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 = R 4 × X where ◮ R 4 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 = R 2 n with basis u 1 , ..., u n , v 1 , ..., v n . Define ω to be � 0 � Id ω := ω ( u i , v i ) = 1 − Id 0 ω ( v i , u i ) = − 1 and to be zero on all other pairs of basis vectors.
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 { ( x 0 , ..., x 4 ) ∈ C 5 \ 0 | x 5 0 + ... + x 5 4 = 0 } / C ∗ = { ( x 0 , ..., x 4 ) ∈ C 5 \ 0 | x 5 0 + ... + x 5 4 = 0 } / ∼ ⊆ CP 4 =: C 5 \ 0 / ∼ where ( x 0 , ..., x 4 ) ∼ ( λ x 0 , ..., λ x 4 ) 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 � Cohomology of X 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 A i ∈ R − mod. d n +2 d n +1 d n − 1 d n ... − − → A n +1 − − → A n − → A n − 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 A i ∈ R − mod. d n +2 d n +1 d n − 1 d n ... − − → A n +1 − − → A n − → A n − 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 of 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 of 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, Favero 1 ’12) ◮ False in dimension 3 (Lesieutre ’13) 1 Reconstruction 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-Katzarkov 2 ’12) 2 Variation 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: Z 1 · · · Z n X X 1 X n 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 X 1 , ..., X n 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 L 1 , L 2 .
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 ) L 1 , L 2 , L 3 �→ 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 L 1 , L 2 , L 3 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 ) .
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