Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras

Sangwook Lee

doi:10.1515/math-2025-0208

Article Open Access

Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras

Sangwook Lee

Published/Copyright: November 27, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

Given a mirror pair of a symplectic manifold X and a Landau-Ginzburg potential W, we are interested in whether the quantum cohomology of X and the Jacobian algebra of W are isomorphic. Since those can be equipped with Frobenius algebra structures, we might ask whether they are isomorphic as Frobenius algebras. We show that the Kodaira-Spencer map gives a Frobenius algebra isomorphism for elliptic orbispheres, under the Floer theoretic modification of the residue pairing.

Keywords: Kodaira-Spencer map; Frobenius algebras; orbifold Jacobian algebras

MSC 2020: 53D37; 55N32

1 Introduction

Closed string mirror symmetry predicts that quantum cohomology of a symplectic manifold and Jacobian ring of the mirror superpotential are isomorphic. There have been results on this problem firstly for toric manifolds: see [1], [2], [3] etc.

Fukaya-Oh-Ohta-Ono also gave a construction of the ring isomorphism in [4], [5], [6], based on the study of closed-open map which is defined more geometrically. Though the codomain of the original closed-open map is Hochschild cohomology which is hard to grasp, Fukaya-Oh-Ohta-Ono proved that the length-0 part of the closed-open map (which was called Kodaira-Spencer map), whose codomain is now Jacobian ring, provides an isomorphism in compact toric case. Their strategy was also employed by Amorim-Cho-Hong-Lau in [7] for proving mirror symmetry for orbifold projective lines.

Since quantum cohomology ring and Jacobian ring are both Frobenius algebras with respect to the Poincaré duality and the residue pairing respectively, we might ask whether their pairings are also related in a suitable way. The question was dealt with in [8], and it was conjectured that if we want the Kodaira-Spencer map to be a Frobenius algebra isomorphism, we need to modify the residue pairing by a constant which is the ratio of ”Floer volume form” and the usual volume form on a Lagrangian submanifold. The rescaling constant appeared naturally when we consider Cardy condition (see [8] for more detail).

In this paper we focus on elliptic orbispheres with three singular points, which are quotients of 2-torus by finite groups Z 3 , Z 4 and Z 6 respectively. Several mirror isomorphisms of Frobenius manifolds for orbifold projective lines have been established. In [9] Satake-Takahashi proved that there is an isomorphism of Frobenius manifolds from Gromov-Witten theory of P 3,3,3 1 and the universal unfolding of the mirror potential, and also for P 2,2,2,2 1 case (where the counterpart is given by the invariant theory of an elliptic Weyl group). For spherical cases, namely for P a , b , c 1 with 1 a + 1 b + 1 c > 1 we refer readers to [10], [11], [12] etc. Towards the ultimate level of closed string mirror symmetry such as above results, we add more elliptic orbisphere examples, namely P 2,4,4 1 and P 2,3,6 1 which are quotients of the elliptic curve by Z 4 and Z 6 respectively. P 3,3,3 1 will be also revisited. Though we work only at the level of Frobenius algebras, we hope that we can find a relationship with previous works, such as the relation between the Floer theoretic rescaling constant and the choice of primitive form.

We summarize the idea. Let W be the mirror superpotential to X, so the Kodaira-Spencer map is given by k s : Q H * ( X ) → Jac ( W ) . The most natural pairings for Frobenius algebra structure are Poincaré pairing and the residue pairing respectively. We point out that the residue pairing ⟨,⟩_res on Jac(W) will be chosen as

(1.1) ⟨ f , g ⟩ res = ( − 1 ) n ( n − 1 ) 2 R e s f g ⋅ d x 1 ∧ ⋯ ∧ d x n ∂ x 1 W , … , ∂ x n W

whose formula appears in [13] as induced from the Mukai pairing on HH _∗(MF(W)), so that it is ”the most canonical” in some sense. It is also remarkable that the sign  ( − 1 ) n ( n − 1 ) 2 also appears in the Cardy condition in [6], Theorem 3.4.1].

Our main theorem is as follows.

Theorem A.

Let X be an elliptic orbisphere and W be its mirror superpotential. Let L ⊂ X be the Seidel Lagrangian with odd degree immersed generators X, Y, Z, and c L be the constant defined by

(1.2) c L ⋅ p = m 2 ( X , m 2 ( Y , Z ) ) ,

where p = m 2 ( X , X ̄ ) . Then the Kodaira-Spencer map

k s : Q H * ( X ) , 〈 ⋅ , ⋅ 〉 P D → Jac ( W ) , 〈 c L ⋅ , c L ⋅ 〉 r e s

is an isomorphism of Frobenius algebras.

Observe that we modified the residue pairing by suitable constant c L as discussed above. To compare pairings we need to compute the residue of k s ( pt X ) , where pt_X is the Poincaré dual of point class of the symplectic manifold X. Though k s ( pt X ) is explicitly computed in [7] for general orbifold sphere X, it is hard to conclude that its (rescaled) residue is indeed 1 = ∫ _Xpt_X as expected. The difficulty arises in the comparison of two different arithmetics of formal power series. We bypass this difficulty by computing k s ( pt X ) in another way. Let G act on the elliptic curve E so that X = E/G. We will use the result of [14] that the Kodaira-Spencer map on X can be lifted to the orbifold Kodaira-Spencer map k s o r b : H * ( E ) → Jac ( W , G ̂ ) , where Jac ( W , G ̂ ) is the orbifold Jacobian algebra ( G ̂ is the character group of G, so it is isomorphic to G if G is abelian). By the relation

k s ( pt X ) = 1 | G | k s o r b ( pt E ) ,

we can recover k s ( pt X ) by k s o r b ( α ∪ β ) = k s o r b ( α ) • k s o r b ( β ) for some α, β ∈ H ¹(E). By a classical result on the residue over an isolated singularity (which will be recalled in Section 4), the following is a rephrasing of Theorem A.

Theorem B.

Let X be an elliptic orbisphere E/G and W be its mirror superpotential. Then

(1.3) c L 2 ⋅ k s o r b ( pt E ) = − det Hess ( W ) | G | ⋅ μ .

The organization of the paper is as follows. In Section 2 we first recall the construction of orbifold Jacobian algebras from Floer theory. Then we briefly review (orbifold) Kodaira-Spencer maps which appear in Fukaya-Oh-Ohta-Ono’s works and also in [14]. In Section 3 we review the product structure on orbifold Jacobian algebras following [15], and explicitly compute the product for the mirror Landau-Ginzburg orbifold to an elliptic curve. Finally, we prove our main result in Section 4, with a remark on nontrivial identities of arithmetics of formal power series.

2 Kodaira-Spencer map to an orbifold Jacobian algebra

2.1 Preliminaries

We briefly recall the deformation theory of Lagrangian submanifolds originally in [16], following every notation in Section 4.1 of [14]. Let X be a symplectic manifold and L be its (possibly immersed) Lagrangian submanifold. Let C F ( L , L ; Λ ) be the Fukaya A _∞-algebra whose underlying space is the sum of de Rham algebra and the module generated by immersed generators. Suppose that L is weakly unobstructed, i.e. the A _∞-structure on C F * ( L , L ; Λ ) can be deformed by weak Maurer-Cartan elements. Assume further that X 1 , … , X n ∈ C F 1 ( L , L ; Λ ) are weak Maurer-Cartan elements such that c ₁ X ₁ + ⋯ + c _n X _n is also a weak Maurer-Cartan element for any c _i ∈ Λ₊. Let x _i be the dual variable of X _i and b : = x ₁ X ₁ + ⋯ + x _n X _n. The weak Maurer-Cartan equation gives rise to the potential W L ∈ R = Λ [ x 1 , … , x n ] . We consider the following A _∞-algebra

B ( L ) : = ( C F * ( L , L ; Λ ) ⊗ Λ 0 R , m k b ) .

