Landau-Ginzburg Mirror Symmetry Conjecture

We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.


Introduction
Mirror symmetry has been a driving force in geometry and physics for the last twenty years. During that time, we have made tremendous progress in our understanding of mirror symmetry, but several important mathematical questions remain unanswered.
Historically, mathematical research focused on mirror symmetry between Calabi-Yau/Calabi-Yau models or Toric/Landau-Ginzburg models, rarely investigating the Landau-Ginzburg pairs. This was mainly due to the lack of a mathematical theory for a Landau-Ginzburg (LG) Amodel, although there were geometric realizations of the Landau-Ginzburg B-model in various contexts. In the mid 2000's, Fan, Jarvis and Ruan invented FJRW theory [13] motivated by the physical work [52] of Witten. This invention is a mathematical theory for a Landau-Ginzburg Amodel, allowing mathematicians to investigate mirror symmetry between two Landau-Ginzburg models. In this paper, we prove a general LG/LG mirror theorem, which can be viewed as a Landau-Ginzburg parallel of the mirror theorem [17,18,[33][34][35][36] between Calabi-Yau manifolds established by Givental and Lian-Liu-Yau. For a survey on the LG/LG mirror symmetry and an outline of the current and related works, see [32]. For several purely algebraic constructions of LG A-model and their relationships with FJRW theory, see [39] and [7].
The LG/LG mirror pairs originate from an old physical construction of Berglund-Hübsch [4] that was completed by Krawitz [27]. Let us briefly review this construction, called the BHK mirror [10]. Let W : C N → C be a quasihomogeneous polynomial with an isolated critical point at the origin. We define its maximal group of diagonal symmetries to be (1) G W = (λ 1 , . . . , λ N ) ∈ (C × ) N W (λ 1 x 1 , . . . , λ N x N ) = W (x 1 , . . . , x N ) .
In the BHK mirror construction, the polynomial W is required to be invertible [10,27], i.e., the number of variables must equal the number of monomials of W . By rescaling the variables, we can always write W as We denote its exponent matrix by E W = (a ij ) N ×N . The mirror polynomial of W is i.e., the exponent matrix E W T of the mirror polynomial is the transpose matrix of E W . The mathematical LG A-model is the FJRW theory of (W, G W ), and one geometry of the LG B-model is the Saito-Givental theory of W T , where the genus zero theory is Saito's theory of primitive forms of W T [43] and the higher genus theory is from the Givental-Teleman's formula [20,49]. There is a longstanding conjecture that these A-and B-models are equivalent. We remark that FJRW theory is defined in [13] for any pair (W, G) where G is an admissible subgroup of G W . The BHK mirror construction applies in this more general situation, yielding a mirror partner (W T , G T ) where G T is a well-defined group dual to G constructed in [3,27]. Although G is never a trivial group, G T could be a trivial group. Further more, G T is a trivial group if and only if G = G W . If G T is nontrivial, we do not know the full mathematical construction of LG B-model for (W T , G T ). For this reason, Conjecture 1.1 is stated only for G W on the A-side.
Krawitz [27] gave a clue to Conjecture 1.1 by finding an explicit isomorphism that matches the FJRW ring of (W, G W ) to the Jacobi algebra of W T for almost all invertible polynomials (when no variables of W have weight 1/2); see Theorem 2.14.
However, it is much harder to work on the whole conjecture, which requires a thorough understanding of both FJRW theory and Saito-Givental theory. Although the powerful Givental-Teleman formula [20,49] reduces Conjecture 1.1 to its genus zero part, Conjecture 1.1 was proved only in a handful of cases previous to our work. These include A-type singularities [12,24], ADE (or simple) singularities [13], simple elliptic singularities [28,38], and exceptional unimodular singularities [31]. All these cases were proved with case-by-case calculations and reconstruction on both sides of the mirror. They have one common feature: the central charge of W is very small (in fact all no greater than 7 6 ). Because of their small central charges, these singularities have special structure, which was used in a critical way to prove Conjecture 1.1. The authors of this paper faced several difficulties in extending the techniques from these earlier special results to general singularities with arbitrarily large central charges. We explain two major difficulties and our solutions here: (1) B-side: a choice of primitive forms.
Genus zero invariants of the Landau-Ginzburg B-model are determined by a choice of primitive forms, which is equivalent to an appropriate choice of splitting of the Hodge filtration associated to the singularity. Different primitive forms/splittings lead to different invariants. This is related to the famous holomorphic anomaly [5]. A standard way to identify primitive forms is in terms of the good basis [43] of the Brieskorn lattice of the singularity. Unfortunately, prior to this work, an explicit good basis was known in only a few cases, due to the difficulty of computing higher residue pairings [44]. This has long been a major difficulty in studing Landau-Ginzburg B-model invariants. For ADE and exceptional unimodular singularities, there is a unique good basis simply by the degree constraints. For simple elliptic singularities, the correct primitive form is checked in [28,38] by hand.
In Theorem 2.10 of our paper, we identify explicitly a good basis for every invertible polynomial that is mentioned in Conjecture 1.1. The good bases defined in Theorem 2.10 produce the correct genus zero invariants to mirror FJRW theory (Theorem 3.2). To establish Theorem 2.10, we find a refinement of the degree constraints using the maximal symmetry group. This allows us to compute higher residue pairings explicitly for all invertible polynomials and study their Hodge-theoretic properties in terms of concrete data.
Aside from the algebraically nice (concave) invariants that can be computed by orbifold Grothendieck-Riemann-Roch formulas, it is very difficult to directly compute FJRW invariants. All previous results have used case-by-case methods to reconstruct genus zero invariants from concave ones. This method grows intractably complex as the central charge increases. In fact, the degree constraints (which is the most powerful tool in small central charges) is no longer useful since in many cases we will inevitably run into some FJRW invariants which are not known how to compute based on the current technique.
In this work, we systematically explore the combinatorial properties of Landau-Ginzburg models. Using the classification from [29] of invertible polynomials in terms of atomic types (see Theorem 2.1), we prove Theorem 3.3, a strong reconstruction and computation theorem for both A-model and B-model invariants. This theorem states that for any invertible polynomial, the full genus zero data is determined by its Frobenius algebra and some special invariants (called 4-point correlators) which are of atomic type only. The A-side special invariants can be exactly calculated by algebraic methods. This solves the A-side computation problem. Our main result in this paper is; Theorem 1.2. The LG mirror symmetry conjecture holds for all invertible polynomials at all genera when no variables of W have weight 1/2.
We highlight key ingredients toward establishing this theorem here: • (Good basis): Krawitz's mirror map sends a natural basis in A-model to a good basis in B-model (Theorem 2.10). In this way, the A-model determines a special good basis and corresponding primitive form, which lead to a concise computation in the B-model. • (Vanishing conditions): Because of its deformation-theoretic origins, the B-model has a certain G W T -symmetry, which forces many invariants to vanish (Lemma 4.1). The corresponding invariants vanish on the A-side for geometric reasons. These vanishing results allow us to state pleasant combinatorial properties of non-vanishing invariants (Lemma 5.3). • (Splitting principle): The vanishing conditions together with the WDVV equations lead to a crucial Splitting Principle (Proposition 5.18): in order to compute all genus zero invariants, it is enough to compute the invariants of the (simpler) atomic polynomials with some special properties. • (Atomic reconstruction): Using exhaustive reconstruction techniques, we reconstruct all genus zero invariants for atomic polynomials from a few special invariants, listed in part (1) of Theorem 3.3. Combined with the Splitting Principle, this shows that all genus-zero invariants are determined by these special invariants. • (Atomic formulas): We evaluate on both the A and B side the special invariants that remain after the atomic reconstruction. This is parts (2) and (3) of Theorem 3.3.
Finally, we remark that the special cases left out in Theorem 1.2 are only the invertible polynomials containing a special chain summand 1 W = x a 1 1 x 2 + x a 2 2 x 3 + · · · + x a N−1 N −1 x N + x a N N with a N = 2. Our reconstruction result in fact shows that Conjecture 1.1 holds once the Frobenius algebras and some genus zero 4-point invariants are identified. Two such examples are the exceptional unimodular singularities W = Z 13 , W 13 , for which Conjecture 1.1 was proved in [31]. The identification of the Frobenius algebras of the other special cases requires the computation of some unknown FJRW invariants.
Outline. In Section 2, we review the A-model FJRW-theory and B-model Saito-Givental theory as well as the mirror construction. In Section 3, we outline the proof of the main theorem via several reconstruction results. In Section 4, we find a good basis using Krawitz's mirror map and explore combinatorial properties of non-vanishing invariants. In Section 5, we develop technical preparations for our reconstruction theorem, including WDVV equations, Splitting principle, and atomic reconstruction of Fermat type. We also prove the conjecture for Fermat polynomials as a warm-up towards the general cases. In Section 6, we prove the atomic formulas via explicit calculations on both sides. In Section 7, we complete a proof of Theorem 1.2 via atomic reconstructions of chain type and loop type.
The numbers q 1 , . . . , q N are called the weights of W . The central charge of W , which can be thought of as the "dimension" of the LG theory, is defined by We call W nondegenerate if it has an isolated critical point at the origin and it contains no monomial of the form x i x j for i = j. This implies that each weight is unique and q j ∈ Q ∩ (0, 1 Finally, we define G W to be the maximal group of diagonal symmetries of W in (1). Since our goal is to prove the LG Mirror Symmetry Conjecture 1.1, in what follows we only discuss the FJRW theory of (W, G W ) for invertible polynomials W with the form in (2).
2.1.1. The state space. The FJRW theory of a pair (W, G W ) is a state space 2 H W and a cohomological field theory {Λ W g,k }, which is a set of linear maps for 2g − 2 + k > 0. Here M g,k is the moduli space of stable k-pointed curves of genus g. The state space is defined as Here Fix(γ) is the fixed locus of γ and N γ is its dimension as a C-vector space. Furthermore, W γ is the restriction of W to Fix(γ), and W ∞ γ is Re(W γ ) −1 ((M, ∞)) for M ≫ 0. Thus, H W is the dual to the space of Lefschetz thimbles.
For each class ξ ∈ H γ , we call γ the sector of ξ. If Fix(γ) = 0 ∈ C N , we say that γ is narrow ; otherwise we say it is broad. Note that if γ is narrow then H γ is 1-dimensional.
There is an alternative expression for H W . Let be the Jacobi algebra of W . It is a theorem of Wall (see [50] and [51]) that the vector space H Nγ (C N , W ∞ γ ; C) is isomorphic to Jac(W γ )dx γ , where dx γ is the product of the differentials of the variables fixed by γ. Thus, With this identification, we write ξ = ⌈m ; γ⌋ where ξ corresponds to the monomial m ∈ Jac(W γ ).
We define a grading on H W as follows. Since G W is a finite abelian group, for any element γ ∈ G W , we may write γ = exp(2π Note that the degree of ξ depends only on its sector. We have a pairing η γ : H γ × H γ −1 → C which is induced by the intersection pairing on Lefschetz thimbles. The direct sum of these pairings gives us a nondegenerate pairing Under the identification of H γ with (Jac(W γ )dx γ ) G W , this pairing is equal to the residue pairing on differential forms. See [6,9] for expositions of this fact.

