Gromov-Witten Invariants and Mirror Symmetry for Non-Fano Varieties Using Scattering Diagrams
Per Berglund Tim Gräfnitz Michael Lathwood


	GW invariants and mirror symmetry for non-Fano toric varieties]Gromov–Witten invariants and mirror symmetry for non-Fano varieties using scattering diagrams



University of New Hampshire 
 Department of Physics and Astronomy 
 105 Main St, Durham, NH 03824 
 USA 
per.berglund@unh.edu


Leibniz-Universität Hannover 
 Institut für Algebraische Geometrie 
 Welfengarten 1, 30167 Hannover 
 Germany 
graefnitz@math.uni-hannover.de


University of New Hampshire 
 Department of Physics and Astronomy 
 105 Main St, Durham, NH 03824 
 USA 
michael.lathwood@unh.edu


	
	
	
	       	 
    Gromov–Witten invariants arise in the topological A-model as counts of worldsheet instantons.
    On the A-side, these invariants can be computed for a Fano or semi-Fano toric variety using generating functions associated to the toric divisors.
    On the B-side, the same invariants can be computed from the periods of the mirror.
	We utilize scattering diagrams (aka wall structures) in the Gross–Siebert mirror symmetry program to extend the calculation of Gromov–Witten invariants to non-Fano toric varieties.
    Following the work of Carl–Pumperla–Siebert, we compute corrected mirror superpotentials ϑ_1(_m) and their periods for the Hirzebruch surfaces _m with m≥ 2.
	
	
	
	[
    Michael Lathwood
    April 29, 2024
====================


	



§ INTRODUCTION

In string compactifications on a Calabi-Yau 3-fold Y, D-branes can wrap non-trivial cycles in Y and this gives rise to a class of 𝒩=1 supersymmetric theories in 4 spacetime dimensions <cit.>. 
A brane which wraps a space-like cycle  in the target spacetime ^1,3× Y is called an instanton. 
The worldvolume of such a brane is localized in the time-direction of ^1,3 (i.e. only appears for an “instant"), hence the name. 
Open Gromov–Witten invariants are the counts of open worldsheet instantons, which are described mathematically by holomorphic maps from a disk to X with the boundary ending on the D-brane.

A local Calabi-Yau 3-fold is given by the canonical bundle K_X over a surface X. The zero section of K_X is isomorphic to X and its normal bundle is equal to K_X. When X is Fano, this is negative, so that curves on the zero section can not deform away from the zero section. This suggests a relation between the open Gromov-Witten invariants of K_X and certain relative (or logarithmic) Gromov-Witten invariants of X. This is made precise by the log-local correspondence of <cit.> combined with the open-closed correspondence of <cit.>. However, this is no longer true in the non-Fano case, because curves can move away from the zero section. In this paper we will study both, open invariants of K_X and logarithmic invariants of X. We will describe different techniques of computing them, including tropical geometry, and study their corresponding objects under mirror symmetry.

Mirror symmetry is a duality of Calabi-Yau manifolds Y and Y̌ such that Type IIA string theory compactified on Y is equivalent to Type IIB string theory compactified on Y̌. Mathematically, the A-model is concerned with symplectic geometry and counts of holomorphic curves, which can be rigorously defined as Gromov-Witten invariants. The B-model is concerned with complex geometry and parameters of the complex structure described by period integrals, pairings of the Kähler form with generators of the group of 1-cycles. Mirror symmetry implies that Gromov-Witten invariants of Y can be computed from the period integrals of Y̌. The latter are usually easier to compute and hence mirror symmetry gives a powerful tool for computing Gromov-Witten invariants. For example, in <cit.> physicists used this method to compute Gromov-Witten invariants of a smooth quintic threefold up to high degree (only limited by computation power). This raised the interest of mathematicians in mirror symmetry, because before these numbers had only be known up to degree 4.

Instead of Calabi-Yau manifolds Y one can consider log Calabi-Yau pairs (X,D), where log Calabi-Yau means that D is an anticanonical divisor of D. We are interested in the case of a smooth divisor D. The “maximal boundary” case with nodal singular divisor has been studied elsewhere <cit.><cit.>. The mirror to a log Calabi-Yau pair is a Landau-Ginzburg model W : X̌→ℂ, that is, a non-compact manifold X̌ and a superpotential function W with compact critical locus. If X is a toric Fano variety, then W is given by the Hori-Vafa potential W_Σ=∑_ρ∈Σ_1z^ρ, a sum of monomials z^ρ corresponding to the rays ρ of the fan Σ of X, with relations among the monomials corresponding to relations among the rays. If X is a non-toric Fano variety, then W is given by the Hori-Vafa potential W_Σ of a toric model X_Σ of X, which means there is a ℚ-Gorenstein deformation between X and X_Σ. We will see in <ref> that in the non-Fano case the superpotential W receives correction terms from the existence of special holomorphic (or tropical) disks. This was noticed in <cit.> and interpreted tropically in <cit.>. In the tropical language, W is the same as the primitive theta function ϑ_1 in the central chamber of a scattering diagram. ϑ_1 is defined as a sum over broken lines, which correspond to 2-marked tropical curves and can be seen as a tropical version of holomorphic disks of Maslov index 2.

The period integrals can be defined as classical periods (Definition <ref>) of the potential W=ϑ_1 or as solutions to Picard-Fuchs type differential equations defined by the Mori vectors (= relations among the rays of the fan). There are three different potential functions one can consider:


  * The Hori-Vafa potential W_Σ.

  * The Hori-Vafa potential of the elliptic fibration over X_Σ given by the canonical bundle K_X_Σ.

  * The corrected potential W=ϑ_1 defined by broken lines.

In the toric case they are all equivalent and their periods are, via mirror symmetry, generating functions of relative (or logarithmic) Gromov-Witten invariants of X or, equivalently, up to some factor, of open Gromov-Witten invariants of K_X. In the non-Fano case, they are not equivalent, and the open-log correspondence of Gromov-Witten invariants does not hold. We conjecture that (2) still computes open Gromov-Witten invariants of K_X and show that (3) still computes logarithmic Gromov-Witten invariants of X. Moreover, we show that the corrected potential W=ϑ_1 is equal to the open mirror map after a change of coordinates given by the closed mirror map, and conjecture that the Newton polytope of ϑ_1 gives a toric model of X and we study this (using the language of mutations) for Hirzebruch surfaces 𝔽_m. To summarize, our main statements and conjectures are the following.


A toric variety X_Σ is a toric model of X if there is a ℚ-Gorenstein deformation between X and X_Σ.


[Conjecture <ref>]
X has toric model X_Σ, where Σ is the spanning fan of the Newton polytope of ϑ_1, where ϑ_1 is defined in an arbitrary chamber. 

Moreover, two (not necessarily Fano) varieties are ℚ-Gorenstein deformation equivalent if and only if their potentials ϑ_1 are mutation equivalent.



The open mirror map is q_0=z_0e^F_W(z), where F_W(z) is the x,y-constant term of log(1-t^-1W),

    F_W(z) = ∑_k>01/k_1(W^k)t^-k.

The closed mirror map is q_i=z_ie^d_iF_W(z), where d_i=β_i· D. Let M_W(q):=e^F_W(z(q)) be the open mirror map after inserting the inverse of the closed mirror map.



The Gross-Siebert slab function is the unique function h_W(z) such that the Gross-Siebert potential

    W_GS = W + h_W(z)

has no constant term. Write h_W(z(q)) after insertion of the inverse of the closed mirror map.


[Theorem <ref>, Theorem <ref>, Proposition <ref>]

We have

    M_ϑ_1(q) = 1+h_ϑ_1(z(-q)) = ϑ_1(y)_∞/y = 1+∑_β∈ NE(X) (β· D-1)R_β· D-1,1(X,β)q^β,

where R_β· D-1,1(X,β) are 2-marked logarithmic Gromov-Witten invariants of (X,D). Here y is related to q by the change of variables q_i=z_i(t/y)^d_i, with d_i=β_i· D.


[Proved in Example <ref>]
    There are two solutions to the recursion relations for the Picard-Fuchs system for K__3.
    
    F_1(z_1,z_2)   =∑_n_1,n_2≥ 0
     n_1≥ 3n_2 (-1)^n_2Γ(2n_1-n_2)/Γ(n_1)Γ(n_1-3n_2+1)Γ^2(n_2+1)z_1^n_1z_2^n_2
    
        	F_2(z_1,z_2)   =∑_n_1,n_2≥ 0
     n_1≤ 3n_2 (-1)^n_1+n_2Γ(3n_2-n_1)/Γ(n_1+1)Γ(n_2-2n_1+1)Γ^2(n_2+1)z_1^n_1z_2^n_2

    The logarithmic solutions are given by t_a=log z_a+S_a(z_1,z_2) where S_1=2F_1-F_2 and S_2=-F_1+3F_2.
    In the language of <cit.>, F_1 is the generating function associated to the canonical divisor and F_2 is the generating function associated to negative the exceptional divisor -E=:E. The open invariants can be obtained from the generating function S_0=-F_1 (displayed in Example <ref>).


We conjecture an analogue of Theorem <ref> to hold in the local setting with open Gromov-Witten invariants. For toric Calabi-Yau 3-folds this was shown in <cit.><cit.><cit.>.

[Conjecture <ref>]

Let Σ' be the fan of K_X_Σ, where X_Σ is a toric model of X. Then

    M_W_Σ'(-q) = 1+h_W_Σ'(z(q)) = 1+∑_β∈ NE(X) N_1(K_X,β)q^β,

where N_1(K_X,β) are winding 1 open Gromov-Witten invariants of K_X.


The paper is organized as follows. 
In <ref> we define toric log Calabi-Yau pairs, toric models, toric degenerations, scattering diagrams, and broken lines. 
We then define a Landau-Ginzburg potential through broken lines. 
In <ref>, we describe the open/closed Picard-Fuchs system following <cit.>.
We then solve this system for K__3 and remark on the case of an elliptic fibration over _3.
The expectation is that elliptic fibration should reproduce the canonical bundle in the large elliptic fiber limit.
In <ref>, we show that these B-side Picard–Fuchs calculations are consistent with the A-side curve counting calculations.
In particular, one needs to take into account a generating function relative to negative the exceptional divisor on the a side, since it will have effective curves for non-Fano _m.
In <ref> we use tropical geometry to show that the theta potential ϑ_1 is a generating function for 2-marked logarithmic Gromov-Witten invariants. This is the last equality of Theorem <ref>.
In <ref> we study the theta potential ϑ_1 inside a central chamber of the scattering diagram and show that it is a correction of the Hori-Vafa potential W_Σ, with correction terms coming from interntal scatterin.
We explicitly compute ϑ_1 for the first Hirzebruch surfaces 𝔽_m and discuss the role of mutations for toric models and more general computations.
In <ref> we show that the open mirror map M(q) and the Gross-Siebert slab function h(z(q)) both equal the theta potential ϑ_1. These are the other equalities of Theorem <ref>.



