Quantum Flips

Quantum Flips

Chin-Lung Wang National Taiwan University (A joint project with Y.-P. Lee and H.-W. Lin)

The 8th ICCM, Beijing June 11, 2019

Contents

1 What is quantum cohomology?

I Example: a toric bundle

2 Quantum motives? The functoriality problem

3 Statement of results on simple flips f : X99KX⁰ Sketch of proof in 3 steps:

4 (i) Irregular singularity of QH(X)along vanishing cycles 5 (ii) Block diagonalizations and BF/GMT over NE(X⁰) 6 (iii) The non-linear F-embedding QH(X⁰) ,→QH(X)

I Example:(2, 1)flips

1. What is Quantum Cohomology?

A: Deformation of (

(

)

∪) by rational curves.

I Let X/C be a projective manifold, Mn(X, β)be the moduli space of stable maps

f :(C, p1, . . . , pn) →X

from n-pointed rational nodal curves to X with image class β∈NE(X), the Mori cone of effective 1-cycles.

I For i∈ [1, n], let ei : Mn(X, β) →X be the evaluation map e_i(f)_:=f(p_i) ∈X.

I Let t∈H=H(X). The g=0 Gromov–Witten potential

F(t) = hh−ii(t):=

∑

n, β

q^β

n!ht^⊗ni^X_n,β

=

∑

n≥0, β∈NE(X)

q^β n!

Z

[Mn(X,β)]^vir

∏

e^∗_it

is a formal function in t and q^β’s (Novikov variables).

I We callR :=C[[q^•]]the (formal) K¨ahler moduli and denote H_R =_H⊗R.

I Let{Tµ}be a basis of H and{T^µ:=_{∑ g}^µνTν}the dual basis with respect to the Poincar´e pairing

gµν= (Tµ.Tν), (g^µν) = (gµν)⁻¹.

I Let t=_{∑ t}^µTµ. The big quantum ring(QH(X),∗)is a t-family of rings QtH(X) = (TtH_R,∗_t):

Tµ∗_tTν:=

∑

e,κ

∂_µ∂_ν∂_eF(t)g^eκTκ ≡

∑

=

∑

e,κ

hhTµ, Tν, Teii(t)g^eκTκ

=

∑

κ, n≥0, β∈NE(X)

q^β

n!hT_µ, T_ν, T^κ, t^⊗ni^X_{n+3, β}T_κ.

I The WDVV associativity equations equip(H_R, g_µν, F_ijk, T0=1)a structure of formal Frobenius manifold overR.

I It is equivalent to the flatness of the Dubrovin connection

∇^z=d−¹

zA :=d−¹ z

∑

µ

dt^µ⊗T_µ∗_t

on the formal relative tangent bundle TH_R for all z∈_C^×:

∂_µA_ν=∂_νA_µ, [A_µ, A_ν] =0, I where the (connection) matrix A_µfor z∇^z_µis z-free:

A_µ(_t) =T_µ∗_t.

I This z-free property uniquely characterizes the constant frame {Tµ}among all frames{˜T_µ}with

˜T_µ(q^•, t, z) ≡T_µ (mod R).

I Let ψ=c1(p^∗₁ω_{C /Mn})be the class of cotangent line at the first marked section p1: Mn →C of C →Mn, then

J(t, z⁻¹):=1+ ^t z+

∑

β,n,µ

q^β n!T_µ

 T^µ z(z−ψ)^{, t}

⊗nX n+1,β

encodes invariants with one descendent insertion.

I The topological recursion relation (TRR):

hhτ_d+1T_i, T_j, T_kii =

∑

z∂µz∂νJ=

∑

I LetD^zbe the ring of differential operators generated by z∂_iwith coefficients inO=_C[z][[q^•, t]]. TheD^z-moduleO^{dim H}associated to z∂_i 7→z∇^z_i is isomorphic to the cyclicD^z-moduleD^zJ.

I In practice, one might be able to find element I(ˆt, z, z⁻¹) ∈_D^zJ(t, z⁻¹)

but only along some restricted variables ˆt∈H1⊂H.

I If H1generates H (either in classical product or quantum product), then often one may compute J(t, z⁻¹)and∇^z.

I For a toric manifold X, such an I function can be found through theC^×-localization data with ˆt∈H^≤2(X)_.

I [Lian–Liu–Yau 1996, Givental 1996] For c₁(X) ≥0, I(ˆt, z⁻¹)can be found and J(ˆt, z⁻¹)is obtained by a mirror transform.

