Mirror Symmetry Through Reflexive Polytopes

Transcription

1 Mirror Symmetry Through Reflexive Polytopes Physical and Mathematical Dualities Ursula Whitcher Harvey Mudd College April 2010

2 Outline String Theory and Mirror Symmetry Some Complex Geometry Reflexive Polytopes From Polytopes to Spaces

3 Where s the Theory of Everything? We understand gravity on a large spatial scale (planets, stars, galaxies). Figure: S. Bush et al. We understand quantum physics on a small spatial scale (electrons, photons, quarks).

4 Are Strings the Answer? Fundamental particles are strings vibrating at different frequencies. Strings wrap other dimensions!

5 T -Duality Pairs of Universes An extra dimension shaped like a circle of radius R and an extra dimension shaped like a circle of radius α /R yield indistinguishable physics! (The slope parameter α has units of length squared.) Figure: Large radius, few windings Figure: Small radius, many windings

6 Heterotic String Theory Supersymmetries Interchange bosons (spin n) with fermions (spin n ) Hybrid Vigor Equations of motion split into left-moving and right-moving solutions. Operators Q and Q act on left-moving and right-moving states, respectively. Q and Q are well-defined up to sign.

7 Building a Model Locally, space-time should look like M 3,1 V. M 3,1 is four-dimensional space-time V is a d-dimensional complex manifold Physicists require d = 3 (6 real dimensions) V is a Calabi-Yau manifold For differential geometers, V is a Ricci-flat Kähler-Einstein manifold Our choice of geometric model fixes a choice of sign for the operators Q and Q

8 Geometry Appears Physical properties of our model for the universe should correspond to geometric properties of the space V. We will be particularly interested in the complex moduli and Kähler moduli of V.

9 The Heterotic Duality Changing the sign of the operator Q corresponds to choosing a new Calabi-Yau manifold V. The geometrical properties of V and V yield equivalent physical theories. Mirror symmetry is the study of the mathematical consequences of this duality.

10 Realizing Mirror Symmetry Geometrically We need: Complex manifolds which are Calabi-Yau and arise in paired or mirror families with dual geometric properties. Varying complex structure in one family should correspond to varying Kähler structure in the other family.

11 Complex Structure An n-dimensional complex manifold is a geometric space which looks locally like C n. Example: Elliptic Curves We can think of varying the parameter τ as either changing the complex manifold, or changing the complex structure on an underlying topological 2-torus.

12 Kähler Structure Standard Product We can pair vectors v and w with their tails at a point z in C using the product for complex numbers: Note that v, v = v 2. v 1 v n v, w = vw More generally, if v =. and w =. w 1 w n are vectors with their tails at a point z = (z 1,..., z n ) in C n, their standard Hermitian product is given by v, w = v i w i = v T w

13 Kähler Structure Hermitian metrics A Hermitian metric H tells us how to pair tangent vectors at any point of a complex manifold and obtain a complex number. H( v, w) = H( w, v)

14 Kähler Structure Hermitian metrics A Hermitian metric H tells us how to pair tangent vectors at any point of a complex manifold and obtain a complex number. H( v, w) = H( w, v) Elliptic Curve Example We can use the standard Hermitian product on C to describe a Hermitian metric for tangent vectors to an elliptic curve.

15 Kähler Structure Kähler Metrics A Kähler metric is a special type of Hermitian metric which can be written in local coordinates as follows. If v and w are vectors with their tails at a point z 0 in C n, κ( v, w) = v T (I + G(z)) w. Here I is the identity matrix and g 11 (z)... g 1n (z) G(z) =..... g n1 (z)... g nn (z) vanishes up to order 2 at z 0.

16 The Geometric Ingredients of Mirror Symmetry We need: Complex manifolds V which are Calabi-Yau and arise in paired or mirror families with dual geometric properties. Varying complex structure in one family should correspond to varying Kähler structure in the other family.

17 Batyrev s Insight We can describe mirror families of Calabi-Yau manifolds using combinatorial objects called reflexive polytopes.