We also denote by B ( L ) a l g the associative algebra with the same underlying space as B ( L ) while equipped with the product v ⋅ w : = ( − 1 ) | v | m 2 b ( v , w ) . The following results evidently reflect the importance of the above definition.

Proposition 2.1

([14]). If L is the Seidel Lagrangian in an orbisphere P a , b , c 1 or the Lagrangian torus at the critical point of W L , then there is an algebra isomorphism

(2.1) Ψ : H * ( B ( L ) a l g ) → ≃ Jac ( W L ) = R / ∂ W L .

Let us now recall the orbifold Jacobian algebra of an isolated singularity equipped with a group action.

Definition 2.2.

Let H be a finite abelian group, which acts on R = K [x ₁, …, x _n] and leaves W invariant. We call the pair (W, H) a Landau-Ginzburg orbifold.

Throughout the paper, we only consider diagonal H-action, i.e. h ⋅ x _i = h _i x _i for some h _i ∈ K *.

Definition 2.3.

Let (W, H) be a Landau-Ginzburg orbifold. Then the twisted Jacobian algebra of (W, H) is defined as

Jac ′ ( W , H ) : = ⨁ h ∈ H Jac ( W h ) ⋅ ξ h ,

where W ^h is the image of W via projection π : R → R/(x _i : hx _i ≠ x _i). The formal generator ξ _h has degree | I h | ∈ Z 2 where I _h = {i ∈ {1, …, n}∣hx _i ≠ x _i}. The H-action on generators is defined by

h ′ ⋅ ξ h = ∏ i ∈ I h h ′ − 1 x i ∏ i ∈ I h x i ξ h ,

and the H-invariant subalgebra

Jac ( W , H ) : = Jac ′ ( W , H ) H

is called the orbifold Jacobian algebra of (W, H).

Theorem 2.4.

Jac′(W, H) is an H-graded algebra, namely
ξ h • ξ h ′ ∈ Jac ( W h ) ⋅ ξ h h ′ .
The product • satisfies braided-commutativity, namely
ξ h • ξ h ′ = ( − 1 ) | ξ h | ⋅ | ξ h ′ | h ⋅ ξ h ′ • ξ h .

In particular, if ξ _h and ξ _h′ are both H-invariant, then
ξ h • ξ h ′ = ( − 1 ) | ξ h | ⋅ | ξ h ′ | ξ h ′ • ξ h ,
which implies that Jac(W, H) is a supercommutative algebra.

We postpone the definition of the product on Jac(W, H) until Section 3. Orbifold Jacobian algebras appear naturally in Floer theory as follows. Let X be a symplectic manifold and a finite abelian group G act on X. Suppose that L ⊂ X / G is a weakly unobstructed Lagrangian submanifold with mirror superpotential W, with an embedded Lagrangian lift L 0 ⊂ X . Then we can construct a new A _∞-algebra structure on B ( L ) ⊗ Λ [ G ̂ ] , whose G ̂ -invariant subalgebra ( B ( L ) ⊗ Λ [ G ̂ ] ) a l g G ̂ is isomorphic to Jac ( W , G ̂ ) .

Proposition 2.5

([14]). Let L ̃ = ⨁ g ∈ G g ⋅ L 0 ⊂ X be a Lagrangian submanifold with lifted weak bounding cochain b ̃ from b. Then

Φ : C F ( L ̃ , L ̃ ) ⊗ R , m k b ̃ → ( B ( L ) ⊗ Λ [ G ̂ ] ) G ̂ , m k b ⊗ 1 , v g ↦ 1 | G | ∑ χ ∈ G ̂ χ ( g ) v ⊗ χ

is an A _∞-isomorphism. Here v _g is an element in C F ( g ⋅ L 0 , h ⋅ L 0 ) and v is the projection of v _g.

Despite being vacuous, Φ can be also defined in the previous nonequivariant setting (namely, endowed with trivial group action) just by the identity.

Proposition 2.6

([14]). Let ( B ( L ) ⊗ Λ [ G ̂ ] ) a l g G ̂ be an associative algebra with

( v ⊗ χ ) ⋅ ( w ⊗ χ ′ ) : = ( − 1 ) | v | m 2 b ⊗ 1 ( v ⊗ χ , w ⊗ χ ′ ) .

Then there is an algebra isomorphism

(2.2) Ψ G ̂ : H * ( ( B ( L ) ⊗ Λ [ G ̂ ] ) a l g G ̂ ) → ≃ Jac ( W , G ̂ ) .

Again, in the nonequivariant setting, Ψ G ̂ is nothing but Ψ above.

Now we recall the construction of Kodaira-Spencer maps. For general weakly unobstructed Lagrangian L ⊂ X with bounding cochain b, we use the similar configuration of holomorphic discs as Fukaya-Oh-Ohta-Ono’s original definition.

Definition 2.7.

For a moduli space of J-holomorphic discs with k + 1 boundary marked points and one interior marked points M k + 1,1 ( β ) for β ∈ H ₂(X, L) and a cycle A ⊂ X, let M k + 1,1 ( β , A ) = M k + 1,1 ( β ) × X A and consider their evaluation maps e v i k : M k + 1,1 ( β , A ) → L at the ith marked point. Consider the length-0 closed open map

(2.3) C O 0 : Q H * ( X ) → H * ( B ( L ) a l g ) , C O 0 ( P D [ A ] ) : = ∑ β ∈ H 2 ( X , L ) ∑ k = 0 ∞ e v 0 k ! ( e v 1 k * b ∧ … ∧ e v k k * b ) .

Definition 2.8.

Let X be a symplectic manifold with finite group G acting effectively on it(G may be trivial), and L be a G-equivariant weakly unobstructed Lagrangian. Then the orbifold Kodaira-Spencer map is k s o r b : = Ψ G ̂ ◦ Φ ◦ C O 0 .

Note that if G is trivial, then k s o r b is just the ordinary Kodaira-Spencer map k s . It was shown in [14] that k s o r b is well-defined, and furthermore it is a ring homomorphism.

2.2 Computation of orbifold Kodaira-Spencer maps from elliptic curves

We choose three different Lagrangian submanifolds on elliptic curves and compute the orbifold Kodaira-Spencer maps with respect to them.

2.2.1 Z 3

Let E = C / ( Z + e 2 π i / 3 Z ) be an elliptic curve and let Z 3 = { 1 , ρ = e 2 π i / 3 , ρ 2 = e 4 π i / 3 } act on E by multiplication, so P 3,3,3 1 = E / Z 3 . Let L 0 ⊂ E be an embedded circle in E, and L 1 : = ρ L 0 , L 2 : = ρ 2 L 0 as in Figure 1. Then L ̃ : = L 0 ⊕ L 1 ⊕ L 2 is a weakly unobstructed Lagrangian on E with potential W ₃₃₃.

$Figure 1: Z 3 ${\mathbb{Z}}_{3}$ -equivariant Lagrangian.$

Figure 1:

Z 3 -equivariant Lagrangian.

Let C _h be a homology cycle representing a class (1,0). Since dim M k + 1,1 ( β , C h ) = k + μ ( β ) , if β ∈ H 2 ( E , L ̃ ) is nontrivial then there is no summand in (2.3) for β. Therefore to compute the Kodaira-Spencer map we only consider M 1,1 ( 0 , C h ) . There is a natural orientation on M 1,1 ( 0 ) because it is diffeomorphic to L ̃ , and the fiber product M 1,1 ( 0 , C h ) is nothing but the intersection of L ̃ and C _h. We conclude that k s ( P D [ C h ] ) is given by (Poincaré dual of) the oriented intersection L ̃ ∩ C h . In Figure 1 we depicted intersection points between L ̃ and C _h whose Poincaré dual 1-forms are a, b, c, d respectively.