 §.§ Acknowledgements

PB and ML are supported in part by the Department of Energy grant DE-SC0020220.
During the Concluding Conference of Simons Collaboration on Homological Mirror Symmetry at SCGP, Helge Ruddat suggested that ML should reach out to TG, which lead to this collaboration.
We thank Helge Ruddat for bringing us together.



§ GEOMETRIC SETUP


We first set notation and define the geometric objects for which we will compute Gromov–Witten invariants. 



 §.§ Toric log Calabi–Yau pairs

A log Calabi–Yau pair (X,D) consists of a smooth projective variety X and a reduced normal crossing divisor D such that D+K_X=0, i.e., D is an anticanonical divisor. There might be less restrictive definitions, e.g. in <cit.> the authors consider the case where D+K_X is numerically equivalent to an effective ℚ-divisor supported on D, but for us this will be the setup. Moreover, we will only consider the case where D is smooth. The “maximal boundary” case with nodal singular divisor has been considered e.g. in <cit.> and <cit.>.

If X is a toric variety, then D has the same class as the toric boundary ∂ X=∑_a D_a, where D_a are the toric divisors. An ample polarization of X gives an embedding into a projective space. This can be described by a polytope Δ and we write X=.
We write n=_ X so that _ = _ = n. 
When thinking of  as a symplectic manifold, the Newton polytope is the moment polytope, namely it is the image of the moment map μ:⟶𝔤^* where 𝔤^*≅^n is the dual Lie algebra of the dense algebraic torus G=T:=(^×)^n in . 
When  is reflexive it is also known as a Delzant polytope.

The combinatorics of  and hence  are encoded in the normal fan Σ, which is constructed as follows. 
The 1-dimensional cones ρ_i∈Σ^(1)⊂Σ are rays normal to[By convention, we choose the inward pointing normals, i.e (ν | m_1 | … | m_n-1)<0 where ν is the normal to the facet spanned by the m_i.] the codimension-1 facets of .
We write r=|Σ^(1)| for the number of rays in the fan.
One can construct the higher dimensional cones σ_i_1… i_k∈Σ^(k)⊂Σ from the rk combinations of the ρ_i bounding a k-simplex.
The primitive vectors that point along the ρ_i form the vertices v_i of the spanning polytope , which is the unique compact top dimensional cone ∈Σ^(n). 

To construct the spanning polytope for the canonical bundle K_, we take the cone over  by placing  in a hyperplane of height 1 above the origin in ^n+1

    Cone()={(cν,c)  | ν∈ , c∈_≥0}⊂^n×


Let v_i∈ be the vertices of the spanning polytope and set v_i=(v_i,1)∈×{1}⊂Cone().
Define the vectors {ℓ^(a)}_a=1^h^1,1() from the linear relations of the v_i

    ∑_i=0^r ℓ^(a)_iv_i=0

where v_0=(0,…,0)∈^n.
By including v_0 we are restricting to the anticanonical hypersurface X↪, and the Calabi-Yau condition is equivalent to ∑_iℓ^(a)_i=0.
Then ℓ^(a) are the generators of the Mori cone, which is dual to the Kähler cone of .
The Mori vectors ℓ^(a) allow us to determine several objects associated to , including but not limited to, the Picard–Fuchs operators (see Equation <ref>), the fundamental period (see Equation <ref>), and the invariant (algebraic) coordinates on the mirror complex structure moduli space z_a∈() <cit.>

    z_a=(-1)^ℓ^(a)_0∏_i=1^r a_i^ℓ^(a)_i.

Here the a_i are coordinates on () that enter the Hori–Vafa superpotential[Here Y^v_i=Y_1^v_i^1… Y_n^v_i^n and v_i=(v_i^1,…,v_i^n)∈∩^n.] <cit.>

    W=∑_i=0^r a_iY^v_i.

We can view W:(^*)^n→ as a torus fibration and compute its period

    ϖ(W)=(1/2π i)^n∫_Γ_0dlog Y_1∧…∧ dlog Y_n/a_0+W
 
One can construct W purely from the vertices v_i of the spanning polytope  associated to the toric variety , and
for this reason the additive relations of the v_i are exactly the multiplicative relations of the terms of W.
We perform a power series expansion on the integrand so that, due to the form of the differential, the residues are given by the terms that are constant with respect to the Y_i.

    ϖ(W)   = (1/2π i)^n∫_Γ_0 (dY_1∧…∧ dY_n/Y_1… Y_n)1/a_0+W
       =∑_k=0^∞(-W/a_0)^k|_Y_i const.
       =∑_k=0^∞(-a_0)^-k∑_ν_1+…+ν_r=k(k!/ν_1!…ν_r!(a_1Y^v_1)^ν_1…(a_rY^v_r)^ν_r)|_Y_i const.
       =∑_k_1 ≥ 0…∑_k_s ≥ 0∏_jΓ(-∑_aℓ_0j^(a)k_a+1)/∏_iΓ(∑_a ℓ_i^(a) k_a+1)z_1^k_1… z_s^k_s


Each vertex v_i∈ corresponds to a prime toric divisor D_i⊂.
The anticanonical divisor is given by the sum of all the toric divisors -K_=∑_i=1^r D_i.
Let s be the number of linearly independent divisors so that the Picard group Pic() has s generators and hence Pic()≅^s.
The linear relations among the v_i generate the Stanley–Reisner ideal ℑ of the Chow ring A_*(), and the toric variety can be constructed as the quotient space

    =^r\ Z_ℑ/Hom(Pic(),^*).

Here Z_ℑ is the variety associated to the 
Stanley–Reisner ideal ℑ. 
One can use ℑ to determine the intersection numbers of the D_i.
[_=^2]

	The spanning polytope for 2-dimensional complex projective space is given by the convex hull of the following 3 vertices, which contains the origin v_0 as its unique interior point.

    _^2=Conv{(0,0),(1,0),(0,1),(-1,-1)}=Conv{v_i}_i=0^3

The spanning polytope for the canoncial bundle over ^2 is given by the cone over _^2

    _K_^2=Conv{(0,0,0),_^2×{1}}

The vertices satisfy the linear relation

    -(v_0,3)+(v_1,1)+(v_2,1)+(v_3,1)=0

We write this relation in terms of the Mori vector ℓ=(-3;1,1,1).
Since there is only one linearly independent relation among the v_i, we have s=1 and Pic(^2)≅ is generated by the class of the line L (hyperplane class). The anticanonical divisor can be thought of as the divisor -D_0 corresponding to the unique interior point (v_0,1)∈×{1}. This is given by -K_^2=-D_0=3L. By Bézout's theorem L^2=1 so the degree of ^2 is (-K_2)^2=9.


[=_m]

     The Hirzebruch surfaces _m=(Ø_^1(m)⊕) are an infinite sequence of toric surfaces which can be thought of as twisted ^1 bundles over ^1, with the parameter m controlling the twisting of the fibers.
     The spanning polytope _m :=__m for _m has the following 4 vertices, see Figure <ref>.

    v_1=(-1,0)   ,   v_2=(1,0)   ,   v_3=(0,1)   ,   v_4=(-m,-1)

For the m>2, we find that _m is not reflexive. The vertices satisfy the linear relations

    ℓ^(1)=(       -2;       1,        1,        0,        0)

    ℓ^(2)=(        -(2-m);        -m,        0,        1,        1)

so for m>2 we have ℓ^(2)_0>0. If E is the exceptional divisor class, F is divisor class corresponding to the fiber, and S is the divisor class corresponding to a section, we have

    E^2=-m    ,    F^2=0    ,    S^2=m

in the Chow ring A_*(_m). 
Since there are two linearly independent relations between the vertices of _m, we know Pic(_m)≅^2. 
We will use both {S,F} and {E,F} as a basis for Pic(_m), with the two bases being related by E=S-mF. 
The anticanonical divisor is given by

    -K__m=2S+(2-m)F=2E+(2+m)F

so that the degree of _m is (-K__m)^2=4m+4(2-m)=8, independent of m.
 




[=^2_(1,1,q)]
	The weighted projective spaces ^2_(1,1,q) will appear in the central fibers of the toric degenerations that we will consider.  
	As a quotient space,
	
    ^2_(1,1,q) = (^*)^3∼       where      [x_0,x_1,x_2]∼ [λ x_0, λ x_1, λ^q x_2] , λ∈^*

	The vertices {v_i}_i=1^3 of the spanning polytope for ^2_(1,1,q) satisfy
	
    v_1+v_2+qv_3=0.





 §.§ Toric models





A toric variety X_Σ is a toric model of a variety X if there is a ℚ-Gorenstein deformation between X and X_Σ.



A toric model of a cubic surface X is given by the blow up of ℙ^2 in six points that lie on a conic. Such a blow up can be performed torically, resulting in the fan shown in Figure <ref>.






A Hirzebruch surface 𝔽_m is a smooth toric variety, but it is not Fano. A toric model that is Fano is given by 𝔽_0=ℙ^1×ℙ^1 if m is even and by 𝔽_1=Bl ℙ^2 if m is odd. This is clear, because there is a smooth and flat family

    ℱ = {x_0^my_1-x_1^my_0+tx_0^m-kx_1^ky_2=0}⊂ℙ^1_x_0,x_1×ℙ^2_y_0,y_1,y_2×𝔸^1_t,

such that the fiber over t=0 is 𝔽_m and the general fiber is 𝔽_m-2k. The family leaves the anticanonical polarization unchanged (i.e. it is ℚ-Gorenstein) and acts on curve classes by

    𝔽_m    →𝔽_m-2k, 
    
    F    ↦   F, 
    
    E    ↦  E-kF, 
    
    S    ↦  E+kF.

Here F, E and S are the classes of a fiber, the exceptional section, and a general section, respectively.



The toric models of a Fano variety are related via mutations of their spanning polytopes, see Proposition <ref>. In <ref> we will study a similar relation for non-Fano varieties and mutations of their potentials.




 §.§ Toric degenerations



In order to construct the mirror  to 
a (log) Calabi-Yau X in the Gross–Siebert mirror symmetry program, one must first place X in a toric degeneration π:𝔛⟶^1. 
This is a family of (log) Calabi-Yaus X_t such that π^-1(t)≅ (X,D) for t≠0.
The family degenerates to a central fiber X_0 := π^-1(0), which is a union of toric varieties, one for each vertex v_i of the spanning polytope

    X_0=∐_v_i X_i.

and the gluing data required to build the central fiber is encoded in .


A toric degeneration of (X,D) is a flat family (𝔛,𝔇) → T over a base T (typically T=𝔸_t^1=Spec ℂ[t] or T=Spec ℂ t) such that the general fiber is isomorphic to (X,D) and the special fiber X_0 is a union of toric varieties glued along toric divisors (and D_0 is a union of toric divisors not involved in the gluing) and such that the family is strictly semistable (i.e. locally of the form x_1⋯ x_k=t^l) away from a codimension 2 subset Z⊂ X_0. The latter condition is equivalent to 𝔛→ T being log smooth with respect to the divisorial log structures defined by X_0⊂𝔛 and {0}⊂ T. In our case we want to keep track of the curve classes of X, so we work over the base T=Spec ℂ[NE(X)] t.


