Quantum Invariance under Flops and Transitions
Chin-Lung WangTaiwan University
August 29, 2008 at Harvard University In celebration of Professor Yau’s 60 birthday
In 1994, Yau suggested the study of finite distance boundary points of the moduli space of Calabi-Yau manifolds, with respect to the Weil-Petersson metric:
ωWP = −∂ ¯∂ log √ −1n Z Xs Ω(s) ∧ Ω(s).
Candelas et. al. 1990: Conifold (ODP) degenerations of Calabi-Yau 3-folds are at finite WP distance (by way of explicit calculations).
—, 1995; MRL 1997:
Schmid’s Nilpotent Orbit Theorem =⇒ A CY degeneration X → ∆ is at finite WP distance iff NF∞n = 0.
Clemens-Schmid exact sequence =⇒ For a semi-stable CY
degeneration X → ∆ with X0=
Sm
i =0Xi, NF∞n = 0 iff there is a
component X0 with hn,0≡ h0(K ) 6= 0.
Corollary: Degenerations of CY acquiring only canonical singularities are at finite WP distance.
—, MRL 2003: Assuming the MMP in dimension n + 1, then the converse holds in dimension n. In particular, for X → ∆ being a finite distance degeneration of CY 3-folds, there exists another
birational model X0→ ∆ such that X00 is a CY with at most
Interesting Geometries occur at finite WP distance: Extremal transitions: Y 7→ X : Y ψ W /o i/o ///oX⊃ Xt = X
where ψ is a crepant (K -equivalent) resolution and i is a smoothing of canonical singularities. Notice that there is a topology change from Y to X .
Flops: Different crepant resolutions Y and Y0 of W are related by
flops. hp,q(Y ) = hp,q(Y0), but they are not homotopy equivalent and the classical cohomology rings are not isomorphic.
K -equivalence: For birational projective complex d -dimensional manifolds f : X 99K X0, X =K X0 if φ∗KX = φ0∗KX0 for some
Y φ ~~~~ ~~~ φ0 A A A A A A A X f //X0
eg. birational Calabi-Yau’s or minimal models.
Conjecture: There existsF = [¯Γf] +P Ti ∈ Ad(X × X0) which
gives isomorphism of Chow motives [X ] ∼= [X0]. F is orthogonal
(preserving the Poincar´e pairing) and
F : QH(X ) ∼= QH(X0)
after an analytic continuation over the K¨ahler moduli. (In general
Gromov-Witten invariants: For α ∈ H(X )⊗n, β ∈ H2(X , Z)
hαig ,n,β =
Z
[Mg ,n(X ,β)]vir
ev∗α
with ev =Q ei : Mg ,n(X , β) → Xn being the evaluation map.
Big quantum ring: Let {Ti} be a basis of H(X ) and t =P tiTi,
Fg(t) := X n,β qβ n!ht ni g ,n,β.
The quantum product uses only g = 0. Let Φ = F0,
Ti∗tTj = X k Φijk(t)Tk = X k,n,β qβ n!hTi, Tj, Tk, t ni 0,n+3,βTk,
K¨ahler moduli: Let KC X = H
1,1
R (X ) × KX be the complexified
K¨ahler cone and let ω = B + iH ∈ KC
X. Then
qβ = e2πi (ω,β), |qβ| = e−2π(H.β) < 1.
It is conjectured that hαi =Phαiβqβ converges in ω ∈ KXC.
Analytic continuation: For X =K X0 and X 6∼= X0,
H2(X ) ∼= H2(X0) but K
X ∩ KX0 = ∅ in H2. If F preserves the
Poincar´e pairing, then F(Ti∗tTj) =FTi∗FtFTj is equivalent to
ΦXijk(ω, t) = ΦXijk0(Fω, Ft).
up to analytic continuations in ω from KC
X to KXC0. Since ω and
Fω are canonically identified and (ω, β)X = (Fω, Fβ)X0, formally
this means
Ordinary Pr flops: Let F , F0 be rank r bundles over S . It is a square E ¯ φ ~~|||||| || CCC ¯ φ0 !!C C C j // Y φ }}{{{{{{ {{ φ0 !!C C C C C C C C Z ¯ ψ A A A A A A A A i // X ψ B B B B B B B B Z0 |||¯ ψ0 ~~||| i0 // X0 ψ0 }}{{{{{{ {{ S j 0 //X
where Z = PS(F ), Z0= PS(F0) and E = Z ×S Z0. Moreover
NE /Y = ¯φ∗OZ(−1) ⊗ ¯φ0∗OZ0(−1),
NZ /X ∼=OZ(−1) ⊗ ¯ψ∗F0.
