CalabiYau Geometry and Mirror Symmetry Conference. CheolHyun Cho (Seoul National Univ.) (based on a joint work with Hansol Hong and SiuCheong Lau)


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1 CalabiYau Geometry and Mirror Symmetry Conference CheolHyun Cho (Seoul National Univ.) (based on a joint work with Hansol Hong and SiuCheong Lau)
2 Mirror Symmetry between two spaces Mirror symmetry explains certain relations between symplectic geometry and complex geometry
3 Mirror Symmetry between two spaces Mirror symmetry explains certain relations between symplectic geometry and complex geometry Very first example, (found by Candelas, de la Ossa, Green, Parkes) is mirror symmetry between CalabiYau hypersurface in P 4 x x x x x 5 5 = 0, and another hypersurface in P 4 /(Z/5) 3 x x x x x 5 5 σx 1 x 2 x 3 x 4 x 5 = 0.
4 Mirror Symmetry between two spaces Mirror symmetry explains certain relations between symplectic geometry and complex geometry Very first example, (found by Candelas, de la Ossa, Green, Parkes) is mirror symmetry between CalabiYau hypersurface in P 4 x x x x x 5 5 = 0, and another hypersurface in P 4 /(Z/5) 3 x x x x x 5 5 σx 1 x 2 x 3 x 4 x 5 = 0. Holomorphic curve counting invariants of X equals certain period integrals of Y.
5 Mirror Symmetry between a space and a function Another version of mirror symmetry relates space X, with a potential function W : Y C on Y. Simplest example is the case of sphere. CP 1 (W : C C) W (z) = z + q z.
6 Mirror Symmetry between a space and a function Another version of mirror symmetry relates space X, with a potential function W : Y C on Y. Simplest example is the case of sphere. CP 1 (W : C C) W (z) = z + q z. Quantum cohomology of CP 1 is isomorphic to the Jacobian ring of W ; C[z, z 1 ]/ W. W = 1 q z 2 gives z 2 = q, which corresponds to [pt] Q [pt] = q[s 2 ]. This example generalizes to all toric manifolds.
7 StromingerYauZaslow StromingerYauZaslow views mirror symmetry as relations between Dual torus fibrations.
8 StromingerYauZaslow StromingerYauZaslow views mirror symmetry as relations between Dual torus fibrations. This point of view been extremely successful in explaining mirror symmetry in basic cases (such as torus bundles, toric manifolds/orbifolds, toric CY s or with mild singular fibers).
9 StromingerYauZaslow StromingerYauZaslow views mirror symmetry as relations between Dual torus fibrations. This point of view been extremely successful in explaining mirror symmetry in basic cases (such as torus bundles, toric manifolds/orbifolds, toric CY s or with mild singular fibers). The singularity of the torus fibration is measured by holomorphic discs, defining W (z) = z + q z.
10 StromingerYauZaslow StromingerYauZaslow views mirror symmetry as relations between Dual torus fibrations. This point of view been extremely successful in explaining mirror symmetry in basic cases (such as torus bundles, toric manifolds/orbifolds, toric CY s or with mild singular fibers). The singularity of the torus fibration is measured by holomorphic discs, defining W (z) = z + q z. It becomes more difficult to use when fibers get more singular.
11 Fukaya category of a surface Σ Objects: Transversal family of embedded Curves L 0, L 1, L 2, (noncontractible)
12 Fukaya category of a surface Σ Objects: Transversal family of embedded Curves L 0, L 1, L 2, (noncontractible) Hom(L i, L j ) is generated by intersection points L i L j. Hom(L i, L i ) is given by a Morse complex of L i = S 1
13 Fukaya category of a surface Σ Objects: Transversal family of embedded Curves L 0, L 1, L 2, (noncontractible) Hom(L i, L j ) is generated by intersection points L i L j. Hom(L i, L i ) is given by a Morse complex of L i = S 1 There are operations on Morphisms: differential m 1, product m 2, homotopy for associativity m 3,, which defines an A category.