[ for (^2,3L)]
	The following algebraic family serves as a toric degeneration for the log Calabi–Yau pair for ^2.
	
    ={x_1x_2x_3=t(y_1+f_3)}

	for (x_1,x_2,x_3,y_1)∈(1,1,1,3), t∈^1, and f_3 is a cubic polynomial that deforms . 
    The zero locus of y_1 defines a family of divisors 𝔇̌⊂. 
    The case f_3=0 corresponds to the toric boundary divisor ∂^2, whereas a nontrivial cubic gives the anticannonical family.
    The central fiber is given by a union of three weighted projective spaces
    
    _0=^2(1,1,3)∐^2(1,1,3)∐^2(1,1,3)

    which are glued along toric divisors as prescribed by the combinatorics of Δ.



There is a way to construct  as a formal scheme over a certain tropical homology group.
Let ι:Sing B̌↪B̌ be the inclusion and let Λ be the sheaf of integral tangent vectors. Then we construct the mirror degeneration as
	
    π̌:⟶Spf  H^1(B̌,ι_*Λ̌)^* t

There is an isomorphism between H^1(B̌,ι_*Λ̌;) and Pic().
The tropicalization of (,) is exactly the triple (B̌,𝒫̌,φ̌).
In <cit.>, it is shown that there exists a perfect pairing between H^1(B̌,ι_*Λ̌) and its dual.


In the Fano case the toric degeneration can be constructed globally as a projective variety (in the so called cone or polytope picture), while for the non-Fano case we have to work more locally (in the fan picture).





  §.§.§ The toric Fano case


For a toric Fano variety ℙ_Δ, one can construct a toric degeneration of ℙ_Δ from its polytope Δ. This is a Fano polytope, i.e., it has a unique interior lattice point. Consider the (unique) convex piecewise linear function φ̌ : Δ→ℝ that takes values 0 at the interior lattice point and 1 along the boundary of Δ. The domains of linearity of φ̌ give the central subdivision of Δ. The upper convex hull of φ̌,

    Δ_φ̌ = {(m,h) ∈Δ×ℝ |  h ≥φ̌(m)},

is an unbounded polytope. It defines a (non-projective) toric variety 𝔛_φ := X_Δ_φ̌. Projection to the last factor (which is the unique unbounded direction) defines a map 𝔛_φ̌→𝔸^1. This is a flat family of projective toric varieties. The general fiber is isomorphic to ℙ_Δ and the central fiber is a union of toric varieties, corresponding to the components of the central subdivision of Δ. One can define a family of divisors 𝔇→𝔸^1 by the vanishing of the coordinate corresponding to the interior lattice point. The general fiber gives the toric boundary of ℙ_Δ. Hence, 𝔛_φ̌→𝔸^1 is a toric degeneration of the log Calabi-Yau pair (ℙ_Δ,∂ℙ_Δ). The polytope B̌=Δ together with its central subdivison 𝒫̌ and the PL function φ̌ is a polarized polyhedral affine manifold  called the intersection complex of (,X)



  §.§.§ The smooth Fano case


The toric degeneration 𝔛_φ̌ constructed above is a subvariety of ℙ^N-1×𝔸^1, where N is the number of lattice points of Δ. Its defining equations correspond to relations among the lattice points of Δ, and in general it is not a complete intersection. Now consider a sufficiently general deformation of the defining equations without changing the central fiber. This yields another toric degeneration 𝔛→𝔸^1. The general fiber is a pair (X,D), where X is a ℚ-Gorenstein smoothing of ℙ_Δ and D is a smoothing of ∂ℙ_Δ. In other words, (ℙ_Δ,∂ℙ_Δ) is a toric model of (X,D), see <ref>.

The smoothing leads to a modification of the intersection complex. Before, Δ was a polytope, and in particular an affine manifold (a topological manifold with affine linear transition maps). The deformation introduces affine singularities, one on each interior edge of the central subdivision 𝒫, such that now B̌=Δ is an affine manifold with singularities.
We have to introduce these singularities to ensure that locally at each vertex of Δ the family 𝔛→𝔸^1 can still be described by the toric construction X_Δ_φ̌ above.



  §.§.§ The fan picture


Dual to the polytope picture, toric varieties can be described by fans, and a union of toric varieties can be described by gluing together such fans, resulting in a polyhedral complex called the dual intersection complex. If X is toric, the total space 𝔛 of the toric degeneration is then simply given by the toric variety obtained by taking the fan over the dual intersection complex. Again, smoothing the general fiber corresponds to changing the affine structure, such that the dual intersection complex becomes an affine manifold with singularities. In this picture, the polarization by the divisor D is given by a multi-valued piecewise linear function φ : B →ℤ^ρ, where ρ is the Picard rank of X.

The dual intersection complex  can be constructed as the discrete Legendre transform of , see <cit.>.

[Dual intersection complex (B,𝒫,φ)]

First, we set B≅^n and construct 𝒫 from the normal fan Σ. 
Let 𝒫 be the polyhedral subdivision of B whose codimension-1 cells are the facets of , called slabs. 
The 1-dimensional cells of 𝒫 are edges of  and ρ \ Int for the rays ρ in the normal fan Σ.
The top dimensional cells consist of a unique[ need not be the unique compact chamber when  is non-Fano. As we will show, this is the case for _m with m>2.] compact chamber ∈𝒫, and several unbounded chambers.
We place affine singularities at points δ∈ B along the edges of  and insert codimension-1 monodromy cuts at δ along the adjacent rays ρ_1,ρ_2.
Then, the affine structure on B is given by the transformation that maps ρ_1↦ρ_2 and leaves the edge invariant. 
Lastly, φ is a piecewise linear function that is 0 on Int and has slope 1 along the 1-dimensional cells of 𝒫.

[^2]
Following Construction <ref>, we plot the dual intersection complex for the anticannonical log Calabi-Yau pair (^2,3L) in Figure <ref>.
	
The transition maps A_i are given by

    A_1=[  0 -1;  1  2 ]   ,     	
     	A_2=[  3 -1;  4 -1 ]   ,   
     	A_3=[  3 -4;  1 -1 ]

and the monodromy M=A_3A_2A_1 is

    M=
        [ 1 9; 0 1 ]
        =
        [         1 (-K_^2)^2;         0         1 ].

For surfaces, the intersection number (-K_)^2 is also the volume of the Newton polytope. That is,
	
    Vol()=∫_ (c_1)^2=[-K_]^2=Vol()

	by the Duistermaat-Heckman theorem <cit.>.




Now, given  one can construct a mirror pair of central fibers X_0,X̌_0. The Gross–Siebert mirror symmetry construction associates a mirror toric degeneration →^1. The central fiber of  plays an important role as the uncorrected mirror to (X,D)


Figure <ref> shows the intersection complex of 𝔽_1, and the dual intersection complex in two different charts. Note that 𝔽_1 is smooth, so the deformation only changes the divisor from the toric boundary to a smooth anticanonical divisor.






  §.§.§ The non-Fano case


In the non-Fano case, we cannot work in the polytope picture, because polytopes correspond to toric varieties with an ample polarization. This means we cannot construct a toric degeneration globally as a projective variety. However, we can still work in the fan picture and construct the toric degeneration by gluing together affine pieces described by the gluing of fans. As before the asymptotic rays are the rays of a toric model of X. But now the spanning polytope Δ^⋆ is non-convex. In order for all fans to be complete, we need to subdivide Δ^⋆. Geometrically this corresponds to a blow up of a torus fixed point. Since the point lies on the central fiber, this does not change the general fiber. The blow up introduces a new component of the central fiber, corresponding to the new vertex. On the other components it acts like to blow up at a point. While Δ^⋆ was non-convex, the cells of the new subdivision are all convex.

Note that 𝔛_φ is not defined as a subvariety of ℙ^N-1×𝔸^1, so we cannot simply deform its defining equations. But 𝔛_φ is defined by gluing toric varieties along toric divisors, and we can perturb this gluing. This is just the same as in the Fano case, and again we introduce affine singularities on the interior edges of the dual intersection complex. So while we don't have an intersection complex for 𝔛→𝔸^1, we indeed have a dual intersection complex.


Figure <ref> shows the intersection complex of 𝔽_3, and the dual intersection complex in two different charts. The spanning polytope Δ^⋆ is non-convex and subdivided.






  §.§.§ Multi-polytopes


If one tries to find a intersection complex, i.e. polytope picture, for a non-Fano variety, one ends up with the notion of a multi-polytope.





 §.§ Scattering and broken lines


In this section we introduce the concepts of tropical curves, scattering diagrams and broken lines, and show that they are all related to each other. Then we show that they give certain relative (or logarithmic) Gromov-Witten invariants, which basically follows from the degeneration formula for logarithmic Gromov-Witten invariants.



  §.§.§ Scattering diagrams


Let B be an affine manifold with singularities and let φ be a piecewise affine function on B. In general we will assume φ to be multi-valued to distinguish curve classes, but taking the total degree (intersection with the divisor D) we obtain a single valued function. Let ι : B^∘↪ B be the complement of the singular locus and let Λ_B=ι_⋆Λ_B^∘ be the pushforward of the sheaf of integral tangent vectors on B^∘. For simplicity we restrict to the 2-dimensional case.


A ray 𝔡 on B consists of a base b_𝔡∈ B, a direction m_𝔡∈Λ_B,b_𝔡 and a function f_𝔡∈ℂ[z^m_𝔡] t. Via parallel transport this defines a section of the sheaf ℛ whose stalk at a point x∈ B is given by R_x=ℂ[P_x]/(t^k), where

    P_x = {p=(m,h)∈Λ_B,x⊕ℤ |  h ≤φ_x(m)}.

Here φ_x is the linear part of φ locally at x and t=z^(0,1), such that the t-order is given by φ_x(m)-h. Note that the t-order of a ray can increase at it propagates.

We demand the following properties:


  * If b_𝔡∈Sing(B), then f_𝔡=1+z^m_𝔡. (This is sometimes called a slab.) Note that in this case Λ_B,x is only 1-dimensional, so m_𝔡 is defined up to sign.

  * If b_𝔡∉Sing(B), then f_𝔡≡ 1  mod  (t). (This is sometimes called a wall.)

Note that we use different sign conventions than many other treatments of this topic. This is to avoid negative signs. For example, <cit.> uses the convention f_𝔡∈ℂ[z^-m_𝔡] t for (“outgoing”) rays.



A scattering diagram 𝒮 on B is a collection of rays such that for each k ≥ 0 there are only finitely many rays with f_𝔡≢1  mod  (t^k). The support |𝒮| of 𝒮 is the union of its rays, considered as subsets of B. A chamber 𝔲 of 𝒮 is a connected component of B∖|𝒮|. 



An affine manifold with singularities B defines a scattering diagram 𝒮_0(B) by taking all possible slabs (2 for each affine singularity).