Taking orientations into account, we have

C O 0 ( P D [ C h ] ) = − a + b + c − d .

To read an orbifold Jacobian algebra element from it, we need to recall the construction of an isomorphism H * ( B ( L ) ⊗ Λ [ Z 3 ̂ ] ) Z 3 ̂ ≅ Jac ( W 333 , Z 3 ̂ ) for the Seidel Lagrangian L . The module C F ( L , L ) is generated by 1 , X , Y , Z , X ∧ Y , Y ∧ Z , Z ∧ X , X ∧ ( Y ∧ Z ) where X, Y and Z are odd degree immersed generators and ∧ is the binary A _∞-product m ₂ without weak bounding cochain insertions. Observe that

X ∧ Y Z ̄ = Y ∧ Z X ̄ = Z ∧ X Y ̄ = X ∧ ( Y ∧ Z ) p = c L

for some constant c L , as in (1.2) (recall that p = m 2 ( X , X ̄ ) ).

Theorem 2.9

([14]). The isomorphism

Ψ Z 3 ̂ : H * ( B ( L ) ⊗ Λ [ Z 3 ̂ ] ) Z 3 ̂ ≅ Jac ( W 333 , Z 3 ̂ )

in (2.2) is given by

(2.4) H * ( B ( L ) ⊗ 1 ) Z 3 ̂ → Jac ( W 333 ) Z 3 ̂ , f ⋅ 1 ↦ f , H * ( B ( L ) ⊗ χ ) Z 3 ̂ → Jac W 333 χ Z 3 ̂ ⋅ ξ χ , X ∧ ( Y ∧ Z ) + ( l o w e r ) ⊗ χ ↦ ξ χ .

For p = m 2 ( X , X ̄ ) ∈ C F ( L , L ) , let p i ∈ C F ( L i , L i ) for i = 0, 1, 2 such that p _i projects to p. Now let us investigate the Poincaré dual of each intersection of L ̃ and C _h, say a for example. Seeing Figure 1 again, p ₁ and a are cohomologous in the de Rham complex Ω * ( L 1 ) , but they are not cohomologous in ( C F ( L ̃ , L ̃ ) , m 1 b ̃ ) (in fact, they are not even cocycles). If we consider a de Rham 0-form I whose de Rham coboundary is p ₁ − a, then m 1 b ̃ ( I ) = p 1 − a + ( l o w e r ) , where (lower) means a linear sum of odd degree immersed generators. In the same vein, we consider de Rham 0-forms J, K and L whose coboundaries are p ₀ − b, p ₂ − c and p ₁ − d respectively. Then

m 1 b ̃ ( I − J − K + L ) = ( − a + b + c − d ) + ( p 1 − p 0 − p 2 + p 1 ) + ( l o w e r ) ,

i.e. −a + b + c − d is cohomologous to −2p ₁ + p ₀ + p ₂ + (lower) in C F ( L ̃ , L ̃ ) with respect to m 1 b ̃ . Therefore, letting Z 3 ̂ = { 1 , χ , χ 2 } , the image via (orbifold) Kodaira-Spencer map is

Φ ( C O 0 ( P D [ C h ] ) ) = 1 3 ( − 2 χ ( ρ ) + 1 + χ ( ρ 2 ) ) p ⊗ χ + ( − 2 χ 2 ( ρ ) + 1 + χ 2 ( ρ 2 ) ) p ⊗ χ 2 + ( l o w e r ) ↦ ( 2.4 ) ( − 2 χ ( ρ ) + 1 + χ ( ρ 2 ) ) ξ χ + ( − 2 χ 2 ( ρ ) + 1 + χ 2 ( ρ 2 ) ) ξ χ 2 3 c L ∈ Jac ( W 333 , Z 3 ̂ ) .

Observe that there is no output on 1-sector due to degree reason. For the cycle C _v of class (0,1),

Φ ( C O 0 ( P D [ C v ] ) ) = 1 3 ( − 2 χ ( ρ 2 ) + 1 + χ ( ρ ) ) p ⊗ χ + ( − 2 χ 2 ( ρ 2 ) + 1 + χ 2 ( ρ ) ) p ⊗ χ 2 + ( l o w e r ) ↦ ( 2.4 ) ( − 2 χ ( ρ 2 ) + 1 + χ ( ρ ) ) ξ χ + ( − 2 χ 2 ( ρ 2 ) + 1 + χ 2 ( ρ ) ) ξ χ 2 3 c L ∈ Jac ( W 333 , Z 3 ̂ ) .

Letting χ(ρ) = ρ, we summarize

(2.5) k s o r b ( P D [ C h ] ) = − ρ c L ξ χ − ρ 2 c L ξ χ 2 , k s o r b ( P D [ C v ] ) = − ρ 2 c L ξ χ − ρ c L ξ χ 2 .

We hope readers notice that c L is involved in the computation.

2.2.2 Z 4

Let E = C / ( Z + i Z ) be an elliptic curve and Z 4 = { 1 , i , i 2 , i 3 } act on E by multiplication. Let L 0 be an embedded circle of homology class (1,1) and L 1 = i L 0 , L 2 = − L 0 , L 3 = − i L 0 as Figure 2.

$Figure 2: Z 4 ${\mathbb{Z}}_{4}$ -equivariant Lagrangian.$

Figure 2:

Z 4 -equivariant Lagrangian.

Then L ̃ = L 0 ⊕ L 1 ⊕ L 2 ⊕ L 3 is weakly unobstructed with potential W ₂₄₄. Every technical detail involved in the computation is just the same as Z 3 -case, so we only note that (Poincaré dual of) the oriented intersection L ̃ ∩ C h is a − b − c + d, and it is cohomologous to p ₃ − p ₀ − p ₁ + p ₂ + (lower) in ( C F ( L ̃ , L ̃ ) , m 1 b ̃ ) . For C _v, we obtain a cycle cohomologous to p ₀ − p ₁ − p ₂ + p ₃ + (lower). If we let Z 4 ̂ = { 1 , χ , χ 2 , χ 3 } such that χ(i) = i, then

(2.6) k s o r b ( P D [ C h ] ) = − 1 − i 2 c L ξ χ + − 1 + i 2 c L ξ χ 3 , k s o r b ( P D [ C v ] ) = 1 − i 2 c L ξ χ + 1 + i 2 c L ξ χ 3 .

2.2.3 Z 6

Let E = C / ( Z + e 2 π i / 3 Z ) and Z 6 = { 1 , ζ = e π i / 3 , ζ 2 , ζ 3 , ζ 4 , ζ 5 } act on E by multiplication. Let L 0 be an embedded circle of class (1,0) and L k = ζ k L 0 for k = 1, …, 5 as in Figure 3.

$Figure 3: Z 6 ${\mathbb{Z}}_{6}$ -equivariant Lagrangian.$

Figure 3:

Z 6 -equivariant Lagrangian.

Let L ̃ : = ⨁ k = 0 5 ζ k L 0 be a weakly unobstructed Lagrangian with potential W ₂₃₆. With the same computation as above, we obtain

C O 0 ( P D [ C h ] ) ∼ − p 2 + p 5 − p 1 + p 4 + ( l o w e r ) , C O 0 ( P D [ C v ] ) ∼ p 1 − p 3 − p 4 + p 0 + ( l o w e r ) ,

hence for Z 6 ̂ = { 1 , χ , … , χ 5 } with χ(ζ) = ζ,

(2.7) k s o r b ( P D [ C h ] ) = − ζ − ζ 2 3 c L ξ χ + ζ + ζ 2 3 c L ξ χ 5 , k s o r b ( P D [ C v ] ) = 1 + ζ 3 c L ξ χ + 1 − ζ 2 3 c L ξ χ 5 .

