mirror symmetry: a brief history. superstring theory ...mg475/melbourne.pdf · with loops moving...
TRANSCRIPT
![Page 1: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/1.jpg)
Mirror symmetry: A brief history.
Superstring theory replaces particles moving through space-timewith loops moving through space-time.
A key prediction of superstring theory is:
The universe is 10 dimensional.
Mark Gross Mirror symmetry and tropical geometry
![Page 2: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/2.jpg)
Mirror symmetry: A brief history.
Superstring theory replaces particles moving through space-timewith loops moving through space-time.
A key prediction of superstring theory is:
The universe is 10 dimensional.
Mark Gross Mirror symmetry and tropical geometry
![Page 3: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/3.jpg)
Mirror symmetry: A brief history.
Superstring theory replaces particles moving through space-timewith loops moving through space-time.
A key prediction of superstring theory is:
The universe is 10 dimensional.
Mark Gross Mirror symmetry and tropical geometry
![Page 4: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/4.jpg)
This is reconciled with real-world observations by positing that theuniverse is of the form
R1,3 × X
where R1,3 is usual Minkowski space-time and X is a (very small!)
six-dimensional compact manifold.
Properties of X should be reflected in properties of the observedworld.
For example, supersymmetry is a desirable phenomenon(unfortunately not yet discovered at the LHC!)
Mark Gross Mirror symmetry and tropical geometry
![Page 5: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/5.jpg)
This is reconciled with real-world observations by positing that theuniverse is of the form
R1,3 × X
where R1,3 is usual Minkowski space-time and X is a (very small!)
six-dimensional compact manifold.
Properties of X should be reflected in properties of the observedworld.
For example, supersymmetry is a desirable phenomenon(unfortunately not yet discovered at the LHC!)
Mark Gross Mirror symmetry and tropical geometry
![Page 6: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/6.jpg)
This is reconciled with real-world observations by positing that theuniverse is of the form
R1,3 × X
where R1,3 is usual Minkowski space-time and X is a (very small!)
six-dimensional compact manifold.
Properties of X should be reflected in properties of the observedworld.
For example, supersymmetry is a desirable phenomenon(unfortunately not yet discovered at the LHC!)
Mark Gross Mirror symmetry and tropical geometry
![Page 7: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/7.jpg)
Supersymmetry suggests (to first approximation) that X should beRicci-flat, i.e., be a Calabi-Yau manifold.
This makes connections between string theory and algebraicgeometry, the study of solution sets to polynomial equations,because Calabi-Yau manifolds can be defined using polynomialequations in projective space.
Mark Gross Mirror symmetry and tropical geometry
![Page 8: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/8.jpg)
Supersymmetry suggests (to first approximation) that X should beRicci-flat, i.e., be a Calabi-Yau manifold.
This makes connections between string theory and algebraicgeometry, the study of solution sets to polynomial equations,because Calabi-Yau manifolds can be defined using polynomialequations in projective space.
Mark Gross Mirror symmetry and tropical geometry
![Page 9: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/9.jpg)
Example
LetCP
4 = (C5 \ (0, 0, 0, 0, 0))/C∗
be four-dimensional complex projective space, with coordinates
x0, . . . , x4.
Let X be the three-dimensional complex manifold defined by theequation
x50 + · · ·+ x54 = 0.
This is a Calabi-Yau manifold, by Yau’s proof of the CalabiConjecture.
Mark Gross Mirror symmetry and tropical geometry
![Page 10: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/10.jpg)
Example
LetCP
4 = (C5 \ (0, 0, 0, 0, 0))/C∗
be four-dimensional complex projective space, with coordinates
x0, . . . , x4.
Let X be the three-dimensional complex manifold defined by theequation
x50 + · · ·+ x54 = 0.
This is a Calabi-Yau manifold, by Yau’s proof of the CalabiConjecture.
Mark Gross Mirror symmetry and tropical geometry
![Page 11: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/11.jpg)
Mirror Symmetry. 1990, Greene and Plesser; Candelas, Lynker andSchimmrigk: Calabi-Yau manifolds should come in pairs, X , X ,inducing the same physics!
One symptom of mirror symmetry:
χ(X ) = −χ(X ).
Mark Gross Mirror symmetry and tropical geometry
![Page 12: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/12.jpg)
Mirror Symmetry. 1990, Greene and Plesser; Candelas, Lynker andSchimmrigk: Calabi-Yau manifolds should come in pairs, X , X ,inducing the same physics!
One symptom of mirror symmetry:
χ(X ) = −χ(X ).
Mark Gross Mirror symmetry and tropical geometry
![Page 13: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/13.jpg)
Example
Let X be the quintic, given by
x50 + · · · x54 = 0
and consider the group action of
G = (a0, . . . , a4)|ai ∈ Z/5Z and∑
i ai = 0
on X given by
(x0, . . . , x4) 7→ (ξa0x0, . . . , ξa4x4), ξ = e2πi/5.
