Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Crepant resolutions and integrable systems
Andrea Brini
Universite de Montpellier 2
SISSA, Sep 20 2013
Context and motivation
Want to: establish relation between Frobenius structures (and quan-
tization thereof) arising from the Gromov–Witten theory of bira-
tionally isomorphic targets.
Punchline:
2D-Toda hierarchy ↔ type A surface resolutions
⇒ Crepant Resolution Conjecture(s)
Based on 1309.4438 with R. Cavalieri, D. Ross; work in progress with
S. Romano and G. Carlet, S. Romano, P. Rossi
2
Context and motivation
X a (Gorenstein) complex algebraic orbifold
π : Y → X a crepant resolution, π∗(KX ) = KY .
Example: its anoni al minimal resolution
Question: relation between geometri al invariants of X and Y ?
[Witten, Aspinwall{Greene{Morrison℄
3
Context and motivation
X a (Gorenstein) complex algebraic orbifold
π : Y → X a crepant resolution, π∗(KX ) = KY .
Example: X = [C2/Γ], Y its canonical minimal resolution
Question: relation between geometri al invariants of X and Y ?
[Witten, Aspinwall{Greene{Morrison℄
4
Context and motivation
X a (Gorenstein) complex algebraic orbifold
π : Y → X a crepant resolution, π∗(KX ) = KY .
Example: X = [C2/Γ], Y its canonical minimal resolution
Question: relation between geometrical invariants of X and Y ?
[Witten, Aspinwall–Greene–Morrison]
5
Context and motivation
• χorb(X)?↔ χ(Y )
� H
�
orb
(X )
?
$ H
�
(Y )
{ as graded ve tor spa es
{ as lassi al ohomology rings
{ as quantum ohomology rings
) genus 0 Gromov{Witten in-
variants
� genus zero GW des endents?
� higher genus GW des en-
dents?
[Roan, Batyrev{Dais, Reid℄
[Yasuda℄
[Ruan℄
[Ruan, Bryan{Graber, Coates{Iritani{Tseng℄
6
Context and motivation
• χorb(X) = χ(Y )
� H
�
orb
(X )
?
$ H
�
(Y )
{ as graded ve tor spa es
{ as lassi al ohomology rings
{ as quantum ohomology rings
) genus 0 Gromov{Witten in-
variants
� genus zero GW des endents?
� higher genus GW des en-
dents?
[Roan, Batyrev–Dais, Reid]
[Yasuda℄
[Ruan℄
[Ruan, Bryan{Graber, Coates{Iritani{Tseng℄
7
Context and motivation
• χorb(X) = χ(Y )
• H•orb(X)?↔ H•(Y )
{ as graded ve tor spa es
{ as lassi al ohomology rings
{ as quantum ohomology rings
) genus 0 Gromov{Witten in-
variants
� genus zero GW des endents?
� higher genus GW des en-
dents?
[Roan, Batyrev–Dais, Reid]
[Yasuda℄
[Ruan℄
[Ruan, Bryan{Graber, Coates{Iritani{Tseng℄
8
Context and motivation
• χorb(X) = χ(Y )
• H•orb(X)∼↔ H•(Y )
– as graded vector spaces
{ as lassi al ohomology rings
{ as quantum ohomology rings
) genus 0 Gromov{Witten in-
variants
� genus zero GW des endents?
� higher genus GW des en-
dents?
[Roan, Batyrev–Dais, Reid]
[Yasuda]
[Ruan℄
[Ruan, Bryan{Graber, Coates{Iritani{Tseng℄
9
Context and motivation
• χorb(X) = χ(Y )
• H•orb(X)?↔ H•(Y )
– as graded vector spaces
– as classical cohomology rings
{ as quantum ohomology rings
) genus 0 Gromov{Witten in-
variants
� genus zero GW des endents?
� higher genus GW des en-
dents?
[Roan, Batyrev–Dais, Reid]
[Yasuda]
[Ruan]
[Ruan, Bryan{Graber, Coates{Iritani{Tseng℄
10
Context and motivation
• χorb(X) = χ(Y )
• H•orb(X)?↔ H•(Y )
– as graded vector spaces
– as classical cohomology rings
– as quantum cohomology rings⇒ genus 0 Gromov–Witten in-variants
� genus zero GW des endents?
� higher genus GW des en-
dents?
