Special quantum invariance under the normalized frame

4.2.1. Compatibility on quantum products via WDVV. In order to deduce con-sequences on quantum products from Proposition 4.5, the following lemma is the starting point.

Lemma 4.6. The isomorphism in (4.14) is compatible with the small quantum D-module structures. Equivalently the quantum products of divisor classes Ta := ΨT_a⁰ on X and of classes σ∗T_a⁰ on X⁰ are compatible along the small parameter ˆs ∈H^≤2(X⁰).

Proof. We first notice that (4.15) σ∗T₀⁰ =

∑

µ

∂σ^µ/∂s⁰T_µ⁰ =

∑

µ

δ^µ₀T_µ⁰ =T₀⁰.

Take two divisor classes T_a⁰, T_b⁰. Then from the WDVV equations, (4.16) hhTa∗T_b, Tⁱ, T_jii^X=

∑

λ

hhTa, Tⁱ, T_λii^XhhT_b, T^λ, T_jii^X.

Along the small parametersΨˆs, by Lemma 4.4, the sum is non-zero only in the non-kernel indices (T_λ 6∈K). By (4.12), the sum then becomes

∑

λ

hhσ∗T_a⁰, T⁰ⁱ, T_λ⁰ii^X0hhσ∗T_b⁰, T^0λ, T_j⁰ii^X0(σ(_ˆs))_. By the WDVV equations on the X⁰side we then conclude

(4.17) hhTa∗T_b, Tⁱ, T_jii^X(_Ψˆs) = hhσ∗T_a⁰∗⁰σ∗T_b⁰, T⁰ⁱ, T_j⁰ii^X0(σ(ˆs)), where the tangent map σ∗ is performed at ˆs and the quantum product on the right-hand-side is on X⁰at σ(ˆs).

By induction on r ∈ N, the equation (4.17) holds for Ta1 ∗. . .∗T_ar and σ∗T_a⁰₁∗. . .∗σ∗T_a⁰_r. The proof is complete.  4.2.2. Pseudo identity and the normalized frame. Recall that T0 ≡ T₀ = _id mod NE(X⁰). The next step is to transform T₀to the identity element (sec-tion) e inT and normalized Ti’s accordingly.

For T_k ∈K^⊥, by Lemma 3.7 we may represent (4.18) T_k∗_Ψˆs= P_k(h∗_Ψˆs, ξ∗_Ψˆs) where P_k is a polynomial with coefficient in x, y.

Definition 4.7. We define theC[[x, y]]-valued R⁰×R⁰matrix(_J^µ_k)by (4.19)

R⁰−1

∑

µ=0

J^µ_kT_µ⁰ := P_k(σ∗(ξ⁰−h⁰)∗⁰, σ∗ξ⁰∗⁰) ∗⁰T₀⁰,

where the quantum product∗⁰is taken at σ(ˆs). Note thatJ^µa =∂σ^µ/∂s^afor a ∈ {_{0, 1, 2}}_andJ^µ_k ∼=δ_k^µ mod NE(X⁰)for all k. Hence(_J^µ_k)is invertible.

Then by Lemma 4.6, or rather equation (4.17), we have hhT_k, Tⁱ, T_jii^X(_Ψˆs) = hhP_k(h∗, ξ∗) ∗T₀, Tⁱ, T_jii^X(_Ψˆs)

= hhP_k(σ∗(ξ⁰−h⁰)∗⁰, σ∗ξ⁰∗⁰) ∗⁰σ∗T₀⁰, T⁰ⁱ, T_j⁰ii^X0(σ(ˆs))

=

∑

µ

J^µ_khhT_µ⁰, T⁰ⁱ, T_j⁰ii^X0(σ(ˆs)). (4.20)

Lemma 4.8. There is a unique element S₀ ∈ T such that S0∗T₀ is the identity element (section) e inT (and so e acts as zero on K).

Proof. By our constructions, the structure constants cⁱ_kj(u, y, z)defined by T_k∗_Tj =

∑

^cikjTi

are series in u, y, z. In particular, by writing S₀ = _∑iw^jT_j, then S₀can be solved explicitly using the relation T₀∗T_j = _∑icⁱ_0jT_i. Indeed, from (4.8), the identity e inT is given by

e= T0−

d−1 j

∑

=0

e_j =

∑

i

ϕⁱT_i

for some series ϕⁱ(u, y, z)in u, y, z. So we need to solve the R⁰×R⁰ linear system of equations

R⁰−1 j

∑

=0

cⁱ_0jw^j = ϕⁱ, i=0, 1, . . . , R⁰−1.

