Unifying Mirror Symmetry Constructionsfavero/slides/MS.pdfUnifying Mirror Symmetry Constructions...

Unifying Mirror Symmetry Constructions


David [email protected]

University of Alberta

May 2016Korean Institute for Advanced Study

Slides available at:www.ualberta.ca/∼favero


String theoretic motivations

String Theoretic Universe

I In physics, there is a desire to unify quantum mechanics andgeneral relativity (gravity).

I String Theory is one such proposal.

I In String Theory, the “fundamental” units of matter arestrings.



Geometric Requirements of String Theory

Physics: locally, space-time must look like

U = R4 × X

where

I R4 is four-dimensional space-time (Minkowski space-time)

I X is a 3-dimensional complex manifold called a Calabi-Yaumanifold.



Symplectic manifolds

DefinitionA symplectic manifold (M, ω) is a manifold M equipped with aclosed non-degenerate differential 2-form ω called the symplecticform 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 :=

[0 Id−Id 0

]



Calabi-Yau manifolds

DefinitionLet 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.

I Equivalent definition: a Ricci-flat, Kahler-Einstein manifold(Yau ’78).



Example of a Calabi-Yau manifold

Example

Consider the set

{(x0, ..., x4) ∈ C5\0 | x50 + ...+ x5

4 = 0}/C∗

= {(x0, ..., x4) ∈ C5\0 | x50 + ...+ 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.


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 anotherCalabi-Yau 3-fold X so that the Type IIB string theory on thespace X gives the same physical theory.

Definition: X is known as the mirror to X .



Geometric ramifications of Mirror Symmetry

Mirror Symmetry


Question:What does this string duality mean geometrically?

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



Geometric ramifications of Mirror Symmetry

Mirror Symmetry


Question:What does this string duality mean geometrically?




Mathematical Mirror Symmetry


Type IIA Type IIB

Symplectic Deformations Complex Deformations

Cohomology of X Cohomology of XEnumerative Geometry Variations of Hodge Structure

Fukaya Category Derived Category of Coherent Sheaves


Derived Categories

Derived Categories

I Derived categories were defined by Verdier in 1967.

I For a ring R, objects of D(R) are formally built from modulesAi ∈ R −mod.

...dn+2−−→ An+1

dn+1−−→ Andn−→ An−1

dn−1−−−→ ...

I The original intent of derived categories was to provide anappropriate setting for homological algebra.


Derived Categories

Derived Categories



...dn+2−−→ An+1


dn−1−−−→ ...



Derived Categories

Derived Categories



...dn+2−−→ An+1


dn−1−−−→ ...



Derived Categories

Derived Categories

I For an algebraic variety X , we can associate a derivedcategory D(X ).

I Objects of D(X ) are roughly vector bundles over submanifoldsof X .

I In the 80s and 90s, Mukai, Beilinson, Bondal, Orlov,Kapranov, and others began to study D(X ) as a geometricinvariant.


Derived Categories

Derived Categories





Derived Categories

Derived Categories





Derived Categories

Derived Categories

I There are 3 conjectures which I consider the most central tothe study of D(X )

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


Derived Categories

Derived Categories

I There are 3 conjectures which I consider the most central tothe study of D(X )

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


Derived Categories

Kawamata’s First Conjecture

Conjecture (Kawamata ’02)

The following set is finite:

{Y | D(Y ) = D(X )}.

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

Kawamata ’02 )I True for varieties with positive or negative curvature

(Bondal-Orlov ’97)I True for complex n-dimensional tori

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


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”

I Standard Flops (Bondal-Orlov ’95)

I Toroidal Flops (Kawamata ’02)

I Flops in dimension 3 (Bridgeland ’02)

I Elementary wall-crossings from variation of GeometricInvariant Theory Quotients(Halpern-Leistner ’12, Ballard-Favero-Katzarkov2 ’12)

2Variation of Geometric Invariant Theory Quotients and DerivedCategories(63 pages). To appear in Journal fur die reine und angewandteMathematik (Crelle’s journal).


