GLUEING QUANTUM D MODULES OVER FLOPS
CHIN-LUNG WANG
ABSTRACT. In this note I summarize my talk at Sanya on December 19, 2011. The main theme is a glueing theorem of two quantum D modules QH(X)and QH(X0)
via analytic continuations over the K¨ahler moduli, where X and X0are local pro-jective models related by an ordinary flop. This is a joint work with Yuan-Pin Lee and Hui-Wen Lin.
1. BF/GMTFOR TORIC BUNDLES
A general framework to determine g=0 GW invariants is to go from certain lo-calization data I to the generating function of one descendent: Let τ=∑µτµTµ∈ H(X), gµν= (Tµ, Tν), Tµ=∑ gµνTν. JX(τ, z−1) =1+τ z +
∑
β∈NE(X),n,µ qβ n!Tµ Tµ z(z−ψ), τ,· · ·, τ 0,n+1,β . Witten’s dilaton, string, and topological recursion relation in 2D gravity had been reformulated by Givental into a symplectic space theory. Let H := H(X),H := H[z, z−1]], H+ := H[z] and H− := z−1H[[z−1]]. H ∼= T∗H+. Let F0(t) be thegenerating function of all descendent invariants. The one form dF0gives a section
of π : H → H+. Givental’s Lagrangian cone L is the graph of dF0. Let R =
\
C[NE(X)]. Denote a=∑ qβa
β(z) ∈R{z}if aβ(z) ∈C[z].
Lemma 1.1. z∇J = (z∂µJν)forms a matrix whose column vectors z∂µJ(τ) generates the tangent space Lτof the Lagrangian coneLas an R{z}-module.
By TRR, z∇J is the fundamental solution matrix of the Dubrovin connection ∇z =d−1
zdτ µ⊗
∑
µ Tµ∗τ
on TH=H×H. Namely we have the quantum differential equation (QDE) z∂µz∂νJ=
∑
C˜κµν(τ, q)z∂κJ.Let ¯p : X→S be a smooth toric bundle with fiber divisor D=∑ tiDi. H(X)is a
free over H(S)with finite generators{De :=∏
iDeii}e∈Λ. Let ¯t :=∑s¯tsT¯s ∈ H(S).
H(X)has basis{Te = T(s,e) = T¯sDe}e∈Λ+. Denote by ∂T¯
s ≡ ∂¯ts the ¯Tsdirectional
derivative on H(S), ∂e=∂(s,e):=∂
¯ts∏i∂etii and the naive quantization
ˆ Te ≡∂ze≡∂z(s,e):=z∂¯ts
∏
i z∂ei ti =z|e|+1∂ (s,e).The Tedirectional derivative is ∂e=∂Te. ∂
zeand z∂eare closely related. 1
2 CHIN-LUNG WANG
Let ¯p : X→S be a split toric bundle quotient fromLL
ρ →S. The hypergeo-metric modification of JSby the ¯p-fibration takes the form
IX(¯t, D, z, z−1):=
∑
β∈NE(X) qβeDz+(D.β)IX/S β (z, z −1)JS βS(¯t, z −1), where IX/Sβ =∏ρ∈4 11/∏ (Dρ+Lρ).βm=1 (Dρ+Lρ+mz)comes from fiber localization, and the product is directed when(Dρ+Lρ).β≤ −1.
In general positive z powers may occur in IX. I is defined only on the subspace ˆt := ¯t+D∈ H(S) ⊕MiCDi ⊂H(X).
Theorem 1.2(J. Brown 2009). (−z)IX(ˆt,−z)lies in the Lagrangian coneL. Definition 1.3 (GMT). For each ˆt, say zI(ˆt)lies in LτofL. The correspondence
ˆt7→τ(ˆt) ∈H(X) ⊗R
is called the generalized mirror transformation.
Proposition 1.4(BF). (1) The GMT: τ=τ(ˆt)satisfies τ(ˆt, q=0) =ˆt.
(2) Under the basis{Te}e∈Λ+, there exists an invertible N×N matrix-valued formal
series B(τ, z), the Birkhoff factorization, such that
∂zeI(ˆt, z, z−1) =z∇J(τ, z−1) B(τ, z),
where(∂zeI)is the N×N matrix with ∂zeI as column vectors. The first column vectors
are I and J respectively (string equation).
2. QUANTUMLERAY-HIRSCH WITH NATURALITY
Consider the local model of a split Prflop f : X99KX0with data(S, F, F0), where F= r M i=0 Li and F0= r M i=0 L0i.
