MIRROR SYMMETRY FOR EXCEPTIONAL UNIMODULAR

arXiv:1405.4530v1 [math.AG] 18 May 2014

MIRROR SYMMETRY FOR EXCEPTIONAL UNIMODULAR SINGULARITIES CHANGZHENG LI, SI LI, KYOJI SAITO, AND YEFENG SHEN

A BSTRACT. In this paper, we prove the mirror symmetry conjecture between the SaitoGivental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by the first three authors recently. This computation matches the orbifold-Grothendieck-Riemann-Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. Our result establishes the first examples of LG-LG mirror symmetry of all genera for weighted homogeneous polynomials of central charge greater than one which contain negative degree deformation parameters.

C ONTENTS 1.

Introduction

1

2.

A-model: FJRW-theory

7

3.

B-model: Saito’s theory of primitive form

19

4.

Mirror Symmetry for exceptional unimodular singularities

28

Appendix A.

34

References

41

1. I NTRODUCTION Mirror symmetry is a fascinating geometric phenomenon discovered in string theory. The rise of mathematical interest dates back to the early 1990s, when Candelas, Ossa, Green and Parkes [6] successfully predicted the number of rational curves on the quintic 3-fold in terms of period integrals on the mirror quintics. Since then, one popular mathematical formulation of mirror symmetry is about the equivalence on the mirror pairs between the Gromov-Witten theory of counting curves and the theory of variation 1

2

CHANGZHENG LI, SI LI, KYOJI SAITO, AND YEFENG SHEN

of Hodge structures. This is proved in [19, 32] for a large class of mirror examples via toric geometry. Mirror symmetry has also deep extensions to open strings incorporating with D-brane constructions [26, 47]. In our paper, we will focus on closed string mirror symmetry. Gromov-Witten theory presents the mathematical counterpart of A-twisted supersymmetric nonlinear σ -models, borrowing the name of A-model in physics terminology. Its mirror theory is called the B-model. On either side, there is a closely related linearized model, called the N=2 Landau-Ginzburg model (or LG model), describing the quantum geometry of singularities. There exist deep connections in physics between nonlinear sigma models on Calabi-Yau manifolds and Landau-Ginzburg models (see [25] for related literature). In this paper, we will study the LG-LG mirror symmetry conjecture, which asserts an equivalence of two nontrivial theories of singularities for mirror pairs (W, G ), (W T , G T ). Here W is an invertible weighted homogeneous polynomial on Cn with an isolated critical point at the origin, and G is a finite abelian symmetry group of W. The mirror weighted homogeneous polynomial W T was introduced by Berglund and Hubsch ¨ [5] ai j n n in early 1990s. For invertible polynomial W = ∑i=1 ∏ j=1 x j , the mirror polynomial is a ji

W T = ∑ni=1 ∏nj=1 x j . The mirror group G T was introduced by Berglund and Henningson [4] and Krawitz [27] independently. Krawitz also constructed a ring isomorphism between two models. Now the mirror symmetry between these LG pairs is also called Berglund-Hubsch-Krawitz ¨ mirror [10]. When G = GW is the group of diagonal symmetries T of W, the dual group GW = {1} is trivial. In order to formulate the conjecture, let us introduce the theories on both sides first. We remark that one of the most general mirror constructions of LG models was proposed by Hori and Vafa [23]. A geometric candidate of LG A-model is the Fan-Jarvis-Ruan-Witten theory (or FJRW theory) constructed by Fan, Jarvis and Ruan [13, 14], based on a proposal of Witten [49]. Several purely algebraic versions of LG A-model have been worked out [7, 35]. The FJRW theory is closely related to the Gromov-Witten theory, in terms of the so-called LandauGinzburg/Calabi-Yau correspondence [9, 36]. The purpose of the FJRW theory is to solve the moduli problem for the Witten equations of a LG model (W, G ) (G is an appropriate subgroup of GW ). The outputs are the FJRW invariants. Analogous to the Gromov-Witten invariants, the FJRW invariants are defined via the intersection theory of appropriate virtual fundamental cycles with tautological classes on the moduli space of stable curves. These invariants virtually count the solutions of the Witten equations on orbifold curves. For our purpose later, we consider G = GW , and summarize the main ingredients of the FJRW theory as follows (see Section 2 for more details):

MIRROR SYMMETRY FOR EXCEPTIONAL UNIMODULAR SINGULARITIES

3

• An FJRW ring ( HW , •). Here HW is the FJRW state space given by the GW -invariant Lefschetz thimbles of W, and the multiplication • is defined by an intersection pairing together with the genus 0 primary FJRW invariants with 3 marked points. FJRW • A prepotential F0,W of a formal Frobenius manifold structure on HW , whose coefficients are all the genus 0 primary FJRW invariants h· · ·iW 0 . FJRW • A total ancestor potential AW that collects the FJRW invariants at all genera. A geometric candidate of the LG B-model of (W T , G T ) is still missing for general G T . When G = GW , then G T = {1} and a candidate comes from the third author’s theory of primitive forms [40]. The starting point here is a germ of holomorphic function ( f = W T to our interest here) f ( x ) : ( Cn , 0 ) → ( C, 0 ) ,

x = { xi }i=1,··· ,n

with an isolated singularity at the origin 0. We consider its universal unfolding

(Cn × Cµ , 0 × 0) → (C × Cµ , 0 × 0),

(x, s) → ( F (x, s), s)

where µ = dimC Jac ( f )0 is the Milnor number, and s = {sα }α =1,··· ,µ parametrize the deformation. Roughly speaking, a primitive form is a relative holomorphic volume form

ζ = P(x, s)dn x,

dn x = dx1 · · · dxn

at the germ (Cn × Cµ , 0 × 0), which induces a Frobenius manifold structure (which is called the flat structure in [40]) at the germ (Cµ , 0). This gives the genus 0 invariants in the LG B-model. At higher genus, Givental [18] proposed a remarkable formula (with its uniqueness established by Teleman [48]) of the total ancestor potential for semi-simple Frobenius manifold structures, which can be extended to some non-semisimple boundary points [11, 33] including s = 0 of our interest. The whole package is now referred to as the Saito-Givental theory. We will call the extended total ancestor potential at s = 0 a Saito-Givental potential and denote it by A fSG . For G = GW , the LG-LG mirror conjecture (of all genera) is well-formulated [10]: Conjecture 1.1. For a mirror pair (W, GW ) and (W T , {1}), there exists a ring isomorphism ( HW , •) ∼ = Jac (W T ) together with a choice of primitive forms ζ , such that the FJRW potential FJRW AW is identified with the Saito-Givental potential AWSGT . For the weighted homogeneous polynomial W = W ( x1 , · · · , xn ), we have W ( λ q 1 x 1 , · · · , λ q n x n ) = λ W ( x 1 , · · · , x n ) , ∀ λ ∈ C∗ ,

4


with each weight qi being a unique rational number satisfying 0 < qi ≤ partial classification of W using the central charge [42] cˆW :=

1 2

[37]. There is a

∑(1 − 2qi ). i

So far, Conjecture 1.1 has only been proved for cˆW < 1 (i.e., ADE singularities) by Fan, Jarvis and Ruan[13] and for cˆW = 1 (i.e., simple elliptic singularities) by Krawitz, Milanov and Shen [29, 34]. However, it was open for cˆW > 1, including exceptional unimodular modular singularities and a wide class of those related to K3 surfaces and CY 3-folds. One of the major obstacle is that computations in the LG B-model require concrete information about the primitive forms. The existence of the primitive forms for a general isolated singulary has been proved by M.Saito [45]. However, explicit formulas were only known for weighted homogeneous polynomials of cˆW ≤ 1 [40]. This is due to the difficulty of mixing between positive and negative degree deformations when cˆW > 1. The main objective of the present paper is to prove that Conjecture 1.1 is true when is one of the exceptional unimodular singularities as in the following table. Here we use variables x, y, z instead of the conventional x1 , x2 , · · · , xn . These polynomials are all of central charge larger than 1, providing the first nontrivial examples with the existence of negative degree deformation (i.e., irrelevant deformation) parameters.

WT

TABLE 1. Exceptional unimodular singularities Polynomial

Polynomial

Polynomial

W12

x4 + y5

U12

x3 + y3 + z4

Q12 x2 y + xy3 + z3

Z12

x3 y + y4 x

S12

x2 y + y2 z + z3 x

E14

x2 + xy4 + z3

E13

x3 + xy5

Z13

x2 + xy3 + yz3

W13 x2 + xy2 + yz4

Q10

x2 y + y4 + z3

Z11

x3 y + y5

Q11

x2 y + y3 z + z3

S11

E12

x3 + y7

Polynomial

x2 y + y2 z + z4

Originally, the 14 exceptional unimodular singularities by Arnold [3] are one parameter families of singularities with three variables. Each family contains a weighted homogenous singularity characterized by the existence of only one negative degree but no zero-degree deformation parameter [42]. In this paper, we consider the stable equivalence class of a singularity, and always choose polynomial representatives of the class with no square terms for additional variables. The FJRW theory with the group of diagonal symmetries is invariant when adding square terms for additional variables.


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LG-LG mirror symmetry for exceptional unimodular singularities. Let us explain how we achieve the goal in more details. Following [27], we can specify a ring isomorphism Jac(W T ) ∼ = ( HW , •). Then we calculate some FJRW invariants, by an orbifoldGrothendieck-Riemann-Roch formula and WDVV equations. More precisely, we have Proposition 1.2. Let W T be one of the 14 singularities, then Ψ : Jac(W T ) → ( HW , •), defined in (2.18) and (2.21), generates a ring isomorphism. Let MiT be the i-th monomial of W T , and φµ be of the highest degree among the specified basis of Jac (W T ) in Table 2. Let qi be the weight of xi with respect to W. For each i, we have genus 0 FJRW invariants (1.1)

h Ψ ( xi ) , Ψ ( xi ) , Ψ (

MiT 2 T ), Ψ(φµ )iW 0 = qi , whenever Mi 6 = xi . x2i

Surprisingly, if W T belongs to Q11 or S11 , then the ring isomorphism Jac (W T ) ∼ = ( HW , •) was not known in the literature. The difficulty comes from that if there is some q j = 21 , then one of the ring generators is a so-called broad element in FJRW theory and invariants with broad generators are hard to compute. We overcome this difficulty for the two cases, using Getzler’s relation on M1,4 . It is quite interesting that the higher genus structure detects the ring structure. We expect that our method works for general unknown cases of ( HW , •) as well. On the B-side, it is already known that there exists a unique primitive form (up to a nonzero scalar) if W T is one of the exceptional singularities [22, 31]. Recently the first three authors have developed a perturbative way to compute the primitive forms for arbitrary weighted homogeneous singularities [31]. This leads to a perturbative algoSG of the associated rithm (as fully developed in section 3.2) for the potential function F0,W T Frobenius manifold structure. This provides the data of LG B-model at genus 0. From the computational results, we show that the A-side FJRW invariants for W in Proposition 1.2 coincide with the B-side corresponding invariants for W T (up to a sign). In the next step, we establish a reconstruction theorem in such cases (Lemma 4.2), showing that the WDVV equations are powerful enough to determine the full prepotentials for both sides from those invariants in (1.1). This gives the main result of our paper: Theorem 1.3. Let W T be one of the 14 exceptional unimodular singularities in Table 1. Then the specified ring isomorphism Ψ induces an isomorphism of Frobenius manifolds between Jac (W T ) (which comes from the primitive form of W T ) and HW (which comes from the FJRW theory of (W, GW )). That is, the prepotentials are equal to each other: (1.2)

FJRW SG F0,W T = F 0,W .

6


In general, the computations of FJRW invariants are challenging due to our very little understanding of virtual fundamental cycles, especially at higher genus. However, according to Teleman [48] and Milanov [33], the non-semisimple limit AWSGT is fully determined by the genus-0 data on the semisimple points nearby. As a consequence, we upgrade our mirror symmetry statement to higher genus and prove Conjecture 1.1 for the exceptional unimodular singularities. Corollary 1.4. Conjecture 1.1 is true for W T being one of the 14 exceptional unimodular singularities in Table 1. The specified ring isomorphism Ψ induces the following coincidence of total ancestor potentials:

(1.3)

AWSGT = AWFJRW .

The choice in Table 1 has the property that the mirror weighted homogeneous polynomials are again representatives of the exceptional unimodular singularities. Arnold discovered a strange duality among the 14 exceptional unimodular singularities, which says the Gabrielov numbers of each coincide with the Dolgachev numbers of its strange dual [2]. The strange duality is also reproved algebraically in [43]. The choices in Table 1 also represent Arnold’s strange duality: the first two rows are strange dual to themselves, and the last two rows are dual to each other. For example, E14 is strange dual to Q10 . Beyond the choices in Table 1, we also discuss the LG-LG mirror symmetry for other invertible polynomial representatives (some of whose mirrors may no longer be exceptional exceptional singularities) where equality (1.3) still holds. The results are summarized in Theorem 4.3 and Remark 4.5. The present paper is organized as follows. In section 2, we give a brief review of the FJRW theory and compute those initial FJRW invariants as in Proposition 1.2. In section 3, we give a self-contained reformulation of the method of [31] and compute the relevant correlation functions in LG B-model. In section 4, we prove Conjecture 1.1 when the B-side is given by one of the exceptional unimodular singularities. We also discuss the more general case when either side is given by an arbitrary weighted homogeneous polynomial representative of the exceptional unimodular singularities. Finally in the appendix, we provide detailed descriptions of the specified isomorphism Ψ as well as a complete list of the genus-zero four-point functions on the B-side for all the exceptional unimodular singularities. We would like to point out that section 2 and section 3 are completely independent of each other. Our readers can choose either sections to start first.


