropical curve instead (see Figure 1b). The dual stratification is a triangulation of the Newton polytope by primitive simplices, so this is an example of a smooth tropical curve with no self intersections. Finally, we look at an example which highlights the difference between self-intersections and smoothness. Consider the tropical polynomial The moment polytope and curve are drawn in Figure 1c. The stratum U 1,x1,x2,x1x2 is not smooth, as it is not a simplex. However, the stratum does not contain any self intersections either, as its interior is lattice point free.

Lagrangian Surgery and Cobordisms
The bulk of this paper will revolve around constructing new Lagrangian submanifolds. Our most-used tool for these constructions is Lagrangian surgery, which allows us to make local modifications to a Lagrangian submanifold to obtain new Lagrangians. By the Weinstein neighborhood theorem, we can locally model the transverse intersection of two Lagrangians by the zero section and cotangent fiber of T * R n . The model Lagrangian neck in T * R n is the Lagrangian parameterized by (c) Non-smooth, self-intersection free sphere with 4 punctures, φ+(x1, x2) = 1 ⊕ x1 ⊕ x2 ⊕ x1x2. Notice that this Lagrangian neck is asymptotic to the disjoint union of the zero section and T * 0 R n . Given a Lagrangian L with a transverse self-intersection, one can replace a neighborhood of the self-intersection with a Lagrangian neck. In summary: Theorem 2.6 ( [31]). Let L be a (not necessarily connected) Lagrangian submanifold with a transverse self-intersection point p. Then there exists a smooth Lagrangian L w which agrees with L outside of the intersection point, and is modeled on the Lagrangian neck in a neighborhood of p.
There are many ways of doing this smoothing depending on the choice of neck inserted, however the Hamiltonian isotopy class of the surgery is dependent only a single parameter w called the neck width, which measures the flux swept by the Lagrangian isotopy as one decreases the neck radius to 0. Given Lagrangians L 0 and L 1 intersecting transversely at a single point q, we denote by L 0 # w q L 1 the Lagrangian connect sum, which is obtained by inserting a neck of width w at a neighborhood of the intersection point q. 3 When the width of the neck inserted is unimportant, we will simply write L 0 # q L 1 . Lagrangian surgery also has an interpretation as an algebraic operation in the Fukaya category.
Theorem 2.7 ( [13]). Let L 0 , L 1 be unobstructed Lagrangians intersecting at a unique point q. Then there is an exact triangle The proof of this theorem comes from a comparison of holomorphic triangles with corner at q, and holomorphic strips in L 0 # w q L 1 which are obtained from rounding this corner. The topological operation of surgery can be understood via cobordisms. In the symplectic world, there is an analogous notion of Lagrangian cobordism relating Lagrangian surgeries.
Compactness: The projection Im z : K → iR ⊂ C is bounded.
We denote such a cobordism K : (L + 0 , . . . , L + k−1 ) L − . Remark 2.9. We follow the cohomological grading of [20], where a cobordism between (L + 0 , . . . , L + k−1 ) L − has input ends L + 0 , . . . , L + k−1 with positive real values {c + i }, and end L − with negative real value c − . This is opposite to the convention from [5]. This can be slightly confusing when considering Lagrangian cobordisms, as the "domain" end of the cobordism is on the right in its projection to C.
Some simple examples of Lagrangian cobordisms include the trivial cobordism L × R, or the suspension of a Hamiltonian isotopy. Given Lagrangians L 0 , L 1 intersecting transversely at a single point q, there exists a surgery trace Lagrangian cobordism (L 0 , L 1 ) L 0 # q L 1 . Just as Lagrangian surgery gave us a way to understand L 0 # q L 1 as a mapping cone, there is a broad-reaching theorem which tells us how to relate cobordant Lagrangians as objects of the Fukaya category.
L − be an embedded monotone Lagrangian cobordism. Then there are k objects Z 0 , . . . , Z k−1 in the Fukaya category, with Z 0 = L + 0 and Z k L − which fit into k exact triangles In particular when k = 2, we have an exact triangle In the case where k = 1, we have an isomorphism This can be restated as a relation between the Lagrangian cobordism category of X, and a category which describes triangular decompositions of objects in the Fukaya category. The proof of this theorem computes for test objects L ∈ Fuk(X) the Floer homology CF • (L × R, K) in two different ways.

Mirror Symmetry for Toric Varieties
LetX Σ be a toric variety given by the fan Σ ⊂ R n = Q. We review some notation and concepts from [10] related to line bundles, divisors and tropical geometry. Each lattice generator v of a ray of Σ gives a torus equivariant divisor D v ofX Σ . In the setting whereX Σ is smooth we may express any linear equivalence class of divisor as a sum v∈Σ a v D v . An integral support function for Σ is a function φ : Q → R which is linear on each cone of Σ, and is integral on the fan in the sense that φ(Σ ∩ Z n ) ⊂ Z. To each divisor class [D], we may associate an integral support function φ [D] which is determined by the values Properties of the line bundle O(D) can be read from the support function: the line bundle is base point-free whenever φ [D] is concave, and ample if and only if φ [D] is strictly concave. The function φ [D] is piecewise linear, so when D is base point free the function φ [D] is a maximally degenerate tropical polynomial (in the sense that every stratum of the tropical variety contains the origin). The tropical variety of the support function of a base point free line bundle can be related to the valuation projection of the corresponding divisor of the line bundle. Let D be a base point free divisor transverse to the toric anticanonical divisor, and let ∆ φ [D] be the Newton polytope of φ [D] . Then over the open torus ofX Σ , each choice of constants c v defines a polynomial which gives a section of the line bundle over the compactification O D →X Σ ⊃ (C * ) n . There exists a choice of constants so that f −1 where the constants c v are elements of the Novikov field. The valuation projection of the variety f D = 0 is a tropical hypersurface defined by a tropical polynomial φ D , which we call the tropicalization of f D . φ D is a deformation of φ [D] . In the complex setting the tropical variety V (φ D ) ⊂ ∆ Σ approximates the image of D under the moment map projection val :X σ → ∆ Σ . The observation that the strata of the tropical variety V (φ) meet the toric boundary of the moment polytope of X Σ transversely is compatible with the existence of a compactification for the variety f −1 (0) ⊂ (C * ) n inside ofX Σ .
Mirror symmetry for toric varieties is based on an understanding of how compactifications modify the mirror construction. It is an expectation in mirror symmetry that compactification onX corresponds to the incorporation of a superpotential W : X → C in the mirror and vice versa. One proposed method for constructing mirror spaces for toric varieties is to considerX Σ as a compactification ofX Σ \ D Σ = (C * ) n , where D Σ is the toric anticanonical divisor. A choice of symplectic form forX Σ picks out the coefficients of a Laurent polynomial, Notation 2.11. For the remainder of this paper, X =X = (C * ) n , Q = R n ,X Σ is a toric variety determined by fan Σ, and W Σ : X → C is the mirror Hori-Vafa superpotential.
This Hori-Vafa superpotential provides a taming condition for non-compact Lagrangian submanifolds of X. We use the notion of a monomial division as a taming condition for its particularly clean description in X.
Definition 2.12 ([19]). Let W Σ : X → C be a Laurent polynomial whose monomials are indexed by the rays of a fan Σ. A monomial division ∆ Σ for W Σ = α∈A c α z α is an assignment of a closed set C α ⊂ Q to each monomial in W Σ such that the following conditions hold: The C α cover the complement of a compact subset of Q = R n There exist constants k α ∈ R >0 so that the maximum of is always achieved by |c α z α | kα for an α such that val(z) ∈ C α . C α is a subset of the open star of the ray α in the fan Σ.
A Lagrangian L ⊂ X is ∆ Σ -monomially admissible if over val −1 (C α ) the argument of c α z α restricted to L is zero outside of a compact set.
Given W Σ , there is often a preferred type of monomial subdivision, the tropical division, with covering regions defined by for some fixed δ ∈ [0, 1]. The data of a monomial division allows the construction of a monomial admissible Fukaya-Seidel category.
Theorem 2.13 ([19]). Given ∆ Σ a monomial division for W Σ , there exists an A ∞ category Fuk ∆Σ (X, W Σ ) whose objects are ∆ Σ -admissible Lagrangians, and whose morphism spaces are defined by localizing an A ∞ pre-category Fuk → (X) with morphisms: Here θ is an admissible Hamiltonian perturbation. The higher composition maps m d in this precategory are given by counts of punctured holomorphic disks.
The Lagrangians considered in the setting of [19] do not bound holomorphic disks.
Theorem 2.14 ( [19]). LetX be a smooth complete toric variety with Hori-Vafa superpotential W Σ . Let T w π P ∆Σ (X, W Σ ) be the category of twisted complexes generated by tropical sections. The A and B-models The equivalence is proven with Theorem 3.14, which characterizes the Floer cohomology between tropical Lagrangian sections (Definition 3.13).