[Roan, Batyrev–Dais, Reid]
[Yasuda]
[Ruan]
[Ruan, Bryan–Graber, Coates–Iritani–Tseng]
11
Context and motivation
• χorb(X) = χ(Y )
• H•orb(X)?↔ H•(Y )
– as graded vector spaces
– as classical cohomology rings
– as quantum cohomology rings⇒ genus 0 Gromov–Witten in-variants
• genus zero GW descendents?
• higher genus GW descen-
dents?
[Roan, Batyrev–Dais, Reid]
[Yasuda]
[Ruan]
[Ruan, Bryan–Graber, Coates–Iritani–Tseng]
[Coates–Iritani–Tseng; see Hiroshi’s talk]
12
The Frobenius manifold/IS viewpoint
• χorb(X) = χ(Y )
• H•orb(X)?↔ H•(Y )
– as graded vector spaces
– as classical cohomology rings
– as quantum cohomology rings⇒ genus 0 Gromov–Witten in-variants
• genus zero GW descendents?
• higher genus GW descen-
dents?
FXcl?←→ FY
cl
↑
FXsmall?←→ FY
small
↑
FX0?←→ FY
0
τX0?←→ τY0
τXǫ?←→ τYǫ
13
Plan
1. Crepant Resolution Conjectures (expectations, speculations,
folklore, existing results)
2. Rational reductions of 2-Toda hierarchy
3. 2 � 1
14
Analytic continuation
(shamelessly ripped from 0809.2749)
Hard Lefschetz orbifolds: age(γ) = age(inv∗γ), ∀γ ∈ HCR(X).
15
Hard Lefschetz orbifolds: conjectures
CRC1 (primaries): there exists a path of analytic continuation γ
and an affine linear map U∞γ : H(Y )→ HCR(X) such that
FX0 (tX ) = FY0 (U∞γ tY )
upon analytic continuation along γ.
CRC2 (descendents): there exists a path of analytic continuation
γ and a triangular automorphism Uγ : LH(Y )→ LHCR(X) such that
τX0 (tX (z)) = τY0 (UγtY (z))
upon analytic continuation along γ. [Bryan–Graber, CIT; Iritani]
16
Hard Lefschetz orbifolds: conjectures
CRC3 (primaries): there exists a path of analytic continuation γ
and an affine linear map U∞γ : H(Y )→ HCR(X) such that
FXg (tX ) = FYg (U∞γ tY )
upon analytic continuation along γ.
CRC4 (descendents): there exists a path of analytic continuation
and a triangular automorphism Uγ : LH(Y )→ LHCR(X) such that
τXǫ (tX (z)) = τYǫ (UγtY (z))
upon analytic continuation along γ. [CIT]
17
The symplectic picture
CRC1–CRC2 can be phrased as the existence of a an isomorphism
of Givental spaces Uγ ∈ HY → HX s.t. Uγ(LY ) = Uγ(LX ), Uα,β ∈
C[[z−1]]; alternatively, as the discrepancy between the S-calibrations
of the two Frobenius manifolds. Proven for toric orbifolds + a handful
of other examples [Iritani, Coates–Corti–Iritani–Tseng]
CRC3–CRC4 are their quantized version:
τXǫ = UγτYǫ
Very few-to-none examples.
18
Type A surface singularities and their resolutions
Main character in the play today: the (fully equivariant) Gromov–
Witten theory of X = [C2/Zn+1] (×C).
This turns out to have a close relation to the two-dimensional Toda
hierarchy.
19
Lax formalism for 2D-Toda hierarchy
Lax operators:
L1 = Λ+∑
j≤0
u(1)j Λj,
L2 = u(2)−1Λ
−1 +∑
j≥0
u(2)j Λj, Λ = eǫ∂x
Lax equations:
∂t(1)k
Li =[Li, (L
k1)+
], ∂
t(2)k
Li =[Li, (L
k2)−
]
20
Lax formalism for 2D-dToda hierarchy
Dispersionless limit is the Ehrenfest limit:
[Takasaki–Takebe]
p = σ(Λ), λi(p) = σLi(p)
dLax equations:
∂t(1)k
λi ={λi, (λ
k1)≥0
}, ∂
t(2)k
λi ={λi, (λ
k2)≤0
}
21
Rational reductions of 2D-(d)Toda
〈{uik}〉 → ∞-dimensional Frobenius manifold structure. Finite dimen-
sional (weak) Frobenius submanifolds arise as symmetry reductions.