14 Operations on Morphisms (Differential m 1 ) m 1 : Hom(L 0, L 1 ) Hom(L 0, L 1 ) counts immersed bigons with boundaries on L 0 and L 1. m 1 (p) = q T Area.
15 Operations on Morphisms (Differential m 1 ) m 1 : Hom(L 0, L 1 ) Hom(L 0, L 1 ) counts immersed bigons with boundaries on L 0 and L 1. m 1 (p) = q T Area. We use Novikov ring Λ = { a i T λ i a i C, lim i λ i = } Corners of the bigon should be locally convex
16 Operations on Morphisms 2 (Product m 2 ) m 2 : Hom(L 0, L 1 ) Hom(L 1, L 2 ) Hom(L 0, L 2 ) counts triangles with boundaries on L 0 and L 1 and L 2. In general (if L i = L j ), we also count with additional Morse trajectory contributions.
17 Differential m 2 1 = 0 m 2 1 = 0 can be proved, since there are two matching diagrams.
18 Operations on Morphisms These satisfy product rule (up to ±): m 1 (m 2 (a, b)) = m 2 (m 1 (a), b) + m 2 (a, m 1 (b))
19 Operations on Morphisms In general, we can define m k counting kgons with areas.
20 Operations on Morphisms In general, we can define m k counting kgons with areas. m 1, m 2, m 3, satisfy A equations. Given ninputs x 1,, x n, m k1 (x 1,, m k2 ( ),, x n ) = 0 k 1 +k 2 =n+1 Fukaya category can be defined in any symplectic manifold: Objects are Lagrangian submanifolds, and Morphisms are their intersection points, and operations by counting Jholomorphic polygons.
21 Matrix Factorization Category Given a (Laurant) polynomial W, for example, W (z) = z n W (x, y, z) = x 3 + y 3 + z 3 3xyz W (z) = z + q z
22 Matrix Factorization Category Given a (Laurant) polynomial W, for example, W (z) = z n W (x, y, z) = x 3 + y 3 + z 3 3xyz W (z) = z + q z We explain a category of Matrix factorization of W. Consider a factorization of the polynomial W : For example, for W = z 5 with critical value 0, W 0 = z i z 5 i
23 Matrix Factorization Category Given a (Laurant) polynomial W, for example, W (z) = z n W (x, y, z) = x 3 + y 3 + z 3 3xyz W (z) = z + q z We explain a category of Matrix factorization of W. Consider a factorization of the polynomial W : For example, for W = z 5 with critical value 0, W 0 = z i z 5 i For W = z + q z, it has a critical value λ = 2 q and W λ = (z q q) (1 z )
24 Matrix Factorizations We can factor W into two matrices as follows. For W = x 3 + y 3 + z 3 3xyz, we have x z y z y x y x z = W x 2 yz z 2 xy y 2 zx z 2 xy y 2 zx x 2 yz y 2 zx x 2 yz z 2 xy = W W W
25 Matrix Factorizations We can factor W into two matrices as follows. For W = x 3 + y 3 + z 3 3xyz, we have x z y z y x y x z = W x 2 yz z 2 xy y 2 zx z 2 xy y 2 zx x 2 yz y 2 zx x 2 yz z 2 xy = W W W We have matrices M 0, M 1 such that M 0 M 1 = W Id, M 1 M 0 = W Id.
26 Category of Matrix Factorizations Let R = C[x, y, z] polynomial ring. Write R R R = R 3 Consider the matrices as maps. R 3 M 0 R 3 M 1 R 3 Matrix factorization of W is two Rmodules P 0, P 1 with maps d : P 0 P 1, d : P 1 P 0 d 2 = (W λ)id.
27 Category of Matrix Factorizations Let R = C[x, y, z] polynomial ring. Write R R R = R 3 Consider the matrices as maps. R 3 M 0 R 3 M 1 R 3 Matrix factorization of W is two Rmodules P 0, P 1 with maps d : P 0 P 1, d : P 1 P 0 d 2 = (W λ)id.