3 Products in orbifold Jacobian algebras

Recall that our goal is to prove

k s o r b c L 2 ⋅ pt E = − det Hess ( W ) | G | ⋅ μ .

For this, we need to compute the product in the orbifold Jacobian algebra. Among various works, we follow the construction of [15]. Other works on orbifold Jacobian algebras include [17], [18], [19] etc.

Let W ∈ K ?[x ₁, …, x _n], and let ( W , G ̂ ) be a Landau-Ginzburg orbifold. For h ∈ G ̂ , let

I h : = { i ∣ h ⋅ x i ≠ x i } ⊂ { 1 , … , n } , I h : = I h c .

Let S = K x 1 , … , x n , x 1 ′ , … , x n ′ . For 0 ≤ j < i ≤ n, define

W ̄ j , i h : = W x 1 ′ , … , x j ′ , x j + 1 , … , x i , h x i + 1 , … , h x n ∈ S ,

W ̃ j , i h : = W x 1 h , … , x j h , x j + 1 , … , x i , h x i + 1 , … , h x n ∈ R .

(we define x i h = x i if hx _i = x _i and x i h = 0 otherwise.) We also define

(3.1) W ̄ i , i h : = W x 1 ′ , … , x i ′ , h x i + 1 , … , h x n ,

(3.2) W ̃ i , i h : = W x 1 h , … , x i h , h x i + 1 , … , h x n .

For i, j ∈ {1, …, n} with j < i, define

(3.3) g j i h : = W ̄ j , i h − W ̄ j − 1 , i h − W ̄ j , i − 1 h − W ̄ j − 1 , i − 1 h x j ′ − x j ( x i − h x i ) i f i ∈ I h , 0 o t h e r w i s e , f j i h : = W ̃ j , i h − W ̃ j − 1 , i h − W ̃ j , i − 1 h − W ̃ j − 1 , i − 1 h ( x j − h x j ) ( x i − h x i ) i f i , j ∈ I h , 0 o t h e r w i s e ,

and

(3.4) g i i h : = 1 x i ′ − x i ⋅ W ̄ i , i h − W ̄ i − 1 , i − 1 h x i ′ − h x i − W ̄ i − 1 , i h − W ̄ i − 1 , i − 1 h x i − h x i i f i ∈ I h , 0 o t h e r w i s e .

Let θ _i and ∂ _i (for i = 1, …, n) be formal variables with |θ _i| = −1, |∂ _i| = 1 and

θ i θ j = − θ j θ i , ∂ i ∂ j = − ∂ j ∂ i , ∂ i θ j = − θ j ∂ i + δ i j .

For an ordered subset I = {i ₁, …, i _k} ⊂ {1, …, n}, we introduce a notation θ I : = θ i 1 … θ i k . Now, define an S-linear map

(3.5) η h : S 〈 θ 1 , … , θ n , ∂ 1 , … , ∂ n 〉 → S 〈 θ 1 , … , θ n , ∂ 1 , … , ∂ n 〉 , θ I ∂ J ↦ ∑ ( − 1 ) | I | g j i h ∂ θ I ∂ θ i ∂ j ∂ J + ∑ f j i h ∂ 2 θ I ∂ θ j ∂ θ i ∂ J .

We also define a map

(3.6) exp ( η h ) : = 1 + η h + η h 2 2 ! + ⋯ : S 〈 θ 1 , ⋯ , θ n , ∂ 1 , ⋯ , ∂ n 〉 → S 〈 θ 1 , … , θ n , ∂ 1 , … , ∂ n 〉 .

For f ( x 1 , … , x n , x 1 ′ , … , x n ′ ) θ I ∂ J ∈ S ⟨ θ 1 , … , θ n , ∂ 1 , … , ∂ n ⟩ , define

h ∗ ( f ⋅ θ I ∂ J ) : = f ( x 1 , … , x n , h − 1 x 1 ′ , … , h − 1 x n ′ ) ⋅ h − 1 ( θ I ∂ J ) ,

where for h _i ∈ K * such that h ⋅ x _i = h _i x _i, h ⋅ θ i = h i − 1 θ i and h ⋅ ∂ _i = h _i ∂ _i. Then for h , h ′ ∈ G ̂ , define

σ ̃ h , h ′ : = ⟨ h ∗ ′ ( exp ( η h ) ( θ I h ) ) ⋅ exp η h ′ θ I h ′ , θ I h h ′ ⟩

which is θ I h h ′ -coefficient of h ∗ ′ ( exp ( η h ) ( θ I h ) ) ⋅ exp η h ′ θ I h ′ . Let

π h : S → S / ( y 1 − h x 1 , … , y n − h x n )

be the “h-twisted” quotient map for h ∈ H. Then

σ h , h ′ : = π h h ′ σ ̃ h , h ′

induces an element of Jac(W ^hh′), and is the structure constant of ξ _h• ξ _h′, i.e.

ξ h • ξ h ′ = σ h , h ′ ⋅ ξ h h ′ .

We present relevant multiplications inside three orbifold Jacobian algebras from elliptic curves.

3.1 ( C / ( Z + e 2 π i / 3 Z ) , Z 3 )

The mirror superpotential is given by

W = ϕ ( x 3 + y 3 + z 3 ) − ψ x y z ∈ Λ [ x , y , z ]

and Z 3 = { 1 , χ , χ 2 } acts on Λ[x, y, z] by

χ ⋅ x = e 2 π i / 3 x , χ ⋅ y = e 2 π i / 3 y , χ ⋅ z = e 2 π i / 3 z .

By I χ = { 1,2,3 } = I χ 2 (we let x ₁ = x, x ₂ = y, x ₃ = z) we deduce that ξ _χ and ξ χ 2 are Z 3 -invariant, hence are elements of Jac ( W , Z 3 ) .

To calculate ξ χ • ξ χ 2 , let us first compute exp(η _χ)(θ ₁ θ ₂ θ ₃) as follows. (Let ρ = e ^2πi/3 for simplicity.)

W ̄ 0,0 χ = W ( x , y , z ) , W ̄ 0,1 χ = ϕ ( x 3 + y 3 + z 3 ) − ρ 2 ψ x y z , W ̄ 0,2 χ = ϕ ( x 3 + y 3 + z 3 ) − ρ ψ x y z , W ̄ 0,3 χ = W ( x , y , z ) , W ̄ 1,1 χ = ϕ ( x ′ 3 + y 3 + z 3 ) − ρ 2 ψ x ′ y z , W ̄ 1,2 χ = ϕ ( x ′ 3 + y 3 + z 3 ) − ρ ψ x ′ y z , W ̄ 1,3 χ = ϕ ( x ′ 3 + y 3 + z 3 ) − ψ x ′ y z , W ̄ 2,2 χ = ϕ ( x ′ 3 + y ′ 3 + z 3 ) − ρ ψ x ′ y ′ z , W ̄ 2,3 χ = ϕ ( x ′ 3 + y ′ 3 + z 3 ) − ψ x ′ y ′ z , W ̄ 3,3 χ = W ( x ′ , y ′ , z ′ ) ,

and

W ̃ 0,0 χ = W ( x , y , z ) , W ̃ 0,1 χ = ϕ ( x 3 + y 3 + z 3 ) − ρ 2 ψ x y z , W ̃ 0,2 χ = ϕ ( x 3 + y 3 + z 3 ) − ρ ψ x y z , W ̃ 0,3 χ = W ( x , y , z ) , W ̃ 1,1 χ = W ̃ 1,2 χ = W ̃ 1,3 χ = ϕ ( y 3 + z 3 ) , W ̃ 2,2 χ = W ̃ 2,3 χ = ϕ z 3 , W ̃ 3,3 χ = 0 .