I [Coates–Givental 2005, Iritani 2008, Brown 2010] I(ˆt, z, z⁻¹)is found for all toric manifolds. However, the structures and computations are far more complicated. Need BF/GMT:

Birkhoff Fatcorizations+Generalized Mirror Transform.

Example: a Fano toric bundle

=

(

(−

) ⊕

) −→

c₁

(

) =

+

>

(

) = _C [

]

(

(

−

))

Let

`

(

) = _Z ` + _Zγ.

QH

(

) =

I {T0, T1, T2, T3} = {1, h, ξ,ξ²},

ˆt=t⁰T₀+D, D=t¹h+t²ξ∈H².

I Let q1=q^`e^t1and q2:=q^γe^t2 (small parameters), then

I(ˆt, z⁻¹):=e^t0T0z

∑

β=d1`+d₂γ

q^βe^Dz+(D.β)I_β =e^ˆtz

∑

q^d₁¹q^d₂²I_d1,d2,

I_d1,d2 := ¹

d₁ m=1∏

(h+mz)²

d₂−d₁ m=1∏

(ξ−h+mz)

d₂ m=1∏

(ξ+mz)

=O(z⁻²).

I [LLY, Givental]=⇒I(ˆt, z⁻¹) =J(ˆt, z⁻¹).However, t³is missing.

I In general, if c₁(X).β<0 for some β, then the z power→ +_∞.

I Technique: use Naive Quantization to replace z∂3J: e.g.

Tb_iI=z∂_iI, i=_{0, 1, 2,} Tb₃I=ξb²I := (z∂₂)²I.

I In general, since I∈D^zJ, we get bT_iI∈D^zJ too. Hence Tb_iI

(ˆt, z, z⁻¹) =z∇J(σ(ˆt), z⁻¹)B(ˆt, z). I The unique gauge transform is called the BF. It implies

J(σ(ˆt)_{, z}⁻¹) =z∂₀J=

∑

I The z⁻¹coefficient of PI gives the GMT: ˆt7→σ(ˆt) ∈H_R.

I In practice, we study B, σ(ˆt)via the Picard–Fuchs equations of I:

` = (z∂1)²−q1(z∂2−z∂1),

γ = (z∂2−z∂1)z∂2−q2.

I This leads to the connection matrix in the frame bTiI:

z∂a(Tb_iI) = (Tb_iI)Ca(ˆt, z), a=1, 2.

I In this example the choice of{Tb_iI}leads to

C₁=









−q₂ q₁q₂

1 −q₁ q₂

q₁ 1







, C₂=









−q₂ q₁q₂+zq₂ q₂

1 q₂

1 1







.

I B(ˆt, z) =I₄+q₂e₀₃( bξ²7→hξ) removes the z-dependence:b

C˜2(ˆt) = −(_z∂2B)_B⁻¹+_BC2(ˆt, z)_B⁻¹=









q₂ q₁q₂ q₂ 1

1 1







 .

I The first column=⇒σ(ˆt) =ˆt. In general ˜C=σ^∗A: i.e.

C˜a(ˆt) =

∑

µ

A_µ(σ(ˆt))^∂σ

µ

∂t^a.

2. Quantum Motives? The Functoriality Problem

Q: Which part of the structure on QH (

) is functorial?

I M_k: the category of Chow motives, k the ground field.

I Objects: ˆX, where X a smooth variety over k.

I Morphisms are correspondences

Γ∈Mor(X, ˆ^ˆ X⁰)_:=A(X×X⁰)_.

I Induced map on Chow groups:[Γ]∗ : A(X) →A(X⁰):

α7→π_∗⁰(_Γ.π^∗α).

I Linear structures: if ˆX∼=X^ˆ0then Aⁱ(X) ∼=Aⁱ(X⁰)for all i. If k is a number field, X and X⁰have the same L functions for each i.

I However, the ring structures are different: A(X) 6∼=A(X⁰)_!

I [Wang 2002] Is there a universal product structure defined on Chow motives? Namely a universal family(_{A ,}∗) →T such that all geometric realizations(A(X),•)correspond to special points.

I Typical examples come from ordinary(r, r⁰)-flops/flips:

E=Z×_SZ⁰⊂Y

φ

uu

φ⁰

))

P¹∼= ` ⊂Z⊂X ^f //

ψ ))

X⁰⊃Z⁰⊃ `⁰ ∼=P¹

ψ⁰

uuS⊂X

I ψ¯ : Z=P_S(F) →S, rk F=r+1, ψ-extremal ray` = [C]. I NZ/X|_ψ¯−1(s)∼=OP^r(−1)^⊕(r0+1)for all s∈S.