28 Category of Matrix Factorizations 2 Given two matrix factorizations, R n M 0 R n M 1 R n M 0 R m N 0 R m N 1 R m M 0
29 Category of Matrix Factorizations 2 Given two matrix factorizations, we define morphisms between them R n M 0 R n M 1 R n M 0 f f f R m N 0 R m N 1 R m M 0 Differential m 1 on Morphisms by N i f f M i m 2 is the composition of maps. This defines differential graded category, which is an A category with m 3 = 0.
30 Category of Matrix Factorizations 2 Given two matrix factorizations, we define morphisms between them R n M 0 R n M 1 R n M 0 f f f R m N 0 R m N 1 R m M 0 Differential m 1 on Morphisms by N i f f M i m 2 is the composition of maps. This defines differential graded category, which is an A category with m 3 = 0. MF (W ) := λ Crit(W ) MF (W, λ) It is also called the category of singularity of W.
31 Homological mirror functor In this setting, Homological mirror symmetry conjectures: There exist a derived equivalence between the geometric A category to the algebraic dgcategory: Fukaya Category of X Matrix Fact.(W )
32 Homological mirror functor In this setting, Homological mirror symmetry conjectures: There exist a derived equivalence between the geometric A category to the algebraic dgcategory: Fukaya Category of X Matrix Fact.(W ) An enhanced version may be an A functor from one to the other. A Lagrangian submanifold L would correspond to a Matrix factorization.
33 Homological mirror functor In this setting, Homological mirror symmetry conjectures: There exist a derived equivalence between the geometric A category to the algebraic dgcategory: Fukaya Category of X Matrix Fact.(W ) An enhanced version may be an A functor from one to the other. A Lagrangian submanifold L would correspond to a Matrix factorization. Intersection between Lagrangian submanifolds gives a Morphism between MF s.
34 Homological mirror functor In this setting, Homological mirror symmetry conjectures: There exist a derived equivalence between the geometric A category to the algebraic dgcategory: Fukaya Category of X Matrix Fact.(W ) An enhanced version may be an A functor from one to the other. A Lagrangian submanifold L would correspond to a Matrix factorization. Intersection between Lagrangian submanifolds gives a Morphism between MF s. There exist an A functor relating A operations on X and dgoperations on MF (W ).
35 Localized Mirror functor We propose a new formalism to understand mirror symmetry: Localized mirror functors
36 Localized Mirror functor We propose a new formalism to understand mirror symmetry: The formalism works as follows. Localized mirror functors 1. A good (weakly unobstructed) Lagrangian immersion L X Potential W L.
37 Localized Mirror functor We propose a new formalism to understand mirror symmetry: The formalism works as follows. Localized mirror functors 1. A good (weakly unobstructed) Lagrangian immersion L X Potential W L. W L is defined by counting Jholomorphic polygons with boundary on L.
38 Localized Mirror functor We propose a new formalism to understand mirror symmetry: The formalism works as follows. Localized mirror functors 1. A good (weakly unobstructed) Lagrangian immersion L X Potential W L. W L is defined by counting Jholomorphic polygons with boundary on L. 2. Each such L gives rise to a canonical functor LM L : LM L : Fuk(X ) Mat.Fact.(W L ) constructed from geometric data (via Fukaya category operations)
39 Localized Mirror functor We propose a new formalism to understand mirror symmetry: The formalism works as follows. Localized mirror functors 1. A good (weakly unobstructed) Lagrangian immersion L X Potential W L. W L is defined by counting Jholomorphic polygons with boundary on L. 2. Each such L gives rise to a canonical functor LM L : LM L : Fuk(X ) Mat.Fact.(W L ) constructed from geometric data (via Fukaya category operations) 3. If LM L is an equivalence, this would prove homological mirror symmetry ( Generalized SYZ ).
40 Potential W L from Lagrangian Immersion L
41 W L as Lagrangian Floer Potential Given a Lagrangian immersion L, consider its A algebra, and degree one immersed generators X 1,, X n.