Notice that, by Lemma 3.7 and the property that B₁ ≡Id (mod x, y), cⁱ_0j = hhT0, Tⁱ, T_jii^X = hhT0, Tⁱ, T_jii^X+

∑

k

f₀^k(u, y)hhT_k, Tⁱ, T_jii^X is a series in u, y with f₀^k(0, 0) =0. AlsohhT0, Tⁱ, T_jii^X= ((T_j, Tⁱ)) =δⁱ_j. This shows that(cⁱ_0j)is invertible and the lemma is proved.  Definition 4.9. We call S0 the pseudo-inverse of T0, which is the inverse of T₀inT, and we define the normalized frame

T˘_µ := T_µ∗S₀ onT.

AlongΨˆs, by setting j=0 in (4.20) we find (4.21) T₀∗T_k =

∑

µ

hhT₀, T_k, T^µii^XT_µ =

∑

µ

J^µ_kT_µ.

Applying S₀∗to (4.21), Lemma 4.8 then leads to the important Proposition 4.10(Basic transformation rule). For k∈ [0, R⁰−1], we have

(4.22) T_k =

R⁰−1 µ

∑

=0

J^µ_kT˘µ (mod K).

In particular, the normalized frame ˘T_µis defined over x, y.

4.2.3. Special quantum invariance. With (4.22), then equation (4.20) becomes (4.23) hhT^˜µ, Tⁱ, T_jii^X(_Ψˆs) = hhT_µ⁰, T⁰ⁱ, T_j⁰ii^X0(σ(ˆs)).

Lemma 4.11. With respect to the pairing ˘g_ij = (T^˘_i, ˘T_j), the dual frame ˘Tⁱ :=

∑j ˘g^ijT˘_j is given by ˘Tⁱ = Tⁱ∗T₀. Proof. Indeed,

(_Tj∗_S0, Tⁱ∗_T0) = (_Tj∗_S0∗_T0, Tⁱ) = (_Tj, Tⁱ) =δⁱ_j.

Here, the Frobenius property on the pairing is used.  Hence for any class a we have

hha, ˘T_j, ˘Tⁱii = (a∗T^˘_j, ˘Tⁱ) = (_a∗T_j∗S0, Tⁱ∗T0)

= (a∗T_j∗S₀∗T₀, Tⁱ) = (a∗T_j, Tⁱ)

= hha, T_j, Tⁱii. (4.24)

Together with (4.23), we arrive at a simple statement:

Theorem 4.12. Under theC[[NE(X⁰)]]-linear map

∑

^{aiT˘i 7→}

∑

^aiTi0,

the quantum product onT at Ψˆs∈ H^≤2(X)is isomorphic to the quantum product on H(X⁰)at σ(ˆs) ∈H(X⁰) ⊗_C[[NE(X⁰)]]. Namely

(4.25) hh_T^˘_µ, ˘Tⁱ, ˘Tjii^X(_Ψˆs) = hhT_µ⁰, T⁰ⁱ, T_j⁰ii^X0(σ(ˆs)) for all 0≤i, j, µ≤ R⁰−1.

The “subring”, or rather “ideal”, (_T,∗)of Q₀H(X)is not isomorphic to Q₀H(X⁰)since σ(0) 6= 0 (cf. Corollary 6.9 for contributions from the ex-tremal ray). Nevertheless a standard induction on Mori cone implies that Corollary 4.13. The big quantum cohomology QH(X⁰) can be effectively com-puted from QH(X)through equation (4.25).

4.3. Non-linear reconstructions. In this subsection we will complete the proof of Theorem 0.1 by constructing the embedding bΨ with the imposed properties.

4.3.1. Remarks on reconstructions over the big parameter spaces. The complica-tion in dealing with the GMT ˆs 7→ σ(ˆs) lies on the fact that it is a graph over the small parameters instead of an invertible transformation. The ba-sic idea to resolve the problem is to apply suitable reconstruction theorems on X and X⁰ respectively and to study the compatibility between them.

When the total cohomology H is generated by H²under cup product, the reconstruction from 3-point genus zero GW invariants to all n-point genus zero invariants follows from the WDVV equations as done by Kontsevich–

Manin. Under the same condition, a version in the setup of abstract quan-tumD-modules was formulated and carried out by Iritani in [6, Theorem 4.9], which says that the “abstract big QDM” is naturally determined by the “abstract small QDM”. The abstract version is suitable in our current context since it does not require inductions on the Mori cone.