I Y=BlZX=Bl_Z⁰X⁰, KY=φ^∗KX+r⁰E=φ^0∗K_X⁰+rE. Hence φ^∗KX =φ^0∗KX⁰+ (r−r⁰)E.

I For flops r=r⁰, we have K-equivalence and ˆX∼=X^ˆ0via Φ := [Γf]∗=φ_∗⁰ ◦φ^∗ : H(X)−→^∼ H(X⁰).

I It preserves the Poincar´e pairing

(_Φa.Φb)^X0 = (φ^0∗Φa.φ^∗b)^Y= ((φ^∗a+ξ).φ^∗b)^Y= (a.b)^X, but NOT the cup product!

I For the simple case (S=_{pt), let αi}∈H^2li(X)_,∑³_i=1l_i=_{dim X,} (_{Φα1.Φα2.Φα3})^X0 = (α₁.α2.α3)^X−

∏

I Solution: use quantum product(QtH,∗_t)instead.

I The effectivity of extremal curve is not preserved:

Φ` = −`⁰ 6∈NE(X⁰).

I It is necessary to consider analytic continuationsQH(X)of QH(X)along the K¨ahler moduli via the partial compactification

Φq^β=q^Φβ toward “q^`=_∞”.

I For flops, the functoriality is simply the canonical isomorphism Φ : QH(X)−→^∼ QH(X⁰).

I In terms of Gromov–Witten invariants: for t∈H(_X)_, ΦhhT_i, T_j, T_kii^X(_t) = hh_ΦTi,ΦTj,ΦTkii^X0(_Φt).

I [Li–Ruan] for 3-folds, [LLW, LLQW] for general ordinary flops.

I The simplest non K-equivalent birational maps preserving the dimension of K¨ahler moduli are smooth ordinary flips.

I Pseudo-abelian completion of Chow motives fM: objects(X, p^ˆ ), where p∈End(X^ˆ) =A(X×X)is a projector: p²=p. Then

Xˆ ≡ (X, 1^ˆ ) = (X, p^ˆ ) ⊕ (X, 1^ˆ −p).

I For flips with r>r⁰,Ψ := [_Γf−1]induces a sub-motive Ψ : ˆX⁰−→(^∼ X, p^ˆ ), p :=_Ψ◦_Φ.

I On cohomology

Ψ : H(X⁰) ,−→H(X),

the Poincar´e pairing is still preserved(_Ψa.Ψb)^X= (a.b)^X0, but not the cup product. Not even the quantum product!

I Solutions?

3. Statements of Results for Simple Flips

99K

I We would like to show that QH(X⁰)can still be regarded as a sub-theory of QH(X)in a canonical, though non-linear, manner.

I First of all, there is a basic split exact sequence 0 // K // H(X) ^Φ // H(X⁰)

ll Ψ // 0 .

I The kernel space (vanishing cycles) K has dimension d :=r−r⁰ and is orthogonal toΨH(X⁰)_:

K=^Mr

j=r⁰+1C[P^j].

I Secondly, the Dubrovin connection∇can be analytically continued along the K¨ahler moduli to a connectionΦ∇under the rule

Φq^β =q^Φβ, β∈NE(X).

I As beforeΦ` = −`⁰and analytic continuations are required.

I We use identification of divisorial coordinates tⁱand Novikov variables q^βi(divisor axiom): let D=_{∑ t}ⁱDi,(Di.βj) =δ_ij,

q_i:=q^βie^ti, ∂_i= ^∂

∂tⁱ

=q_i ∂

∂q_i. I Hence

∇_µ=∂_µ−¹ zT_µ∗ has only (formal) regular singularities at q_i =0.

I The resulting connectionΦ∇turns out to be analytic in the extremal ray variable q^{`</sup>and contains irregular singularities in the K directions along q<sup>`}=∞, that is q^`0 =0.

I This suggests to extract the Dubrovin connection∇⁰on TH⁰_R0, where H⁰ =H(X⁰)andR⁰=_C[[NE(X⁰)]], fromΦ∇

by removing the K directions

— since∇⁰is (formlly) regular.

I We will show that there is a bundle-decomposition

TH⊗_R⁰[1/q^{`0</sup>] =<sub>T</sub> ⊕<sub>K</sub> (∗) into irregular eigenbundleK which extends K over R<sup>0</sup>[1/q<sup>`0}] and the regular eigenbundleT =K^⊥.