Plugging W ̄ j , i χ and W ̃ j , i χ into (3.3) and (3.4), we have

g 11 χ = ϕ ( x ′ − ρ 2 x ) , g 12 χ = − ρ ψ z , g 13 χ = − ψ y , g 22 χ = ϕ ( y ′ − ρ 2 y ) , g 23 χ = − ψ x ′ , g 33 χ = ϕ ( z ′ − ρ 2 z ) , f 12 χ = ρ ψ z 1 − ρ , f 13 χ = ψ y 1 − ρ , f 23 χ = 0 .

By definitions (3.5) and (3.6),

exp ( η χ ) ( θ 1 θ 2 θ 3 ) = ∑ I ≠ ∅ ∑ J f I J θ I ∂ J + g 11 χ f 23 χ − g 12 χ f 13 χ + g 13 χ f 12 χ ∂ 1 + − g 22 χ f 13 χ + g 23 χ f 12 χ ∂ 2 + g 33 χ f 12 χ ∂ 3 + g 11 χ g 22 χ g 33 χ ∂ 3 ∂ 2 ∂ 1 = ∑ I ≠ ∅ ∑ J f I J θ I ∂ J − ϕ ψ ( y ′ y − ρ 2 y 2 ) + ρ ψ 2 x ′ z 1 − ρ ∂ 2 + ρ ϕ ψ ( z ′ z − ρ 2 z 2 ) 1 − ρ ∂ 3 + ϕ 3 ( x ′ − ρ 2 x ) ( y ′ − ρ 2 y ) ( z ′ − ρ 2 z ) ∂ 3 ∂ 2 ∂ 1

and

π χ χ 2 χ ∗ 2 ⁡ exp ( η χ ) ( θ 1 θ 2 θ 3 ) = χ ∗ 2 ⁡ exp ( η χ ) ( θ 1 θ 2 θ 3 ) | x ′ = x , y ′ = y , z ′ = z = χ ∗ 2 ∑ I ≠ ∅ ∑ J f I J θ I ∂ J | x ′ = x , y ′ = y , z ′ = z − ϕ ψ ( ρ 2 − 1 ) y 2 + ψ 2 x z 1 − ρ ∂ 2 + ϕ ψ ( 1 − ρ ) z 2 1 − ρ ∂ 3 + 3 ϕ 3 ( ρ 2 − ρ ) x y z ⋅ ∂ 3 ∂ 2 ∂ 1 .

We do not have to compute ∑ _I≠∅ ∑ _J f _IJ θ _I ∂ _J explicitly, because it does not contribute to σ ̃ χ , χ 2 which is the coefficient of θ I χ χ 2 = θ I 1 = θ ∅ .

The same computations as above gives exp η χ 2 ( θ 1 θ 2 θ 3 ) as follows.

exp η χ 2 ( θ 1 θ 2 θ 3 ) = ∑ I ∑ J ≠ ∅ f I J θ I ∂ J + θ 1 θ 2 θ 3 − f 12 χ 2 θ 3 + f 13 χ 2 θ 2 − f 23 χ 2 θ 1 = ∑ I ∑ J ≠ ∅ f I J θ I ∂ J + θ 1 θ 2 θ 3 − ρ 2 ψ z 1 − ρ 2 θ 3 + ψ y 1 − ρ 2 θ 2 .

This time, we do not have to compute ∑ _I ∑ _J≠∅ f _IJ θ _I ∂ _J explicitly because it does not contribute to σ ̃ χ , χ 3 . We conclude that

(3.7) σ χ , χ 3 = 3 ϕ 3 ( ρ 2 − ρ ) − ψ 3 3 x y z − ϕ ψ 2 ( ρ 2 − 1 ) 3 y 3 − ϕ ψ 2 ( ρ 2 − 1 ) 3 z 3 .

3.2 ( C / ( Z + i Z ) , Z 4 )

The mirror superpotential is given by

W = − q x y z + q 6 x 2 + a ( y 4 + z 4 ) + b y 2 z 2 ∈ Λ [ x , y , z ]

and Z 4 = { 1 , χ , χ 2 , χ 3 } acts on Λ[x, y, z] by

χ ⋅ x = − x , χ ⋅ y = i y , χ ⋅ z = i z .

By I χ = { 1,2,3 } = I χ 3 (we let x ₁ = x, x ₂ = y, x ₃ = z) we deduce that ξ _χ and ξ χ 3 are Z 4 -invariant, hence are elements of Jac ( W , Z 4 ) .

As above we compute exp(η _χ)(θ ₁ θ ₂ θ ₃) from

W ̄ 0,0 χ = W ( x , y , z ) , W ̄ 0,1 χ = q 6 x 2 + q x y z + a y 4 + a z 4 + b y 2 z 2 , W ̄ 0,2 χ = q 6 x 2 − q i x y z + a y 4 + a z 4 − b y 2 z 2 , W ̄ 0,3 χ = W ( x , y , z ) , W ̄ 1,1 χ = q 6 x ′ 2 + q x ′ y z + a y 4 + a z 4 + b y 2 z 2 , W ̄ 1,2 χ = q 6 x ′ 2 − q i x y z + a y 4 + a z 4 − b y 2 z 2 , W ̄ 1,3 χ = q 6 x ′ 2 − q x ′ y z + a y 4 + a z 4 + b y 2 z 2 , W ̄ 2,2 χ = q 6 x ′ 2 − q i x ′ y ′ z + a y ′ 4 + a z 4 − b y ′ 2 z 2 , W ̄ 2,3 χ = q 6 x ′ 2 − q x ′ y ′ z + a y ′ 4 + a z 4 + b y ′ 2 z 2 , W ̄ 3,3 χ = W ( x ′ , y ′ , z ′ ) ,

and

W ̃ 0,0 χ = W ( x , y , z ) , W ̃ 0,1 χ = q 6 x 2 + q x y z + a y 4 + a z 4 + b y 2 z 2 , W ̃ 0,2 χ = q 6 x 2 − q i x y z + a y 4 + a z 4 − b y 2 z 2 , W ̃ 0,3 χ = W ( x , y , z ) , W ̃ 1,1 χ = a y 4 + a z 4 + b y 2 z 2 , W ̃ 1,2 χ = a y 4 + a z 4 − b y 2 z 2 , W ̃ 1,3 χ = a y 4 + a z 4 + b y 2 z 2 , W ̃ 2,2 χ = W ̃ 2,3 χ = a z 4 , W ̃ 3,3 χ = 0 .

Plugging W ̄ j , i χ and W ̃ j , i χ into (3.3) and (3.4), we have

g 11 χ = q 6 , g 12 χ = − q i z , g 13 χ = − q y , g 22 χ = a ( y ′ + y ) ( y ′ + i y ) − b z 2 , g 23 χ = − q x ′ + b ( y ′ + y ) ( z + i z ) , g 33 χ = a ( z ′ + z ) ( z ′ + i z ) + b y ′ 2 , f 12 χ = − q i z 2 , f 13 χ = q y 2 , f 23 χ = − i b y z .

