Unifying Mirror Symmetry Constructions David Favero - - PowerPoint PPT Presentation

Unifying Mirror Symmetry Constructions

Unifying Mirror Symmetry Constructions

David Favero favero@ualberta.ca

University of Alberta

May 2016 Korean Institute for Advanced Study Slides available at: www.ualberta.ca/∼favero

Unifying Mirror Symmetry Constructions String theoretic motivations

String Theoretic Universe

◮ In physics, there is a desire to unify quantum mechanics and

general relativity (gravity).

◮ String Theory is one such proposal. ◮ In String Theory, the “fundamental” units of matter are

strings.

Unifying Mirror Symmetry Constructions String theoretic motivations

Geometric Requirements of String Theory

Physics: locally, space-time must look like U = R4 × X where

◮ R4 is four-dimensional space-time (Minkowski space-time) ◮ X is a 3-dimensional complex manifold called a Calabi-Yau

manifold.

Unifying Mirror Symmetry Constructions String theoretic motivations

Symplectic manifolds

Definition

A symplectic manifold (M, ω) is a manifold M equipped with a closed non-degenerate differential 2-form ω called the symplectic form i.e., locally, an alternating nondegenerate bilinear form.

Example (The local picture)

Let M = R2n with basis u1, ..., un, v1, ..., vn. Define ω to be ω(ui, vi) = 1 ω(vi, ui) = −1 and to be zero on all other pairs of basis vectors. ω := Id −Id

Unifying Mirror Symmetry Constructions String theoretic motivations

Calabi-Yau manifolds

Definition

Let M be a complex manifold with a compatible symplectic form. We say that M is Calabi-Yau if it is simply-connected, compact, and admits a non-vanishing holomorphic n-form.

◮ Equivalent definition: a Ricci-flat, K¨

ahler-Einstein manifold (Yau ’78).

Unifying Mirror Symmetry Constructions String theoretic motivations

Example of a Calabi-Yau manifold

Example

Consider the set {(x0, ..., x4) ∈ C5\0 | x5

0 + ... + x5 4 = 0}/C∗

= {(x0, ..., x4) ∈ C5\0 | x5

0 + ... + x5 4 = 0}/ ∼

⊆ CP4 =: C5\0/ ∼ where (x0, ..., x4) ∼ (λx0, ..., λx4) for all λ ∈ C∗. Remark: The Calabi-Yau condition is that 5 = 4 + 1.

Unifying Mirror Symmetry Constructions Introduction to Mirror Symmetry

Types of String Theories

Let X be a three dimensional Calabi-Yau manifold.

Mirror Symmetry

Given Type IIA string theory on the space X, there is another Calabi-Yau 3-fold X so that the Type IIB string theory on the space X gives the same physical theory. Definition: X is known as the mirror to X.

Unifying Mirror Symmetry Constructions Introduction to Mirror Symmetry

Geometric ramifications of Mirror Symmetry

Mirror Symmetry

Given Type IIA string theory on the space X, there is another Calabi-Yau 3-fold X so that the Type IIB string theory on the space X gives the same physical theory.

Question:

What does this string duality mean geometrically?

Unifying Mirror Symmetry Constructions Introduction to Mirror Symmetry

Geometric ramifications of Mirror Symmetry

Mirror Symmetry

Given Type IIA string theory on the space X, there is another Calabi-Yau 3-fold X so that the Type IIB string theory on the space X gives the same physical theory.

Question:

What does this string duality mean geometrically?

Mantra:

Mirror symmetry is a duality between the symplectic geometry of X and the complex/algebraic geometry of X.

Unifying Mirror Symmetry Constructions Introduction to Mirror Symmetry

Mathematical Mirror Symmetry

Mantra:

Mirror symmetry is a duality between the symplectic geometry of X and the complex/algebraic geometry of X. Type IIA Type IIB Symplectic Deformations Complex Deformations Cohomology of X Cohomology of X Enumerative Geometry Variations of Hodge Structure Fukaya Category Derived Category of Coherent Sheaves

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ Derived categories were defined by Verdier in 1967.

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ Derived categories were defined by Verdier in 1967. ◮ For a ring R, objects of D(R) are formally built from modules

Ai ∈ R − mod. ...

dn+2

− − → An+1

dn+1

− − → An

dn

− → An−1

dn−1

− − − → ...

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ Derived categories were defined by Verdier in 1967. ◮ For a ring R, objects of D(R) are formally built from modules

Ai ∈ R − mod. ...

dn+2

− − → An+1

dn+1

− − → An

dn

− → An−1

dn−1

− − − → ...

◮ The original intent of derived categories was to provide an

appropriate setting for homological algebra.

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ For an algebraic variety X, we can associate a derived

category D(X).

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ For an algebraic variety X, we can associate a derived

category D(X).

