HMS Seminar  Talk 1. Netanel Blaier (Brandeis) September 26, 2016


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1 HMS Seminar  Talk 1 Netanel Blaier (Brandeis) September 26, 2016
2 Overview Fukaya categories : (naive) Lagrangian Floer homology, A structures Introduction : what is mirror symmetry? The physical story Restatement 1: Classical MS Restatement 2: Homological Mirror Symmetry The SYZ conjecture
3 Outline for section 1 Fukaya categories : (naive) Lagrangian Floer homology, A structures Introduction : what is mirror symmetry? The physical story Restatement 1: Classical MS Restatement 2: Homological Mirror Symmetry The SYZ conjecture
4 15 minutes of symplectic topology A symplectic manifold (M 2n, ω) is a smooth manifold with a closed, nondegenerate 2form. A symplectomorphism is a diffeomorphism φ : M M such that φ ω = ω. Darboux theorem: there exists an open neighbourhood D of every point p M that is symplectomorphic to (R 2n, ω std := dx i dy i ). A Lagrangian L n M is a half dimensional submanifold with ω L = 0. Weinstein tubular neighbourhood: Every Lagrangian L M has an open neighbourhood that is symplectomorphic to T L. Everything is a Lagrangian! (Weinstein s creed) Unfortunately, very hard to classify and understand directly: 1. Possibly known for R 4 (2016, Georgios D. Rizell). 2. Not even a conjecture for R 6!
5 15 minutes of symplectic topology A symplectic manifold (M 2n, ω) is a smooth manifold with a closed, nondegenerate 2form. A symplectomorphism is a diffeomorphism φ : M M such that φ ω = ω. Darboux theorem: there exists an open neighbourhood D of every point p M that is symplectomorphic to (R 2n, ω std := dx i dy i ). A Lagrangian L n M is a half dimensional submanifold with ω L = 0. Weinstein tubular neighbourhood: Every Lagrangian L M has an open neighbourhood that is symplectomorphic to T L. Everything is a Lagrangian! (Weinstein s creed) Unfortunately, very hard to classify and understand directly: 1. Possibly known for R 4 (2016, Georgios D. Rizell). 2. Not even a conjecture for R 6!
6 15 minutes of symplectic topology A Hamiltonian is just a function H : M R. The Hamiltonian vector field X H of H is defined by ι XH ω = dh. Thus, every function generates a 1parameter family of symplectomorphism { φ t H}. The time1 flow is called a Hamiltonian isotopy. Corollary: symplectic topology is very flabby. Remarkably, Lagrangians have an interesting intersection theory.
7 15 minutes of symplectic topology A Hamiltonian is just a function H : M R. The Hamiltonian vector field X H of H is defined by ι XH ω = dh. Thus, every function generates a 1parameter family of symplectomorphism { φ t H}. The time1 flow is called a Hamiltonian isotopy. Corollary: symplectic topology is very flabby. Remarkably, Lagrangians have an interesting intersection theory.
8 15 minutes of symplectic topology A Hamiltonian is just a function H : M R. The Hamiltonian vector field X H of H is defined by ι XH ω = dh. Thus, every function generates a 1parameter family of symplectomorphism { φ t H}. The time1 flow is called a Hamiltonian isotopy. Corollary: symplectic topology is very flabby. Remarkably, Lagrangians have an interesting intersection theory.
9 Floer cohomology  formal properties In nice cases (not always!), we can associate to L 0, L 1 M a group HF (L 0, L 1 ) which attempts to measure (in a Hamiltonian isotopy invariant way) how many times must L 0 and L 1 intersect. Properties: 1. χ(hf (L 0, L 1 )) = L 0 L HF (φ 1 H L 0, L 1 ) = HF (L 0, L 1 ) = HF (L 0, φ 1 H L 1). 3. HF (L, L) = H sing (L). 4. If L 0 L 1, then HF (L 0, L 1 ) = H (CF (L 0, L 1 ), d), where CF (L 0, L 1 ) is freely generated by the intersection points. Thus, for example, we have a refined lower bound φ H L L =: rkcf (φ H L, L) rkhf (L, L) = rkh (L). There are cases where we can define CF (L 0, L 1 ) and d, but d 2 0. In that case, we say that (L 0, L 1 ) is obstructed.