42 W L as Lagrangian Floer Potential Given a Lagrangian immersion L, consider its A algebra, and degree one immersed generators X 1,, X n. Suppose that b = x 1 X 1 + x 2 X x n X n becomes weak bounding cochains, or m 1 (b) + m 2 (b, b) + m 3 (b, b, b) + = W (b) e is a multiple of unit.
43 W L as Lagrangian Floer Potential Given a Lagrangian immersion L, consider its A algebra, and degree one immersed generators X 1,, X n. Suppose that b = x 1 X 1 + x 2 X x n X n becomes weak bounding cochains, or m 1 (b) + m 2 (b, b) + m 3 (b, b, b) + = W (b) e is a multiple of unit. Then, the coefficient W (b) is a function in Λ[[x 1,, x n ]], defines localized mirror of L.
44 Matrix Factorization corresponding to L1
45 Matrix Factorization corresponding to L 1 In general, consider L 1 transversely intersecting the immersed Lagrangian L. Intersection L 1 L consists of even and odd intersection points. We count bigons allowing odd degree turns at the upper boundary. m b,0 1 (p) = m k+1 (b, b, b, b, b,, b, p) k=0 counting the bigon gives an element Λ 0 [x 1,, x n ]. This deformed complex (CF ((L, b), L 1 ), m b,0 1 ) satisfies (m b,0 1 )2 = W (b), providing the matrix factorization of W (b).
46 Localized mirror functor We define an A functor φ L : Fuk(X ) MF (W L ) Say φ L sends Lagrangian L j to Matrix factorizations M j. In general we define (φ L ) 1 : CF (L i, L j ) Mor(M i, M j ) (φ L ) 1 (p ij )( ) = ±m b,0,0 2 (, p ij ) (φ L ) k : CF (L 1, L 2 ) CF (L k 1, L k ) Mor(M 1, M k ) (φ L ) k (p 12, p 23,, p (k 1)k )( ) = ±m b,0,,0 k+1 (, p 12, p 23,, p (k 1)k )
47 Example of orbispheres Cubic mirror symmetry: x 3 + y 3 + z 3 = 0 in P 2 x 3 + y 3 + z 3 σxyz = 0 in P 2 /(Z/3)
48 Example of orbispheres Cubic mirror symmetry: x 3 + y 3 + z 3 = 0 in P 2 x 3 + y 3 + z 3 σxyz = 0 in P 2 /(Z/3) 1. Zero set of x 3 + y 3 + z 3 = 0 in P 2 gives the elliptic curve E (three copy of E). 2. The relationship between the Kähler parameter q and complex parameter σ was explained by SiuCheong Lau.
49 Example of orbispheres Cubic mirror symmetry: x 3 + y 3 + z 3 = 0 in P 2 x 3 + y 3 + z 3 σxyz = 0 in P 2 /(Z/3) 1. Zero set of x 3 + y 3 + z 3 = 0 in P 2 gives the elliptic curve E (three copy of E). 2. The relationship between the Kähler parameter q and complex parameter σ was explained by SiuCheong Lau. 3. Further quotient by Z/3action, gives an orbisphere P 1 3,3,3.
50 Quotient orbisphere P 1 3,3,3 We take a Z 3 quotient of the elliptic curve
51 Main example Our first goal is to define mirror potential W of the form W (x, y, z) = x 3 + y 3 + z 3 σxyz so that we have mirror symmetry or Eliptic curve (W : (C 3 /Z 3 ) C) Eliptic curve/z 3 (W : C 3 C) In general, we consider X = P 1 a,b,c and prove homological mirror symmetry between X = P 1 a,b,c and W = x a + y b + z c σxyz + when 1 a + 1 b + 1 c 1, by constructing a mirror functor.
52 Immersed Lagrangian Our main object is the following immersed curve in the orbisphere (Seidel considered such Lagrangian immersion to prove HMS for genus two curve)
53 Weakly unobstruced Proposition Seidel Lagrangian in P 1 a,b,c is weakly unobstructed
54 Weakly unobstruced Proposition Seidel Lagrangian in P 1 a,b,c is weakly unobstructed To prove this, consider a (k + 1)gon P used in m k (b,, b), whose corners are given by odd degree generators and output is an even degree immersed generator.