◮ Objects of D(X) are roughly vector bundles over submanifolds

• f X.

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ For an algebraic variety X, we can associate a derived

category D(X).

◮ Objects of D(X) are roughly vector bundles over submanifolds

• f X.

◮ In the 80s and 90s, Mukai, Beilinson, Bondal, Orlov,

Kapranov, and others began to study D(X) as a geometric invariant.

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ There are 3 conjectures which I consider the most central to

the study of D(X)

Unifying Mirror Symmetry Constructions Derived Categories

Derived Categories

◮ There are 3 conjectures which I consider the most central to

the study of D(X)

◮ Two are due to Kawamata and one is due to Kontsevich.

Unifying Mirror Symmetry Constructions Derived Categories

Kawamata’s First Conjecture

Conjecture (Kawamata ’02)

The following set is finite: {Y | D(Y ) = D(X)}.

◮ True in dimension 1 (easy) ◮ True in dimension 2 (Orlov ’96, Bridgeland-Macocia ’01,

Kawamata ’02 )

◮ True for varieties with positive or negative curvature

(Bondal-Orlov ’97)

◮ True for complex n-dimensional tori

(Huybrechts—Nieper–Wisskirchen ’11, Favero1 ’12)

◮ False in dimension 3 (Lesieutre ’13)

1Reconstruction and Finiteness Results for Fourier-Mukai Partners,

Advances in Mathematics, V. 229, I. 1, pgs 1955-1971, 2012.

Unifying Mirror Symmetry Constructions Derived Categories

Kawamata’s Second Conjecture

Conjecture (Kawamata ’02)

If X and Y are Calabi-Yau and have isomorphic open (dense) subsets, then, their derived categories are equivalent. Known for the following types of “algebraic surgeries”

◮ Standard Flops (Bondal-Orlov ’95) ◮ Toroidal Flops (Kawamata ’02) ◮ Flops in dimension 3 (Bridgeland ’02) ◮ Elementary wall-crossings from variation of Geometric

Invariant Theory Quotients (Halpern-Leistner ’12, Ballard-Favero-Katzarkov2 ’12)

2Variation of Geometric Invariant Theory Quotients and Derived

Categories(63 pages). To appear in Journal f¨ ur die reine und angewandte Mathematik (Crelle’s journal).

Unifying Mirror Symmetry Constructions Derived Categories

Weak Factorization Theorem

Theorem (Weak Factorization Theorem, Wlodarczyk ’03)

Suppose X and Y are compact algebraic varieties which agree on a (dense) open subset. Then, there exists a diagram of morphisms: Z1

• · · ·
• Zn
• X

X1 Xn Y such that each triangle is an elementary wall-crossing.

Remark

Assuming X, Y are Calabi-Yau, then all that remains to know for Kawamata’s Conjecture is that we can choose X1, ..., Xn to be Calabi-Yau.

Unifying Mirror Symmetry Constructions Derived Categories

Boundary Conditions

Recall that when strings move though time they create surfaces (worldsheets). When we discuss strings in X, we think of our surfaces as mapping into spacetime X. Type IIA string theory requires that the end points of our strings move in some subspaces L1, L2.

Unifying Mirror Symmetry Constructions Derived Categories

The Fukaya Category

Konsevich proposed that the target for the topological quantum field theory associated to Type IIA string theory is called the Fukaya category Type IIA TQFT : Strings → Fuk(X) L1, L2, L3 → A Lagrangian Subspaces of X (objects) A, B, C → Intersection points (morphisms)

Unifying Mirror Symmetry Constructions Derived Categories

Boundary Conditions

Kontsevich proposed D(X) as the natural target for the topological quantum field theory associated to Type IIB string theory: Type IIB TQFT : Strings → D(X) which takes the L1, L2, L3 to objects in D(X) and A, B, C to morphisms in D(X).

Unifying Mirror Symmetry Constructions Derived Categories

Homological Mirror Symmetry

Mirror symmetry exchanges Type IIA and Type IIB string theories between X and its mirror X. Type IIA TQFT : Strings → Fuk( X) Type IIB TQFT : Strings → D(X)

Conjecture (Homological Mirror Symmetry, Kontsevich)

Let X be a Calabi-Yau manifold and X be its mirror. There is an equivalence of categories Fuk( X) = D(X).

Unifying Mirror Symmetry Constructions Derived Categories

Conjecture (Homological Mirror Symmetry, Kontsevich)

Let X be a Calabi-Yau manifold and X be its mirror. There is an equivalence of categories Fuk( X) = D(X). Known for

◮ Dimension 1 (Polishchuk-Zaslow ’98) ◮ Dimension 2 (Seidel ’03) ◮ Hypersurfaces in Projective Space (Sheridan ’11) ◮ non-Calabi-Yau cases:

◮ Fano Toric Varieties (Abouzaid ’06) ◮ Del Pezzo Surfaces (Auroux-Katzarkov-Kontsevich ’05) ◮ Abelian Surfaces (Abouzaid-Smith ’10) ◮ some non-Fano toric varieties

(Ballard-Diemer-Favero-Kerr-Katzarkov ’15)

◮ Many non-compact cases

Unifying Mirror Symmetry Constructions Derived Categories

What is the mirror?