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Acknowledgement: The authors would like to thank Kentaro Hori, Hiroshi Iritani, Todor Eliseev Milanov, Atsushi Takahashi, and especially Yongbin Ruan, for helpful discussions and valuable comments. S.L. is partially supported by NSF DMS-1309118. K.S. is partially supported by JSPS Grant-in-Aid for Scientific Research (A) No. 25247004. In addition, Y.S. would like to thank Alessandro Chiodo, Jeremy Guéré, Weiqiang He for helpful discussions. S.L. would like to thank Kavli IPMU, and Y.S. would like to thank IAS of HKUST for their hospitality. Part of the work was done during their visit. 2. A- MODEL : FJRW- THEORY 2.1. FJRW-theory. In this section, we give a brief review of FJRW theory. For more details, we refer the readers to [13, 14]. We start with a nondegenerate weighted homogeneous polynomial W = W ( x1 , · · · , xn ), where the nondegeneracy means that W has isolated critical point at the origin 0 ∈ Cn and contains no monomial of the form xi x j for i 6= j. This implies that each xi has a unique weight qi ∈ Q ∩ (0, 21 ] [37]. Let GW be the group of diagonal symmetries, n o (2.1) GW := (λ1 , . . . , λn ) ∈ (C∗ )n W (λ1 x1 , . . . , λn xn ) = W ( x1 , . . . , xn ) . In this paper, we will only consider the FJRW theory for the pair (W, GW ). In general, the FJRW theory also works for any subgroup G ⊂ GW where G contains the exponential grading element √ √ (2.2) J = exp(2π −1q1 ), · · · , exp(2π −1qn ) ∈ GW . Definition 2.1. The FJRW state space HW for (W, GW ) is defined to be the direct sum of all GW -invariant relative cohomology:

(2.3)

HW :=

M

Hγ ,

Hγ := H Nγ (Fix(γ ); Wγ∞ ; C) GW .

γ ∈ GW

Here Fix(γ ) is the fixed points set of γ, and CNγ ∼ = Fix (γ ) ⊂ Cn . Wγ is the restriction of W to Fix (γ ). Wγ∞ := (ReWγ )−1 ( M, ∞) with M ≫ 0, where ReWγ is the real part of Wγ . Each γ ∈ GW has a unique form √ √ (2.4) γ = exp(2π −1Θγ1 ), · · · , exp(2π −1Θγn ) ∈ (C∗ )n ,

Θγi ∈ [0, 1) ∩ Q.

Thus HW is a graded vector space, where for each nonzero α ∈ Hγ , we assign its degree n

deg α = Nγ +

∑ (Θγi − qi ).

i= 1

We call Hγ a narrow sector if Fix(γ ) consists of 0 ∈ Cn only, or a broad sector otherwise.

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The FJRW vector space HW is equipped with a symmetric and nondegenerate pairing

h , i :=

∑

γ ∈ GW

h , iγ

from the intersection pairing h , iγ : Hγ × Hγ −1 → C of Lefchetz thimbles, and we have (see section 5.1 of [13] and references therein) ! M G (2.5) (Jac (Wγ )ωγ ) W , Residue . ( HW , h , i) ∼ = γ ∈ GW

Here ωγ is a volume form on Fix (γ ) of the type dx j1 ∧ · · · ∧ dx j Nγ , where we mean ωγ = 1 if Nγ = 0. GW acts on both xi and dxi . Let (Jac(Wγ )ωγ ) GW be the GW -invariant part of the action. We choose a generator 1γ := ωγ ∈ H Nγ (Fix(γ ); Wγ∞ ; C).

(2.6)

If Hγ is narrow, then Hγ ∼ = C is generated by 1γ . If Hγ is broad, we = (Jac (Wγ )ωγ ) GW ∼ denote generators of Hγ by φ 1γ via φ ωγ ∈ (Jac (Wγ )ωγ ) GW . It is highly nontrivial to construct a virtual cycle for the moduli of solutions of Witten equations. For the details of the construction, we refer to the original paper of Fan, Jarvis and Ruan [14]. Let C := Cg,k be a stable genus-g orbifold curve with marked points p1 , . . . , pk (where 2g − 2 + k > 0). We only allow orbifold points at marked points and nodals. Near each orbifold point p, a local chart is given by C/G p with G p ∼ = Z/mZ for some positive integer m. Let L1 , . . . , Ln be orbifold line bundles over C . Let σi be a C ∞ section of Li . We consider the W-structures, which can be thought of as the background data to be used to set up the Witten equations ∂W = 0. ∂¯σi + ∂σi a

For simplicity, we only discuss cases that W = M1 + · · · + Mn , with Mi = ∏nj=1 x j i j . Let KC be the canonical bundle for the underlying curve C and ρ : C → C be the forgetful morphism. A W-structure L consists of (C , L1 , · · · , Ln , ϕ1 , · · · , ϕn ) where ϕi is an isomorphism of orbifold line bundles

ϕi :

n O j= 1

⊗a L j i, j

∗

−→ ρ (KC,log ),

KC,log := KC ⊗

k O

O ( p j ).

j= 1

A W-structure induces a representation r p : G p → GW at each point p ∈ C . We require it to be faithful. The moduli space of pairs C = (C , L) is called the moduli of stable Worbicurves and denoted by W g,k . According to [13], W g,k is a Deligne-Mumford stack, and the forgetful morphism st : W g,k → M g,k to the moduli space of stable curves is


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flat, proper and quasi-finite. W g,k can be decomposed into open and closed stacks by decorations on each marked point,

∑

W g,k =

γ j : = r p j ( 1) .

W g,k (γ1 , . . . , γk ),

(γ1 ,...,γk )∈( GW ) k

Furthermore, let Γ be the dual graph of the underlying curve C. Each vertex of Γ represents an irreducible component of C, each edge represents a node, and each half-edge represents a marked point. Let ♯ E(Γ) be the number of edges in Γ . We decorate the halfedge representing the point p j by an element γ j ∈ GW . We denote the decoration by Γγ1 ,...,γk and call it a GW -decorated dual graph. We further call it fully GW -decorated if we also assign some γ+ ∈ GW and γ− = (γ+ )−1 on the two sides of each edge. Let us denote the moduli of stable W-orbicurves with fixed decorations (γ1 , . . . , γk ) on Γ by W g,k (Γγ1 ,...,γk ). We have a decomposition W

g,k (γ1 , . . . , γk )

= ∑ W g,k (Γγ1 ,...,γk ). Γ

Let ρ : C → C be the forgetful morphism to the underlying curve. If W nonempty, then the Line bundle criterion follows([13, Proposition 2.2.8]): k

(2.7)

deg(ρ∗ Li ) = (2g − 2 + k)qi −

∑ Θi

γj

j= 1

g,k (γ1 , · · ·

, γk ) is

∈ Z, i = 1, · · · , n.

In [14], Fan, Jarvis and Ruan perturb the polynomial W to polynomials of Morse type and construct virtual cycles from the solutions of the perturbed Witten equations. Those virtual cycles transform in the same way as the Lefschetz thimbles attched to the critical points of the perturbed polynomials. As a consequence, they construct a virtual cycle k

[W g,k (Γγ1 ,...,γk )]vir ∈ H∗ (W g,k (Γγ1 ,...,γk ), C) ⊗ ∏ HNγ j (Fix(γ j ), Wγ∞j , C) GW , j= 1

which has total degree k

(2.8)

2

(cˆW − 3)(1 − g) + k − ♯ E(Γ) −

n

∑∑

j= 1 i= 1

γ (Θi j

!

− qi ) .

⊗ k −→ H ∗ (M , C)} Based on this, they obtain a cohomological field theory {ΛW g,k g,k : ( HW )

with a flat identity. Each ΛW g,k is defined by extending the following map linearly to HW , k |GW |g vir [ ) : = PD st )] ∩ ΛW ( α , . . . , α W ( γ , . . . , γ αj ∗ 1 1 k k g,k ∏ g,k deg(st) j= 1

!

,

α j ∈ Hγ j .

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Definition 2.2. Let ψ j be the j-th psi class in H ∗ (M g,k ). Define FJRW invariants (or correlators) (2.9)

hτℓ1 (α1 ), . . . , τℓk (αk )iW g,k

=

Z

k

M g,k

j ΛW g,k (α 1 , . . . , α k ) ∏ ψ j ,

ℓ

j= 1

α j ∈ Hγ j .

The FJRW invariants in (2.9) are called primary if all ℓ j = 0. We simply denoted them by hα1 , . . . , αk iW g . We call ( HW , •) an FJRW ring where the multiplication • on HW is defined by

hα • β, γ i = hα , β, γ iW 0 .

(2.10)

Let us fix a basis {α j }µj=1 of HW , with α1 being the identity. Let t( z) = ∑m≥0 ∑µj=1 tm,α j α j zm . The FJRW total ancestor potential is defined to be ! 1 FJRW g−1 (2.11) AW = exp ∑ h¯ ∑ k! ht(ψ1 ) + ψ1 , . . . , t(ψk ) + ψk iWg,k . g≥0 k≥ 0 There is a formal Frobenius manifold structure on HW , in the sense of Dubrovin [12]. Its prepotential is given by FJRW F0,W =

1 ∑ k! ht0 , . . . , t0 iW0,k , k≥ 3

µ

t0 =

∑ t0,α α j . j

j= 1

The prepotential satisfies the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations: (2.12)

FJRW ∂3 F0,W

∑ ∂tα ∂tα ∂tα η i, j

a

d

i

ij

FJRW ∂3 F0,W

∂tα j ∂tαb ∂tαc

=∑ i, j

FJRW ∂3 F0,W

∂tα a ∂tαb ∂tαi

η

ij

FJRW ∂3 F0,W

∂tα j ∂tαc ∂tαd

,

tα := t0,α ,

where ηi j is the inverse of the matrix hαi , α j i . It implies ([13, Lemma 6.2.6])

h. . . , α a , αb • αc , αd i0,k = Sk + h. . . , α a • αb , αc , αd i0,k + h. . . , α a , αb , αc • αd i0,k (2.13) − h. . . , α a • αd , αb , αc i0,k . where k ≥ 3, Sk is a linear combination of products of correlators with number of marked points no greater than k − 1. Moreover, both S3 = S4 = 0. Another important tool is the Concavity Axiom [13, Theorem 4.1.8]. Consider the universal W-structure (L1 , · · · , Ln ) on the universal curve π : C → W g,k (Γγ1 ,...,γk ). (2.14)

If all Hγi are narrow and π∗ (⊕ni=1 Li ) = 0,

then R1 π∗ (⊕ni=1 Li ) is a vector bundle of constant rank, denoted by D, and (2.15)

k [W g,k (Γγ1 ,...,γk )]vir ∩ ∏ 1γi = (−1) D c D R1 π∗ (⊕ni=1 Li ) ∩ [W g,k (Γγ1 ,...,γk )]. i= 1


11

It can be calculated by the orbifold Grothendieck-Riemann-Roch formula [8, Theorem 1.1.1]. As a consequence, if the codimention D = 1, we have ! γΓ 4 B ( Θγ j ) n B ( Θ ) B2 ( q i ) 2 2 W i i [Γ] . κ1 − ∑ ψj + ∑ (2.16) Λ0,4 (1γ1 , . . . , 1γ4 ) = ∑ 2 2 2 Γ j= 1 i= 1 Here B2 ( x) := x2 − x + 61 is the second Bernoulli polynomial. κ1 is the 1-st kappa class on M0,4 . Here the graphs Γ are fully GW -decorated on the boundary of W 0,4 (γ1 , . . . , γ4 ) such that each component of Γ satisfies the line bundle criterion (2.7). We call a correlator concave if it satisfies (2.14). Otherwise we call it is nonconcave. Nonconcave correlator may contain broad sectors. In this paper, we will use WDVV to compute the nonconcave correlators. Some other methods are described in [7, 21]. 2.2. FJRW invariants. In this subsection, we will prove Proposition 1.2. Let us first describe the construction of the mirror polynomial W T . Let W = M1 + · · · + Mn , with Mi = ai j ∏nj=1 x j . We call such a polynomial W invertible because its exponent matrix EW := ai j is invertible. Berglund and Hubsch ¨ [5] introduced a mirror polynomial W T , W T :=

(2.17)

n

n

∑ ∏ xj

a ji

.

i= 1 j= 1

Its exponent matrix EW T is just the transpose matrix of EW , i.e. EW T = ( EW ) T . In [30], Kreuzer and Skarke proved that every invertible W is a direct sum of three atomic types of singularities: Fermat, chain and loop. If W is of atomic type, then W T belongs to the same atomic type. We list the three atomic types (with qi ≤ 21 ) and a C-basis of their Jacobi algebra as follows. The table also contains an element φµ of highest degree. TABLE 2. Invertible singularities

m-Fermat m-Chain: m-Loop:

Polynomial f

C-basis of Jac ( f )

am x1a1 + · · · + xm

ki ∏m i= 1 xi , ki < ai − 1

am x1 x1a1 x2 + x2a2 x3 + · · · + xm

ki ∏m i= 1 xi , ki < ai

am x1a1 x2 + x2a2 x3 + · · · + xm

ki {∏ m i= 1 xi }k

φµ a −2

i ∏m i= 1 xi

ai − 1 x1a1 −2 ∏m i= 2 xi a −1

i ∏m i= 1 xi

Here in the case of m-Chain, k = (k1 , · · · , km ) satisfies (1) k j ≤ a j − 1 for all j and (2) the property that k is not of the form ( a1 − 1, 0, a3 − 1, 0, · · · , a2l −1 − 1, i, ∗, · · · , ∗) with i ≥ 1. A first step towards the LG-LG mirror symetry Conjecture 1.1 is a ring isomorphism between ( HW , •) and Jac (W T ). The ring isomorphism has been studied in [1, 13, 15, 27, 28] for various examples. According to the Axiom of Sums of singularities [13, Theorem 4.1.8

12


(8)] in FJRW theory, the FJRW ring ( HW , •) is a tensor product of the FJRW ring of each direct summand. Krawitz constructed a ring isomorphism for each atomic type if all qi < 12 [27]. For our purpose, if W is a polynomial in Table 1, then it is already known that ( HW , •) is isomorphic to Jac (W T ) except for W = x2 + xyq + yzr , (q, r) = (3, 3), (2, 4). We will give the new constructions for the two exceptional cases, and will also briefly introduce the earlier constructions for the other 12 cases. −1 Since EW is invertible, we can write EW using column vectors ρk ,

−1 EW = ρ1 | · · · |ρn ,

( k) ( k) T , ρ k : = ϕ1 , · · · , ϕn

( k)

∈ Q.

