Towards A + B Theory in Conifold Transitions for Calabi–Yau Threefolds

Towards A + _{B Theory} in Conifold Transitions for Calabi–Yau Threefolds

Chin-Lung Wang National Taiwan University

(Joint work with Yuan-Pin Lee and Hui-Wen Lin)

Second Annual Meeting of ICCM Taipei, Taiwan

December 27, 2018

Contents

1 Calabi–Yau 3-folds 2 A model and B model 3 Mirror, flops, and transitions

4 An observation from ordinary k-fold singularity 5 Statements for conifold transitions

6 Linked GW invariants for A model 7 Linked B model via VMHS

I. Calabi–Yau 3-folds

I A projective manifold X/C is Calabi–Yau if π1(X)is finite and KX=_{0 (or c1}(X) =_0).

I Yau’s solution to the Calabi conjecture=⇒for any cpt K¨ahler X with c₁(X)_R=0,∃finite cover eX→X:

Xe =A×B×C.

A∼=_C^g/Λ (flat), B is hyperk¨ahler (SU(m)), and C is CY (SU(n)). Also π₁(B) =π₁(C) =0,C is projective.

I The first new case appears in dim=3. We have

h¹(_O) =h²(_O) =0. WLOG we assume that π1(X) =_0.

I Question:classification of CY 3-folds?

I What is the global structure (symmetries?) ofMCY3?

I Examples.Adjunction formula for hypersurfaces X⊂Y:

K_X= (K_Y+X)|_X =0⇐⇒X is anti-canonical in Fano Y.

I X= (n+1) ⊂Pⁿ. E.g. the Fermat hypersurfaces xⁿ₀⁺¹+ · · · +xⁿ_n⁺¹ =0

is a CY(n−1)-fold. E.g. quintic 3-folds.

I X= (~d₁, . . . ,~_d

k) ⊂_∏^m_i=1Pⁿⁱ with~_d

j = (d_ji)^m_i=1and

∑

d_ji=n_i+1, 1≤i≤m.

This is a CICY of dimension D=_{∑ ni}−m.

I Let NDbe the numbers of them. ThenN3=7890.

I Toric CY.A lattice polytope4 ⊂M_R, M∼=_Zⁿ⁺¹is reflexive if 0∈int4and its polar (dual) polytope

4^◦ := {w∈N :=M^∨ | hw, vi ≥ −1, ∀v∈ 4}

is also a lattice polytope, in N_R.

I Number of them [Kruezer–Skarke, 2000]:

N₁=16, N₂ =4319, N₃ =473800776, . . . .

I For a reflexive pair(4,4^◦), the toric variety P₄ :=Proj(^M

k≥0C^k4∩M) is Fano withH⁰(K⁻_P¹

4) =^L_{v∈4∩MC t}^v; similarly for P₄^◦.

I For a general section f , X_f := {f =0}is a CY n-fold.

II. Classical A model and B model

I The Hodge numbers of a CY 3-fold X are 1

0 0

0 h²² 0

1 h²¹ h¹² 1

0 h¹¹ 0

0 0

1

I h¹¹ =h¹(X,ΩX) =h²parametrizes K¨ahler classes.

I A(X) =QH(X)is the g=0 Gromov–Witten theory in ω=B+iH ∈_KX^C=H²(X,R) ⊕√

−1Amp(X).

I h²¹ =h¹(X, TX)parametrizes complex deformations.

I B(X) = (H³,∇^GM)is the VHS on the complex moduliMX

under the Gauss–Manin connection with lattice H³(X,Z).

I A model. For a CY 3-fold X, let β∈ H₂(X,Z),

M(X, β) = {h : C→X stable|C is nodal, pa(C) =0, h∗[C] =β}/∼. I Virtual dim=0: the essential genus 0 GW invariants are

n^X_β = h−i^X_β = Z

[M(X,β)]^virt1∈_Q.

I Toric example. Let X_f ⊂P4with f ∈H⁰(K⁻_P4¹).

I A(X_f)is determined byC^×-localizationdata [LLY, G 1999]:

I^X(q^•, z⁻¹) =

∑

β∈H2(X_f,Z)

q^β ∏^K_m=1^−1.β(K⁻¹_P4+mz)

∏ρ∈Σ1∏^D_m=1^ρ.β(Dρ+mz) .

I Σ is the normal fan of P4and D_ρis the torus invariant divisor corresponding to the one-edge ρ∈_Σ1,q^β =e^2πi(β.ω).

I B model. For the CY family π :X →S :=_MX, H³ :=R³π∗C⊗_OS→S,

F^p =π∗Ω^p_{X /S},H^pq=F^p∩F^q,Ω∈ _Γ(S, F³)_{. Then}

∇^GMF^p,→F^p−1⊗_Ω¹_S, h ∇^GM_∂/∂x

jΩi^h_j=²¹₁ = H²¹.