2.1.2.
The cohomological field theory. The construction of the cohomological field theory {Λ W g,k } is highly nontrivial. We will only summarize it here, and refer the interested reader to the original papers [13] and [14] for more details.
The construction uses the moduli space of stable W -orbicurves. Let C be a stable orbicurve of genus g with marked points p 1 , . . . , p k . At each marked point and node we have a local chart C/Z m for some positive interger m. We require that the actions on the two branches of a node be inverses.
Let ρ : C → C be the forgetful morphism from the orbifold curve C to the underlying coarse curve.
where K C is the canonical bundle of C and O(p j ) is the holomorphic line bundle of degree one whose sections may have a simple pole at p j . If the local group at a marked point of an orbicurve is Z m , the line bundles L 1 , . . . L N induce a representation Z m → (C × ) N . The representation is required to be faithful. The image of this representation will always be in G W . The image of 1 ∈ Z m singles out some γ ∈ G W at each marked point; these group elements are called the decorations.
Given an invertible polynomial W , the moduli space of pairs (C , L) is called the moduli space of stable W -orbicurves and denoted by W g,k . According to [13], it is a Deligne-Mumford stack, and there is a forgetful morphism st : W g,k → M g,k . The forgetful morphism is flat, proper, and quasi-finite (see Theorem 2.2.6 of [13]). The decorations γ i at the marked points p i decompose W g,k into open and closed substacks W g,k (γ 1 , . . . , γ k ). Furthermore, the stack W g,k (γ 1 , . . . , γ k ) is stratified, and each closure in it is denoted by W g,k (Γ γ 1 ,...,γ k ) for some Γ γ 1 ,...,γ k . Here Γ γ 1 ,...,γ k is called a G W -decorated dual graph of an underlying stable curve of genus g and k marked points. We call Γ γ 1 ,...,γ k fully G W -decorated if we assign some γ + ∈ G W and γ − = γ −1 + on two sides of each node.
In [14] the authors perturb the polynomial W to polynomials of Morse type and construct virtual cycles from the solutions of perturbed Witten equations. That is, they construct As a consequence, they obtain a cohomological field theory {Λ W g,k : Here PD is the Poincaré dual and J W is the exponential grading operator, defined by The FJRW potential. The cohomological field theory allows us to define FJRW invariants (or genus-g k-point correlators) as Here ψ i := c 1 (L i ) is the i-th psi class, where L i is the i-th tautological line bundle on M g,k .
The invariant is primary if there are no psi classes, i.e., ℓ i = 0 for all 1 ≤ i ≤ k. We call the classes ξ 1 , . . . , ξ k the insertions of the correlator.
The FJRW invariants induce various structures on H W . The pairing , and the primary genus-zero 3-point correlators define a product ⋆ on H W , by (6) α ⋆ β, γ = α, β, γ W 0 , where α, β, γ ∈ H W . This definition makes the pairing Frobenius with respect to ⋆, so that the FJRW ring (H W , ⋆) is a commutative and associative Frobenius algebra with the unit ⌈1 ; J W ⌋.
The primary genus-zero correlators define a Frobenius manifold structure on H W . Let B be a set whose elements are a basis for H W . The pre-potential of the Frobenius manifold is The Frobenius manifold pre-potential encodes the genus-0 data of the FJRW theory of (W, G). The FJRW invariants of all genera are encoded in the total ancestor FJRW-potential via the isomorphism (⌈m ; γ⌋, ⌈n ; δ⌋) → ⌈mn ; γδ⌋. Moreover, Second, certain vanishing properties of the FJRW correlators will be critical when we reconstruct the pre-potential in (7). In the A-model, these come from two of the so-called correlator "axioms", which are summarized in the following proposition. γ i be the j-th phase of γ i ∈ G W . If ξ 1 , . . . , ξ k W 0 = 0, then the following equalities hold: Formula (8) is called the Dimension Axiom because it is a consequence of the degree of the class Λ W 0 (ξ 1 , . . . , ξ k ). Formula (9) is called the Integer Degree Axiom because l j is the degree of the line bundle ρ * L j on the underlying coarse curve, when that curve is smooth. Formula (9) follows from the fact that line bundles must have integer degrees, so if l j ∈ Z then the corresponding component of W 0,k is empty. We call l j the j th line bundle degree of ξ 1 , . . . , ξ k W 0 .
Remark 2.5. One useful application of formula (9) is due to Krawitz: if the correlator ξ 1 , ξ 2 , ξ 3 W 0 is nonzero and ξ i ∈ H γ i , then γ 3 = J W (γ 1 γ 2 ) −1 . Then from (6) and the definition of the pairing, In the remainder of this paper, we will only use primary genus-zero correlators, so we will drop the genus-subscript g from the correlator notation. Moreover, when context makes the polynomial clear we will suppress W , writing a genus-0 A-model correlator as ξ 1 , . . . , ξ k .
2.2. B-model: Saito-Givental theory. In this section, we follow the B-model convention and use f for a quasihomogeneous polynomial with isolated singularity at the origin: Outside of this section, f ≡ W T , and The Frobenius algebra structure of the B-model is simply Jac(f ) with the grading coming from the quasihomogeneous weights, equipped with the residue pairing. Note that Jac(f 1 ⊕f 2 ) = Jac(f 1 ) ⊗ Jac(f 2 ) (compare Theorem 2.2).
The genus zero invariants (or the Frobenius manifold structure) are induced from Saito's theory of primitive forms [43]. Since the Frobenius manifold is generically semisimple, the higher genus invariants are given by the famous Givental-Teleman formula [20,49].
2.2.1. Saito's triplet for primitive forms: Brieskorn lattice, higher residue pairing and the good basis. Here we review the basics of Saito's theory of primitive forms. Because we wish to prove Conjecture 1.1, we will only discuss the theory for quasihomogeneous f . See [43,47,48] for discussions of arbitrary isolated singularities.
Let Ω k C N ,0 be the space of germs of holomorphic k-forms at the origin in C N . Define which is a formally completed version of the Brieskorn lattice associated to f (see [44]). Here z is a formal variable. There exists a natural semi-infinite Hodge filtration on H (0) f given by H We define a natural Q-grading, or weight, on Jac(f ), on H (0) f , and on Ω f which is generated by (10) wt In [44], K. Saito constructs a higher residue pairing K f : ] satisfying the following properties.
(4) The leading z-order of K f defines a pairing which coincides with the usual residue pairing Ω f ⊗ Ω f → C.
The last property implies that K f defines a semi-infinite extension of the residue pairing, which explains the name "higher residue". Following [43], we define a good section and a good basis for f .