Then X/G is very singular, but there is a resolution X → X/G .
X is the mirror of the quintic discovered by Greene and Plesser.
χ(X ) = −200, χ(X ) = 200.
Mark Gross Mirror symmetry and tropical geometry
![Page 14: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/14.jpg)
Example
Let X be the quintic, given by
x50 + · · · x54 = 0
and consider the group action of
G = (a0, . . . , a4)|ai ∈ Z/5Z and∑
i ai = 0
on X given by
(x0, . . . , x4) 7→ (ξa0x0, . . . , ξa4x4), ξ = e2πi/5.
Then X/G is very singular, but there is a resolution X → X/G .
X is the mirror of the quintic discovered by Greene and Plesser.
χ(X ) = −200, χ(X ) = 200.
Mark Gross Mirror symmetry and tropical geometry
![Page 15: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/15.jpg)
Example
Let X be the quintic, given by
x50 + · · · x54 = 0
and consider the group action of
G = (a0, . . . , a4)|ai ∈ Z/5Z and∑
i ai = 0
on X given by
(x0, . . . , x4) 7→ (ξa0x0, . . . , ξa4x4), ξ = e2πi/5.
Then X/G is very singular, but there is a resolution X → X/G .
X is the mirror of the quintic discovered by Greene and Plesser.
χ(X ) = −200, χ(X ) = 200.
Mark Gross Mirror symmetry and tropical geometry
![Page 16: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/16.jpg)
Example
Let X be the quintic, given by
x50 + · · · x54 = 0
and consider the group action of
G = (a0, . . . , a4)|ai ∈ Z/5Z and∑
i ai = 0
on X given by
(x0, . . . , x4) 7→ (ξa0x0, . . . , ξa4x4), ξ = e2πi/5.
Then X/G is very singular, but there is a resolution X → X/G .
X is the mirror of the quintic discovered by Greene and Plesser.
χ(X ) = −200, χ(X ) = 200.
Mark Gross Mirror symmetry and tropical geometry
![Page 17: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/17.jpg)
Enumerative geometry (19th century).This is the study of questions of the flavor: “How many geometricgadgets of a given sort are contained in some other gadget, orintersect some collection of gadgets.”For example, given two points in CP
2, there is precisely one line (asubset defined by a linear equation) passing through two points.(Cayley-Salmon) A smooth cubic surface in CP
3 always containsprecisely 27 lines.e.g., the Clebsch diagonal surface
x30 + x31 + x32 + x33 = (x0 + x1 + x2 + x3)3.
Mark Gross Mirror symmetry and tropical geometry
![Page 18: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/18.jpg)
Enumerative geometry (19th century).This is the study of questions of the flavor: “How many geometricgadgets of a given sort are contained in some other gadget, orintersect some collection of gadgets.”For example, given two points in CP
2, there is precisely one line (asubset defined by a linear equation) passing through two points.(Cayley-Salmon) A smooth cubic surface in CP
3 always containsprecisely 27 lines.e.g., the Clebsch diagonal surface
x30 + x31 + x32 + x33 = (x0 + x1 + x2 + x3)3.
Mark Gross Mirror symmetry and tropical geometry
![Page 19: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/19.jpg)
Enumerative geometry (19th century).This is the study of questions of the flavor: “How many geometricgadgets of a given sort are contained in some other gadget, orintersect some collection of gadgets.”For example, given two points in CP
2, there is precisely one line (asubset defined by a linear equation) passing through two points.(Cayley-Salmon) A smooth cubic surface in CP
3 always containsprecisely 27 lines.e.g., the Clebsch diagonal surface
x30 + x31 + x32 + x33 = (x0 + x1 + x2 + x3)3.
Mark Gross Mirror symmetry and tropical geometry
![Page 20: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/20.jpg)
Enumerative geometry (19th century).This is the study of questions of the flavor: “How many geometricgadgets of a given sort are contained in some other gadget, orintersect some collection of gadgets.”For example, given two points in CP
2, there is precisely one line (asubset defined by a linear equation) passing through two points.(Cayley-Salmon) A smooth cubic surface in CP
3 always containsprecisely 27 lines.e.g., the Clebsch diagonal surface
x30 + x31 + x32 + x33 = (x0 + x1 + x2 + x3)3.
Mark Gross Mirror symmetry and tropical geometry
![Page 21: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/21.jpg)
Enumerative geometry (19th century).This is the study of questions of the flavor: “How many geometricgadgets of a given sort are contained in some other gadget, orintersect some collection of gadgets.”For example, given two points in CP
2, there is precisely one line (asubset defined by a linear equation) passing through two points.(Cayley-Salmon) A smooth cubic surface in CP
3 always containsprecisely 27 lines.e.g., the Clebsch diagonal surface
x30 + x31 + x32 + x33 = (x0 + x1 + x2 + x3)3.