I Periods. Let δm ∈H₃(X)be a basis with dual δ^∗_m∈ H³(X). For η ∈_Γ(S,H³), since∇^GMδ_m^∗ =0, we have

∇^GM_∂/∂x

jη=

∑

m

δ_m^∗ ∂

∂x_j Z

δm

η, j∈ [1, h²¹]. GM⇐⇒Picard–Fuchs equations of period integralsR

δmΩ.

I Toric example: B(X_f)is determined by the GKZ* system:

(1) symmetry operators;

(2) for`a relation ofm<sub>i</sub> ∈ 4 ∩Mwith∑`_i=0,

` :=

∏

∏

III. Mirror, flops, and transitions

I Mirror symmetry.

I Topological MS:(Y, Y^◦)is a mirror pair of CY 3-folds if h²¹(Y) =h¹¹(Y^◦), h¹¹(Y) =h²¹(Y^◦).

I Classical MS, or A↔B MS:B(Y) ∼=A(Y^◦),A(Y) ∼=B(Y^◦).

I Toric Example: Consider 2 families of CY 3-folds X_f ⊂P₄, X_g^◦ ⊂P₄^◦.

I Topological MS holds [Batyrev ’94].

I A↔B MS holds for “many cases”.

I Observation: Σ1=rays from 0 to Vert(4^◦).

I [HLY 1998]∃max-deg-point (⇒mirror transform).

I Flops.A D-flop between CY 3-folds is a birational diagram

` ⊂Y ^{f //}

ψ

""

Y⁰ ⊃ `⁰

ψ⁰

{{X

where ψ is D-negative (log-extremal) and ψ⁰is D⁰-positive.

I [Koll´ar, Kawamata 1988] Birational CY 3-folds are connected by flops.3D flops are classified.

I [Koll´ar–Mori 1992] Birational CY 3-folds Y and Y⁰ have MX∼=_MX⁰ =⇒B(Y) ∼=B(Y⁰)

since flops can be performed in flat families.

I [Li–Ruan 2000]A(Y) ∼=A(Y⁰)under q^β 7→q^f∗β(` 7→ −`⁰).

I Transitions.

I Geometric transition X%Y (or Y&X) of CY 3-folds:

Y

ψ

K_Y=ψ^∗K_X,

X ^//X NF³_∞ =0.

I X%Y is a conifold transition if X_singhas only ODPs (X, p_i):= {x²₁+x₂²+x²₃+x²₄ =0}.

I Q1[Reid 1987] CanALL CY 3-foldsbe connected through (possibly non-projective) conifold transitions?

I Q2[W 2009] Does(A(X), B(X))determines(A(Y), B(Y)) and vice versa? NoticeA(X) <A(Y)andB(X) >B(Y).

IV. An observation from ordinary k-fold singularity

A + B Model in Quantum Geometry

Oct. 12, 2009 NTU Math

Dragon

LOCAL EXAMPLES: Consider the dim k hyper-surface X 0  C ^k+1 : x 0 k + x 1 k + … + x k k = 0

with p = 0  X 0 being an ordinary k-fold singularity. The blow-up f: X = Bl p (X 0 ) → X 0 is crepant with exceptional divisor

E = (k)  P ^k , N E/X = O(-1)| E .

The local structure of E  X, namely the germ (E, X) is equivalent to P ^k “cut out” by the rank 2 vector bundle:

V k = O(k)⊕O(-1) → P ^k .

X 0 can be smoothed into a flat family M →  with general smooth

fiber X’ = M t . The semi-stable reduction : W →  is used to compare

X and X’ since W t = X’ and W 0 = X∪E’ for some Fano E’.

Quantum Transition from A to B:

The Gromov-Witten extremal function f(a) =  dN <a> dL q ^dL attached to the extremal ray L  NE(X) can be calculated, using the quantum Serre duality principle, by the bundle

V k + = O(k)⊕O(1) → P ^k .

This is in turn reduced to O(k) → P ^k-1 , the Calabi-Yau CY k ! Where is the Picard-Fuchs operator P k for f(a)?

Since dim CY k = k – 2, we must have deg P = k – 2. But dim X’ =

k. It must be the case that there is a “sub-VHS of R ^k  _* C of weight

k – 2” which starts at Ω  H ^n-1,1 = H ¹ (X’, T). Let Γ be the vanishing

cycle along , then P k is the Picard-Fuchs op for 

Ω .