Construction of Tropical Lagrangians in X
The goal of this section is to construct for each tropical polynomial φ : Q → R a Lagrangian L(φ) whose projection val(L(φ)) is -close to V (φ) in the Hausdorff metric. Our construction will be rooted in the language of Lagrangian cobordisms, giving us a path to prove a homological mirror symmetry statement for L(φ). We additionally prove that the Lagrangian L(φ) is an unobstructed object of the Fukaya category.

Surgery Profiles
We will need an explicit surgery profile to build L(φ).
Proposition 3.1. Let f 0 : R n → R be the constant function f 0 = 0, and let f 1 : R n → R be a smooth convex function. Let U be the region where df 1 = 0 and suppose we have normalized f 1 so that f 1 (U ) = 0. Additionally, suppose that for given > 0, there exists a constant c so that f 1 (x) < 2c implies that x ∈ B (U ).
Consider the Lagrangian sections L 0 = df 0 , and L 1 = df 1 in T * R n . There exists a Lagrangian L 0 # U L 1 in a small neighborhood of the symmetric difference Furthermore, there exists a Lagrangian cobordism K with ends (L 0 , L 1 ) L 0 # U L 1 . We call L 0 # U L 1 the Lagrangian surgery at U .
Proof. We first give a description of a Lagrangian L 0 # U L 1 which satisfies the desired properties. By convexity, f 1 > 0 on the complement of U . Let r : R > → R and s : R > → R be functions satisfying the following properties: The concatenation of curves (t, r (t)) and (t, s (t)) is a smooth plane curve.
The profiles of these functions are drawn in Figure 2. A quantity which we will later use is the neck width of the surgery profile curves, which is defined as w r ,s := −r (c ) + s (c ).
Consider the Lagrangian submanifolds given by the graphs defined as sections over the domain f 1 ≥ c . The union of these two charts is a smooth Lagrangian submanifold, which is our definition of L 0 # U L 1 . The profile is only defined where f 1 ≥ c , so L 0 # U L 1 is disjoint from the set L 0 ∩ L 1 . As r is the identity for f 1 > 2c , we have that d(r • f 1 ) = L 1 on the region where f 1 (x) > 2c . A similar statement can be made about s . These observations (along with the choice of c ) give us that L 0 # U L 1 is contained a small neighborhood of the symmetric difference.
It remains to show that L 0 , L 1 and L 0 # U L 1 fit into a cobordism. This cobordism will be constructed as a Lagrangian surgery in one dimension higher. Letf 0 : R n+1 → R be the constant zero function, and let functionf  We will now take the surgery of sectionsL 0 = df 0 andL 1 = df 1 . Let U be the region where df 1 = 0.L 0 and L 1 agree on the regionŨ = U × (−∞, − ). As the intersection over this region is defined by the intersection of convex primitives, we may use our previous construction to define the surgery cobordism K :=L 0 #ŨL 1 .
Since K agrees withL 0 andL 1 outside of a small neighborhood ofŨ , As the function g(t) is constant on each t-fiber, we obtain that K| t=− = L 0 # U L 1 and conclude These are the conditions required for K to be a Lagrangian cobordism between L 0 , L 1 and L 0 # U L 1 .

Remark 3.2.
We could have chosen f 1 to be concave and still have had a surgery construction. However the existence of a surgery cobordism depends on the convexity of f 1 . If instead we had started with a concave primitive f 1 , the functionf is neither concave or convex, and we are unable to construct our cobordism using the surgery profile previously described. This manifests itself as an ordering on the ends of the cobordism. In particular, there is usually no Lagrangian cobordism with ends (L 1 , L 0 ) and (L 0 # U L 1 ).
In the setting where U is a closed ball, this is nothing different than a special choice of neck on the standard Lagrangian connect sum cobordism. The construction above provides an alternative definition to Figure 3: The projection of the surgery cobordism to the C-parameter. The curve defining the upper boundary of this projection is parameterized by z = t + i dg dt .
the surgery trace cobordism considered in [6]. In the non-compact setting, we get an interpretation of what it means to take the connect sum of Lagrangians which agree on a non-compact set. This operation was dependent on a number of choices-in particular, a choice of profile function r and s . Different choices for these parameters give isotopic Lagrangian submanifolds. However, just as in the setting of transverse surgery of Lagrangian submanifolds, the Hamiltonian isotopy class of these Lagrangians can be made independent of choices up to the neck width.
Recall, if i t : L × [0, t 0 ] → X is a Lagrangian isotopy, the flux of the isotopy is the class We can use this to associate a flux class to a Lagrangian surgery. Fix choices of profile functions r 0 , s 0 . Extend this to a smooth family of surgery profiles r , s , dependent on the choice of parameter ∈ (0, 0 ). As approaches zero, r approaches the identity and s approaches the constant zero function. This smooth family of surgery profiles defines a Lagrangian homotopy i : L 0 #L 1 × (0, 0 ) → X. Proof. The flux cohomology class is only dependent on r 0 , s 0 , and not the extension of these to families r , s , as we can interpolate between any two choices of extensions, and the Flux class is invariant under isotopies of isotopies relative ends. We now give an explicit computation of this Flux class in terms of the profile function. Parameterize the neck of the surgery N with a map ψ : ∂U × [−t 0 , t 0 ] → L 0 # U L 1 . We define this map by specifying it on the charts given by d(r • f 1 ) and d(s • f 1 ) so that: On these two charts, we ask that As the isotopy i : L 0 #L 1 ×[0, ) → X is constant outside of surgery neck N , the Flux class descends to a class in H 1 (N, ∂N ). In dimension greater than 2, U is the zero set of a convex function and therefore ∂U is simply connected. In this case, H 1 (N, ∂N ) is generated by a single curve γ : [−t 0 , t 0 ] → N with γ(−t 0 ) ∈ ∂U ×{−t 0 } and γ(t 0 ) ∈ ∂U × {t 0 }. In dimension equal to 2, there is another generator c ∈ H 1 (N, ∂N ), however the integral of the Louiville form on this cycle vanishes and so Flux i (c) = 0. Therefore, the flux is characterized by Flux i (γ).
Since we are in the exact setting (with ω = dη), the flux on a path c : [0, 1] → L can be computed instead as Flux i (γ) : When the Lagrangian is parameterized by primitive g, so that i(x) = (x, dg), then i•c η = g(c(1)) − g(c(0)).
Using the decomposition of N into two charts parameterized by r Proof. This follows from the observation that the set of profiles (r , s ) of fixed neck width is path connected, so we may always find a flux-free isotopy between two surgery profiles of the same neck-width.
As the Hamiltonian isotopy class of the surgery is only dependent on the neck-width of the surgery, we will write L 0 # w U L 1 to denote a surgery of L 0 and L 1 at U with a choice of profile functions for the surgery with neck width w. By iterating Proposition 3.1 at each intersection point we get the following statement about symmetric differences of Lagrangians.
Corollary 3.5. Let L 0 and L 1 be two Lagrangian submanifolds of X. Suppose that for each connected component U k of the intersection U = L 0 ∩ L 1 , there is a neighborhood U k ⊂ V k ⊂ L 0 which may be identified with a subset V k ⊂ R n . Consider the Weinstein charts B * V k ⊂ X. Suppose that L 1 restricted to this chart B * V k is the graph of an exact differential form df k : V k → B * V k which vanishes on U k . Suppose additionally that the primitives f k are all convex functions on V k .
There exists a Lagrangian L 0 # w k U k L 1 in a small neighborhood of the symmetric difference and a Lagrangian cobordism K : Here, {w k } is the sequence of neck-widths chosen at for each of the surgeries.
Example 3.6. We compare our Lagrangian surgery with fixed neck in the non-compact setting to ordinary Lagrangian surgery as drawn in Figure 4. Let L 0 , L 1 ⊂ T * S 1 be a cotangent fiber and its image under inverse Dehn twist around the zero section (see Figure 4a and Figure 4b). An application of Proposition 3.1 shows that L 0 L 1 is cobordant to the zero section of T * S 1 by applying surgery on the overlapping regions outside a neighborhood of the zero section (see Figure 4c) Let us compare this to the surgery obtained by first perturbing L 1 by the wrapping Hamiltonian θ and then taking the Lagrangian connect sum. Then L 0 and θ(L 1 ) intersect at two points, which we can resolve in the usual way. The resulting Lagrangian L 0 #(θ(L 1 )) has three connected components, two of which are non-compact (see Figure 4d). Despite this, L 0 #(θ(L 1 )) and L 0 # U L 1 agree as objects of the Fukaya category, as an additional argument shows that the non-compact components of L 0 #(θ(L 1 )) are trivial as objects of the Fukaya category. This example will become the simplest example of a construction of a tropical Lagrangian submanifold. Figure 4: The difference between surgery with neck U and ordinary Lagrangian surgery.
In the setting of the cotangent bundle T * R n , the connect sum has image under the projection to R n which lives in a neighborhood of the complement of the regions U . We now examine what the projection to the fiber of T * 0 R n of this surgery looks like. Let arg : T * R n → T * 0 R n be projection to the cotangent fiber of zero, and for any set C, let arg(C) be the image of this set under the projection. Suppose that f has minimal value 0. By the convexity of f , whenever {x | df (x) = 0} is compact and c > 0, then 0 is an interior , which is a ball containing the origin.
Choose c small enough so that B c (0) is an open ball in arg(df (R n )). Choose small enough so that f (x) ≤ 2 implies df (x) ⊂ B c (0). Consider the following three parameterized subsets of T * 0 R n : Since s , r ≤ 1, the chain rule for the compositions r • f, s • f gives the inclusion (C r ∪ C s ) ⊂ B c (0). As topological chains, C r and C s have boundary components corresponding to where f (x) = and f (x) = 2 . The boundary components have the following identifications: As the inner boundaries of C r and C s match, we may glue these two charts into a single chain C r+s , with boundary ∂(C r+s ) = ∂B + {0}. The chain C r+s provides a contraction of the boundary sphere of B to the point 0 ∈ B. We therefore obtain that B ⊂ C r+s , completing the proof.
In the setting where U is non-compact, we still obtain that for sufficiently small surgery parameters Let f : R n → R be a convex function. Suppose there exists a (not necessarily compact) polytope V ⊂ R n , with faces F . Define the chains with domains restricted to faces by The key insight in the proof of Proposition 3.7 was that the chain C r,F + C s,F describes a nullhomotopy of ∂ 2 B F . However, since ∂ 2 B F is no longer a sphere, this will not be sufficient to show that the image of B F is equal to the image of C r,F + C s,F . Instead, we consider the sphere: We would like to induct on the dimension of F to show that C r,F + C s,F defines a nullhomotopy between S F and 0. In order to glue together these nullhomotopies, we need an additional condition.
Proposition 3.8. Let f : R n → R and V be defined as above. Suppose that at each q contained in face F , the differential df vanishes on the normal plane N q F . Then C r,F + C s,F defines a nullhomotopy between S F and 0.
Proof. We induct on the dimension of F . Suppose that for all G < F , there exists a reparameterization of C r,G + C s,G which describes a nullhomotopy between S G and 0. Because of the normality condition, this nullhomotopy occurs in the normal plane to G, and so describes a nullhomotopy of a sphere of dimension dim G − 1 in a dimension dim G linear subspace. Therefore, the image of C r,G + C s,G surjects over B G . We now look at the image of the boundary of ∂(C r,F + C s,F ) Therefore, C r,F + C s,F defines a nullhomotopy between S F and 0.
With this extension of the proof of Proposition 3.8, we get the following corollary.
Corollary 3.9. Let L 0 and L 1 be sections of T * R n as in Proposition 3.1. Let V be a polytope with faces F . Suppose that at each q contained in face F , the differential df vanishes on the normal plane N q F . Then for choice of construction parameter for surgery sufficiently small, In particular, if there exists polytope V which is parallel to df at all face and arg(L 1 | V ) = arg(L 1 ), then arg(L 0 # U L 1 ) = arg(L 1 ).

