Polytopes, Polynomials, and String Theory

Transcription

1 Polytopes, Polynomials, and String Theory Ursula Whitcher Harvey Mudd College August 2010

2 Outline The Group String Theory and Mathematics Polytopes, Fans, and Toric Varieties

3 The Group Dagan Karp Dhruv Ranganathan 12 Paul Riggins 12 Ursula Whitcher Daniel Moore 11 Dmitri Skjorshammer 11

4 String Theory and Mathematics String Theory Fundamental particles are strings vibrating at different frequencies. Strings wrap extra, compact dimensions.

5 String Theory and Mathematics Gromov-Witten Theory Worldsheets of strings Moduli spaces of maps of curves Strong enumerative results

6 String Theory and Mathematics Mirror Symmetry Extra, compact dimensions are Calabi-Yau varieties. Mirror symmetry predicts that Calabi-Yau varieties should occur in paired or mirror families. Varying the complex structure of one family corresponds to varying the Kähler structure of the other family.

7 Polytopes, Fans, and Toric Varieties Lattices Let N be a lattice isomorphic to Z n. The dual lattice M of N is given by Hom(N, Z); it is also isomorphic to Z n. We write the pairing of v N and w M as v, w.

8 Polytopes, Fans, and Toric Varieties Cones A cone in N is a subset of the real vector space N R = N R generated by nonnegative R-linear combinations of a set of vectors {v 1,..., v m } N. We assume that cones are strongly convex, that is, they contain no line through the origin. Figure: Cox, Little, and Schenk

9 Polytopes, Fans, and Toric Varieties Fans A fan Σ consists of a finite collection of cones such that: Each face of a cone in the fan is also in the fan Any pair of cones in the fan intersects in a common face. Figure: Cox, Little, and Schenk

10 Polytopes, Fans, and Toric Varieties Simplicial Fans We say a fan Σ is simplicial if the generators of each cone in Σ are linearly independent over R.

11 Polytopes, Fans, and Toric Varieties Lattice Polytopes A lattice polytope is the convex hull of a finite set of points in a lattice. We assume that our lattice polytopes contain the origin. Definition Let be a lattice polytope in N which contains (0, 0). The polar polytope is the polytope in M given by: {(m 1,..., m k ) : (n 1,..., n k ) (m 1,..., m k ) 1 for all (n 1, n 2 ) }

12 Polytopes, Fans, and Toric Varieties Reflexive Polytopes Definition A lattice polytope is reflexive if is also a lattice polytope. If is reflexive, ( ) =.

13 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 of one-dimensional cones. 3. Take the fan S over the faces of M.

14 Polytopes, Fans, and Toric Varieties Toric Varieties as Quotients Let Σ be a fan in R n. Let {v 1,..., v q } be generators for the one-dimensional cones of Σ. Σ defines an n-dimensional toric variety V Σ. V Σ is the quotient of a subset C q Z(Σ) of C q by a subgroup of (C ) q.

15 Polytopes, Fans, and Toric Varieties Example Figure: Polygon Let R be the fan obtained by taking cones over the faces of. Z(Σ) consists of points of the form (0, 0, z 3, z 4 ) or (z 1, z 2, 0, 0). V R = (C 4 Z(Σ))/ (z 1, z 2, z 3, z 4 ) (λ 1 z 1, λ 1 z 2, z 3, z 4 ) (z 1, z 2, z 3, z 4 ) (z 1, z 2, λ 2 z 3, λ 2 z 4 ) where λ 1, λ 2 C. Thus, V R = P 1 P 1.

16 Polytopes, Fans, and Toric Varieties Blowing Up Adding cones to a fan Σ corresponds to blowing up subvarieties of V Σ We can use blow-ups to resolve singularities or create new varieties of interest Dhruv Ranganathan and Paul Riggins classified the symmetries of all toric blowups of P 3, including varieties corresponding to the associahedron, the cyclohedron, and the graph associahedra.

17 Calabi-Yau Hypersurfaces Let be a reflexive polytope, with polar polytope. Let R be the fan over the faces of Let Σ be a refinement of R Let {v k } N generate the one-dimensional cones of Σ

18 Calabi-Yau Hypersurfaces Let be a reflexive polytope, with polar polytope. Let R be the fan over the faces of Let Σ be a refinement of R Let {v k } N generate the one-dimensional cones of Σ The following polynomial defines a Calabi-Yau hypersurface in V Σ : f = x M If n = 3, f defines a K3 surface. c x q k=1 z v k,x +1 k

19 Quasismooth Hypersurfaces Let Σ be a simplicial fan, and let X be a hypersurface in V Σ. Suppose that X is described by a polynomial f in homogeneous coordinates. Definition If the products f / z i, i = 1... q do not vanish simultaneously on X, we say X is quasismooth.

20 Picard-Fuchs Equations A period is the integral of a differential form with respect to a specified homology class. Periods of holomorphic forms encode the complex structure of varieties. The Picard-Fuchs differential equation of a family of varieties is a differential equation that describes the way the value of a period changes as we move through the family.

21 The Residue Map in P n Let X be a smooth hypersurface in in P n described by a homogeneous polynomial f. Then there exists a residue map Res : H n (P n X ) H n 1 (X ).

