Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and...
Transcript of Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and...
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Polytopes, Polynomials, and String Theory
Polytopes, Polynomials, and String Theory
Ursula [email protected]
Harvey Mudd College
August 2010
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Polytopes, Polynomials, and String Theory
Outline
The Group
String Theory and Mathematics
Polytopes, Fans, and Toric Varieties
Hypersurfaces and Picard-Fuchs Equations
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Polytopes, Polynomials, and String Theory
The Group
I Dagan KarpI Dhruv Ranganathan ’12I Paul Riggins ’12
I Ursula WhitcherI Daniel Moore ’11I Dmitri Skjorshammer ’11
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Polytopes, Polynomials, and String Theory
String Theory and Mathematics
String TheoryI “Fundamental” particles are strings vibrating at different
frequencies.I Strings wrap extra, compact dimensions.
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Polytopes, Polynomials, and String Theory
String Theory and Mathematics
Gromov-Witten Theory
I Worldsheets of strings
I Moduli spaces of maps of curves
I Strong enumerative results
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Polytopes, Polynomials, and String Theory
String Theory and Mathematics
Mirror Symmetry
I Extra, compact dimensions are Calabi-Yau varieties.
I Mirror symmetry predicts that Calabi-Yau varieties shouldoccur in paired or mirror families.
I Varying the complex structure of one family corresponds tovarying the Kahler structure of the other family.
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Lattices
Let N be a lattice isomorphic to Zn. The dual lattice M of N isgiven by Hom(N,Z); it is also isomorphic to Zn. We write thepairing of v ∈ N and w ∈ M as 〈v ,w〉.
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Cones
A cone in N is a subset of the real vector space NR = N ⊗ Rgenerated by nonnegative R-linear combinations of a set of vectorsv1, . . . , vm ⊂ N. We assume that cones are strongly convex, thatis, they contain no line through the origin.
Figure: Cox, Little, and Schenk
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Fans
A fan Σ consists of a finite collection of cones such that:
I Each face of a cone in the fan is also in the fan
I Any pair of cones in the fan intersects in a common face.
Figure: Cox, Little, and Schenk
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Simplicial Fans
We say a fan Σ is simplicial if the generators of each cone in Σ arelinearly independent over R.
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Lattice Polytopes
A lattice polytope is the convex hull of a finiteset of points in a lattice. We assume that ourlattice polytopes contain the origin.
DefinitionLet ∆ be a lattice polytope in N which contains (0, 0). The polarpolytope ∆ is the polytope in M given by:
(m1, . . . ,mk) : (n1, . . . , nk)·(m1, . . . ,mk) ≥ −1 for all (n1, n2) ∈ ∆
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Reflexive Polytopes
DefinitionA lattice polytope ∆ is reflexive if ∆ is also a lattice polytope.
If ∆ is reflexive, (∆) = ∆.
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Fans from Polytopes
Figure: Cox, Little, and Schenk
We may define a fan using a polytope in several ways:
1. Take the fan R over the faces of ⊂ N.
2. Refine R by using other lattice points in as generators ofone-dimensional cones.
3. Take the fan S over the faces of ⊂ M.
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Toric Varieties as Quotients
I Let Σ be a fan in Rn.
I Let v1, . . . , vq be generators for the one-dimensional conesof Σ.
I Σ defines an n-dimensional toric variety VΣ.
I VΣ is the quotient of a subset Cq − Z (Σ) of Cq by asubgroup of (C∗)q.
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Example
Figure: Polygon
Let R be the fan obtained by taking cones overthe faces of . Z (Σ) consists of points of theform (0, 0, z3, z4) or (z1, z2, 0, 0).
VR = (C4 − Z (Σ))/ ∼
(z1, z2, z3, z4) ∼ (λ1z1, λ1z2, z3, z4)
(z1, z2, z3, z4) ∼ (z1, z2, λ2z3, λ2z4)
where λ1, λ2 ∈ C∗. Thus, VR = P1 × P1.
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Polytopes, Polynomials, and String Theory
Polytopes, Fans, and Toric Varieties
Blowing UpI Adding cones to a fan Σ corresponds to blowing up
subvarieties of VΣ
I We can use blow-ups to resolve singularities or create newvarieties of interest
I Dhruv Ranganathan and Paul Riggins classified thesymmetries of all toric blowups of P3, including varietiescorresponding to the associahedron, the cyclohedron, and thegraph associahedra.
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
Calabi-Yau Hypersurfaces
I Let be a reflexive polytope, with polar polytope .I Let R be the fan over the faces of I Let Σ be a refinement of R
I Let vk ⊂ ∩ N generate the one-dimensional cones of Σ
The following polynomial defines a Calabi-Yau hypersurface in VΣ:
f =∑
x∈∩M
cx
q∏k=1
z〈vk ,x〉+1k
If n = 3, f defines a K3 surface.