18 Lattice Polygons The points in the plane with integer coordinates form a lattice N. A lattice polygon is a polygon in the plane which has vertices in the lattice.

19 Reflexive Polygons We say a lattice polygon is reflexive if it has only one lattice point, the origin, in its interior. Figure: A reflexive triangle

20 Describing a Reflexive Polygon List the vertices

21 Describing a Reflexive Polygon List the vertices {(0, 1), (1, 0), ( 1, 1)}

22 Describing a Reflexive Polygon List the vertices {(0, 1), (1, 0), ( 1, 1)} List the equations of the edges

23 Describing a Reflexive Polygon List the vertices List the equations of the edges {(0, 1), (1, 0), ( 1, 1)} x y 2x y x + 2y = 1 = 1 = 1

24 A Dual Lattice Let M be another copy of the points in the plane with integer coordinates. The dot product lets us pair points in N with points in M: (n 1, n 2 ) (m 1, m 2 ) = n 1 m 1 + n 2 m 2

25 Polar Polygons Edge equations define new polygons Let be a lattice polygon in N which contains (0, 0). The polar polygon is the polygon in M given by: {(m 1, m 2 ) : (n 1, n 2 ) (m 1, m 2 ) 1 for all (n 1, n 2 ) }

26 Polar Polygons Edge equations define new polygons Let be a lattice polygon in N which contains (0, 0). The polar polygon is the polygon in M given by: {(m 1, m 2 ) : (n 1, n 2 ) (m 1, m 2 ) 1 for all (n 1, n 2 ) } (x, y) ( 1, 1) (x, y) (2, 1) (x, y) ( 1, 2) = 1 = 1 = 1

27 Polar Polygons Edge equations define new polygons Let be a lattice polygon in N which contains (0, 0). The polar polygon is the polygon in M given by: {(m 1, m 2 ) : (n 1, n 2 ) (m 1, m 2 ) 1 for all (n 1, n 2 ) } (x, y) ( 1, 1) (x, y) (2, 1) (x, y) ( 1, 2) = 1 = 1 = 1 Figure: Our triangle s polar polygon

28 Mirror Pairs If is a reflexive polygon, then: is also a reflexive polygon ( ) =. and are a mirror pair.

29 A Polygon Duality Mirror pair of triangles Figure: 3 boundary lattice points Figure: 9 boundary lattice points = 12

30 Mirror Pairs of Polygons Figure: F. Rohsiepe, Elliptic Toric K3 Surfaces and Gauge Algebras

31 Other Dimensions Definition Let { v 1, v 2,..., v q } be a set of points in R k. The polytope with vertices { v 1, v 2,..., v q } is the convex hull of these points.

32 Polar Polytopes Let N be the lattice of points with integer coordinates in R k. A lattice polytope has vertices in N. As before, we have a dual lattice M and a dot product (n 1,..., n k ) (m 1,..., m k ) = n 1 m n k m k Definition Let be a lattice polygon 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 ) }

33 Reflexive Polytopes Definition A lattice polytope is reflexive if is also a lattice polytope. If is reflexive, ( ) =. and are a mirror pair.

34 Mirror Polytopes Yield Mirror Spaces polytope polar polytope Laurent polynomial mirror Laurent polynomial space mirror space

35 From Polytopes to Polynomials Standard basis vectors in N variables z i (1, 0,..., 0) z 1 (0, 1,..., 0) z 2... (0, 0,..., 1) z n Lattice points in monomials defined on (C ) n (m 1,..., m k ) z (1,0,...,0) (m 1,...,m k ) 1 z (0,1,...,0) (m 1,...,m k ) 2 z (0,0,...,1) (m 1,...,m k ) k Laurent polynomials p α defined on (C ) n

36 From Polynomials to Spaces The solutions to the Laurent polynomials p α describe geometric spaces. z z 1 = 0 z z 1 = 0 z z5 2 + z5 3 + z = 0 Figure: Slice of a Calabi-Yau threefold

37 Compactifying Our Laurent polynomials p α define spaces which are not compact: z i can be infinitely large. We can solve this problem by adding in some points at infinity using a standard procedure from algebraic geometry. The resulting compact spaces V α are Calabi-Yau varieties of dimension d = k 1. When k = 2, for generic choice of α, the V α are elliptic curves. When k = 4, for generic choice of α, the V α are smooth 3-dimensional Calabi-Yau manifolds.