Mark Gross Mirror symmetry and tropical geometry
![Page 22: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/22.jpg)
Mark Gross Mirror symmetry and tropical geometry
![Page 23: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/23.jpg)
1990: Candelas-de la Ossa-Green-Parkes: Amazing calculation,following predictions of string theory.
Let X be the quintic three-fold.
Let N1 be the number of lines in X .
Let N2 be the number of conics in X .
Let Nd be “the number of rational curves of degree d in X”. Sucha curve is the image of a map CP
1 → CP4 defined by
(u : t) 7→ (f0(u, t), . . . , f4(u, t))
where f0, . . . , f4 are polynomials of degree d without commonzeroes.
Mark Gross Mirror symmetry and tropical geometry
![Page 24: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/24.jpg)
1990: Candelas-de la Ossa-Green-Parkes: Amazing calculation,following predictions of string theory.
Let X be the quintic three-fold.
Let N1 be the number of lines in X .
Let N2 be the number of conics in X .
Let Nd be “the number of rational curves of degree d in X”. Sucha curve is the image of a map CP
1 → CP4 defined by
(u : t) 7→ (f0(u, t), . . . , f4(u, t))
where f0, . . . , f4 are polynomials of degree d without commonzeroes.
Mark Gross Mirror symmetry and tropical geometry
![Page 25: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/25.jpg)
1990: Candelas-de la Ossa-Green-Parkes: Amazing calculation,following predictions of string theory.
Let X be the quintic three-fold.
Let N1 be the number of lines in X .
Let N2 be the number of conics in X .
Let Nd be “the number of rational curves of degree d in X”. Sucha curve is the image of a map CP
1 → CP4 defined by
(u : t) 7→ (f0(u, t), . . . , f4(u, t))
where f0, . . . , f4 are polynomials of degree d without commonzeroes.
Mark Gross Mirror symmetry and tropical geometry
![Page 26: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/26.jpg)
1990: Candelas-de la Ossa-Green-Parkes: Amazing calculation,following predictions of string theory.
Let X be the quintic three-fold.
Let N1 be the number of lines in X .
Let N2 be the number of conics in X .
Let Nd be “the number of rational curves of degree d in X”. Sucha curve is the image of a map CP
1 → CP4 defined by
(u : t) 7→ (f0(u, t), . . . , f4(u, t))
where f0, . . . , f4 are polynomials of degree d without commonzeroes.
Mark Gross Mirror symmetry and tropical geometry
![Page 27: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/27.jpg)
1990: Candelas-de la Ossa-Green-Parkes: Amazing calculation,following predictions of string theory.
Let X be the quintic three-fold.
Let N1 be the number of lines in X .
Let N2 be the number of conics in X .
Let Nd be “the number of rational curves of degree d in X”. Sucha curve is the image of a map CP
1 → CP4 defined by
(u : t) 7→ (f0(u, t), . . . , f4(u, t))
where f0, . . . , f4 are polynomials of degree d without commonzeroes.
Mark Gross Mirror symmetry and tropical geometry
![Page 28: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/28.jpg)
N1 = 2875 (19th century, H. Schubert.)
N2 = 609250, (1986, Sheldon Katz).
N3 = 317206375, (1990, Ellingsrud and Strømme)Candelas, de la Ossa, Green and Parkes proposed that thesenumbers Nd could be computed via a completely differentcalculation on X . This calculation involves period integrals,expressions of the form ∫
αΩ,
where α is a 3-cycle in X and Ω is a holomorphic 3-form on X .
In this way, they gave a prediction, motivated entirely by stringtheory, for all the numbers Nd .
Mark Gross Mirror symmetry and tropical geometry
![Page 29: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/29.jpg)
N1 = 2875 (19th century, H. Schubert.)
N2 = 609250, (1986, Sheldon Katz).
N3 = 317206375, (1990, Ellingsrud and Strømme)Candelas, de la Ossa, Green and Parkes proposed that thesenumbers Nd could be computed via a completely differentcalculation on X . This calculation involves period integrals,expressions of the form ∫
αΩ,
where α is a 3-cycle in X and Ω is a holomorphic 3-form on X .
In this way, they gave a prediction, motivated entirely by stringtheory, for all the numbers Nd .
Mark Gross Mirror symmetry and tropical geometry
![Page 30: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/30.jpg)
N1 = 2875 (19th century, H. Schubert.)
N2 = 609250, (1986, Sheldon Katz).
N3 = 317206375, (1990, Ellingsrud and Strømme)Candelas, de la Ossa, Green and Parkes proposed that thesenumbers Nd could be computed via a completely differentcalculation on X . This calculation involves period integrals,expressions of the form ∫
αΩ,
where α is a 3-cycle in X and Ω is a holomorphic 3-form on X .
In this way, they gave a prediction, motivated entirely by stringtheory, for all the numbers Nd .
Mark Gross Mirror symmetry and tropical geometry
![Page 31: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/31.jpg)
N1 = 2875 (19th century, H. Schubert.)
