the gromov-witten potential of the local p(1,2)todorov/banff.pdfthe crepant resolution conjecture....
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![Page 1: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/1.jpg)
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)
![Page 2: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/2.jpg)
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)
![Page 3: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/3.jpg)
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)
![Page 4: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/4.jpg)
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)
![Page 5: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/5.jpg)
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)
![Page 6: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/6.jpg)
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)
![Page 7: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/7.jpg)
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)
![Page 8: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/8.jpg)
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)
![Page 9: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/9.jpg)
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)
![Page 10: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/10.jpg)
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)
![Page 11: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/11.jpg)
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)
![Page 12: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/12.jpg)
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)
![Page 13: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/13.jpg)
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)
![Page 14: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/14.jpg)
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)
![Page 15: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/15.jpg)
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)
![Page 16: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/16.jpg)
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)
![Page 17: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/17.jpg)
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)
![Page 18: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/18.jpg)
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)
![Page 19: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/19.jpg)
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)
![Page 20: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/20.jpg)
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)
![Page 21: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/21.jpg)
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)
![Page 22: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/22.jpg)
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)
![Page 23: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/23.jpg)
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)
![Page 24: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/24.jpg)
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)
![Page 25: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/25.jpg)
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)
![Page 26: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/26.jpg)
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)
![Page 27: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/27.jpg)
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)
![Page 28: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/28.jpg)
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)
![Page 29: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/29.jpg)
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)
![Page 30: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/30.jpg)
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)
![Page 31: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/31.jpg)
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)
![Page 32: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/32.jpg)
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)
![Page 33: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/33.jpg)
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)
![Page 34: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/34.jpg)
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)
![Page 35: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/35.jpg)
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)
![Page 36: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/36.jpg)
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)
![Page 37: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/37.jpg)
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)
![Page 38: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/38.jpg)
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)
![Page 39: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/39.jpg)
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)
![Page 40: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/40.jpg)
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)
![Page 41: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/41.jpg)
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)
![Page 42: The Gromov-Witten potential of the local P(1,2)todorov/banff.pdfThe Crepant Resolution Conjecture. Examples Local P(1,2) The Gromov-Witten potential of the local P(1,2) Gueorgui Todorov](https://reader030.vdocuments.us/reader030/viewer/2022040304/5e99ac78910b6878716e44e6/html5/thumbnails/42.jpg)
The Crepant Resolution Conjecture.Examples
Local P(1, 2)
Set-upThe invariantsPositive degreeThe Potential
Gueorgui Todorov Local P(1, 2)