ϕi

We can view ρk as an element in GW by defining the action √ √ ( k) ( k) ρk = (exp(2π −1ϕ1 ), · · · , exp(2π −1ϕn )) ∈ GW . Thus ρi J ∈ GW , with J the exponential grading element in (2.2). Proposition 2.3 ([27]). For any n-variable invertible polynomial W with each degree qi < 21 , there is a degree-preserving ring isomorphism Ψ : Jac (W T ) → ( HW , •). In particular, if ρi J is narrow for i = 1, · · · , n, then Ψ is generated by Ψ( xi ) = 1ρi J ,

(2.18)

i = 1, · · · , n

Example 2.4. Let W = x p + yq , p, q > 2. Denote γi, j =

exp( 2π

√ 2π −1 j −1 i ) , exp ( ) . p q

√

The FJRW ring ( HW , •) is generated by {1γ2,1 , 1γ1,2 }. Then W T = W and the ring isomorphism ∼ =

Ψ : Jac(W T ) → ( HW , •) generated by (2.18) extends as Ψ( xi−1 y j−1 ) = 1γi, j ,

(2.19)

1 ≤ i ≤ p, 1 ≤ j ≤ q.

For 2-Loop singularities, ρi J may not be narrow. However, ring isomorphisms still exist. According to [1, 27], we have Example 2.5. For W = x2 y + xy3 + z3 ∈ Q12 , GW ∼ = µ15 . A ring isomorphism Ψ : ∼ = T Jac (W ) → ( HW , •) is obtained by extending (2.18) from Ψ( x) = x1 J 10 .

(2.20)

The corresponding vector space isomorphism Ψ : Jac (W T ) → HW is as follows: HW

1J

1 J 13

1 J 11

x1 J 10

y2 1 J 10

1 J8

1 J7

x1 J 5

y2 1 J 5

1 J4

1 J2

Jac (W T )

1

y

z

x

y2

yz

xy

xz

y2 z

xy2

xyz xy2 z

1 J 14


13

Now we discuss if there exists qi = 21 for W. Without loss of generality, we assume W am . Then Fix(ρ1 J ) = {( x1 , · · · , xm ) ∈ is of the atomic type: W = x21 + x1 x2a2 + · · · + xm−1 xm k C | xi = 0, i > 2}. Thus Hρ1 J is generated by a broad element x2a2 −1 1ρ1 J , which is a ring generator of HW . If m = 2, it is known [15] that Ψ : Jac (W T ) → ( HW , •) generates a ring isomorphism, by Ψ( x1 ) = a2 x2a2 −1 1ρ1 J and Ψ( x2 ) = 1ρ2 J . The key point is that the residue formula in (2.5) implies a2 − 1 1ρ1 J , x2a2 −1 1ρ1 J , 1 J iW h x2a2 −1 1ρ1 J , x2a2 −1 1ρ1 J , 1ρ1−a2 J −1 iW 0 = h x2 0 = − 2

1 . a2

Inspired from this, for m ≥ 3, we consider

K := h x2a2 −1 1ρ1 J , x2a2 −1 1ρ1 J , 1ρ1−a2 ρ−1 J −1 iW 0 . 2

If K 6= 0, then it is possible to define Ψ ( x1 ) =

(2.21)

r

a − 2 K

3

x2a2 −1 1ρ1 J .

In Section 2.3, using Getzler’s relation, we will prove the following nonvanishing lemma, Lemma 2.6. Let W = x2 + xyq + yzr , (q, r) = (3, 3), (2, 4). Then Kq,r := h yq−1 1ρ1 J , yq−1 1ρ1 J , 1ρ1−a2 ρ−1 J −1 iW 0 6 = 0.

(2.22)

2

3

As a direct consequence of Lemma 2.6, it is not hard to check the following statement. Proposition 2.7. Let W T be one of the exceptional unimodular singularities in Table 1, then the map Ψ in (2.18) and (2.21) generates a degree-preserving ring isomorphism Ψ : Jac (W T ) ∼ = ( HW , •). Proof. We only need to consider W = x2 + xyq + yzr , (q, r) = (3, 3), (2, 4). Lemma 2.6 allows us to extend Ψ by defining Ψ( x) as in (2.21). Then we can check directly that q Ψ ( x) • Ψ ( x) = − h yq−1 1ρ1 J , yq−1 1ρ1 J , 1ρ1−qρ−1 J −1 iW 1ρq−1ρ J = −qΨ( yq−1 z). 0 3 Kq,r 2 3 2 Other relations are easily checked.

∼ =

Explicitly, if W = x2 + xy3 + yz3 , the vector space isomorphism Ψ : Jac(W T ) → HW is HW

1J

1 J 15

1 J 13

Jac(W T )

1

y

z

q

− K33,3 y2 1 J 12 1 J 11 1 J 9 1 J 7 x

y2

yz

z2

q

− K33,3 y2 1 J 6 xz

If W = x2 + xy2 + yz4 , then the vector space isomorphism is given by

1 J5

1 J3

1 J 17

y2 z yz2

y2 z2

14


HW

1J

1 J 13

Jac (W T )

1

z

q

− K22,4 y1 J 12 1 J 11 1 J 9

y1 J 8

1 J7

1 J5

z2

xz

yz

z3

x

y

q

− K22,4 y1 J 4 xz2

1 J3

1 J 15

yz2

yz3

We will give explicit formulas of the isomorphism Ψ of all other cases in the appendix. Those isomorphisms Ψ turn out to identify the ancestor total potential of the FJRW theory of (W, GW ) with that of Saito-Givental theory of W T up to a rescaling. Next we compute the FJRW invariants in Proposition 1.2. We introduce a new notation 1φ := Ψ(φ),

(2.23)

φ ∈ Jac (W T ).

Due to the above conventions, the second part of Proposition 1.2 is simplified as follows. Proposition 2.8. Let MiT be the i-th monomial of W T with the ordering in Table 1. We have

h1xi , 1xi , 1 M T / x2 , 1φµ iW 0 = qi ,

(2.24)

i

i

∀i = 1, · · · , n.

Proof. We classify all the correlators in (2.24) into concave correlators and nonconcave correlators. For the concave correlators, we use (2.16) to compute. For the nonconcave correlators, we use WDVV to reconstruct them from concave correlators and again use (2.16). We will freely interchange the notation

( x1 , x2 , x3 ) = ( x, y, z).

(2.25)

Let us start with concave correlators. As an example, we compute h1x , 1x , 1x p−2 , 1φµ iW 0 for W = x p + yq . The computation of all the other concave corrlators in (2.24) follows γi, j similarly. For W = x p + yq , we recall that for γi, j ∈ GW ∼ = µ p × µq , we have Θ1 = γi, j j i p , Θ 2 = q . All the sectors are narrow and 1γi, j = 1 xi −1 y j −1 with our notation conventions. According to the line bundle criterion (2.7), we know for h1x , 1x , 1x p−2 , 1φµ iW 0 , deg ρ∗ L1 = −2, deg ρ∗ L2 = −1. Thus π∗ L1 = π∗ L2 = 0 and the correlator is concave. Moreover, R1 π∗ L2 = 0 and the nonzero contribution of the virtual cycle only comes from R1 π∗ L1 . Now we can apply (2.16). There are three decorated dual graphs in Γ0,4 (1x , 1x , 1x p−2 , 1φµ ), where we simply denote 1i, j := 1γi, j , 12,1

❙ γ ❙ Γ1 ✓ ✓

12,1

1 p−1,1

γΓ−1 1 ✓ ✓ ❙ ❙

1 p−1,q−1

12,1

❙ γ ❙ Γ2 ✓ ✓

1 p−1,1

12,1

γΓ−2 1 ✓ ✓ ❙ ❙

1 p−1,q−1

12,1

❙ γ ❙ Γ3 ✓ ✓

1 p−1,q−1

12,1

γΓ−3 1 ✓ ✓ ❙ ❙

1 p−1,1


γΓ

The decorations of the boundary classes are Θ1 i =

p−3 p , 0, 0

15

for i = 1, 2, 3. We obtain

h12,1 , 12,1 , 1 p−1,1 , 1 p−1,q−1iW 0 = = =

Z

M0,4

ΛW 0,4 ( 12,1 , 12,1 , 1 p− 1,1 , 1 p− 1,q − 1 )

2 p−1 p−3 1 1 ) + 2B2 (0) + B2 ( ) B2 ( ) − 2B2 ( ) − 2B2 ( 2 p p p p 1 . p

All the nonconcave correlators in (2.24) are listed as follows:

• • • • • •

2 2 4 h1 y , 1 y , 1z , 1φµ iW 0 for 3-Chain W = x + xy + yz . 2 q r h1x , 1x , 1 y , 1φµ iW 0 for 3-Chain W = x + xy + yz , ( q, r) = ( 3, 3) or ( 2, 4). 2 2 3 W h1x , 1x , 1 y , 1φµ iW 0 and h 1 y , 1 y , 1 z , 1φµ i 0 for 3-Loop W = x z + xy + yz . 2 4 3 h1x , 1x , 1 y , 1φµ iW 0 for W = x + xy + z . 2 3 3 h1x , 1x , 1 y , 1φµ iW 0 for W = x y + xy + z . 2 2 4 h1x , 1 y , 1 y , 1φµ iW 0 for W = x y + y + z .

For the nonconcave correlators, we will use the WDVV equations and the ring relations to reconstruct them from concave correlators. Let us start with the value of 2 2 4 3 T h1 y , 1 y , 1 yq−2 z , 1φµ iW 0 in a 3-Chain W = x + xy + yz . Since φµ = yz ∈ Jac (W ) and 1 y • 1 yz = 0, we get 1 1 = . 16 4 The first equality follows from the WDVV equation(2.13). We also use 1 y • 1 yz = 0. W Both h1z , 1 y , 1 yz , 1z2 • 1 y iW 0 and h 1 z , 1 y • 1 y , 1 yz , 1 z2 i 0 are concave correlators and can be computed by (2.16). For other nonconcave correlators, we will list the WDVV equations. The concavity computation is checked easily. For 3-Chain W = x2 + xyq + yzr , (q, r) = (3, 3) or (2, 4), 1φµ = 1 yq−1 zr−1 . W W h1z , 1 y , 1 yz • 1z2 , 1 y iW 0 = h 1 z , 1 y , 1 yz , 1 z2 • 1 y i 0 − h 1 z , 1 y • 1 y , 1 yz , 1 z2 i 0 = 0 − (− 4)

W W h1 y , 1x , 1φµ , 1x iW 0 = −h 1 y , 1 x • 1 x , 1 y , 1 yq−2 zr−1 i 0 = qh 1 y , 1 yq−1 z , 1 y , 1 yq−2 zr−1 i 0 =

1 . 2

For 3-Loop W = x2 z + xy2 + yz3 , 1φµ = 1xyz2 . We get W W h1 y , 1x , 1xy • 1z2 , 1x iW 0 = h 1 y , 1 x , 1 xy , 1 z2 • 1 x i 0 − h 1 y , 1 x • 1 x , 1 xy , 1 z2 i 0 = W W h1z , 1 y , 1z • 1xyz , 1 y iW 0 = h 1 z , 1 y • 1 z , 1 xyz , 1 y i 0 − h 1 z , 1 y • 1 y , 1 z , 1 xyz i 0 =

1 13 1 13

2 − (−2) 13 =

1 − (−3) 13 =

For W = x2 + xy4 + z3 , 1x is broad. However, 1 W W h1 y , 1x , 1φµ , 1x iW 0 = −h 1 y , 1 x • 1 x , 1 y , 1 y2 z i 0 = 4h 1 y , 1 y3 , 1 y , 1 y2 z i 0 = . 2

5 13 . 4 13 .

16


For W = x2 y + xy3 + z3 , we get W W h1 y , 1x , 1xy • 1 yz , 1x iW 0 + h 1 y , 1 x • 1 x , 1 xy , 1 yz i 0 = h 1 y , 1 x , 1 xy , 1 yz • 1 x i 0

1 1 W W = − 21 h1 y , 1x , 1 y • 1 y2 z , 1xy iW 0 = − 2 h 1 y , 1 xy , 1 xy , 1 y2 z i 0 = − 2 h 1 y , 1 xy , 1 y • 1 yz , 1 xy i 0

= −h1 y , 1xy , 1 yz , 1xy2 iW 0 .

The first, third and last equalities are WDVV equations. Finally, we get 1 1 2 W W h1x , 1x , 1 y , 1xy2 z iW 0 = −h 1 y , 1 x • 1 x , 1 xy , 1 yz i 0 − h 1 y , 1 xy , 1 yz , 1 xy2 i 0 = −(− ) − (− ) = . 5 5 5 For W = x2 y + y2 + z4 , we get W 1x , 1 y • 1 y , 1xy , 1z2 iW h1x , 1 y , 1xy • 1z2 , 1 y iW 0 0 = h 1 x , 1 y , 1 xy , 1 z2 • 1 y i 0 − h W W = h1 y , 1x , 1x , 1 y • 1 yz2 i0 − h1 y , 1x • 1 yz2 , 1x , 1 y i0 − h1x , 1 y • 1 y , 1xy , 1z2 iW 0 .

Combining this equation and y2 = −2x, we get

1 1 3 W W h1x , 1 y , 1xyz2 , 1 y iW = . 0 = −h 1 y , 1 x , 1 x , 1 xz2 i 0 + h 1 x , 1 x , 1 xy , 1 z2 i 0 = −(− ) + 8 4 8

2.3. Nonvanishing invariants. In this subsection, we will prove Lemma 2.6. Our tool is the Getzler’s relation [16], which is a linear relation between codimension two cycles in H∗ (M1,4 , Q). Let us briefly introduce this relation here. Consider the dual graph, ∆0 · ∆{234} :=

✛✘ 1 ✟✟ ✟ ❍ ✟2 ❍❍✟✟ ✚✙ 3 ❍ ❍❍ 4

This graph represents a codimension-two stratum in M1,4 : A vertex represents a genus-0 component. An edge connecting two vertices (including a circle connecting the same vertex) represents a node, a tail (or half-edge) represents a marked point on the component of the corresponding vertex. Let ∆0,3 be the S4 -invariant of the codimension-two stratum in M1,4 , ∆0,3 = ∆0 · ∆{123} + ∆0 · ∆{124} + ∆0 · ∆{134} + ∆0 · ∆{234} . We denote δ0,3 = [∆0,3 ] the corresponding cycle in H4 (M1,4 , Q). We list the corresponding unordered dual graph for other strata below. A filled circle (as a vertex) represents a genus-1 component. See [16] for more details.