N2 = 609250, (1986, Sheldon Katz).
N3 = 317206375, (1990, Ellingsrud and Strømme)Candelas, de la Ossa, Green and Parkes proposed that thesenumbers Nd could be computed via a completely differentcalculation on X . This calculation involves period integrals,expressions of the form ∫
αΩ,
where α is a 3-cycle in X and Ω is a holomorphic 3-form on X .
In this way, they gave a prediction, motivated entirely by stringtheory, for all the numbers Nd .
Mark Gross Mirror symmetry and tropical geometry
![Page 32: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/32.jpg)
N1 = 2875 (19th century, H. Schubert.)
N2 = 609250, (1986, Sheldon Katz).
N3 = 317206375, (1990, Ellingsrud and Strømme)Candelas, de la Ossa, Green and Parkes proposed that thesenumbers Nd could be computed via a completely differentcalculation on X . This calculation involves period integrals,expressions of the form ∫
αΩ,
where α is a 3-cycle in X and Ω is a holomorphic 3-form on X .
In this way, they gave a prediction, motivated entirely by stringtheory, for all the numbers Nd .
Mark Gross Mirror symmetry and tropical geometry
![Page 33: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/33.jpg)
The formulas for Nd of Candelas et al were not proved until1996-7, by Givental and Lian,Liu,Yau.
This work involved a direct calculation of the numbers Nd . Butwhat is the basic underlying geometry of mirror symmetry?
Mantra: Mirror symmetry should be a duality which interchangessymplectic geometry (A-model) and complex geometry (B-model).
Mark Gross Mirror symmetry and tropical geometry
![Page 34: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/34.jpg)
The formulas for Nd of Candelas et al were not proved until1996-7, by Givental and Lian,Liu,Yau.
This work involved a direct calculation of the numbers Nd . Butwhat is the basic underlying geometry of mirror symmetry?
Mantra: Mirror symmetry should be a duality which interchangessymplectic geometry (A-model) and complex geometry (B-model).
Mark Gross Mirror symmetry and tropical geometry
![Page 35: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/35.jpg)
The formulas for Nd of Candelas et al were not proved until1996-7, by Givental and Lian,Liu,Yau.
This work involved a direct calculation of the numbers Nd . Butwhat is the basic underlying geometry of mirror symmetry?
Mantra: Mirror symmetry should be a duality which interchangessymplectic geometry (A-model) and complex geometry (B-model).
Mark Gross Mirror symmetry and tropical geometry
![Page 36: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/36.jpg)
Try 1:
V a real finite dim’l vector space V ∗ = Hom(V ,R) the dual space.
Try 2:
V × V with complex structure V × V ∗ withJ(v1, v2) = (−v2, v1) symplectic structure
ω((v1,w1), (v2,w2))= 〈w1, v2〉 − 〈w2, v1〉
This is a very simple example of mirror symmetry.
Mark Gross Mirror symmetry and tropical geometry
![Page 37: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/37.jpg)
Try 1:
V a real finite dim’l vector space V ∗ = Hom(V ,R) the dual space.
Try 2:
V × V with complex structure V × V ∗ withJ(v1, v2) = (−v2, v1) symplectic structure
ω((v1,w1), (v2,w2))= 〈w1, v2〉 − 〈w2, v1〉
This is a very simple example of mirror symmetry.
Mark Gross Mirror symmetry and tropical geometry
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Try 1:
V a real finite dim’l vector space V ∗ = Hom(V ,R) the dual space.
Try 2:
V × V with complex structure V × V ∗ withJ(v1, v2) = (−v2, v1) symplectic structure
ω((v1,w1), (v2,w2))= 〈w1, v2〉 − 〈w2, v1〉
This is a very simple example of mirror symmetry.
Mark Gross Mirror symmetry and tropical geometry
![Page 39: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/39.jpg)
Try 1:
V a real finite dim’l vector space V ∗ = Hom(V ,R) the dual space.
Try 2:
V × V with complex structure V × V ∗ withJ(v1, v2) = (−v2, v1) symplectic structure
ω((v1,w1), (v2,w2))= 〈w1, v2〉 − 〈w2, v1〉
This is a very simple example of mirror symmetry.
Mark Gross Mirror symmetry and tropical geometry
![Page 40: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/40.jpg)
Try 1:
V a real finite dim’l vector space V ∗ = Hom(V ,R) the dual space.
Try 2:
V × V with complex structure V × V ∗ withJ(v1, v2) = (−v2, v1) symplectic structure
ω((v1,w1), (v2,w2))= 〈w1, v2〉 − 〈w2, v1〉
This is a very simple example of mirror symmetry.
Mark Gross Mirror symmetry and tropical geometry
![Page 41: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/41.jpg)
A vector space is not a particularly interesting example.We can make this more interesting by choosing V to have anintegral structure, i.e.,
V = Λ⊗Z R
where Λ ∼= Zn. Set
Λ := w ∈ V ∗ | 〈w ,Λ〉 ⊆ Z ⊆ V ∗
X (V ) := V × V /Λ with complex X (V ) := V × V ∗/Λ withstructure J as before. symplectic structure as before.