I From WDVV equations, bothT and K are shown to be integrable distributions.

I The integrable submanifold passing through the section M_q0 ⊃ {(q⁰6=0, t=0)}

is then the proposed manifold corresponding to QH(X⁰). I However, to relateT , and henceM_q0, to QH(X⁰), we need to

work on the connection (z-dependent) version of (∗).

I Hence there are non-trivial BF/GMT involved, and it is unclear what kind of functoriality should exist.

I The end result turns out to be quite satisfactory — the product structure is preserved but not the metric (Poincar´e pairing)!

Theorem (Lee–Lin–Wang, 2017)

For thelocal modelf : X99KX⁰of simple(r, r⁰)flips, there is a unique R⁰-point σ0(q⁰) ∈H⁰_R0and a unique embedding bΨ(q⁰, s)overR⁰:

Ψ : Hb (X⁰)_R0−→ M ,−→H(X)_R0, σ₀(q⁰) +s7−→Ψb(q⁰, s).

where s∈H(X⁰), such that

(1) (Ψ, σb 0)restricts to(_{Ψ : H}⁰ ,−→_{H, 0})when modulo q^{`0</sup>, (2) Ψ induces an F-embedding over Rb <sup>0</sup>[1/q<sup>`0}]:

(TH⁰_R0

[1/q^{`0</sup>],∇<sup>0</sup>)<sup>  d bΨ //</sup>(TH<sub>R</sub>0[1/q<sup>`0}],∇)|_{M // K} ∼=N_Ψb .

I In particular, outside the divisor q^`0=0, the big quantum products on the corresponding tangent spaces are preserved.

I Denote the tangent frame by bΨi=∂_iΨ and the induced metric byb gij= (Ψbi, bΨj), Ψbⁱ :=

∑

I Then bΨ is an F-embedding:

hhΨbµ, bΨⁱ, bΨjii^X(Ψb(q⁰, s)) = hhT⁰_µ, T⁰ⁱ, T_j⁰ii^X0(σ₀(q⁰) +s).

I Hence there is a family of ring isomorphisms/decompositions:

QΨ(qb ⁰,s)H(X) ∼=Q_σ0(q⁰_)+sH(X⁰) ×_C^r−r0, which depend on the points(q⁰, s).

4. STEP (i)

Irregular Singularity of QH (

) along Vanishing Cycles

I Small parameters ˆt=t⁰T0+D∈H^≤2(X), ˆs=s⁰T⁰₀+D⁰. D=t¹h+t²ξ=_ΨD⁰ =_Ψ(s¹h⁰+s²ξ⁰) =s¹(ξ−h) +s²ξ.

s¹= −t¹, s²=t²+t¹.

I K¨ahler moduli: NE(X) =_Z` ⊕<sub>Zγ, NE</sub>(X<sup>0</sup>) =<sub>Z</sub>`⁰⊕_Zγ⁰. Φ` = −`⁰, Φγ=γ⁰+ `⁰,

q₁=q^`e^t1, q₂=q^γe^t2,

x=q₁⁰ =q^`0e^s1 =1/q₁, y=q⁰₂=q^γ0e^s2 =q₁q2.

I Naive quantization, for i∈ [0, r], j∈ [0, r⁰+1], a=hⁱξ^j, ˆa≡∂^za:= ˆhⁱξˆ^j= (z∂_h)ⁱ(z∂_ξ)^j= (z∂1)ⁱ(z∂2)^j. I X is Fano, c₁(X) = (r−r⁰)h+ (r⁰+2)ξis ample,

I X⁰is bad, c₁(X⁰) = (r⁰−r)h⁰+ (r+2)ξ⁰has no fixed sign.

I For β=d₁` +d₂γ∈NE(X),

I_β = ¹

∏^d_m=1¹ (h+mz)^r+1_∏^d_m=1^2−d1(ξ−h+mz)^r0+1_∏^d_m=1² (ξ+mz)

I I=e^ˆt/z∑βe^D.βq^βI_βis annihilated by Picard–Fuchs equations:

`= (z∂_h)^r+1−q₁(z∂_ξ−h)^r0+1_,

γ=z∂_ξ(z∂_ξ−h)^r0+1−q2.