Tropical Lagrangian Sections
There is a nice collection of admissible Lagrangians inside of Fuk ∆Σ (X, W Σ ) called the tropical Lagrangian sections of X → Q which we will use as building blocks in our construction. These were introduced in [2]. For our model of the Fukaya Seidel category, we use the monomial admissible Fukaya Seidel category Fuk ∆Σ (X, W Σ ). This version of the FS category is due to [19].
Let φ : Q → R be a tropical polynomial. Choose small enough so that for any point q ∈ U {vi} the convex hull of dφ(B (q)) is either all of U {vi} , or contains U {vi} as a boundary component. This means that is small enough that for all q ∈ Q the induced stratification of B (q) from the tropical variety has at most one vertex. See Figure 5 for a non-example.
Defineφ to be the smoothing of φ by convolution with a symmetric non-negative bump function ρ with support B (0), a small ball around the origin.
When we have fixed a size , we will simplify notation and refer to this as a smoothingφ. The smoothing φ enjoys many of the same properties of φ.
Proof. The first property comes from the preservation of linear functions under symmetric smoothing.
The second property is a general statement about convolutions of concave functions against non-negative functions. Let ψ be a smooth concave function. Then A = Hess(ψ) is the matrix with entries given by a ij = (∂ i ∂ j ψ). Concavity of ψ is equivalent to checking that A is positive semi-definite i.e. for all vectors v, v T · A · v ≥ 0. Letψ = ρ * ψ, and letÃ = Hess(ψ).
. Since ρ ≥ 0, and the convolution of non-negative functions is again non-negative, we obtain thatψ is concave. 4 The third property requires a lemma about convolution and the argument of a function.
We delay the proof of the lemma. With the lemma, arg(dφ(R n )) ⊂ Hull(arg(dφ(x))) = ∆ φ . For the reverse direction, we remark that for every v ∈ ∆ Z φ , the set U {vi} on which dφ = v is non-empty. Therefore, of Lemma 3.11. Let A ⊂ R n be a compact set. Let ρ : R n → R be a function with R n ρd vol = 1 and ρ ≥ 0. The ρ-weighted center of mass of A is avg ρ (A) := ( A x 1 ρd vol, . . . , A x n ρd vol). Let {avg ρ (A)} be the closure over all ρ-weighted centers of mass. We first sketch an argument showing that To show that Hull(A) ⊂ {avg ρ (A)}, we use Carathéodory's theorem to write any p ∈ Hull(A) as p = n+1 i=1 α i p i as a convex combination of points p i ∈ A. Let ρ B (x) be a non-negative smooth bump function with support on B (x) and total integral 1. Let ρ ,αi,pi := n+1 i=1 α i ρ B (pi) be a sum of such bump functions. Then lim →0 avg ρ ,α i ,p i (A) = p. The reverse inclusion is obtained by taking a sequence of approximations for the weighting ρ by We now prove a generalization. Given A ⊂ R n a compact set, and η : A → R k a function, define Hull(η(A)) to be the convex hull of the image of A under η. We expand η in coordinates as η = (η 1 , . . . , η k ). Let ρ : R n → R be a function with R n ρd vol = 1. The ρ-weighted average of η over A is the value we obtain a sequence of ρ-weighted averages of η which converges to p. For the reverse inclusion, we approximate a weighting ρ by a sum of bump functions with small support.
With this generalization we can prove Lemma 3.11, as These sets U {vi} can be characterized in terms of the smoothing ball B and the combinatorics of φ.
Proposition 3.12. The set U {vi} is the set of all points q ∈ Q such that dφ| B (q) belongs to U {vi} , and is not contained in the boundary of U {vi} .
Proof. By Lemma 3.11, if dφ| B (q) belongs to U {vi} then dφ(q) ∈ U {vi} . The only concern may be thatdφ| q is not in the interior of U {vi} , however the requirement that dφ| B (q) is not contained in the boundary of U {vi} rules out this possibility. Therefore, q ∈ U {vi} . Suppose now that q ∈ U {vi} . We would like to show that within an -radius of q, dφ(q) belongs to U {vi} , and that at least one point is not contained in the boundary. We first show containment. Suppose for contradiction that dφ| B (q) ⊂ U {vi} . Then take additional vertices w k so that dφ| B (q) ⊂ U {vi}∪{w k } . We break into two cases.
The set {w k } only contains one element. In this case, the weighted average over arguments defining dφ(q) cannot possibly lie in U {vi} .
The set {w k } contains at least 2 elements. This contradicts our assumption on the size of , as we now see top dimensional strata corresponding to two different boundaries of U {vi} .
This proves that dφ| B (q) ⊂ U {vi} . As the value of dφ(q) is an interior point of U {vi} by the definition of q ∈ U vi , it cannot be the case all of dφ| B (q) is contained in the boundary of U {vi} .
This proposition gives us a clean description of the sets U {vi} , and additional information on the restriction of φ to each of these subsets. In the setting of top dimensional strata, we have an inclusion The graph of dφ is a Lagrangian in T * Q rather than in X = (C * ) n , but after periodizing the cotangent bundle, we get sections of the SYZ fibration.  2]). The tropical Lagrangian section σ φ : Q → X associated to φ is the composition When the smoothing radius is not important, we will drop it and simply write σ φ . The key observation is that for each lattice point Given a monomial admissibility condition (W Σ , ∆ Σ ), we say that φ is an admissible tropical polynomial if σ φ is an ∆ Σ -monomially admissible Lagrangian. From here on out, we will only work with admissible tropical polynomials.
Take an admissible perturbation of σ −φ so that arg(σ −φ ) is locally a diffeomorphism onto its image. This can be arranged by adding a small strictly convex function toφ. After incorporating this Hamiltonian perturbation, the intersections between σ −φ and σ 0 become transverse. The intersections are in bijection with the points where arg(σ −φ ) = 0. Since arg(σ −φ ) is (after considering the small Hamiltonian perturbation) a slight enlargement of the Newton polytope ∆ φ , the intersections between σ φ and σ 0 are identified with the lattice points of ∆ φ .
This observation, along with a characterization of holomorphic strips on convex functions allows one to describe the Floer cohomology of the tropical Lagrangian sections combinatorially.
Theorem 3.14 ( [2,19]). LetX Σ be a toric variety, and let (X, W Σ ) be its mirror Landau-Ginzburg model. Let ∆ Σ be a monomial admissibility condition. Let φ 1 , φ 2 be the support functions for line bundles onX Σ (see Subsection 2.3). Then after appropriately localizing, there is a quasi-isomorphism Furthermore, the A ∞ structure on the subcategory of the Fukaya-Seidel category generated by tropical Lagrangian sections is quasi-isomorphic to the dg-structure on the dg-enhancement of derived category of coherent sheaves onX Σ .
AssumingX Σ is smooth and projective, line bundles generate the derived category of coherent sheaves onX Σ . This proves that the subcategory of Fuk ∆Σ (X, W Σ ) generated by tropical Lagrangian sections is equivalent to D b Coh(X Σ ).