Fundamental question:

Given a Calabi-Yau variety X, what is its mirror?

Unifying Mirror Symmetry Constructions Derived Categories

What is the mirror?

Fundamental question:

Given a Calabi-Yau variety X, what is its mirror?

Example

Consider the Fermat quintic X5 given by x5

0 + x5 1 + x5 2 + x5 3 + x5 4 = 0.

This Fermat quintic is symmetric by scaling the xi by fifth roots of unity:

(x0, x1, x2, x3, x4) → (ζx0, ζ−1x1, x2, x3, x4) (x0, x1, x2, x3, x4) → (ζx0, x1, ζ−1x2, x3, x4) (x0, x1, x2, x3, x4) → (ζx0, x1, x2, ζ−1x3, x4)

Symmetry group G = (Z/5Z)3. Take X5 to be the quotient X5/G.

Unifying Mirror Symmetry Constructions Derived Categories

Mirror Constructions

There are many constructions of mirrors, each having different

• contexts. They don’t always agree and have internal

inconsistencies! Examples:

◮ Greene-Plesser-Roan ’90 ◮ Berglund-H¨

ubsch ’93

◮ Batyrev-Borisov ’95 ◮ Strominger-Yau-Zaslow ’96 ◮ Hori-Vafa ’00 ◮ Clarke ’08

Unifying Mirror Symmetry Constructions The Berglund-H¨ ubsch Construction of Mirror Symmetry

Invertible Polynomials

Start with an invertible matrix A = (aij)n

i,j=0 with all nonnegative

integer entries. Take the polynomial, FA :=

n

• i=0

n

• j=0

xaij

j

Assume that:

◮ FA : Cn+1 → C has a unique critical point at the origin. ◮ FA is quasihomogeneous of degree d: there exists d ∈ N and

(q0, ..., qn) ∈ Nn+1 such that FA(λq0x0, ..., λqnxn) = λdFA(x0, ..., xn) for all λ ∈ C∗.

◮ Calabi-Yau condition: n i=0 qi = d.

Unifying Mirror Symmetry Constructions The Berglund-H¨ ubsch Construction of Mirror Symmetry

Two running examples

Consider the following examples: n = 2 d = 3 (q0, q1, q2) = (1, 1, 1). A1 : =   3 3 3   FA1 = x3 + y3 + z3 A2 : =   2 1 3 3   FA2 = x2y + y3 + z3

Unifying Mirror Symmetry Constructions The Berglund-H¨ ubsch Construction of Mirror Symmetry

Groups of Symmetries of FA

◮ Diagonal automorphisms:

Aut(FA) := {(λ0, ..., λn) | FA(λixi) = FA(xi)} ⊆ (C∗)n+1 ⊆ Gln+1(C) Let A−1 := B = (bij) Fact: this is generated by ρj := (e2πib0j, . . . , e2πibnj) for 0 ≤ j ≤ n.

◮ Special Linear Automorphisms:

Sl(FA) := Sln(C) ∩ Aut(FA) =

• (λ0, ..., λn) ∈ Aut(FA)
• i

λi = 1

• ◮ Exponential grading group: JFA := ρ0 · · · ρn.

Unifying Mirror Symmetry Constructions The Berglund-H¨ ubsch Construction of Mirror Symmetry

Group Duality

Choose a group G so that JFA ⊆ G ⊆ Sl(FA). Given the data A, G we can associate a hypersurface in a quotient

• f weighted projective space

ZA,G := {(x0, ...., xn) ∈ Cn+1\0 | FA(x0, ..., xn) = 0}/GC∗ ⊆ P(q0, ..., qn)/(G/JFA) := (Cn+1\0)/ ∼ where (x0, ..., xn) ∼ (λq0x0, ..., λqnxn) for all λ ∈ C∗ (x0, ..., xn) ∼ (λ0x0, ..., λnxn) for all (λ0, ..., λn) ∈ G The choice of JFA ⊆ G ⊆ Sl(FA) ensures that ZA,G is Calabi-Yau.

Unifying Mirror Symmetry Constructions The Berglund-H¨ ubsch Construction of Mirror Symmetry

Mirror Symmetry

Define ρT

j := (e2πibj0, . . . , e2πibjn)

and a “dual group” by G T

A :=

  

n

• j=0

(ρT

j )mj

• n
• j=0

xmj

j

is G-invariant    . Berglund and H¨ ubsch proposed the following basic duality: (A, G) ← → (AT, G T

A ).

