Mirror symmetry at higher genus
Transcript of Mirror symmetry at higher genus
![Page 1: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/1.jpg)
Mirror symmetry at higher genus
Kevin Costello
Northwestern
06/10/2011
![Page 2: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/2.jpg)
Overview
1 Generalities on non-homological mirror symmetry: Gromov-Witteninvariants (A-model) are related to the “closed-string B-model”.
2 Introduce a new approach to the higher-genus closed string B-model(joint with Si Li, Harvard). This is based on theBershadsky-Cecotti-Ooguri-Vafa quantum field theory, and my workon renormalization. (Preprint available on my homepage, also Li’sthesis).
3 State a theorem of Li, that mirror symmetry holds for the ellipticcurve: the generating function of Gromov-Witten invariants of theelliptic curve coincides with the partition function of the BCOVquantum field theory of the mirror elliptic curve.
![Page 3: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/3.jpg)
Overview
1 Generalities on non-homological mirror symmetry: Gromov-Witteninvariants (A-model) are related to the “closed-string B-model”.
2 Introduce a new approach to the higher-genus closed string B-model(joint with Si Li, Harvard). This is based on theBershadsky-Cecotti-Ooguri-Vafa quantum field theory, and my workon renormalization. (Preprint available on my homepage, also Li’sthesis).
3 State a theorem of Li, that mirror symmetry holds for the ellipticcurve: the generating function of Gromov-Witten invariants of theelliptic curve coincides with the partition function of the BCOVquantum field theory of the mirror elliptic curve.
![Page 4: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/4.jpg)
Overview
1 Generalities on non-homological mirror symmetry: Gromov-Witteninvariants (A-model) are related to the “closed-string B-model”.
2 Introduce a new approach to the higher-genus closed string B-model(joint with Si Li, Harvard). This is based on theBershadsky-Cecotti-Ooguri-Vafa quantum field theory, and my workon renormalization. (Preprint available on my homepage, also Li’sthesis).
3 State a theorem of Li, that mirror symmetry holds for the ellipticcurve: the generating function of Gromov-Witten invariants of theelliptic curve coincides with the partition function of the BCOVquantum field theory of the mirror elliptic curve.
![Page 5: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/5.jpg)
Mirror symmetry pre-Kontsevich
Mirror symmetry was first formulated around 1990 (Candelas, de la Ossa,Green, and Greene-Plesser).
Original form of the conjecture: X ,X∨ a mirror pair of Calabi-Yauthree-folds. Then, the conjecture states
Numbers of rational curves on X ←→variations of Hodge structure of X∨
What does this mean? Rational curves and variations of Hodge structureare objects of a completely different nature.
Answer : (Givental, Barannikov). Both sides are encoded in a pair (V , L)where
V is a symplectic vector space.
L ⊂ V is a conic Lagrangian submanifold.
![Page 6: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/6.jpg)
Mirror symmetry pre-Kontsevich
Mirror symmetry was first formulated around 1990 (Candelas, de la Ossa,Green, and Greene-Plesser).
Original form of the conjecture: X ,X∨ a mirror pair of Calabi-Yauthree-folds. Then, the conjecture states
Numbers of rational curves on X ←→variations of Hodge structure of X∨
What does this mean? Rational curves and variations of Hodge structureare objects of a completely different nature.
Answer : (Givental, Barannikov). Both sides are encoded in a pair (V , L)where
V is a symplectic vector space.
L ⊂ V is a conic Lagrangian submanifold.
![Page 7: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/7.jpg)
Mirror symmetry pre-Kontsevich
Mirror symmetry was first formulated around 1990 (Candelas, de la Ossa,Green, and Greene-Plesser).
Original form of the conjecture: X ,X∨ a mirror pair of Calabi-Yauthree-folds. Then, the conjecture states
Numbers of rational curves on X ←→variations of Hodge structure of X∨
What does this mean? Rational curves and variations of Hodge structureare objects of a completely different nature.
Answer : (Givental, Barannikov). Both sides are encoded in a pair (V , L)where
V is a symplectic vector space.
L ⊂ V is a conic Lagrangian submanifold.
![Page 8: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/8.jpg)
B-model small Lagrangian cone
There are two versions of the story: small (without descendents) and large(includes descendents).
Let X be a Calabi-Yau 3-fold (equipped with holomorphic volume form).Let
V smallB (X ) = H3(X ,C)
MX = formal moduli space of Calabi-Yau manifolds near X
There’s a map
MX → V smallB (X )
Y 7→ [ΩY ] ∈ H3(X ).
LsmallB (X ) is the image of this map.
![Page 9: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/9.jpg)
B-model small Lagrangian cone
There are two versions of the story: small (without descendents) and large(includes descendents).
Let X be a Calabi-Yau 3-fold (equipped with holomorphic volume form).Let
V smallB (X ) = H3(X ,C)
MX = formal moduli space of Calabi-Yau manifolds near X
There’s a map
MX → V smallB (X )
Y 7→ [ΩY ] ∈ H3(X ).
LsmallB (X ) is the image of this map.
![Page 10: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/10.jpg)
A-model small Lagrangian cone (Givental)
Let
V smallA = ⊕3
p=0Hp,p(X )⊗ C((q))⟨αp,p, β3−p,3−p
⟩= (−1)p
∫Xα ∧ β
Let F0 be the generating function of genus 0 Gromov-Witten invariants:
F0 : H0,0 ⊕ H1,1 → C((q))(∂
∂α1. . .
∂
∂αkF0
)(0) =
∑qd
∫[M0,k,d (X )]virt
ev∗1 α1 . . . ev∗k αk .
Note:VA = T ∗(H0,0 ⊕ H1,1)⊗ C((q)).
LetLsmall
A (X ) = 1 + graph of dF0 ⊂ V smallA
Formal germ of Lagrangian cone at 1 ∈ V smallA .
![Page 11: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/11.jpg)
A-model small Lagrangian cone (Givental)
Let
V smallA = ⊕3
p=0Hp,p(X )⊗ C((q))⟨αp,p, β3−p,3−p
⟩= (−1)p
∫Xα ∧ β
Let F0 be the generating function of genus 0 Gromov-Witten invariants:
F0 : H0,0 ⊕ H1,1 → C((q))(∂
∂α1. . .
∂
∂αkF0
)(0) =
∑qd
∫[M0,k,d (X )]virt
ev∗1 α1 . . . ev∗k αk .
Note:VA = T ∗(H0,0 ⊕ H1,1)⊗ C((q)).
LetLsmall
A (X ) = 1 + graph of dF0 ⊂ V smallA
Formal germ of Lagrangian cone at 1 ∈ V smallA .
![Page 12: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/12.jpg)
A-model small Lagrangian cone (Givental)
Let
V smallA = ⊕3
p=0Hp,p(X )⊗ C((q))⟨αp,p, β3−p,3−p
⟩= (−1)p
∫Xα ∧ β
Let F0 be the generating function of genus 0 Gromov-Witten invariants:
F0 : H0,0 ⊕ H1,1 → C((q))(∂
∂α1. . .
∂
∂αkF0
)(0) =
∑qd
∫[M0,k,d (X )]virt
ev∗1 α1 . . . ev∗k αk .
Note:VA = T ∗(H0,0 ⊕ H1,1)⊗ C((q)).
LetLsmall
A (X ) = 1 + graph of dF0 ⊂ V smallA
Formal germ of Lagrangian cone at 1 ∈ V smallA .
![Page 13: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/13.jpg)
A-model small Lagrangian cone (Givental)
Let
V smallA = ⊕3
p=0Hp,p(X )⊗ C((q))⟨αp,p, β3−p,3−p
⟩= (−1)p
∫Xα ∧ β
Let F0 be the generating function of genus 0 Gromov-Witten invariants:
F0 : H0,0 ⊕ H1,1 → C((q))(∂
∂α1. . .