A point x∈ B where two or more rays intersect is called a vertex. (This is sometimes called a joint.) Locally at a vertex x we produce new rays with base x by the following scattering procedure. For each ray 𝔡 containing x consider the complement 𝔡∖{x}. This consists of one component (if x=m_𝔡) or two components (otherwise). We cyclically order all such components (with respect to a simple loop around x that is disjoint from Sing(B)) to obtain a sequence 𝔡_1,…,𝔡_s. Each 𝔡_i defines a ℂ t-automorphism θ_𝔡_i of the localized ring (R_x)_∏ f_𝔡_i by

    θ_𝔡_i(z^m) = f_𝔡_i^⟨ n_i,m ⟩,

where n_i is the primitive normal vector to m_𝔡_i, positive with respect to the chosen ordering (i.e. the orientation coming from the loop around x). Now, for k ≥ 0, define

    θ^k = θ_𝔡_s∘…∘θ_𝔡_1 mod  (t^k).

It turns out that this acts on z^m by multiplication with a polynomial. Hence, to make θ^k the identity, we have to add finitely many rays. Doing this iteratively for all k≥ 0, we obtain scattering diagrams 𝒮_0,𝒮_1,… that are consistent to order k. Taking the formal limit we get a consistent scattering diagram 𝒮_∞.




  §.§.§ Broken lines




A broken line for a wall structure 𝒮 on B is a proper continuous map 𝔟 : (-∞,0] → B_0 with image disjoint from any vertices of 𝒮, along with a sequence -∞=t_0<t_1<⋯<t_r=0 for some r≥ 1 with 𝔟(t_i)∈|𝒮| for i≤ r-1, and for each i=1,…,r an expression a_iz^m_i with a_i∈ℂ∖{0}, m_i∈Λ_𝔟(t) for any t∈(t_i-1,t_i), defined at all points of 𝔟([t_i-1,t_i]), and subject to the following conditions:


  * 𝔟|_(t_i-1,t_i) is a non-constant affine map with image contained in a unique chamber 𝔲_i of 𝒮, and 𝔟'(t)=-m_i for all t∈(t_i-1,t_i).

  * For each i=1,…,r-1 the expression a_i+1z^m_i+1 is a result of transport of a_iz^m_i from 𝔲_i to 𝔲_i+1, i.e., is a term in the expansion of θ_𝔡(a_iz^m_i), where 𝔡 is the ray that contains the intersection of closures 𝔲̅_̅i̅∩𝔲̅_̅i̅+̅1̅.

  * a_1=1 and (m_1,h) has t-order zero at 𝔟(t_1), i.e., h=φ(m_1).

Write a_𝔟z^m_𝔟 for the ending monomial a_rz^m_r.



Figure <ref> shows the scattering diagram and some broken lines for 𝔽_3, in two different charts.




The idea of the Gross-Siebert program is that the dual intersection complex of X after scattering, i.e. 𝒮_∞(B), is the intersection complex of its mirror X̌. One can construct X̌ from B by gluing thickenings of affine models <cit.>. The consistency of 𝒮_∞(B) ensures that the gluing does not depend on the affine chart.



 §.§ Landau–Ginzburg models via theta functions


Supersymmetric sigma models with Fano target X are mirror to Landau–Ginzburg models W:(^*)^n→ <cit.>.

The superpotential W is given by the sum of ending monomials of all broken lines on 𝒮_∞(B) <cit.>. This is made precise with the definition of theta functions <cit.>.

Theta functions are sections of bundles over toric degenerations; in our case we would like to consider theta functions for .


Let X be a variety with dual intersection complex (of a toric model) B. Let m be an asymptotic direction on B. Let 𝔅_m(B)_P be the set of broken lines 𝔟 on 𝒮_∞(B) with initial monomial a_1z^m_1=z^m and endpoint 𝔟(0)=P. Define the theta function

    ϑ_m(X)_P = ∑_𝔟∈𝔅_m(B)_P a_𝔟 z^m_𝔟.



In our case, with smooth divisor D, the dual intersection complex B has exactly one unbounded direction m_, so asymptotic directions on B are just multiples of m_. We write ϑ_q(P) for ϑ_q· m_(P) with q∈ℕ.


The proper Landau–Ginzburg superpotential is given by the primitive theta function W=ϑ_1(x), and, as shown in <cit.>, this gives the open mirror map. 



Consistency of the scattering diagram ∞(X) (see <ref>) guarantees that the local description ϑ_p(x) for theta functions patch together to give global functions on , i.e. give an element of Γ(,𝒪_).
In other words, the ϑ_p form coordinates on the toric degeneration  and we can form the coordinate ring.
The central idea of intrinsic mirror symmetry <cit.>, is that the mirror is the spectrum of this ring.


Theta functions generate a commutative ring with unit ϑ_0 by the multiplication rule

    ϑ_p(X)_P ·ϑ_q(X)_P = ∑_r=0^∞α_p,q^r ϑ_r(X)_P

with structure constants

    α_p,q^r = ∑_(𝔟_1,𝔟_2)∈𝔅_p(B)_P×𝔅_q(B)_P 
     m_𝔟_1+m_𝔟_2=  r a_𝔟_1a_𝔟_2.

The theta functions ϑ_p(X)_P only depend on the chamber containing P and the structure constants α_p,q^r do not depend on P.


The structure constants α_pq^r are equal to certain punctured Gromov-Witten invariants <cit.>, which can be seen as an algebraic analogue of holomorphic disks. The idea of intrinsic mirror symmetry <cit.> is that one can circumvent the gluing procedure above and directly define the mirror to X as the spectrum of the ring of theta functions, with structure constants α_pq^r given punctured Gromov-Witten invariants.



§ B-MODEL: PERIODS AND PICARD–FUCHS EQUATIONS


Curve counting invariants of a Calabi–Yau manifold X are encoded in the periods of its mirror  <cit.>.
The periods satisfy a system of differential equations of generalized hypergeometric type called the Picard–Fuchs equations. 
The Picard–Fuchs operators Ł_a are determined by the Mori vectors ℓ^(a) as follows

    Ł_a=∏_ℓ_i^(a)>0∏_j=0^ℓ^(a)_i-1(∑_b=1^sℓ^(b)_iθ_b-j)-z_a∏_ℓ_i^(a)<0∏_j=0^-ℓ^(a)_i-1(∑_b=1^sℓ^(b)_iθ_b-j)

where θ_a=z_a∂_z_a is the logarithmic derivative with respect to z_a.

The solutions of the closed Picard–Fuchs system L_c={Ł_a}_a=1^s are given by the periods of .
There is a solution ϖ^(0)(z) to L_c called the fundamental period, see Equation <ref>.
In the compact case, all of the solutions to the Picard–Fuchs system can be obtained from appropriate linear combinations of derivatives of ϖ^(0) <cit.>.
This allows us to construct the period vector, which for a Calabi–Yau 3-fold is of the form

    =[   ϖ^(0); ϖ_a^(1); ϖ_a^(2);   ϖ^(3) ]


where ϖ_a^(1),ϖ_a^(2),ϖ^(3) are obtained using the Frobenius method.
Since the roots of the Ł_a are separated by integers, the additional solutions will have (log z_a)^k terms for 0≤ k≤ 3, and a power series term.
If κ_ijk is the intersection matrix of X, then  

    ϖ_a^(1)   =∂_n_aϖ^(0)
    ϖ_a^(2)   =1/2κ_ajk∂_n_j∂_n_kϖ^(0)
    ϖ^(3)   =-1/6κ_ijk∂_n_i∂_n_j∂_n_kϖ^(0)

The closed mirror maps t_a(z) are given by the ratio of the log periods to the fundamental period. 

    t_a(z)=ϖ^(1)_a(z)/ϖ^(0)(z)=log(z_a)+∑_n S_a(z)

Define the closed string symplectic parameters

    q_a=e^t_a=z_a e^S_a(z).

There is a linear combination ϖ^(2)_𝒲 of the ϖ^(2)_a solutions which, after inverting the mirror map to obtain z_a(t_a), can be used to define a 4-dimensional =1 superpotential  <cit.>

    𝒲(t)=ϖ^(2)_𝒲(z(t))/ϖ^(0)(z(t))

The associated Yukawa couplings can be computed by differentiating the ϖ^(2)_a solutions

    Y_ijk=∂/∂ t_i∂/∂ t_jϖ^(2)_k(z(t))/ϖ^(0)(z(t))

The Yukawa couplings can be expanded to obtain the counts of rational curves on the anticanonical Calabi–Yau hypersurface X↪.

[Elliptic fibrations over _m]
    Elliptic fibrations over toric surfaces can be constructed as a hypersurface in weighted projective space ^4(w_1,w_2,w_3,w_4,1) <cit.>.
    The weights are chosen so that one can obtain a Weierstrass model over the chosen toric base. 
    Here we construct an elliptic fibrations over the Hirzebruch surfaces _m for m≤ 3.
    Consider the spanning polytope ⊂^4 with the vertices
    
    v_1=(1,0,0,0),  v_2=(0,1,0,0),  v_3=(0,0,1,0),  v_4=(0,0,0,1)
     v_5 =(-w_1,-w_2,-w_3,-w_4), 
            v_6=(-3,-2,0,0),
     v_7=(-3(2+m),-2(2+m),-m,-1)

    For m≤ 2, the Mori vectors can be written directly by using the Mori vectors from Example <ref>
    
        ℓ^(1)=(       -6;       2,       3,       1,       0,       0,       0,       0) 

        ℓ^(2)=(        0;       0,       0,       -2,       1,       1,       0,       0) 

        ℓ^(3)=(        0;        0,        0,        m-2,       -m,       0,       1,       1) 
    For m=3, the Mori vectors are
    
        ℓ^(1)=(       -6;       2,       3,       1,       0,       0,       0,       0) 

        ℓ^(2)=(       0;       0,       0,       -2,       1,       1,       0,       0) 

        ℓ^(3)=(       6;       -2,       -3,       0,       -3,       0,       1,       1) 
    and the associated Picard–Fuchs operators are
    
    Ł_1   =(θ_1-θ_3)(θ_1-2θ_2)θ_1-12z_1θ_1(6(θ_3-θ_1)+1)(6(θ_3-θ_1)+5)
    Ł_2   =(θ_2-3θ_3)θ_2-z_2(θ_1-2θ_2-1)(θ_1-2θ_2)
    Ł_3   =θ_3^2(6(θ_3-θ_1)+1)(6(θ_3-θ_1)+5)-1/12z_3(θ_1-θ_3)∏_j=0^2(θ_2-3θ_3-j)

The fundamental period is given by

    ϖ^(0)=∑_n_1≥ 2n_2≥ 6n_3Γ(6(n_1-n_3)+1)/Γ_e(n_1,n_2,n_3)Γ_b(n_2,n_3)z_1^n_1z_2^n_2z_3^n_3

where

    Γ_e(n_1,n_2,n_3)   =Γ(2(n_1-n_3)+1)Γ(3(n_1-n_3)+1)Γ(n_1-2n_2+1)
    Γ_b(n_2,n_3)   =Γ(n_2-3n_3+1)Γ(n_2+1)Γ^2(n_3+1)

are Gamma function factors coming from the elliptic fibers and the base, respectively.
By taking the large elliptic fiber limit z_1→ 0 (and hence θ_1→ 0), one can obtain the Picard–Fuchs operators for the canonical bundle K__3

    Ł̃_2   =(θ_2-3θ_3)θ_2-4z_2θ_2(θ_2+1/2)
    Ł̃_3   =θ_3^2(6θ_3-1)(6θ_3-5)+1/12z_3θ_3∏_j=0^2(θ_2-3θ_3-j)



In the non-compact case, the fundamental period becomes a constant which we normalize to 1.
This can be seen in the above example: to obtain the non-compact space we take z_1→ 0 and the only surviving term in the sum <ref> is n_1=n_2=n_3=0, so ϖ^(0)=1.
Hence we cannot obtain the log and the log squared solutions by differentiating ϖ^(0). 
We illustrate our approach in the non-compact case with the following example.