10 Floer cohomology  formal properties In nice cases (not always!), we can associate to L 0, L 1 M a group HF (L 0, L 1 ) which attempts to measure (in a Hamiltonian isotopy invariant way) how many times must L 0 and L 1 intersect. Properties: 1. χ(hf (L 0, L 1 )) = L 0 L HF (φ 1 H L 0, L 1 ) = HF (L 0, L 1 ) = HF (L 0, φ 1 H L 1). 3. HF (L, L) = H sing (L). 4. If L 0 L 1, then HF (L 0, L 1 ) = H (CF (L 0, L 1 ), d), where CF (L 0, L 1 ) is freely generated by the intersection points. Thus, for example, we have a refined lower bound φ H L L =: rkcf (φ H L, L) rkhf (L, L) = rkh (L). There are cases where we can define CF (L 0, L 1 ) and d, but d 2 0. In that case, we say that (L 0, L 1 ) is obstructed.
11 Floer cohomology  formal properties In nice cases (not always!), we can associate to L 0, L 1 M a group HF (L 0, L 1 ) which attempts to measure (in a Hamiltonian isotopy invariant way) how many times must L 0 and L 1 intersect. Properties: 1. χ(hf (L 0, L 1 )) = L 0 L HF (φ 1 H L 0, L 1 ) = HF (L 0, L 1 ) = HF (L 0, φ 1 H L 1). 3. HF (L, L) = H sing (L). 4. If L 0 L 1, then HF (L 0, L 1 ) = H (CF (L 0, L 1 ), d), where CF (L 0, L 1 ) is freely generated by the intersection points. Thus, for example, we have a refined lower bound φ H L L =: rkcf (φ H L, L) rkhf (L, L) = rkh (L). There are cases where we can define CF (L 0, L 1 ) and d, but d 2 0. In that case, we say that (L 0, L 1 ) is obstructed.
12 Floer cohomology  opening the blackbox (gently) Coefficients: a small field k (= Z 2 for now, maybe Q or C later). Novikov field K = k whose elements are formal sums { } K = c i q λ i c i k, λ i R, lim λ i = + i i=0 Keeps track of areas of holomorphic curves. Morphism spaces are Kvector spaces freely generated by intersection points: CF (L 0, L 1 ) := K L 0 L 1.
13 There is a differential on the morphism spaces, d : CF (L 0, L 1 ) CF (L 0, L 1 ). Given p, q L 0 L 1, the coefficient of q in d(p) is the number of Jholomorphic strips u like this: Floer differential L 1 t s q p L 0 weighted by q ω (u). Morally, HF (L 0, L 1 ) is Morse homology of the action functional defined on the space of paths ([0, 1], {0}, {1}) (M, L 0, L 1 ).
14 There is a differential on the morphism spaces, d : CF (L 0, L 1 ) CF (L 0, L 1 ). Given p, q L 0 L 1, the coefficient of q in d(p) is the number of Jholomorphic strips u like this: Floer differential L 1 t s q p L 0 weighted by q ω (u). Morally, HF (L 0, L 1 ) is Morse homology of the action functional defined on the space of paths ([0, 1], {0}, {1}) (M, L 0, L 1 ).
15 Floer cohomology  opening the blackbox (gently) Analytically, it is very hard to actually define it in this way. Floer s Brilliant idea: replace an ODE on the path space by a PDE on the manifold! Solve the gradient equation u + J(u(s, t)) u s t = 0 for u : R [0, 1] M subject to: u(s, 0) L 0, u(s, 1) L 1, lim u(s, t) = p, lim u(s, t) = q. s s +
16 Floer cohomology  opening the blackbox (gently) Analytically, it is very hard to actually define it in this way. Floer s Brilliant idea: replace an ODE on the path space by a PDE on the manifold! Solve the gradient equation u + J(u(s, t)) u s t = 0 for u : R [0, 1] M subject to: u(s, 0) L 0, u(s, 1) L 1, lim u(s, t) = p, lim u(s, t) = q. s s +
17 Donaldson observed that there is a composition Triangle product [µ 2 ] : HF (L 1, L 2 ) HF (L 0, L 1 ) HF (L 0, L 2 ) where the coefficient of r in [µ 2 ](p, q) is the number of holomorphic triangles u like this: p L 0 L 1 r q L 2 weighted by q ω (u).