Definition 2.6 (Good basis). A good section σ is a splitting of the projection H
f , such that σ preserves the Q-grading, and K f (Im(σ), Im(σ)) ⊂ z N C. A basis of the image Im(σ) of a good section σ is a good basis of H Equivalently, a good basis consists of homogeneous elements {η α } ⊂ H (0) f such that {η α } represents a basis of Ω f and K f (η α , η β ) ∈ z N C for all α and β.
Example 2.7. The ADE singularities are those for whichĉ f < 1. For these singularities any homogeneous basis of Ω f is a good basis, and any two such choices are "equivalent" (i.e. there exists a unique good section) [43].
Proof. It follows from the construction of the higher residue pairing in [44] that The proposition is a direct consequence of this equality.
A good basis is not unique in general. Landau-Ginzburg mirror symmetry favors a particular choice of good basis, which we call the standard basis. This basis was used by Krawitz in [27] to describe the mirror map between Frobenius algebras. We define the standard basis for an atomic polynomial below, and we get a basis for a general invertible polynomial with Proposition 2.8.
In this definition and later, we use φ f to denote the element of the standard basis that spans the 1-dimensional subspace of Jac(f ) of highest degree. It is a fact that wt(φ f ) =ĉ f . Definition 2.9. The standard basis of an atomic polynomial f is {φ α } µ α=1 , where µ = dim C Jac(f ), φ µ = φ f , and the monomials φ α are defined as follows. • Because we are interested in mirror symmetry, the forms here are dual to the forms in Theorem 2.1 See Example 2.13 for further clarification. This theorem will be proved in Section 4.1.
As shown in [43], a good basis of f gives rise to a primitive form, which is a certain family of holomorphic volume forms with respect to a universal unfolding of f . The primitive form induces a Frobenius manifold structure on Jac(f ) (which was called a flat structure in [43]). We will not give the precise definition of primitive form here. Instead, we present a perturbative description developed in [30,31] which is a formal solution of the Riemann-Hilbert-Birkhoff problem described in [43]. We also use the perturbative description of the primitive form to compute the invariants of our Landau-Ginzburg B-model.

2.2.2.
A perturbative formula. Given a polynomial g(x), we will denote [g(x)d N x] its class in H ). Let s = {s α } be the linear coordinates on Jac(f ) dual to the basis {φ α }, so the coordinates s parametrize a local universal deformation F = f + α s α φ α of f . The following formula gives a perturbative way to compute the associated primitive form.
Furthermore, ζ is the series expansion in s of the primitive form associated to the good basis B, and J plays the role of the FJRW J-function in the following sense. By Theorem 2.12, we may write

Let
(13) t α (s) = J α −1 (s) ∈ C s . We call t = {t α } the flat coordinates for Jac(f ). In fact (14) t α = s α + O(s 2 ), and we may write each s α as a function of t. Then in terms of the flat coordinates, the Frobenius manifold prepotential F SG 0,f,ζ associated to the primitive form ζ satisfies where η is the matrix in Definition 2.11. The B-model correlators are defined via The proof of Theorem 2.12 in [31] outlines an algorithm for recursively solving ζ and J as follows. Let ζ (≤k) be the k-th Taylor expansion in terms of s. To zeroth order (in s), equation (11) is Because ζ has only positive powers of z, this is uniquely solved by ζ (≤0) = [d N x]. Suppose we have solved for ζ (≤k) , which satisfies k+1 is the part with nonnegative powers of z. Then ζ (≤k+1) = ζ (≤k) − R + k+1 uniquely solves Equation (11) up to order k + 1 in s.

B-model Saito-Givental potential.
Saito's theory of primitive forms gives the genus zero invariants (see Formula (16)) in the LG B-model. For higher genus, Givental [20] proposed a remarkable formula for the total ancestor potential of a semi-simple Frobenius manifold. The uniqueness of Givental's formula was established by Teleman [49]. According to the work of Milanov [37], the total ancestor potential can be extended uniquely to the origin, which is a nonsemisimple point we are interested in.
Saito's genus zero theory together with the total ancestor potential is now referred to as the Saito-Givental theory of a singularity. We will call the extended total ancestor potential at the origin a Saito-Givental potential and denote it by A SG f,ζ , where the subscript ζ shows its dependence on the chosen primitive form ζ.

2.3.
Krawitz's mirror map. Recall that given an invertible polynomial its exponent matrix is E W := (a ij ) N ×N , and the mirror polynomial (also called the transpose polynomial) W T is defined by The inverse matrix E −1 W plays an important role in the mirror map constructed by Krawitz in [27]. Let us write and define According to [27], the group G W is generated by {ρ j } N j=1 and G W T is generated by {ρ T j } N j=1 . Recall q j is the weight of x j in W . Let q T j be the weight of x j in W T . We remark that Example 2.13. The transpose of the chain polynomial in Theorem 2.1 is is a degree-preserving isomorphism of Frobenius algebras, in the sense that wt(φ) = deg W (Ψ(φ)) for every monomial φ ∈ Jac(W T ). Furthermore, We will call Ψ "Krawitz's mirror map", or simply "the mirror map." In this paper, we show that by appropriate rescaling 3 , Krawitz's mirror map identifies the FJRW and Saito-Givental potentials of all genus, proving mirror symmetry. From now on, for any monomial φ ∈ Jac(W T ), we will use the following notation for the degree: Remark 2.15. When W = j W j , both the A and the B-model Frobenius algebras decompose as tensor products of the Frobenius algebras of the W j , and in this case the mirror map is a tensor product of mirror maps.

Main results
The main result of this paper is Theorem 1.2, which can be more precisely stated as In fact, it suffices to prove this theorem at the level of Frobenius manifolds, i.e., at genus zero. This is because in the cases we deal with, the work of Teleman [49] and Milanov [37] shows that the genus zero data completely determines the higher genus data of the LG models. Thus, in the remainder of this article, we only need to prove the following theorem.  (22) F SG 0,W T ,ζ = F FJRW 0,W . As explained in Section 2.2, a primitive form is associated to a good basis. The good basis yielding mirror symmetry in Theorem 3.2 is the standard basis of Definition 2.9.
Theorem 3.2 is proved by showing that F SG 0,W T ,ζ and F FJRW 0,W are completely determined by a handful of 4-point correlators. We then explicitly compute these correlators to show they differ only by a sign. We may exactly match the potentials by rescaling the primitive form and the B-model ring generators as in [13, Section 6.5], (23) x Thus, Theorem 3.2 is a consequence of the following theorem.  3 The rescaling consists of Formula (23) and Formula (102).
Here we use B-model notation, and φ W T is the element in Jac(W T ) of highest degree, normalized as in Definition 2.9. The A-model correlators are obtained by mapping the insertions via Krawitz's mirror map in Theorem 2.14.
(2) The values of these correlators are q i on the A-side. (3) The values of these correlators are −q i on the B-side.
Remark 3.4. The correlators in Theorem 3.3 may be described as is any monomial of a Fermat or loop summand, or the final monomial of a chain summand.
. Similar notation will be used throughout the paper. Such a formulation of correlators and their values was first discovered for simple elliptic singularities in [38] and then verified for exceptional unimodular singularities in [31].

Good basis and B-model vanishing
In this section, we introduce some tools in the B-model for the proof of Part (1) of Theorem 3.3. More explicitly, we will use the symmetries of an invertible polynomial to prove Theorem 2.10 and establish the Dimension Axiom and Integer Degree Axiom in the B-model (Lemma 4.1).

Good basis of invertible polynomials.
In this subsection we prove Theorem 2.10. As a consequence, we obtain a Frobenius manifold structure on the base space of the universal unfolding of the corresponding singularity. This structure can be computed perturbatively as described in Section 2.2.2, furnishing the genus zero data in the B-model. We will adopt the same notation as in Section 2.2 and write f instead of W T for the mirror polynomial.
We only need to prove Theorem 2.10 for chains and loops, since Fermat polynomials are the A-type singularities discussed in Example 2.7. We will use the following notation: (2) The linear coordinates on C N are x 1 , · · · , x N and x N +k ≡ x k .
In the notation of Section 2.3, the inverse of the exponent matrix of f = W T is Let ρ T j be the linear transformation f . Moreover, it is easy to see that the higher residue pairing K f is ρ T j -invariant. These symmetries are enough to prove that the standard basis is a good basis. Let N be monomials in the standard basis for either the chain or loop type. Let (m 1 , · · · , m N ) = (r 1 + r ′ 1 , · · · , r N + r ′ N ).
for all j (the extra 2 comes from two copies of d N x). This is equivalent to The remainder of the proof splits into two cases, corresponding to the possible types of f .
The exponent matrix has the form Equation (24) in this case becomes This can not appear if both x r 1 1 · · · x r N N and x N are in the standard basis. For cases (1) and (2), we check directly that With the convention k 1 ≡ k N +1 , equation (24) above implies Let h i = k i − 1 for each i. Equation (25) becomes If there is some h i+1 = 0, then the above equation implies h i = 0, and recursively, Otherwise, we can assume none of the h i is zero. There are two situations. Either there is one h i with |h i | = 1 or all |h i | ≥ 2. For the first case, we assume some h i+1 = ±1. Since h i = 0 by assumption, the inequality (26) implies h i = ∓1. We can repeat this process and get the following solution when N is an even number: Finally we prove it is impossible to have all |h i | ≥ 2. Equation (26) implies If all |h i+1 | ≥ 2, this implies In fact, if h i+1 ≥ 2, then the RHS of inequality (27) implies h i < 1. By assumption, we know inequality (28) follows from the LHS of (27). A similar argument works for h i+1 ≤ −2. We repeat this process and we find which is impossible. Thus the only possibilities for the k i 's are By the same degree reason as in the chain case, we know

4.2.
Vanishing conditions in B-model. We will now prove the B-model properties that are the analogs of the Dimension Axiom (8) and Integer Degrees Axiom (9) on the A-side. These give us vanishing conditions for B-model correlators which we will later use to reconstruct the potential F SG 0,W T ,ζ . Lemma 4.1. Let Ψ be Krawitz's mirror map (Theorem 2.14). The A-model correlator and Ψ(X) satisfies the Integer Degrees Axiom (9) if and only if Moreover, if the B-model correlator x e k,i i is nonzero, then both (30) and (31) hold.
Proof. The equivalence of (8) and (30) follows from the fact that Ψ is degree-preserving (Theorem 2.14) andĉ W =ĉ W T . Also from Theorem 2.14 we know By directly calculating the quantity l j in (9) using (18) and (5), we get This is exactly Equation (31). Now assume X = 0. Then (30) holds because the potential F SG 0,W T ,ζ has eigenvalueĉ W T − 3 with respect to the Euler vector field µ α=1 (1−deg(φ α ))s α ∂ ∂sα . This well-known fact also follows explicitly from the perturbative formula (11) which respects the Q-grading.
Finally, we prove that if X = 0 then (31) holds. To do so we introduce a G W T -action on the B-model. Since G W T is generated by {ρ T j } N j=1 , it suffices to define each ρ T j -action as follows.
We can check that the action of ρ T j is compatible with the relations Thus the perturbative formula (11) shows that f, F , ζ, and J are all invariant under the G W T -action. Furthermore, according to (12) and (13), ρ T j acts on t α by a factor of c −1 α . Each ρ T j acts on the ν-th insertion of X by Therefore ρ T j acts on the corresponding monomial in the prepotential (16) by a factor of On the other hand, the (higher) residue pairing is invariant under the ρ T j -action. (18), it follows that the pairing Matching the above two factors, we find This is exactly (31).