By definitions (3.5) and (3.6),

exp ( η χ ) ( θ 1 θ 2 θ 3 ) = ∑ I ≠ ∅ ∑ J f I J θ I ∂ J + g 11 χ f 23 χ − g 12 χ f 13 χ + g 13 χ f 12 χ ∂ 1 + − g 22 χ f 13 χ + g 23 χ f 12 χ ∂ 2 + g 33 χ f 12 χ ∂ 3 + g 11 χ g 22 χ g 33 χ ∂ 3 ∂ 2 ∂ 1 = ∑ I ≠ ∅ ∑ J f I J θ I ∂ J − q 6 b i y z ∂ 1 + 1 2 q b y z 2 − q a ( y ′ + y ) ( y ′ + i y ) y + ( − q x ′ + b ( y ′ + y ) ( z + i z ) ) q i z ∂ 2 + ( a ( z ′ + i z ) ( z ′ + z ) + b y ′ 2 ) ⋅ q i z 2 ∂ 3 + q 6 ( a ( y ′ + y ) ( y ′ + i y ) − b z 2 ) ( a ( z ′ + z ) ( z ′ + i z ) + b y ′ 2 ) ∂ 3 ∂ 2 ∂ 1 ,

and

π χ χ 3 χ ∗ 3 ⁡ exp ( η χ ) ( θ 1 θ 2 θ 3 ) = χ ∗ 3 ⁡ exp ( η χ ) ( θ 1 θ 2 θ 3 ) | x ′ = x , y ′ = y , z ′ = z = χ ∗ 3 ∑ I ≠ ∅ ∑ J f I J θ I ∂ J | x ′ = x , y ′ = y , z ′ = z + q 6 b i y z ∂ 1 + − q b y z 2 − 2 i ( i + 1 ) q a y 3 + q 2 i x z − 2 ( i + 1 ) q b y z 2 i 2 ∂ 2 + − q a ( i + 1 ) z 3 − q i b y 2 z 2 i ∂ 3 + q 6 ( ( 2 i − 2 ) a y 2 − b z 2 ) ( ( 2 i − 2 ) a z 2 − b y 2 ) ∂ 3 ∂ 2 ∂ 1 .

We do not have to compute ∑ _I≠∅ ∑ _J f _IJ θ _I ∂ _J explicitly, because it does not contribute to σ ̃ χ , χ 3 which is the coefficient of θ I χ χ 3 = θ I 1 = θ ∅ .

The same computations as above gives exp η χ 3 ( θ 1 θ 2 θ 3 ) as follows.

exp η χ 3 ( θ 1 θ 2 θ 3 ) = ∑ I ∑ J ≠ ∅ f I J θ I ∂ J + θ 1 θ 2 θ 3 − f 12 χ 3 θ 3 + f 13 χ 3 θ 2 − f 23 χ 3 θ 1 = ∑ I ∑ J ≠ ∅ f I J θ I ∂ J + θ 1 θ 2 θ 3 + q i z 2 θ 3 + q y 2 θ 2 − b i y z θ 1 .

This time, we do not have to compute ∑ _I ∑ _J≠∅ f _IJ θ _I ∂ _J explicitly because it does not contribute to σ ̃ χ , χ 3 . We conclude that

(3.8) σ χ , χ 3 = − q 3 x y z 4 + q 2 a ( i + 1 ) 2 − 4 q 6 i ( i + 1 ) a b y 4 + q 2 a ( i + 1 ) z 4 2 + 2 q 6 b 2 − 8 q 2 a 2 i − q 2 b ( i − 1 ) 2 y 2 z 2 .

3.3 ( C / ( Z + e 2 π i / 3 Z ) , Z 6 )

The mirror superpotential is given by

W = − q x y z + q 6 x 2 + a 1 y 3 + a 2 z 6 + a 3 y 2 z 2 + a 4 y z 4 ∈ Λ [ x , y , z ]

and Z 6 = { 1 , χ , χ 2 , χ 3 , χ 4 , χ 5 } acts on Λ[x, y, z] by

χ ⋅ x = − x , χ ⋅ y = e 2 π i / 3 y , χ ⋅ z = e π i / 3 z .

By I χ = { 1,2,3 } = I χ 5 we deduce that ξ _χ and ξ χ 5 are Z 6 -invariant, so they are elements of Jac ( W , Z 6 ) . Let ζ = e^πi/3 for simplicity. To compute ξ χ • ξ χ 5 , we need

W ̄ 0,0 χ = W ( x , y , z ) , W ̄ 0,1 χ = q 6 x 2 + q x y z + a 1 y 3 + a 2 z 6 + a 3 y 2 z 2 + a 4 y z 4 , W ̄ 0,2 χ = q 6 x 2 − ζ q x y z + a 1 y 3 + a 2 z 6 + ζ 2 a 3 y 2 z 2 + a 4 y z 4 , W ̄ 0,3 χ = W ( x , y , z ) , W ̄ 1,1 χ = q 6 x ′ 2 + q x ′ y z + a 1 y 3 + a 2 z 6 + a 3 y 2 z 2 + a 4 y z 4 , W ̄ 1,2 χ = q 6 x ′ 2 − ζ q x ′ y z + a 1 y 3 + a 2 z 6 + ζ 2 a 3 y 2 z 2 + ζ 4 a 4 y z 4 , W ̄ 1,3 χ = q 6 x ′ 2 − q x ′ y z + a 1 y 3 + a 2 z 6 + a 3 y 2 z 2 + a 4 y z 4 , W ̄ 2,2 χ = q 6 x ′ 2 − ζ q x ′ y ′ z + a 1 y ′ 3 + a 2 z 6 + ζ 2 a 3 y ′ 2 z 2 + ζ 4 a 4 y ′ z 4 , W ̄ 2,3 χ = q 6 x ′ 2 + q x ′ y ′ z + a 1 y ′ 3 + a 2 z 6 + a 3 y ′ 2 z 2 + a 4 y ′ z 4 , W ̄ 3,3 χ = W ( x ′ , y ′ , z ′ ) ,

and

W ̃ 0,0 χ = W ( x , y , z ) , W ̃ 0,1 χ = q 6 x 2 + q x y z + a 1 y 3 + a 2 z 6 + a 3 y 2 z 2 + a 4 y z 4 , W ̃ 0,2 χ = q 6 x 2 − ζ q x y z + a 1 y 3 + a 2 z 6 + ζ 2 a 3 y 2 z 2 + ζ 4 a 4 y z 4 , W ̃ 0,3 χ = W ( x , y , z ) , W ̃ 1,1 χ = a 1 y 3 + a 2 z 6 + a 3 y 2 z 2 + a 4 y z 4 , W ̃ 1,2 χ = a 1 y 3 + a 2 z 6 + ζ 2 a 3 y 2 z 2 + ζ 4 a 4 y z 4 , W ̃ 1,3 χ = a 1 y 3 + a 2 z 6 + a 3 y 2 z 2 + a 4 y z 4 , W ̃ 2,2 χ = W ̃ 2,3 χ = a z 4 , W ̃ 3,3 χ = 0 .