I I=I(z⁻¹) =⇒I=Jsmalland Q0H(X)is “easy”. It is still non-trivial to write down the Dubrovin connection∇^X. I The naive frame, for e=hⁱξ^j(or even hⁱ(ξ−h)^j_),

∂^zeI≡ˆhⁱξˆ^jI := (z∂_h)ⁱ(z∂_ξ)^jI

does notlead to z-free connection matrices for z∂1, z∂2!

Example: the case of(2, 1)flips.

I For the naive frame respecting H(X) =_ΨH(X⁰) ⊕^⊥K, with v6= ˆκ0= (ξ^ˆ−ˆh)², we have

z∂₁(∂^zeI) =z q₁ ∂

∂q₁(∂^zeI) = (∂^zeI)C₁(q, z)_,

C1(q, z) =





























q1q2 −zq1q2

1 −q1q2

q1q2

1 1

1 −1 q₁ −zq₁ z²q₁ 1

−1 1 1 −q1 2zq1

1 q1



























 .

I It is even unclearwhere the irregular singularities at q₁=_{∞ are} located. (Not just in the K directions?)

TheΨ-corrected quantum frame

I The quantized basis corresponding to kerΦ is chosen to be ˆκ_iI= _ˆhⁱ(ξ^ˆ−_ˆh)^r0+1I, i∈ [0, r−r⁰−1]. I For e1∈ [0, r+1], e2∈ [0, r⁰], we define

ve:= _ˆh^e1(ξ^ˆ−ˆh)^e2I+δ_{(e1, e2)}(−₁)^r0−e2ˆκ_e1+e2−(r0+1),

where (

δ_{(e1, e2)}=0 if e₁+e₂∈ [0, r⁰], and δ_{(e1, e2)}=1 otherwise.

I The added term comes from kerΦ⇐⇒e₁+e₂∈ [r⁰+_{1, r}]_.

I But H^2j(X⁰)with j≥r+1 are also corrected accordingly.

I The frame reduces to a classical basis when modulo NE(X).

The connection matrices for z∂1and z∂2.

I For i=1, 2, the connection matrix C_i(q1, q2)in theΨ corrected frame is independent of z. Moreover, Ai(ˆt) =Ci.

I Write C_i=^C

11 i C¹²_i C²¹_i C²²_i



wrt. H(X) =_ΨH(X⁰) ⊕^⊥K.

I Let d=dim K=r−r⁰.

I For C1, the d×d block corresponding to kerΦ is given by

C²²₁ =











(−1)^r0+1q1

1 . ..

1









 .

I Other entries in C1and C2have “good properties”!

I Corollary 1.TheΨ-corrected frame corresponds to the constant frame for∇^X.

I Corollary 2.Under the analytic continuation in the K¨ahler moduli over NE(X⁰),∇^Xis irregular in the divisor(x=0) precisely in the kernel block.

I To proceed, we denote

R=dim H(X) = (r+1)(r⁰+2), R⁰=dim H(X⁰) = (r+2)(r⁰+1). And then d=R−R⁰ =r−r⁰ =dim K.

5. STEP (ii)

Block Diagonalizations and BF/GMT over NE (

)

I We have Aj(ˆt) =Cj, j=1, 2:

C²²₁ =











0 0 · · · (−1)^r0+1q₁ 1 0 · · · 0

. ..

0 · · · 1 0











= ¹ x











0 0 · · · (−1)^r0+1 x 0 · · · 0

. ..

0 · · · x 0









 .

I We will now work on the irregular system of PDE in variables (x, y)with a parameter z.

I The irregularity comes only from x, and it is thus necessary to keep track of the lowest order entries in x in Cj’s.

I A transformation is needed to bring C²²₁ into its “semisimple”

form:let u=x^1/d, we modify the constant frame to{Ti}with {T_i}^R_i=0⁰⁻¹= {Te}, {T_R⁰_+i}^d−1_i=0 = {uⁱk_i}^d−1_i=0.

Lemma on shearing (=base change inD-modules).

I Let Y(x) =diag(1^R0, u⁰, u¹,· · ·, u^d−1). After substitutions S=YW and x=u^d, the equation zx_∂x^∂S=C₁S becomes

zu ∂

∂uW=D₁(u, z)W, (∗∗) D¹¹₁ =d·C¹¹₁ ,

D¹²₁ =d·C¹²₁ ·diag(u⁰, u¹,· · ·, u^d−1), D²¹₁ =d·diag(u⁰, u⁻¹, . . . , u^−d+1) ·C²¹₁ ,

D²²₁ = ^d u ·











0 0 · · · (−1)^r0+1 1 −z¹_du · · · 0

. .. . ..