Reconstruction
In this section, we will introduce the key lemma that turns the WDVV equations into a powerful tool for reconstructing genus zero potentials. Finally, we will completely reconstruct an arbitrary sum of Fermat polynomials as an example of our proof strategy in the general case.

A reconstruction lemma from WDVV equations.
We introduce a powerful reconstruction lemma that follows from the WDVV equations. The statement of this lemma requires the following definition.
It is easy to see that for Jac(W T ), the set of primitive elements is a subset of {x 1 , . . . , x N }. By mirror symmetry, the set of primitive elements in H W is a subset of {Ψ(x 1 ), . . . , Ψ(x N )}. The next lemma says that the prepotential F 0 in each theory is completely determined by correlators with mostly primitive insertions.
where S is a linear combination of correlators with fewer than k insertions. If k = 4, then there are no such terms in the equation, i.e., S = 0. In addition, the k-point correlators are uniquely determined by the pairing, the three-point correlators, and by correlators of the form ξ 1 , . . . ξ n with n < k where ξ i is primitive for i ≤ n − 2.
Since the proof of Lemma 5.2 uses only the WDVV equations, it holds for both F SG 0,W T ,ζ and F FJRW 0,W . This lemma implies that to compare F SG 0,W T ,ζ and F FJRW 0,W , it suffices to compare correlators of the form Here we are using B-side notation; the corresponding A-model correlator is The indicies of the primitive inserstions (including α and β if they are primitive) in the correlator X in (34) are arranged in decreasing order.
In the remainder of this paper, we will apply the vanishing conditions of Lemma 4.1 to correlators of the form (34). In this context, let and let ℓ X i be the number of insertions in X equal to x i , ignoring α and β. Thus Now let b X i be the real numbers defined by the equation and let When there is no possibility of confusion, we will drop the superscript X from the notation. When we apply Lemma 4.1 to X, we produce the following lemma. (34) can be written so it satisfies the following properties:

Lemma 5.3. Any nonvanishing A-or B-model correlator of the form in
(P1). All the numbers K i are integers, i,e., (P2). The following equation holds: (P3). The maximum values for m i and n i are as follows: Let X be a nonvanishing correlator of the form in (34). After we write the insertions of X in the standard basis, this correlator satisfies (P3).
We will prove (P1) and (P2) for the B-model only. The same proof works for the A-model because the A-model Axioms (8) and (9) correspond to the B-model vanishing conditions by Lemma 4.1.
Since X is as in (34), in the context of Lemma 4.1 we have First we will show that (P1) is equivalent to the Integer Degrees Axiom. On the B-side, this axiom says that X is zero unless Then using (18) and (39), we have The last equality follows from (35). So X satisfies the Integer Degrees Axiom if and only if b j ∈ Z for all j, which is true if and only if K j ∈ Z for all j.
Next we derive (P2) from the Dimension Axiom. Let q T i be the i th weight of W T . Then by (30), the correlator X vanishes unless According to Equation (39), the left hand side of (40) is This implies Here the last equality uses (18), (35), and (36).
Lemmas 5.2 and 5.3 tell us when correlators are in a particularly nice form. We make this precise with the following definition.
(1) it has at least four insertions, (2) it is in the form of (34), and

5.2.
A warm up example: the Fermat polynomial. In this section we prove Part (1) of Theorem 3.3 in the special case where W = x a 1 1 +x a 2 2 +. . .+x a N N is a sum of Fermat polynomials, as a way to illustrate our general proof strategy. According to Remark 2.3, we can assume a i > 2 for all i.
First, we reduce the reconstruction problem to the summands of W . We only need to consider correlators of type X −1 .
Proof. By Definition (35), we have Since K i ∈ Z, we have K i ≥ 0. Then Property (38) in Lemma 5.3 implies that there is a unique j ∈ {1, . . . , N } such that K j = 1 and K i = 0 for i = j. Moreover, by Equation (42) we know ℓ i = 0 for i = j. Then since we know ℓ j ≥ 2. Furthermore, using (42), we have Plugging in K j = 1 it is easy to show that ℓ j ≤ 2, so ℓ j = 2. Then (41) shows that m j + n j = 2a j − 4 and the result follows.
Now we complete the proof of Part (1) of Theorem 3.3 in the Fermat case.
Proof. By Lemma 5.5, we only need to reconstruct x j , x j , x a j −2 j α, x a j −2 β from the correlator above. Apply the Reconstruction Lemma 5.
Then γǫ and γδ both vanish because they have a factor of x . , x 1 , α, β . To prove Theorem 3.3, it suffices to show that any correlator of type X −1 can be reconstructed from the correlators in Theorem 3.3. In this section we reconstruct correlators of type X −1 from correlators of "type X 0 " (see Definition 5.16), which are associated to a particular atomic summand of W . This is called Splitting Principle.
We will use the following notation. Suppose that W = W j is a disjoint sum, where each summand W j is of atomic type as described in Theorem 2.1. If x i is a variable appearing in W j , we say that x i ∈ W j , or simply i ∈ W j . Likewise, if α is a monomial in variables appearing in W j , we say α ∈ W j . We define K W := i∈W K i . For any ordered subset of indices S ⊆ {1, 2, · · · , N }, we define that is, K S is a vector of the K i such that i is in S and K S is the sum of the components of this vector. We define ℓ S , m S , and n S similarly. The goal of this section is to reduce the proof of Theorem 3.3 Part (1) to a reconstruction for each atomic type. More specifically, in this section we prove Proposition 5.18, which says that any correlator of type X −1 can be reconstructed from correlators satisfying i∈W j ℓ i ≥ 2, for some j. That is, we reconstruct from correlators with at least two primitive insertions coming from some summand W j of W . We say these correlators have type X 0 .
Throughout the remainder of this paper, we take i + N ≡ i whenever i is in a length-N loop summand of W .

Preliminaries on loop indices.
Let X be a correlator of type X −1 . We will prove the main result of this section, Proposition 5.18, by analyzing possible values for K W .
For each summand W j , we say a set of loop indices S ⊂ W j obeys the Negative-Positive rule (the NP-rule) if it has the property that for any index i ∈ S, if K i < 0, then the index i + 1 ∈ S.
Note that if W is a loop, every i ∈ W is a loop index, and if W is an N -variable chain, every i < N is a loop index. The following lemma summarizes some useful inequalities for loop indices.
Lemma 5.8. If i ∈ W is a loop index, then the following inequalities hold: Proof. We obtain (44) by substituting b i = ℓ i − K i + 1 into (43). Then (45) follows from using Property (P3) in Lemma 5.3 which says that n i + m i ≤ 2a i − 2. Rearranging slightly, we get (46). Then the last inequality follows by using ℓ i ≥ 0 and adding (1 − a i )K i to both sides of (45).
From inequality (47), we get the following corollary.
The lemma and corollary above will be used repeatedly in our reconstruction for the loop and chain polynomials. In addition, they determine K S when S is a set of loop indices that obeys the NP-rule.  Furthermore, we have the following cases: • If K S = 0, then K S is a concatenation of (0)s and (-1,1)s.
Proof. If there exists some index i ∈ S such that K i < 0, then i + 1 ∈ S by assumption and Corollary 5.
If (48) fails, we can repeat the process above for all negative K i and eventually get a contradiction. Thus K S ≥ 0.
Let A = {i ∈ S | K i ≥ 0, K i−1 ≥ 0}. Then Corollary 5.9 implies If K S = 0, then we get K i = 0 for each i ∈ A. Another application of Corollary 5.9 shows that the rest of the K i 's are pairs of (−1, 1).
If K S = 1, then (49) implies there is at most one j ∈ A such that K j = 1. If there is one such j ∈ A, then as the same discussion as above shows K i = 0 for i ∈ A, i = j and the rest of the K i 's are pairs of (−1, 1). If there is no such j ∈ A, then K i = 0 for all i ∈ A. For the rest of the K i 's, besides pairs of (−1, 1), there will be exactly one pair (K i , K i+1 ) such that K i < 0, K i + K i+1 = 1. Then the statement follows from (47) and a i ≥ 2.
Once we know K W , we can often solve for ℓ W and m W +n W , as in the following two lemmas.
Lemma 5.11. Let i ∈ W be a loop index. Then Proof. We can check this by using Lemma 5.8. More explicitly, we obtain the values of (ℓ i , ℓ i+1 ) by plugging the values of (K i , K i+1 ) into (46). Then the values of (m i , n i ) will follow from (44). For (53), we get it from (44) and m i + n i ≥ 0. For the last property, we apply (45) to obtain 0 ≥ K i+1 ≥ ℓ i+1 . This implies K i+1 = ℓ i+1 = 0 and the statement follows again from (44). Lemma 5.12. Let i ∈ S be a loop index where K S is a concatenation of (0)s and (-1,1)s, and suppose i + 1 ∈ S or K i+1 ≤ 0. Then ℓ i ≤ 1, and ℓ i = 1 implies m i + n i = 2a i − 2.
So ℓ i ≤ 1 as desired. If ℓ i = 1, then we saw in the previous paragraph that K i = 0. Since K i+1 ≤ 0, the remainder of the result follows from (44) and (51).