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
Calabi-Yau Hypersurfaces
I Let be a reflexive polytope, with polar polytope .I Let R be the fan over the faces of I Let Σ be a refinement of R
I Let vk ⊂ ∩ N generate the one-dimensional cones of Σ
The following polynomial defines a Calabi-Yau hypersurface in VΣ:
f =∑
x∈∩M
cx
q∏k=1
z〈vk ,x〉+1k
If n = 3, f defines a K3 surface.
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
Quasismooth Hypersurfaces
Let Σ be a simplicial fan, and let X be a hypersurface in VΣ.Suppose that X is described by a polynomial f in homogeneouscoordinates.
DefinitionIf the products ∂f /∂zi , i = 1 . . . q do not vanish simultaneously onX , we say X is quasismooth.
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
Picard-Fuchs Equations
I A period is the integral of a differential form with respect to aspecified homology class.
I Periods of holomorphic forms encode the complex structure ofvarieties.
I The Picard-Fuchs differential equation of a family of varietiesis a differential equation that describes the way the value of aperiod changes as we move through the family.
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Residue Map in Pn
Let X be a smooth hypersurface in in Pn described by ahomogeneous polynomial f . Then there exists a residue map
Res : Hn(Pn − X )→ Hn−1(X ).
Let Ω0 be a homogeneous holomorphic n-form on Pn. We mayrepresent elements of Hn(Pn − X ) by forms PΩ0
f k ,where P is ahomogeneous polynomial.
Let J(f ) =< ∂f∂z1, . . . , ∂f
∂zn+1>. We have an induced residue map
Res : C[z1, . . . , zn+1]/J → Hn−1(X ).
![Page 22: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/22.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Residue Map in Pn
Let X be a smooth hypersurface in in Pn described by ahomogeneous polynomial f . Then there exists a residue map
Res : Hn(Pn − X )→ Hn−1(X ).
Let Ω0 be a homogeneous holomorphic n-form on Pn. We mayrepresent elements of Hn(Pn − X ) by forms PΩ0
f k ,where P is ahomogeneous polynomial.
Let J(f ) =< ∂f∂z1, . . . , ∂f
∂zn+1>. We have an induced residue map
Res : C[z1, . . . , zn+1]/J → Hn−1(X ).
![Page 23: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/23.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Griffiths-Dwork Technique in Pn
We want to compute the Picard-Fuchs equation for aone-parameter family of Calabi-Yau hypersurfaces Xt in Pn.
Plan
I ddt
∫α =
∫ddt (α)
I Since H∗(Xt ,C) is a finite-dimensional vector space, onlyfinitely many derivatives can be linearly independent
I Thus, there must be C(t)-linear relationships betweenderivatives of periods of the holomorphic form
I Use Res to convert to a polynomial algebra problem inC(t)[z1, . . . , zn+1]/J
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Griffiths-Dwork Technique in Pn
We want to compute the Picard-Fuchs equation for aone-parameter family of Calabi-Yau hypersurfaces Xt in Pn.
Plan
I ddt
∫α =
∫ddt (α)
I Since H∗(Xt ,C) is a finite-dimensional vector space, onlyfinitely many derivatives can be linearly independent
I Thus, there must be C(t)-linear relationships betweenderivatives of periods of the holomorphic form
I Use Res to convert to a polynomial algebra problem inC(t)[z1, . . . , zn+1]/J
![Page 25: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/25.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Griffiths-Dwork Technique in Pn
We want to compute the Picard-Fuchs equation for aone-parameter family of Calabi-Yau hypersurfaces Xt in Pn.
Plan
I ddt
∫α =
∫ddt (α)
I Since H∗(Xt ,C) is a finite-dimensional vector space, onlyfinitely many derivatives can be linearly independent
I Thus, there must be C(t)-linear relationships betweenderivatives of periods of the holomorphic form
I Use Res to convert to a polynomial algebra problem inC(t)[z1, . . . , zn+1]/J
![Page 26: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/26.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Griffiths-Dwork Technique in Pn
We want to compute the Picard-Fuchs equation for aone-parameter family of Calabi-Yau hypersurfaces Xt in Pn.
Plan
I ddt
∫α =
∫ddt (α)
I Since H∗(Xt ,C) is a finite-dimensional vector space, onlyfinitely many derivatives can be linearly independent
I Thus, there must be C(t)-linear relationships betweenderivatives of periods of the holomorphic form
I Use Res to convert to a polynomial algebra problem inC(t)[z1, . . . , zn+1]/J
![Page 27: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/27.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Griffiths-Dwork Technique in Pn
We want to compute the Picard-Fuchs equation for aone-parameter family of Calabi-Yau hypersurfaces Xt in Pn.