0 · · · 1 −z^d−1_d u









 .

I D²¹₁ is polynomial in u.Thus, (∗∗) is irregular of Poincar´e rank 1 in u, and the irregular part only appears in the(2, 2)block D²²₁ .

I Therefore, D1(z=0)has R eigenvalues, including 0^R0and d distinct nonzero eigenvalues from D²²₁ (0)as solutions to

ω^d= (−1)^r0+1.

I By theclassical proceduredue to Wasow/Shibuya, together with theflatnessof the Dubrovin connection, we conclude that (i) The connection matrices C1, C2can be simultaneously block

diagonalized to ˜C1, ˜C2, such that the(2, 2)blocks are diagonalized.

(ii) Furthermore, the block-diagonalization frame (gauge matrix) P= [˜T0, . . . , ˜T_R⁰₋₁, ˜T_R⁰, . . . , ˜TR−1] =^IR0 ∗

∗ I_d



can be chosen so that ˜Tihas the initial term Tiin u.

(iii) T spanned by ˜T0, . . . , ˜TR⁰−1andK spanned by ˜TR⁰, . . . , ˜TR−1

lead to reduction of connection and are orthogonal to each other.

I Extract QH(X⁰)_{from QH}(X)_{:On X}⁰_{, let β}⁰=d⁰₁`⁰+d⁰₂γ⁰, then

I^X_β0⁰= ¹

∏^d₁⁰¹(h⁰+mz)^r0+1_∏1^d02−d01(ξ⁰−h⁰+mz)^r+1_∏^d₁⁰²(ξ⁰+mz) .

I It has Picard–Fuchs equations

`⁰:= (z∂₂−z∂₁)^r0+1−q⁰₁(z∂₁)^r+1_,

_γ⁰:= (z∂2)(z∂1)^r+1−q⁰₂.

I Since`<sup>0</sup> =q<sup>−1</sup><sub>1</sub> `andγ⁰ =z∂2`−q1γ, we get the I Key Lemma.OverC[q1, q⁻¹₁ , q2] ∼=C[q⁰₁, q⁰₁⁻¹, q⁰₂], we have

h`,γi ∼= h`⁰,γ⁰i.

I Corollary.The matrices ˜C¹¹₁ , ˜C¹¹₂ can be used to compute∇^X0.

I For a, b∈H(X)we have ab=a∗b+_∑βq^βc_βfor some c_β∈H(X). By induction on the Mori cone we conclude that

Tµ∗ =

∑

β∈NE(X)

q^βP_β(h∗, ξ∗)

where P_βis a polynomial. Since X is Fano, the sum is finite.

I So the block diagonalization in u=x^1/d, y, z extends to all Tµ∗. I In fact ˜C¹¹₁ and ˜C¹¹₂ , henceall ˜C¹¹_µ, are expressible in x, y, z.

I Two technical problems:

(i) Remove the NEW z-dependence in ˜C¹¹_µ(x, y, z)introduced in the block-diagonalization.(Sol. BF/GMT.)

(ii) Since Tµ∗is generated by h∗and ξ∗over NE(X)instead of over NE(X⁰),will ˜C¹¹_µ(x, y, z)contain negative powers in x?(Sol. No!)

(i) Let B₁=B₁(x, y, z)be the BF matrix and B₁(0):=B₁(x, y, 0). [T0, . . . , T_R⁰₋₁]:=^[˜T0, . . . , ˜T_R⁰₋₁]B⁻¹₁ 

(z=0).

I Under x=q^`0e^s1, y=q^γ0e^s2, a=0, 1, 2, the “z-free” matrix C⁰_a(_ˆs) = −(z∂aB₁)B⁻¹₁ +B₁C˜¹¹_a B⁻¹₁ =B₁(₀)C^˜11_a;0B₁(₀)⁻¹(x, y) is related to A⁰_µ(σ)for T⁰_µ∗⁰atσ=σ(ˆs) ∈H(X⁰)[[x, y]]via

C⁰_a(_ˆs) =

∑

µ

∂s^a(_ˆs)_, a=_{0, 1, 2,} hhTa, Tj, Tⁱii^X(ˆs) =

∑

∂σ^µ

∂s^a(ˆs)hhT⁰_µ, T⁰_j, T⁰ⁱii^X0(σ(ˆs)).