δ2,2 :

❙ ❙ ✓ ✓

✉

δ2,3 :

✓ ✓ ❙ ❙

✉

δ3,4 : ✉

δ2,4 :

✟✟ ✟ ❍ ✟ ❍❍✟✟ ❍❍ ❍

✉

δ0,4 :

✟✟ ✟ ❍ ✟ ❍❍✟✟ ❍❍ ❍

δβ :

✛✘ ✚ ✚✏ ✏ ✚✏ P ❩PP ✚✙ ❩❩

✟✟ ✟ ❍ ❍❍✟✟✟ ❍❍ ❍

17

✛✘ ❍❍ ✟✟ ❍ ✟ ✟ ❍ ❍❍ ✟✟ ✚✙

In [16], Getzler found the following identity: 12δ2,2 + 4δ2,3 − 2δ2,4 + 6δ3,4 + δ0,3 + δ0,4 − 2δβ = 0 ∈ H4 (M1,4 , Q).

(2.26)

Proof of Lemma 2.6: We start with W = x2 + xy2 + yz4 . We normalize √ u := y1 J 12 , v = −2y1 J 8 , w = −2y1 J 4 . The nonvanishing pairings between these broad elements are hu, wi = 1, hv, vi = 1.

We integrate ΛW 1,4 ( 1 J 9 , 1 J 9 , 1 J 9 , 1 J 9 ) over the Getzler’s relation (2.26). The Composition law [13, Theorem 4.1.8 (6)] in FJRW theory implies Z

δ0,3

=

ΛW 1,4 ( 1 J 9 , 1 J 9 , 1 J 9 , 1 J 9 )

4h 1 J 9 , 1 J 9 , 1 J 9 , 1 J 7 iW 0

∑η

α ,β

α ,β



 = 4h 1 J 9 , 1 J 9 , 1 J 9 , 1 J 7 iW 0

h 1 J 9 , 1 J 9 , α , β iW 0

!

W 2h1 J 9 , 1 J 9 , 1 J 13 , 1 J 3 iW 0 + 2h 1 J 9 , 1 J 9 , 1 J 11 , 1 J 5 i 0 + W W 2h 1 J 9 , 1 J 9 , 1 J 9 , 1 J 7 iW 0 + 2h 1 J 9 , 1 J 9 , u, w i 0 + h 1 J 9 , 1 J 9 , v, vi 0



.

The factor 4 comes from that there are 4 strata in ∆0,3 which contribute. We have the factor 2 for h1 J 9 , 1 J 9 , 1 J 13 , 1 J 3 iW 0 since both α = 1 J 13 and α = 1 J 3 give the same correlator. Finally, 1 J is the identity, and the string equation implies h1 J 9 , 1 J 9 , 1 J 15 , 1 J iW 0 = 0. There are two correlators contain broad sectors, we simply denote C1 := h1 J 9 , 1 J 9 , v, viW 0 ,

C2 := h1 J 9 , 1 J 9 , u, wiW 0 .

We can calculate the concave correlators using orbifold-GRR formula in (2.16) and get 1 h1 J 9 , 1 J 9 , 1 J 13 , 1 J 3 iW 0 = , 4 This implies Z

δ0,3

1 h1 J 9 , 1 J 9 , 1 J 11 , 1 J 5 iW 0 = − , 8

ΛW 1,4 ( 1 J 9 , 1 J 9 , 1 J 9 , 1 J 9 ) = C2 +

h 1 J 9 , 1 J 9 , 1 J 9 , 1 J 7 iW 0 =

C1 1 + . 2 4

1 . 8

18


Similarly, we get Z

δβ

2 2 ΛW 1,4 ( 1 J 9 , 1 J 9 , 1 J 9 , 1 J 9 ) = 6C2 + 3C1 +

9 , 16

Z

δ0,4

ΛW 1,4 ( 1 J 9 , 1 J 9 , 1 J 9 , 1 J 9 ) =

165 . 128

The last equality requires the computation for a genus-0 correlator with 5 marked points. It is reconstructed from some known 4-point correlators by WDVV equations. On the other hand, using the homological degree (2.8), we conclude the vanishing of the integration of ΛW 1,4 ( 1 J 9 , 1 J 9 , 1 J 9 , 1 J 9 ) over those strata which contain genus-1 component. Thus Z

12δ2,2 + 4δ2,3 − 2δ2,4 + 6δ3,4

ΛW 1,4 ( 1 J 9 , 1 J 9 , 1 J 9 , 1 J 9 ) = 0.

Now apply Getzler’s relation (2.26), we get

− 12C22 + C2 − 6C12 +

(2.27)

C1 53 + = 0. 2 128

On the other hand, since 1 J 9 = 1 J 13 • 1 J 13 , we apply WDVV equations and get  2  W = h 1 , u, viW  , h u, u, 1 i 13 9  J J 0 0 

W W W h1 J 9 , 1 J 9 , v, viW 0 + h 1 J 13 , 1 J 13 , 1 J 9 , 1 J 15 i 0 = 2h 1 J 9 , 1 J 13 , v, w i 0 h 1 J 13 , u, vi 0 ,     h1 9 , 1 9 , u, wiW + h1 13 , 1 13 , 1 9 , 1 15 iW = h1 9 , 1 13 , v, wiW h1 13 , u, viW . J J J J J J J J J 0 0 0 0

W If hu, u, 1 J 9 iW 0 = 0, then h 1 J 13 , u, vi 0 = 0 and the rest two equations above implies

C1 = C2 = −h1 J 13 , 1 J 13 , 1 J 9 , 1 J 15 iW 0 = −

3 , 16

where the last equality follows from (2.16). However, this contradicts with formula (2.27). Next we consider W = x2 + xy3 + yz3 . We denote   u := y2 1 12 , w = −3y12 1 4 , J J  C1 := h1 13 , 1 13 , w, wiW , C2 := h1 7 , 1 13 , u, wiW , J

J

J

J

0

0

C3 := h1 J 7 , 1 J 7 , u, uiW 0 .

We integrate ΛW 1,4 ( 1 J 13 , 1 J 13 , 1 J 7 , 1 J 7 ) over the Getzler’s relation (2.26) and get

− 8C22 −

(2.28)

2C2 8 − 2C1 C3 + =0 3 81

On the other hand, since 1 J 7 = 1 J 13 • 1 J 13 , the WDVV equations imply   h1 7 , 1 13 , u, wiW + h1 13 , 1 13 , 1 13 , 1 17 iW = h1 13 , 1 13 , w, wiW h1 13 , u, uiW , J J J J J J J J J 0 0 0 0  h1 7 , 1 7 , u, uiW = h1 7 , 1 13 , u, wiW h1 13 , u, uiW . J

J

0

J

J

0

J

0

5 Now h1 J 13 , u, uiW 0 = 0 implies C2 = − 18 and C3 = 0. This contradicts with (2.28).


3. B- MODEL : S AITO ’ S

19

THEORY OF PRIMITIVE FORM

Throughout this section, we consider the Landau-Ginzburg B-model defined by f : X = Cn → C, where f is a weighted homogeneous polynomial with isolated singularity at the origin: f ( λ q 1 x1 , · · · , λ q n xn ) = λ f ( x1 , · · · , xn ) . Recall that qi are called the weights of xi , and the central charge of f is defined by cˆ f =

∑(1 − 2qi ). i

Associated to f , the third author has introduced the concept of a primitive form [40], which, in particular, induces a Frobenius manifold structure (sometimes called a flat structure) on the local universal deformation space of f . This gives rise to the genus zero correlation functions in the Landau-Ginzburg B-model, which are conjectured to be equivalent to the FJRW-invariants on the mirror singularities. The general existence of primitive forms for local isolated singularities is proved by M.Saito [45] via Deligne’s mixed Hodge theory. In the case for f being a weighted homogeneous polynomial, the existence problem is greatly simplified due to the semisimplicity of the monodromy [40, 45]. However, explicit formulas of primitive forms were only known for ADE and simple elliptic singularities [40] (i.e., for cˆ f ≤ 1). This led to the difficulty of computing correlation functions in the Landau-Ginzburg B-model, and has become one of the main obstacles toward proving mirror symmetry between Landau-Ginzburg models. Recently there has developed a perturbative method [31] to compute the primitive forms for arbitrary weighted homogeneous singularities. With the help of certain reconstruction type theorem from the WDVV equation (see e.g. Lemma 4.2), it completely solves the computation problem in the Landau-Ginzburg B-model at genus zero. In this section, we will give a self-contained exposition of the method of [31] for the purpose of applications to mirror symmetry, and compute the relevant correlation functions for the 14 exceptional unimodular singularities. 3.1. Higher residue and good basis. Let 0 ∈ X = Cn be the origin. Let ΩkX,0 be the germ of holomorphic k-forms at 0. In this paper we will work with the following space [41] (0)

−1 H f := ΩnX,0 [[ z]]/(df + zd)ΩnX,0

20


which is a formally completed version of the Brieskorn lattice associated to f . Given a (0) differential form ϕ ∈ ΩnX,0 , we will use [ϕ] to represent its class in H f . (0)

(− k)

There is a natural semi-infinite Hodge filtration on H f given by H f graded pieces (− k)

Hf

(− k− 1)

/H f

∼ = Ωf,

(0)

:= zk H f , with

−1 where Ω f := ΩnX,0 /df ∧ ΩnX,0 .

(0)

In particular, H f is a free C[[ z]]-module of rank µ = dimC Jac ( f )o , the Milnor number of f . We will also denote the extension to Laurent series by (0)

H f := H f ⊗C[[z]] C(( z)). (0)

There is a natural Q-grading on H f defined by assigning the degrees deg( xi ) = qi ,

deg(dxi ) = qi ,

deg( z) = 1.

Then for a homogeneous element of the form ϕ = zk g( xi )dx1 ∧ · · · ∧ dxn ,, we have deg(ϕ) = deg( g) + k + ∑ qi . i

In [41], the third author constructed a higher residue pairing (0)

(0)

Kf : Hf ⊗ Hf

→ zn C[[ z]]

which satisfies the following properties (1) K f is equivariant with respect to the Q-grading, i.e., deg(K f (α , β)) = deg(α ) + deg(β) (0)

for homogeneous elements α , β ∈ H f .

(2) K f (α , β) = (−1)n K f (β, α ), where the − operator takes z → − z. (3) K f (v( z)α , β) = K f (α , v(− z)β) = v( z)K f (α , β) for v( z) ∈ C[[ z]]. (4) The leading z-order of K f defines a pairing (0)

(0)

(0)

(0)

H f / zH f ⊗ H f / zH f

→ C,

α ⊗ β 7→ lim z−n K f (α , β) z→0

which coincides with the usual residue pairing

η f : 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”. It is naturally extended to K f : H f ⊗ H f → C(( z))


21

which we denote by the same symbol. This defines a symplectic pairing ω f on H f by

ω f (α , β) := Resz=0 z−n K f (α , β)dz, (0)

with H f being a maximal isotropic subspace. Following [40], (0)

Definition 3.1. A good section σ is defined by a splitting of the quotient H f

→ Ωf,

(0)

σ : Ωf → Hf , such that: (1) σ preserves the Q-grading; (2) K f (Im(σ ), Im (σ )) ⊂ zn C.

(0)

A basis of the image Im (σ ) of a good section σ will be referred to as a good basis of H f . Definition 3.2. A good opposite filtration L is defined by a splitting (0)

Hf = Hf ⊕ L such that: (1) L preserves the Q-grading; (2) L is an isotropic subspace; (3) z−1 : L → L. Remark 3.3. Here for f being weighted homogeneous, (1) is a convenient and equivalent statement to the conventional condition that ∇ GM z∂ z preserves L (see e.g. [31] for an exposition). The above two definitions are equivalent. In fact, a good opposite filtration L defines ∼ =

(0)

the splitting σ : Ω f → H f ∩ zL. Conversely, a good section σ gives rise to the good opposite filtration L = z−1 Im(σ )[ z−1 ]. As shown in [40, 45], the primitive forms associated to the weighted homogeneous singularities are in one-to-one correspondence with good sections (up to a nonzero scalar). Therefore, we only introduce the notion of good sections, and refer our readers to loc. cite for precise notion of the primitive forms. We remark that for general isolated singularities, we need the notion of very good sections [45, 46] in order to incorporate with the monodromy. (0)

3.2. The perturbative equation. We start with a good basis {[φα dn x]}αµ =1 of H f , where dn x := dx1 · · · dxn . In this subsection, we will formulate the perturbative method of [31] for computing its associated primitive form, flat coordinates and the potential function. The construction works for general f after the replacement of a good basis by a very good one (see also [46]). We will focus on f being weighted homogeneous since in such case it leads to a very effective computation algorithm in practice. In the following discussion we will then assume {φα }αµ =1 to be weighted homogeneous polynomials in C[x] that represent a basis of the Jacobi algebra Jac ( f ) and φ1 = 1.

22


3.2.1. The exponential map. Let F be a local universal unfolding of f (x) around 0 ∈ Cµ : µ

F : Cn × Cµ → C,

F (x, s) := f (x) +

∑ sαφα (x),

α =1

s = (s1 , · · · , sµ ).

The polynomial F becomes weighted homogeneous of total degree 1 after the assignment deg(sα ) := 1 − deg(φα ). The higher residue pairing is also defined for F as the family version, but we will not use it explicitly in our discussion (although implicitly used essentially). (0)

Let B := SpanC {[φα dn x]} ⊂ H f be spanned by the chosen good basis. Then (0)

H f = B[[ z]],

H f = B(( z)).

Let BF := SpanC {φα dn x} be another copy of the vector space spanned by the forms φα dn x. We use a different notation to distinguish it with B, since BF should be viewed as a subspace of the Brieskorn lattice for the unfolding F. See [31] for more details. Consider the following exponential operator [31] e( F− f )/ z : BF → B(( z))[[s]] defined as a C-linear map on the basis of BF as follows. Let C[s]k := Symk (SpanC {s1 , · · · , sµ }) denote the space of k-homogeneous polynomial in s (not to be confused with the weighted homogeneous polynomials). As elements in H f ⊗ C[s]k , we can decompose

[ z−k ( F − f )kφα dn x] =

∑ ∑ hαβ,m zm [φβ dn x], ( k)

m ≥− k β

( k)

where hαβ,m ∈ C[s]k . Then we define ∞

e( F− f )/ z (φα dn x) :=

∑∑ ∑

k= 0 β m ≥− k

( k)

hαβ,m

zm [φβ dn x] ∈ B(( z))[[s]] k!