18 Triangle product As a result, the L i are objects of the DonaldsonFukaya category: Objects Unobstructed Lagrangians L i M. Morphism Hom(L i, L j ) = HF (L i, L j ). Unfortunately, this does not capture the full story (e.g. insufficient for LES in Floer theory, functors,...) The solution are Fukaya categories  which keep track of the relations between the different CF (L 0, L 1 ). There is a price to pay  Fukaya categories are not, well, categories,... The DonaldsonFukaya category can be obtained from the Fukaya category by taking cohomology.
19 Triangle product As a result, the L i are objects of the DonaldsonFukaya category: Objects Unobstructed Lagrangians L i M. Morphism Hom(L i, L j ) = HF (L i, L j ). Unfortunately, this does not capture the full story (e.g. insufficient for LES in Floer theory, functors,...) The solution are Fukaya categories  which keep track of the relations between the different CF (L 0, L 1 ). There is a price to pay  Fukaya categories are not, well, categories,... The DonaldsonFukaya category can be obtained from the Fukaya category by taking cohomology.
20 Triangle product As a result, the L i are objects of the DonaldsonFukaya category: Objects Unobstructed Lagrangians L i M. Morphism Hom(L i, L j ) = HF (L i, L j ). Unfortunately, this does not capture the full story (e.g. insufficient for LES in Floer theory, functors,...) The solution are Fukaya categories  which keep track of the relations between the different CF (L 0, L 1 ). There is a price to pay  Fukaya categories are not, well, categories,... The DonaldsonFukaya category can be obtained from the Fukaya category by taking cohomology.
21 Chainlevel refinement We define the morphism spaces as the chaincomplexes We denote µ 1 = d. Now, there is a bilinear map Hom(L i, L j ) := (CF (L i, L j ), d) µ 2 : CF (L 1, L 2 ) CF (L 0, L 1 ) CF (L 0, L 2 ) which is not associative. Instead, the associator satisfies µ 2 (, µ 2 (, )) µ 2 (µ 2 (, ), ) = µ 3 (µ 1 ( ),, ) + µ 3 (, µ 1 ( ), ) +..., i.e. it is chainhomotopic to zero, and the homotopy defines another trilinear operation µ 3. Q. What does this hierarchy µ 1, µ 2, µ 3, µ 4,..., µ d,... form??
22 Chainlevel refinement We define the morphism spaces as the chaincomplexes We denote µ 1 = d. Now, there is a bilinear map Hom(L i, L j ) := (CF (L i, L j ), d) µ 2 : CF (L 1, L 2 ) CF (L 0, L 1 ) CF (L 0, L 2 ) which is not associative. Instead, the associator satisfies µ 2 (, µ 2 (, )) µ 2 (µ 2 (, ), ) = µ 3 (µ 1 ( ),, ) + µ 3 (, µ 1 ( ), ) +..., i.e. it is chainhomotopic to zero, and the homotopy defines another trilinear operation µ 3. Q. What does this hierarchy µ 1, µ 2, µ 3, µ 4,..., µ d,... form??
23 Chainlevel refinement We define the morphism spaces as the chaincomplexes We denote µ 1 = d. Now, there is a bilinear map Hom(L i, L j ) := (CF (L i, L j ), d) µ 2 : CF (L 1, L 2 ) CF (L 0, L 1 ) CF (L 0, L 2 ) which is not associative. Instead, the associator satisfies µ 2 (, µ 2 (, )) µ 2 (µ 2 (, ), ) = µ 3 (µ 1 ( ),, ) + µ 3 (, µ 1 ( ), ) +..., i.e. it is chainhomotopic to zero, and the homotopy defines another trilinear operation µ 3. Q. What does this hierarchy µ 1, µ 2, µ 3, µ 4,..., µ d,... form??
24 Chainlevel refinement We define the morphism spaces as the chaincomplexes We denote µ 1 = d. Now, there is a bilinear map Hom(L i, L j ) := (CF (L i, L j ), d) µ 2 : CF (L 1, L 2 ) CF (L 0, L 1 ) CF (L 0, L 2 ) which is not associative. Instead, the associator satisfies µ 2 (, µ 2 (, )) µ 2 (µ 2 (, ), ) = µ 3 (µ 1 ( ),, ) + µ 3 (, µ 1 ( ), ) +..., i.e. it is chainhomotopic to zero, and the homotopy defines another trilinear operation µ 3. Q. What does this hierarchy µ 1, µ 2, µ 3, µ 4,..., µ d,... form??