V. Statements for conifold transitions

Let X%Y be a projective conifold transition of CY 3-folds through X with k ODPs p₁, . . . , p_k, π :X →_{∆, ψ : Y}→X:

C_i⊂Y

ψ

N_Ci/Y=_OP1(−1)^⊕2

N_Si/X =T^∗S³ S_i⊂X ^π // p_i ∈X

Let µ :=h^2,1(X) −h^2,1(Y) >0 and ρ :=h^1,1(Y) −h^1,1(X) >0.

χ(X) −kχ(S³) =χ(Y) −kχ(S²) =⇒µ+ρ=k.

Hence there arenon-trivial relationsbetween the “vanishing cycles”:

A= (a_ij) ∈M_k×µ,

∑

∑

Let 0→V_Z,→H₃(X,Z) →H₃(X,^¯ Z) →_{0 and V :}=C_Z⊗C.

Theorem (Basic exact sequence)

We have an exact sequence ofweight two pure Hodge structures:

0→H²(Y)/H²(X)−→^B C^{k A}−→^t V→0.

Since ψ : Y→X deforms in families, this identifiesMYas a codimenison µ boundary strata inM_Xand locallyM_X ∼=∆^µ×MY. Write V=ChΓ1, . . . ,Γµiin terms of a basisΓj’s. Then the α-periods

rj= Z

Γj

Ω, 1≤j≤µ

form the degeneration coordinates around[X]. The discriminant loci of M_Xis described by acentral hyperplane arrangementDB=^Sk_i=1Di: Proposition (Friedman 1986)

Let wi=ai1r1+ · · · +aiµrµ, then the divisor Di:= {wi=0} ⊂M_Xis the loci where the sphere Sishrinks to an ODP pi.

I The β-periods in transversal directions are given by a function u:

up=∂pu= Z

βp

Ω

I The Bryant–Griffiths–Yukawa couplings extend over DBand

upmn:=∂³_pmnu=O(1) +

∑

1 2π√

−1

a_ipa_ima_in w_i =

Z

∂p∂m∂nΩ∧_Ω

for 1≤p, m, n≤µ. It is holomorphic outside this index range.

I Let y=_∑^k_i=1y_ie_i∈_C^k, with eⁱ’s being the dual basis on(_C^k)^∨. Thetrivial logarithmic connectiononC^k⊕ (C^k)^∨−→C^kis

∇^k=d+¹ z

∑

dyi

y_i ⊗ (eⁱ⊗e^∗_i).

Theorem (Local invariance: Exc(_A) +Exc(_B) =trivial)

(1) ∇^krestricts to the logarithmic part of∇^GMon V^∗.

(2) ∇^krestricts to the logarithmic part of∇^Dubrovinon H²(Y)/H²(X).

Theorem (LinkedA +_{B theory)}

Let[X]be a nearby point of[X]inM_X,

(1) A(X)is a sub-theory ofA(Y)(i.e. quantum sub-ring).

(2) B(Y)is a sub-theory ofB(X)(sub-moduli, invariant sub-VHS).

(3) A(Y)can be reconstructed from a “refinedA theory” on

X^◦:=X\^[k

i=1Si

“linked” by the vanishing 3-spheres inB(X).

(4) B(X)can be reconstructed from the variations of MHS on H³(Y^◦),

Y^◦ :=Y\^[k

i=1C_i,

“linked” by the exceptional curves inA(Y).

For (3) and (4),effective methodsare under developed.

VI. Linked GW invariants for A model

I It is easy to see thatA(X) ⊂_A(Y): for 06=β∈H₂(X), by degeneration formula in GW theory [Li] we can show

n^X_β =

∑

I To determineA(Y)fromA(X) +_B(X), it is equivalent to find a definition of each individual term n^Y_γ with the same βin terms of a refined data in X.

Lemma

H₂(X^◦) ∼=H₂(Y^◦) ∼=H₂(Y). In particular, for a map h : C→X^◦, γ:=h∗[C] ∈H₂(Y)is well-defined.

I This γ is calleda linking data(β, L). It encodes the link between h(C)(2D) and S_i’s (3D) inside a 6D space X.

I If the stable maps h : C→Xdo not touch^S_iS³_i, then the linked GW invariants n^X_(β,L)is defined and n^X_(β,L) =n^Y_γ.

I In general this is not true. However, it istrue in the virtual sense, which is all we need:

Proposition

For X_twith t∈_A¹\ {0}small in the degenerating family π:X →_A¹arising from the semi-stable reduction, we have a decomposition of the virtual class[M(Xt, β)]^virtinto a finite disjoint union of cycles

[M(Xt, β)]^virt =

ä

where[M(Y, γ)]^virt ∼ [M(X_t, γ)]^virt ∈A_vdim M(X_t, β)
is a cycle class corresponding to the linking data γ of Xt.