Theorem (Y.-P. Lee, H.-W. Lin, —; 2006–2008)
(1) For Pr flops f : X 99K X0, the graph closure F = [¯Γf] induces
canonical isomorphism of Chow motives.
(2) For simple Pr flops, the full Gromov-Witten theory in the
stable range 2g + n ≥ 3 can be analytic continued to each other under the graph correspondence.
(3) For Pr flops, the Gromov-Witten theory in the stable range
2g + n ≥ 3 attached the the extremal rays are invariant up to analytic continuations.
(4) For Pr flops with split bundles F =L Li and F0=L L0i, the
big quantum cohomology rings are analytic continuations of each other under the graph correspondence.
Genus zero theory: The Conjecture for 3-folds was previously solved by A. Li and Y. Ruan in 1998. 3 ingredients of their proof:
(1) Symplectic deformations and decompositions of K equivalent
maps into P1 flops. (Kawamata, Koll´ar, Friedman.)
(2) Multiple cover formula for P1 = C ⊂ X , NC /X =O(−1)⊕2:
h−iX0,dC = 1 d3.
(Aspinwall-Morrison, Voisin, Lian-Liu-Yau.)
Witten 1992: The defect of classical cup product is corrected by the 3-point functions on C .
(3) Relative GW invariants and the degeneration formula.
(Li-Ruan, Inoel-Parker, J. Li.) For β 6∈ Z[C ],
We make progresses on (2) and (3). The defect of product structure:
f : X 99K X0 a simple Pr flop, S = pt,F = [¯Γf],
h = hyperplane class of Z = Pr, h0 = hyperplane class of Z0, ` := [C ] = hr −1 line class in Z (extremal ray) etc.. Then
F[hs] = (−1)r −s[h0s
]. In particularF` = −`0.
Lemma. For α ∈ Ai(X ), β ∈ Aj(X ), γ ∈ Ak(X ) with i + j + k = dim X = 2r + 1,
Quantum corrections attached to the extremal rays: dim [Mg ,n(X , β)]virt = −(KX.β) + (dim X − 3)(1 − g ) + n.
Theorem. For all αi ∈ Ali(X ) with 1 ≤ li ≤ r and
Pn
i =1li = 2r + 1 + (n − 3), there are recursively determined
universal constants Nl1,...,ln, such that for n ≤ 3, N∗ ≡ 1 and
hα1, . . . , αni0,n,d = (−1)
(d −1)(r +1)N l1,...,lnd
n−3(α
1.hr −l1) · · · (αn.hr −ln).
Consider the basic geometric series f(q) := q
1 − (−1)r +1q. Then
3-Point functions (small quantum product): hα1, α2, α3i := X β∈NE (X )hα1, α2, α3i0,3,βq β = (α1.α2.α3) + X d ∈Nhα1, α2, α3id `q d `+X β6∈Z`hα1, α2, α3iβq β. Since (Fαi.h0(r −li)) = (−1)li(Fαi.Fhr −li) = (−1)li(αi.hr −li), hFα1,Fα2,Fα3i − hα1, α2, α3i = (−1)r(α1.hr −l1)(α2.hr −l2)(α3.hr −l3) + (α1.hr −l1)(α2.hr −l2)(α3.hr −l3)((−1)2r +1f(q`0) − f(q`)) = 0, modulo the 3rd (non-extremal) terms.
Unlike the r = 1 case, analytic continuations for the 3rd terms are needed!
Big quantum product: For n = 3 + k point extremal invariants with k ≥ 1, we get hα1, . . . , αni = Nl1,...,ln(α1.h r −l1) · · · (α n.hr −ln) q` d dq` k f(q`)
Since (−1)Pli = (−1)k+1, taking into account of
q−` d
dq−` = −q
` d
dq`
Sketch of proof:
The virtual fundamental class: [ ¯M0,n(X , d `)]virt is represented
by the Euler class of Ud = R1ft∗en+1∗ N, where N = NZ /X:
M0,n+1(Pr, d ) en+1 // ft Pr M0,n(Pr, d ) .
That is, [M0,n(X , d `)]virt = e(Ud) ∩ [M0,n(Pr, d `)] and
Z [ ¯M0,n(X ,d `)]virt ev∗α = Z ¯ M0,n(Pr,d ) ev∗(α| Pr).e(Ud).
The theorem is equivalent to Z ¯ M0,n(Pr,d ) e1∗hl1· · · e∗ nhln.e(Ud) = (−1)(d −1)(r +1)Nl1,...,lnd n−3.