∂
∂αkF0
)(0) =
∑qd
∫[M0,k,d (X )]virt
ev∗1 α1 . . . ev∗k αk .
Note:VA = T ∗(H0,0 ⊕ H1,1)⊗ C((q)).
LetLsmall
A (X ) = 1 + graph of dF0 ⊂ V smallA
Formal germ of Lagrangian cone at 1 ∈ V smallA .
![Page 14: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/14.jpg)
Polarizations
A-model: Lagrangian cone in VA is defined over Spec C((q)).
B model to match this, we need to take a family of varieties overSpec C((q)).
Then VB = H3(Xq) is a symplectic vector space over C((q)).
A-model: symplectic vector space VA is polarized
VA =(H0,0 ⊕ H1,1
)⊕(H2,2 ⊕ H3,3
)(direct sum of Lagrangian subspaces).
B-model: polarization is subtle. One Lagrangian subspace is
F 2H3(X ) ⊂ H3(X ).
Complementary Lagrangian: naive guess is complex conjugate F2H3(X ).
Correct polarization: X → Spec C((q)), M : H3(X )→ H3(X ) monordomy.Look at Ker(M − 1)2 ⊂ H3(X ).
![Page 15: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/15.jpg)
Polarizations
A-model: Lagrangian cone in VA is defined over Spec C((q)).
B model to match this, we need to take a family of varieties overSpec C((q)).
Then VB = H3(Xq) is a symplectic vector space over C((q)).
A-model: symplectic vector space VA is polarized
VA =(H0,0 ⊕ H1,1
)⊕(H2,2 ⊕ H3,3
)(direct sum of Lagrangian subspaces).
B-model: polarization is subtle. One Lagrangian subspace is
F 2H3(X ) ⊂ H3(X ).
Complementary Lagrangian: naive guess is complex conjugate F2H3(X ).
Correct polarization: X → Spec C((q)), M : H3(X )→ H3(X ) monordomy.Look at Ker(M − 1)2 ⊂ H3(X ).
![Page 16: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/16.jpg)
Polarizations
A-model: Lagrangian cone in VA is defined over Spec C((q)).
B model to match this, we need to take a family of varieties overSpec C((q)).
Then VB = H3(Xq) is a symplectic vector space over C((q)).
A-model: symplectic vector space VA is polarized
VA =(H0,0 ⊕ H1,1
)⊕(H2,2 ⊕ H3,3
)(direct sum of Lagrangian subspaces).
B-model: polarization is subtle. One Lagrangian subspace is
F 2H3(X ) ⊂ H3(X ).
Complementary Lagrangian: naive guess is complex conjugate F2H3(X ).
Correct polarization: X → Spec C((q)), M : H3(X )→ H3(X ) monordomy.Look at Ker(M − 1)2 ⊂ H3(X ).
![Page 17: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/17.jpg)
Polarizations
A-model: Lagrangian cone in VA is defined over Spec C((q)).
B model to match this, we need to take a family of varieties overSpec C((q)).
Then VB = H3(Xq) is a symplectic vector space over C((q)).
A-model: symplectic vector space VA is polarized
VA =(H0,0 ⊕ H1,1
)⊕(H2,2 ⊕ H3,3
)(direct sum of Lagrangian subspaces).
B-model: polarization is subtle. One Lagrangian subspace is
F 2H3(X ) ⊂ H3(X ).
Complementary Lagrangian: naive guess is complex conjugate F2H3(X ).
Correct polarization: X → Spec C((q)), M : H3(X )→ H3(X ) monordomy.Look at Ker(M − 1)2 ⊂ H3(X ).
![Page 18: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/18.jpg)
Descendents
If α1, . . . , αn ∈ H∗(X ), define
〈τk1(α1), . . . , τkn(αn)〉g ,n,d =
∫[Mg,n,d (X )]virt
ψk11 ev∗1(α1) . . . ψkn
n ev∗n(αn).
The generating function
Fg ∈ O(H∗(X )[[t]])⊗ C[[q]]
is defined by(∂
∂(tk1α1). . .
∂
∂(tknαn)Fg
)(0) =
∑qd 〈τk1(α1), . . . , τkn(αn)〉g ,n,d
![Page 19: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/19.jpg)
Descendents
If α1, . . . , αn ∈ H∗(X ), define
〈τk1(α1), . . . , τkn(αn)〉g ,n,d =
∫[Mg,n,d (X )]virt
ψk11 ev∗1(α1) . . . ψkn
n ev∗n(αn).
The generating function
Fg ∈ O(H∗(X )[[t]])⊗ C[[q]]
is defined by(∂
∂(tk1α1). . .
∂
∂(tknαn)Fg
)(0) =
∑qd 〈τk1(α1), . . . , τkn(αn)〉g ,n,d
![Page 20: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/20.jpg)
A-model symplectic formalism with descendents (Givental)
LetV big
A (X ) = H∗(X )((t))
with symplectic pairing
〈αf (t), βg(t)〉 =
(∫Xαβ
)Res f (t)g(−t)dt.
IdentifyV big
A (X ) = T ∗(H(X )[[t]]).
DefineLbig
A = 1 + Graph(dF0) ⊂ H∗(X )((t))
![Page 21: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/21.jpg)
A-model symplectic formalism with descendents (Givental)
LetV big
A (X ) = H∗(X )((t))
with symplectic pairing
〈αf (t), βg(t)〉 =
(∫Xαβ
)Res f (t)g(−t)dt.
IdentifyV big
A (X ) = T ∗(H(X )[[t]]).
DefineLbig
A = 1 + Graph(dF0) ⊂ H∗(X )((t))
![Page 22: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/22.jpg)
Polyvector fields
X a CY of dimension d . Let
PVi ,j(X ) = Ω0,j(X ,∧iTX ).
Contracting with Ω ∈ H0(X ,KX ) gives an isomorphsim
PVi ,j(X ) ∼= Ωd−i ,j(X ).
Define
∂ : PVi ,j(X )→ PVi ,j+1(X ) ∂ : PVi ,j(X )→ PVi ,j−1(X )
corresponding to usual ∂, ∂ operators on Ω∗,∗(X ).
PV∗,∗(X ) has graded-commutative product, and trace
Tr : PV3,3(X )→ C Tr(α) =∫X Ω(α ∨ Ω).
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Polyvector fields
X a CY of dimension d . Let
PVi ,j(X ) = Ω0,j(X ,∧iTX ).
Contracting with Ω ∈ H0(X ,KX ) gives an isomorphsim
PVi ,j(X ) ∼= Ωd−i ,j(X ).
Define
∂ : PVi ,j(X )→ PVi ,j+1(X ) ∂ : PVi ,j(X )→ PVi ,j−1(X )
corresponding to usual ∂, ∂ operators on Ω∗,∗(X ).
PV∗,∗(X ) has graded-commutative product, and trace
Tr : PV3,3(X )→ C Tr(α) =∫X Ω(α ∨ Ω).
![Page 24: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/24.jpg)
Polyvector fields
X a CY of dimension d . Let
PVi ,j(X ) = Ω0,j(X ,∧iTX ).
Contracting with Ω ∈ H0(X ,KX ) gives an isomorphsim
PVi ,j(X ) ∼= Ωd−i ,j(X ).
Define
∂ : PVi ,j(X )→ PVi ,j+1(X ) ∂ : PVi ,j(X )→ PVi ,j−1(X )
corresponding to usual ∂, ∂ operators on Ω∗,∗(X ).
PV∗,∗(X ) has graded-commutative product, and trace
Tr : PV3,3(X )→ C Tr(α) =∫X Ω(α ∨ Ω).
![Page 25: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/25.jpg)
B-model Lagrangian cone with descendents (Barannikov)
LetV big
B = PV(X )((t)).