[Closed mirror map for K__3]

    From Example <ref>, we have the Mori vectors ℓ^(1), ℓ^(2) for K__3.
    We use these to construct the Picard–Fuchs operators following Equation <ref>
    
    Ł_1   =θ_1(θ_1-3θ_2)-z_1(-2θ_1+θ_2)(-2θ_1+θ_2-1)
    Ł_2   =(-2θ_1+θ_2)θ_2-z_2(θ_1-3θ_2)(θ_1-3θ_2-1)(θ_1-3θ_2-2)

    The fundamental period is again given by the constant solution ϖ^(0)=1, and hence we look for a solution of the form in Equation <ref>. Let
    
    F_a(z_1,z_2)=∑_n_1,n_2≥ 0 c^(a)_n_1,n_2z_1^n_1z_2^n_2

    be power series in z_1 and z_2. 
    The power series terms S_a of the closed mirror maps t_a=log z_a + S_a(z_1,z_2) will be written as linear combinations of the F_a.
    By imposing Ł_b F_a=0, we get recurrence relations on the coefficients c^(a)_n_1,n_2.
    
    Ł_1F_a=0 c^(a)_n_1+1,n_2   =(2n_1-n_2)(2n_1-n_2+1)/(n_1+1)(n_1-3n_2+1)c^(a)_n_1,n_2
    Ł_2F_a=0 c^(a)_n_1,n_2+1   =-(n_1-3n_2)(n_1-3n_2-1)(n_1-3n_2-2)/(n_2+1)^2(2n_1-n_2-1)c^(a)_n_1,n_2

    For the first solution, we solve Equation <ref> with n_2=0
    
    c^(1)_n_1+1,0=2n_1(2n_1+1)/(n_1+1)^2c^(1)_n_1,0 c^(1)_n_1,0=(2n_1-1)!/(n_1!)^2

    With similar factorial recursion identities (coming from the properties of the Gamma function), one can substitute c^(1)_n_1,0 into Equation <ref> to solve for c^(1)_n_1,n_2.
    
    c^(1)_n_1,n_2   =(-1)^n_2(2n_1-n_2-1)!/(2n_1-1)!n_1!/(n_1-3n_2)!1/(n_2!)^2c^(1)_n_1,0
       =(-1)^n_2(2n_1-n_2-1)!/n_1!(n_1-3n_2)!(n_2!)^2.

    For the second solution, we solve Equation <ref> with n_1=0
    
    c^(2)_0,n_2+1=-3n_2(3n_2+1)(3n_2+2)/(n_2+1)^3c^(2)_0,n_2 c^(2)_0,n_2=(-1)^n_2(3n_2-1)!/(n_2!)^3.

    After factoring out a minus sign to apply more factorial identities, one can substitute c^(2)_0,n_2 into Equation <ref> to solve for c^(2)_n_1,n_2.
    
    c^(2)_n_1,n_2   =(-1)^n_1(3n_2-n_1-1)!/(3n_2-1)!n_2!/(n_2-2n_1)!1/n_1!c^(2)_0,n_2
       =(-1)^n_1+n_2(3n_2-n_1-1)!/n_1!(n_2-2n_1)!(n_2!)^2

    One can check that c^(1)_n_1,n_2 and c^(2)_n_1,n_2 solve the recurrence relations <ref> and <ref>.
    Now to obtain the log solutions we note that Ł_a t_b=Ł_a(log z_b+S_b)=0 and hence Ł_a S_b=-Ł_a log z_b. We calculate
    
    Ł_1 S_1   =2z_1         Ł_1 S_2=-z_1
    Ł_2 S_1   =2z_2         Ł_2 S_2=-6z_2

    	Looking at the lowest order terms of F_1 and F_2 we have
    
    c^(1)_1,0z_1   =z_1         c^(1)_0,1z_2=0 
    
        		c^(2)_1,0z_1   =0         c^(2)_0,1z_2=-2z_2.

    This fixes S_1 and S_2
    
    S_1=2F_1-F_2         S_2=-F_1+3F_2

    where
    
    F_1(z_1,z_2)   =∑_n_1,n_2≥ 0
     n_1≥ 3n_2 (-1)^n_2Γ(2n_1-n_2)/Γ(n_1)Γ(n_1-3n_2+1)Γ^2(n_2+1)z_1^n_1z_2^n_2
    
        	F_2(z_1,z_2)   =∑_n_1,n_2≥ 0
     n_1≤ 3n_2 (-1)^n_1+n_2Γ(3n_2-n_1)/Γ(n_1+1)Γ(n_2-2n_1+1)Γ^2(n_2+1)z_1^n_1z_2^n_2

    The z_1 parameter is mirror to the fiber class parameter
    
    q_1=z_1 e^S_1

    and the z_2 parameter is mirror to the exceptional class parameter
    
    q_2=z_2 e^S_2



The work of Lerche and Mayr <cit.> studies the 𝒩=1 moduli space for the B-model with D-branes by deriving a system of differential equations that relates the open and closed moduli spaces L_oc={Ł_a^(oc)}_a=0^s.
The operators in L_oc involve logarithmic derivatives θ_0=z_0∂_z_0 with respect to an additional open string parameter z_0.
We have L_c=L_oc|_θ_0→0, so the solutions to the closed system with no z_0 dependence are also solutions to the open-closed system.
The parameters z_0 together with z_1,…,z_s define flat coordinates on the open string moduli space ^o(). 
We now explain how to obtain the additional Mori vector ℓ^(0) with an explicit D-brane construction, following <cit.>, and give an example for K__3.

The open-string sector of the B-model consists of D5-branes wrapped on a 2-cycle C∈ H_2(X,). These are mirror to D6-branes on the mirror manifold  with a non-perturbative superpotential W. The superpotential for the D5-brane on C can been calculated by integrating over a 3-chain Γ(z_0) with ∂Γ(z_0)=C(z_0)-C_*

    W(z_0)=∫_Γ(z_0)Ω=∫_C(z_0)-C_*ω

Define the ^* coordinates ỹ_i=Y^v_i and form the conical bundle

    {W(z)+uv=0}

 and use this defining equation to write ỹ_i=ỹ_i(z_0) so that the curve C is defined by the equations

    ỹ_i(0)=z_0ỹ_j(0)

This yields an additional Mori vector

    ℓ^(0)=(0,…,0,1,0,…,0,-1,0)

where 1 is in the ith spot and -1 is in the jth spot.
Since the v_i and hence the ỹ_i correspond to a choice of divisor, the choice of ℓ^(0) specifies the D-brane that an open string starts and ends on.

[Open mirror map for K__3]
	We include the additional Mori vector ℓ^(0) to obtain
	
    ℓ^(1)=(       -2,       1,        1,        0,        0;        0,        0)

    ℓ^(2)=(        -(2-m),        -m,        0,        1,        1;        0,        0)

    ℓ^(0)=(        1,        0,        0,        -1,        0;        -1,        1)

Now the Picard–Fuchs operators are modified

    Ł_1   =θ_1(θ_1-3θ_2)-z_1(-2θ_1+θ_2+θ_0)(-2θ_1+θ_2+θ_0-1)
    Ł_2   =(-2θ_1+θ_2-θ_0)(θ_2-θ_0)-z_2(θ_1-3θ_2)(θ_1-3θ_2-1)(θ_1-3θ_2-2)
    Ł_0   =(θ_0-2θ_1+θ_2)θ_0+z_0(θ_2-θ_0)θ_0

    The log solutions t_1,t_2 still solve this open-closed Picard–Fuchs system since S_1,S_2 are only functions of the closed string moduli and hence θ_0 t_a=0. 
    However, we have an additional solution t_0=log z_0 + S_0(z_1,z_2), and we solve for S_0 in terms of F_1 and F_2 the same way we solved for S_1 and S_2.
    Namely, we compute Ł_b S_0 = -Ł_b log z_0 and choose the appropriate linear combination which reproduces this lowest order term. 
    
    .
        Ł_1 S_0    = -Ł_1log z_0 = -z_1
    Ł_2 S_0    = -Ł_2log z_0 = 0
        } S_0=-F_1

    Hence we can compute the open mirror map
    
    t_0=z_0e^S_0





§ A-MODEL: METHODS FOR COMPUTING GROMOV–WITTEN INVARIANTS


Here we describe four methods for computing Gromov–Witten invariants of log Calabi-Yau pairs (X,D), using ^2 and _m as the running examples.
The toric data that we will need to do these computations is given in Examples <ref> and <ref>, respectively. 
We will later compare the methods in <ref> and <ref> for the Fano and semi-Fano examples. 
The method in <ref> utilizes intrinsic mirror symmetry which is not dependent on the ampleness of -K_, so we solely employ this method in the non-Fano cases.


 §.§ Symplectic geometry


In <cit.>, it is shown that the corrections to the slab function f_0 associated to the canonical divisor D_0:=[K_]=-∑_i D_i in a (semi-Fano) toric variety  are given by the wall-crossing generating function of open Gromov–Witten invariants exp(g_0) for the canonical bundle K_ over . One can compute exp(g_0) explicitly in terms of the intersection data of  as follows.


  * To each toric divisor D_i one can associate a hypergeometric series

    g_j(z)=∑_α∈j(-1)^-D_j·α-1Γ(-D_j·α)/∏_i≠ jΓ(D_i·α+1)z^α

where the sum is over all numerically effective curve classes α∈j=H_2(,D_j;)_eff relative to D_j

    H_2(,D_j;)_eff={α∈ H_2() | α· D_j <0    and   α· D_i ≥ 0   ∀ i≠ j }

and the intersection products are taken in the Chow ring A_*() by using the divisor class dual to α in an appropriate basis of effective curves {ℓ_1,…,ℓ_s} in j.
These effective curves are dual to the divisors {D_1,…,D_s} that generate the Picard group Pic(_), so the divisor classes will be used to denote effective curves.
 The complex structure moduli z=(z_1,…,z_s)=(z^ℓ_1,…,z^ℓ_s) form algebraic coordinates[The number of Kähler moduli of a Calabi-Yau n-fold X is the same as the number of complex structure moduli of its mirror, i.e.  s=h^1,1(X)=h^n-1,1()] 
 on the complex structure moduli space of the mirror ().
 