As above, we calculate the following:

g 11 χ = q 6 , g 12 χ = − ζ q z , g 13 χ = − q y , g 22 χ = a 1 ( y ′ + ζ y ) + ζ 2 a 3 z 2 , g 23 χ = − q x ′ + a 3 ( y ′ + y ) ( z + ζ z ) + a 4 3 i z 3 , g 33 χ = a 2 ( z ′ − ζ 2 z ) ( z ′ − ζ 3 z ) ( z ′ − ζ 4 z ) ( z ′ − ζ 5 z ) + a 3 y ′ 2 + a 4 y ′ ( z ′ 2 + z ′ z + z 2 + ( z ′ + z ) ζ z + ζ 2 z 2 ) , f 12 χ = − ζ q z 2 , f 13 χ = q y 2 , f 23 χ = − a 3 y z − ζ a 4 z 3 1 − ζ

and

π χ χ 5 χ ∗ 5 ⁡ exp ( η χ ) ( θ 1 θ 2 θ 3 ) = χ ∗ 5 ∑ I ≠ ∅ ∑ J f I J θ I ∂ J | x ′ = x , y ′ = y , z ′ = z + q 6 ⋅ a 3 y z + ζ a 4 z 3 1 − ζ ∂ 1 + − a 1 3 i y − ζ 2 a 3 z 2 q y 2 + q x + a 3 3 i y z + a 4 3 i z 3 ζ q z 2 ζ 2 ∂ 2 + − 6 a 2 z 4 + a 3 ζ 4 y 2 + a 4 ( 3 ζ 4 − 1 ) y z 2 ζ 2 q z 2 ∂ 3 + q 6 ( − 6 a 2 z 4 + a 3 ζ 4 y 2 + a 4 ( 3 ζ 4 − 1 ) y z 2 ) a 1 3 i y + ζ 2 a 3 z 2 ∂ 3 ∂ 2 ∂ 1 .

Also, we have

exp η χ 5 ( θ 1 θ 2 θ 3 ) = ∑ I ∑ J ≠ ∅ f I J θ I ∂ J + θ 1 θ 2 θ 3 − ζ − 1 q z 2 θ 3 + q y 2 θ 2 + a 3 y z + ζ − 1 a 4 z 3 1 − ζ − 1 θ 1 .

In conclusion,

σ χ , χ 5 = − a 1 3 i q 2 ζ 2 4 + q 6 a 1 a 3 ζ 4 3 i y 3 + q 6 a 4 2 + 3 q 2 a 2 ζ 2 − 6 q 6 ζ 2 a 2 a 3 z 6 + ( 9 − 3 i ) q 6 a 3 a 4 2 + ( − 1 + 3 i ) q 2 a 4 4 − 6 q 6 a 1 a 2 3 i y z 4 + 2 q 6 a 3 2 + q 6 a 1 a 4 ( 3 ζ 4 − 1 ) 3 i y 2 z 2 − q 3 x y z 4 .

4 Main results

A Frobenius algebra is a unital associative algebra over K together with a nondegenerate bilinear form ⟨⋅, ⋅⟩ such that ⟨xy, z⟩ = ⟨x, yz⟩. Observe that given a nondegenerate pairing ⟨⋅, ⋅⟩ on a Frobenius algebra, the rescaling ⟨c⋅, c⋅⟩ for c ≠ 0 can be still used to define a new Frobenius algebra with the same ring structure. Given two Frobenius algebras (A, ⟨,⟩_A) and (B, ⟨,⟩_B), f : A → B is an isomorphism of Frobenius algebras iff it is a ring isomorphism and an isometry with respect to nondegenerate pairings. Given unities 1_A ∈ A and 1_B ∈ B, let tr_A : A → K and tr_B : B → K be maps (a.k.a. traces) given by ⟨ ⋅ , 1 A ⟩ A and ⟨ ⋅ , 1 B ⟩ B respectively. Then f : A → B is an isometry if and only if it preserves traces.

For a compact symplectic manifold, the quantum cup product together with the Poincaré pairing ⟨⋅,⋅⟩_PD gives rise to a Frobenius algebra structure on QH*(X). For an isolated singularity W ∈ K ?[x ₁, …, x _n], we equip Jac(W) with the residue pairing

⟨ f , g ⟩ r e s = ( − 1 ) n ( n − 1 ) 2 R e s f g ⋅ d x 1 ∧ ⋯ ∧ d x n ∂ x 1 W , … , ∂ x n W .

As pointed out in [8], the Kodaira-Spencer map does not intertwine pairings ⟨⋅,⋅⟩_PD and ⟨⋅,⋅⟩_res in general. Examining the Cardy condition, we are led to consider a modification of the residue pairing by the constant given by ratio of Floer volume form and the usual volume form on the reference Lagrangian L . In Section 2.2 we denoted such ratios by c L .

Theorem 4.1.

Let X be an elliptic orbisphere and W ∈ Λ[x, y, z] be the LG mirror to X. Let A = (QH*(X), ⟨⋅,⋅⟩_PD) and B = ( Jac ( W ) , ⟨ c L ⋅ , c L ⋅ ⟩ r e s ) be Frobenius algebras where c L is the rescaling constant of Seidel Lagrangian L ⊂ X . Then

(4.1) c L 2 ⋅ k s ( pt X ) = − det H e s s ( W ) μ ,

where μ = dim Jac(W). As a result, k s : A → B is an isomorphism of Frobenius algebras, i.e. it preserves traces as follows:

〈 pt X , 1 X 〉 P D = 〈 c L ⋅ k s ( pt X ) , c L ⋅ 1 〉 r e s .

Provided (4.1), the second statement follows immediately from a classical result on the residue over an isolated singularity (the author thanks Cheol-Hyun Cho for pointing out this fact).

Theorem 4.2

(Section 17, [20]).

R e s det H e s s ( W ) ⋅ d x 1 ∧ ⋯ ∧ d x n ∂ x 1 W , … , ∂ x n W = μ .

We can easily compare k s : Q H * ( X ) → Jac ( W ) and k s orb : H * ( E ) → Jac ( W , G ̂ ) as follows. Basically, holomorphic discs employed in the computation of k s o r b ( pt E ) are same as those used for k s ( pt X ) , but the discs which are related by G-action on E are identified as the same disc on X. Thus, |G| different discs which are used for k s o r b ( pt E ) project to a single disc for k s ( pt X ) . We conclude that

k s o r b ( pt E ) = 1 | G | ⋅ k s ( pt X ) .

The rest of the paper is thus devoted to prove

c L 2 ⋅ k s o r b ( pt E ) = − det H e s s ( W ) | G | ⋅ μ ,

where G = Z 3 , Z 4 or Z 6 acts on E, with W given accordingly with respect to G-equivariant Seidel Lagrangian.

4.1 Z 3 -case

Recall from (2.5) that (with ρ = e ^2πi/3)

c L ⋅ k s o r b ( P D [ C h ] ) = − ρ ξ χ − ρ 2 ξ χ 2 , c L ⋅ k s o r b ( P D [ C v ] ) = − ρ 2 ξ χ − ρ ξ χ 2 ,

hence by (3.7)

c L 2 ⋅ k s o r b ( pt E ) = c L 2 k s o r b ( P D [ C h ] ) • k s o r b ( P D [ C v ] ) = ( ρ 2 − ρ ) ξ χ • ξ χ 2 = ( ρ 2 − ρ ) ⋅ 3 ϕ 3 ( ρ 2 − ρ ) − ψ 3 3 x y z − ϕ ψ 2 ( ρ 2 − 1 ) 3 y 3 − ϕ ψ 2 ( ρ 2 − 1 ) 3 z 3 .

Observe that we used supercommutativity of the product for the second identity. The Jacobian ideal ∂W is given by

∂ W = ( 3 ϕ x 2 − ψ y z , 3 ϕ y 2 − ψ x z , 3 ϕ z 2 − ψ x y ) ,

so y 3 = z 3 = ψ x y z 3 ϕ modulo ∂W, and

c L 2 ⋅ k s o r b ( pt E ) = − 9 ϕ 3 + ψ 3 3 x y z .

On the other hand,

H e s s ( W ) = 6 ϕ x − ψ x − ψ y − ψ z 6 ϕ y − ψ x − ψ y − ψ x 6 ϕ z ,

hence det Hess(W) = (216ϕ ³ − 8ψ ³)xyz in Jac(W). By μ = 8 and |G| = 3, we easily verify (4.1).