Tropical Lagrangian Hypersurfaces
For this section, we fix (W Σ , ∆ Σ ) some monomial division and work in the setting where Theorem 3.14 holds.
It is usually desirable for Lagrangians to have transverse intersections. However, the highly non-transverse configuration of unperturbed tropical Lagrangian sections will work in our favor as locally the intersection of σ 0 and σ −φ is given by the graphs of one-forms with convex primitives. This allows us to apply our surgery profile from Proposition 3.1. Proof. The set σ 0 ∩ σ −φ is homeomorphic to arg(−dφ) −1 (Z n ). Since −φ −1 is convex, we can compute the set on which it has a fixed derivative as the minimal locus of a appropriately shifted convex function Since the minimal locus of a convex function is contractible,  Example 3.17. The simple example of a surgery given in Example 3.6 is the tropical Lagrangian. In this setting X = C * , and Q = R. The tropical hypersurfaces of Q are simply points, and so we expect that the tropical Lagrangian associated to a point will be an SYZ fiber. The two sections σ 0 and σ −(0⊕x1) are the line and twisted line of Figure 4 which are surgered together. The resulting Lagrangian is Hamiltonian isotopic to the SYZ fiber with valuation 0.
We will later show that the choice of data D does not change the exact isotopy class of L, and will therefore usually write L(φ) instead. In the definition of a tropical Lagrangian submanifold, we've taken the connect sum along each strata of U v corresponding to the non-self-intersection strata of the tropical polynomial φ. As a result, a tropical variety with self-intersections only lifts to an immersed Lagrangian. As an example, the Lagrangian lift of the tropical polynomial φ 0 Figure 1a is an immersed sphere with 3 punctures and 1 transverse self intersection.
These intersections may be transverse, but they need not be -an example is φ = x 1 ⊕ x 2 ⊕ (x 1 x 2 ) −1 as a tropical function on Q = R 3 , where the self-intersection is a clean intersection R ⊂ L(φ).
The distinction in terminology between the smoothness and self-intersections becomes important here. For example, the tropical curve exhibited in Figure 1c lifts to an embedded tropical Lagrangian submanifold, as the tropical curve has no self-intersections. However the tropical curve from Figure 1c is not an example of a smooth tropical curve. Additionally, the tropical Lagrangian submanifolds satisfy these technical requirements giving them well defined Floer cohomology.
Admissibility: Let (W Σ , ∆ Σ ) be a monomial admissibility condition as defined in [19]. Then whenever σ −φ is monomially admissible, so are L D (φ) and K D (φ). To prove that the argument projection matches, we use Corollary 3.9 where the polytope V chosen is a large scaling of the Newton polytope. This polytope intersects which intersects each tropical stratum at infinity perpendicularly, and therefore for each p ∈ F , dφ| NpF = 0.
The the proof of admissibility is Proposition 3.20. The proof of unobstructedness is left to Section 4.

Independence from choices
In our definition of L D (φ), we've made choices for the data D = (ρ , r , s ). Fortunately, these choices do not modify the exact isotopy class of L(φ). Proof. These two Lagrangians are Lagrangian isotopic as we can smoothly interpolate between the parameters D 1 and D 2 in our construction. Let L D t (φ) be a Lagrangian isotopy between the choices of data. Since (C * ) n is exact, we can prove the proposition by showing that the integral of the Liouville form η (satisfying dη = ω) on cycles of L D t (φ) is independent of the choices that we've made. For simplicity, let L(φ) be the tropical Lagrangian constructed with any choice of data D = {ρ, s, r}, and we'll show that the Liouville form on H 1 (L(φ)) only depends on the choice of φ. For this computation, we will choose a convenient spanning set for H 1 (L(φ)). We now assume that L(φ) is connected, and that 0 is a lattice point of ∆ φ . To compute H 1 (L(φ)), decompose L(φ) = L r ∪ L s , where L r and L s are the charts corresponding to the profiles r, s in Figure  2. These charts are homotopic to the tropical amoeba R n \ {U v }, so H 1 (L s ) = H 1 (L r ) = H 1 (V (φ)). The inclusion i s : L s ∩ L r → L s is also an isomorphism on the first homology. The Meyer-Vietoris sequence allows us to decompose the homology of L(φ) as The map (i r , i s ) 1 is injective, so the kernel of (j r ) 1 − (j s ) 1 has dimension b 1 (L r ∩ L s ). Therefore, Im ((j r ) 1 − (j s ) 1 ) = Im (j r • i r ) 1 , identifying a copy of H 1 (L r ∩ L s ) ⊂ H 1 (L(φ)). To characterize H 1 (L(φ))/H 1 (L r ∩ L s ), note that dim ker((i r , i The connected components of L r ∩ L s are in bijection with the surgery regions {U v } for σ 0 and σ −φ , which (by Proposition 3.15) are in bijection with the non-self intersection lattice points ∆ Z φ . This makes {e v } v∈Z n ∩∆ φ a basis for H 0 (L r ∩ L s ). The kernel of (i r , i s ) 0 , which may be thought of as the image of ∂ 1 , is spanned by pairs {e vi − e vj }.
We take the cycles from H 1 (L r ∩ L s ) and cycles {c {vi,vj } } to be a generating set for H 1 (L(φ)), and compute the integral of η on these generators.
To compute the integral of the Liouville form on a cycle c included via i : H 1 (L r ∩ L s ) → H 1 (L(φ)), we may always select a representative for c which lies completely inside of the chart L r . Since L r is exact, the Liouville form vanishes, η(c) = 0 and we may conclude η(i(H 1 (V (φ))) = 0. Therefore, the integral of the Liouville form on this portion of homology is independent of the choices.
To compute the integral of the Liouville form on cycles which are of the form c {vi,vj } , we give a local description of L(φ) containing the cycle c {vi,vj } . We now need to make an assumption on the size of , so that at each facet U {vi,vj } there exists a point p ∈ U {vi,vj } with a sufficiently small neighborhood We assume that 1 , 2 are chosen to be smaller than this .
When restricted to B (p), the tropical polynomial φ has only 2 domains of linearity, and so the restriction may be written as φ| B = a vi x vi ⊕ a vj x vj . Because of the extra of room from the larger neighborhood B 2 (p), the smoothingφ| B is only dependent on the local combinatorics of our strata. The functionφ| B is invariant with respect to translations in the (v i − v j ) ⊥ hyperplane. This means that the functionφ| B factors asψ(x vi−vj ) whereψ is the smoothing of a tropical function a 0 ⊕ a 1 t. This function has the symmetry We now will construct a cycle representing the homology class [c {vi,vj } ]. Consider the holomorphic cylinder C which is the lift of the line The intersection of C with L(φ)| B is a representative for the class c {vi,vj } . Consider now the cycleĉ {vi,vj } ⊂ C which is given by the intersection C ∩ N * If we can show that η(c {vi,vj } ) = η(ĉ {vi,vj } ) , we will be finished as we will have shown that η(c {vi,vj } ) is independent of choices of α i , β i . The difference between these two quantities c {v i ,v j } )−ĉ {v i ,v j } η is now equated with the symplectic area between these two cycles on the holomorphic cylinder C. We now recenter our coordinates so that p is at the origin. In these coordinates, the symmetry from Equation 4 translates into the odd symmetry of the tropical section The cycle c {vi,vj } ⊂ C also inherits this symmetry. We may use this symmetry to decompose the integral into two equal cancelling components, and conclude that η(c {vi,vj } ) = η(ĉ {vi,vj } ). Proposition 3.19 shows we can take an exact isotopy to bring val(L(φ)) arbitrarily close to V (φ) if desired. A difference between the valuation of a complex hypersurface (an amoeba) and the valuation of L(φ) is that the tentacles of the Lagrangian do not winnow off to zero radius as we increase valuation.