5.3.2.
Reduction to atomic types. We are now ready to prove the first big lemma.
Lemma 5.13. Let X be a correlator of type X −1 for W = W i . There is a unique j such that Proof. We will prove that K W j ≥ 0 for each j. Then the result follows from (38). We have three cases, depending on the atomic type of W j . If W j is a Fermat, then K W j ≥ 0 by Lemma 5.5. If W j is a loop, then K W j ≥ 0 by Lemma 5.10.
So assume W j is a chain with variables x 1 , . . . , x N . We know This implies We use Lemma 5.10 to find three possibilities for K W j . In each case we use (46) to compute ℓ W j and (44) to compute m W j + n W j . Also recall that (55) implies ℓ N = 0 so m N + n N = 2a N − 2.
We list all the possibilities here, using the notation K = K W j and so forth. Also we let M i = 2a i − 2. We will omit the subscript in M i in the way that the M which appears in the i th spot represents M i . Now we know what (K W 1 , K W 2 , . . .) looks like: it is a tuple of zeros with a single 1. The next lemma investigates the form of K W j when K W j = 0.
Lemma 5.14. Let X be a correlator of type X −1 . If K W = 0, then (1) If W is a Fermat then ℓ = 0.
(2) If W is a loop then for all i ∈ W we have ℓ i ≤ 1.
Proof. The claim for the Fermat type follows from (42) when we substitute K = 0.
If W is a loop, by Lemma 5.10 the tuple K W is some concatenation of of (−1, 1)s and (0)s. Also, for every i ∈ W , certainly i + 1 ∈ W . So this result follows from Lemma 5.12.
Finally, let W be a chain. We will show that K is a concatenation of (0)s and (-1,1)s. If the set {1, . . . , N −1} obeys the NP-rule, then i<N K i ≥ 0. Since K N ≥ 0 and i K i = 0, we must have i<N K i = K N = 0 and by Lemma 5.10, the vector (K 1 , . . . , K N −1 ) is a concatentation of (0)s and (-1,1)s.
On the other hand, if {1, . . . , N − 1} does not obey the NP-rule, then K N −1 ≤ −1. But then (47) shows that K N −2 cannot be negative, so the set {1, . . . , N − 2} obeys the NP-rule. Also (47) shows that K N −1 + K N ≥ 0, so i≤N −2 K i ≤ 0. Then Lemma 5.10 tells us that is a concatentation of (0)s and (-1,1)s. Also K N = −K N −1 , and plugging into (47) tells us that 0 ≥ (1 − a i )(1 + K N −1 ). This means that Thus K is a concatenation of (0)s and (-1,1)s. Then Lemma 5.12 proves this lemma for i < N . Thus we only need to check when i = N . We've seen above that K N is 0 or 1. If K N = 1, we saw above that K N −1 = −1 and so by Lemma 5.11, ℓ N = 0. If K N = 0, then (55) says  (46) and (44) to compute ℓ and m + n, respectively. In both cases, the form of m + n contradicts Property (P3) in Lemma 5.3.
Then ℓ N ≤ 1, as desired. In fact, we will show that when a N = 2, we have ℓ N = 0, so the remainder of the lemma is vacuously true in this case. For if ℓ N = 1 then there are three possibilities: • In the remainder of this paper we will repeatedly reconstruct correlators using Lemma 5.2. This lemma allows us to write X = A + B + C + S where A, B, and C are k-point correlators and S is a linear combination of correlators with fewer than k insertions. If X is a correlator of type X −1 it is critical to understand when A, B, and C have type X −1 and how K A , K B , and K C relate to K X .
If A = 0 has the form of (34) then by Lemma 5.3 it satisfies (P1) and (P2). Moreover, if A satisfies (P3) then it is of type X −1 , and in this case b X i = b A i because the changes in ℓ X i , m X i , and n X i cancel each other out. Hence K X i − K A i = ℓ X i − ℓ A i . If A does not satisfy (P3), then we reduce its insertions so they are in the standard basis, yielding an equivalent correlator A ′ of type X −1 . Suppose the reconstruction only affected variables in the direct summand W j of W ; i.e., ℓ X W j as well. The same argument above works for the other two correlators B and C as well. These observations lead to the following remark. (

2) If A ′ is obtained from A by writing its insertions in the standard basis, and if ℓ
Furthermore, if A is any nonvanishing correlator of type X −1 , then (3) If there exists i ∈ W j with ℓ i ≥ 2 and W j is a chain with K A N ≥ 0, a Fermat, or a loop, then by Lemma 5.14 we have K A W j = 1. The same results above are true for correlators B and C as well.
Definition 5. 16. A correlator X is called of type X 0 for W = W j if X is of type X −1 with K W 1 = 1 and K W j = 0 for j > 1, and The main result of this section is to reconstruct correlators of type X −1 from correlators of type X 0 , see Proposition 5.18. By using the Jacobi relations, it is not hard to get the following lemma. Proof. Let X be a correlator of type X −1 . Using Lemma 5.13 and reordering the summands of W if necessary, we may assume K W 1 = 1 and K W j = 0 for j > 1. If for all j > 1, the summand W j is a Fermat, then Lemma 5.14 shows ℓ i = 0 for i ∈ W j for j > 1. Then since i∈W ℓ i ≥ 2, we know (56) holds. Now assume that (56) does not hold for X. Then we can assume that W 2 is a loop or chain polynomial and that there is i ∈ W 2 with ℓ i ≥ 1.
If W 2 is a chain, we do some preparatory reconstruction so Lemma 5.14 is applicable. Let us label the last variable of W 2 by N 2 . We know from (55) that K N 2 ≥ −1. If K N 2 = −1, we saw in the proof of Lemma 5.13 that ℓ N 2 = 0 (so in particular i = N 2 ) and m N 2 = n N 2 = a N 2 − 1, so X = x i , . . . , x N 2 α, β . Now apply the Reconstruction Lemma 5.2 with γ = β, δ = x i , ǫ = x N 2 , and φ = α. Then If these correlators are nonvanishing, by Remark 5.15(2) they each have K W 1 = 1. Also K N 2 ≥ 0 for the first two since ℓ N 2 ≥ 1. If K N 2 = −1 for the last correlator, then it vanishes because m N 2 = a N 2 − 2 = a N 2 − 1. Thus we may assume K N 2 ≥ 0. Now we return to the general case where W 2 is a chain or a loop. By Lemma 5.14, we know ℓ i = 1 and X = x i , x k , . . . , x i α, β for some k = i. Apply the Reconstruction Lemma with γ = β, δ = x k , ǫ = x i , and φ = α, yielding We need to check that if W 2 is a chain, Lemma 5.14 is still applicable to each of these correlators; i.e., K N 2 ≥ 0. Now if k ∈ W 2 and k = N 2 , swap the values of i and k. This way we can assume k = N 2 (since i and k were distinct). There are two cases: • If i = N 2 then all three of the correlators above have ℓ N 2 > 0, so each has K N 2 ≥ 0. • If i = N 2 , by Lemma 5.17, the exponents m N 2 and n N 2 do not change when we write the correlator insertions in the standard basis. Thus K N 2 is unaffected, and so is still nonnegative for each correlator. Now we apply Lemma 5.14 to the first two correlators in (58): if they do not vanish, K W 2 = 1, since ℓ i ≥ 2. Thus these correlators have the desired form. Now, the third correlator still has K W 2 = 0 by Remark 5.15 (2). Therefore, we can repeat this reconstruction on the third correlator. Eventually the third correlator will have m i + n i ≤ a i − 1, which contradicts Lemma 5.14 (and thus this final correlator vanishes).

Atomic reconstruction and Fermat type. Now let us restate Part (1) of Theorem 3.3.
Proposition 5.19. Let W be an invertible polynomial and write W T as the sum of monomials W T = M 1 + . . . + M N . Then the potential F SG 0,W T ,ζ is completely determined by the Frobenius algebra structure and the correlators Fermat summand x a with a > 2; any monomial of a loop summand; or the final monomial of a chain summand.
Moreover, given an isomorphism of graded Frobenius algebras Ψ : Jac(W T ) ∼ = (H W , ⋆) satisfying (20), the potential F FJRW 0,W is similarly determind by the correlators . We will give a complete proof of this proposition in this section and in Section 7. The proof uses the WDVV equations, Jacobi relations (FJRW ring relations), and properties shared by correlators in both models. Since our proof of the first claim in Proposition 5.19 essentially uses only Lemma 4.1 and Lemma 5.2, the second claim is an immediate corollary.