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.

RemarkAssuming X ,Y are Calabi-Yau, then all that remains to know forKawamata’s Conjecture is that we can choose X1, ...,Xn to beCalabi-Yau.


Derived Categories

Boundary Conditions

Recall that when strings move though time they create surfaces(worldsheets). When we discuss strings in X , we think of oursurfaces as mapping into spacetime X .

Type IIA string theory requires that the end points of our stringsmove in some subspaces L1, L2.


Derived Categories

The Fukaya Category

Konsevich proposed that the target for the topological quantumfield theory associated to Type IIA string theory is called theFukaya category

Type IIA TQFT : Strings → Fuk(X )

L1, L2, L3 7→ A Lagrangian Subspaces of X (objects)

A,B,C 7→ Intersection points (morphisms)


Derived Categories

Boundary Conditions

Kontsevich proposed D(X ) as the natural target for the topologicalquantum 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 tomorphisms in D(X ).


Derived Categories

Homological Mirror Symmetry

Mirror symmetry exchanges Type IIA and Type IIB string theoriesbetween 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 anequivalence of categories

Fuk(X ) = D(X ).


Derived Categories

Conjecture (Homological Mirror Symmetry, Kontsevich)

Let X be a Calabi-Yau manifold and X be its mirror. There is anequivalence of categories

Fuk(X ) = D(X ).

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

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

(Ballard-Diemer-Favero-Kerr-Katzarkov ’15)I Many non-compact cases


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

x50 + x5

1 + x52 + x5

3 + x54 = 0.

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

(x0, x1, x2, x3, x4) 7→ (ζx0, ζ−1x1, x2, x3, x4)

(x0, x1, x2, x3, x4) 7→ (ζx0, x1, ζ−1x2, x3, x4)

(x0, x1, x2, x3, x4) 7→ (ζx0, x1, x2, ζ−1x3, x4)

Symmetry group G = (Z/5Z)3.

Take X5 to be the quotient X5/G .


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

x50 + x5

1 + x52 + x5

3 + x54 = 0.

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

(x0, x1, x2, x3, x4) 7→ (ζx0, ζ−1x1, x2, x3, x4)

(x0, x1, x2, x3, x4) 7→ (ζx0, x1, ζ−1x2, x3, x4)

(x0, x1, x2, x3, x4) 7→ (ζx0, x1, x2, ζ−1x3, x4)

Symmetry group G = (Z/5Z)3.

Take X5 to be the quotient X5/G .


Derived Categories

Mirror Constructions

There are many constructions of mirrors, each having differentcontexts. They don’t always agree and have internalinconsistencies!Examples:

I Greene-Plesser-Roan ’90

I Berglund-Hubsch ’93

I Batyrev-Borisov ’95

I Strominger-Yau-Zaslow ’96

I Hori-Vafa ’00

I Clarke ’08


The Berglund-Hubsch Construction of Mirror Symmetry

Invertible Polynomials

Start with an invertible matrix A = (aij)ni ,j=0 with all nonnegative

integer entries. Take the polynomial,

FA :=n∑

i=0

n∏j=0

xaijj

Assume that:

I FA : Cn+1 → C has a unique critical point at the origin.

I 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∗.

I Calabi-Yau condition:∑n

i=0 qi = d .



Two running examples

Consider the following examples:

n = 2

d = 3

(q0, q1, q2) = (1, 1, 1).

A1 : =

3 0 00 3 00 0 3

FA1 = x3 + y3 + z3

A2 : =

2 1 00 3 00 0 3

FA2 = x2y + y3 + z3



Groups of Symmetries of FAI Diagonal automorphisms:

Aut(FA) := {(λ0, ..., λn) | FA(λixi ) = FA(xi )}⊆ (C∗)n+1 ⊆ Gln+1(C)

LetA−1 := B = (bij)

Fact: this is generated by ρj := (e2πib0j , . . . , e2πibnj ) for0 ≤ j ≤ n.