Admissibility of Tropical Lagrangians
In order for us to study this object in the Fukaya category, we will have to show that it satisfies the admissibility conditions of Fuk ∆Σ (X, W Σ ), and show that this Lagrangian is unobstructed.
Proof. To show that L(φ) is admissible, we need show that over each region C α from Definition 2.12, the argument of c α z α is zero outside of a compact set. If we associate the exponent α to a vector α ∈ Z n , the admissibility condition can be restated as the argument of L lying inside the T n−1 α ⊥ subtorus of the fibers F q of (C * ) n over the regions of C α . Both σ −φ and σ 0 are constrained within this subtorus as arg(σ −φ ), arg(σ 0 ) ⊂ T n−1 α ⊥ on the region C α . Proposition 3.7 allows us to conclude that arg(σ 0 # U σ −φ ) = arg(L(φ)) is similarly contained within the region C α . Corollary 4.2. For n = 1, 2, the Lagrangians L(φ) ⊂ (C * ) n bound no holomorphic disks for regular perturbations.
In the general case, we cannot hope for this kind of unobstructedness result. In [21], we show that there exists almost complex structures so that L(φ) bounds disks of Maslov index 0 by finding a mutation structure on Lagrangian submanifolds. See [33,Remark 1.4] for a similar discussion on the presence of holomorphic disks on tropical Lagrangians. Instead, we show that the pearly A ∞ algebra of this Lagrangian cobordism is unobstructed by bounding cochain.

Pearly Model for Lagrangian Floer Theory
To define a bounding cochain, one needs to associate to a Lagrangian L ⊂ X a filtered A ∞ algebra CF • (L). We choose to use the pearly model for the A ∞ algebra, where the Morse A ∞ algebra CM • (L, h) is deformed by insertion of holomorphic disks at the vertices of flow trees [16]. The construction of this algebra has been carried out with a variety of conditions placed for Lagrangians L and symplectic manifolds X [7,9,25]. Unfortunately, the setting which we are interested in -pearly Floer cohomology for non-monotone and non-compact Lagrangian submanifolds -has not to our knowledge been constructed in the literature. Rather, existing definitions of the pearly complex each cover a portion of these requirements.

Existing Constructions and Their Properties:
We give an example of a model where all the properties of the Floer cohomology we will need have been checked, albeit with stricter requirements on the Lagrangian submanifolds than we will have in our application. In [7] and the related work [6], a quantum homology algebra QH • (L, h) is constructed for Lagrangians L equipped with Morse functions h. These Lagrangians L ⊂ X are allowed to be non-compact, provided that the Lagrangians L are monotone and convex at infinity. The Lagrangian Floer theory that they construct is well defined in Lefschetz fibrations, and exhibits a decomposition rule.
The corollary can be used to prove a Gromov-compactness result for W -admissible monotone Lagrangian submanifolds of X, as every pearly trajectory with output on a critical point will have holomorphic disk insertions with boundary contained in W    With these examples from the monotone setting in mind, we take a look at constructions of the pearly model in the non-monotone setting. These constructions are generally more involved, as global geometric perturbations of Floer data are insufficient to achieve transversality of the perturbed Cauchy-Riemann equation. We will look at the approach to the pearly model which achieves regularity of moduli spaces of holomorphic treed disks through the use of domain dependent perturbations.  [9]) Let X be a compact symplectic manifold with rational symplectic form. Let L ⊂ X be an embedded Lagrangian submanifold, and h : L → R a Morse function. There exists a filtered A ∞ algebra CF • (L, h) whose construction involves choices of stabilizing divisor and domain dependent perturbation data. Different choices of perturbation data and divisors produce filtered A ∞ homotopic algebras.
A key ingredient in the construction is the stabilizing divisor D ⊂ X \ L, which has the property that every disk with positive symplectic area and boundary on L intersects the divisor D. This provides interior marked points on every disk with boundary on L, which can be used to stabilize the domain of the disk and construct a domain dependent perturbed Cauchy-Riemann equation. This allows [9] to achieve transversality, as the space of domain dependent perturbations is much larger (and in particular can handle the cases of non-injective points).

Needed constructions and properties of CF • (L, h):
However, to our knowledge, an analogue of Lemma 4.4 has not been proved in the non-monotone setting, nor has stabilizing divisor perturbation method been extended to the non-compact setting. As the construction of this category is beyond the scope of this paper, we will provide a set of properties of the pearly model which we will be using throughout this paper, and an argument sketching how to augment the proof of [9, Theorem 4.1] to these assumptions.   Justification of Assumption 4.11. We outline how the compactness result [9,Theorem 4.27] could be extended to the case where (L, h) is admissible and X is non-compact. Pick a connected subset K ⊂ C such that K ⊂ K , z ∞ ∈ K , and the gradient of h points outward along ∂K. A perturbation system [9, following Definition 4.26] for a combinatorial type Γ of flow tree is a choice of domain-dependent Morse functions and domain-dependent almost complex structures We say that a perturbation system is W -admissible if (in addition to satisfying the requirements of [9, Definition 4.10]) these perturbations are trivial outside of the compact set K , The space of such perturbations is denoted P l Γ (X, D). As outside of the region K the perturbations are not domain dependent, the open mapping principle may still be applied to show that the disk components of treed disks may not escape the region K . As the flow of h points outward of K , we obtain that every W -admissibly perturbed holomorphic treed disk with boundary on L has projection contained within K .
It remains to show that we can extend the results of [9,Theorem 4.19] to find a co-meager subset of W -admissible perturbation data for which the domain-perturbed Cauchy-Riemann equations are regular. The idea, following the original proof, is to consider the space of maps of treed disks and W -admissible perturbation datum, is the space of maps with prescribed incidences at boundaries and interior marked points and W k,p Sobelov regularity. We then consider a Banach bundle E i k,p,l,Γ → B i k,p,l,Γ with a section∂ Γ :    D) is empty, and (therefore trivially!) cut out transversely. It remains to show that this can be extended to a coherent choice of perturbations for the entire moduli space. The construction of a coherent perturbation is iterative, using the order induced on the Γ's by graph morphisms, (see statement of [9,Theorem 4.19]). Whenever Γ ≺ Γ in this ordering, a split labelling on Γ induces a split labelling on Γ . The most important of these cases proves that perturbation produces coherent boundary strata: if Γ is obtain from Γ by gluing at a breaking, either the portion of Γ above the break or the portion of Γ below the break has split label. Since the split label condition is downward closed under the iteration order used to construct coherent perturbation schemes, we can construct a coherent perturbation by initially choosing the trivial perturbation for all the split labels, and then proceeding to choose perturbations for the trees whose labels are not split using [9,Theorem 4.19].
We will use this corollary as a substitute for the stronger statement Corollary 4.5 to prove an analogue of Proposition 4.8.
Justification of Assumption 4.12. The condition that CF • (L + , ∂L + , h| L + ) be an A ∞ ideal (Definition A.3) is that m k (x 1 , . . . , x k ) ∈ CF • (L + , ∂L + , h| L + ) whenever at least one of the x i ∈ CF • (L + , ∂L + , h| L + ). This can be rephrased as m k (x 1 , . . . , x k ), y = 0 whenever {y; x 1 , . . . , x k } is a split label. This is equivalent to the condition that the moduli space of treed disks with split labels is empty.
Remark 4.15. On the abstract perturbation side, the work of [25] and [12] construct methods for regularizing the moduli space of holomorphic disks using polyfolds and Kuranishi structures respectively. Both of these approaches associate to a general compact Lagrangian submanifold L ⊂ X an A ∞ algebra. In the second approach, the A ∞ homotopy class of the algebra is shown to be free of choices and invariant under Hamiltonian isotopy. In both constructions, the A ∞ relations are a result of a coherent choice of abstract perturbations. As in the stabilizing divisor case, these coherent abstract perturbation data are constructed iteratively (see, for instance, [11,Section 6.2] or [25,Section 11.5],) which should allow the strategy of proof outlined above for the domain-dependent case to be extended to these regularization techniques as well. The employment of these techniques to prove Assumption 4.11 and Assumption 4.12 is an ongoing project of the author.
When the Morse function is unimportant, we will simply write CF • (L).

Proof of unobstructedness
As noted before, to prove that the pearly algebra is unobstructed, we will require the extension of the pearly model to the non-compact setting with properties satisfying Assumption 4.11. We will also need the extension of the splitting of Floer theory in symplectic fibrations, as stated by Assumption 4.12. The proof is in three steps. We first describe a geometric relation between L(φ) and L tr (φ), the Lagrangian obtained by taking a surgery between σ 0 and σ −φ after applying a perturbation to make intersections transverse (see Proposition 4.27). We then show that CF • (L(φ)) may be expressed as a quotient of the Floer theory of CF • (L tr (φ)). Finally, we show that CF • (L tr (φ)) is unobstructed by bounding cochain.