55 Weakly unobstruced Proposition Seidel Lagrangian in P 1 a,b,c is weakly unobstructed To prove this, consider a (k + 1)gon P used in m k (b,, b), whose corners are given by odd degree generators and output is an even degree immersed generator. By reflection across equator, we get another polygon P op, which also contribute to m k (b,, b).
56 Weakly unobstruced Proposition Seidel Lagrangian in P 1 a,b,c is weakly unobstructed To prove this, consider a (k + 1)gon P used in m k (b,, b), whose corners are given by odd degree generators and output is an even degree immersed generator. By reflection across equator, we get another polygon P op, which also contribute to m k (b,, b). 1. Reflection preserves the orientation of the Lagrangian, but the complex orientation is the opposite, which gives rise to ( 1) k
57 Weakly unobstruced Proposition Seidel Lagrangian in P 1 a,b,c is weakly unobstructed To prove this, consider a (k + 1)gon P used in m k (b,, b), whose corners are given by odd degree generators and output is an even degree immersed generator. By reflection across equator, we get another polygon P op, which also contribute to m k (b,, b). 1. Reflection preserves the orientation of the Lagrangian, but the complex orientation is the opposite, which gives rise to ( 1) k 2. The number of times P or P op passes through generic point (for nonstandard spin structure) is k + 1 mod 2. (If P and P op covers minimal edges 6l times, then P covers, 3l times. But each edge of P consists of odd number of minimal edges. Hence l = k + 1 mod 2.)
58 Immersed Lagrangian Upstairs Immersed curve are straight lines in the universal cover.
59
60 Construction of mirror potential where φ = W = qxyz q 9 x 3 q 9 y 3 + q 9 z 9 + = φ(q)(x 3 + y 3 z 3 ) + ψ(q)xyz. ( 1) 3k+1 (2k + 1)q 3(12k2 +12k+3) k=0 ψ = q α + α ( ( 1) 3k (6k + 1)q (6k+1)2 k=1 α ) + (6k 1)q α (6k 1)2.
61 Construction of mirror potential where φ = W = qxyz q 9 x 3 q 9 y 3 + q 9 z 9 + = φ(q)(x 3 + y 3 z 3 ) + ψ(q)xyz. ( 1) 3k+1 (2k + 1)q 3(12k2 +12k+3) k=0 ψ = q α + α ( ( 1) 3k (6k + 1)q (6k+1)2 k=1 After change of coordinate, we have α ) + (6k 1)q α (6k 1)2. x 3 + y 3 + z 3 ψ φ xyz
62 General a, b, c case In general, one can consider P 1 a,b,c, where each orbifold point has Z a, Z b, Z c singularity. We have W (x, y, z) = x a + y b + z c σxyz +
63 General a, b, c case In general, one can consider P 1 a,b,c, where each orbifold point has Z a, Z b, Z c singularity. We have W (x, y, z) = x a + y b + z c σxyz + We can count all necessary polygons to compute W explicitly (joint work with Sanghyun Kim)
64 The case of (2, 3, 5)
65 Homological Mirror functor in action
66 Equator (3 3) matrix factorization of W Equator (straight line passing through 3 orbifold points upstairs) corresponds to (3 3) matrix factorization of W explained before.
67 Mirror matrix factorization J = x 1 x 2 x 3 y 1 β 1 x 2 + γ 1 yz β 3 z 2 + γ 3 xy β 2 y 2 + γ 2 zx y 2 β 2 z 2 + γ 2 xy β 1 y 2 + γ 1 zx β 3 x 2 + γ 3 yz y 3 β 3 y 2 + γ 3 zx β 2 x 2 + γ 2 yz β 1 z 2 + γ 1 xy This satisfies E = y 1 y 2 y 3 x 1 α 1 x α 2 z α 3 y x 2 α 3 z α 1 y α 2 x. x 3 α 2 y α 3 x α 1 z J E = E J = W Id
68 Matrix factorization of wedgecontraction type Consider W R = C[x 1,, x n ] with isolated singularity at the origin. Consider odd variables θ 1,, θ n so that we have θ i θ j = θ j θ i, θi θj = θj θi
69 Matrix factorization of wedgecontraction type Consider W R = C[x 1,, x n ] with isolated singularity at the origin. Consider odd variables θ 1,, θ n so that we have θ i θ j = θ j θ i, θi θj = θj θi Consider the algebra R < θ 1,, θ n > of rank 2 n. Define w i so that W = x 1 w x n w n.