25 15 minutes of homological algebra Let K be a field, and A = A a graded vector space. The Hochschild cochains of length s 0 are CC s+t (A) s := Hom t (A s, A), and the Hochschild cochain complex is CC d (A) := CC s+t (A) s. d=s+t,s 0 ψ = ψ 0 + ψ 1 + ψ CC d (A). It admits a Gerstenhaber product and bracket: φ ψ(x s,..., x 1 ) := i,j ±φ(x s,..., ψ(x i+j,..., x i+1 ),..., x 1 ), [φ, ψ] := φ ψ ± ψ φ.
26 15 minutes of homological algebra An A structure is µ CC 2 (A), µ 0 = 0 such that 1 2 [µ, µ] = 0. Explicitly, the first few equations µ = (0, µ 1, µ 2, µ 3,...) has to satisfy are exactly d 2 = 0, the Leibnitz rule, and an associator relation: µ 1 (µ 1 (x 1 )) = 0 ± µ 2 (x 2, µ 1 (x 1 )) ± µ 2 (µ 1 (x 2 ), x 1 ) = µ 1 (µ 2 (x 2, x 1 )) ± µ 2 (x 2, µ 2 (x 1, x 0 )) ± µ 2 (µ 2 (x 2, x 1 ), x 0 ) = ±µ 3 (µ 1 (x 2 ), x 1, x 0 ) ±... so we can define an associative cohomology algebra H(A, µ). Finally, we note that given any such µ, we can define a differential µ (φ) = [µ, φ] on CC (A). Then HH (A) := H(CC (A), µ ) is called Hochschild cohomology.
27 The Fukaya category Metatheorem (Fukaya): making some choices, we can define an A structure µ d on F uk(m) extending the Floer differential and triangle product. The category (F uk(m), µ d ) is independent of all choices up to an A quasiisomorphism. F uk(m) is the right setting to discuss various relations between Lagrangians, or rather, the splitclosure derived category D π F uk(m) which is a formal enlargement with objects {L 0 L 1... L k }. This is much more then a bookkeeping device...
28 Outline for section 2 Fukaya categories : (naive) Lagrangian Floer homology, A structures Introduction : what is mirror symmetry? The physical story Restatement 1: Classical MS Restatement 2: Homological Mirror Symmetry The SYZ conjecture
29 Supersymmetric string theory (very roughly...) Strings propagate in spacetime span a worldsheet (=surface Σ). Come in two flavors: open string ( Σ φ) and closed string. Get a 2D QFT on Σ, with some fields taking value in spacetime (=manifolds). The theory is required to be supersymmetric and conformal. Most common example is a nonlinear σmodel  where the input data is a CalabiYau manifold. 1. (X, J) is a smooth, complex manifold. 2. K X := n T X = holo O X, and thus there exists a nonvanishing holomorphic volume form Ω. 3. ω C = B + iω Ω 1,1 (X) is a complexified Kähler form.
30 The Holy Grail To every such SCFT one can associate two N = 1supersymmetric TFT s: the A and Bmodel. There exists supersymmetric operators (Q, Q ) whose simultanuous eigenspaces are H q ( p T X) and H q (X, Ω p X ). If there exists another CY ( ˆX, Ĵ, ˆωC ) such that H q ( and the TFT s are equivalent: p T X) H q (X, Ω p X ) A(X) = B( ˆX), A( ˆX) = B(X). We will say that X and ˆX are mirror symmetric. The Problem: the words I have just written have no (mathematical) meaning!
31 The Holy Grail To every such SCFT one can associate two N = 1supersymmetric TFT s: the A and Bmodel. There exists supersymmetric operators (Q, Q ) whose simultanuous eigenspaces are H q ( p T X) and H q (X, Ω p X ). If there exists another CY ( ˆX, Ĵ, ˆωC ) such that H q ( and the TFT s are equivalent: p T X) H q (X, Ω p X ) A(X) = B( ˆX), A( ˆX) = B(X). We will say that X and ˆX are mirror symmetric. The Problem: the words I have just written have no (mathematical) meaning!
32 Hodge theory First thing we expect from CY mirrors: H q (X, p TX ) = H q ( ˆX, Ω p ˆX ). We can always define a map v 1... v p ι v1... v p Ω. Because of the CY assumption this is an isomorphism So there is an equality p TX = Ω n p X. H n p,q (X) = H p,q ( ˆX).