  * Under the mirror map M:()→(X), the image of the algebraic coordinates are the flat coordinates q_a=M(z_a) on the Kähler moduli space of X.
 The mirror map takes the form

    q_a=z_aexp(∑_i=0^sQ_ia g_i(z) )

 where Q_ia is the U(1)^s charge matrix of a gauged linear sigma model (GLSM) for which  is the space of supersymmetric ground states[The Q_ia can be obtained by writing the toric divisors in the {ℓ_i} basis D_a=∑_i=1^s Q_iaℓ_i]. 
 One can invert this power series to obtain the inverse mirror map M^-1:(X)→() so that M^-1(q_a)=z_a.

  * The open Gromov–Witten invariants n_β_j+α are then given by writing g_0 in the flat coordinates and taking an exponential series

    exp(g_j(M^-1(z)))=∑_α∈j n_β_j+αq^α

Each toric prime divisor D_i corresponds to a basic disc class β_i∈π_2(,T) bounded by a Lagrangian torus fiber T∈. We write

    n_β_0+k_1ℓ_1+k_2ℓ_2+…+k_sℓ_s=n_k_1k_2… k_s



[^n]
The spanning polytope for n-dimensional complex projective space is given by the convex hull of the following n+1 vertices.

    =Conv{(1,0,…,0),(0,1,…,0),…,(0,0,…,1),(-1,-1,…,-1)}

Therefore ℓ=(-(n+1);1,1,…,1) and hence by Equation <ref>, we have

    g_0(z)=∑_k≥0(-1)^(n+1)k((n+1)k-1)!/(k!)^n+1z^k



[^2]
The effective curve classes relative to D_0=-3L are given by multiples of the hyperplane class 

    NE_0(^2)={kL |  k∈_>0}.

From Equation <ref>, we have the generating function

    g_0(z)=-∑_k≥0(-1)^3kΓ(3k)/Γ^3(k+1)z^k

Then the open Gromov–Witten invariants are given by Equation <ref>.

    exp(g_0(M^-1(z)))=1-2q+5q^2-32q^3+286q^4-3038q^5+…

We tabulate these below



[_m]

    We choose the basis span_{E,F}≅ H_2(_m)⊂ A_*(_m) so that an effective curve may be written β=n_F F+n_E E for some integers n_F,n_E.
    The effective curves relative to D_0=-2S-(m-2)F=-2E-(2+m)F are given by imposing constraints on n_F and n_E from Equation <ref>
    
    NE_0(_m)={ n_F F + n_E E  |  n_F,n_E≥0 , n_F≥ mn_E}

    Using the intersection numbers in Example <ref>, we have the generating function associated to the canonical divisor.
    
    g_0(z)=∑_n_F ≥ m n_E(-1)^mn_EΓ(2n_F+(2-m)n_E)/Γ(n_F+1)Γ(n_F-mn_E+1)Γ^2(n_E+1)z_F^n_Fz_E^n_E

    and we can also write down the generating function associated to the fiber class
    
    g_F(z)=∑_n_E≥ 0Γ(2n_E)/Γ^2(n_E+1)z_E^n_E

    Since (-E)·(-K__m)=m-2, there are numerically effective curves relative to -E=:E for m>2, and hence in the non-Fano case we obtain an additional generating function
    
    g_E(z)=∑_n_F ≤ m n_E(-1)^n_F+n_EΓ(mn_E-n_F)/Γ(n_F+1)Γ((m-2)n_E-2n_F+1)Γ^2(n_E+1)z_F^n_Fz_E^n_E

    By applying Equation <ref>, we can obtain the closed mirror maps. For m=0,1, we have
    
    q_F   =z_Fe^Q_F0g_0=z_Fe^2g_0
    
        	q_E   =z_Ee^Q_E0g_0=z_Ee^(2-m)g_0.

    For m=2, the charge on the exceptional divisor is zero Q_E0=0, so we have
    
    q_F   =z_Fe^2g_0-g_F
    
        	q_E   =z_Ee^2g_F.

    Lastly, for m>2 we include the generating function g_E associated to negative the exceptional divisor, where Q_E0=2-m and Q_EE=m.
    
    q_F   =z_Fe^2g_0+(2-m)g_E
    
        	q_E   =z_Ee^(2-m)g_0+mg_E

    Again, the open Gromov–Witten invariants are given by Equation <ref>.
    Below we tabulate these invariants for _m with m≤ 3.
    This matches the results obtained from the Picard-Fuchs method described in Example <ref>.
  
    








 
    


 §.§ Mirror Landau–Ginzburg models

 In the previous section we discussed how the open Gromov–Witten invariants of can be calculated by using the intersection data encoded in the spanning polytope , which for reflexive polytopes is the polar dual of the polar polytope =()^∘.
 By Batryev–Borisov mirror symmetry <cit.>, the mirror to a Calabi–Yau hypersurface in  is given by a Calabi-Yau hypersurface in . 
 Therefore g_0 is closely related to the fundamental period ϖ^(0)() of the mirror family _z to the anticanonical hypersurface X∈[-K_]↪ (see <cit.>).

    ϖ^(0)(_z)=∫_γ_0Ω̌_z=∑_α∈0(Γ(-D_0·α+1)/∏_i Γ(D_i·α+1))z^α

 The holomorphic volume form Ω̌_z∈ H^n,0_∂̅(_z) is given by
 
    Ω̌_z=d x_1 ∧…∧ dx_n/dW_z(x)

The mirror family can be thought of as being fibered over () near the large complex structure limit z_a→ 0 with the fiber over z=(z_1,…,z_s) given by

    _z=W_z^-1(1)

where W_z:(^*)^n+1→ is the Landau–Ginzburg superpotential mirror to the ambient toric variety .




 §.§ Slab function normalization

The slab function normalization condition states that any slab function cannot contain pure q powers in its log expansion.
Applying this condition to the slab function associated to the canonical divisor f_0 allows one to compute g_0 order-by-order since f_0 is given by

    f_0=e^g_0(q)+∑_v_i∈∩^nz^ℓ_ix^v_i,xy^v_i,y

where v_i=(v_i,x,v_i,y)∈ are the integral points of . We illustrate this with the following examples.

[^2, following <cit.>]
	The slab function for two dimensional complex projective space is given by
	
    f_0=e^g_0+x+y+z/xy

	Taking the log, we have 
	
    log f_0=g_0+log(x̃+ỹ+q/x̃ỹ)

	where x̃=e^-g_0x, ỹ=e^-g_0y, and q=ze^-3g_0.
	Therefore we can expand the log and the constant terms (with respect to x̃,ỹ) will give us g_0. 
	Since ℓ_0=3, constant terms can only occur at in the expansion at orders that are a multiple of 3
	
    log(x̃+ỹ+q/x̃ỹ)|_x̃,ỹ const.   =∑_k=0^∞(-1)^k-1/k(x̃+ỹ+q/x̃ỹ)^k|_x̃,ỹ const.
       =∑_k=0^∞(3k-1)!/(k!)^3

	This is the same result we obtained previously.


[^1×^1]
	The slab function normalization condition was applied to ^1×^1 in <cit.>.
	This reproduces our result from Example <ref>.





 §.§ Scattering diagrams with 


Scattering diagrams (see <ref>) are an important tool to calculate the instanton corrections to the central fiber X̌_0 of the mirror toric degeneration π:⟶^s. 
In particular, scattering diagrams carry the invariants that allow us to “smoothen" X̌_0 into a mirror family where a general fiber is the mirror to (X,D).  

The second author developed a code to compute scattering diagrams and broken lines for 
log Calabi–Yau pairs (X,D), see <ref> for definitions. 
When the cited review paper was published, this was only done for very ample log Calabi–Yau pairs i.e. when X is a surface with K_X^2≥ 3.
However, we have since developed the code to include non-Fano cases of interest.


	A wall structure consistent to all orders, called the canonical wall structure ∞(X), was constructed by Gross and Siebert <cit.>. Our code computes the canonical wall structure k(X) to finite order k if given enough time and memory.


[^2]
   The initial scattering diagram 𝒮_0(X) is given by defining pairs of antiparallel outgoing walls 𝔡_± at each affine singularity. 
   One can then iterate the consistency condition to obtain the higher order scattering diagrams 𝒮_k(X).
   See below for a plot produced by the code for ^2.








§ TROPICAL CURVES AND TROPICAL CORRESPONDENCE





 §.§ Tropical curves


Tropical curves live on the dual intersection complex, so this doesn't depend on the Fano property.



A tropical curve on an affine manifold with singularities B is a map h : Γ→ B from a graph Γ, without bivalent vertices but possibly with some legs (non-compact edges with only 1 vertex), together with


  * a non-negative integer g_V (genus) for each vertex V;

  * a non-negative integer ℓ_E (length) for each compact edge E;

  * a weight vector u_(V,E)∈Λ_B,h(V) for each flag (V,E);

such that h respects the graph structure of Γ and


  * if E is a compact edge with vertices V_1, V_2, then h maps E affine linearly to the line segment connecting h(V_1) and h(V_2), and h(V_2)-h(V_1)=ℓ_Eu_(V_1,E). In particular, u_(V_1,E) = -u_(V_2,E). Here the affine structure on Γ is given by the lengths ℓ_E of its edges;

  * if E is a leg with vertex V, then h maps E affine linearly either to the ray h(V)+ℝ_≥ 0u_(V,E) or to the line segment [h(V),δ) for δ an affine singularity of B such that δ-h(V)∈ℝ_>0 u_(V,E), i.e., u_(V,E) points from h(V) to δ. Note that in the latter case the direction of u_(V,E) is determined by Λ_B,δ≃ℤ.

  * at each vertex V we have the balancing condition

    ∑_E∋ Vu_(V,E) = 0.


We write the set of compact edges of Γ as E(Γ), the set of legs as L(Γ), the set of legs mapped to a ray (unbounded legs) as L_∞(Γ) and the set of legs mapped to an open line segment (bounded legs) as L_Δ(Γ) (since such edges end at the singular locus Δ of B).



All our tropical curves will be genus 0, so that g_V=0 for all vertices and Γ is a tree.




Let h : Γ→ B be a tropical curve. For a trivalent vertex V∈ V(Γ) define m_V=| u_(V,E_1)∧ u_(V,E_2)|=|det(u_(V,E_1)|u_(V,E_2))|, where E_1,E_2 are any two edges adjacent to V. For a vertex V∈ V(Γ) of valency ν_V>3 let h_V be the one-vertex tropical curve describing h locally at V and let h'_V be a deformation of h_V to a trivalent tropical curve. It has ν_V-2 vertices. Define m_V=∏_V'∈ V(h'_V)m_V'. For a bounded leg E∈ L_Δ(Γ) define m_E=(-1)^w_E+1/w_E^2. Then define the multiplicity of h to be

    Mult(h) = 1/|Aut(h)|·∏_V m_V ·∏_E∈ L_Δ(Γ) m_E.





 §.§ Tropical disks, scattering and broken lines



A tropical disk h^∘ : Γ→ B is a tropical curve with the choice of univalent vertex V_∞, adjacent to a unique edge E_∞, such that the balancing condition (Definition <ref>, (iii)) only holds for vertices V ≠ V_∞. Define the multiplicity m_V_∞=1, such that Mult(h^∘) is the multiplicity of the balanced tropical curve obtained by forgetting V_∞.