38 Mirror Symmetry polytope polar polytope Laurent polynomials p α mirror Laurent polynomials pα spaces V α mirror spaces Vα

39 Counting Complex Moduli The possible deformations of complex structure of V α form a complex vector space of dimension h d 1,1 (V α ). For k 4, h d 1,1 (V α ) = l( ) k 1 Γ l (Γ ) + Θ l (Θ )l ( ˆΘ ) l() = number of lattice points l () = number of lattice points in the relative interior of a polytope or face The Γ are codimension 1 faces of The Θ are codimension 2 faces of Θ ˆΘ is the face of dual to Θ

40 Counting Kähler Moduli For k 4, h 1,1 (V α ) = l( ) k 1 Γ l (Γ) + Θ l (Θ)l ( ˆΘ) l() = number of lattice points l () = number of lattice points in the relative interior of a polytope or face The Γ are codimension 1 faces of The Θ are codimension 2 faces of Θ ˆΘ is the face of dual to Θ

41 Comparing V and V For k 4, h 1,1 (V α ) = l( ) k 1 Γ l (Γ) + Θ l (Θ)l ( ˆΘ) h d 1,1 (V α ) = l( ) k 1 Γ l (Γ ) + Θ l (Θ )l ( ˆΘ )

42 Comparing V and V For k 4, h 1,1 (V α ) = l( ) k 1 Γ l (Γ) + Θ l (Θ)l ( ˆΘ) h d 1,1 (V α ) = l( ) k 1 Γ l (Γ ) + Θ l (Θ )l ( ˆΘ ) h 1,1 (V α) = l( ) k 1 Γ l (Γ ) + Θ l (Θ )l ( ˆΘ ) h d 1,1 (V α) = l( ) k 1 Γ l (Γ) + Θ l (Θ)l ( ˆΘ)

43 Mirror Symmetry from Mirror Polytopes We have mirror families of Calabi-Yau varieties V α and V α of dimension d = k 1. h 1,1 (V α ) = h d 1,1 (V α) h d 1,1 (V α ) = h 1,1 (V α)

44 An Example Four-dimensional analogue: has vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), and ( 1, 1, 1, 1). has vertices ( 1, 1, 1, 1), (4, 1, 1, 1), ( 1, 4, 1, 1), ( 1, 1, 4, 1), and ( 1, 1, 1, 4).

45 An Example Four-dimensional analogue: has vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), and ( 1, 1, 1, 1). has vertices ( 1, 1, 1, 1), (4, 1, 1, 1), ( 1, 4, 1, 1), ( 1, 1, 4, 1), and ( 1, 1, 1, 4). h 1,1 (V α ) = l( ) n 1 Γ l (Γ) + Θ l (Θ)l ( ˆΘ) = = 1.

46 Example (Continued) has vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), and ( 1, 1, 1, 1). has vertices ( 1, 1, 1, 1), (4, 1, 1, 1), ( 1, 4, 1, 1), ( 1, 1, 4, 1), and ( 1, 1, 1, 4). h 1,1 (V α ) = 1 h 3 1,1 (V α ) = l( ) n 1 Γ l (Γ ) + Θ l (Θ )l ( ˆΘ ) = = 101.

47 The Hodge Diamond Calabi-Yau Threefolds h 1,1 (V ) 0 1 h 2,1 (V ) h 2,1 (V ) 1 0 h 1,1 (V )

48 The Hodge Diamond Calabi-Yau Threefolds h 1,1 (V ) 0 1 h 2,1 (V ) h 2,1 (V ) 1 0 h 1,1 (V ) V α V α