Proposition 3.4. The exponential map extends to a C(( z))[[s]]-linear isomorphism e( F− f )/ z : BF (( z))[[s]] → B(( z))[[s]]. Proof. Clearly, e( F− f )/ z extends to a C(( z))[[s]]-linear map on BF (( z))[[s]]. The statement follows by noticing e( F− f )/ z ≡ 1 mod (s) under the manifest identification between B and BF . We will use the same symbol K f : B(( z))[[s]] × B(( z))[[s]] → C(( z))[[s]] to denote the C[[s]]-linear extension of the higher residue pairing to H f [[s]] = B(( z))[[s]].


23

Lemma 3.5. For any ϕ1 , ϕ2 ∈ BF , we have K f (e( F− f )/ zϕ1 , e( F− f )/ zϕ2 ) ∈ zn C[[ z, s]] In particular, e( F− f )/ z maps BF [[ z]] to an isotropic subspace of H f [[s]]. Proof. Let KF denote the higher residue pairing for the unfolding F [41]. The exponential operator e( F− f )/ z gives an isometry (with respect to the higher residue pairing) between the Brieskorn lattice for the unfolding F and the trivial unfolding f [31, 46]. That is, K f (e( F− f )/ zϕ1 , e( F− f )/ zϕ2 ) = KF (ϕ1 , ϕ2 ) ∈ zn C[[ z, s]], where ϕ1 , ϕ2 are treated as elements of Brieskorn lattice for the unfolding F. Remark 3.6. The above lemma can also be proved directly via an explicit formula of K f described in [31]. By such a formula, there exists a compactly supported differential operator P( ∂∂x¯i , z ∂x∂ i , y∂ xi , ∧d x¯ i ) on smooth differential forms composed of ∂∂x¯i , z ∂x∂ i , y∂ xi , ∧d x¯ i and some cut-off function such that Z

K f (e( F− f )/ zϕ1 , e( F− f )/ zϕ2 ) = zn

X

e( F− f )/ zϕ1 ∧ P(

∂ ∂ ,z , y∂ xi , ∧d x¯ i )(e−( F− f )/ zϕ2 ). ∂ x¯i ∂xi

Since P will not introduce negative powers of z when passing through e( f − F)/ z, the lemma follows. (0)

Theorem 3.7. Given a good basis {[φα dn x]}αµ =1 ⊂ H f , there exists a unique pair (ζ , J ) satisfying the following: (1) ζ ∈ BF [[ z]][[s]],

(2) J ∈ [dn x] + z−1 B[ z−1 ][[s]] ⊂ H f [[s]], and

e( F− f )/ zζ = J .

(⋆)

Moreover, both ζ and J are weighted homogeneous. Proof. We will solve ζ (s) recursively with respect to the order in s. Let ∞

ζ=

∞

∑ ζ(k) = ∑ ∑ ζ(αk)φα dn x,

k= 0

k= 0 α

ζ(αk) ∈ C[[ z]] ⊗C C[s]k .

Since e( F− f )/ z ≡ 1 mod (s), the leading order of (⋆) is

ζ ( 0) ∈ [dn x] + z− 1 B [ z− 1 ] which is uniquely solved by ζ(0) = φ1 dn x. Suppose we have solved (⋆) up to order N, i.,e, ζ(≤ N ) := ∑kN=0 ζ(k) such that e( F− f )/tζ(≤ N ) ∈ [dn x] + z−1 B[ z−1 ][[s]]

mod (s N +1 ).

Let R N +1 ∈ B(( z)) ⊗C C[s]( N +1) be the ( N + 1)-th order component of e( F− f )/tζ(≤ N ) . Let − R N +1 = R+ N +1 + R N +1

24


− −1 −1 ˜+ where R+ N + 1 ∈ B [[ z]] ⊗C C[s]( N + 1) , R N + 1 ∈ z B [ z ] ⊗C C[s]( N + 1) . Let R N + 1 ∈ B F [[ z]] ⊗C C[s]( N +1) correspond to R+ N + 1 under the manifest identification between B and B F . Then

ζ(≤ N +1) := ζ(≤ N ) − R˜ + N +1 gives the unique solution of (⋆) up to order N + 1. This algorithm allows us to solve ζ , J perturbatively to arbitrary order. The weighted homogeneity follows from the fact that (⋆) respects the weighted degree. Remark 3.8. The above theorem has been essentially proved in [31]. Therein it is shown that the volume form ∞

∑ ∑ ζ(αk)φα dn x

k= 0 α

gives the power series expansion of a representative of the primitive form associated to the good basis {[φα dn x]}αµ =1 . In particular, this is a perturbative way to compute the primitive form via a formal solution of the Riemann-Hilbert-Birkhoff problem. 3.2.2. Flat coordinates and potential function. Let (ζ , J ) be the unique solution of (⋆). As shown in [31], ζ represents the power series expansion of a primitive form. However for the purpose of mirror symmetry, it is more convenient to work with J , which plays the role of Givental’s J-function (see [20] for an introduction). This allows us to read off the flat coordinates and the potential function of the associated Frobenius manifold structure. With the natural embedding z−1 C[ z−1 ][[s]] ֒→ z−1 C[[ z−1 ]][[s]], we decompose

J = [dn x] +

−∞

∑

m =− 1

zm Jm ,

where Jm =

∑ Jmα [φα dn x], Jmα ∈ C[[s]]. α

We denote the z−1 -term by tα (s) := J−α1 (s). It is easy to see that tα is weighted homogeneous of the same degree as sα such that tα = sα + O(s2 ). Therefore tα defines a set of new homogeneous local coordinates on the (formal) deformation space of f . Proposition 3.9. The function J = J (s(t)) in coordinates tα satisfies γ ∂tα ∂tβ J = z−1 ∑ Aαβ (t)∂tγ J γ

γ (t) ∈ C[[t]] of weighted degree deg φα + deg φβ − deg φγ . Morefor some homogeneous Aαβ over, for any α , β, γ , δ, δ δ = ∂tβ Aαγ , ∂tα Aβγ

δ δ σ Aσβγ = ∑ Aβσ Aαγ ∑ Aασ σ

σ


25

Proof. Consider the splitting

H f [[s]] = B(( z))[[s]] = H+ ⊕ H− , where

H+ := e( F− f )/ z ( BF [[ z]][[s]]) ⊂ B(( z))[[s]],

H− := z−1 B[ z−1 ][[s]].

Let BF := H+ ∩ zH− . Equation (⋆) implies that z∂tα J ∈ B F , with z-leading term of constant coefficient z∂tα J ∈ [φα dn x] + H− .

In particular, { z∂tα J } form a C[[s]]-basis of BF .

Similarly, z2 ∂tα ∂tβ J = z2 ∂tα ∂tβ (e( F− f )/ zζ ) ∈ H+ , and z2 ∂tα ∂tβ J ∈ zH− by the above property of leading constant coefficient. Therefore z2 ∂tα ∂tβ J ∈ BF . This implies the γ γ = Aαβ (s(t)) such that existence of functions Aαβ z2 ∂tα ∂tβ J =

γ ( t) ∂t ∑ zAαβ

γ

γ

J

The homogeneous degree follows from the fact that J is weighted homogeneous. Let Aα denote the linear transformation on BF by

γ Aα : z∂β J → ∑ Aαβ z∂tγ J . γ

We can rewrite the above equation as (∂tα − z−1 Aα )∂tβ J = 0. We notice that ∂tα − z−1 Aα , ∂tβ − z−1 Aβ = 0 on BF , ∀α , β.

Therefore the last equations in the proposition hold.

Lemma 3.10. In terms of the coordinates tα , we have K f ( z∂tα J , z∂tβ J ) = zn gαβ . Here gαβ is the constant equal to the residue pairing η f (φα dn x, φβ dn x). Proof. We adopt the same notations as in the above proof. Since z∂tα J ∈ H+ , K f ( z∂tα J , z∂tβ J ) ∈ zn C[[ z]][[s]] by Lemma 3.5. Since also z∂tα J = [φα dn x] + H− ∈ zH− , we have K f ( z∂tα J , z∂tβ J ) ∈ zn gαβ + zn−1 C[ z−1 ][[s]]. The lemma follows from the above two properties. δ g . Then A Corollary 3.11. Let Aαβγ (t) := ∑δ Aαβ δγ αβγ is symmetric in α , β, γ .

26


Proof. By the previous lemma, ∂tγ K f ( z∂tα J , z∂tβ J ) = 0. The corollary now follows from Proposition 3.9. The properties in the propositions of this subsection can be summarized as follows. γ The triple (∂tα , Aαβ , gαβ ) defines a (formal) Frobenius manifold structure on a neighborhood S of the origin with {tα } being the flat coordinates, together with the potential function F0 (t) satisfying Aαβγ (t) = ∂tα ∂tβ ∂tγ F0 (t).

It is not hard to see that F0 (t) is homogeneous of degree 3 − cˆ f . As in the next proposition, the potential function F0 (t) can also be computed perturbatively. Let

F0 (t) = F0,(≤ N ) (t) + O(t N +1 ).

Proposition 3.12. The potential function F0 associated to the unique pair (ζ , J ) satisfies ∂tα F0 (t) =

Moreover,

(≤ N ) ( t) F0

∑ gαβ J−α2 (s(t)). β

is determined by ζ(≤ N −3) (s).

Proof. The first statement follows directly from Proposition 3.9. α N + 1 ). It is easy to Recall ζ (s) = ζ(≤ N ) (s) + O(s N +1 ). Let Jmα (s) = Jm, (≤ N ) ( s) + O ( s

(≤ N )

α see that F0 (t) only depends on J−α1,(≤ N −2)(s), J−α2,(≤ N −1)(s), and Jm, (≤ N ) ( s) only depends on ζ(≤ N +m) (s). Hence, the second statement follows.

Remark 3.13. By Remark 3.8, ζ is in fact an analytic primitive form. Therefore, both tα and F0 (t) are in fact analytic functions of s at the germ s = 0. 3.3. Computation for exceptional unimodular singularities. We start with the next proposition, which follows from a related statement for Brieskorn lattices [22]. An explicit calculation of the moduli space of good sections for general weighted homogenous polynomials is also given in [31, 46]. For exposition, we include a proof here. Proposition 3.14. If f is one of the 14 exceptional unimodular singularities, then there exists a unique good section {[φα dn x]}αµ =1 , where {φα } ⊂ C[x] are (arbitrary) weighted homogeneous representatives of a basis of the Jacobi algebra Jac ( f ). Proof. We give the details for E12 -singularity. The other 13 types are established similarly. The E12 -singularity is given by f = x3 + y7 with deg x = 31 , deg y = 22 charge cˆ f = 21 . We consider the weighted homogeneous monomials

1 7,

and central

{φ1 , · · · , φ12 } = {1, y, y2 , x, y3 , xy, y4 , xy2 , y5 , xy3 , xy4 , xy5 } ⊂ C[ x, y]


27

which represent a basis of Jac ( f ). The classical residue pairing gαβ between φα , φβ is equal to 1 if α + β = 13, and 0 otherwise. Since K f preserves the Q-grading, deg K f ([φα dxdy], [φβ dxdy]) = deg φα + deg φβ + 2 − cˆ f , which has to be an integer for a non-zero pairing. A simple degree counting implies that K f ([φα dxdy], [φβ dxdy]) = z2 gαβ and therefore {[φα dxdy]} constitutes a good basis.

Let {φα′ } be another set of weighted homogeneous polynomials such that {[φα′ dxdy]} gives a good basis. We can assume φα′ ≡ φα as elements in Jac( f ) and deg φα′ = deg φα . (0) Since [φα dxdy] forms a C[[ z]]-basis of H f , we can decompose

[φα′ dxdy] = ∑ Rαβ [φβ dxdy], β

Rαβ ∈ C[[ z]].

By the weighted homogeneity, Rαβ is homogeneous of degree deg φα − deg φβ , which is not an integer unless α = β. Thus [φα′ dxdy] = [φα dxdy], and hence the uniqueness. Let F0 be the potential function of the associated Frobenius manifold structure. Then F0 is an analytic function, as an immediate consequence of the above uniqueness together with the existence of the (analytic) primitive form. As will be shown in Lemma 4.2, we only need to compute F0,(≤4) to prove mirror symmetry. We illustrate the perturbative calculation for the E12 -singularity f = x3 + y7 . The full result is summarized in the appendix by similar calculations. We adopt the same notations as in the proof of Proposition 3.14. By Proposition 3.12, we only need ζ(≤1) to compute F0,(≤4), which is

ζ(≤1) = dxdy.

Using the equivalence relation in H f , we can expand e( F− f )/ z (ζ(≤1) ) =

3

( F − f )k −k z ζ(≤1) + O(s4 ) k! k= 0

∑

in terms of the good basis {φα }. We find the flat coordinates up to order 2 . t1 = s1 − s57s7 − s37s9 , . t4 = s4 − s87s9 − s77s10 − . t7 = s7 , . t10 = s10 − 4s97s12 ,

s 5 s 11 7

. t2 = s2 − . − s37s12 , t5 = s5 − . t8 = s8 − . t11 = s11 ,

s 27 2s 5 s 9 7 − 7 , 2s 29 7 , 3s 9 s 11 − 3s77s12 , 7

. t3 = s3 − . t6 = s6 − . t9 = s9 , . t12 = s12 .

3s 7 s 9 7 , 2s 9 s 10 7

−

2s 7 s 11 7

−

2s 5 s 12 7 ,

28


This allows us to solve the inverse function sα = sα (t) up to order 2. An straightforward but tedious computation of the z−2 -term shows that in terms of flat coordinates (3)

(4)

F0,(≤4) = F0 + F0 , (3)

where F0

is the third order term representing the algebraic structure of Jac( f ) (3)

∂tα ∂tβ ∂tγ F0

= η f ([φα φβφγ dxdy], [dxdy]).

