the gromov-witten potential of the local p(1,2)todorov/banff.pdfthe crepant resolution conjecture....
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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)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Outline
1 The Crepant Resolution Conjecture.Motivations and originsGromov-Witten invariantsVersions of the Conjecture
2 ExamplesC
2/Z2
C2/Z3
3 Local P(1, 2)Set-upThe invariantsPositive degreeThe Potential
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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”
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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”
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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)
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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)
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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)
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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)
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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)
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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
∑
β
⟨
γn00 · · · γna
a
⟩Yβ
yn00
n0!· · · yna
a
na!qd1
1 · · ·qdrr
where β = d1β1 + · · · + drβr is summed over allnon-negative β.
2 Analogously for an orbifold X can define a FX .
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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
∑
β
⟨
γn00 · · · γna
a
⟩Yβ
yn00
n0!· · · yna
a
na!qd1
1 · · ·qdrr
where β = d1β1 + · · · + drβr is summed over allnon-negative β.
2 Analogously for an orbifold X can define a FX .
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Motivations and originsGromov-Witten invariantsVersions of the Conjecture
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
C2/Z2
C2/Z3
The first example (Bryan-Graber)
For the pair:
X = [C2/Z2] Y = OP1(−2)
FXxxx = −1
2 tan( x
2
)
F Yst =
∑
d1
d3 edyqd
After the change of variables:
y = ixq = −1
We have that
F Yxxx =
12i
[
1 − eix
1 + eix
]
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
C2/Z2
C2/Z3
The first example (Bryan-Graber)
For the pair:
X = [C2/Z2] Y = OP1(−2)
FXxxx = −1
2 tan( x
2
)
F Yst =
∑
d1
d3 edyqd
After the change of variables:
y = ixq = −1
We have that
F Yxxx =
12i
[
1 − eix
1 + eix
]
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
C2/Z2
C2/Z3
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
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
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
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
Degree zero
Given by the triple intersection in (equivariant) cohomology
〈a, b, c〉0 =
Z
Za ∪ b ∪ c
which are computed by Atiyah-Bott localization formula. Except the degreezero invariant with all stacky insertions. We have
˙
Sn¸
0= −
Z
Admg
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
2
”
.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
Degree zero
Given by the triple intersection in (equivariant) cohomology
〈a, b, c〉0 =
Z
Za ∪ b ∪ c
which are computed by Atiyah-Bott localization formula. Except the degreezero invariant with all stacky insertions. We have
˙
Sn¸
0= −
Z
Admg
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
2
”
.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
Degree zero
Given by the triple intersection in (equivariant) cohomology
〈a, b, c〉0 =
Z
Za ∪ b ∪ c
which are computed by Atiyah-Bott localization formula. Except the degreezero invariant with all stacky insertions. We have
˙
Sn¸
0= −
Z
Admg
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
2
”
.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
Degree zero
Given by the triple intersection in (equivariant) cohomology
〈a, b, c〉0 =
Z
Za ∪ b ∪ c
which are computed by Atiyah-Bott localization formula. Except the degreezero invariant with all stacky insertions. We have
˙
Sn¸
0= −
Z
Admg
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
2
”
.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
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.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
...and the winner is...
Reduce everything to computing integrals of the type∫
Admg
d→0,(t1,...,t2g+2)
λgλg−iψi−1d i−1
which in generating function by Cavalieri contributes
cos2d(z2
2
)
.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
The Potential
Theorem
F Z =z3
0
36t1t2+
2t1 + t212t1t2
z20 z1 +
(t1 − t2)(4t1 − t2)3tt t2
z0z21
+14
z0z22 +
t2 − t14
z1z22 +
(t1 − t2)3(8t1 + t2)36t1t2
z31 − (2t1 + t2)G
+(2t1 + t2)X
d odd
14d2
(d − 2)!!
(d − 1)!!edz1 qd
Z
cos2d“ z2
2
”
dz2
+(2t1 + t2)X
d even
12d3
edZ1 qd.
where G′′′ = 12 tan( z2
2 ).
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
But the third partial derivatives...
After analytic continuation and specialization we have
F Zz1z1z2
=α√
1 + α2,
whereα = ez1 cos2
(z2
2
)
.
Gueorgui Todorov Local P(1, 2)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
Gueorgui Todorov Local P(1, 2)