The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture....

The Crepant Resolution Conjecture.Examples

Local P(1, 2)

The Gromov-Witten potential of the localP(1, 2)

Gueorgui Todorov

University of Utah

Recent Progress on the Moduli Space of Curves, BIRS

Gueorgui Todorov Local P(1, 2)

Local P(1, 2)

Outline

1 The Crepant Resolution Conjecture.Motivations and originsGromov-Witten invariantsVersions of the Conjecture

2 ExamplesC

3 Local P(1, 2)Set-upThe invariantsPositive degreeThe Potential

Local P(1, 2)

Motivations and originsGromov-Witten invariantsVersions of the Conjecture

Physics

McKay Philosophy“Geometry on a quotient orbifold [X/G] (that is theG-equivariant geometry on X ) is equivalent to geometry on acrepant resolution Y of X/G”

Physics“Propagating strings can’t tell the difference between anorbifold and its crepant resolution”

Local P(1, 2)

Physics

McKay Philosophy“Geometry on a quotient orbifold [X/G] (that is theG-equivariant geometry on X ) is equivalent to geometry on acrepant resolution Y of X/G”

Physics“Propagating strings can’t tell the difference between anorbifold and its crepant resolution”

Local P(1, 2)

Gromov-Witten invariants

The Gromov-Witten invariants of a projective manifold Y are

multilinear functions 〈...〉Yg,β on its cohomology H∗(Y ).

involve integrals over the space of stable maps Mg,n(Y , β)

The Gromov-Witten invariants of an orbifold X are

multilinear functions 〈...〉Xg,β of the orbifold cohomologyH∗

orb(X ).

involve integrals over the space of twisted stable mapsMg,n(X , β)

Local P(1, 2)

orb(X ).

Local P(1, 2)

orb(X ).

Local P(1, 2)

orb(X ).

Local P(1, 2)

orb(X ).

Local P(1, 2)

Ruan’s first version

Let X be a Gorenstein orbifold satisfying the hard Lefschetzcondition and admitting a crepant resolution Y .

ConjectureYongbin Ruan (2002)

There is an isomorphism between H∗(Y ) and H∗

orb(X ) thatidentifies the Gromov-Witten theories.

Local P(1, 2)

Ruan’s first version

Let X be a Gorenstein orbifold satisfying the hard Lefschetzcondition and admitting a crepant resolution Y .

ConjectureYongbin Ruan (2002)

There is an isomorphism between H∗(Y ) and H∗

orb(X ) thatidentifies the Gromov-Witten theories.

Local P(1, 2)

Gromov-Witten Potential

1 Let {β1, . . . , βr} be a positive basis of H2(Y ). Encodegenus zero Gromov-Witten invariants of Y in the Potentialfunction

F Y (y0, . . . , ya, q1, . . . , qr ) =

∞∑

n0,...,na=0

γn00 · · · γna

⟩Yβ

n0!· · · yna

na!qd1

1 · · ·qdrr

where β = d1β1 + · · · + drβr is summed over allnon-negative β.

2 Analogously for an orbifold X can define a FX .

Local P(1, 2)

Gromov-Witten Potential

1 Let {β1, . . . , βr} be a positive basis of H2(Y ). Encodegenus zero Gromov-Witten invariants of Y in the Potentialfunction

F Y (y0, . . . , ya, q1, . . . , qr ) =

∞∑

n0,...,na=0

γn00 · · · γna

⟩Yβ

n0!· · · yna

na!qd1

1 · · ·qdrr

where β = d1β1 + · · · + drβr is summed over allnon-negative β.

2 Analogously for an orbifold X can define a FX .

Local P(1, 2)

Bryan-Graber’s precise version

ConjectureBryan-Graber (2005)

The orbifold Gromov-Witten potential FX is equal to the(ordinary) Gromov-Witten potential FY of a crepant resolutionafter a change of variables induced by:

a linear isomorphism

L : H∗

orb(X ) → H∗(Y );

an analytic continuation of the potential to a specializationof the excess quantum parameters (on the resolution side)to roots of unity.

Local P(1, 2)

L : H∗

orb(X ) → H∗(Y );

Local P(1, 2)

L : H∗

orb(X ) → H∗(Y );

Local P(1, 2)

The first example (Bryan-Graber)

For the pair:

X = [C2/Z2] Y = OP1(−2)

FXxxx = −1

2 tan( x

F Yst =

d3 edyqd

After the change of variables:

y = ixq = −1

We have that

F Yxxx =

1 − eix

1 + eix

Local P(1, 2)

The first example (Bryan-Graber)

For the pair:

X = [C2/Z2] Y = OP1(−2)

FXxxx = −1

2 tan( x

F Yst =

d3 edyqd

After the change of variables:

y = ixq = −1

We have that

F Yxxx =

1 − eix

1 + eix

Local P(1, 2)

Another example (Bryan-Graber-Pandharipande)