Differential Q = ∂ + t∂, symplectic pairing
〈f (t)α, g(t)β〉 = Tr(αβ) Res f (t)g(−t)dt.
Define Lagrangian cone LbigB ⊂ VB by
LbigB = tef /t | f ∈ PV(X )[[t]].
Formal germ of cone defined near 1 ∈ PV(X )((t)).
LsmallB is moduli of deformations of CY X .
LbigB is moduli of “extended” CY deformations of X . (Equivalent to
deformations of Perf(X ) as a dg Calabi-Yau category).
Easy to verify: LbigB is preserved by the differential (and satisfies Givental’s
other axioms).
![Page 26: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/26.jpg)
B-model Lagrangian cone with descendents (Barannikov)
LetV big
B = PV(X )((t)).
Differential Q = ∂ + t∂, symplectic pairing
〈f (t)α, g(t)β〉 = Tr(αβ) Res f (t)g(−t)dt.
Define Lagrangian cone LbigB ⊂ VB by
LbigB = tef /t | f ∈ PV(X )[[t]].
Formal germ of cone defined near 1 ∈ PV(X )((t)).
LsmallB is moduli of deformations of CY X .
LbigB is moduli of “extended” CY deformations of X . (Equivalent to
deformations of Perf(X ) as a dg Calabi-Yau category).
Easy to verify: LbigB is preserved by the differential (and satisfies Givental’s
other axioms).
![Page 27: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/27.jpg)
B-model Lagrangian cone with descendents (Barannikov)
LetV big
B = PV(X )((t)).
Differential Q = ∂ + t∂, symplectic pairing
〈f (t)α, g(t)β〉 = Tr(αβ) Res f (t)g(−t)dt.
Define Lagrangian cone LbigB ⊂ VB by
LbigB = tef /t | f ∈ PV(X )[[t]].
Formal germ of cone defined near 1 ∈ PV(X )((t)).
LsmallB is moduli of deformations of CY X .
LbigB is moduli of “extended” CY deformations of X . (Equivalent to
deformations of Perf(X ) as a dg Calabi-Yau category).
Easy to verify: LbigB is preserved by the differential (and satisfies Givental’s
other axioms).
![Page 28: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/28.jpg)
B-model Lagrangian cone with descendents (Barannikov)
LetV big
B = PV(X )((t)).
Differential Q = ∂ + t∂, symplectic pairing
〈f (t)α, g(t)β〉 = Tr(αβ) Res f (t)g(−t)dt.
Define Lagrangian cone LbigB ⊂ VB by
LbigB = tef /t | f ∈ PV(X )[[t]].
Formal germ of cone defined near 1 ∈ PV(X )((t)).
LsmallB is moduli of deformations of CY X .
LbigB is moduli of “extended” CY deformations of X . (Equivalent to
deformations of Perf(X ) as a dg Calabi-Yau category).
Easy to verify: LbigB is preserved by the differential (and satisfies Givental’s
other axioms).
![Page 29: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/29.jpg)
Genus 0 mirror symmetry conjecture with descendents
Conjecture
X a Calabi-Yau, X∨ → Spec C((q)) the mirror family.Then there is a quasi-isomorphism of symplectic vector spaces
V bigA (X ) = H∗(X )((t)) ' PV(X∨)((t)) = V big
B (X )
taking LbigA to Lbig
B .
Proved in many cases by Givental, Lian-Liu-Yau, Barannikov.
![Page 30: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/30.jpg)
Higher genus picture
Genus 0 A and B-model: a Lagrangian submanifold L in a symplecticvector space V .
Higher genus: we should quantize this picture. Symplectic vector space Vquantizes to the Weyl algebra
W(V ) = free algebra over C[[~]] generated by α ∈ V ∨
with relations [α, β] = ~ 〈α, β〉 .
Lagrangian submanifold L ⊂ V quantizes to a vector in Fock(V ), theFock module for W(V ).
![Page 31: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/31.jpg)
Higher genus picture
Genus 0 A and B-model: a Lagrangian submanifold L in a symplecticvector space V .
Higher genus: we should quantize this picture. Symplectic vector space Vquantizes to the Weyl algebra
W(V ) = free algebra over C[[~]] generated by α ∈ V ∨
with relations [α, β] = ~ 〈α, β〉 .
Lagrangian submanifold L ⊂ V quantizes to a vector in Fock(V ), theFock module for W(V ).
![Page 32: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/32.jpg)
Higher genus picture
Genus 0 A and B-model: a Lagrangian submanifold L in a symplecticvector space V .
Higher genus: we should quantize this picture. Symplectic vector space Vquantizes to the Weyl algebra
W(V ) = free algebra over C[[~]] generated by α ∈ V ∨
with relations [α, β] = ~ 〈α, β〉 .
Lagrangian submanifold L ⊂ V quantizes to a vector in Fock(V ), theFock module for W(V ).
![Page 33: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/33.jpg)
A-model at higher genus (Givental)
VA = T ∗H∗(X )[[t]].
So,Fock(VA) = O(H∗(X )[[t]])
algebra of functions on H∗(X )[[t]].
Let
Fg ∈ O(H∗(X )[[t]])⊗ C[[q]]
be the generating function for genus g Gromov-Witten invariants withdescendents. Then
ZA = exp(∑
~g−1Fg
)∈ Fock(VA)[[q]]
A-model partition function.
Vector in Fock space which in ~→ 0 limit becomes LA ⊂ VA.
![Page 34: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/34.jpg)
A-model at higher genus (Givental)
VA = T ∗H∗(X )[[t]].
So,Fock(VA) = O(H∗(X )[[t]])
algebra of functions on H∗(X )[[t]]. Let
Fg ∈ O(H∗(X )[[t]])⊗ C[[q]]
be the generating function for genus g Gromov-Witten invariants withdescendents.
Then
ZA = exp(∑
~g−1Fg
)∈ Fock(VA)[[q]]
A-model partition function.
Vector in Fock space which in ~→ 0 limit becomes LA ⊂ VA.
![Page 35: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/35.jpg)
A-model at higher genus (Givental)
VA = T ∗H∗(X )[[t]].
So,Fock(VA) = O(H∗(X )[[t]])
algebra of functions on H∗(X )[[t]]. Let
Fg ∈ O(H∗(X )[[t]])⊗ C[[q]]
be the generating function for genus g Gromov-Witten invariants withdescendents. Then
ZA = exp(∑
~g−1Fg
)∈ Fock(VA)[[q]]
A-model partition function.
Vector in Fock space which in ~→ 0 limit becomes LA ⊂ VA.
![Page 36: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/36.jpg)
B-model partition function?
“Small” B-model partition function Z smallB should be a state in the Fock
space for H3(X ,C).
Large B-model partition function should be a state in the Fock space forPV(X )((t)).
RecallLB = tef /t | f ∈ PV(X )[[t]] ⊂ PV(X )((t)).
LB is the extended moduli of deformations of X .
Problem
Quantize the Lagrangian submanifold LB ⊂ PV(X )((t)).
We will discuss how to do this using QFT.
![Page 37: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/37.jpg)
B-model partition function?
“Small” B-model partition function Z smallB should be a state in the Fock
space for H3(X ,C).
Large B-model partition function should be a state in the Fock space forPV(X )((t)).
RecallLB = tef /t | f ∈ PV(X )[[t]] ⊂ PV(X )((t)).
LB is the extended moduli of deformations of X .
Problem
Quantize the Lagrangian submanifold LB ⊂ PV(X )((t)).
We will discuss how to do this using QFT.
![Page 38: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/38.jpg)
B-model partition function?
“Small” B-model partition function Z smallB should be a state in the Fock
space for H3(X ,C).