Let 𝔡∈𝒮_∞(B) be a ray and choose a point P∈(𝔡). 

Define ℌ^∘_𝔭,w(B) to be the set of all tropical disks h^∘ : Γ→ B with no unbounded legs and such that h^∘(V_∞)=P and u_(V_∞,E_∞)=-w· m_𝔡.




For a ray 𝔡∈𝒮_∞(B) we have

    f_𝔡 = ∑_w=1^∞∑_h^∘∈ℌ^∘_𝔭,w(B) w Mult(h^∘) t^w z^wm_𝔭.




Let ℌ^∘_q(B)_P be the set of tropical disks h^∘ : Γ→ B with one unbounded leg E_out and such that h^∘(V_∞)=P and u_(V_out,E_out)=qm_out and u_(V_∞,E_∞)=-pm_.




Let 𝔟 be a broken line in 𝔅_q^(k)(B)_P. If P lies in an unbounded chamber of 𝒮_k, then m̅_𝔟 is parallel to m_.



This is <cit.>, Proposition 3.5, or <cit.>, Proposition 2.3.




There is a surjective map with finite preimages

    μ : ℌ^∘_q(B)_P →𝔅_q(B)_P

defined by taking the path from V_∞ to V_out. We say a tropical disk in μ^-1(𝔟) is obtained from 𝔟 by disk completion. We have

    a_𝔟 = ∑_h^∘∈μ^-1(𝔟)(h^∘).




This is <cit.>, Lemma 6.4 and Proposition 6.15, with some more details given in <cit.>, Proposition 3.2 and Proposition 4.18.




  §.§.§ Leg extension


If h^∘ is a tropical disk with u_(V_∞,E_∞)=-wm_, then we can forget V_∞ and extend E_∞ to an unbounded leg E_out with u_(V_out,E_out)=wm_out. Hence, the above statements for tropical disks can be translated to tropical curves as follows.


Let ℌ_w(B) be the set of tropical curves on B with one unbounded leg E_out of weight w. This implies u_(V_out,E_out)=wm_out. Further, let ℌ_p,q(B)_P be the set of tropical curves on B having two unbounded legs, of weights p and q, and such that the image of the unbounded leg of weight p contains P.



We have

    ∏_𝔡 : m_𝔡 = wm_out f_𝔡 = ∑_w=1^∞∑_h∈ℌ_w(B) w Mult(h) z^(wm_𝔭,0).





Let P ∈ B be a point in an unbounded chamber of 𝒮_p+q(B). Then there is a surjective map with finite preimages

    μ : ℌ_p,q(B) →𝔅_p,q(B)_P,

such that

    a_𝔟 = ∑_h∈μ^-1(𝔟)(h).

A preimage is given by disk completion and leg extension.




 §.§ Tropical curve classes



The curve class of a tropical curve is defined by its intersection with tropical cocycles defined by elementary tropical curves. For more details see <cit.>, 3, and <cit.>, 4.2.

Write ℌ_w(B,β) and ℌ_p,q(B,β)_P for the set of tropical curves of class β in ℌ_w(B) and ℌ_p,q(B)_P, respectively.



Figure <ref> shows some tropical curves on 𝔽_3 of class 2F+E:




 §.§ The degeneration formula and tropical correspondence


The idea is that tropical curves describe the components of the curve on the central fiber of the toric degeneration. Formally we have to work with a log smooth degeneration. This is obtained by successively blowing up components on the central fiber of the degeneration. The multiplicities m_L of bounded legs (Definition <ref>) account for components mapping onto or intersecting the exceptional locus. The degeneration formula works locally, so the Fano condition doesn't matter.



Let B be the dual intersection complex of a toric model of X. Then

    R_w(X,β) = ∑_h∈ℌ_w(B,β)Mult(h),

and, for any point P ∈ B_0,

    R_p,q(X,β) = 1/p∑_h∈ℌ_p,q(B,β)_PMult(h).





At infinity we have

    ϑ_q(X)_∞ = y^q + ∑_p≥ 1∑_β : β.E=p+q p R_p,q(X,β) s^β t^p+1y^-p.




Figure <ref> shows some tropical curves contributing to R_3(𝔽_3,2F+E)=9 and R_2,1(𝔽_3,2F+E)=1.






 §.§ Relating scattering diagrams of Hirzebruch surfaces


We know that the Hirzebruch surfaces of even/odd parity have the same relative Gromov-Witten invariants. So we expect their scattering diagrams to be isomorphic. Figure <ref> shows how the 𝔽_1-diagram and the 𝔽_3-diagram are related. The tropical curves of class F+E in 𝔽_1, i.e. class 2F+E in 𝔽_3, are shown in red and blue.





§ THE CORRECTED POTENTIAL AND MUTATIONS





 §.§ The potential



The theta potential of X is the first theta function ϑ_1(X)=ϑ_m_out(X). It depends on the chamber of the scattering diagram 𝒮_∞(X) in which it is defined. We write ϑ_1(X)_0 for the potential close to the origin, i.e. in some chamber inside the (possibly non-convex) polytope Δ^⋆, which is the spanning polytope of the fan of a toric model of X. We write ϑ_(X)_∞ for the potential at infinity, i.e. in an unbounded chamber. Note that ϑ(X)_∞ only depends on one variable y, while ϑ_1(X)_0 depends on several variables.



In the Fano case, ϑ_1(X)_0 is given by the toric potential

    ϑ_1(X)_0 = W_Σ = t·∑_ρ∈Σ^[1] z^m_ρ.

Here Σ is the fan of any toric model of X, see Definition <ref>.


It was noticed in <cit.>, Theorems 4.5 and 4.6, that in the non-Fano case there are additional correction terms. For the Hirzebruch surfaces 𝔽_2 and 𝔽_3 they have been computed explicitly in <cit.> using wall crossing (in the cone picture) and in <cit.> using broken lines (in the fan picture). Here we explain the latter approach for 𝔽_2, 𝔽_3 and additionally for 𝔽_4, 𝔽_4, and make some remarks about the t-order grading. In <cit.> this grading differs from the other approaches due to a different choice of polarization. We argue that one can indeed take the anticanonical polarization to obtain the same t-grading as in <cit.>, and we explain how to recognize curves classes from the broken lines. Moreover, we show that the potentials for 𝔽_0, 𝔽_2, 𝔽_4, and for 𝔽_1, 𝔽_3, indeed have the same periods under the change of curve classes F↦ F, S ↦ S-F.


We have

    ϑ_1(X)_0 = W_Σ + W',

where W_Σ is the toric potential of a toric model of X and W' is a correction term that comes from scattering inside the convex hull Conv(Δ^⋆). In particular, if X is Fano, then W'=0.



We compute the potentials ϑ_1(𝔽_m) of Hirzebruch surfaces 𝔽_m in the chamber below the non-convex vertex of Δ^⋆. To do so, we can choose the point P=(-1,-ϵ) with ϵ>0 sufficiently small. This gives the following potentials, where z_1=z^F and z_2=z^E,

    ϑ_1(𝔽_0)_(-1,-ϵ)   = t ·( x+y+z_1/x+z_2/y) 
    ϑ_1(𝔽_1)_(-1,-ϵ)   = t ·( x+y+z_1/x+z_1z_2/xy) 
    ϑ_1(𝔽_2)_(-1,-ϵ)   = t ·( x+y+z_1/x+z_1z_2/x(1+z_1/xy) ) 
    ϑ_1(𝔽_3)_(-1,-ϵ)   = t ·( x+y+z_1/x+z_1z_2y/x(1+z_1/xy)^2 ) 
    ϑ_1(𝔽_4)_(-1,-ϵ)   = t ·( x+y+z_1/x+z_1z_2y(1+z_1/xy)+z_1z_2y^2/x(1+z_1/xy)^3 )

Note that 𝔽_0 and 𝔽_1 are Fano, so there are no correction terms and the potential is simply the toric one. For 𝔽_2 and 𝔽_3 the potential was computed in <cit.> using symplectic methods and in <cit.> using broken lines. Note the different conventions. In <cit.> they don't give classes and have the wrong t-order. Moreover, they have one broken line that actually isn't there. Instead, one broken line appears with coefficient 2.

For 𝔽_3, 𝔽_4, 
the computation is shown in Figures <ref>, <ref>, 
respectively. For m≥ 4 there is infinite scattering inside Δ^⋆. All rays have t-order zero inside Δ^⋆, but the t-order increases as they leave Δ^⋆, so only a very limited number of them (actually only the initial rays) are involved in the broken line calculation.

One can easily check that the classical periods of ϑ_1(𝔽_m)_0 agree for m even/odd, as expected.







We conjecture that

    ϑ_1(𝔽_5)_(-1,-ϵ) = t ·( x+y+z_1/x+z_1z_2xy+z_1z_2y^2(1+z_1/xy)^2+z_1z_2y^3/x(1+z_1/xy)^4 ).

This is what we get from the mutations in the next section, and we verified that this gives the same period as ϑ_1(𝔽_3)_(-1,-ϵ) after z_2↦ z_1z_2 and as ϑ_1(𝔽_1)_(-1,-ϵ) after z_2↦ z_1^2z_2. However, we could not find the broken line for the monomial z_1z_2xy. See Figure <ref> for the other broken lines. The shape of a broken line with positive x-coordinate has to be different, because all rays in the interior have positive x-coordinate, so can only increase the x-coordinate. For ϑ_1(𝔽_7)_(-1,-ϵ) we could find a broken line with positive x-coordinate, as shown in Figure <ref>.







In the calculations above we have omitted the variables z_i, i.e. the curve classes. One can find the curve class corresponding to a broken line by shifting the endpoint to infinity and completing the broken lines to a tropical curve, as in Proposition <ref>. Then one can calculate the curve class of the tropical curve by intersection with tropical cocycles.
 
See Figure <ref> for an example calculation.






 §.§ Mutations


Mutations of Laurent polynomials were introduced in <cit.><cit.><cit.> and related to mirror symmetry in <cit.>.


Let N≃ℤ^2 be a lattice and let w∈ M be a primitive vector in the dual lattice. Then w induces a grading of ℂ[N]. Let a∈ℂ[w^∩ N] be a Laurent polynomial in the zeroth piece of ℂ[N], where w^∩ N={v ∈ N  |  w(v)=0}. The pair (w,a) defines an automorphism of the Laurent polynomial ring ℂ(N) by

    μ_w,a : ℂ(N) →ℂ(N),    x^v ↦ x^va^w(v).

Let f∈ℂ[N]. We say f is mutable with respect to (w,a) if mut_w,a(f) ∈ℂ[N], in which case we call mut_w,a(f) a mutation of f.



The wall crossing morphisms θ_𝔡 are equal to mutations mut_w,a with w=n_𝔡 and a=f_𝔡. The Laurent phenomenon of <cit.> in our case means that in any chamber inside Δ^⋆ the potential ϑ_1(X) is a Laurent polynomial and mutable with respect to its boundary walls, so that it stays a Laurent polynomial after mutation.