Mirror Symmetry can be viewed as exchanging the spaces: ZA,G ← → ZAT ,G T

A

Unifying Mirror Symmetry Constructions Unification of Mirror Constructions

Back to the example

A1 : =   3 3 3   FA1= x3 + y 3 + z3 FAT

1 = x3 + y 3 + z3

A2 : =   2 1 3 3   FA2= x2y + y 3 + z3 FAT

2 = x2 + xy 3 + z3

Set G = JA1 = (ζ3, ζ3, ζ3) = JA2. Notice that ZA1,G ∼ = ZA2,G are actually just symplectomorphic tori. Therefore, they should have the same mirror. However, G T

A1= (Z/3Z)⊕2

ZAT

1 ,G T A1⊆ P3/(Z/3Z)

G T

A2= Z/6Z

ZAT

2 ,G T A2⊆ P(3 : 1 : 2)

Unifying Mirror Symmetry Constructions Unification of Mirror Constructions

We have ZA1,G ∼ = ZA2,G are symplectomorphic tori. Hence, we have Fuk(ZA1,G) ∼ = Fuk(ZA2,G) and by Homological mirror symmetry we expect Fuk(ZA1,G) ∼ = Fuk(ZA2,G) = D(ZAT

1 ,G T A1) = D(ZAT 2 ,G T A2)

Theorem (Favero-Kelly ’14)

Given any two FA1,G and FA2,G that give hypersurfaces in the same quotient of weighted projective space, their Berglund-H¨ ubsch mirrors have equivalent derived categories.

Unifying Mirror Symmetry Constructions Unification of Mirror Constructions

Some Theorems

Theorem (Favero-Kelly ’14)

Given any two FA1,G and FA2,G that give hypersurfaces in the same quotient of weighted projective space, their Berglund-H¨ ubsch mirrors have equivalent derived categories. Since Homological Mirror Symmetry is known for the Fermat mirror in projective space, we get the following Corollary:

Corollary

Homological Mirror Symmetry holds for Berglund-H¨ ubsch mirrors to projective hypersurfaces i.e. given any ZA,G ⊆ Pn Fuk(ZA,G) = D(ZAT ,G T

A ).

Unifying Mirror Symmetry Constructions Unification of Mirror Constructions

Some Theorems

Conjecture (Batyrev/Nill ’08)/Theorem(Favero-Kelly3 ’14)

Batyrev and Nill’s conjecture holds: multiple mirrors in the Batyrev-Borisov construction of mirror symmetry (for Calabi-Yau complete intersections in toric varieties) have equivalent derived categories.

Theorem (Doran-Favero-Kelly ’15)

Multiple mirrors in Clarke’s construction of mirror symmetry for hypersurfaces have equivalent derived categories.

3Proof of a Conjecture of Batyrev and Nill, (23 pages). To appear in

American Journal of Mathematics.

Unifying Mirror Symmetry Constructions Unification of Mirror Constructions

Invariants from the Derived Category

Many invariants descend from derived categories:

◮ Cohomology ◮ Algebraic K-theory (Thomason-Trobaugh ’90) ◮ Geometric motives for certain equivalences (Orlov ’05) ◮ Griffiths Groups (Favero-Iliev-Katzarkov4 ’14)

4Griffiths Groups for Derived Categories with applications to

Fano-Calabi-Yaus, Pure and Applied Mathematics Quarterly, V. 10 N. 1 pgs 1-55, 2014.

Unifying Mirror Symmetry Constructions Unification of Mirror Constructions

More invariants of the Derived Category

◮ Rouquier Dimension/Orlov Spectra (Rouquier ’08, Orlov ’09)

◮ Related to relations in the symplectic mapping class group and

“algebraic surgeries” (birational geometry) (Ballard-Favero-Katzarkov5 ’12)

◮ Related to Algebraic Cycles/The Hodge Conjecture

(Ballard-Favero-Katzarkov6 ’14)

◮ (local) Zeta Functions, dim 2, abelian varieties (Honigs ’13)

5Orlov Spectra: Gaps and Bounds Inventiones Mathematicae, V. 189 I. 2,

pgs 359-430, 2012.

6A Category of Kernels for Equivariant Factorizations, Publications

Math´ ematiques de l’IH´ ES, V. 120 I. 1, pgs 1-111, 2014.

Unifying Mirror Symmetry Constructions Future Directions

Future directions

◮ Formulate Cohomological Field Theories (e.g. GW Theory,

FJRW Theory) using derived categories of pairs (joint with Ciocan-Fontaine, Kim)

◮ Towards a solution to Kawamata’s Conjecture (joint with

Ballard, Diemer, Katzarkov, Kontsevich)

◮ Give decompositions of derived categories for special Fano

linear systems (joint with Kelly)