Large B-model partition function should be a state in the Fock space forPV(X )((t)).
RecallLB = tef /t | f ∈ PV(X )[[t]] ⊂ PV(X )((t)).
LB is the extended moduli of deformations of X .
Problem
Quantize the Lagrangian submanifold LB ⊂ PV(X )((t)).
We will discuss how to do this using QFT.
![Page 39: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/39.jpg)
B-model partition function?
“Small” B-model partition function Z smallB should be a state in the Fock
space for H3(X ,C).
Large B-model partition function should be a state in the Fock space forPV(X )((t)).
RecallLB = tef /t | f ∈ PV(X )[[t]] ⊂ PV(X )((t)).
LB is the extended moduli of deformations of X .
Problem
Quantize the Lagrangian submanifold LB ⊂ PV(X )((t)).
We will discuss how to do this using QFT.
![Page 40: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/40.jpg)
B-model partition function?
“Small” B-model partition function Z smallB should be a state in the Fock
space for H3(X ,C).
Large B-model partition function should be a state in the Fock space forPV(X )((t)).
RecallLB = tef /t | f ∈ PV(X )[[t]] ⊂ PV(X )((t)).
LB is the extended moduli of deformations of X .
Problem
Quantize the Lagrangian submanifold LB ⊂ PV(X )((t)).
We will discuss how to do this using QFT.
![Page 41: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/41.jpg)
Quantum field theory and the B-model
Small B-model partition function should be a “quantization” of moduli ofCalabi-Yaus
MX ⊂ H3(X ,C).
(Lagrangian submanifold).
Bershadsky-Cecotti-Ooguri-Vafa consider a “gravitational” quantum fieldtheory on X , a Calabi-Yau three-fold.
Fields include Beltrami differentials Ω0,1(X ,TX ), the gauge group is thediffeomorphism group.
The space of solutions to the equation of motion is MX , formal modulispace of Calabi-Yaus near X .
They argue that the partition function of this theory is the B-modelpartition function.
Witten : the BCOV partition function is a state in Fock(H3(X )).
![Page 42: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/42.jpg)
Quantum field theory and the B-model
Small B-model partition function should be a “quantization” of moduli ofCalabi-Yaus
MX ⊂ H3(X ,C).
(Lagrangian submanifold).
Bershadsky-Cecotti-Ooguri-Vafa consider a “gravitational” quantum fieldtheory on X , a Calabi-Yau three-fold.
Fields include Beltrami differentials Ω0,1(X ,TX ), the gauge group is thediffeomorphism group.
The space of solutions to the equation of motion is MX , formal modulispace of Calabi-Yaus near X .
They argue that the partition function of this theory is the B-modelpartition function.
Witten : the BCOV partition function is a state in Fock(H3(X )).
![Page 43: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/43.jpg)
Quantum field theory and the B-model
Small B-model partition function should be a “quantization” of moduli ofCalabi-Yaus
MX ⊂ H3(X ,C).
(Lagrangian submanifold).
Bershadsky-Cecotti-Ooguri-Vafa consider a “gravitational” quantum fieldtheory on X , a Calabi-Yau three-fold.
Fields include Beltrami differentials Ω0,1(X ,TX ), the gauge group is thediffeomorphism group.
The space of solutions to the equation of motion is MX , formal modulispace of Calabi-Yaus near X .
They argue that the partition function of this theory is the B-modelpartition function.
Witten : the BCOV partition function is a state in Fock(H3(X )).
![Page 44: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/44.jpg)
Quantum field theory and the B-model
Small B-model partition function should be a “quantization” of moduli ofCalabi-Yaus
MX ⊂ H3(X ,C).
(Lagrangian submanifold).
Bershadsky-Cecotti-Ooguri-Vafa consider a “gravitational” quantum fieldtheory on X , a Calabi-Yau three-fold.
Fields include Beltrami differentials Ω0,1(X ,TX ), the gauge group is thediffeomorphism group.
The space of solutions to the equation of motion is MX , formal modulispace of Calabi-Yaus near X .
They argue that the partition function of this theory is the B-modelpartition function.
Witten : the BCOV partition function is a state in Fock(H3(X )).
![Page 45: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/45.jpg)
Extended BCOV theory
We want to quantize extended moduli space
LbigB ⊂ PV(X )((t))
to produce a state in the Fock space for PV(X )((t)).
NotePV(X )((t)) ∼= T ∗ PV(X )[[t]]
as graded vector spaces (not as cochain complexes).
Fields of extended BCOV theory are
PV(X )[[t]].
Extended BCOV action is the functional
F0 ∈ O(PV(X )[[t]])
such thatGraph(dF0) = LB .
![Page 46: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/46.jpg)
Extended BCOV theory
We want to quantize extended moduli space
LbigB ⊂ PV(X )((t))
to produce a state in the Fock space for PV(X )((t)).
NotePV(X )((t)) ∼= T ∗ PV(X )[[t]]
as graded vector spaces (not as cochain complexes).
Fields of extended BCOV theory are
PV(X )[[t]].
Extended BCOV action is the functional
F0 ∈ O(PV(X )[[t]])
such thatGraph(dF0) = LB .
![Page 47: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/47.jpg)
Extended BCOV theory
We want to quantize extended moduli space
LbigB ⊂ PV(X )((t))
to produce a state in the Fock space for PV(X )((t)).
NotePV(X )((t)) ∼= T ∗ PV(X )[[t]]
as graded vector spaces (not as cochain complexes).
Fields of extended BCOV theory are
PV(X )[[t]].
Extended BCOV action is the functional
F0 ∈ O(PV(X )[[t]])
such thatGraph(dF0) = LB .
![Page 48: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/48.jpg)
Extended BCOV theory
We want to quantize extended moduli space
LbigB ⊂ PV(X )((t))
to produce a state in the Fock space for PV(X )((t)).
NotePV(X )((t)) ∼= T ∗ PV(X )[[t]]
as graded vector spaces (not as cochain complexes).
Fields of extended BCOV theory are
PV(X )[[t]].
Extended BCOV action is the functional
F0 ∈ O(PV(X )[[t]])
such thatGraph(dF0) = LB .
![Page 49: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/49.jpg)
Extended BCOV theory
Concretely:F0 ∈ O(PV(X )[[t]])
satisfies(∂
∂(α1tk1). . .
∂
∂(αntkn)F0
)(0) = Tr(α1 . . . αn)
∫M0,n
ψk11 . . . ψkn
n .
(αi ∈ PV(X )).
This a degenerate QFT: quadratic term is ill-defined as a functional, butit’s inverse (Green’s kernel/propagator) makes sense.
BV formalism: classical field theories are given by differential gradedsymplectic manifolds. Symplectic form is of cohomology degree −1.Action functional S , Poisson bracket −,− and differential are related byS ,− = d.
Here: dg Poisson manifold, with a potential F0 satisfying F0,− = d.Can still be treated using usual techniques.
![Page 50: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/50.jpg)
Extended BCOV theory
Concretely:F0 ∈ O(PV(X )[[t]])
satisfies(∂
∂(α1tk1). . .
∂
∂(αntkn)F0
)(0) = Tr(α1 . . . αn)
∫M0,n
ψk11 . . . ψkn
n .
(αi ∈ PV(X )).
This a degenerate QFT: quadratic term is ill-defined as a functional, butit’s inverse (Green’s kernel/propagator) makes sense.
BV formalism: classical field theories are given by differential gradedsymplectic manifolds. Symplectic form is of cohomology degree −1.Action functional S , Poisson bracket −,− and differential are related byS ,− = d.
Here: dg Poisson manifold, with a potential F0 satisfying F0,− = d.Can still be treated using usual techniques.
![Page 51: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/51.jpg)
Extended BCOV theory
Concretely:F0 ∈ O(PV(X )[[t]])
satisfies(∂
∂(α1tk1). . .