I Special Linear Automorphisms:

Sl(FA) := Sln(C) ∩ Aut(FA)

=

{(λ0, ..., λn) ∈ Aut(FA)

∣∣∣∣∣∏i

λi = 1

}I Exponential grading group: JFA

:= 〈ρ0 · · · ρn〉.



Group Duality

Choose a group G so that JFA⊆ G ⊆ Sl(FA).

Given the data A,G we can associate a hypersurface in a quotientof 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.



Mirror SymmetryDefine

ρTj := (e2πibj0 , . . . , e2πibjn)

and a “dual group” by

GTA :=

n∏

j=0

(ρTj )mj

∣∣∣∣∣∣n∏

j=0

xmj

j is G -invariant

.

Berglund and Hubsch proposed the following basic duality:

(A,G )←→ (AT ,GTA ).

Mirror Symmetry can be viewed as exchanging the spaces:

ZA,G ←→ ZAT ,GTA


Unification of Mirror Constructions

Back to the example

A1 : =

3 0 00 3 00 0 3

FA1 = x3 + y3 + z3

FAT1

= x3 + y3 + z3

A2 : =

2 1 00 3 00 0 3

FA2 = x2y + y3 + z3

FAT2

= x2 + xy3 + z3

Set G = JA1 = (ζ3, ζ3, ζ3) = JA2 . Notice that ZA1,G∼= ZA2,G are

actually just symplectomorphic tori. Therefore, they should havethe same mirror. However,

GTA1

= (Z/3Z)⊕2

ZAT1 ,G

TA1

⊆ P3/(Z/3Z)

GTA2

= Z/6ZZAT

2 ,GTA2

⊆ P(3 : 1 : 2)



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(ZAT1 ,G

TA1

) = D(ZAT2 ,G

TA2

)

Theorem (Favero-Kelly ’14)

Given any two FA1,G and FA2,G that give hypersurfaces in the samequotient of weighted projective space, their Berglund-Hubschmirrors have equivalent derived categories.



Some Theorems

Theorem (Favero-Kelly ’14)

Given any two FA1,G and FA2,G that give hypersurfaces in the samequotient of weighted projective space, their Berglund-Hubschmirrors have equivalent derived categories.

Since Homological Mirror Symmetry is known for the Fermatmirror in projective space, we get the following Corollary:

Corollary

Homological Mirror Symmetry holds for Berglund-Hubsch mirrorsto projective hypersurfaces i.e. given any ZA,G ⊆ Pn

Fuk(ZA,G ) = D(ZAT ,GTA

).



Some Theorems

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

Batyrev and Nill’s conjecture holds: multiple mirrors in theBatyrev-Borisov construction of mirror symmetry (for Calabi-Yaucomplete intersections in toric varieties) have equivalent derivedcategories.

Theorem (Doran-Favero-Kelly ’15)

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

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



Invariants from the Derived Category

Many invariants descend from derived categories:

I Cohomology

I Algebraic K-theory (Thomason-Trobaugh ’90)

I Geometric motives for certain equivalences (Orlov ’05)

I Griffiths Groups (Favero-Iliev-Katzarkov4 ’14)

4Griffiths Groups for Derived Categories with applications toFano-Calabi-Yaus, Pure and Applied Mathematics Quarterly, V. 10 N. 1 pgs1-55, 2014.



More invariants of the Derived Category

I Rouquier Dimension/Orlov Spectra (Rouquier ’08, Orlov ’09)I Related to relations in the symplectic mapping class group and

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

I Related to Algebraic Cycles/The Hodge Conjecture(Ballard-Favero-Katzarkov6 ’14)

I (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, PublicationsMathematiques de l’IHES, V. 120 I. 1, pgs 1-111, 2014.


Future Directions

Future directions

I Formulate Cohomological Field Theories (e.g. GW Theory,FJRW Theory) using derived categories of pairs (joint withCiocan-Fontaine, Kim)

I Towards a solution to Kawamata’s Conjecture (joint withBallard, Diemer, Katzarkov, Kontsevich)

I Give decompositions of derived categories for special Fanolinear systems (joint with Kelly)

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