4.2 Z 4 -case

From (2.6) we have

c L 2 ⋅ k s o r b ( pt E ) = − i ξ χ • ξ χ 3 = − i σ χ , χ 3 ,

where σ χ , χ 3 as (3.8). Modulo ∂W = (2q ⁶ x − qyz, 4ay ³ − qxz + 2byz ², 4az ³ − qxy + 2by ² z), we have relations

x y z = 2 q 5 x 2 , y 2 z 2 = 4 q 10 x 2 , 2 a y 4 = 2 a z 4 = ( q 6 − 4 b q 10 ) x 2 ,

hence in Jac(W),

− i σ χ , χ 3 = − i i q 8 2 − 4 i q 12 b + 8 i q 16 b 2 − 32 i q 16 a 2 x 2 = q 8 2 − 4 q 12 b + 8 q 16 b 2 − 32 q 16 a 2 x 2 .

H e s s ( W ) = 2 q 6 − q z − q y − q z 12 a y 2 + 2 b z 2 − q x + 4 b y z − q y − q x + 4 b y z 12 a z 2 + 2 b y 2

we have

det H e s s ( W ) = ( 48 q 6 a b − 12 q 2 a ) ( y 4 + z 4 ) − 2 q 8 x 2 + ( 288 q 6 a 2 − 24 q 6 b 2 + 4 q 2 b ) y 2 z 2 + ( 16 q 7 b − 2 q 3 ) x y z = ( − 18 q 8 + 144 q 12 b − 288 q 16 + 1152 q 16 a 2 ) x 2 .

This time, |G| = 4 and μ = 9, and it is straightforward that (4.1) holds.

4.3 Z 6 -case

From (2.7) we have

c L 2 ⋅ k s o r b ( pt E ) = − 3 i 3 ξ χ • ξ χ 5 = − 3 i 3 σ χ , χ 5 .

We also compute

det H e s s ( W ) = 120 q 6 a 2 a 3 − 32 q 6 a 4 2 − 30 q 2 a 2 z 6 + 360 q 6 a 1 a 2 − 16 q 6 a 3 a 4 − 4 q 2 a 4 y z 4 + 144 q 6 a 1 a 4 − 24 q 6 a 3 2 + 4 q 2 a 3 y 2 z 2 + 16 q 7 a 4 x z 3 + 24 q 6 a 1 a 3 − 6 q 2 a 1 y 3 + 16 q 7 a 3 − 2 q 3 x y z − 2 q 8 x 2 .

The relations modulo ∂W needed for our purpose are as follows:

x y z = 2 q 5 x 2 , a 1 y 3 = − 8 q 14 a 3 a 4 2 − 16 q 10 a 2 a 3 + 32 q 14 a 2 a 3 2 − 24 q 14 a 1 a 2 a 4 + 2 q 10 a 4 2 + 2 q 6 a 2 4 q 4 a 4 2 + 3 a 2 − 12 q 4 a 2 a 3 x 2 ,

z 6 = q 6 − 8 q 4 a 3 − 48 q 8 a 1 a 4 + 16 q 8 a 3 2 + 1 4 q 4 a 4 2 + 3 a 2 − 12 q 4 a 2 a 3 x 2 , y z 4 = 2 q 10 36 q 4 a 1 a 2 − 4 q 4 a 3 a 4 + a 4 4 q 4 a 4 2 + 3 a 2 − 12 q 4 a 2 a 3 x 2 .

The result (4.1) follows from |G| = 6 and μ = 10. We omit the tedious computation which involves substitutions of above relations to det Hess(W) and σ χ , χ 5 .

4.4 Nontrivial identitites of formal power series

Recall from [7] that

k s ( pt X ) = 1 8 q ⋅ ∂ W ∂ q ,

where X is an orbisphere P a , b , c 1 with area 1 (If the area is A, then we take 1 8 A instead of 1 8 ). Hence we also have

k s o r b ( pt E ) = 1 8 | G | q ⋅ ∂ W ∂ q .

We computed k s o r b ( pt E ) in another way, namely by orbifold Jacobian algebra structure. Let W ₃₃₃, W ₂₄₄ and W ₂₃₆ be mirror superpotentials from Seidel Lagrangians L 333 ⊂ P 3,3,3 1 , L 244 ⊂ P 2,4,4 1 and L 236 ⊂ P 2,3,6 1 respectively.

Let us consider P 3,3,3 1 . Recall that

W 333 = ϕ ( x 3 + y 3 + z 3 ) − ψ x y z

with its coefficients are

ϕ = ∑ k ∈ Z ( − 1 ) k + 1 k + 1 2 q ( 6 k + 3 ) 2 , ψ = ∑ k ∈ Z ( − 1 ) k + 1 ( 6 k + 1 ) q ( 6 k + 1 ) 2 .

We also have

c L 333 = ∑ k ∈ Z ( − 1 ) k q ( 6 k + 1 ) 2 .

By the following

c L 333 2 24 q ⋅ ∂ ϕ ∂ q ( x 3 + y 3 + z 3 ) − ∂ ψ ∂ q x y z = c L 333 2 24 q ⋅ ψ ϕ ∂ ϕ ∂ q − ∂ ψ ∂ q x y z = − 9 ϕ 3 + ψ 3 3 x y z ,

we derive an identity of two arithmetics of formal power series

c L 333 2 24 q ⋅ ψ ϕ ∂ ϕ ∂ q − ∂ ψ ∂ q = − 9 ϕ 3 + ψ 3 3 .

Note that the identity was first proved in [8] using theory of modular forms, and now we have a new proof without appealing to number theory.

The same argument can be applied to other two cases. For W ₂₄₄, we have

q 3 32 6 q 5 x 2 − x y z + ∂ a ∂ q ( y 4 + z 4 ) + ∂ b ∂ q y 2 z 2 = q 3 32 4 q 5 + 1 a ∂ a ∂ q ⋅ ( q 6 − 4 b q 10 ) + 4 q 10 ∂ b ∂ q x 2 = q 8 2 − 4 q 12 b + 8 q 16 b 2 − 32 q 16 a 2 x 2

which implies

q 3 32 4 q 5 + 1 a ∂ a ∂ q ⋅ ( q 6 − 4 b q 10 ) + 4 q 10 ∂ b ∂ q = q 8 2 − 4 q 12 b + 8 q 16 b 2 − 32 q 16 a 2 ,

where a and b are formal power series given by

a = ∑ r ≥ 0 ( 2 r + 1 ) q 16 ( 2 r + 1 ) 2 − 4 + ∑ s > r ≥ 0 ( 2 r + 2 s + 2 ) q 16 ( 2 r + 1 ) ( 2 s + 1 ) − 4 ,

b = ∑ r ≥ 1 , s ≥ 1 − ( 4 r + 4 s − 2 ) q 16 ( 2 r − 1 ) 2 s − 4 + ( 2 r + 2 s ) q 64rs−4 .

We note that c L 2,4,4 = ± q , so its square is q ². For W ₂₃₆ we can obtain a similar type of identity which is omitted.

Corresponding author: Sangwook Lee, Department of Mathematics and Integrative Institute of Basic Science, Soongsil University, 369 Sangdo-ro, Dongjak-gu, Seoul, Korea, E-mail: sangwook@ssu.ac.kr

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The author states no conflict of interest.
Research funding: This work was funded by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2021R1A6A1A10044154).
Data availability: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-09-12

Accepted: 2025-09-30

Published Online: 2025-11-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0208

Keywords for this article

Kodaira-Spencer map; Frobenius algebras; orbifold Jacobian algebras

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