Descendent invariants: Let Li be the line bundle on ¯Mg ,n(X , β)
whose fiber at (f ; C , (x1, . . . , xn)) is Tx∗iC . Let ψi = c1(Li).
D τk1(h l1), · · · , τ kn(h ln)E d = Z ¯ M0,n(Pr,d ) Yn i =1ψ ki i e ∗ i hli .e(Ud).
Step 1. One point invariants. For l + k = 2r − 1, 1 ≤ l ≤ r ,
D τkhl E d = (−1)d (r +1)+k dk+2 C k+1 r .
Consider a C× action on P1 with weight z. By the localization theorem and the work of Lian-Liu-Yau (1996, Mirror Principle I),
J(d `, z−1) ≡ e1∗
e(Ud)
z(z − ψ) = Pd ≡ (−1)
(d −1)(r +1) 1
(h + dz)r +1.
No mirror transformations are needed since r + 1 ≥ 2.
Step 2. Divisor relation for g = 0. [Lee-Pandharipande 2003] For L ∈ Pic(X ) and i 6= j ,
ei∗L = ej∗L + (β, L)ψj − X β1+β2=β
(β1, L)[Di ,β1|j,β2]
vir.
Also ψi + ψj = [Di |j]vir and for n ≥ 3, ψj = [Dj |ik]vir.
Deformations to the Normal Cone
X = X × A1
Φ : W →X is the blowing-up along Z × {0}
Wt ∼= X for all t 6= 0
W0 = Y ∪ ˜E with ˜E = PZ(NZ /X ⊕O)
φ = Φ|Y : Y → X is the blowing-up along Z
p = Φ|E˜ : ˜E → Z ⊂ X is the compactified normal bundle.
Y ∩ ˜E = E = PZ(NZ /X) is the φ − exceptional divisor
By similar constructions we also have Φ0 : W0 →X0= X0× A1 and
W00 = Y0∪ ˜E0. By definition of ordinary flips we have Y = Y0 and E = E0.
Representatives of Classes on W0
All cohomology classes α ∈ H∗(X , Z)⊕n are global and the
restriction α(t) on Wt is defined for all t.
Let j1 : Y ,→ W0, j2 : ˜E ,→ W0, j : E ,→ Y and j+: E ,→ ˜E .
The class α(0) can be represented by explicit data (j1∗α(0), j2∗α(0)) = (α1, α2)
such that
j∗α1 = j+∗α2 and φ∗α1+ p∗α2 = α.
Such representatives are not unique. For e being a class in E ,
Cohomology Reduction to Local Models
For a simple flop f : X 99K X0, let α ∩ Z 6= ∅ with representatives α(0) = (α1, α2) andFα(0) = (α01, α02).
If α1 = α01 then Fα2 = α 0 2.
Degeneration formula: 4(E ) =P
iSi ⊗ Si. hαiXβ =X I X η∈Ωβ Cηhα1; SIi(Y ,E )Γ1 hα2; S Ii( ˜E ,E ) Γ2 . LethαiX =P
βhαiXβqβ andFf (qβ) = f (qFβ) be the change of
variables.
To prove the functional equationFhαiX ∼= hFαiX0, it is enough to show that
Fhα2; SIi( ˜E ,E )
µ ∼= hFα2; SIi( ˜E
0,E )
Apply deformation to normal cone to ˜E , W0 = ˜Y ∪ ˜E , . the
degeneration formula (with descendent) implies that hα1, . . . , αn, τµ1−1Si1, . . . , τµρ−1Siρi ˜ E β = hα1, . . . , αn; Si1, . . . , Siρi ( ˜E ,E ) µ,β + X h· · · i( ˜Y ,E0)h∗ ∗ ∗i( ˜E ,E )
where ∗ ∗ ∗ is of lower order in cohomology degree and contact order. May apply induction.
So in order to proveFhαiX ∼= hFαiX0, it is enough to show that
Fhα1, . . . , αn, τk1Si1, . . . , τkρSiρi ˜ E ∼ = hFα1, . . . ,Fαn, τk1FSi1, . . . , τkρFSiρi ˜ E0
Setup on Local Models
The ordinary cohomology ring of ˜E = PPr(O(−1)
⊕(r +1)⊕O) is
given by
H∗( ˜E ) = Z[h, ξ]/(hr +1, (ξ − h)r +1ξ).
where h = c1(OPr(1)) and ξ = c1(OE˜(1)).
Since c1( ˜E ) = (r + 2)ξ is semi-positive,E is a semi-Fano toric˜
variety.