(4)

The fourth order term F0 , which we call the 4-point function, is computed by (4)

−F0

=

1 1 1 1 1 1 1 t5 t6 t27 + t36 t8 + t25 t7 t8 + t3 t27 t8 + t4 t6 t28 + t25 t6 t9 + t3 t6 t7 t9 14 18 7 7 6 14 7 1 1 2 1 1 1 1 + t3 t5 t8 t9 + t2 t7 t8 t9 + t2 t6 t29 + t35 t10 + t4 t26 t10 + t3 t5 t7 t10 + t2 t27 t10 7 7 14 14 6 7 14 1 2 1 2 1 1 2 1 2 1 2 1 + t4 t8 t10 + t3 t9 t10 + t2 t5 t9 t10 + t3 t5 t11 + t4 t6 t11 + t3 t7 t11 + t2 t5 t7 t11 6 14 7 7 6 7 7 1 3 1 2 1 1 1 1 + t2 t3 t9 t11 + t4 t12 + t3 t5 t12 + t2 t25 t12 + t2 t3 t7 t12 + t22 t9 t12 . 7 18 14 14 7 14

In particular, for our later use, we can read off 1 ∂t4 ∂t4 ∂t4 ∂t12 F0 |t=0 = − , 3 4. M IRROR S YMMETRY

1 ∂t2 ∂t2 ∂t9 ∂t12 F0 |t=0 = − . 7

FOR EXCEPTIONAL UNIMODULAR SINGULARITIES

In this section, we use two reconstruction results to prove the mirror symmetry conjecture between the 14 exceptional unimodular singularities and their FJRW mirrors both at genus 0 and higher genera. 4.1. Mirror symmetry at genus zero. Throughout this subsection, we assume W T to be one of the 14 exceptional unimodular singularities in Table 1. We will consider the ring isomorphism Ψ : Jac (W T ) → ( HW , •) defined in Proposition 2.7. We will also denote the specified basis of Jac (W T ) therein by {φ1 , · · · , φµ } such that deg φ1 ≤ deg φ2 ≤ · · · ≤ deg φµ . As have mentioned, there is a formal Frobenius manifold structure on the FJRW FJRW ring ( HW , •) with a prepotential F0,W . We have also shown in the previous section that there is a Frobenius manifold structure with flat coordinates (t1 , · · · , tµ ) associated to SG from now on. (the primitive form) ζ therein, whose prepotential will be denoted as F0,W T T

,SG We introduce the primary correlators h· · · iW associated to the Frobenius manifold 0,k structure on B-side. The primary correlators, up to linear combinations, are given by

(4.1)

T

,SG hφi1 , · · · , φik iW = 0,k

SG ∂k F0,W T

∂ti1 · · · ∂tik

( 0) .


29

As from the specified ring isomorphism Ψ and (2.10), we have T

W ,SG . h1φi , 1φ j , 1φk iW 0,3 = hφi , φ j , φk i 0,3

As from Proposition 2.8 and the computation in section 3.3 and in the appendix, we have T

W ,SG T 2 h1xi , 1xi , 1 M T / x2 , 1φµ iW . 0,4 = −h xi , xi , Mi / xi , φµ i 0,4 i

i

To deal with the sign, we will do the following modifications, as in [13, section 6.5]. We √ π − 1r . Let F r ˜ SG denote the potential function of the Frobesimply denote (−1) := e 0 ˆ − c nius manifold structure ζ˜ := (−1) W T ζ . Set φ˜ j := (−1)− deg φ j φ j and define a map ˜ : Jac(W T ) → HW by Ψ ˜ (φ˜ j ) := Ψ(φ j ). Let ˜t denote the flat coordinate of F˜ SG , namely Ψ 0 t˜ j = (−1)1−deg t j t j .

(4.2)

( 4) ,SG ( 4) ,SG ( 3) ,SG ( 3) ,SG As a consequence, we have F˜ 0,W T = F0,W T and F˜ 0,W T = −F0,W T . Denote 1˜ φ˜ j := ˜ (φ˜ j ). Then Ψ ˜ defines a pairing-preserving ring isomorphism, which is read off from the Ψ T

˜

˜ ˜ ˜ W ,ζ ,SG , Moreover, identities h1˜ φ˜ i , 1˜ φ˜ j , 1˜ φ˜ k iW 0,3 = hφi , φ j , φk i 0,3 (4.3)

T ^ T 2 ˜ W ,ζ˜ ,SG . ei , xei , M , 1˜ φ˜ µ iW h1˜ xei , 1˜ xei , 1˜ ^ 0,4 = h x i / xi , φµ i 0,4 2 T

Mi / x i

From now on, we will simplify the notations by ignoring the symbolãnd the superscript ζ˜ . In addition, we will simply denote both HW and Jac (W T ) as H, and simply denote the correlators on both sides as hφi1 , · · · , φik i0,k (or hφi1 , · · · , φik i), whenever there is no risk of confusion. We have the following ”Selection rule” for primary correlators. Lemma 4.1. A primary correlator hφi1 , · · · , φik i0,k on either A-side or B-side is nonzero only if k

(4.4)

∑ deg φi

j= 1

j

= cˆW T − 3 + k.

Proof. The A-side case follows from formula (2.8) and cˆW = cˆW T . The primary correlator SG ( 0), where deg φ = 1 − deg t . Then the statement on B-side is given by ∂ti1 · · · ∂tik F0,W ij ij T SG ( 0) is weighted homogenous of degree 3 − cˆ follows, by noticing that F0,W T WT . A homogeneous α ∈ H is called a primitive class with respect to the specified basis {φ j }, if it cannot be written as α = α1 • α2 for 0 < deg αi < deg α . A primary correlator hφi1 , · · · , φik i0,k is called basic if at least k − 2 insertions φi j are primitive classes. Now Theorem 1.3 is a direct consequence of the equalities (4.3) and the following statement: Lemma 4.2 (Reconstruction Lemma). If W T is one of the 14 exceptional singularities, then all the following hold. (1) The prepotential F0 is uniquely determined from basic correlators h. . .i0,k with k ≤ 5.

30


(2) All basic correlators hφi1 , · · · , φi5 i0,5 vanish. (3) All the 4-point basic correlators are uniquely determined from the formula (1.1). Proof of (1): The potential function F0 satisfies the WDVV equation (2.12) (hence the formula (2.13)). We can assume that h· · ·i0,k is not of type h1, · · ·i0,k , k ≥ 4, (otherwise it vanishes according to string equation, or the invariance of the primitive form along the φ1 -direction where we notice φ1 = 1). Consider a correlator h· · · , α a , αb • αc , αd i0,k , with last three insertions non-primitive. By formula (2.13), such correlator is the sum of Sk together with three terms whose insertion replaces αb • αc with lower degree ones αb or αc at the same position. Repeating this will turn αb • αc into a primitive class, up to product of correlators with less number of insertions. By induction both on the degree of non-primitive classes and k, we can reduce any correlator to a linear combination of basic correlators. Now we assume that hφi1 , · · · , φik i0,k is a nonzero basic correlator. Then we can write φi1 • · · · • φik = xa yb zc . It follows from the degree constriant (4.4) that k

(4.5)

cˆW T − 3 + k =

∑ deg φi

j

= aq x + bq y + cq z ,

j= 1

Let us denote by P the maximal numbers among the degree of a generator x, y and z (or x and y if W T = W T ( x, y) is in two variables x, y only). By direct calculations, we conclude k≤

cˆW T + 1 + 2 < 6. 1−P

Proof of (2): For W T = x p + yq , x, y are generators for the ring structure H. The multiplications for all the insertions will be in a form of xa • yb . By the degree constraint, a nonzero basic correlator hφi1 , · · · , φik i0,k satisfies (4.6)

aq + bp = (k − 1) pq − 2p − 2q,

On the other hand, we assume the first k − 2 insertions to be primitive classes, so that they are either x or y. The top degree class φµ = x p−2 • yq−2 is of degree 2 − 2p − 2q . Therefore we have the following inequalities required for non-vanishing correlator: (4.7) a ≤ k − 2 + 2( p − 2),

b ≤ k − 2 + 2( q − 2) ,

a + b ≤ k − 2 + 2( p − 2 + q − 2) .

It is easy to see that there is no ( a, b) satisfying both (4.6) and (4.7) if k = 5. Hence hφi1 , · · · , φi5 i0,5 = 0. The arguments for the remaining W T on B-side and all the W on A-side are all similar and elementary, details of which are left to the readers. Proof of (3): Let us start with W T = x p + yq , where we notice that p, q are coprime. The degree constraint (4.6) with k = 4 implies that ( a, b) = (2p − 2, q − 2) or ( p − 2, 2q − 2).


31

Thus the possibly nonzero basic correlators are h x, x, x p−2 yi , x p−2 yq−2−i i0,4 , i = 0, . . . , q − 2. On the other hands, if formula (1.1) holds, then by WDVV equation (2.13), we have

h x, x, x p−2 • yi , x p−2 yq−2−i i

= −h x, x p−2 , yi , x • x p−2 yq−2−i i + h x, x • x p−2 , yi , x p−2 yq−2−i i + h x, x, x p−2 , yi • x p−2 yq−2−i i 1 = h x, x, x p−2 , yi • x p−2 yq−2−i i = . p For 2-Chain W T = x p y + yq , the degree constraint (4.5) tells us a

q−1 1 + b = cˆW − 3 + k. pq q

For k = 4, this implies that ( a, b) = (2p − 2, q) or ( p − 2, 2q − 1). The basic correlators are h x, x, x p−2 y1+i , x p−2 yq−1−i i with 0 ≤ i ≤ q − 1, h y, y, xi yq−2 , x p−2−i yq−1 i with 0 ≤ i ≤ p − 2 and h x, y, xi yq−1 , x p−3−i yq−1 i with 0 ≤ i ≤ p − 3. The first two types are uniquely determined from the correlators which are listed in Proposition 1.2. For example, if 0 < i < q − 1, since p x p−1 y = ∂ x W T = 0 in Jac (W T ), we have

h x, x, x p−2 y • yi , x p−2 yq−1−i i = h x, x, x p−2 y, x p−2 yq−1−i • yi i.

The last type is determined by

h x, y, xi yq−1 , x p−3−i yq−1 i 1 = − h x, y, xi yq−1 , x p−2−i • x p−1 i q 1 = − h x, y, x p−1 , x p−2 yq−1 i + h x, y • x p−1 , x p−2−i , xi yq−1 i q 1 1 1 = − h x, y, x • x p−2 , x p−2 yq−1 i = − h x, x, x p−2 • y, x p−2 yq−1 i = − q q pq Here we use the relation x p + qyq−1 = ∂ y W T = 0 in Jac (W T ) in the first equality. For 2-Loop W T = x3 y + xy4 , the degree constraint (4.6) with k = 4 implies that ( a, b) = (5, 4) or (3, 7). If the formula (1.1) holds, namely if

h x, x, xy, x2 y3 i =

3 , 11

h y, y, xy2 , x2 y3 i =

2 , 11

2 2 3 , h x, y, y2 , x2 y3 i = 11 and h x, x, x2 y2 , xy2 i = 11 then we conclude h x, y, x2 , x2 y3 i = 11 3 2 from a single WDVV equation for each correlator. For the rest, we conclude h x, x, xy , x yi = 1 1 3 3 11 and h x, y, xy , xy i = − 11 by solving the following linear equations which come from the WDVV equation,   −3h x, x, x2 y, xy3 i + h x, x • xy3 , y3 , yi = h x, x • y3 , y, xy3 i,  −4h x, y, xy3 , xy3 i = h x, y, x2 , x • xy3 i + h x, y • x2 , x, xy3 i.

32


Here the coefficient −3 (resp. −4) comes from 3x2 y + y4 = 0 (resp. x3 + 4xy3 = 0) in 1 1 and h y, y, x2 y2 , xy3 i = 11 . Jac (W T ). Similarly, we conclude h x, y, x2 y2 , x2 yi = − 11

For W T = x2 y + y4 + z3 ∈ Q10 , the number of 4-point basic correlators is 10. Three of them are the initial correlators in (1.1), h x, x, y, y3 zi, h y, y, y2 , y3 zi, h z, z, z, y3 zi, the rest are h y, y, y2 z, y3 i, h y, z, y3 , y3 i, h z, z, yz, y2 zi, h x, x, yz, y3 i, h x, x, y2 , y2 zi, h x, y, xz, y3 i, and h z, z, xz, xzi. We have 7 WDVV equations to reconstruct them from the initial correlators,  4h y, y, y2 z, y3 i = h x, x, y, y3 zi, h y, z, y3 , y3 i = h y, y, y2 z, y3 i − h y, y, y2 , y3 zi,    h z, z, yz, y2 zi = h z, z, z, y3 zi, h x, x, yz, y3 i = h x, x, y, y3 zi, h x, x, y2 , y2 zi = h x, x, y, y3 zi,    h x, y, xz, y3 i = h x, x, y, y3 zi, h z, z, xz, xzi = −4h z, z, z, y3 zi.

For other singularities of 3-variables, all the basic 4-point correlators are uniquely determined from the initial correlators in formula (1.1), by the same technique. However, the discussion is more tedius. For example, there are 21 of 4-point basic correlators for type S12 singularity W T = x2 y + y2 z + z3 x. We can write down 18 WDVV equations carefully to determine all the 21 basic correlators from 3 correlators in the formula (1.1). The details are skipped here.

4.2. Mirror symmetry at higher genus. In Section 2, we already constructed the total ancestor FJRW potential AWFJRW for a pair (W, GW ). Now we give the B-model total ancestor Saito-Givental potential AWSGT . Let S be the universal unfolding of the isolated singularity W T . For a semisimple point s ∈ S, Givental [18] constructed the following formula containing higher genus information of the Landau-Ginzburg B-model of f , (see [11, 17, 18] for more details) ! 1 µ i SG cs R cs (T ). log ∆ (s) Ψ A f (s) := exp − 48 i∑ =1 Here T is the product of µ -copies of Witten-Kontsevich τ -function. ∆i (s), Ψs and Rs are data coming from the Frobenius manifold. The operators b· are the so-called quantization operators. We call A fSG (s) the Saito-Givental potential for f at point s. Teleman

[48] proved that A fSG (s) is uniquely determined by the genus 0 data on the Frobenius manifold. By definition, the coefficients in each genus-g generating function of A fSG (s) is just meromorphic near the non-semisimple point s = 0. Recently, using Eynard-Orantin recursion, Milanov [33] proved AWSGT (t) extends holomorphically at t = 0. We denote such an extension by AWSGT and the Corollary 1.4 follows from Theorem 1.3 and Teleman’s theorem.


33

4.3. Alternative representatives and the other direction. Although the theory of primitive forms depends only on the stable equivalence class of the singularity, the FJRW theory definitely depends on the choice of the polynomial together with the group. For the exceptional unimodular singularities, in the following we list all the additional invertible weighted homogeneous polynomial representatives without quadratic terms x2k in additional variables xk as follows (up to permutation symmetry among variables):  3 8  W12 : x2 y + y2 + z5 , W13 : x4 y + y4 ;   E14 : x + y , (4.8) Q12 : x2 y + y5 + z3 , Z13 : x3 y + y6 , U12 : x2 y + y3 + z4 ;    U12 : x2 y + xy2 + z4 .