∂
∂(αntkn)F0
)(0) = Tr(α1 . . . αn)
∫M0,n
ψk11 . . . ψkn
n .
(αi ∈ PV(X )).
This a degenerate QFT: quadratic term is ill-defined as a functional, butit’s inverse (Green’s kernel/propagator) makes sense.
BV formalism: classical field theories are given by differential gradedsymplectic manifolds. Symplectic form is of cohomology degree −1.Action functional S , Poisson bracket −,− and differential are related byS ,− = d.
Here: dg Poisson manifold, with a potential F0 satisfying F0,− = d.Can still be treated using usual techniques.
![Page 52: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/52.jpg)
Extended BCOV theory
Concretely:F0 ∈ O(PV(X )[[t]])
satisfies(∂
∂(α1tk1). . .
∂
∂(αntkn)F0
)(0) = Tr(α1 . . . αn)
∫M0,n
ψk11 . . . ψkn
n .
(αi ∈ PV(X )).
This a degenerate QFT: quadratic term is ill-defined as a functional, butit’s inverse (Green’s kernel/propagator) makes sense.
BV formalism: classical field theories are given by differential gradedsymplectic manifolds. Symplectic form is of cohomology degree −1.Action functional S , Poisson bracket −,− and differential are related byS ,− = d.
Here: dg Poisson manifold, with a potential F0 satisfying F0,− = d.Can still be treated using usual techniques.
![Page 53: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/53.jpg)
The classical master equation
RecallPV(X )((t)) = T ∗ (PV(X )[[t]])
as graded vector space but not as a cochain complex.
If Φ ∈ O(PV(X )[[t]]) then
Graph(dΦ) ⊂ PV(X )((t))
is preserved by the differential on PV(X )((t)) ⇐⇒
QΦ + 12Φ,Φ = 0
−,− a Poisson bracket on O(PV(X )[[t]]) of degree 1.
This equation is called classical master equation.
Since LbigB is preserved by the differential, F0 satisfies classical master
equation.
![Page 54: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/54.jpg)
The classical master equation
RecallPV(X )((t)) = T ∗ (PV(X )[[t]])
as graded vector space but not as a cochain complex.
If Φ ∈ O(PV(X )[[t]]) then
Graph(dΦ) ⊂ PV(X )((t))
is preserved by the differential on PV(X )((t)) ⇐⇒
QΦ + 12Φ,Φ = 0
−,− a Poisson bracket on O(PV(X )[[t]]) of degree 1.
This equation is called classical master equation.
Since LbigB is preserved by the differential, F0 satisfies classical master
equation.
![Page 55: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/55.jpg)
Interpreting the classical master equation
Two interpretations of F0 ∈ O(PV(X )[[t]]):
1 Classical action functional for generalized BCOV theory.
2 Generating function for Lagrangian submanifold ofPV(X )((t)) = V big
B .
Classical master equation has interpretation in both settings:
1 Consistency condition for classical gauge theory (usual interpretation).
2 Lagrangian submanifold is preserved by the differential.
Aim : quantize this classical field theory. My book Renormalization andeffective field theory gives the definition of quantization we use, and allowsone to construct quantizations by obstruction theory (term by term in ~).
![Page 56: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/56.jpg)
Interpreting the classical master equation
Two interpretations of F0 ∈ O(PV(X )[[t]]):
1 Classical action functional for generalized BCOV theory.
2 Generating function for Lagrangian submanifold ofPV(X )((t)) = V big
B .
Classical master equation has interpretation in both settings:
1 Consistency condition for classical gauge theory (usual interpretation).
2 Lagrangian submanifold is preserved by the differential.
Aim : quantize this classical field theory. My book Renormalization andeffective field theory gives the definition of quantization we use, and allowsone to construct quantizations by obstruction theory (term by term in ~).
![Page 57: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/57.jpg)
Interpreting the classical master equation
Two interpretations of F0 ∈ O(PV(X )[[t]]):
1 Classical action functional for generalized BCOV theory.
2 Generating function for Lagrangian submanifold ofPV(X )((t)) = V big
B .
Classical master equation has interpretation in both settings:
1 Consistency condition for classical gauge theory (usual interpretation).
2 Lagrangian submanifold is preserved by the differential.
Aim : quantize this classical field theory. My book Renormalization andeffective field theory gives the definition of quantization we use, and allowsone to construct quantizations by obstruction theory (term by term in ~).
![Page 58: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/58.jpg)
Interpreting the classical master equation
Two interpretations of F0 ∈ O(PV(X )[[t]]):
1 Classical action functional for generalized BCOV theory.
2 Generating function for Lagrangian submanifold ofPV(X )((t)) = V big
B .
Classical master equation has interpretation in both settings:
1 Consistency condition for classical gauge theory (usual interpretation).
2 Lagrangian submanifold is preserved by the differential.
Aim : quantize this classical field theory. My book Renormalization andeffective field theory gives the definition of quantization we use, and allowsone to construct quantizations by obstruction theory (term by term in ~).
![Page 59: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/59.jpg)
Interpreting the classical master equation
Two interpretations of F0 ∈ O(PV(X )[[t]]):
1 Classical action functional for generalized BCOV theory.
2 Generating function for Lagrangian submanifold ofPV(X )((t)) = V big
B .
Classical master equation has interpretation in both settings:
1 Consistency condition for classical gauge theory (usual interpretation).
2 Lagrangian submanifold is preserved by the differential.
Aim : quantize this classical field theory. My book Renormalization andeffective field theory gives the definition of quantization we use, and allowsone to construct quantizations by obstruction theory (term by term in ~).
![Page 60: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/60.jpg)
Interpreting the classical master equation
Two interpretations of F0 ∈ O(PV(X )[[t]]):
1 Classical action functional for generalized BCOV theory.
2 Generating function for Lagrangian submanifold ofPV(X )((t)) = V big
B .
Classical master equation has interpretation in both settings:
1 Consistency condition for classical gauge theory (usual interpretation).
2 Lagrangian submanifold is preserved by the differential.
Aim : quantize this classical field theory. My book Renormalization andeffective field theory gives the definition of quantization we use, and allowsone to construct quantizations by obstruction theory (term by term in ~).
![Page 61: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/61.jpg)
Quantization (naive approach)
Naive idea: look for a series
F =∑
~gFg ∈ O(PV(X )[[t]])[[~]]
satisfying quantum master equation
QF + 12F,F+ ~∆F = 0.
QME has two interpretations:
1 Consistency condition for quantum gauge theory.
2 exp(F/~) is killed by the differential in the Fock space for PV(X )((t)).
Problem : ∆ is not defined (because of ultraviolet divergences of quantumfield theory).
![Page 62: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/62.jpg)
Quantization (naive approach)
Naive idea: look for a series
F =∑
~gFg ∈ O(PV(X )[[t]])[[~]]
satisfying quantum master equation
QF + 12F,F+ ~∆F = 0.
QME has two interpretations:
1 Consistency condition for quantum gauge theory.
2 exp(F/~) is killed by the differential in the Fock space for PV(X )((t)).
Problem : ∆ is not defined (because of ultraviolet divergences of quantumfield theory).
![Page 63: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/63.jpg)
Quantization (naive approach)
Naive idea: look for a series
F =∑
~gFg ∈ O(PV(X )[[t]])[[~]]
satisfying quantum master equation
QF + 12F,F+ ~∆F = 0.
QME has two interpretations:
1 Consistency condition for quantum gauge theory.
2 exp(F/~) is killed by the differential in the Fock space for PV(X )((t)).
Problem : ∆ is not defined (because of ultraviolet divergences of quantumfield theory).