Plan
I ddt
∫α =
∫ddt (α)
I Since H∗(Xt ,C) is a finite-dimensional vector space, onlyfinitely many derivatives can be linearly independent
I Thus, there must be C(t)-linear relationships betweenderivatives of periods of the holomorphic form
I Use Res to convert to a polynomial algebra problem inC(t)[z1, . . . , zn+1]/J
![Page 28: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/28.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Griffiths-Dwork TechniqueAdvantages and Disadvantages
Advantages
We can work with arbitrary polynomial parametrizations ofhypersurfaces.
Disadvantages
We need powerful computer algebra systems to work withC(t)[z1, . . . , zn+1]/J.
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
Extending the Griffiths-Dwork Technique
Cox and Katz discuss the Griffiths-Dwork technique for:
I Weighted projective spaces
I Ample hypersurfaces in varieties obtained from simplicial fans
I Discrete group quotients of these spaces
ProblemWhat if the fan over the faces of a polytope is not simplicial?
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
Extending the Residue Map
I Anvar Mavlyutov showed that Res can be defined forsemiample, quasismooth hypersurfaces in simplicial toricvarieties.
Res : C[z1, . . . , zq]/J → Hn−1(X ).
I In this case, Res may not be injective.
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Skew Octahedron
p1
p2
p3
p4
p5
6p
p7
p8
v1
v2
v3
v4
v5
v6
I Let be the reflexive octahedron with vertices v1 = (1, 0, 0),v2 = (1, 2, 0), v3 = (1, 0, 2), v4 = (−1, 0, 0), v5 = (−1,−2, 0),and v6 = (−1, 0,−2).
I contains 19 lattice points.I Let R be the fan obtained by taking cones over the faces of .
Then R defines a toric variety VR which is isomorphic to(P1 × P1 × P1)/(Z2 × Z2 × Z2).
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
A Symmetric Hypersurface Family
f (t) = t · z0z1z2z3z4z5z6z7z8z9z10z11z12z13z14z15z16z17
+ z21 z2
2 z25 z2
6 z27 z2
8 z9z10z11z12z13z14z15
+ z210z2
2 z24 z2
5 z26 z7z8z2
9 z11z12z13z16z17
+ z20 z2
1 z22 z6z7z2
8 z10z12z213z14z2
15z17
+ z20 z2
2 z24 z6z8z9z2
10z12z213z15z16z2
17
+ z21 z2
3 z25 z6z2
7 z8z9z211z12z2
14z15z16
+ z23 z2
4 z25 z6z7z2
9 z10z211z12z14z2
16z17
+ z20 z2
1 z23 z7z8z11z12z13z2
14z215z16z17
+ z20 z2
3 z24 z9z10z11z12z13z14z15z2
16z217
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Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Picard-Fuchs Equation
Theorem (MSW)
Let ω =∫
Res(
Ω0f
). Then ω is the period of a holomorphic form,
and satisfies the Picard-Fuchs equation
∂3ω
∂t3+
6(t2 − 32)
t(t2 − 64)
∂2ω
∂t2+
7t2 − 64
t2(t2 − 64)
∂ω
∂t+
1
t(t2 − 64)ω = 0
I As expected, the differential equation is third-order
I The differential equation is a symmetric square
![Page 34: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/34.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
The Picard-Fuchs Equation
Theorem (MSW)
Let ω =∫
Res(
Ω0f
). Then ω is the period of a holomorphic form,
and satisfies the Picard-Fuchs equation
∂3ω
∂t3+
6(t2 − 32)
t(t2 − 64)
∂2ω
∂t2+
7t2 − 64
t2(t2 − 64)
∂ω
∂t+
1
t(t2 − 64)ω = 0
I As expected, the differential equation is third-order
I The differential equation is a symmetric square
![Page 35: Polytopes, Polynomials, and String Theoryuaw/notes/mathfest.pdf · Polytopes, Polynomials, and String Theory Polytopes, Polynomials, and String Theory Ursula Whitcher ursula@math.hmc.edu](https://reader033.fdocuments.in/reader033/viewer/2022051606/6017abf8bb502a633d02dd22/html5/thumbnails/35.jpg)
Polytopes, Polynomials, and String Theory
Hypersurfaces and Picard-Fuchs Equations
Polytopes, Polynomials, and String Theory
Ursula [email protected]
Harvey Mudd College
August 2010