X = [C2/Z3] Y = two −2 curves

The third derivatives of the Potential for XSum of tan

The third derivatives of the Potential for Y

Sum of expChange of variables:

y0 = x0

y1 =i√3

(ωx1 + ωx2)

y2 =i√3

(ωx1 + ωx2)

qi = ω

Local P(1, 2)

Set-upThe invariantsPositive degreeThe Potential

Intermediate case

Let Z be the weighted blow-up of C2/Z3 at the origin with

weights one and two.1 Z ∼= O(KP(1,2)) is Local P(1, 2)

2 Z is not a global quotient3 Possible insertions

The fundamental class 1 with dual z0

Divisorial class H with dual z1

Stacky class S with dual z2

Local P(1, 2)

Intermediate case

Local P(1, 2)

Intermediate case

Local P(1, 2)

Intermediate case

Local P(1, 2)

Intermediate case

Local P(1, 2)

Intermediate case

Local P(1, 2)

Degree zero

Given by the triple intersection in (equivariant) cohomology

〈a, b, c〉0 =

Za ∪ b ∪ c

which are computed by Atiyah-Bott localization formula. Except the degreezero invariant with all stacky insertions. We have

0= −

2→0,(t1,...,t2g+2)

λgλg−1.

Theorem

Faber and Pandharipande (Bryan and Pandharipande, and also Bertram,Cavalieri, –) If G is the contribution to the potential then we have that

G′′′ =12

tan“ z2

Local P(1, 2)

Degree zero

〈a, b, c〉0 =

Za ∪ b ∪ c

0= −

2→0,(t1,...,t2g+2)

λgλg−1.

Theorem

G′′′ =12

tan“ z2

Local P(1, 2)

Degree zero

〈a, b, c〉0 =

Za ∪ b ∪ c

0= −

2→0,(t1,...,t2g+2)

λgλg−1.

Theorem

G′′′ =12

tan“ z2

Local P(1, 2)

Degree zero

〈a, b, c〉0 =

Za ∪ b ∪ c

0= −

2→0,(t1,...,t2g+2)

λgλg−1.

Theorem

G′′′ =12

tan“ z2

Local P(1, 2)

Reductions

Use the Euler sequence on P(1, 2) to reduce from Z to the totalspace of a CY threefold Z ′ = L1 ⊕ L2 (the total space of abundle over P(1, 2)).The image of non-constant maps to Z ′ must lie in the zerosection E ∼= P(1, 2). We can identify

M0,n(Z′, d [E ]) ∼= M0,n(P(1, 2), d)

BUT keep in mind the difference of virtual fundamental classesgiven by

e(R•π∗f ∗NE/Z ) = e(R•π∗f ∗L1 ⊕ L2)

where f and π are the universal map and family.

Local P(1, 2)

Reductions

M0,n(Z′, d [E ]) ∼= M0,n(P(1, 2), d)

e(R•π∗f ∗NE/Z ) = e(R•π∗f ∗L1 ⊕ L2)

Local P(1, 2)

Reductions

M0,n(Z′, d [E ]) ∼= M0,n(P(1, 2), d)

e(R•π∗f ∗NE/Z ) = e(R•π∗f ∗L1 ⊕ L2)

Local P(1, 2)

Odd d case

Problem 1

The moduli spaces M0,n(P(1, 2), d) are very complicated.

Solution: Use localization under a lifting of a C∗ action on

P(1, 2).

Problem 2

The fixed loci are very complicated.

Solution: Use different lifting of the action to L1 ⊕ L2 to reduceto a single fixed loci.

Local P(1, 2)

Odd d case

Problem 1

P(1, 2).

Problem 2

Local P(1, 2)

Odd d case

Problem 1

P(1, 2).

Problem 2

Local P(1, 2)

Odd d case

Problem 1

P(1, 2).

Problem 2

Local P(1, 2)

Odd d case

Problem 1

P(1, 2).

Problem 2

Local P(1, 2)

Odd d case

Problem 1

P(1, 2).

Problem 2

Local P(1, 2)

...and the winner is...

Reduce everything to computing integrals of the type∫

d→0,(t1,...,t2g+2)

λgλg−iψi−1d i−1

which in generating function by Cavalieri contributes

cos2d(z2

Local P(1, 2)

The Potential

Theorem

F Z =z3

36t1t2+

2t1 + t212t1t2

z20 z1 +

(t1 − t2)(4t1 − t2)3tt t2

z0z22 +

t2 − t14

z1z22 +

(t1 − t2)3(8t1 + t2)36t1t2

z31 − (2t1 + t2)G

+(2t1 + t2)X

(d − 2)!!

(d − 1)!!edz1 qd

cos2d“ z2

+(2t1 + t2)X

d even

edZ1 qd.

where G′′′ = 12 tan( z2

Local P(1, 2)

But the third partial derivatives...

After analytic continuation and specialization we have

F Zz1z1z2

=α√

1 + α2,

whereα = ez1 cos2

Local P(1, 2)

The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture....

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