It is quite natural to investigate Conjecture 1.1 for all the weighted homogeneous polynomial representatives on the B-side. Theorem 4.3. Conjecture 1.1 is true if W T is given by any weighted homogenous polynomial representative of the exceptional unimodular singularities that W T is not x2 y + xy2 + z4 . That is, there exists a mirror map, such that AWFJRW = AWSGT .

Sketch of the proof. Thanks to Corollary 1.4, it remains to show the case when W T is given by (4.8). By Proposition 3.14, there is a unique good section. Let us specify a weighted homogeneous basis {φ1 , · · · , φµ } of Jac(W T ) as in Table 2 for each atomic type and take product of such bases for mixed types. Then we could obtain the four-point function by direct calculations (see the link in the appendix for precise output). An isomorphism Ψ : Jac(W T ) → HW is chosen similarly as in Section 2. We compute the corresponding four-point FJRW correlators as in Proposition 2.8 by the same proof therein. If W T is not x2 y + xy2 + z4 , then the four-point FJRW correlators turn out to be the same as the B-side four-point correlator up to a sign. These invariants completely determine the full data of the generating function at all genera on both sides, by exactly the same reconstruction technique as in the previous two subsections. Therefore, we conclude the statement. Remark 4.4. If W T = x2 y + xy2 + z4 , HW has broad ring generators x1 J 8 and y1 J 8 . Our method does not apply to compute

h x1 J 8 , x1 J 8 , y1 J 8 , 1 J 15 iW 0 ,

h y1 J 8 , y1 J 8 , x1 J 8 , 1 J 15 iW 0 ,

1φµ = 1 J 15 .

If W T = x2 y + y2 + z5 , we may need a further rescaling on Ψ( x) since we only know 2 1 = 2hΨ( y), Ψ( y), Ψ( y), Ψ( y), Ψ( yz3 )iW hΨ( x), Ψ( x), Ψ( y), Ψ( yz3 )iW 0 = . 0 4

34


The first equality is a consequence of the WDVV equation and the second equality is a consequence of the orbifold GRR calculation with codimension D = 2 (i.e., formula (2.15)). The other direction. Among all the representatives W on the A-side, there are in total three cases for which W T is no longer exceptional unimodular. The corresponding W T is given by x3 + xy6 , x2 + xy5 + z3 , or x2 + xy3 + z4 . Let us end this section by the following remark, which gives a positive answer to Conjecture 1.1 for those representatives. Remark 4.5. (1) For the remaining three cases, W T is no longer given by any one of the exceptional unimodular singularities. (2) A similar calculation as Proposition 3.14 shows that there exists a unique primitive form (up to a constant) for x2 + xy5 + z3 . However, for the other two cases x3 + xy6 and x2 + xy3 + z4 , there is a whole one-dimensional family of choices of primitive forms. (3) Let us specify a basis {φ1 , · · · , φµ } of Jac (W T ) following Table 2. It is easy to check that {[φ1 dn x], · · · , [φµ dn x]} form a good basis and specifies a choice of primitive form. A similar calculation shows that the B-side four-point function coincides with the A-side one (up to a sign as before), and they completely determine the full data of the generating functions at all genera by the same reconstruction technique again. A PPENDIX A. A.1. The vector space isomorphism. Here we list the vector space isomorphism Ψ : Jac (W T ) → ( HW ) for the remaining cases of W in Table 1.

(1) 3-Fermat type. W = W T = x3 + y3 + z4 ∈ U12 . We denote 1i, j,k := 1 ∈ Hγ for √ √ √ γ = exp( 2π 3−1 i ), exp( 2π 3−1 j , exp( 2π 4−1 k ) ∈ GW . The isomorphism Ψ is given by Ψ( xi−1 y j−1 zk−1 ) = 1i, j,k ,

1 ≤ i ≤ 3, 1 ≤ j ≤ 3, 1 ≤ k ≤ 4.

(2) Chain type. Let W = x3 + xy5 . The mirror W T is of type Z11 . Note GW ∼ = µ15 . HW

1J

1 J 13

1 J 11

1 J 10

1 J8

Jac (W T )

1

y

x

y2

xy

∓5y4 1 J 0 1 J 7 x2

y3

1 J5

1 J4

1 J2

1 J 14

xy2

y4

xy3

xy4

1 J2

Let W = x3 y + y5 . The mirror W T is of type E13 . Note GW ∼ = µ15 . HW

1J

1 J 13

1 J 12

1 J 11

1 J9

1 J8

Jac(W T )

1

y

y2

x

y3

xy

∓3y2 1 J 0 y4

1 J7

1 J6

1 J5

1 J4

xy2

x2

xy3

x2 y x2 y2

1 J 14 x2 y3


35

Let W = x2 y + y3 z + z3 . The mirror W T is of type Z13 . Note GW ∼ = µ18 . HW

1J

1 J 16

1 J 14

1 J 13

1 J 11

1 J 10

Jac (W T )

1

y

z

y2

yz

x

∓3y2 1 J 9 z2

1 J8

1 J7

1 J5

1 J4

1 J2

1 J 17

y2 z

xy

xz

xy2

xyz xy2 z

Let W = x2 y + y2 z + z4 . The mirror W T is of type W13 . Note GW ∼ = µ16 . HW

1J

1 J 14

1 J 13

1 J 11

1 J 10

1 J9

Jac (W T )

1

z

y

z2

yz

x

∓2y1 J 8 z3

1 J7

1 J6

1 J5

1 J3

1 J2

1 J 15

yz2

xz

xy

xz2

xyz xyz2

(3) Loop type. There is one 2-Loop of type Z12 : W = W T = x3 y + xy4 with GW ∼ = µ11 . HW

1J

Jac(W T )

1 J8

1 J6

1 J4

x

y2

y

1

1 J2

x2 1 J 0

y3 1 J 0

1 J9

1 J7

1 J5

1 J3

1 J 10

xy

x2

y3

xy2

x2 y

xy3

x2 y2

x2 y3

There is one 3-Loop with W T of type S12 : W = x2 z + xy2 + yz3 with GW ∼ = µ13 . HW

1 J 1 J 11

Jac (W T )

1 J 10

1 J9

1 J8

1 J7

1 J6

1 J5

1 J4

1 J3

x

y

z2

xz

yz

xy

xz2

yz2 xyz xyz2

z

1

1 J2

1 J 12

(4) Mixed type. Let W = x2 + xy4 + z3 . The mirror W T is of type Q10 . Note GW ∼ = µ24 . HW

1J

Jac (W T )

1 J 19

1 J 17

y

1

∓4y3 1 J 16 1 J 13 1 J 11 ∓4y3 1 J 8 1 J 7

z

y2

x

yz

xz

y3

1 J5

1 J 23

y2 z

y3 z

Let W = x2 y + y4 + z3 . The mirror W T is of type E14 . Note GW ∼ = µ24 . HW

1J

1 J 22

1 J 19

1 J 17

Jac(W T )

1

z

z2

x

∓2y1 J 16 1 J 14 1 J 13 1 J 11 1 J 10 ∓2y1 J 8 y2

xz

x

yz2

yz

y2 z

1 J7

1 J5

1 J2

1 J 23

yz2

xy

xyz xyz2

36


A.2. Four-point functions for exceptional unimodular singularities. In the following, (4) we provide the four-point functions F0 (t) of the Frobenius manifold structure associated to the primitive form ζ for all the remaining 13 cases in Table 1. We mark the terms that give the B-side four-point invariants corresponding to (1.1) by using boxes. We also (4) (4) provide the expression of ζ up to order 3. We remind our readers of F˜ 0 (˜t) = −F0 (t) as discussed in section 4.1. We obtain the list with the help of a computer. The codes are written in mathematica 8, available at http://member.ipmu.jp/changzheng.li/index.htm • Type E13 : f = x3 + xy5 . {φi }i = 1, y, y2 , x, y3 , xy, y4 , y2 x, x2 , y3 x, yx2 , x2 y2 , y3 x2 .

ζ = 1−

(4)

−F0

=− + − + + + + +

4 1 2 s12 s13 − xs + O(s4 ). 75 25 13

3 3 1 1 1 3 1 2 t6 t37 − t5 t27 t8 + t5 t6 t28 + t3 t38 + t36 t9 + t5 t6 t7 t9 + t4 t27 t9 + t25 t8 t9 10 5 10 15 90 5 5 5 2 1 1 1 1 1 2 2 2 2 t4 t6 t8 t9 + t3 t7 t8 t9 − t4 t5 t9 − t3 t6 t9 − t2 t8 t9 + t5 t6 t10 15 5 10 15 30 10 3 1 1 1 1 3 2 t t7 t10 − t3 t27 t10 + t3 t6 t8 t10 + t2 t28 t10 + t24 t9 t10 + t3 t5 t9 t10 10 5 10 5 10 30 5 3 1 3 1 2 1 t2 t7 t9 t10 + t2 t6 t210 + t25 t6 t11 + t4 t26 t11 + t4 t5 t7 t11 + t3 t6 t7 t11 5 10 10 15 5 5 1 2 2 1 2 1 1 t4 t8 t11 + t3 t5 t8 t11 + t2 t7 t8 t11 − t3 t4 t9 t11 − t2 t6 t9 t11 + t23 t10 t11 15 5 5 15 15 10 1 1 1 2 2 1 t2 t5 t10 t11 − t2 t4 t211 + t4 t25 t12 + t24 t6 t12 + t3 t5 t6 t12 + t3 t4 t7 t12 5 30 5 10 5 5 1 2 1 1 1 1 2 3 t4 t13 t2 t6 t7 t12 + t3 t8 t12 + t2 t5 t8 t12 − t2 t4 t9 t12 + t2 t3 t10 t12 + 45 5 5 5 15 5 1 1 1 1 1 1 2 t2 t10 t13 t3 t4 t5 t13 + t23 t6 t13 + t2 t5 t6 t13 + t2 t4 t7 t13 + t2 t3 t8 t13 + 10 5 10 5 5 5

• Type E14 : f = x2 + xy4 + z3 . {φi }i = 1, y, x, y2 , xy, y3 , xy2 , z, yz, xz, y2 z, xyz, y3 z, xy2 z .

ζ = 1+

(4)

−F0

1 2 1 1 1 2 3 s12 s14 + s10 s214 + ys12 s214 + y s14 + O(s4 ). 64 64 48 192

1 1 1 1 1 1 1 2 t3 t5 t9 + t25 t6 t9 + t4 t5 t7 t9 + t3 t6 t7 t9 + t2 t27 t9 − t23 t4 t10 − t2 t3 t5 t10 16 8 4 4 8 8 8 3 2 1 2 1 1 2 1 3 1 1 + t4 t5 t6 t10 + t3 t6 t10 + t4 t7 t10 + t2 t6 t7 t10 + t8 t9 t10 − t3 t11 + t4 t25 t11 2 8 8 4 6 24 4 1 1 2 1 2 2 1 1 1 2 + t3 t5 t6 t11 + t3 t4 t7 t11 + t2 t5 t7 t11 − t6 t7 t11 + t8 t10 t11 − t9 t11 − 1/9t8 t311 2 4 4 4 6 6 1 2 1 1 1 1 3 1 1 2 − t2 t3 t12 + t4 t5 t12 + t3 t4 t6 t12 + t2 t5 t6 t12 − t6 t12 + t2 t4 t7 t12 + t28 t9 t12 16 4 2 4 6 4 6

=−


37

1 3 1 1 1 1 1 1 3 t8 t14 + t3 t4 t5 t13 + t2 t25 t13 + t23 t6 t13 − t5 t26 t13 + t2 t3 t7 t13 − t39 t13 − t28 t213 + 18 2 8 8 2 4 9 6 2 1 1 1 1 1 − t8 t9 t11 t13 + t3 t24 t14 + t2 t4 t5 t14 + t2 t3 t6 t14 − t4 t26 t14 + 18 t22 t7 t14 − t4 t6 t7 t13 3 8 4 4 4 2 • Type Z11 : f = x3 y + y5 . {φi }i = 1, y, x, y2 , xy, x2 , y3 , xy2 , y4 , xy3 , xy4 .

ζ = 1+

(4)

−F0

17 2 3 s10 s211 + ys + O(s4 ). 675 81 11

5 1 1 1 1 2 1 t5 t3 + t4 t2 t7 + t4 t5 t27 − t3 t37 + t35 t8 + t24 t7 t8 + t3 t6 t7 t8 18 6 3 6 15 90 18 15 3 1 1 1 1 1 1 1 + t2 t27 t8 + t3 t5 t28 + t24 t5 t9 + t3 t5 t6 t9 + t2 t26 t9 − t3 t4 t7 t9 + t2 t5 t7 t9 10 6 30 3 3 15 15 1 1 1 1 1 1 + t2 t4 t8 t9 − t2 t3 t29 + t34 t10 + t3 t25 t10 + t3 t4 t6 t10 + t2 t4 t7 t10 15 30 18 6 3 5 1 2 1 1 2 1 1 2 t2 t7 t11 + t3 t8 t10 + t2 t9 t10 + t2 t24 t11 + 16 t23 t5 t11 + t2 t3 t6 t11 + 15 6 30 15 3

=−

• Type Z12 : f = x3 y + xy4 . {φi }i = 1, y, x, y2 , xy, x2 , y3 , xy2 , x2 y, xy3 , x2 y2 , x2 y3 .