Unobstructedness: some tools
Before proving Proposition 4.16, we look at three tools largely independent from the discussion of tropical Lagrangians. The first is a comparison between our tropical Lagrangians and the geometry of symplectic fibrations. The second is a statement about bottlenecked Lagrangians and their bounding cochains. The third is an existence result for bounding cochains on sequences of Lagrangians converging to tautologically unobstructed Lagrangians.

Tropical Symplectic Fibrations.
We now summarize a discussion in [19] relating the monomial admissibility condition (Definition 2.12) the bottleneck condition Definition 4.6 by a construction from [2] called the tropically localized superpotential. Starting with an ample divisor D = v α D α on X with polytope P , the Newton polytope of the support function φ D , Abouzaid constructs a family of superpotentials W t,1 : (C * ) n → C deforming W Σ . The fiber M t,1 := W −1 t,1 (1) has valuation projection val(M t,1 ) which lives close to a tropical variety. Additionally, Q \ val(M t,1 ) has a distinguished connected component P t,1 which can be rescaled to lie close to P . Near the boundary of P , the tropically localized superpotential can be explicitly written as where the ρ α are smooth functions. The functions ρ α (z) only depend on val(z), and are constructed so that Whenever val(z) is outside a small neighborhood of the dual facet of α, φ(z) = 1.
The upshot of working with the tropically localized superpotential is the following. With the usual superpotential, the fiber W −1 Σ (1) should roughly have a decomposition into regions where the subsets of monomials of W Σ codominate the other terms. On each of these regions, W Σ is approximately equal to the dominating monomials. With the tropically localized superpotential the hypersurface M t,1 similarly admits a decomposition, however W t,1 honestly matches the dominating monomials on each region of domination.
The monomial admissibility condition for W = c α z α only requires that each monomial term z α dominates in the region C α after possibly being raised to some power k α . We may assume that the k α are rational, and therefore find an integer N and rescalings c N α of c α defining a new Laurent polynomial Associated to thisW N Σ , we obtain a Newton polytope P N ⊂ Q containing the valuation of points val(W −1 N Σ (B 1 (0))). As we increase N , the polytopes P N scale to cover all of Q. Therefore, we may additionally assume that N is chosen large enough so that a given monomially admissible Lagrangian L satisfies the monomial admissibility condition in a neighborhood of the boundary of val −1 (P N ).
Set W t,1 to be the tropically localized W N Σ . Since the tropically localized superpotential only involves the monomials z α for which val(z) ∈ C α , and z α (L) ∈ R + over the region where L meets val −1 (P N ), we may conclude: See Figure 7a for a diagram of W t,1 (σ −φ ) and W t,1 (L(φ)).

Bottlenecked Lagrangians and Bounding Cochains.
Bottlenecks not only provide a method for producing admissible Lagrangians in the non-compact setting, but they also give a several useful decompositions of the Floer cohomology which can be used to produce bounding cochains. This is where we will employ the expected property of the pearly algebra stated in Assumption 4.12. Here are three observations on bottlenecked Lagrangians and bounding cochains.
Proof. If L + = L z0 × R >0 , the L + can be given a Morse function with no critical points. Proof. Following the justification of Assumption 4.12, pick perturbations for both CF • (L 1 ) and CF • (L 2 ) by first picking perturbations for the non-split labels, and then for the split labels. Since L 1 , L 2 and h 1 , h 2 match completely upon restriction to L − 1 = L − 2 , we can choose regularizing perturbations for non-split labels belonging completely to Crit(h)| L − 1 = Crit(h)| L + 1 so that the moduli spaces of disks exactly match. Therefore, the A ∞ operations agree for CF • (L 1 , h 1 ) and CF • (L 2 , h 2 ) when restricted to inputs which lie entirely in the negative components.
) is unobstructed as well.
Proof. Follows from Lemma A.11, which states that if the domain of an A ∞ map in unobstructed, then the codomain can be unobstructed by a pushforward bounding cochain. Applying this to the projection map from ) proves the proposition.

Eventually Unobstructed Lagrangians and Bounding Cochains.
This is the portion of the proof where we use the continuation map property of Assumption 4.11.
Definition 4.22. Let {L α } α∈N be a sequence of Hamiltonian isotopic Lagrangian submanifolds. We say that this sequence is eventually unobstructed if for every energy level λ, there exists α λ so that β ≥ α λ implies that L β bounds no holomorphic disks of energy less than λ belonging to treed disks contributing to CF • (L β ). Proof. The suspension cobordism K α,β given by the concatenation of our Lagrangian cobordisms correspond to continuation maps f α,β : as defined in Assumption 4.11, which satisfy the property that We note that the valuation of f 0 α+1,α goes to infinity. This is because the valuation of f 0 α+1,α can be bounded below by the energy of the smallest holomorphic disk which occurs in the Hamiltonian suspension cobordism K α,α+1 between L α and L α+1 . By hypothesis of the lemma, the minimal energy of these holomorphic disk goes to ∞ as α goes to ∞.
For Lagrangian L α , we let m k α be the A ∞ structure on CF • (L α ). For a deforming chain d ∈ CF • (L α ), we let (m k α ) d be the deformed curved A ∞ structure. By our assumption, for each λ there is a α λ so that β ≥ α λ implies that val(m 0 β ) will be greater than λ. Given a deformation b α ∈ CF • (L α ) and a filtered A ∞ homomorphism f α,β as above, we get a pushforward map on deformations If b α is a bounding cochain for CF • (L α ), then this pushforward is again a bounding cochain. The same is true for deformations which are bounding cochains up to a low valuation.
In the simplest example, we define b α = (f α,0 ) * (0) ∈ CF • (L 0 ) to be the pushforward of the trivial deformation of L α . This deformation may be rewritten using the quadratic A ∞ relations as The condition that our Lagrangians successively only bound disks of increasing energy means that the m 0 α have increasing valuation, so the sequence of cochains (m 0 ) 0 bα = (f 0 α,0 ) * (0) unobstruct CF (L 0 ) to higher and higher valuations. We now show that this sequence {f 0 α,0 } of deforming cochains converge to an actual bounding cochain.
From the quadratic relation for composition of filtered A ∞ homomorphisms f α+1,0 = f α,0 • f α+1,α we obtain To prove the convergence, it suffices to show that the differences converge to zero (as we are proving convergence in an ultrametric space). As the valuation of the f α,α+1 goes off to infinity, the sequence of cochains (f 0 α,0 ) * (0) converge in CF • (L 0 ).

Unobstructedness: returning to the proof
We now compare the surgery profile defined in Proposition 3.1, and the standard transversal surgery. Proof. We first describe a family of Lagrangians L α 1 . Let U 1 ⊃ U 0 ⊃ U be small collared neighborhoods of U , and let h α : [0, 1] α × U 1 → R be a family of smooth functions satisfying the following: As a section, dh 0 = L 1 on all of U 1 .
As a section, dh α = L 1 on U 1 \ U 0 for all α. dh 1 (x) = (dist q (x)) 2 in a small neighborhood of q. h α is convex for every α.
Let L α 1 be the Lagrangian obtained by removing the portion of L 1 which lives above U 0 , and gluing in dh α instead. Clearly L 1 1 and L 1 are Lagrangian isotopic. By construction L 1 1 and L 0 intersect transversely. For each α, the Lagrangians L 0 and L α 1 have convex intersection region U α . We may construct the surgeries L 0 # U α L α 1 in a smooth family. Since (with appropriate choices of surgery neck) L 0 # U 1 L 1 1 = L 0 # q L 1 1 , we may conclude that there is Lagrangian isotopy between our generalized Lagrangian surgery and the standard transverse surgery.
We will use this comparison for our tropical Lagrangians. We define the set  This follows from observing that the gradient flow of a monomially admissible Morse function at the boundary ∂L − (φ) points outward, and that one can pick Morse function for L(φ) which only has critical points in the overlapping region with L − (φ) and applying Proposition 4.18.
Proposition 4.27. Suppose that φ is a smooth tropical polynomial. Let W t,1 be a tropically localized superpotential so that L(φ)| P N is W t,1 admissible. There exists a monomial admissible Hamiltonian wrapping isotopy (see Definition 2.12) θ so that σ 0 and θ(σ −φ ) have transverse intersections q v for each v ∈ ∆ φ ∩ Z n . Furthermore there exists a Lagrangian L tr (φ) satisfying the following properties: ). Note here that we are only performing Lagrangian surgery at the transverse intersections corresponding to non-self intersection points of φ.
L tr (φ) agrees with L(φ) on the set C 1 .
The Lagrangians L tr (φ) and L(φ) are compared in Figure 7. Since L tr (φ) matches L(φ) on the compact set C 1 , L tr (φ) is similarly bottlenecked by this symplectic fibration at the point z bn = 1. Let L tr,− (φ) and L tr,+ (φ) be the negative and positive ends of the bottleneck. Since L tr,− ⊂ C 1 , we obtain that L tr,− = L − (φ). It remains to prove that L tr (φ) is unobstructed by bounding cochain. We do this by constructing an eventually unobstructed sequence starting at L tr (φ). We now describe a sequence of Hamiltonian isotopic Lagrangian submanifolds {L tr α } α∈N , with L tr 0 = L tr (φ). For notation, we denote the union of two tropical sections which have been made transverse by an infinitesimal wrapping Hamiltonian as L tr ∞ := σ 0 ∪ (θ (σ −φ )). For each v ∈ ∆ Z φ , let q v ∈ L tr ∞ be the corresponding self-intersection point. Around each q v there is a standard symplectic neighborhood B (q v ), which we identify with a neighborhood of the origin in C n . We take a Hamiltonian isotopy of L tr ∞ so that its restriction to each B (q v ) matches R n ∪ iR n . The sequence of Hamiltonian isotopic Lagrangian submanifolds L tr α are constructed by replacing with a standard surgery neck of radius r α . The constants r α are chosen so that lim α→∞ r α = 0. In order to make this a sequence of Hamiltonian isotopic Lagrangian submanifolds, we cancel out the small amount of Lagrangian flux swept out by the surgery necks with an equal amount of Lagrangian isotopy on L tr . These Hamiltonian isotopies are chosen so that L tr By Proposition 4.25, the first member of this sequence L tr 0 can be constructed in such a way that it is Hamiltonian isotopic to L tr (φ). [13] gives us a relation between disks on the L tr (φ) and the disks on σ 0 ∪ (θ (σ −φ )).