I Since(A⁰_µ)ⁱ₀=δⁱ_µ, σ(ˆs)is determined by the first column:

(C_a⁰)^µ₀(ˆs) = hhTa, T0, T^µii^X(ˆs) = ^∂σ

µ

∂s^a(ˆs).

6. STEP (iii)

The Non-Linear F-Embedding QH (

) ,→

(

)

(ii) The next step is totransform T0to the identity element (section) e∈T and normalized Ti’s toT˜i’saccordingly.

I Lemma.There is a unique element S0∈T such that S0∗T0=e,

and so e acts as zero onK .(This requires delicate calculations!) I Define the normalized frame onT by

Teµ:=Tµ∗S0.

I

Theorem (Initial quantum invariance up to a shifting)

LetTi(q⁰) =Tei(q⁰, ˆs=0, z=0)and σ₀(q⁰) =σ(q⁰, ˆs=0). Then we have h_Tµ,Tⁱ,Tji^X= hhT⁰_µ, T⁰ⁱ, T_j⁰ii^X0(σ₀(q⁰)).

I An F-manifold M is a complex manifold with a commutative product structure on each TpM, such that a WDVV-type integrability condition is forced when p∈M varies.

I In QH(X), this is the structure which remembers∗_pbut forgets the metric gij. Hertling and Manin showed that the WDVV equations can be rewritten as

LX∗Y∗ =X∗LY∗ +Y∗LX∗ for any local vector fields X and Y.

I I.e., for any local vector fields X, Y, Z, W:

[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.

I Denote byKthe irregular eigenbundle andT := K^⊥the regular eigenbundle, which extendK and T from s=0 to big s.

I

Lemma

T is an integrable distribution of the relative tangent bundle TH_R⁰. In particular, Im bΨ is the integral submanifoldM(overR⁰) containing the slice(q^`06=0, t=0)which contains ImΨ when modulo R⁰.

I

Proof.

Let X, Z be any local vector fields inT = K^⊥. Let Y=eiand W=ej

be idempotents inK. Since a∗b=0 for a∈ K, b∈ K^⊥, 0= −X∗Z∗ [ei, ej] −δijej∗ [X, Z]. Let i=j we get e_j∗ [X, Z] =0 for all j. Hence[X, Z] ∈ K^⊥.

I The quantum product on theFrobenius manifoldH(X⁰) ⊗R⁰is semi-simple. Let v₀⁰, . . . , v⁰_R0−1be the idempotent vector fields.

I Dubrovin 1996:[v⁰_i, v⁰_j] =0 for all 0≤i, j≤R⁰−1. Hence the corresponding canonical coordinates u⁰⁰, . . . , u^0R0−1satisfying

(u⁰ⁱ(q⁰, s=0)) =σ₀(q⁰) and v⁰_i=_∂/∂u⁰ⁱ_exist.

I This was extended toF-manifoldsby Hertling. The F-manifold Mis semi-simple in the sense that∗_pon TpMfor p∈ Mis semi-simple. Denote the idempotent vector fields by v₁. . . . , v_R⁰. I Hertling 2002: [v_i, v_j] =0 for all 0≤i, j≤R⁰−1. Hence the

canonical coordinates u⁰, . . . , u^R0−1near each p∈ Mexist in the sense that v_i=∂/∂uⁱ.

I Fixing the initial correspondence of frames:

I We have constructed an analytic family of coordinate systems (u⁰(q⁰, p), . . . , u^R0−1(q⁰, p))parametrized by q⁰∈R⁰. Write

Ti(q⁰) =

∑

I

hTµ,Tⁱ,Tji^X= hhT_µ⁰, T⁰ⁱ, T_j⁰ii^X0(σ₀(q⁰))_{. (1)} From this relation, we see easily that:

I

Lemma

After a possible reordering of{v_j⁰}, we have for all i=0, . . . , R⁰−1:

T_i⁰=

∑

I Now we define the map ˆΨ by matching the canonical coordinates.

Namely, ˆΨ(q⁰, s) ∈ Mis the unique point onMso that uⁱ(_Ψ^ˆ(q⁰, s)) =u⁰ⁱ(q⁰, s) =u⁰ⁱ(σ₀(q⁰) +_s) for i=0, . . . , R⁰−1.

I Since the tangent map ˆΨ∗matches the idempotents Ψˆ∗∂/∂u⁰ⁱ=∂/∂uⁱ,

it induces a product structure isomorphism, and hence an F-structure isomorphism by “coordinates-free WDVV”.

I Also along s=0, by Lemma we have Ψˆ∗T_i⁰=Ti

which matches the initial condition along theR⁰-axis.