If Δ_1^⋆ and Δ_2^⋆ are related by a mutation of Laurent polynomials supported on them, then they are ℚ-Gorenstein deformation equivalent. In particular, mutation exchanges toric models of a variety X.



For a Hirzebruch surface 𝔽_m, the potential ϑ_1(𝔽_m) is a mutation of ϑ_1(𝔽_m-2) with respect to w=(-1,1) and a=1+z_1/xy under the change of variables 

    ϑ_1(𝔽_m)    mut_(-1,1),1+z_1/xyϑ_1(𝔽_m-2) 
    
    x    ↦ y^-1
    
    y    ↦ x 
    
    z_1    ↦ z_1 
    
    z_2    ↦ z_1^-1z_2

The map (x,y) ↦ (y^-1,x) rotates the Newton polytope of ϑ_1(X)_0, but doesn't change the potential. The map (z_1,z_2) ↦ (z_1,z_1^-1z_2), gives the change of curve classes F,E ↦ F,E-F, as in <ref>. For m≤ 5 the mutations are shown in Figures <ref>, <ref>, and from this the potentials for m≤ 9 are anticipated in Figures <ref>, <ref>.












 §.§ Toric models of non-Fano varieties: a conjecture




X has toric model X_Σ, where Σ is the spanning fan of the Newton polytope of ϑ_1(X)_0. In particular, two (not necessarily Fano) varieties X and Y are ℚ-deformation equivalent if and only if their potentials ϑ_1 are mutation equivalent.



Note that if the conjecture is true, then X has a Fano toric model if and only if ϑ_1(X)_0 is mutation equivalent to a Laurent polynomial whose Newton polytope is reflexive, as in the case of Hirzebruch surfaces above.



If two (not necessarily Fano) varieties X and Y have mutation equivalent potentials ϑ_1(X)_0 and ϑ_1(Y)_0, then their periods, disk invariants, and mirror maps agree.




 §.§ Mirror symmetry and rigid MMLPs


It was proposed in <cit.> that the mirror partner of a Fano variety X is a Laurent polynomial f such that the classical period of f equals the regularized quantum period of X.



The classical period of a Laurent polynomial f∈ℂ[x_1^±1,…,x_n^±1] is defined as

    π_f(t) = (1/2π i)^n ∫_|x_1|=…=|x_n|=11/1-tfdx_1/x_1⋯dx_n/x_n,    t ∈ℂ, |t| ≪∞.

This has Taylor expansion

    π_f(t) = ∑_k=0^∞coeff_1(f^k)t^k.





The period π_ϑ_1(X) does not depend on the chamber of 𝒮_∞(X) in which ϑ_1(X) is defined.



The regularized quantum period of a Fano manifold X is defined as

    Ĝ_X(t) = 1+∑_k=2^∞ k!c_kt^k,

where c_0=1, c_1=1, and for k≥ 2,

    c_k = ∫_ℳ_0,1,(k)ψ^k-2ev^⋆[pt]

are the descendant Gromov-Witten invariants of X.



A Laurent polynomial f is a mirror partner for X if π_f=Ĝ_X.


It was conjectured in <cit.> that Fano manifolds correspond under mirror symmetry to mutation equivalence classes of rigid maximally mutable Laurent polynomials (rigid MMLPs). This leads us to the following conjecture.


A Laurent polynomial f is maximally mutable if it is maximal with respect to the partial ordering defined by label-preserving injections of mutation graphs. Then it is called rigid if it is uniquely determined by its Newton polytope, mutation graph and the coefficient of its constant term being 0.




The potential ϑ_1(X)_0 is a rigid maximally mutable Laurent polynomial.



Conjecture <ref> is true for Fano surfaces and Fano threefolds. In this case ϑ_1(X)_0 is the toric potential and the conjecture of <cit.> is known for these cases.



If Conjecture <ref> is true, then Conjecture <ref> is true for all surfaces and threefolds that admit a Fano toric model, for the same reasons.



Conjecture <ref> is true for all Hirzebruch surfaces 𝔽_m, since they have a Fano toric model 𝔽_0 or 𝔽_1.




§ MIRROR SYMMETRY





 §.§ The LG potential equals the open mirror map



The open mirror map of X with respect to a potential function W is defined as q_i=z_ie^F_W(z), where F_0(z) is the x,y-constant term of -log(1-W), that is,

    F_W(z) = ∑_k>01/k_1(W^k).

Write M_W(q):=e^F_W(z(q)) for the open mirror map after inserting the inverses of the closed mirror maps q_i=z_ie^d_iF_W(z), where d_i=β_i· D.



Note that coeff_1(ϑ_1^k) has a t^k-factor, so M_t^-1ϑ_1(q) does not depend on t. Further note that by Proposition <ref> M_t^-1ϑ_1(q) does not depend on the chamber of 𝒮_∞(X) in which ϑ_1 is defined.



Let Σ' be the fan of K_X_Σ, where X_Σ is a toric model of X. Then
log(z_0)+F_W_Σ'(z) is a solution to the Picard-Fuchs equation defined by the Mori vectors coming from K_X_Σ.




Under the change of variables q_i=z_i(t/y)^d_i, with d_i=β_i· D, we have

    ϑ_1(y)_∞ = yM_t^-1ϑ_1(q).



To prove Theorem <ref> we first we need two lemmas.


For a Laurent series f(x)=∑_k≥ k_min f_kx^k write [x^k]f=f_k.




Let f(y)=y^-1+∑_k≥ 0f_ky^k be a Laurent series with only simple poles. Then the composition inverse g(z) satisfying f(g(z))=z is a power series in z^-1 given by

    g(z^-1) = ∑_k> 0z^-k/k[x^-1]f^k.




Look at the multiplicative inverse

    1/f(y)=y/h(y),    h(y)=yf(y)=1+∑_k≥ 0f_ky^k+1.

This is a power series with nonzero linear term, and by the Laurent inversion theorem we have 1/f(g(z))=z for

    g(z) = ∑_k>0z^k/k[x^k-1]h^k = ∑_k>0z^k/k[x^-1]f^k.

The equation 1/f(g(z))=z is equivalent to f(g(z))=z^-1 and to f(g(z^-1))=z. So the inverse of f(y) is g(z^-1) as claimed.



Note that the Laurent series f(y) is not bijective, so there is no global inverse. The inverse is around z=0 for which 1/f(g(z)) is finite.




Let f(x)=1+∑_k≥ 1f_kx^k be a power series. Then

    exp(∑_k>01/k[x^k]f^kz^k) = ∑_k>01/k[x^k-1]f^kz^k-1.




This is proved in <cit.> as a consquence of certain Bell polynomial identities.



Note that the closed mirror map is q_i=z_iM_t^-1ϑ_1(q)^d_i, so that the change of variables q_i=z_i(t/y)^d_i means t/y=M_t^-1ϑ_1(q) or

    y(t) = tM_t^-1ϑ_1(q)^-1 = te^-F_0(z) = te^-∑_k>01/k[y^0]ϑ_1^kt^-k.

Note that [y^0]ϑ_y^k has a t^k-factor and F_0(z) does not depend on t. Now the claimed equation is

    ϑ_1(te^-F_0(z)) = t.

Consider the Laurent series f(y)=ϑ_1(y^-1). By Lemma <ref>, the function g(t) satisfying f(g(t))=t is given by

    g(t) = ∑_k>01/k[y^-1]f^kt^-k.

The claimed formula is equivalent to y(t)=g(t)^-1, hence to y(t^-1)^-1=g(t^-1), which is

    texp(∑_k>01/k[y^0]f^kt^k) = ∑_k>01/k[y^-1]f^kt^k.

This follows from Lemma <ref> applied to the power series yf(y).



We have

    M_t^-1ϑ_1(q) = 1+∑_β∈ NE(X) (β· D-1)R_β· D-1,1(X,β)q^β.




Combine Theorem <ref> and Corollary <ref>.




 §.§ Local mirror symmetry and slab functions



The following idea is taken from <cit.>. Let W be a potential, e.g. the Hori-Vafa potential

    W_Σ = 1 + ∑_i=1^n x_i + ∑_i=1^ρ z_ix^v_i

The Gross-Siebert potential <cit.> is the correction

    W_GS = 1 + ∑_i=1^n x_i + ∑_s=1^ρ z_ix^v_i + h(z),

where h(z) is defined by the normalization condition that log(W_GS) has no x_i-constant terms. Let x̌_i=e^F_W(z)x_i be the open mirror map. The closed mirror map is q_i(z) = z_ie^d_iF_W(z), hence

    z_i(q) = q_ie^-d_iF_W(z(q)) = q_ie^(∑_j=1^n v_i,j-1)F_W(z(q))

Then the Gross-Siebert potential becomes

    W_GS   = 1+ h_W(z(q)) + ∑_i=1^n x_i + ∑_i=1^ρ e^-F_W(z(q))q_i(e^F_W(z(q))x)^v_i
       = e^-F_W(z)((1+h_W(z(q)))e^F_W(z(q)) + ∑_i=1^n e^F_W(z(q))x_i + ∑_i=1^ρ q_i(e^F_W(z(q))x)^v_i).

By the normalization condition the constant term of the logarithm of the term in the large bracket must be F_W(z), but this means (1+h_W(z(q)))e^F_W(z(q))=1. Combining this with Theorem <ref> we see that the slab function h_W(q) is (up to sign) equal to the open mirror map. Note that we could have started with another potential W instead of the Hori-Vafa potential W_Σ. Hence, we have the following.



We have

    1+h_W(z(q)) = M_W(-q).





 §.§ The open mirror map and disk counting


It was shown in <cit.> that in the Fano case the Gross-Siebert slab function h(z(q)) is a generating function for open Gromov-Witten invariants on the local Calabi-Yau K_X, the total space of the canonical bundle of X. The discussion in <ref> together with Corollary <ref> implies that h(z(q)) is a generating function for 2-marked logarithmic Gromov-Witten invariants (up to sign and some factor). This is consistent with the log-open correspondence (see e.g. <cit.>).

In the non Fano case, if one uses the corrected potential ϑ_1(X)_0 instead of the toric potential W_Σ to define h(z(q)), then h(z(q)) is still a generating function for 2-marked logarithmic Gromov-Witten invariants for the same reason. This gives the following.


We have h_t^-1ϑ_1(z(-q))=ϑ_1(y)_∞ under the change of variables q_i=z_i(t/y)^d_i, hence

    h_t^-1ϑ_1(z(q)) = ∑_β∈ NE(X) (-1)^β· D(β· D-1)R_β· D-1,1(X,β)q^β.



The log-open correspondence is not valid in the non-Fano case, because curves can move away from the zero section of K_X. We conjecture, that the result of <cit.> is still valid in the non-Fano case if one uses the Hori-Vafa potential W_Σ', where Σ' is the fan of K_X_Σ for some toric model X_Σ of X.



We have

    h_W_Σ'(z(q)) = ∑_β∈ NE(X) N_1(X,β)q^β,

where N_1(X,β) is the open Gromov-Witten invariant of K_X with winding one.


halpha