22 The Residue Map in P n Let X be a smooth hypersurface in in P n described by a homogeneous polynomial f. Then there exists a residue map Res : H n (P n X ) H n 1 (X ). Let Ω 0 be a homogeneous holomorphic n-form on P n. We may represent elements of H n (P n X ) by forms PΩ 0,where P is a f k homogeneous polynomial. Let J(f ) =< f z 1,..., f z n+1 >. We have an induced residue map Res : C[z 1,..., z n+1 ]/J H n 1 (X ).

23 The Griffiths-Dwork Technique in P n We want to compute the Picard-Fuchs equation for a one-parameter family of Calabi-Yau hypersurfaces X t in P n.

24 The Griffiths-Dwork Technique in P n We want to compute the Picard-Fuchs equation for a one-parameter family of Calabi-Yau hypersurfaces X t in P n. Plan d dt α = d dt (α)

25 The Griffiths-Dwork Technique in P n We want to compute the Picard-Fuchs equation for a one-parameter family of Calabi-Yau hypersurfaces X t in P n. Plan d dt α = d dt (α) Since H (X t, C) is a finite-dimensional vector space, only finitely many derivatives can be linearly independent

26 The Griffiths-Dwork Technique in P n We want to compute the Picard-Fuchs equation for a one-parameter family of Calabi-Yau hypersurfaces X t in P n. Plan d dt α = d dt (α) Since H (X t, C) is a finite-dimensional vector space, only finitely many derivatives can be linearly independent Thus, there must be C(t)-linear relationships between derivatives of periods of the holomorphic form

27 The Griffiths-Dwork Technique in P n We want to compute the Picard-Fuchs equation for a one-parameter family of Calabi-Yau hypersurfaces X t in P n. Plan d dt α = d dt (α) Since H (X t, C) is a finite-dimensional vector space, only finitely many derivatives can be linearly independent Thus, there must be C(t)-linear relationships between derivatives of periods of the holomorphic form Use Res to convert to a polynomial algebra problem in C(t)[z 1,..., z n+1 ]/J

28 The Griffiths-Dwork Technique Advantages and Disadvantages Advantages We can work with arbitrary polynomial parametrizations of hypersurfaces. Disadvantages We need powerful computer algebra systems to work with C(t)[z 1,..., z n+1 ]/J.

29 Extending the Griffiths-Dwork Technique Cox and Katz discuss the Griffiths-Dwork technique for: Weighted projective spaces Ample hypersurfaces in varieties obtained from simplicial fans Discrete group quotients of these spaces Problem What if the fan over the faces of a polytope is not simplicial?

30 Extending the Residue Map Anvar Mavlyutov showed that Res can be defined for semiample, quasismooth hypersurfaces in simplicial toric varieties. Res : C[z 1,..., z q ]/J H n 1 (X ). In this case, Res may not be injective.

31 v5 v4 v6 v2 v1 v3 p 4 Polytopes, Polynomials, and String Theory The Skew Octahedron p 2 p3 p1 p 6 p8 p7 p5 Let be the reflexive octahedron with vertices v 1 = (1, 0, 0), v 2 = (1, 2, 0), v 3 = (1, 0, 2), v 4 = ( 1, 0, 0), v 5 = ( 1, 2, 0), and v 6 = ( 1, 0, 2). contains 19 lattice points. Let R be the fan obtained by taking cones over the faces of. Then R defines a toric variety V R which is isomorphic to (P 1 P 1 P 1 )/(Z 2 Z 2 Z 2 ).

32 A Symmetric Hypersurface Family f (t) = t z 0 z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 z 9 z 10 z 11 z 12 z 13 z 14 z 15 z 16 z 17 + z1 2 z2 2 z5 2 z6 2 z7 2 z8 2 z 9 z 10 z 11 z 12 z 13 z 14 z 15 + z10z z4 2 z5 2 z6 2 z 7 z 8 z9 2 z 11 z 12 z 13 z 16 z 17 + z0 2 z1 2 z2 2 z 6 z 7 z8 2 z 10 z 12 z13z 2 14 z15z z0 2 z2 2 z4 2 z 6 z 8 z 9 z10z 2 12 z13z 2 15 z 16 z z1 2 z3 2 z5 2 z 6 z7 2 z 8 z 9 z11z 2 12 z14z 2 15 z 16 + z3 2 z4 2 z5 2 z 6 z 7 z9 2 z 10 z11z 2 12 z 14 z16z z0 2 z1 2 z3 2 z 7 z 8 z 11 z 12 z 13 z14z 2 15z 2 16 z 17 + z0 2 z3 2 z4 2 z 9 z 10 z 11 z 12 z 13 z 14 z 15 z16z

33 The Picard-Fuchs Equation Theorem (MSW) Let ω = ( ) Res Ω0 f. Then ω is the period of a holomorphic form, and satisfies the Picard-Fuchs equation 3 ω t 3 + 6(t2 32) 2 ω t(t 2 64) t 2 + 7t2 64 ω t 2 (t 2 64) t + 1 t(t 2 64) ω = 0

34 The Picard-Fuchs Equation Theorem (MSW) Let ω = ( ) Res Ω0 f. Then ω is the period of a holomorphic form, and satisfies the Picard-Fuchs equation 3 ω t 3 + 6(t2 32) 2 ω t(t 2 64) t 2 + 7t2 64 ω t 2 (t 2 64) t + 1 t(t 2 64) ω = 0 As expected, the differential equation is third-order The differential equation is a symmetric square

35 Polytopes, Polynomials, and String Theory Ursula Whitcher Harvey Mudd College August 2010