NE ( ˜E ) = R+` ⊕ R+γ with ` the line class in Z (= Pr) and γ the
fiber line class of ˜E → Z . Denote
β = d1` + d2γ.
The virtual dimension = c1( ˜E ).β + · · · = (r + 2)d2+ ·. Soevery
hαi =P
Two Important Special Cases
CASE I: d2 = 0. Then FhαiX ∼= hFαiX0 has been proved before by the generalized multiple cover formula.
CASE II: One-point descendent invariant for any d2 ∈ N.
By the theory of Euler data of Lian-Liu-Yau on semi-Fano smooth toric varieties we get (here no mirror transform is needed)
e∗v 1 z(z − ψ) = Pβ= 0 Y m=−∞ (ξ − h + mz)r +1 d1 Y m=1 (h + mz)r +1 d2−d1 Y m=−∞ (ξ − h + mz)r +1 d2 Y m=1 (ξ + mz) .
One-point Descendent Invariant of special type
Notation: Denote by X = ˜E , X0 = ˜E0. It is convenient to consider the generating series (Givental’s J function)
JX := X β∈NE (X ) qβe∗v 1 z(z − ψ) = 1 z2 X β∈NE (X ) qβX k≥0 e∗v ψk zk. Theorem
For any α ∈ H∗(X ), the one point function hτkξαiX satisfies the
functional equation (without analytic continuation):
F hτkξ.αiX = hτkF(ξ.α)iX0 =τkξ0.Fα X0
.
Equivalently,F is linear in Jξ:
Corrections for higher genus: Let dim X ≥ 3, ` ∈ NE (X ) with (KX.`) = 0, the virtual dimension of Mg ,n(X , d `) is given by
Dg ,n= (dim X − 3)(1 − g ) + n.
If ` is of flopping type, hαig ,n,d ` depends only on (Z , NZ /X) for
d ≥ 1. (But not for d = 0.). If Dg ,n < 0, all GW invariants vanish.
Genus one: If g = 1 then D1,n = n and each insertion is a divisor.
Hence if d ≥ 1 the n-point invariants are determined by
h−i1,d =
Z
[M1,0(X ,d `)]vir
1.
For d = 0 and n ≥ 2, the divisor axiom shows that hαi1,n,0= 0.
Indeed Mg ,n(X , 0) ∼= X × Mg ,n and
[Mg ,n(X , 0)]vir = e(E ) ∩ [X × Mg ,n]
where E = π1∗TX ⊗ π2∗Hg∨ with Hg the Hodge bundle. Let
λi = ci(Hg). For (g , n) = (1, 1), e(E ) = ctop(X ) − ctop−1(X ).λ1,
hαiX1,0 = −(ctop−1(X ).α)X · Z M1,1 λ1 = − 1 24(ctop−1(X ).α)X,
For simple Pr flops, we verify thatFhαiX1 = hFαiX10 by proving h−i1,d = (−1)d (r +1)r + 1
24d
and by calculating (c2r(X ).h) − (c2r(X0).Fh) in local models.
For dim X = 3 and g ≥ 2, Dg ,n= n. It is reduced to n = 0. For
simple P1 flop with d ≥ 1, Faber-Pandharipande 2000 showed
h−ig ,d := Z
[Mg ,0(X ,d `)]vir
1 = Cgd2g −3
where Cg = |χ(Mg)|/(2g − 3)!.
The generating function h−ig :=
∞
X
d =0
h−ig ,dqd = h−ig ,0+ Cgδ2g −3f,
is invariant underF since 2g − 3 ≥ 1. For h−iXg ,0= h−iXg ,00 , the degeneration analysis reduces the proof to local models, which are both isomorphic to PP1(O(−1)2⊕O).
Formal loop space: H+:=L∞k=0H zk = H[z]. Fg(t) is a
function onHt, t =Pµ,kt µ kTµz
k. The formal loop space over H
is
H := T∗H
+ = H[z, z−1]].
(H, Ω) is symplectic. Letb· be the Heisenberg quantization. Ancestor potential: In the stable range 2g + m ≥ 3, let π = ft ◦ st : Mg ,m+l(X , β) → Mg ,m. ¯ψi := π∗c1(Li). ¯ Fg(¯t, s) := X β,m,l qβ m!l !h¯t m, sli g ,m+l ,β is a function onH+× H where ¯t =P¯tkµTµψ¯k, s =P sµTµ. AX(¯t, s) := exp ∞ X g =0 ~g −1FXg(¯t, s).