![Page 64: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/64.jpg)
Quantization (naive approach)
Naive idea: look for a series
F =∑
~gFg ∈ O(PV(X )[[t]])[[~]]
satisfying quantum master equation
QF + 12F,F+ ~∆F = 0.
QME has two interpretations:
1 Consistency condition for quantum gauge theory.
2 exp(F/~) is killed by the differential in the Fock space for PV(X )((t)).
Problem : ∆ is not defined (because of ultraviolet divergences of quantumfield theory).
![Page 65: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/65.jpg)
Quantization (naive approach)
Naive idea: look for a series
F =∑
~gFg ∈ O(PV(X )[[t]])[[~]]
satisfying quantum master equation
QF + 12F,F+ ~∆F = 0.
QME has two interpretations:
1 Consistency condition for quantum gauge theory.
2 exp(F/~) is killed by the differential in the Fock space for PV(X )((t)).
Problem : ∆ is not defined (because of ultraviolet divergences of quantumfield theory).
![Page 66: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/66.jpg)
Definition of quantization
Solution (Renormalization and effective field theory, C. 2011): givesgeneral definition of a perturbative QFT.
Definition
A quantization of the BCOV theory is a family of action functionals
F[L] ∈ O(PV(X )[[t]])[[~]]
(F[L] is “scale L effective action”). These must satisfy:
Renormalization group equation: F[L] expressed in terms of F[ε] by(roughly) “integrating out modes of wave-length between ε and L”.
Each F[L] satisfies quantum master equation
QF[L] + 12F[L],F[L]L + ~∆LF[L] = 0.
Locality axiom : as L→ 0, F[L] approximated by the integral of aLagrangian.
![Page 67: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/67.jpg)
Definition of quantization
Solution (Renormalization and effective field theory, C. 2011): givesgeneral definition of a perturbative QFT.
Definition
A quantization of the BCOV theory is a family of action functionals
F[L] ∈ O(PV(X )[[t]])[[~]]
(F[L] is “scale L effective action”). These must satisfy:
Renormalization group equation: F[L] expressed in terms of F[ε] by(roughly) “integrating out modes of wave-length between ε and L”.
Each F[L] satisfies quantum master equation
QF[L] + 12F[L],F[L]L + ~∆LF[L] = 0.
Locality axiom : as L→ 0, F[L] approximated by the integral of aLagrangian.
![Page 68: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/68.jpg)
Definition of quantization
Solution (Renormalization and effective field theory, C. 2011): givesgeneral definition of a perturbative QFT.
Definition
A quantization of the BCOV theory is a family of action functionals
F[L] ∈ O(PV(X )[[t]])[[~]]
(F[L] is “scale L effective action”). These must satisfy:
Renormalization group equation: F[L] expressed in terms of F[ε] by(roughly) “integrating out modes of wave-length between ε and L”.
Each F[L] satisfies quantum master equation
QF[L] + 12F[L],F[L]L + ~∆LF[L] = 0.
Locality axiom : as L→ 0, F[L] approximated by the integral of aLagrangian.
![Page 69: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/69.jpg)
Definition of quantization
Solution (Renormalization and effective field theory, C. 2011): givesgeneral definition of a perturbative QFT.
Definition
A quantization of the BCOV theory is a family of action functionals
F[L] ∈ O(PV(X )[[t]])[[~]]
(F[L] is “scale L effective action”). These must satisfy:
Renormalization group equation: F[L] expressed in terms of F[ε] by(roughly) “integrating out modes of wave-length between ε and L”.
Each F[L] satisfies quantum master equation
QF[L] + 12F[L],F[L]L + ~∆LF[L] = 0.
Locality axiom : as L→ 0, F[L] approximated by the integral of aLagrangian.
![Page 70: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/70.jpg)
Quantizing the BCOV theory
In general, one can construct quantizations of a classical theory (in thissense) using obstruction theory, term by term in ~.
Theorem (C., Si Li )
The BCOV theory admits a (canonical) quantization on any complex torus.
Proof: obstruction theory/ cohomological calculations.
Best results in the case of an elliptic curve: there the quantization isunique. The situation in higher dimensions is not so satisfactory (yet!)
Quantum master equation and RGE imply we can construct a cohomologyclass
[exp(F[L]/~)] ∈ H∗(Fock(PV(X )((t))))
independent of L.
This will be the partition function of the BCOV theory.
![Page 71: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/71.jpg)
Quantizing the BCOV theory
In general, one can construct quantizations of a classical theory (in thissense) using obstruction theory, term by term in ~.
Theorem (C., Si Li )
The BCOV theory admits a (canonical) quantization on any complex torus.
Proof: obstruction theory/ cohomological calculations.
Best results in the case of an elliptic curve: there the quantization isunique. The situation in higher dimensions is not so satisfactory (yet!)
Quantum master equation and RGE imply we can construct a cohomologyclass
[exp(F[L]/~)] ∈ H∗(Fock(PV(X )((t))))
independent of L.
This will be the partition function of the BCOV theory.
![Page 72: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/72.jpg)
Quantizing the BCOV theory
In general, one can construct quantizations of a classical theory (in thissense) using obstruction theory, term by term in ~.
Theorem (C., Si Li )
The BCOV theory admits a (canonical) quantization on any complex torus.
Proof: obstruction theory/ cohomological calculations.
Best results in the case of an elliptic curve: there the quantization isunique. The situation in higher dimensions is not so satisfactory (yet!)
Quantum master equation and RGE imply we can construct a cohomologyclass
[exp(F[L]/~)] ∈ H∗(Fock(PV(X )((t))))
independent of L.
This will be the partition function of the BCOV theory.
![Page 73: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/73.jpg)
Quantizing the BCOV theory
In general, one can construct quantizations of a classical theory (in thissense) using obstruction theory, term by term in ~.
Theorem (C., Si Li )
The BCOV theory admits a (canonical) quantization on any complex torus.
Proof: obstruction theory/ cohomological calculations.
Best results in the case of an elliptic curve: there the quantization isunique. The situation in higher dimensions is not so satisfactory (yet!)
Quantum master equation and RGE imply we can construct a cohomologyclass
[exp(F[L]/~)] ∈ H∗(Fock(PV(X )((t))))
independent of L.
This will be the partition function of the BCOV theory.
![Page 74: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/74.jpg)
Mirror symmetry for the elliptic curve
E elliptic curve. Mirror family: E∨τ , τ ∈ H, q = e2πiτ .
Symplectic vector spaces:
V bigA (E ) = H∗(E )((t))⊗ C((q))
V bigB (Eτ ) = H∗
(Ω0,∗(E ,∧∗TE )((t))
), differential ∂ + t∂.
There’s a natural isomorphism of symplectic vector spaces
V bigA (E ) ∼= V big
B (Eτ ).
Theorem (Li)
Under this isomorphism, the A-model partition functionZA(E ) ∈ Fock(V big
A (E )) corresponds to ZB(E∨) ∈ Fock(V bigB (E∨)).
This means all GW invariants of an elliptic curve E can be computed fromquantum BCOV theory on the mirror elliptic curve E∨.
![Page 75: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/75.jpg)
Mirror symmetry for the elliptic curve
E elliptic curve. Mirror family: E∨τ , τ ∈ H, q = e2πiτ .
Symplectic vector spaces:
V bigA (E ) = H∗(E )((t))⊗ C((q))
V bigB (Eτ ) = H∗
(Ω0,∗(E ,∧∗TE )((t))
), differential ∂ + t∂.
There’s a natural isomorphism of symplectic vector spaces
V bigA (E ) ∼= V big
B (Eτ ).
Theorem (Li)
Under this isomorphism, the A-model partition functionZA(E ) ∈ Fock(V big
A (E )) corresponds to ZB(E∨) ∈ Fock(V bigB (E∨)).
This means all GW invariants of an elliptic curve E can be computed fromquantum BCOV theory on the mirror elliptic curve E∨.