To trace the reconstruction procedure in all directions of H⁰consistently, we set ˆs = 0 and keep only the Novikov variables{q⁰¹, q⁰²} = {q^`0, q^γ0}in equation (4.25) as the starting point. Namely we decouple the roles played by ˆs and q^0a’s back to the status they are in the definition in (0.1).

Denote the resulting frames byTi(q⁰)and let σ₀ =σ₀(q⁰)be the general-ized mirror point at ˆs=0. Equation (4.12), via (4.24), takes the form (4.26) hTa,Tj,Tⁱi^X =

∑

µ



δ_a^µ+q^0a∂σ₀^µ

∂q^0a

hhT_µ⁰, T_j⁰, T⁰ⁱii^X0(σ₀).

Here δ_a^µis inserted since ∂σ^µ/∂s^a ≡ δ^µ_a mod NE(X⁰). Also (4.25) becomes (4.27) h_Tµ,Tj,Tⁱi^X = hhT_µ⁰, T_j⁰, T⁰ⁱii^X0(σ₀).

We regard (4.26) as the connection matrix A_a(q⁰, s)at s=0 for z∇_a = z∂a−Aa ≡ zq^0a ∂

∂q^0a −Aa, and (4.27) as the connection matrixΩµ(q⁰, s)_{at s}=_{0 for}

z∇_µ =z∂_µ−_Ωµ ≡z ∂

∂s^µ −_Ωµ.

Notice that the coordinates s⁰, . . . , s^R0−1are centered at σ₀when viewing on the H(X⁰)side and centered at 0 on the H(X)side. Also while the matrices A_a andΩµare identically the same for X and X⁰, their meaning in quantum product are taken in completely different manners.

The flatness of∇is equivalent to the WDVV equations [Aa, A_b] = [Aa,Ωµ] = [_Ωµ,Ων] =0,

∂_aA_b =∂_bA_a, ∂_aΩµ =∂_µA_a, ∂_µΩν=∂_νΩµ. (4.28)

Consider the ideal m= (s⁰, s¹, . . . , s^R0−1). By induction on k∈_{N, we may} (i) solve A_a(q⁰, s) (mod m^k)from ∂_µA_a =∂_aΩµ (mod m^k−1), and then (ii) solveΩµ(q⁰, s) (mod m^k)as a polynomial inΩj(q⁰, s)’s (mod m^k−1)

and Aa’s (mod m^k).

The starting case k = 1 for (ii) is essentially Theorem 4.12. The relevant formulas are (4.19) and (4.20) used in the proof of Lemma 4.8. Indeed, let

I(q⁰) = (_I^k_µ):=_J⁻¹

be the inverse matrix of(_J^µ_k)which depends only on q⁰’s. Then at σ₀, T_µ⁰∗⁰ =

∑

k

I^kµP_k(σ∗(ξ⁰−h⁰)∗⁰, σ∗ξ⁰∗⁰)

=

∑

k

I^k_µP_k(A2−A₁, A2), (4.29)

andΩµ(q⁰, 0)is given by (4.25) via (4.22).

Thus it remains to understand the geometric meanings on both sides under the WDVV reconstruction. On X⁰this is standard and it leads to

(_Ωµ)ⁱ_j(q⁰, s) = hhT_µ⁰, T_j⁰, T⁰ⁱii^X0(σ₀+s). In particular(_Ωµ)ⁱ₀(q⁰, s) =δ_µⁱ since T₀⁰ is the identity.

On X the reconstruction is not linear—in each step of (ii) the identity sec-tionT0(q⁰, s) (mod m^k)receives new correction terms. With this modifi-cation been done for each k, which is hard, the resulting structure should then lead to deformations of the embeddingΨ : H(X⁰) ,→ H(X)to certain Ψb(q⁰, s)which relates quantum products of X and X⁰.

When the GW theory under consideration is analytic, alternatively we may view WDVV as a Frobenius integrability condition in the context of integrable distributions and to construct bΨ through certain “canonical co-ordinates”. We will take this approach in the next section, and it is best described in terms of the notion of F-manifolds.

4.3.2. Integrable distribution and the canonical coordinates. Recall that an F-manifold M is a complex F-manifold equipped with a commutative and as-sociative product structure on each tangent space TpM, such that a WDVV-type integrability condition is forced when p varies. In the context of tum cohomology, this is simply the structure which remembers the quan-tum product but forgets the metric g_ij, and with a coordinate-free form of the WDVV (integrability) equations.