While this seems like a very simplistic point of view, in fact this toyexample already exhibits rich features of mirror symmetry, whichwe will explore.A more general point of view replaces V with a more generalmanifold with an affine structure, and this leads to an extensiveprogram (G.-Siebert) for understanding mirror symmetry ingeneral. We will not go down this route today.
Mark Gross Mirror symmetry and tropical geometry
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A vector space is not a particularly interesting example.We can make this more interesting by choosing V to have anintegral structure, i.e.,
V = Λ⊗Z R
where Λ ∼= Zn. Set
Λ := w ∈ V ∗ | 〈w ,Λ〉 ⊆ Z ⊆ V ∗
X (V ) := V × V /Λ with complex X (V ) := V × V ∗/Λ withstructure J as before. symplectic structure as before.
While this seems like a very simplistic point of view, in fact this toyexample already exhibits rich features of mirror symmetry, whichwe will explore.A more general point of view replaces V with a more generalmanifold with an affine structure, and this leads to an extensiveprogram (G.-Siebert) for understanding mirror symmetry ingeneral. We will not go down this route today.
Mark Gross Mirror symmetry and tropical geometry
![Page 43: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/43.jpg)
A vector space is not a particularly interesting example.We can make this more interesting by choosing V to have anintegral structure, i.e.,
V = Λ⊗Z R
where Λ ∼= Zn. Set
Λ := w ∈ V ∗ | 〈w ,Λ〉 ⊆ Z ⊆ V ∗
X (V ) := V × V /Λ with complex X (V ) := V × V ∗/Λ withstructure J as before. symplectic structure as before.
While this seems like a very simplistic point of view, in fact this toyexample already exhibits rich features of mirror symmetry, whichwe will explore.A more general point of view replaces V with a more generalmanifold with an affine structure, and this leads to an extensiveprogram (G.-Siebert) for understanding mirror symmetry ingeneral. We will not go down this route today.
Mark Gross Mirror symmetry and tropical geometry
![Page 44: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/44.jpg)
A vector space is not a particularly interesting example.We can make this more interesting by choosing V to have anintegral structure, i.e.,
V = Λ⊗Z R
where Λ ∼= Zn. Set
Λ := w ∈ V ∗ | 〈w ,Λ〉 ⊆ Z ⊆ V ∗
X (V ) := V × V /Λ with complex X (V ) := V × V ∗/Λ withstructure J as before. symplectic structure as before.
While this seems like a very simplistic point of view, in fact this toyexample already exhibits rich features of mirror symmetry, whichwe will explore.A more general point of view replaces V with a more generalmanifold with an affine structure, and this leads to an extensiveprogram (G.-Siebert) for understanding mirror symmetry ingeneral. We will not go down this route today.
Mark Gross Mirror symmetry and tropical geometry
![Page 45: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/45.jpg)
A vector space is not a particularly interesting example.We can make this more interesting by choosing V to have anintegral structure, i.e.,
V = Λ⊗Z R
where Λ ∼= Zn. Set
Λ := w ∈ V ∗ | 〈w ,Λ〉 ⊆ Z ⊆ V ∗
X (V ) := V × V /Λ with complex X (V ) := V × V ∗/Λ withstructure J as before. symplectic structure as before.
While this seems like a very simplistic point of view, in fact this toyexample already exhibits rich features of mirror symmetry, whichwe will explore.A more general point of view replaces V with a more generalmanifold with an affine structure, and this leads to an extensiveprogram (G.-Siebert) for understanding mirror symmetry ingeneral. We will not go down this route today.
Mark Gross Mirror symmetry and tropical geometry
![Page 46: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/46.jpg)
A vector space is not a particularly interesting example.We can make this more interesting by choosing V to have anintegral structure, i.e.,
V = Λ⊗Z R
where Λ ∼= Zn. Set
Λ := w ∈ V ∗ | 〈w ,Λ〉 ⊆ Z ⊆ V ∗
X (V ) := V × V /Λ with complex X (V ) := V × V ∗/Λ withstructure J as before. symplectic structure as before.
While this seems like a very simplistic point of view, in fact this toyexample already exhibits rich features of mirror symmetry, whichwe will explore.A more general point of view replaces V with a more generalmanifold with an affine structure, and this leads to an extensiveprogram (G.-Siebert) for understanding mirror symmetry ingeneral. We will not go down this route today.
Mark Gross Mirror symmetry and tropical geometry
![Page 47: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/47.jpg)
To continue our exploration, we need to travel to the tropics...
Mark Gross Mirror symmetry and tropical geometry
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Suppose L ⊆ V is a rationally defined affine linear subspace.
