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