![Page 76: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/76.jpg)
Mirror symmetry for the elliptic curve
E elliptic curve. Mirror family: E∨τ , τ ∈ H, q = e2πiτ .
Symplectic vector spaces:
V bigA (E ) = H∗(E )((t))⊗ C((q))
V bigB (Eτ ) = H∗
(Ω0,∗(E ,∧∗TE )((t))
), differential ∂ + t∂.
There’s a natural isomorphism of symplectic vector spaces
V bigA (E ) ∼= V big
B (Eτ ).
Theorem (Li)
Under this isomorphism, the A-model partition functionZA(E ) ∈ Fock(V big
A (E )) corresponds to ZB(E∨) ∈ Fock(V bigB (E∨)).
This means all GW invariants of an elliptic curve E can be computed fromquantum BCOV theory on the mirror elliptic curve E∨.
![Page 77: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/77.jpg)
Mirror symmetry for the elliptic curve
E elliptic curve. Mirror family: E∨τ , τ ∈ H, q = e2πiτ .
Symplectic vector spaces:
V bigA (E ) = H∗(E )((t))⊗ C((q))
V bigB (Eτ ) = H∗
(Ω0,∗(E ,∧∗TE )((t))
), differential ∂ + t∂.
There’s a natural isomorphism of symplectic vector spaces
V bigA (E ) ∼= V big
B (Eτ ).
Theorem (Li)
Under this isomorphism, the A-model partition functionZA(E ) ∈ Fock(V big
A (E )) corresponds to ZB(E∨) ∈ Fock(V bigB (E∨)).
This means all GW invariants of an elliptic curve E can be computed fromquantum BCOV theory on the mirror elliptic curve E∨.
![Page 78: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/78.jpg)
Correlators and the Hodge filtration
Choice of splitting of the Hodge filtration on H1(E ) leads to a polarizationof the symplectic vector space H∗(PV(X )((t))).
This leads to B-model correlators⟨α1tk1 , . . . , αntkn
⟩E ,S⊂H1(E)
g ,n∈ C
for αi ∈ H∗(PV(E ), ∂).
The correlators depend holomorphically on E and on choice S of splittingof Hodge filtration. They are also SL2(Z) invariant (i.e. modular).
But naive splitting F1
(complex conjugate to Hodge filtration) does notvary holomorphically with E . “Holomorphic anomaly”.
![Page 79: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/79.jpg)
Correlators and the Hodge filtration
Choice of splitting of the Hodge filtration on H1(E ) leads to a polarizationof the symplectic vector space H∗(PV(X )((t))).
This leads to B-model correlators⟨α1tk1 , . . . , αntkn
⟩E ,S⊂H1(E)
g ,n∈ C
for αi ∈ H∗(PV(E ), ∂).
The correlators depend holomorphically on E and on choice S of splittingof Hodge filtration. They are also SL2(Z) invariant (i.e. modular).
But naive splitting F1
(complex conjugate to Hodge filtration) does notvary holomorphically with E . “Holomorphic anomaly”.
![Page 80: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/80.jpg)
Correlators and the Hodge filtration
Choice of splitting of the Hodge filtration on H1(E ) leads to a polarizationof the symplectic vector space H∗(PV(X )((t))).
This leads to B-model correlators⟨α1tk1 , . . . , αntkn
⟩E ,S⊂H1(E)
g ,n∈ C
for αi ∈ H∗(PV(E ), ∂).
The correlators depend holomorphically on E and on choice S of splittingof Hodge filtration. They are also SL2(Z) invariant (i.e. modular).
But naive splitting F1
(complex conjugate to Hodge filtration) does notvary holomorphically with E . “Holomorphic anomaly”.
![Page 81: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/81.jpg)
Correlators and the Hodge filtration
Choice of splitting of the Hodge filtration on H1(E ) leads to a polarizationof the symplectic vector space H∗(PV(X )((t))).
This leads to B-model correlators⟨α1tk1 , . . . , αntkn
⟩E ,S⊂H1(E)
g ,n∈ C
for αi ∈ H∗(PV(E ), ∂).
The correlators depend holomorphically on E and on choice S of splittingof Hodge filtration. They are also SL2(Z) invariant (i.e. modular).
But naive splitting F1
(complex conjugate to Hodge filtration) does notvary holomorphically with E . “Holomorphic anomaly”.
![Page 82: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/82.jpg)
Large complex structure splitting
If τ ∈ H, let Eτ be the elliptic curve. If σ ∈ H let F 1σH1(Eτ ) be Hodge
filtration for structure σ: then F1σH1(Eτ ) splits Hodge filtration on H1(Eτ ).
To match the A-model use splitting limσ→i∞ F1σH1(Eτ ).
Physicists say: “Fix τ and let τ go to ∞”. With this splitting correlatorsare quasi-modular forms.
Theorem (Li)
B-model correlators on Eτ with this splitting of the Hodge filtration areequal to A-model correlators on mirror curve E∨, with q = e2πiτ .
1 ∈ H0(E ,OE )↔ 1 ∈ H0(E∨)
dz ∈ H1(E ,OE )↔ dz ∈ H0,1(E∨)
∂z ∈ H0(E ,TE )↔ dz ∈ H1,0(E∨)
∂zdz ∈ H1(E ,TE )↔ dzdz ∈ H2(E∨).
![Page 83: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/83.jpg)
Large complex structure splitting
If τ ∈ H, let Eτ be the elliptic curve. If σ ∈ H let F 1σH1(Eτ ) be Hodge
filtration for structure σ: then F1σH1(Eτ ) splits Hodge filtration on H1(Eτ ).
To match the A-model use splitting limσ→i∞ F1σH1(Eτ ).
Physicists say: “Fix τ and let τ go to ∞”. With this splitting correlatorsare quasi-modular forms.
Theorem (Li)
B-model correlators on Eτ with this splitting of the Hodge filtration areequal to A-model correlators on mirror curve E∨, with q = e2πiτ .
1 ∈ H0(E ,OE )↔ 1 ∈ H0(E∨)
dz ∈ H1(E ,OE )↔ dz ∈ H0,1(E∨)
∂z ∈ H0(E ,TE )↔ dz ∈ H1,0(E∨)
∂zdz ∈ H1(E ,TE )↔ dzdz ∈ H2(E∨).
![Page 84: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/84.jpg)
Large complex structure splitting
If τ ∈ H, let Eτ be the elliptic curve. If σ ∈ H let F 1σH1(Eτ ) be Hodge
filtration for structure σ: then F1σH1(Eτ ) splits Hodge filtration on H1(Eτ ).
To match the A-model use splitting limσ→i∞ F1σH1(Eτ ).
Physicists say: “Fix τ and let τ go to ∞”. With this splitting correlatorsare quasi-modular forms.
Theorem (Li)
B-model correlators on Eτ with this splitting of the Hodge filtration areequal to A-model correlators on mirror curve E∨, with q = e2πiτ .
1 ∈ H0(E ,OE )↔ 1 ∈ H0(E∨)
dz ∈ H1(E ,OE )↔ dz ∈ H0,1(E∨)
∂z ∈ H0(E ,TE )↔ dz ∈ H1,0(E∨)
∂zdz ∈ H1(E ,TE )↔ dzdz ∈ H2(E∨).
![Page 85: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/85.jpg)
Large complex structure splitting
If τ ∈ H, let Eτ be the elliptic curve. If σ ∈ H let F 1σH1(Eτ ) be Hodge
filtration for structure σ: then F1σH1(Eτ ) splits Hodge filtration on H1(Eτ ).
To match the A-model use splitting limσ→i∞ F1σH1(Eτ ).
Physicists say: “Fix τ and let τ go to ∞”. With this splitting correlatorsare quasi-modular forms.