Atomic reconstruction.
After reordering the summands of W so that x i is in W 1 , the correlators in (59) are all of type X 0 . We say the correlators in (59) have final type. According to Splitting Principle Proposition 5.18, in order to prove Proposition 5.19, it suffices to reconstruct correlators of type X 0 from correlators of final type. We will prove this reconstruction in three cases, depending on whether the atomic W 1 is a Fermat, chain, or loop. This is called atomic reconstruction.
In each case, we filter the correlators with several types, denoted by F k , C k , and L k , respectively. Correlators of type F 0 (or C 0 , L 0 ) are correlators of type X 0 where W 1 is a Fermat (or chain, loop) polynomial. The types with the largest values of k are correlators of final type.
For each atomic type, we prove Proposition 5.19 by induction on k. In the k-th step, we reconstruct a correlator of type F k−1 (or C k−1 , L k−1 ) from correlators of type F ≥k (or C ≥k , L ≥k ), correlators that vanish, and correlators with fewer insertions.
Remark 5.20. Let X be a correlator of type X −1 with K W j = 1 and K W i = 0 for i = j and r∈W j ℓ r ≥ 2. By reordering the summands of W we can assume j = 1, so X is of type X 0 . In the remainder of our reconstruction argument we will make this assumption whenever possible. When X is of type X 0 , we let K = K W 1 and we use ℓ, m, and n similarly.

Atomic reconstruction of Fermat type. This subsection proves Proposition 5.19 for W =
W i when W 1 is a Fermat polynomial W 1 = x a with a > 2. We start with the following definition.
Now we prove Proposition 5.19 in two steps.
If not, there is some insertion x i where i ∈ W 1 . Apply the Reconstruction Lemma 5.2 with δ = x i , ǫ = x, φ = x a−3 α, and γ = x a−2 β. Then ǫγ has a factor of x a−1 which is zero in Jac(W T ). But the two remaining terms (with γδ and φδ) have ℓ = 3, and so these correlators must also vanish (if K W 1 = 1 then ℓ 1 ≤ 2; if K W j = 0 then ℓ j = 0 by Lemma 5.14). So we can reconstruct X from correlators with strictly fewer insertions.
Step 2. Let X = x, x, x a−2 α, x a−2 β be a correlator of type F 1 . Apply the Reconstruction Lemma 5.2 with γ = x, ǫ = x a−2 , φ = α, and δ = x a−2 β. Then γǫ and γδ both vanish because they have a factor of x a−1 , and we get X = x, x, x a−2 , x a−2 αβ . Now by the Dimension Axiom in Lemma 4.1, if X = 0, the product αβ must be proportional to the unique element of top degree in Jac(W T − W T 1 ). Hence X is a scalar multiple of a correlator of type F 2 .
The strategies to prove Proposition 5.19 for chain and loop types are similar to the one for Fermat type, but much more complicated. We leave a complete proof in Section 7.

Computation
The goal of this section is to compute the correlators in Theorem 3.3. In the A-model side, the most powerful tool is from an orbifold Grothendieck-Riemann-Roch formula. When the correlator in Theorem 3.3 is concave, then the virtual cycle can be extracted from a top Chern class (75), which will imply the very useful formula (76) by [8]. By analyzing the combinatorical aspect of the insertions in the A-model correlators, we will show that most of them in Theorem 3.3 are concave. We will compute these concave correlators in this section and leave the computations of the nonconcave cases in Appendix A. In the B-model side, the values of the correlators in Theorem 3.3 follow directly from Li-Li-Saito's perturbative formula [30].
Proof. By Theorem 4.2.2 of [13], we have Here Because of this result, in the remainder of this section we will assume that W is an atomic polynomial. Before we start the computation, let us state some useful formulas for each atomic type. Recall that ρ Since Here we use the convention that an empty product is 1. These formulas lead to the following expression for the i th weight of W : 6.1.1. Combinatorial preparation. Let c be an integer such that c ∈ [−2, 2], we define The following results are useful later.
Proof. From (66), (63), and (62), we have and the result follows since a N > 2. If i < N , then since q i in (63) is an alternating series, with strictly decreasing absolute value for each term, and since |c| ≤ 2, the result follows from Here Y i,c = 0 if and only if the first three equalities hold. That happens if and only if a N = 3, c = 2, and i = N − 1.
Proof. (1) N − i is odd. In this case, by (64) and (65), we first write Y i,c as If N = 2, then i = 1 and the result follows from In order to prove the other side of the inequality, if i < N − 1, we rewrite Y i,c as Since If i = N − 1, the result follows from a similar discussion by rewriting Y N −1,c from (65), (2) N − i is even. In this case, the result follows from a similar discussion by rewriting Y i,c as

Now we continue with our computation of
when W is a chain or loop polynomial as above. We will sometimes use θ N , S N and H to denote the correponding sector and use the symbols θ Recall that L i is the i-th orbifold line bundle in the W -structure. If (N, a N ) = (2, 2), then on each smooth fiber, the degree of L i is Proof. The proof is a direct computation using the ring isomorphism in Theorem 2.14. Lemma 6.3, Lemma 6.4 and Equation (61) show that the quantities listed are in [0, 1). In particular, if N = a N = 2, i.e., W = x a 1 1 x 2 + x 2 2 x 1 , then 1). Then (69) follows, since The following G W -decorated graphs will be useful in the computation of X.
Note that the two graphs on the right are the same. Here the element γ k,± ∈ G W is chosen uniquely such that the Interger Degree Axiom (9) is satisfied for each component. It is possible that H γ k,± = ∅. Let γ In particular, if (N, a N ) = (2, 2), we can use the symbol in (66) to rewrite the following numbers 6.1.2. Concavity Axiom. Now we introduce the Concavity Axiom from [13] to compute the necessary FJRW invariants. We recall the universal W -structure (L 1 , . . . , L N ) on the universal curve π : C → W g,k (γ 1 , . . . , γ k ). A correlator ξ 1 , . . . , ξ k g is called concave if all the insertions ξ j are narrow and for each geometric point [C] ∈ W g,k (γ 1 , . . . , γ k ), In this case, π * ( N i=1 L i ) = 0, R 1 π * ( N i=1 L i ) is locally free, and the Concavity Axiom (see Theorem 4.1.8 in [13]) implies Here c top is the top Chern class and [W g,k (Γ γ 1 ,...,γ k )] is the fundamental cycle. Then Theorem 1.1.1 in [8] expresses the FJRW virtual cycles in terms of tautological classes on M g,k . In particular, on M 0,4 we have R 1 π * L i = 0 for some unique L i and

 
Here κ 1 is the first kappa class, ψ j is the j-th psi class, B 2 is the second Bernoulli polynomial that B 2 (x) = x 2 − x + 1 6 , and Γ cut are all the fully G W -decorated graphs on the boundary. For the correlator X = θ N , θ N , S N , H in (60), the graphs are listed in Figure 1.
Lemma 6.6. Consider the correlator X = θ N , θ N , S N , H in (60). Assume a N > 2. If for k = 1, 2, the unordered pairs (ℓ and the sectors of Θ N , S N and H are narrow, then Proof. For a singular curve [C] ∈ W 0,4 (θ N , θ N , S N , H), from (70) and (71), we know where δ narrow is 1 when the local isotropy group at the node acts nontrivially on the fiber and 0 otherwise. Thus we can check (77)  k,− ) = (−1, 0) and i < N . According to (79), the unique node n ∈ C must be broad. We denote the normalization of C by p : C 1 C 2 → C, and get a long exact sequence Let us focus on the first line. Since (ℓ (i) k,+ , ℓ (i) k,− ) = (−1, 0), the third term is just C. The broadness implies that the last arrow is an isomorphism. Thus (74) follows. Now we apply Riemann-Roch formula to (69) and (77). Then R 1 π * L i<N = 0, and R 1 π * L N is a vector bundle of rank 1. Now formula (78) follows from Equation (76) and Combine Lemma 6.3 with the notation in (73), we get the following corollary.
We plug these numbers into (78) and get Thus the correlator X satisfies the condition in Lemma 6.6 and we can apply Formula (78) to compute its value. A direct computation shows As a consequence, we have If a N = 2, then X is never concave. We can classify them into three exceptional families. The computation of each family is known. We list the computations in Appendix A.
Since the variables are symmetric in loop case, we only need to deal with i = N .
Let f = W T . We recall that the perturbative formula (15) takes the following form By definition (16) and Equation (15), we know Here we denote S := M N /x 2 N , H := φ W T and t α is the flat coordinate dual to φ α . Following the notations in Section 2.2.2, we use the subscript (≤ k) to denote the k-th Taylor expansion in terms of s (or t). As shown in Proposition 3.12 in [31], the perturbative formula implies that (F SG 0,f,ζ ) (≤4) (t) depends on ζ (≤1) (s), the primitive form up to first order, only. The algorithm described in Section 2.2.2 shows that Therefore we only need to expand the LHS of (81) using ζ (≤1) (s) to compute the 4-point function. The term J 1 −2 corresponds to the coefficient in front of 2 . This has no contribution to the RHS of Equation (82), since the Equation (14) shows .
.). On the other hand, since to obtain (80), we only need to prove the following equation:

For both Fermat polynomial and chain polynomial, Equation (83) is true because
For the loop polynomial, Equation (83) follows from Equation (65) and by cancelling M 1 , · · · , M N −1 among the relations

A proof of Proposition 5.19
In this section, we complete a proof of Proposition 5.19 for W = W i when W 1 is a chain or a loop. 7.1. Atomic reconstruction of Chain type. This subsection proves Proposition 5.19 for 7.1.1. Preliminary facts about chain polynomials. We will repeatedly use the following relations in Jac(W T 1 ): These relations imply (85) x i−1 x a i i = 0, i < N. Additionally, the following lemma tells us what K looks like in most cases.
Thus We introduce the following definition.
Definition 7.2. Let X and X ′ be correlators of type C 0 with the same number of insertions. Assume . . , ℓ r+1 = ℓ ′ r+1 , and ℓ r > ℓ ′ r , for some r ∈ W 1 . We say that X is maximal if there does not exist X ′ > X, or equivalently, if The relation > is well-defined because of the ordering of primitive insertions in correlators of type X −1 (see (34)). Also, this relation is transitive. We immediately have Lemma 7.3. Let X be a correlator of type C 0 . If there is i ∈ W 1 such that we can rewrite X as X = . . . , x i , . . . , x a i+1 i+1 α, β with i + 1 ∈ W 1 , then X can be reconstructed from correlators with fewer insertions and correlators Z of type C 0 satisfying Z > X. . By repeating the process, we can reduce the exponent of x i+1 and eventually, we will have determined X from correlators with fewer insertions and correlators of the form Y = . . . , x i+1 , . . . , * , * with ℓ Y i = ℓ X i − 1 and ℓ Y i+1 = ℓ X i+1 + 1. After reducing insertions to the standard basis, all the nonvanishing correlators we get from this process are of type X −1 . Furthermore, if each such Y is of type C 0 , then the result follows since Y > X. Thus we only need to reconstruct those correlators that are not of type C 0 . Then we must have K W 1 = 0 for such a correlator Y .
On the other hand, since X is of type C 0 , besides x i , X must have at least one more insertion x k , with x k ∈ W 1 . Since the Reconstruction Lemma 5.2 does not change the insertions in dotted positions of X = . . . , x i , . . . , x a i+1 i+1 α, β , we know Y could be rewritten in the form of Y = x i+1 , x k , . . . , * , * . Since K W 1 = 0, we may assume K N 1 ≥ 0, otherwise we do a preparatory reconstruction as (57) to get K N 1 ≥ 0. Thus by Lemma 5.14, we know i + 1 = k and ℓ i+1 = ℓ k = 1. Then we can repeat the process as in Proposition 5.18 to reconstruct Y from type C 0 correlators Z such that Z > X, and correlators with fewer insertions. Such correlators Z will be of the form Z = x i+1 , x i+1 , . . . , α Z , β Z if k ≤ i, or Z = x k , x k , . . . , α Z , β Z if k > i. We remark that during the process, the ordered pair of inserions (x i , x k ) in (58) are replaced by the ordered pair (x i+1 , x k ) if k ≤ i, or by (x k , x i+1 ) if k > i. This guarantees that we have Z > X.
By the above lemma, we have Proposition 7.4. Let X be a correlator of type C 0 which is not maximal. Then X can be reconstructed from correlators with fewer insertions and correlators Z of type C 0 satisfying Z > X.
Proof. Since X is not maximal, we can choose i to be the largest index such that i ∈ W 1 with i < N and ℓ i ≥ 1. So X = . . . , x i , x m N N α X , β X for some m N ≥ 0. If m N ≥ 1, then we apply Reconstruction Lemma 5.2 with γ = β X , δ = x i , ǫ = x N , and φ = x m N −1 N α X . By Remark 5.15, the correlators with δφ and δγ are type C 0 -correlators of the form . . . , x N , * , * . The correlator with ǫγ equals . . . , x i , x m N −1 N α X , x N β X . By induction reconstruct X from C 0 -correlators of the form . . . , x N , * , * and the C 0 -correlator Similarly, we move all x N −1 from α Y to β Y , and so on, until we move all x i+1 from α to β. Thus we reconstruct X from correlators X ′ of type C 0 with X ′ > X, and the correlator Z = . . . , After reducing to the standard basis, Z is of type X 0 and m Z k + n Z k ≤ a k − 1 for k > i. From here on we will speak only of the correlator Z and drop the Z-superscript from our notation. By definition, Thus K N ≥ 0 and we may use Lemma 7.1. This lemma gives us a list of possible vectors K which we analyze case by case. In each case, if the correlator is not in the desired form, we write the insertions in a nonstandard basis and so that there is some k with ℓ k ≥ 1 and m k + n k ≥ a k . Then we use Lemma 7.3 to finish the reconstruction.
• The second and fourth case here are same as Case (1) when K N = 1.
• For the third case, (K i−1 , K i ) = (0, 1). As we did for (K N −1 , K N ) = (0, 1), we find a factor of α equal to x a i i (note the different index). All three cases above are possible. Since ℓ i−1 = 1, we can apply Lemma 7.3.
• For the fifth case, let r be the last index before the (−1, 1) pairs of K. If K r = 0, then we can find a factor of α equal to x a i+1 i+1 (like in Cases (2)-(3) of (K N −1 , K N ) = (0, 1)). If K r = 1 (but not as part of a (-1,1) pair), then m r + n r = ℓ r (a r − 1) is 0 only if ℓ r = 0.
Similarly, for the last case, let r = i − 1. Then m r + n r = ℓ r (a r − 1), which will be 0 only if ℓ r = 0.
If ℓ r > 0, then m r + n r > 0 and as usual we can find that α has a factor x a i+1 i+1 . Thus we only need to check for both of these cases when ℓ r = 0. Since X was of type X 0 and the reconstruction from X to Z did not change the number of insertions in W 1 , there is some k < r < i such that ℓ k ≥ 1. But in the indices less than r, the vector K is a concatenation of (-1,1)s and (0)s. This means that if we truncate the vector K before the k-th place, then the truncated vectors will look like the fourth case or the fifth case with K r = 0. So we can use (84) to find a factor of α equal to x a k+1 k+1 . 7.1.2. Reconstruction procedure. The remainder of this subsection proves Proposition 5.19 when W 1 = x a 1 1 x 2 + x a 2 2 x 3 + . . . + x a N N is a chain. Definition 7.5. Let X be a correlator of type C 0 . We say We will do the reconstruction in four steps. In k-th step, we will reconstruct a correlator of type C k−1 from correlators of type C ≥k , correlators that vanish, and correlators with fewer insertions.
Step 1. We do this by applying Proposition 7.4.
Step 2. Let X be a correlator of type C 1 . Our discussion breaks into three cases: (1) m N n N = 0. For case (1), we will reconstruct X from correlators with fewer insertions and correlators of type C 2 . We may assume m N = 0. Let S be the set of insertions x i in X such that x i ∈ W 1 . If S = ∅, choose some x i ∈ S and use the Reconstruction Lemma 5.2 with γ = β, δ = x i , ǫ = x N , and φ = α/x N . Then all correlators coming from the reconstruction have ℓ N ≥ ℓ X N ≥ 2, so K N ≥ 0 (we saw in the proof of Lemma 5.13 that when K N < 0, we have ℓ N − 0). Then Remark 5.15(3) implies that all the correlators have type C 0 or they vanish. Moreover, all these correlators have m N = m X N − 1. Repeat this same reconstruction until either m N = 0 in all correlators or S = ∅.
In cases (2) and (3) we must have m N + n N ≤ a N − 1 so (87) holds, and K is in the form of Lemma 7.1. So K N is 0 or 1; if K N = 0 then (87) shows that ℓ N = 0, and if K N = 1 then (87) shows that ℓ N is 1, 2 or 3. But since X is of type C 1 we know ℓ N ≥ 2, so in fact K N = 1 and (ℓ N , a N ) = (2, 2), (2,3) or (3,2).
If (ℓ N , a N ) = (2, 2), then by (86), we get We If (ℓ N , a N ) = (2, 2), then X = x i 1 ∈ W 1 , . . . , x is ∈ W 1 , x N , x N , α, β . Since K N = 1 and ℓ = (0, . . . , 0, 2), a calculation using (44) and Lemma 7.1 shows that there is some j ∈ W 1 with m j + n j ≥ a j . In particular m j > 0. Reconstruct X as in case (1), beginning with δ = x i 1 , γ = β, ǫ = x j , and φ = α/x j . All resulting correlators have type C 1 as discussed in case (1). By repeating this reconstruction, we can determine X from correlators with fewer insertions, type C 1 correlators of the form x N , x N , x j , . . . , and a type C 1 correlator with the same primitive insertions as X but with m j = 0 for all j ∈ W 1 . The correlators x N , x N , x j , . . . can be reconstructed from correlators x N , x N , x N , . . . as in Proposition 7.4. The final correlator vanishes by the discussion at the start of this paragraph.
Step 3. Let X be a correlator of type C 2 . From (55), since ℓ N ≥ 2, we find Thus K N ≥ 0 and equality is possible only if a N = 2. If K N = 0 and a N = 2, then (55) shows that ℓ N ≤ 2, so in fact X is of type C 3 . If K N = 0, then K N = 1 by Lemma 7.1. Then (55) shows ℓ N ≤ 2a N /(a N − 1), so ℓ N = 2 (and X is of type C 3 ), or ℓ N = a N = 3, or a N = 2 and ℓ N = 3 or 4. We will show that in each case where ℓ N > 2, the correlator does not satisfy (P3) in Lemma 5.3, a contradiction.