Idea of Proof
The proof follows the methods used in [30,Section 6], [32,Theorem 1.2], or [13], which all make comparisons between disks with boundary on the surgery to polygons with boundary on transversely intersecting Lagrangians. Let {u α } be a sequence of holomorphic disks of bounded energy and boundary on L tr α contributing to CF • (L tr α ). Then the images of the {u α } are mutually contained within a compact set of X. We would like to apply a Gromov-compactness argument on the sequence of u α but cannot as the family L tr α does not converge to L tr ∞ in a strong enough sense. However, it is the case that In [13,Section 62] it is shown that for such a sequence of disks u α : (D 2 , ∂D) → (X, L tr α ), one may construct a family of approximate solutions u α,app : (D 2 , ∂D) → (X, L tr ∞ ) by replacing the regions of the curve u α which intersect B (q v ) with holomorphic corners based on a standard model from [13,Section 59].
The following neck stretching argument is used to show that these approximate solutions approach an honest solution. As α → ∞, the restriction . As a result, the holomorphic maps {u α } converge to cylindrical maps in the neck region B (q v ) \ {q v } [13,Section 62.4]. This provides an error bound on the failure of u α,app to being a holomorphic polygon. As the {u α } converge to cylindrical maps this error approaches zero.
Since the {u α } have images confined a compact set of X,the maps {u α,app } are similarly constrained. We can apply Arzela-Ascoli to take a subsequence of {u α,app } which converge to a holomorphic map u ∞ with boundary on L tr ∞ . In this case, we can rule out the existence of holomorphic polygons with boundary on L tr ∞ .
Proof. This follows from an index computation. A holomorphic polygon with boundary contained in σ 0 ∪ (θ (σ −φ )) has 2k − 1 inputs and 1 output. The inputs must alternate between being an element of CF • (σ 0 , θ (σ −φ )) and CF • (θ (σ −φ ), σ 0 ). We will look at the case where output p lies in p ∈ CF • (σ 0 , θ (σ −φ )) and the inputs x i , y j are in The dimension of moduli space of regular polygons with these boundary conditions can be explicitly computed based on the index of the points x i and y j . The degree of the input intersections of the form x i is n, and the degree of each intersection of the form y j is 0. The output intersection p has degree n. The dimension of this space of disks is which is negative whenever n ≥ 2.
The argument for when the output marked point p is in CF • (θ (σ −φ ), σ 0 ) is the same.  We additionally need to prove that the Lagrangians K tr α,α+1 ⊂ X × C given by the suspension of the Hamiltonian isotopy between L tr α and L tr α+1 are an eventually unobstructed sequence. This follows from the same argument. A sequence of holomorphic disks with boundary on K tr α,α+1 produces a holomorphic disk with boundary on K ∞ = L tr ∞ × R. Since L tr ∞ × R is a trivial cobordism and the complex structure was chosen to be the standard product structure, every holomorphic disk with boundary on L tr ∞ × R gives us a holomorphic disk with boundary on L tr ∞ . By Proposition 4.30, there are no such disks. Therefore, the Lagrangian cobordisms K α,α+1 are eventually unobstructed.
As both {L tr α } α∈N and {K tr α,α+1 } α∈N are eventually unobstructed sequences of Lagrangians, it follows from Lemma 4.23 that L tr 0 = L tr (φ) is unobstructed, completing the proof of Proposition 4.16.
Remark 4.31. Note that in dimension 1, the Lagrangian sections θ (σ −φ ) ∪ σ 0 may still bound interesting holomorphic disks. In dimension 1, see Figure 8a for an example of a disk which has boundary on tropical sections. We now provide some evidence that these disks correspond to higher genus open Gromov-Witten invariants of the tropical Lagrangian. In the 1 dimensional example, the disk in Figure 8a becomes a holomorphic annulus with boundary on L(x 2 1 ). We can replicate this phenomenon in higher dimensions. Let φ E (x 1 , x 2 ) be the tropical polynomial describing a tropical elliptic curve V (φ E ) as drawn in Figure 8b. Then the Lagrangian L(φ) bounds holomorphic annuli which are modelled on the previous example in one dimension higher. See Figure 8b for this example (in red) and an example of holomorphic genus 0 curve with four boundaries on L(φ E ). At this point, it is unclear what the presence of these higher genus open Gromov-Witten invariants entail.

Homological Mirror Symmetry for L(φ)
In this last section we look at some applications of our construction to homological mirror symmetry. Our Lagrangian submanifolds will have the additional structure of a Lagrangian brane, meaning that they are equipped with a choice of Morse function, spin structure, and bounding cochain. To simplify notation, we will often refer to data of a Lagrangian brane by the Lagrangian submanifold L. Since the Lagrangians L(φ) that we study are not exact, they do not fit into the framework of [19], and additionally we are required to work with the Fukaya category defined over Novikov coefficients.
Assumption 5.1. We assume that the monomially admissible Fukaya Seidel category can be extended to include unobstructed Lagrangian submanifolds, and that the appropriate analogues of Theorem 2.10 and Theorem 2.14 hold in this setting.
The mirror to the Landau-Ginzburg model (X, W Σ ) is the rigid analytic spaceX Λ Σ . The intuition for our constructions should be understood independently of the requirements of Novikov coefficients.
The Novikov toric varietyX Λ Σ comes with a valuation map val :X Λ Σ → Q using the valuation on the Novikov ring. A difference between complex geometry and geometry over the Novikov ring is that in the Figure 9: SYZ mirror symmetry predicts that Lagrangians are swapped with complex subvarieties by fiberwise duality over a tropical curve in the base.
non-Archimedean setting the valuation of a divisor is described exactly by its tropicalization, as opposed to living in the amoeba of the tropicalization. We first give a family Floer argument motivating this mirror statement. From SYZ mirror symmetry we know that the mirror to a point in the complement of the anticanonical divisor z ∈X Λ Σ \ Df is a fiber of the SYZ fibration equipped with local system. One method to compute the mirror sheaf to L(φ) is to compute CF • (L(φ), F q ) and assemble this data into a sheaf over X using techniques from family Floer theory. This line of proof is rooted in a long-known geometric intuition for mirror symmetry via tropical degeneration (see Figure 9). However, the precise computation of the support is difficult due to the need to count holomorphic strips contributing to the Floer differential. In [21], we compute the support in a few fundamental examples.