X (L) := L× L/(L ∩ Λ) ⊆ X (V ) X (L) := L× L⊥/(L⊥ ∩ Λ) ⊆ X (V )holomorphic submanifold. Lagrangian submanifold.
These are not topologically very interesting. For example, ifdim L = 1, we obtain holomorphic curves which are cylinders.
Mark Gross Mirror symmetry and tropical geometry
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Suppose L ⊆ V is a rationally defined affine linear subspace.
X (L) := L× L/(L ∩ Λ) ⊆ X (V ) X (L) := L× L⊥/(L⊥ ∩ Λ) ⊆ X (V )holomorphic submanifold. Lagrangian submanifold.
These are not topologically very interesting. For example, ifdim L = 1, we obtain holomorphic curves which are cylinders.
Mark Gross Mirror symmetry and tropical geometry
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Suppose L ⊆ V is a rationally defined affine linear subspace.
X (L) := L× L/(L ∩ Λ) ⊆ X (V ) X (L) := L× L⊥/(L⊥ ∩ Λ) ⊆ X (V )holomorphic submanifold. Lagrangian submanifold.
These are not topologically very interesting. For example, ifdim L = 1, we obtain holomorphic curves which are cylinders.
Mark Gross Mirror symmetry and tropical geometry
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Let’s try to get a more interesting “approximate” holomorphiccurve by gluing together cylinders, taking three rays meeting atb ∈ V :
v1
v2v3
Mark Gross Mirror symmetry and tropical geometry
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We can try to glue the three cylinders by gluing in a surfacecontained in the fibre f −1(b).
Noting that H1(f−1(b),Z) = Λb, the tangent vectors v1, v2 and v3
represent the boundaries of the three cylinders in H1(f−1(b),Z).
Thus the three circles bound a surface if
v1 + v2 + v3 = 0.
This is the tropical balancing condition.
Mark Gross Mirror symmetry and tropical geometry
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We can try to glue the three cylinders by gluing in a surfacecontained in the fibre f −1(b).
Noting that H1(f−1(b),Z) = Λb, the tangent vectors v1, v2 and v3
represent the boundaries of the three cylinders in H1(f−1(b),Z).
Thus the three circles bound a surface if
v1 + v2 + v3 = 0.
This is the tropical balancing condition.
Mark Gross Mirror symmetry and tropical geometry
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We can try to glue the three cylinders by gluing in a surfacecontained in the fibre f −1(b).
Noting that H1(f−1(b),Z) = Λb, the tangent vectors v1, v2 and v3
represent the boundaries of the three cylinders in H1(f−1(b),Z).
Thus the three circles bound a surface if
v1 + v2 + v3 = 0.
This is the tropical balancing condition.
Mark Gross Mirror symmetry and tropical geometry
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This leads us to the notion of a tropical curve in V :
Definition
A parameterized tropical curve in V is a graph Γ (possibly withnon-compact edges with zero or one adjacent vertices) along with
a weight function w associating a non-negative integer toeach edge;
a proper continuous map h : Γ → V
satisfying the following properties:
Mark Gross Mirror symmetry and tropical geometry
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This leads us to the notion of a tropical curve in V :
Definition
A parameterized tropical curve in V is a graph Γ (possibly withnon-compact edges with zero or one adjacent vertices) along with
a weight function w associating a non-negative integer toeach edge;
a proper continuous map h : Γ → V
satisfying the following properties:
Mark Gross Mirror symmetry and tropical geometry
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This leads us to the notion of a tropical curve in V :
Definition
A parameterized tropical curve in V is a graph Γ (possibly withnon-compact edges with zero or one adjacent vertices) along with
a weight function w associating a non-negative integer toeach edge;
a proper continuous map h : Γ → V
satisfying the following properties:
Mark Gross Mirror symmetry and tropical geometry
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Definition
(cont’d.)
1 If E is an edge of Γ and w(E ) = 0, then h|E is constant;otherwise h|E is a proper embedding of E into V as a linesegment, ray or line of rational slope.
2 The balancing condition. For every vertex of Γ with adjacentedges E1, . . . ,En, let v1, . . . , vn ∈ Λ be primitive tangentvectors to h(E1), . . . , h(En) pointing away from h(V ). Then
n∑i=1
w(Ei )vi = 0.
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Definition
(cont’d.)
1 If E is an edge of Γ and w(E ) = 0, then h|E is constant;otherwise h|E is a proper embedding of E into V as a linesegment, ray or line of rational slope.
2 The balancing condition. For every vertex of Γ with adjacentedges E1, . . . ,En, let v1, . . . , vn ∈ Λ be primitive tangentvectors to h(E1), . . . , h(En) pointing away from h(V ). Then
n∑i=1
w(Ei )vi = 0.
Mark Gross Mirror symmetry and tropical geometry
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Example
Take V = R2. Here is a tropical curve:
This can be interpreted as a curve of genus 1 or genus 0,depending on which domain we use to paramaterize the curve.