VII. Linked B model via VMHS

I B(Y)is a sub-theory ofB(X)by viewingMY,→_MXas a boundary strata ofMX.

I We will show thatB(Y), together with the knowledge of extremal curves Z :=^S_iC_i⊂Y determinesB(X).

I Proposition

There is a short exact sequence ofmixed Hodge structures

0→V→H³(X) →H³(U) →0, (1) where H³(X)is equipped with thelimiting MHS of Schmid,

V∼=H^1,1_∞ H³(X),

and H³(U)is equipped with thecanonical mixed Hodge structure of Deligne. In particular, F³H³(X) ∼=F³H³(U), F²H³(X) ∼=F²H³(U).

I We have on X, U∼=X^◦ :=X\p, where p=^S{p_i}_:

· · ·H¹_p(_ΘX¯) →H¹(_ΘX¯) →H¹(U, T_U) →H²_p(_ΘX¯) → · · ·.

I [Schlessinge] p_i is a hypersurface singularity=⇒

depthOpi =3=⇒H_p¹(_ΘX¯) =0 and H²_p(_ΘX¯) ∼= ^Lk_i=1Cpi: 0→H¹(_ΘX¯) →H¹(U, T_U) →H²_p(_ΘX¯) → · · ·.

I Comparing with the local to global spectral sequence

0→H¹(_ΘX¯)→^λ Ext¹(_ΩX¯,OX¯) →H⁰(_{E xt}¹(_ΩX¯,OX¯))→^κ H²(_ΘX¯),

I ⇒Def(X^¯) ∼=H¹(U, T_U). Similarly, for Y⊃Z=^SC_iwe get Def(Y) =H¹(T_Y) ⊂H¹(U, T_U) ∼=Def(X^¯)_,

and thenMY ,→_MX¯ (unobstructedness theorem).

I WriteI :=_I as the ideal sheaf ofM ⊂_M .

I Since H²(U, T_U) 6=0, the deformation of U could be obstructed. Nevertheless, the first-order deformation of U exists and is parameterized by H¹(U, T_U) ⊃Def(Y).

I Therefore, we have the following smooth family π: U→ Z_1:=Z_MX(_I²) ⊃_MY, whereZ₁ =Z_MX¯(_I²)stands for the nonreduced subscheme ofM_Xas the first jet extension ofMYinM_X.

I [Katz]∇^GMfor π : U→ Z₁is defined by the lattice

H³(U,Z) ⊂H³(U,C).It underlies VMHSinstead of VHS.

I The proposition implies

W_iH³(U) =0, i≤2; W₃ ⊂W₄ with Gr^W₃ H³(U) ∼=H³(Y)and Gr^W₄ H³(U) ∼=V^∗.

I The Hodge filtration of the local system F⁰=H³(U,C): F^• = {F³ ⊂F²⊂F¹⊂F⁰}

satisfiesGriffiths’ transversality.

I Since KU ∼=_OU, F³is a line bundle overZ_{1spanned by} Ω∈_Ω³_U/Z

1. Near[Y] ∈ Z₁,

F²is then spanned byΩ and v(_Ω) where v runs through a basis of H¹(U, T_U).

I Notice that v(_Ω) ∈W₃precisely when v∈H¹(Y, T_Y).

I Proposition⇒F³⊂F²on H³(U)overZ₁lifts uniquely to

˜F³ ⊂ _˜F²on H³(X)overZ₁with

˜F³ ∼=F³, ˜F² ∼=F².

I The complete lifting ˜F^•is then determined since

˜F¹= (˜F³)^⊥

bythe first Hodge–Riemann relationon H³(X).

I Now ˜F^•overZ₁uniquely determines a horizontal map Z₁→_D.^ˇ

I Since it has maximal tangent dimension h¹(U, T_U) =h¹(X, T_X), it determinesthe maximal horizontal slice

ψ:M →_D^ˇ withM ∼=M_XnearMY.

I LetΓ be the monodromy group generated by the local monodromy T⁽ⁱ⁾=_{exp N}⁽ⁱ⁾around the divisor

D_i:= {w_i=

∑

I Under the coordinates t= (r, s), the period map φ:M_X= MX¯\^[k

i=1Dⁱ →_D/Γ is then given (by an extension of Schmid’s NOT) as

φ(r, s) =exp

∑

log w_i 2π√

−1N⁽ⁱ⁾

!

ψ(r, s),

I Since N⁽ⁱ⁾is determined by the Picard–Lefschetz formula, the period map φ is completely determined by A and C_i’s.

I Hence the refinedB model on Y\Z=U determines theB