The contraction ψ : X → X has exceptional loci ¯¯ ψ : Z = PS(F) → S with N =
NZ/X =ψ¯∗F0⊗OZ(−1). Similarly Z0 ⊂X0, N0. ¯p : X=PZ(N⊕O) p
→Z→ψ¯ S is a double projective bundle. For h, ξ being the relative hyperplane classes,
H(X) =H(S)[h, ξ]/(fF, fN⊕O), fF = r
∏
i=0 ai:=∏
(h+Li), fN⊕O =br+1 r∏
i=0 bi:=ξ∏
(ξ−h+L0i).The graph correspondence F = [Γ¯f] ∈ A(X×X0)induces an isomorphism F : H(X) ∼= H(X0)as groups:F ¯thiξj = ¯t(F h)i(F ξ)j= ¯t(ξ0−h0)iξ0jif i≤r. F
also preserves the Poincar´e pairing, but not the ring structure.
Theorem 2.1 (LLW 2010). F induces an isomorphism of quantum rings QH(X) ∼= QH(X0)under analytic continuations in the K¨ahler moduli formally defined by
GLUEING QUANTUM D MODULES OVER FLOPS 3 Let γ,`be the fiber line class in X → Z →S. ThenF γ = γ0+ `0, butF` =
−`0 6∈ NE(X0). So analytic continuations are necessary. Any β∈ A
1(X)is of the
form β = βS+d` +d2γ where βS ∈ A1(S)is identified with its canonical lift in
A1(Z)with(βS.h) =0= (βS.ξ). h, ξ are dual to`, γ hence β.h=d, β.ξ=d2. Lemma 2.2(Minimal lift). Given a primitive βS ∈NE(S), β∈ NE(X)if and only if
d≥ −µ and d2≥ −ν,
where µ=maxi{(βS.Li)}, µ0=maxi{(βS.Li0)}, and ν=max{µ+µ0, 0}.
For general βS, the above numerical condition defines NEI(X). The minimal
one βIS is called the I-minimal lift. Back to ¯p : X → S where D = t1h+t2ξ and
¯t∈H(S). IβX/S =IβZ/SIβX/Zis given by r
∏
i=0 1 β.ai ∏ m=1 (ai+mz) r∏
i=0 1 β.bi ∏ m=1 (bi+mz) 1 β.ξ ∏ m=1 (ξ+mz) .Although IβX/Smakes sense for any β∈N1(X), it is non-trivial only if β∈ NEI(X). Proposition 2.3(Picard–Fuchs system on X/S). `IX =0 andγIX =0, where
` = r
∏
j=0 z∂aj−q `et1∏
r j=0 z∂bj, γ=z∂ξ r∏
j=0 z∂bj−q γet2. Proposition 2.4(F -invariance of PF ideal).FhX `,Xγi ∼= h X0 `0,X 0 γ0i.
Theorem 2.5(Quantum Leray–Hirsch). (1) (I-Lifting) The QDE on QH(S)can be lifted to H(X)as
z∂iz∂jI=
∑
k, ¯βq¯βIe(D. ¯βI)C¯kij, ¯β(¯t)z∂kD¯βI(z)I,
where D¯βI(z)is an operator depending only on ¯βI. Any other lifting is related to
it modulo the Picard–Fuchs system.
(2) Together with the Picard–Fuchs`andγ, they determine a first order matrix system under the naive quantization basis:
z∂a(∂zeI) = (∂zeI)Ca(z, q), where ta=t1, t2or ¯ti.
(3) For ¯β∈NE(S), its coefficients in Caare polynomial in qγet
2
, q`et1 and f(q`et1), and formal in ¯t. Here f(q) := q/(1− (−1)r+1q)is the “origin of analytic
con-tinuation” satisfying f(q) +f(q−1) = (−1)r.
(4) The system isF -invariant, though in general F ¯βI6= ¯βI0.
Finally we construct a gauge transformation B to eliminate all the z dependence of Cain theF -invariant system z∂a(∂zeI) = (∂zeI)Ca. B is nothing more than the
Birkhoff factorization matrix in ∂zeI(ˆt) = (z∇J)(τ)B(τ)valid at the generalized
mirror point τ = τ(ˆt). The aboveF -invariance leads to F τ = τ0 andF B(τ) =
B0(τ0), hence the glueing of Dubrovin connections under analytic continuations. C.-L. WANG: TAIDAINSTITUTE OFMATHEMATICALSCIENCES(TIMS), NATIONALTAIWANUNI -VERSITY, TAIPEI10617, TAIWAN