Theorem (Li)
B-model correlators on Eτ with this splitting of the Hodge filtration areequal to A-model correlators on mirror curve E∨, with q = e2πiτ .
1 ∈ H0(E ,OE )↔ 1 ∈ H0(E∨)
dz ∈ H1(E ,OE )↔ dz ∈ H0,1(E∨)
∂z ∈ H0(E ,TE )↔ dz ∈ H1,0(E∨)
∂zdz ∈ H1(E ,TE )↔ dzdz ∈ H2(E∨).
![Page 86: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/86.jpg)
Sketch of proof
1 Prove that Virasoro constraints hold on the B-model. Obstructiontheory argument: they hold classically, there is a unique quantization,so they hold at the quantum level.
2 Reduces calculation to “stationary sector”: need to computecorrelators ⟨
ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
where ω ∈ H1(E ,TE ) has Tr(ω) = 1, so ω = ∂zdz/2 Im τ .
3 Localization: in limiting splitting of Hodge filtration, ω becomessupported on an a-cycle.
4 Implies there are operators Ok | k ≥ 0 in the chiral free boson suchthat
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
![Page 87: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/87.jpg)
Sketch of proof
1 Prove that Virasoro constraints hold on the B-model. Obstructiontheory argument: they hold classically, there is a unique quantization,so they hold at the quantum level.
2 Reduces calculation to “stationary sector”: need to computecorrelators ⟨
ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
where ω ∈ H1(E ,TE ) has Tr(ω) = 1, so ω = ∂zdz/2 Im τ .
3 Localization: in limiting splitting of Hodge filtration, ω becomessupported on an a-cycle.
4 Implies there are operators Ok | k ≥ 0 in the chiral free boson suchthat
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
![Page 88: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/88.jpg)
Sketch of proof
1 Prove that Virasoro constraints hold on the B-model. Obstructiontheory argument: they hold classically, there is a unique quantization,so they hold at the quantum level.
2 Reduces calculation to “stationary sector”: need to computecorrelators ⟨
ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
where ω ∈ H1(E ,TE ) has Tr(ω) = 1, so ω = ∂zdz/2 Im τ .
3 Localization: in limiting splitting of Hodge filtration, ω becomessupported on an a-cycle.
4 Implies there are operators Ok | k ≥ 0 in the chiral free boson suchthat
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
![Page 89: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/89.jpg)
Sketch of proof
1 Prove that Virasoro constraints hold on the B-model. Obstructiontheory argument: they hold classically, there is a unique quantization,so they hold at the quantum level.
2 Reduces calculation to “stationary sector”: need to computecorrelators ⟨
ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
where ω ∈ H1(E ,TE ) has Tr(ω) = 1, so ω = ∂zdz/2 Im τ .
3 Localization: in limiting splitting of Hodge filtration, ω becomessupported on an a-cycle.
4 Implies there are operators Ok | k ≥ 0 in the chiral free boson suchthat
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
![Page 90: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/90.jpg)
Sketch of proof
1 Operators Ok | k ≥ 0 in the chiral free boson such that
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
2 Right hand side: symmetric under permutation of ki . So operatorsOki
commute.
3 So we have a completely integrable system. Commutativity, classicalbehaviour, and scaling behaviour completely determines the Oki
.
4 Apply boson-fermion correspondence, Okibecoming commuting
operators in system of 2 free chiral fermions.
5 Okounkov-Pandharipande: A-model correlators are expectation valuesof a family of commuting operators in a system of 2 chiral freefermions. The operators are the same: essentially characterized bycommutativity.
![Page 91: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/91.jpg)
Sketch of proof
1 Operators Ok | k ≥ 0 in the chiral free boson such that
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
2 Right hand side: symmetric under permutation of ki . So operatorsOki
commute.
3 So we have a completely integrable system. Commutativity, classicalbehaviour, and scaling behaviour completely determines the Oki
.
4 Apply boson-fermion correspondence, Okibecoming commuting
operators in system of 2 free chiral fermions.
5 Okounkov-Pandharipande: A-model correlators are expectation valuesof a family of commuting operators in a system of 2 chiral freefermions. The operators are the same: essentially characterized bycommutativity.
![Page 92: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/92.jpg)
Sketch of proof
1 Operators Ok | k ≥ 0 in the chiral free boson such that
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
2 Right hand side: symmetric under permutation of ki . So operatorsOki
commute.
3 So we have a completely integrable system. Commutativity, classicalbehaviour, and scaling behaviour completely determines the Oki
.
4 Apply boson-fermion correspondence, Okibecoming commuting
operators in system of 2 free chiral fermions.
5 Okounkov-Pandharipande: A-model correlators are expectation valuesof a family of commuting operators in a system of 2 chiral freefermions. The operators are the same: essentially characterized bycommutativity.
![Page 93: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/93.jpg)
Sketch of proof
1 Operators Ok | k ≥ 0 in the chiral free boson such that
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
2 Right hand side: symmetric under permutation of ki . So operatorsOki
commute.
3 So we have a completely integrable system. Commutativity, classicalbehaviour, and scaling behaviour completely determines the Oki
.
4 Apply boson-fermion correspondence, Okibecoming commuting
operators in system of 2 free chiral fermions.
5 Okounkov-Pandharipande: A-model correlators are expectation valuesof a family of commuting operators in a system of 2 chiral freefermions. The operators are the same: essentially characterized bycommutativity.
![Page 94: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/94.jpg)
Sketch of proof
1 Operators Ok | k ≥ 0 in the chiral free boson such that
TrFock
(e2πiτHOk1 . . .Okn
)=⟨ωtk1 , . . . , ωtkn
⟩τ,∞g ,n
.
2 Right hand side: symmetric under permutation of ki . So operatorsOki
commute.
3 So we have a completely integrable system. Commutativity, classicalbehaviour, and scaling behaviour completely determines the Oki
.
4 Apply boson-fermion correspondence, Okibecoming commuting
operators in system of 2 free chiral fermions.
5 Okounkov-Pandharipande: A-model correlators are expectation valuesof a family of commuting operators in a system of 2 chiral freefermions. The operators are the same: essentially characterized bycommutativity.
![Page 95: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/95.jpg)
GW invariants of an elliptic curve are complicated (determined byOkounkov-Pandharipande, 2002).
If
F E (z1, . . . , zn; q)
= z1 . . . zn
∞∏m=1
(1− qm) exp(∑
qd 〈τk1(ω), . . . , τkn(ω)〉g ,n,d zk11 . . . zkn
)and
θ(z) = θ(z , q) =∑n∈Z
(−1)nq(n+12)2/2e(n+
12)z
then
F E (z1, . . . , zn; q) =∑
permutations ofz1,...,zn
det[θ(j−i+1)(z1+···+zn−j )
(j−i+1)!
]ni ,j=1
θ(z1)θ(z1 + z2) . . . θ(z1 + · · ·+ zn)
![Page 96: Mirror symmetry at higher genus](https://reader030.fdocuments.net/reader030/viewer/2022020706/61fc881c8d33c02b785e4385/html5/thumbnails/96.jpg)
GW invariants of an elliptic curve are complicated (determined byOkounkov-Pandharipande, 2002).
If
F E (z1, . . . , zn; q)
= z1 . . . zn
∞∏m=1
(1− qm) exp(∑
qd 〈τk1(ω), . . . , τkn(ω)〉g ,n,d zk11 . . . zkn
)and
θ(z) = θ(z , q) =∑n∈Z
(−1)nq(n+12)2/2e(n+
12)z
then
F E (z1, . . . , zn; q) =∑
permutations ofz1,...,zn
det[θ(j−i+1)(z1+···+zn−j )
(j−i+1)!
]ni ,j=1
θ(z1)θ(z1 + z2) . . . θ(z1 + · · ·+ zn)