Rational reductions:
λ(p) = λn1(p) =Pn(p)
Qm(p−1), λ1(p)
nλ2(p)m = 1
Landau–Ginzburg formulas on Hλ = (Pn, Qm):
η(X, Y ) =∑
pi∈Crit(λ)
Resp=piX(λ)Y (λ)
λ′dp
p
η(X, Y · Z) =∑
pi∈Crit(λ)
Resp=piX(λ)Y (λ)Z(λ)
λ′dp
p
[Dubrovin, Krichever]
22
LG formalism for dToda
Theorem: this induces a quasi-homogeneous, charge d = 1 Frobenius
structure on Hλ, with a non-horizontal unit vector field, semi-simple
when Pn, Qm have simple zeroes.
Corollary: Dual Frobenius structure is non-homogeneous with flat
unit:
g(X, Y ) =∑
pi∈Crit(λ)
Resp=piX(log λ)Y (log λ)λ
λ′dp
p
g(X, Y ⋆ Z) =∑
pi∈Crit(λ)
Resp=piX(logλ)Y (logλ)Z(log λ)λ
λ′dp
p
23
Twisted periods and hypergeometric integrals
Remark 1: zeroes of Pn, Qm are exponentiated flat coordinates for
g.
Remark 2: flat coordinates for the ⋆-deformed connection are given
by the Euler–Pochhammer periods of λ1/zd log p in the twisted ho-
mology of the line [Deligne–Mostow]
⇒ generalized (Lauricella) hypergeometric functions
24
GW ↔ IS
• OP1(−1)⊕O
P1(−1) ↔ Ablowitz–Ladik hierarchy [A.B.]
• Ablowitz–Ladik hierarchy is the (1,1) rational reduction of 2D-
Toda [Carlet, Rossi, A.B.]
• educated guess (after [Milanov–Tseng, Getzler, Carlet]):
OP(n,m)(−n)⊕OP(n,m)(−m) ↔ (n,m) rational reduction
[Carlet, Romano, Rossi, A.B.℄
� m = 0; n > 1: q-deformed Gelfand{Di key
[Frenkel{Reshetikhin℄
25
GW ↔ IS
• OP1(−1)⊕O
P1(−1) ↔ Ablowitz–Ladik hierarchy [A.B.]
• Ablowitz–Ladik hierarchy is the (1,1) rational reduction of 2D-
Toda [Carlet, Rossi, A.B.]
• educated guess: ✓
OP(n,m)(−n)⊕OP(n,m)(−m) ↔ (n,m) rational reduction
[Carlet, Romano, Rossi, A.B.]
• m = 0, n > 1: [C2/Zn−1]× C ↔ q-deformed Gelfand–Dickey
[Frenkel–Reshetikhin]
26
1- Wall-crossings in genus zero
• Mirror symmetry picture as a 1-dimensional logarithmic LG
model (6= Givental’s toric mirror)
• Analytic continuation (quite massively) simplified; closed form
expressions for Uγ
– (Re-)proves CRC1
– Verifies (the fully equivariant version of) Iritani’s K-group CRC
27
2- Toric Lagrangian branes and the open CRC
Recently: Crepant Resolution Conjecture for open GW invariants
of semi-projective CY 3-orbifolds with toric Lagrangian branes
[Cavalieri, Ross, A.B.]
Enumerative information on disk invariants ↔ sections of Givental
space FdiskZ : HT (Z)→HZ. The OCRC:
O : HX → HY , FdiskY = OFdisk
X
Corollaries:
• prove OCRC
• Bryan–Graber-type theorem for effective legs (cf. [Ke–Zhou])
• a generalized BG theorem for ineffective legs
28
3- Monodromy of B-branes
▲�✁
▲�✂
❈✄❖✄ ✚
Monodromy group of the global A-model quantum D-module ↔
Burau–Gassner representation of the colored braid group in n+1
strands
29
4- The quantized CRC
Theorem: the quantized CRC holds for An resolutions.
Idea of the proof:
1. quantized CRC for HL orbifolds holds ⇔ RX = RY ;
2. pick Stokes wedge s.t. twisted period map integrals over Morse-theoreticcycles;
3. all order asymptotics is computable/controllable parametrically in the basepoint
30
5- non-toric examples (in progress)
Analogy of GW([C2/Zn+1]) with An Frobenius manifold
→ generalization to full ADE series
31
Thank you
32