Frobenius formalism: The Dubrovin connection on TH is ∇z = d − 1 z X µds µ◦ T µ∗ . Recall that ∇2
z = 0 ⇐⇒ WDVV. The fundamental solution N × N
matrix S (N = dim H) is found at ∞ by S = J(s, 1/z). Semi-simple Frobenius manifolds: If (QH, ∗) is semi-simple, i.e. there exist eigen-vector fields i with i∗ j = δiji, let ui be
the dual (canonical) coordinates and U = diag(u1, · · · , nN). Let
Ψ−1 be the transition matrix from {i} to {Tµ}. Then Givental
shows that ∇zS = 0 near z = 0 for
S = Ψ−1(s)R(s, z)eU/z
Ancestor potentials via quantization, the s.s. case: Let
DN(t) =
QN
i =1Dpt(ti) be the descendent potential of N points.
C. Teleman 2007 classified all semi-simple TFT’s. In particular the following formula (conjectured by Givental) holds:
AX(¯t, s) = e¯c(s)Ψb−1(s)bRX(s, z)e
d U/z(s)D
N(t),
where ¯c(s) = 481 ln det(i, j).
Semi-simplicity for local models: For X = PPr(O(−1)r +1⊕O),
QH∗(X ) is semi-simple. Indeed, for q1= q` and q2= qγ,
QHsmall∗ (X ) ∼= C[h, ξ][q1, q2]/(hr +1−q1(ξ −h)r +1, (ξ −h)r +1ξ −q2).
The eigenvalues of h∗ and ξ∗ are all different, hence (QH∗, ∗) is semi-simple at the origin s = 0. Since semi-simplicity is an open condition, QH∗(X ) is also semi-simple.
From descendent to ancestors: Let Dj be the (virtual) divisor on
Mg ,m+l(X , β) as the image of the gluing morphism
X
β0+β00=β
X
l0+l00=l
M0,{j }+l0+•(X , β0)×XMg ,(m−1)+l00+•(X , β00) → Mg ,m+l(X , β),
Then ψj − ¯ψj = [Dj]. The j -th point is in the g = 0 component.
In the stable range 2g + n ≥ 3, hτk+1,¯lα1, · · · ig(¯t, s)
= hτk,l +1α1, · · · ig(¯t, s) +
X
ν
hτkα1, Tνi0(s) hτlTν, · · · ig(¯t, s).
This reduces all descendent of special type to ancestors. The proof for higher genus is complete by the degeneration analysis.
Simple extremal transition in dimension k:
Let ¯X be a k dimensional variety contain only a hypersurface
canonical singularity (p, ¯X ) defined by x0k+ · · · + xkk = 0. A crepant resolution can be obtained by a standard blow-up
φ : Y = BlpX → ¯¯ X . ¯X can be smoothed into a flat family X → ∆
with general smooth fiber X = Xt with t 6= 0 and X0 = ¯X . We
Semi-stable degenerations W → ∆ attached to X → ∆: Notice that the total space X is a smooth variety and W can be achieved by taking a degree k base change
X0 // X ∆ t7→t k //∆
and then set W = Blp0X0. Here p0 ∈ X0 is now a k + 1 dimensional
simple hypersurface singularity of order k in Ck+2. Thus
W0 = Y ∪ ˜E with ˜E ⊂ Pk+1 being a degree k Fano hypersurface.
The intersection E = Y ∩ ˜E , which is the φ exceptional divisor, can
NE / ˜E =O(1) and NE /Y =O(−1). (E, Y ) is equivalent to Pk “cut
out” by a rank 2 split bundle Vk =O(k) ⊕ O(−1).
The A model on X can be compared with the one on Y through the degeneration analysis on the semi-stable family
W
π
∆
thanks to the description of ˜E as a toric Fano hypersurface.
haiX =X
µ
ha1 | µi(Y ,E )∗ ha2 | µi( ˜E ,E )
Let ` ∈ NE (Y ) be the φ extremal ray, which is of flopping type. The genus 0 extremal function is defined by
f (a) = haiYextr := X
d ∈Z+
haiY0,d `qd `.
By the localization calculation in local mirror symmetry or rather the quantum Serre duality principle, the calculation of f (a) may be transformed into a calculation on
Vk+=O(k) ⊕ O(1)
Pk
which in turn reduced toO(k) over Pk−1, that is the case of
From A model of Y to B model of X :
The key observation is to “observe” the appearance of CYk in the
degeneration family π : W → ∆. In fact, Theorem
There is a sub-degeneration of VHS which corresponds the the vanishing cycle along π, whose Picard-Fuchs equation turns out have f (a) as its solution, up to a mirror change of variable!