7.2.2.
Reconstruction procedure. As in the previous sections, we define progressively simpler correlator types and perform the reconstruction in a number of steps.
Definition 7.7. Let X be a correlator of type L 0 .
• X is of type L 1 if X = x p , x p , . . . , α, β for some p ∈ W 1 .
• X is of type L 2 if X = x p , . . . , x p , α, β for some p ∈ W 1 .
In k-th step, we will reconstruct a correlator of type L k−1 from correlators of type L ≥k and correlators with fewer insertions. Each type has K W 1 = 1 by Lemma 5.14(3), so we will generally not need to check this condition.
Step 1. Let X be a correlator of type L 0 . If there exists some ℓ i ∈ W 1 such that ℓ i ≥ 2, then we are done. If ℓ i ≤ 1 for all i ∈ W 1 , then we will show that for some i ∈ W 1 , we have ℓ i = 1 and m i + n i ≥ a i . By Proposition 5.18 we can assume there exist distinct i, k ∈ W 1 such that ℓ i = ℓ k = 1. Now assume K W 1 is in the form of Lemma 7.6. If some ℓ i = 1 for i ≤ N − 2, then Lemma 5.12 shows Otherwise, ℓ N −1 = ℓ N = 1, and by Lemma 7.6 we must have K N = 1. In this case K N −1 is 1 or 0. But if K N −1 = 1 then K N −2 = −1 and ℓ N −1 = 0 by (50). So K N −1 = 0 and (44) shows Now we do the reconstruction part of this step. Let k ∈ W 1 be any index such that ℓ k = 1 and m k + n k ≥ a k . Then X = . . . , x k , x i , x k α, β where m k ≥ 1 and i = k (we do not require i ∈ W 1 ). Using the Reconstruction Lemma 5.2 with γ = β, δ = x i , ǫ = x k , and φ = α, we find X = . . . , x k , x k , β, x i α − . . . , x k , x k , α, x i β − . . . , x k , x i , α, x k β + S.
After rewriting the last insertion in the standard basis, the first two correlators are of type L 1 . We use the same reconstruction on the third correlator until it has the form . . . , where α is a monomial in the standard basis with no factor of x k . When we rewrite the last insertion in the standard basis, we find that m k + n k < a k . This contradicts Lemma 5.14 if K W 1 = 0, so K W 1 = 1. Then either the third correlator is of type L 1 , or it has some k ′ such that ℓ k ′ = 1 and m k ′ + n k ′ ≥ a k ′ . We can repeat the same reconstruction, moving all x ′ k from α to β. Eventually, we will run out of such indices, showing that the third form eventually vanishes.
Step 2. Let X be a correlator of type L 1 . This step is similar to Step 1. Since ℓ p ≥ 2, we know that p is N or N − 1 by Lemma 5.12. Then K N = 1 by Lemma 7.6. We will show that m p + n p ≥ a p . Then the same reconstruction argument as in Step 1 gives the desired result.
Now we address this exceptional case.
Next we claim that we can find α ′ ∝ α and β ′ ∝ β in Jac(W T ) such that This α ′ and β ′ may not be in the standard basis.
If K N −1 = 1, then (K N −2 , K N −1 ) = (−1, 1) and K = (. . . , 0, −1, 1, . . . , −1, 1, 1). Here the 0 is the last 0 before the (−1, 1)-sequence; say it occurs at the r th spot. We know from (44) After reducing the insertions of the first three correlators, by Remark 5.15(3) they all are of type L 1 if they are nonvanishing. By Lemma 7.6 they have K N = 1. As discussed earlier, a nonvanishing correlator in the form of (91) with K N = 1 must have ℓ N −1 = 0. Thus the first two correlators vanish. Use the same reconstruction on the third correlator until n N −1 = a N −1 . We still have n N −2 > 0, so now β contains the factor x N −2 x a N−1 N −1 , which is 0 by (90). So the third correlator is also zero. Thus we have reconstructed X from correlators with fewer insertions.
Step 3. Let X be a correlator of type L 2 , so ℓ i = 0 if i = p. Since X is also of type L 1 , we know K N = 1, and p = N − 1 or N from Step 2.
The last step will use the following lemmas.
is of type X −1 then it can be reconstructed from correlators of type L 4 .
Proof. If (N, a i ) = (2, 2), then up to symmetry W 1 = x a 1 x 2 +x 1 x 2 2 and X = x 1 , x 2 , x a−1 1 x 2 , αx a−1 1 . If a = 2, then X is already of final type. If a > 2, the result follows from a reconstruction as in (106) and (107) in Section A. Now we assume (N, a i ) = (2, 2). We apply the Reconstruction Lemma 5.2 to the correlator By Dimension Axiom in Lemma 4.1, if A i is nonzero, then it is a correlator of type L 4 . Now let us start the reconstruction. Let X k,i be the correlator in (93). We replace k+1 /a k in X k,i using the relation (89), and then apply the Reconstruction Lemma 5.2 with δ = x k , ǫ = x k+1 , φ = x a k+1 −1 k+1 , and γ = α φ W T 1 /x i . If i = k, k + 1, the terms with γδ and γǫ are both zero by (90), and (95) For k = i − 1 or k = i, one of γδ or γǫ is nonzero, and we get Combining (95), (96), and (97), we find Using (94) we know X k,i can be reconstructed from correlators of type L 4 .
Proof. In X k,i , replace the insertion x k−1 x a k −1 k with −x a k+1 k+1 /a k in X k,i using the relations (89), and then apply the Reconstruction Lemma 5.2 with δ = x k , ǫ = x k+1 , φ = x a k+1 −1 k+1 , and γ = β φ W T 1 . Via a similar argument to the one used in the proof of Lemma 7.8, we have Then X k = 0 as desired. Now apply the Reconstruction Lemma to Y k with δ = x k , ǫ = α, φ = x a k+1 −1 k+1 , and γ = β φ W T

1
. We know γδ = 0 for degree reasons. The term with ǫγ is of type L 4 . The term with δφ is X k+1 , which we have seen is 0.
Thus both m N −1 and n N −1 are at least 1, and we can assume m N = a N − 2 and n N = a N − 1. If m = (0, . . . , 0, 1, a N − 2) we are done; otherwise there is some i such that m i is larger than it should be.
Apply the Reconstruction Lemma with δ = x N , γ = β, ǫ = x i , and φ = α/ǫ. Then γδ has a factor equal to x N −1 x a N N , which is 0. We can check that the correlator with γǫ is of type L 3 and K = (0, . . . , 0, 0, 1), but deg(α) < deg(α X ). So if we use the same reconstruction on this correlator, eventually the second form in the reconstruction will be , if a N = 2, then X = θ N , θ N , S N , H is never concave. They can be classified into three exceptional families listed below. For the first two families, we use WDVV equations to solve X from concave correlators. For the last family, we apply a result of Guéré [23].
We can use Lemma 6.4 to compute the phases. Now we list all the cases as follows: Case 1. N = 2 and a N −1 = 2. In this case, θ N and θ N −1 are broad. Case 2. N = 2 and a N −1 > 2. In this case, θ N is broad. A similar discussion using the normalization exact sequence as in Lemma 6.6 implies there is a singular curve [C] ∈ W 0,4 (θ N , θ N , S N , H), such that H 0 (C, L N −1 | C ) = C. Thus the correlator is not concave. Now we compute the correlator X = θ N , θ N , S N , H for each case as shown above.
Both θ 1 and θ 2 are broad. It is very difficult for us to compute X directly. However, all the correlators can still be determined by WDVV equations and the correlator X 0 := J 2 , J 2 , J 2 , J 2 , J 2 , J 2 , J 2 .
We can check that X 0 is concave and This correlator can be calculated by the Concavity Axiom using Theorem 1.1.1 in [8]. The computation of X 0 is exactly the same as the computation of J 2 and γ(ℓ, k) is defined by the generating function ℓ≥0 γ(ℓ, k) z ℓ ℓ! = (e z − 1) k k! .
On the other hand, when all insertions in the correlator are narrow, Chang-Li-Li [7] showed that the Polishchuk-Vaintrob and Fan-Jarvis-Ruan-Witten virtual classes are equal. Thus we can use Formula (108) to compute the correlator X in Case 3, where t(j) = j + 1. According to (69), on a generic fiber, the line bundle degrees are deg L j = −1 for j < N, and deg L N = −2.
Also, since we are working on M 0,4 , by degree considerations the sum over ℓ has only the summand ℓ = 1, and the power series defined by the exponential terminates after the linear part. Thus, plugging in the function s j , Formula (108) becomes where λ N = λ −a N−1 and λ j = λ for j < N . Because L j is concave and l j = −1 for j < N − 1, so Ch 1 (Rπ * L j ) = 0. Thus = a N −1 Ch 1 (Rπ * L N −1 ) − Ch 1 (Rπ * L N ).
As in the derivation of (78), we can apply Theorem 1.1.1 in [8] to compute