ζ = 1− (4)

−F0

6 29 9 5 s11 s12 − ys212 + s10 s212 + xs3 + O(s4 ). 121 121 1331 1331 12

5 7 7 3 4 10 3 t5 t6 + t5 t26 t7 + t5 t6 t27 − t5 t37 + t4 t26 t8 + t4 t6 t7 t8 33 22 22 22 11 11 1 2 1 1 1 6 − t4 t27 t8 + t4 t5 t28 + t3 t6 t28 − t3 t7 t28 + t2 t38 + t35 t9 11 11 11 22 22 66 2 2 6 1 4 2 2 − t4 t5 t6 t9 − t3 t6 t9 + t4 t5 t7 t9 + t3 t6 t7 t9 + t3 t27 t9 + t24 t8 t9 11 11 11 11 11 11 1 3 1 1 1 1 + t3 t5 t8 t9 − t2 t6 t8 t9 + t2 t7 t8 t9 − t3 t4 t29 − t2 t5 t29 + t4 t25 t10 11 11 11 11 22 22 1 4 3 1 7 1 + t24 t6 t10 + t3 t5 t6 t10 + t2 t26 t10 − t24 t7 t10 − t3 t5 t7 t10 + t2 t6 t7 t10 22 11 22 22 11 11 1 1 1 1 1 3 − t2 t27 t10 − t3 t4 t8 t10 + t2 t5 t8 t10 + t23 t9 t10 + t2 t4 t9 t10 − t2 t3 t210 22 11 11 22 11 22 1 2 1 5 3 5 2 + t4 t5 t11 + t3 t25 t11 + t3 t4 t6 t11 − t2 t5 t6 t11 + t3 t4 t7 t11 + t2 t5 t7 t11 22 11 11 11 11 11 1 2 3 1 1 1 2 + t3 t8 t11 + t2 t4 t8 t11 − t2 t3 t9 t11 + t2 t10 t11 + t3 t24 t12 11 11 11 22 11 2 3 2 3 2 1 2 + 22 t3 t5 t12 + t2 t4 t5 t12 + t2 t3 t6 t12 + t2 t3 t7 t12 + 11 t2 t8 t12 11 11 11

=−

38


• Type Z13 : f = x2 + xy3 + yz3 . {φi }i = 1, y, z, y2 , yz, x, z2 , y2 z, xy, xz, xy2 , xyz, xy2 z .

ζ = 1+

(4)

−F0

=− + + + + +

7 2 7 5 2 2 3 s s13 + s10 s213 + ys12 s213 + y s13 + O(s4 ). 486 12 486 243 729

1 2 2 1 3 5 3 1 1 1 1 4 t t − t5 t − t t8 − t25 t28 − t3 t38 − t5 t26 t9 + t4 t27 t9 + t4 t6 t8 t9 12 6 7 6 7 108 6 6 9 18 3 9 1 1 1 1 1 1 1 t3 t7 t8 t9 + t4 t5 t29 + t3 t6 t29 + t2 t8 t29 + t35 t10 − t4 t26 t10 − t3 t6 t7 t10 3 9 36 6 18 6 6 1 1 1 1 2 1 t3 t5 t8 t10 − t2 t6 t9 t10 + t4 t5 t6 t11 + t3 t26 t11 + t3 t5 t7 t11 + t2 t4 t6 t13 3 6 9 36 3 9 1 1 1 2 1 1 1 t2 t4 t9 t12 − t4 t8 t11 + t2 t6 t8 t11 − t3 t4 t9 t11 + t2 t5 t9 t11 + t2 t4 t10 t11 3 9 9 9 9 9 1 2 2 1 2 5 2 1 1 1 1 2 t2 t7 t11 − t3 t10 + t3 t5 t12 + t4 t6 t12 − t2 t26 t12 + t3 t4 t7 t12 + t23 t8 t12 3 24 6 18 12 3 6 2 3 2 1 2 1 1 t2 t11 t12 − t4 t13 + 16 t23 t5 t13 + t2 t3 t7 t13 + 19 t22 t9 t13 − t2 t3 t211 + t24 t9 t10 18 27 3 18 9

• Type W12 : f = x4 + y5 . {φi }i = 1, y, x, y2 , xy, x2 , y3 , xy2 , x2 y, xy3 , x2 y2 , x2 y3 .

ζ = 1−

(4)

−F0

=

1 1 2 s11 s12 − ys + O(s4 ). 20 20 12

1 1 1 1 1 2 2 1 2 t5 t7 + t5 t6 t8 + t4 t5 t7 t8 + t24 t28 + t2 t7 t28 + t25 t6 t9 20 8 5 10 10 8 1 2 1 1 1 2 1 1 2 2 + t4 t7 t9 + t2 t7 t9 + t3 t6 t8 t9 + t3 t5 t9 + t4 t5 t10 + t3 t26 t10 10 10 4 8 10 8 1 1 1 1 1 + t2 t5 t7 t10 + t2 t4 t8 t10 + t22 t210 + t34 t11 + t3 t5 t6 t11 5 5 20 15 4 1 1 1 1 2 t2 t7 t12 + t2 t4 t7 t11 + t23 t9 t11 + t2 t24 t12 + 18 t23 t6 t12 + 10 5 8 10

• Type W13 : f = x2 + xy2 + yz4 . {φi }i = 1, z, y, z2 , yz, x, z3 , yz2 , xz, xy, xz2 , xyz, xyz2 .

ζ = 1−

(4)

−F0

ys2 3s10 s213 11zs12 s213 3z2 s313 15s212 s13 5 s11 s13 + − 13 + + + + O ( s4 ) . 64 1024 128 256 512 512

3 2 2 1 3 3 1 1 1 1 1 t6 t7 − t5 t7 − t36 t8 − t4 t27 t8 − t25 t28 − t5 t26 t9 − t4 t6 t7 t9 + t4 t5 t8 t9 32 6 48 4 8 32 4 4 1 2 2 1 1 1 2 1 1 1 2 2 2 2 + t2 t8 t9 − t4 t9 − t3 t6 t9 − t2 t7 t9 + t5 t6 t10 + t4 t6 t10 + t4 t5 t7 t10 8 16 8 16 16 16 2

=−


39

1 1 1 1 1 3 + t3 t27 t10 + t24 t8 t10 + t3 t6 t8 t10 + t2 t7 t8 t10 + t3 t5 t9 t10 + t2 t6 t9 t10 8 8 8 4 8 16 1 1 1 2 1 2 1 2 1 2 2 − t3 t4 t10 − t2 t5 t10 + t4 t5 t11 − t4 t6 t11 − t3 t6 t11 − t2 t6 t7 t11 8 16 8 16 8 8 1 1 1 1 1 1 + t2 t5 t8 t11 − t2 t4 t9 t11 + t23 t10 t11 − t22 t211 + t24 t5 t12 + t3 t5 t6 t12 4 8 16 32 4 4 1 1 1 1 1 1 + t2 t26 t12 + t3 t4 t7 t12 + t2 t5 t7 t12 + t2 t4 t8 t12 + t23 t9 t12 − t2 t3 t10 t12 32 2 4 4 8 8 1 1 2 1 3 2 t3 t6 t13 + t2 t3 t7 t13 + 18 t22 t8 t13 + t3 t4 t13 + t2 t4 t5 t13 + 16 8 4 4 • Type Q10 : f = x2 y + y4 + z3 . {φi }i = 1, y, z, x, y2 , yz, xz, y3 , y2 z, y3 z .

ζ = 1+

(4)

−F0

=

11 3 3 s9 s210 + ys + O(s4 ). 128 384 10

1 3 1 1 1 1 1 1 t t6 + t3 t36 + t4 t25 t7 − t23 t27 + t24 t6 t8 + t2 t5 t6 t8 + t2 t4 t7 t8 24 5 18 4 3 4 8 2 1 2 1 2 1 2 1 2 1 3 1 2 t3 t10 + 14 t2 t24 t10 + 16 t2 t5 t10 + t4 t5 t9 + t2 t5 t9 + t3 t6 t9 + t2 t8 t9 + 18 4 8 6 16

• Type Q11 : f = x2 y + y3 z + z3 . {φi }i = 1, y, z, x, y2 , yz, z2 , xz, y2 z, yz2 , y2 z2 .

ζ = 1−

(4)

−F0

=

1 2 13 25 5 s10 s11 − ys11 + s9 s211 + zs3 + O(s4 ). 108 24 648 1944 11

1 1 1 1 1 1 1 t5 t3 + t2 t6 t7 + t3 t26 t7 − t24 t27 − t3 t5 t27 − t2 t6 t27 + t4 t5 t6 t8 36 6 4 5 36 24 9 18 2 1 1 1 1 1 1 − t3 t4 t7 t8 − t23 t28 − t35 t9 + t24 t6 t9 + t2 t26 t9 + t23 t7 t9 6 4 12 4 12 36 1 1 1 1 1 1 + t2 t5 t7 t9 + t2 t4 t8 t9 + t24 t5 t10 + t3 t25 t10 + t23 t6 t10 + t2 t5 t6 t10 6 2 4 4 12 3 1 2 1 1 5 3 1 2 − t2 t3 t7 t10 + t2 t9 t10 + 108 t3 t11 + 14 t2 t24 t11 + t2 t3 t5 t11 + 12 t2 t6 t11 9 12 6

• Type Q12 : f = x2 y + xy3 + z3 . {φi }i = 1, x, y, xy, y2 , xy2 , z, xz, yz, xyz, y2 z, xy2 z .

ζ = 1+

(4)

−F0

3 2 t t4 t8 − 10 2 1 − t2 t24 t9 + 10

=−

1 2 1 1 1 s10 s12 + s8 s212 + ys10 s212 + s11 s212 + O(s4 ). 75 75 50 25

1 1 2 3 1 1 2 t3 t4 t12 t3 t24 t8 + t2 t4 t5 t8 + t4 t25 t8 + t2 t3 t6 t8 + t3 t5 t6 t8 + 10 10 5 10 5 5 1 2 1 1 1 1 1 1 t t5 t9 + t22 t6 t9 + t3 t4 t6 t9 + t2 t5 t6 t9 − t25 t6 t9 + t7 t8 t29 − t49 5 4 5 5 5 5 6 36

40


1 1 2 3 1 1 1 + t2 t3 t5 t12 − t2 t3 t4 t10 + t22 t5 t10 + t3 t4 t5 t10 + t2 t25 t10 − t35 t10 + t23 t6 t10 5 5 10 5 10 5 10 1 3 1 2 1 2 3 1 3 2 2 − t2 t10 + t2 t4 t11 + t3 t4 t11 + t2 t4 t5 t11 − t4 t5 t11 + t2 t3 t6 t11 − t3 t5 t6 t11 10 10 5 5 5 5 5 1 1 1 1 1 1 1 3 t7 t12 + t27 t9 t10 − t27 t28 + t27 t8 t11 − t7 t29 t11 − t27 t211 + 15 t22 t3 t12 − t3 t25 t12 + 18 6 3 6 5 6 4 • Type S11 : f = x2 y + y2 z + z4 . {φi }i = 1, z, x, y, z2 , xz, yz, z3 , xz2 , yz2 , yz3 .

ζ = 1−

(4)

−F0

3 9 5 7 s10 s11 − zs211 + s8 s211 + ys3 + O(s4 ). 64 128 512 1024 11

1 1 1 1 1 5 2 2 t5 t6 + t35 t7 + t4 t26 t7 + t3 t6 t27 − t4 t25 t8 − t3 t5 t6 t8 32 48 4 4 16 8 1 1 1 1 1 1 − t2 t26 t8 + t24 t7 t8 + t2 t5 t7 t8 − t23 t28 − t2 t4 t28 − t3 t25 t9 16 8 16 32 16 16 1 1 1 1 2 2 1 2 1 2 − t2 t5 t6 t9 + t3 t4 t7 t9 − t2 t3 t8 t9 − t2 t9 + t4 t5 t10 + t2 t5 t10 2 2 8 8 8 8 1 2 1 2 1 1 2 1 3 2 t2 t5 t11 + t3 t4 t6 t10 + t3 t7 t10 + t2 t8 t10 + 4 t3 t4 t11 + 8 t2 t24 t11 + 32 2 4 32

=−

• Type S12 : f = x2 y + y2 z + xz3 . {φi }i = 1, z, x, y, z2 , xz, yz, xy, xz2 , yz2 , xyz, xyz2 .

ζ = 1− (4)

−F0

9z2 s312 30s211 s12 2xs212 20s8 s212 93zs11 s212 12s10 s12 + − + + + + O ( s4 ) . 169 2197 169 2197 4394 2197

5 4 1 1 1 1 5 t + t5 t2 t7 − t25 t27 − t4 t6 t27 − t3 t37 + t25 t6 t8 156 6 13 6 13 13 26 26 2 2 3 1 1 1 + t4 t26 t8 + t4 t5 t7 t8 + t3 t6 t7 t8 + t2 t27 t8 − t24 t28 − t3 t5 t28 26 13 13 26 52 13 1 2 1 1 1 1 − t2 t6 t28 − t35 t9 − t4 t5 t6 t9 − t3 t26 t9 + t24 t7 t9 + t3 t5 t7 t9 13 39 13 13 13 13 1 1 1 2 2 1 1 1 + t2 t6 t7 t9 + t3 t4 t8 t9 + t2 t5 t8 t9 − t3 t9 − t2 t4 t29 − t4 t25 t10 13 13 13 26 13 13 1 2 1 2 3 2 1 + t4 t6 t10 + t3 t5 t6 t10 + t2 t26 t10 − t3 t4 t7 t10 − t2 t5 t7 t10 + t23 t8 t10 26 13 13 13 13 26 5 2 7 3 1 1 2 2 1 + t2 t4 t8 t10 + t2 t3 t9 t10 − t2 t10 + t4 t5 t11 + t3 t25 t11 + t3 t4 t6 t11 13 13 26 26 26 13 1 3 2 2 1 2 4 5 2 t3 t4 t12 + t2 t5 t6 t11 + t3 t7 t11 + t2 t4 t7 t11 − t2 t3 t8 t11 + t2 t9 t11 + 26 13 26 13 13 26 3 2 3 2 + 13 t2 t24 t12 + t2 t3 t5 t12 + 26 t2 t6 t12 13

=−


41

• Type U12 : f = x3 + y3 + z4 . {φi }i = 1, z, x, y, z2 , xz, yz, xy, xz2 , yz2 , xyz, xyz2 .

ζ = 1+ (4)

−F0

1 1 1 2 3 1 2 s11 s12 + s8 s212 + zs11 s212 + z s + O ( s4 ) . 72 72 36 72 12

1 1 1 1 1 1 1 = t25 t6 t7 + t3 t26 t8 + t4 t27 t8 + t2 t5 t7 t9 + t23 t8 t9 + t2 t5 t6 t10 + t24 t8 t10 8 6 6 4 6 4 6 1 2 1 2 1 2 1 2 1 3 1 3 t3 t12 + 18 t4 t12 + 18 t22 t5 t12 + t2 t9 t10 + t2 t5 t11 + t3 t6 t11 + t4 t7 t11 + 18 8 8 6 6 R EFERENCES

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