Indeed, viewing the quantum product∗as a(2, 1)tensor, Hertling and Manin (cf. [5, Definition 2.8, Theorem 2.14, 2.15] had shown that the WDVV equations can be rewritten as

(4.30) L_X∗Y∗ =X∗L_Y∗ +Y∗L_X∗

for any local vector fields X and Y, where L denotes the Lie derivatives. In explicit terms this means that for any local vector fields X, Y, Z, W we have

[X∗Y, Z∗W] − [X∗Y, Z] ∗W− [X∗Y, W] ∗Z

=X∗ [Y, Z∗W] −X∗ [Y, Z] ∗W−X∗ [Y, W] ∗Z +Y∗ [X, Z∗W] −Y∗ [X, Z] ∗W−Y∗ [X, W] ∗Z.

(4.31)

To apply it to our flip situation, we denote byKthe irregular eigenbun-dle and its orthogonal complementT = K^⊥the regular eigenbundle which extend the correspondingK and T from s=0 to the big parameter space.

Lemma 4.14. Both KandT and the irregular/regular decomposition of the big quantum product on TH_R⁰ are defined over the big parameter space H_R⁰ over a punctured neighborhood of q^`0 =0.

Proposition 4.15. The regular eigenbundleT is an integrable distribution of the relative tangent bundle TH_R⁰.

In particular, the image of bΨ is the integrable submanifoldM(overR⁰) contain-ing the slice(q^`0 6= 0, t = 0)which contains the classical correspondence when moduloR⁰.

Proof. Let X, Z be any two local vector fields valued inT = K^⊥. Let Y=e_i and W =e_j be two idempotents valued inK_{. Since a}∗b=0 for any a∈ K and b∈ K^⊥, (4.31) becomes

(4.32) 0= −X∗Z∗ [e_i, e_j] −δ_ije_j∗ [X, Z]_.

Let i= j we conclude that e_j∗ [X, Z] =0 for all j. Hence[X, Z] ∈ K^⊥.  Remark 4.16. The above proof requires only thatK contains no nilpotent sections, i.e. generically semi-simple. Hence Proposition 4.15 works in the global case as well, though in the formal setting. In the local case all the local models are toric and the analyticity is known (by Iritani), thus the Frobenius theorem needed is the classical one. In the global case we need to invoke the Frobenius theorem in the formal setting.

Now we use the full strength of the local model structure. The quantum product on the Frobenius manifold H(X⁰) ⊗_R⁰ is semi-simple. Deonte by the idempotent vector fields on H(X⁰) ⊗_R⁰by v⁰₀, . . . , v⁰_R0−1. A well-known result of Dubrovin [2, Main Lemma (3.47)] says that canonical coordinates exist. In our setting, we apply it in a family in q⁰ with center at σ₀(q⁰): Lemma 4.17. We have[v⁰_i, v⁰_j] = 0 for all 0 ≤ i, j ≤ R⁰−1. Hence the corre-sponding canonical coordinates u⁰⁰, . . . , u^0R0−1satisfying

(u⁰ⁱ(q⁰, s=0)) =σ₀(q⁰) and v⁰_i = ∂/∂u⁰ⁱexist.

Dubrovin’s result was extended to F-manifolds by Hertling [5, Theorem 2.11]. In our setting, the F-manifoldM is semi-simple (or massive) in the sense that the quantum product on T_pMfor p∈ Mis semi-simple. Denote the idempotent vector field be v1. . . . , v_R⁰.

Lemma 4.18. We have[v_i, v_j] =0 for all 0≤i, j≤ R⁰−1. Hence the canonical coordinates u⁰, . . . , u^R0−1onMexist in the sense that v_i =∂/∂uⁱ.

We emphasize that we have constructed an analytic family of coordinate systems(u⁰(q⁰, p), . . . , u^R0−1(q⁰, p))parametrized by q⁰ ∈_R⁰_{. Write}

(4.33) Ti(q⁰) =

R⁰−1

∑

j=₀

a^j_i(q⁰)v_j(q⁰, s=0)

for an invertible R⁰×R⁰ matrix(a^j_i(q⁰)). From Theorem 4.12 (or (4.27)), we see easily that the same linear combination passes to the X⁰ side:

在文檔中 1. From Picard–Fuchs to small D (頁 27-33)