Mark Gross Mirror symmetry and tropical geometry
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Example
Take V = R2. Here is a tropical curve:
This can be interpreted as a curve of genus 1 or genus 0,depending on which domain we use to paramaterize the curve.
Mark Gross Mirror symmetry and tropical geometry
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Example
This curve can be viewed as an approximation to a curve of degree3 in CP
2.Mikhalkin showed that curves in CP
2 through a given number ofpoints can in fact be counted by counting tropical curves of thisnature.This gave the first hint that curve-counting can really beaccomplished using tropical geometry.
Mark Gross Mirror symmetry and tropical geometry
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Example
This curve can be viewed as an approximation to a curve of degree3 in CP
2.Mikhalkin showed that curves in CP
2 through a given number ofpoints can in fact be counted by counting tropical curves of thisnature.This gave the first hint that curve-counting can really beaccomplished using tropical geometry.
Mark Gross Mirror symmetry and tropical geometry
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Example
This curve can be viewed as an approximation to a curve of degree3 in CP
2.Mikhalkin showed that curves in CP
2 through a given number ofpoints can in fact be counted by counting tropical curves of thisnature.This gave the first hint that curve-counting can really beaccomplished using tropical geometry.
Mark Gross Mirror symmetry and tropical geometry
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Mark Gross Mirror symmetry and tropical geometry
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Mirror symmetry for CP2.
In 1994, Givental gave a mirror for CP2. This description wasenhanced by Barannikov in 1999 to allow mirror calculations toanswer the question: “How many rational curves of degree d passthrough 3d − 1 points in the complex plane?”
The mirror is a Landau-Ginzburg model, the variety (C∗)2 alongwith a function
W : (C∗)2 → C
given byW = x + y + z ,
where x , y are coordinates on (C∗)2 and xyz = 1.
Mark Gross Mirror symmetry and tropical geometry
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Mirror symmetry for CP2.
In 1994, Givental gave a mirror for CP2. This description wasenhanced by Barannikov in 1999 to allow mirror calculations toanswer the question: “How many rational curves of degree d passthrough 3d − 1 points in the complex plane?”
The mirror is a Landau-Ginzburg model, the variety (C∗)2 alongwith a function
W : (C∗)2 → C
given byW = x + y + z ,
where x , y are coordinates on (C∗)2 and xyz = 1.
Mark Gross Mirror symmetry and tropical geometry
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Mirror symmetry for CP2.
In 1994, Givental gave a mirror for CP2. This description wasenhanced by Barannikov in 1999 to allow mirror calculations toanswer the question: “How many rational curves of degree d passthrough 3d − 1 points in the complex plane?”
The mirror is a Landau-Ginzburg model, the variety (C∗)2 alongwith a function
W : (C∗)2 → C
given byW = x + y + z ,
where x , y are coordinates on (C∗)2 and xyz = 1.
Mark Gross Mirror symmetry and tropical geometry
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To extract enumerative predictions, one needs to consider a familyof potentials which are perturbations of the above potential, e.g.,
Wt := t0 + (1 + t1)W + t2W2,
and calculate oscillatory integrals of the form
∫Γ
eWt/~f (x , y , t, ~)dx ∧ dy
xy,
where Γ runs over suitably chosen (possibly unbounded) 2-cycles in(C∗)2, f is a carefully chosen function which puts the aboveintegrals in some “normalized” form, and the result needs to beexpanded in a power series of some specially chosen coordinates ont-space.
The desired numbers will appear as some of the coefficients of thispower series.
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To extract enumerative predictions, one needs to consider a familyof potentials which are perturbations of the above potential, e.g.,
Wt := t0 + (1 + t1)W + t2W2,
and calculate oscillatory integrals of the form
∫Γ
eWt/~f (x , y , t, ~)dx ∧ dy
xy,
where Γ runs over suitably chosen (possibly unbounded) 2-cycles in(C∗)2, f is a carefully chosen function which puts the aboveintegrals in some “normalized” form, and the result needs to beexpanded in a power series of some specially chosen coordinates ont-space.
The desired numbers will appear as some of the coefficients of thispower series.
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A better conceptual approach (G., 2009) uses tropical techniquesto construct the “right” perturbation of W directly, so that theintegral manifestly is counting curves.
Construct infinitesimal perturbations of the potential by countingtropical disks; these are genus zero tropical curves which just endat a point:
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A better conceptual approach (G., 2009) uses tropical techniquesto construct the “right” perturbation of W directly, so that theintegral manifestly is counting curves.
Construct infinitesimal perturbations of the potential by countingtropical disks; these are genus zero tropical curves which just endat a point:
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Choose points P1, . . . ,Pk ,Q ∈ R2 general, and consider all rigid
tropical disks passing through some subset of P1, . . . ,Pk andterminating at Q.Label each end with the variable x , y or z , and each Pi with avariable ui with u2i = 0. Build potential Wk as a sum ofmonomials over all tropical disks.