I H(X⁰)is contractible=⇒Ψ exists globally.^ˆ QED

Ending Remarks

I Work in progress by LLW:

(1) Globalization to simple(r, r⁰)flips.

(2) Generalizations to ordinary flips with non-trivial base.

(3) Reconstruction of QH(X)_{from QH}(X⁰)and “the K-block”.

I Other approaches to quantum flips:

(4) [Woodward et. al.] studying wall crossing of GW invariants in different GIT quotients.

(5) [Shoemaker et. al] studying asymptotic of I functions in the toric setup.

I Would be interesting to compare their approaches with ours.

Example:(_{2, 1})_flip

R=9, R⁰ =8. The following frame (recall I=J_small) v1= ˆ1J=J,

v₂= ˆhJ, v₃= (ξ^ˆ−ˆh)J,

v4= ˆh²J− (ξ^ˆ−ˆh)²J, v5= ˆh(ξ^ˆ−ˆh)J+ (ξ^ˆ−ˆh)²J, v6= ˆh³J−ˆh(ξ^ˆ−ˆh)²J, v7= ˆh²(ξ^ˆ−ˆh)J+ˆh(ξ^ˆ−ˆh)²J, v₈= ˆh³(ξ^ˆ−ˆh)J+ˆh²(ξ^ˆ−ˆh)²J,

v9= ˆκ₀J= (ξ^ˆ−ˆh)²J,

respects H(X) =_Φ⁻¹H(X⁰) ⊕^⊥K when modulo q₁, q2. They are precisely

z∂iJ at t∈H⁰⊕H², 1≤i≤9, and we get the Dubrovin connection:

A₁=h∗_small=





























q1q2

1

q₁q₂ 1

1

1 −1

1

−1 1

1 −1 q1



























 ,

A2=ξ∗_small=





























−q2 q2 q1q2 q2

1 −q2 q2

1 q1q2

1 q2

1 1 1

1 1

1 q₂



























 .

x :=q⁰₁=1/q₁, y :=q₂⁰ =q₁q2. Chain rule: y ∂y=xy ∂q₂ =∂2, and

x ∂x=x(−x⁻²∂_q1+y ∂q₂) = −∂₁+∂₂=∂_ξ−h. Further simplification: Let wi=_∑jvjTji

T :=



























 1

1

1

2 1

1

1

2 1

1

1

2 1

1 1



























 .

gij := (wi, wi)^X=δ_9,i+j, 1≤i, j≤8,

A1=































−¹₂xy xy xy

−¹₂xy xy

1 ¹₄xy −¹₂xy

xy

1 −¹₂xy

1

1 −¹₂

1

−¹₂ 1 xy −1/x





























 ,

A₂=





























−¹₂xy xy y xy

1 −¹₂xy xy

1

2 1

4xy −¹₂xy y

1 xy

1 1 −¹₂xy

1

1 1

1

2 1

xy



























 .

Block diagonalization w.r.t. H(X) =_Φ⁻¹H(X⁰) ⊕^⊥K (Wasow 1960’s) + flatness of∇^X=⇒

∃! formal gauge transformation S=PZ

P(x, y, z) =I+^{ 0 g}

•

f• 0



=











1 g₁

. .. ... 1 g₈ f₁ · · · f₈ 1









 ,

such that

z(x ∂x)Z=E1Z, z(y ∂y)Z=E2Z with E1, E2beingblock diagonalized. Also, for i⁰ :=9−i,

fi(x, y, z) = −¯g_i⁰ := −g9−i(x, y,−z). Get the deformed,(x, y, z)-dependent, frame

we_i=w_i+f_iˆκ0, 1≤i≤8, eˆκ0= ˆκ0+

∑

From

−z∂_kP+A_kP=PE_k, the block decomposition is equivalent to

 A¹¹_k +A¹²_k f• −z∂_kg^•+A¹¹_k g^•+A¹²_k

−z∂_kf•+A²¹_k +A_k²²f• A²¹_k g^•+A²²_k



=

 E¹¹_k g^•E²²_k f•E¹¹_k E²²_k

 .

In particular we get the equation for fi: z∂_kf_i=A²²_k f_i+ (A²¹_k )_i−

∑

= −^δk1

x f_i+ (A_k)_9i−

∑

f_j(A_k)_ji+f_j(A_k)_j9f_i . k=1: system of inhomogeneous non-linear perturbation of

zx ∂xh= −¹ xh.