70 Matrix factorization of wedgecontraction type Consider W R = C[x 1,, x n ] with isolated singularity at the origin. Consider odd variables θ 1,, θ n so that we have θ i θ j = θ j θ i, θi θj = θj θi Consider the algebra R < θ 1,, θ n > of rank 2 n. Define w i so that W = x 1 w x n w n. Define δ = x i θ i + w i θi. i δ 2 = W
71 Matrix factorization of wedgecontraction type II (R < θ 1,, θ n >, δ) defines a matrix factorization of W. If n = 3, then 2 3 = 8, and even and odd parts of R < θ 1,, θ n > is 4 dimensional. Hence provides (4 4) matrix factorization of W.
72 Matrix factorization of wedgecontraction type II (R < θ 1,, θ n >, δ) defines a matrix factorization of W. If n = 3, then 2 3 = 8, and even and odd parts of R < θ 1,, θ n > is 4 dimensional. Hence provides (4 4) matrix factorization of W. Dyckerhoff proved that this matrix factorization generates the Matrix factorization category if W has only isolated singularity at the origin. This matrix factorization corresponds to the skyscraper sheaf at the singular point.
73 Seidel Lagrangian maps to wedgecontraction matrix factorization To prove homological mirror symmetry, we prove that Seidel Lagrangian L maps to (4 4) matrix factorization. First CF (L, L) has 8 generators. (min, max, X, X, Y, Ȳ, Z, Z)
74 Seidel Lagrangian maps to wedgecontraction matrix factorization To prove homological mirror symmetry, we prove that Seidel Lagrangian L maps to (4 4) matrix factorization. First CF (L, L) has 8 generators. (min, max, X, X, Y, Ȳ, Z, Z) We can identify θ 1 = X, θ 2 = Y, θ 3 = Z. But X Y Z. In fact, we need quantum corrections. After this quantum corrections, this become wedgecontraction type.
75 Seidel Lagrangian maps to wedgecontraction matrix factorization To prove homological mirror symmetry, we prove that Seidel Lagrangian L maps to (4 4) matrix factorization. First CF (L, L) has 8 generators. (min, max, X, X, Y, Ȳ, Z, Z) We can identify θ 1 = X, θ 2 = Y, θ 3 = Z. But X Y Z. In fact, we need quantum corrections. After this quantum corrections, this become wedgecontraction type. We show that W a,b,c has isolated singularities at the origin for 1 a + 1 b + 1 c 1. (joint with S.H. Kim) In such case, Seidel Lagrangian also splitgenerates derived Fukaya category of P a,b,c (using Abouzaid, AFOOO), This proves the homological mirror symmetry.
76 Equivariant theory In elliptic curve quotient E/Z 3, Lagrangian is given by Z 3 equivariant brane in E. Given a character ρ of Z 3, we have an orbibundle.
77 Equivariant theory In elliptic curve quotient E/Z 3, Lagrangian is given by Z 3 equivariant brane in E. Given a character ρ of Z 3, we have an orbibundle. Ǧ naturally act on x, y, z, and W is invariant under Ǧaction. We get LandauGinzburg orbifold (W, Ǧ).
78 Equivariant theory In elliptic curve quotient E/Z 3, Lagrangian is given by Z 3 equivariant brane in E. Given a character ρ of Z 3, we have an orbibundle. Ǧ naturally act on x, y, z, and W is invariant under Ǧaction. We get LandauGinzburg orbifold (W, Ǧ). We get upstairs functor to equivariant Matrix factorization category. Fuk(E) MF Ǧ (W )
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