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Choose points P1, . . . ,Pk ,Q ∈ R2 general, and consider all rigid
tropical disks passing through some subset of P1, . . . ,Pk andterminating at Q.Label each end with the variable x , y or z , and each Pi with avariable ui with u2i = 0. Build potential Wk as a sum ofmonomials over all tropical disks.
Mark Gross Mirror symmetry and tropical geometry
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xQ
P1
P2
xyz = κ
W2 = x
Mark Gross Mirror symmetry and tropical geometry
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Q
P1
P2
xyz = κ
y
W2 = x + y
Mark Gross Mirror symmetry and tropical geometry
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Q
P1
P2
xyz = κ
z
W2 = x + y + z
Mark Gross Mirror symmetry and tropical geometry
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Q
P1
P2
xyz = κ
xu1
z
W2 = x + y + z + u1xz
Mark Gross Mirror symmetry and tropical geometry
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Q
P1
P2
xyz = κ
z
xu2
W2 = x + y + z + u1xz + u2xz
Mark Gross Mirror symmetry and tropical geometry
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Q
P1
P2
xyz = κ
z
xu2
xu1
W2 = x + y + z + u1xz + u2xz + u1u2x2z
Mark Gross Mirror symmetry and tropical geometry
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Take Γ ⊂ (C∗)2 to be the compact torus
Γ = |x | = |y | = 1.
Calculate the integral
1
(2πi)2
∫Γ
eWk/~dx ∧ dy
xy
via a Taylor series expansion of the exponential and residues.Via residues, the only terms which contribute are constant on(C∗)2, i.e., with the same power of x , y and z , using xyz = κ.The power series expansion selects a set of tropical disks whichthen must glue at Q to give a tropical curve, the balancingcondition being enforced by the integration.
Mark Gross Mirror symmetry and tropical geometry
![Page 82: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/82.jpg)
Take Γ ⊂ (C∗)2 to be the compact torus
Γ = |x | = |y | = 1.
Calculate the integral
1
(2πi)2
∫Γ
eWk/~dx ∧ dy
xy
via a Taylor series expansion of the exponential and residues.Via residues, the only terms which contribute are constant on(C∗)2, i.e., with the same power of x , y and z , using xyz = κ.The power series expansion selects a set of tropical disks whichthen must glue at Q to give a tropical curve, the balancingcondition being enforced by the integration.
Mark Gross Mirror symmetry and tropical geometry
![Page 83: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/83.jpg)
Take Γ ⊂ (C∗)2 to be the compact torus
Γ = |x | = |y | = 1.
Calculate the integral
1
(2πi)2
∫Γ
eWk/~dx ∧ dy
xy
via a Taylor series expansion of the exponential and residues.Via residues, the only terms which contribute are constant on(C∗)2, i.e., with the same power of x , y and z , using xyz = κ.The power series expansion selects a set of tropical disks whichthen must glue at Q to give a tropical curve, the balancingcondition being enforced by the integration.
Mark Gross Mirror symmetry and tropical geometry
![Page 84: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/84.jpg)
Take Γ ⊂ (C∗)2 to be the compact torus
Γ = |x | = |y | = 1.
Calculate the integral
1
(2πi)2
∫Γ
eWk/~dx ∧ dy
xy
via a Taylor series expansion of the exponential and residues.Via residues, the only terms which contribute are constant on(C∗)2, i.e., with the same power of x , y and z , using xyz = κ.The power series expansion selects a set of tropical disks whichthen must glue at Q to give a tropical curve, the balancingcondition being enforced by the integration.
Mark Gross Mirror symmetry and tropical geometry
![Page 85: Mirror symmetry: A brief history. Superstring theory ...mg475/melbourne.pdf · with loops moving through space-time. A key prediction of superstring theory is: The universe is 10](https://reader034.vdocument.in/reader034/viewer/2022050218/5f63de4120e2087332302712/html5/thumbnails/85.jpg)
e.g., k = 2:
1
(2πi)2
∫Γ
(1 + ~−1(x + y + z + (u1 + u2)xz + u1u2x
2z)
+ ~−2(x + y + z + (u1 + u2)xz + u1u2x
2z)2/2 + · · · )dx ∧ dy
xy
= 1 + κ~−2(u1 + u2) + O(~−3).
Mark Gross Mirror symmetry and tropical geometry
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e.g., k = 2:
1
(2πi)2
∫Γ
(1 + ~−1(x + y + z + (u1 + u2)xz + u1u2x
2z)
+ ~−2(x + y + z + (u1 + u2)xz + u1u2x
2z)2/2 + · · · )dx ∧ dy
xy
= 1 + κ~−2(u1 + u2) + O(~−3).
Mark Gross Mirror symmetry and tropical geometry
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So we see precisely the contribution from one line (the coefficientof κ~−2(u1 + u2)).
Q
P1
P2
z
xu2
xu1
y
Mark Gross Mirror symmetry and tropical geometry