Mirror Symmetry for Tropical Lagrangian hypersurfaces
of Theorem 5.2. We use Lagrangian cobordisms to prove this theorem. The functionf associated to the effective divisor Df ⊂X defines a section of the line bundle OX Σ (Df ), giving us an exact triangle This gives us a description of O Df in terms of line bundles onX Σ . By Theorem 2.14, we have an identification of Fuk((C * ) n , W Σ ) with D b Coh(X Λ Σ ) giving us the following mirror correspondences between sheaves and Lagrangian submanifolds: where φ D is the support function of the divisor D. The Lagrangians σ −φ and σ −φ [D] are Hamiltonian isotopic.
Using an extension of Theorem 2.10 to the unobstructed setting, we obtain an exact triangle for some mapǧ. L(φ) is therefore identified under the mirror functor to a sheaf O Dǧ supported on Dǧ, an effective divisor for the bundle O(D)f . The divisors Dǧ and Df are rationally equivalent.
If we wish to prove that Dǧ and Df match up exactly, we need to better understand the mapǧ in Equation 6. Though we cannot determine this map without making a computation of holomorphic strips with boundary on the cobordism K, we conjecture

Twisting by Line Bundles
Let ψ : Q → R be a piecewise linear function, and let θ ψ be the time 1 Hamiltonian flow associated to the pullback of the smoothing,ψ • val : X → R. This Hamiltonian flow can be compared to fiberwise sum [35] with the section σ ψ θ ψ (L) = L + σ ψ .
While wrapping is not an admissible Hamiltonian isotopy, it still sends admissible Lagrangian branes to admissible Lagrangian branes, giving an automorphism of the Fukaya category. Provided that V (φ) and V (ψ) have a pair of pants decompositions with no codimension 2 strata intersecting, the Lagrangian L(φ) + σ ψ can be given an explicit description in terms of the pair of pants decomposition. We outline this construction in complex dimension 2, but the higher dimensional constructions are analogous. When V (φ) and V (ψ) have locally planar intersection, the support of V (ψ) is contained in the cylindrical region between each of the pants in the decomposition of V (φ). This means that if the smoothing and construction parameters for the tropical Lagrangians are chosen small enough, the strata are disjoint from one another. Therefore, the Lagrangian L(φ) matches L(φ) + σ ψ over the charts near the vertices of the tropical curve, U φ {vi,vj ,v k } . To construct L(φ) + σ ψ from this pair of pants decomposition, it suffices to modify the cylinders living over regions U φ {vi,vj } . We construct this modification in a local model where φ = 0 ⊕ x 1 and ψ = 0 ⊕ x 2 . Topologically L(φ)+σ ψ | U φ {v i ,v j } is a cylinder, with an additional twist in the argument direction perpendicular to V (ψ) at the point of intersection between the two tropical varieties, as drawn in Figure 10a, which shows L(φ E ) + σ φpants , where V (φ E ) is the tropical elliptic curve, and V (φ pants ) is a tropical pair of pants meeting the tropical elliptic transversely at 3 points. This kind of modification to our tropical Lagrangian was remarked upon in [27, Remark 5.2] as a more general way to construct tropical Lagrangians. This discussion shows that if L(φ) is mirror to O D , then the twisted Lagrangian L(φ) + σ ψ is mirror to O D ⊗ L ψ . This can also be understood as the mirror to the pushforward of the pullback of L ψ .
From the pair of pants description, it is clear that we can "twist" our Lagrangian in the argument along edges in ways that do not arise from adding on a section σ ψ -see for instance, Figure 10b. These too should be mirror to pushforwards of line bundles on D, however these line bundles are not the pullbacks of line bundles on X.
This local twisting can be defined more rigorously by working with the sheaf of affine differentials. On U ⊂ R n these are the sections σ : U → val −1 (U ) which are locally described as the differential of tropical polynomials. However, such a section need not be defined globally as the differential of a tropical polynomial. As an example, in the mirror to CP 2 we consider the open set U as drawn in Figure 10b. There exists a tropical differential on U whose critical locus intersects the tropical elliptic curve V (φ E ) at a single point.  The Lagrangian given by twisting along this tropical differential is expected to be mirror to the direct image of a degree 1 line bundle on E, an elliptic curve whose tropicalization is V (φ E ). We expect that we can understand these twistings by employing tropical geometry on the affine structure of val(φ E ) itself. This tropical differential does not extend to a section over the entire base, as degree 1 line bundles on E do not arise from pullback of a line bundle on CP 2 .
Conjecture 5.6. The twisted tropical Lagrangians L(φ; σ) are mirror to the direct image of line bundles on the mirror divisor D.

A A ∞ algebras and Bounding Cochains
In this appendix we review some statements on filtered A ∞ algebras. In Subsection A.1, we fix notation for filtered A ∞ algebras. Subsection A.2 looks at properties of filtered A ∞ homomorphisms, and Subsection A.3 reviews the definition and construction of bounding cochains.

A.1 Notation: Filtered A ∞ algebras
We review curved A ∞ algebras. In order to ensure convergence of the deformations we develop, we work with filtered A ∞ algebras. This will mean working over the Novikov field.
Definition A.1 ( [15]). Let R be a commutative ring with unit. The universal Novikov ring over R is the set of formal sums Let k be a field. The Novikov Field is the set of formal sums The valuation of an element is the smallest exponent of T which appears with non-zero coefficient. An energy filtration on a graded Λ-module A • is a filtration F λi A k so that Each A k is complete with respect to the filtration, and has a basis with zero valuation over Λ.
Multiplication by T λ increases the filtration by λ.
The energy filtration will play an important role in the algebraic setting where many of our constructions will either induct on the energy filtration.
Definition A.2. Let A • have an energy filtration. A filtered A ∞ structure (A, m k ) is an enhancement of A • with Λ ≥0 linear graded higher products for each k ≥ 0 satisfying the following properties: Energy: The product respects the energy filtration in the sense that : Non-Zero Energy Curvature: The obstructing curvature term has positive energy, m 0 ∈ F λ>0 C.
We say that (A, m k ) is uncurved or tautologically unobstructed if m 0 = 0.
For the purposes of exposition, we will work up to signs from here on out.
Definition A.3. Let A be a filtered A ∞ algebra. An ideal of A is a subspace I ⊂ A so that for every b ∈ I and a 1 , . . . , a k−1 ∈ A, m k (a 1 ⊗ · · · ⊗ a j ⊗ b ⊗ a j+1 ⊗ · · · ⊗ a k−1 ) ∈ I.
Note that we do not require m 0 ∈ I.
The quotient of an A ∞ algebra by an ideal is again a filtered A ∞ algebra. Given (A, m k ) a filtered A ∞ algebra, define the positive filtration ideal A >0 := {a ∈ A : val(a) > 0}.
We may recover an uncurved A ∞ algebra by taking the quotient, This is always an uncurved as the m 0 term is required to have positive valuation.

A.2 Morphisms of filtered A ∞ algebras
The definition of morphisms between filtered A ∞ algebras is similar to the definition of morphisms of differential graded algebras, except that the homomorphism relation is relaxed by homotopies. for k ≥ 0 satisfying the following conditions:

Filtered:
The maps preserve energy Quadratic A ∞ relations: The f k , m k A and m k B mutually satisfy the quadratic filtered A ∞ homomorphism relations for k ≥ 0 (j1+j+j2=k) j,j1,j2≥0 ±m l B (f i1 ⊗ · · · ⊗ f i l ) Proposition A.5. Let f k : A ⊗k → B and g k : B ⊗k → C be two filtered A ∞ homomorphisms. Then (g • f ) k := j1+···+j l =k ji≥0 g l (f j1 ⊗ · · · ⊗ f j l ) is an A ∞ homomorphism.

A.3 Deformations of A ∞ algebras
The presence of higher product structures gives us additional wiggle room to deform the product structures on a filtered A ∞ algebra. We will be mainly interested in the case when we can deform a given filtered A ∞ algebra into an uncurved one to obtain a well defined cohomology theory.
Notation A.6. As a shorthand, we write for the sum over all monomials containing n + k terms, n of which are a and k for which are id.
Definition A.7. Let a ∈ A be an element of positive valuation. Define the a-deformed product m k a : A ⊗k → A by the sum m k a := n m k+n (id ⊕a) ( n+k n ) a .
We call this a graded deformation if the element a has homological degree 1.
The convergence of this sum is guaranteed by the positive valuation of the deforming element.
Proposition A.8. (A, m k a ) is again a filtered curved A ∞ algebra. We are interested in the cases where (A, m a ) gives us a well defined homology theory even though A itself may be curved.
Definition A.9. We say that a ∈ A is a bounding cochain or Maurer-Cartan solution if m 0 a = k m k (a ⊗k ) = 0.
If A has a bounding cochain, we say that A is unobstructed.
When m 0 = 0, when we say that A is tautologically unobstructed or uncurved. In the unobstructed setting, we have a well defined cohomology theory of (A, m k b ). Definition A. 10. Let A be an A ∞ algebra. The space of Maurer-Cartan elements is defined as The Maurer-Cartan equation is non-linear. In the event that the Maurer-Cartan space contains a linear subspace, then 0 is a Maurer-Cartan element and the algebra A is uncurved.
Lemma A.11. Let f : A → B be a filtered A ∞ morphism. Then there exists a pushforward map between the bounding cochains on A and the bounding cochains of B given by Proof. We want to show that b B := k f k (b ⊗k A ) satisfies the Maurer-Cartan equation Surprisingly, deformations commute with each other in the following sense: is an A ∞ homomorphism.
One may use the previous two claims to construct the pushforward map on bounding cochains, as This, along with the statement on the pushforward of a bounding cochain, proves the following characterization of unobstructed A ∞ algebras.