33 Counting is fun! Let (M, ω, J) be a Kähler manifold. GromovWitten invariants count the number of isolated rational curves in a given homology class, subject to some generic point constraints. Examples: 1. How many lines (d = 1) pass through two generic points? How many conics (d = 2) pass through five generic points? Number of lines (d = 1) on a quintic threefold: Number of conics (d = 2) on a quintic threefold: Gromov (1985). GromovWitten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count Jholomorphic curves: solutions to a perturbed CauchyRiemann PDE (with a possibly nonintegrable J!)
34 Counting is fun! Let (M, ω, J) be a Kähler manifold. GromovWitten invariants count the number of isolated rational curves in a given homology class, subject to some generic point constraints. Examples: 1. How many lines (d = 1) pass through two generic points? How many conics (d = 2) pass through five generic points? Number of lines (d = 1) on a quintic threefold: Number of conics (d = 2) on a quintic threefold: Gromov (1985). GromovWitten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count Jholomorphic curves: solutions to a perturbed CauchyRiemann PDE (with a possibly nonintegrable J!)
35 Counting is fun! Let (M, ω, J) be a Kähler manifold. GromovWitten invariants count the number of isolated rational curves in a given homology class, subject to some generic point constraints. Examples: 1. How many lines (d = 1) pass through two generic points? How many conics (d = 2) pass through five generic points? Number of lines (d = 1) on a quintic threefold: Number of conics (d = 2) on a quintic threefold: Gromov (1985). GromovWitten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count Jholomorphic curves: solutions to a perturbed CauchyRiemann PDE (with a possibly nonintegrable J!)
36 Counting is fun! Let (M, ω, J) be a Kähler manifold. GromovWitten invariants count the number of isolated rational curves in a given homology class, subject to some generic point constraints. Examples: 1. How many lines (d = 1) pass through two generic points? How many conics (d = 2) pass through five generic points? Number of lines (d = 1) on a quintic threefold: Number of conics (d = 2) on a quintic threefold: Gromov (1985). GromovWitten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count Jholomorphic curves: solutions to a perturbed CauchyRiemann PDE (with a possibly nonintegrable J!)
37 Yukawa coupling Idea: we instead look at the correlation functions (=expectation values for observables). These will have a mathematical formulation. This defines an additional structure called Yukawa coupling,, : On H 1 (X, T X ) =H 2,1 (X) this is connected to periods and the GaussManin connection. On H 1,1 (X) this is the GW invariant! Classical MS: if X and ˆX are mirrors then (H 1,1 (X),,, A ) = (H 2,1 (X),,, B )
38 Early success Idea (Candelas, de la Ossa, Green and Parkes, 1991): We can use the periods of the holomorphic volume form on the mirror to the to quintic threefold predict GromovWitten invariants! (agreed with known results for d = 1, 2, 3). Proved mathematically by Givental and LianLiYau in many improtant cases (including all CY and Fano complete intersections in toric varieties)... and so a new field in mathematics was born!
39 DBranes Open strings propagate the worldsheet is a surface with boundary. Constraints on the values of the fields at the boundary are called Dbranes. Field theory axioms there is a category of Dbranes. Amodel branes are Lagrangian manifolds + flat bundles. Bmodel branes are complex analytic manifolds + holomorphic bundles. Kontsevich (1994): If X and ˆX are mirrors, then D π F uk(x) = D b Coh( ˆX), D π F uk( ˆX) = D b Coh(X). Morphism in D b Coh(X) are Hom s and Ext s (mirror symmetry related the intersection theories!)
40 Why is that AWESOME Can now understand the conjecture is simple cases like surfaces (CY 3folds are complicated!) Can (finally!) compute things on the Aside, find autoequivalences on the Bside,... Openstring mirror symmetry implies closedstring mirror symmetry by taking Hochschild cohomology! (GPS,2016): QH (X) = HH (D π F uk(x)) H (X, T X ) = HKR HH (D b Coh(X)) F uk(x) and Coh(X) turns out to be very important mathematical objects (Heegaard Floer theory, SeidelSmith Khovanov homology, representation theory, birational geometry,...)
41 Why is that AWESOME Can now understand the conjecture is simple cases like surfaces (CY 3folds are complicated!) Can (finally!) compute things on the Aside, find autoequivalences on the Bside,... Openstring mirror symmetry implies closedstring mirror symmetry by taking Hochschild cohomology! (GPS,2016): QH (X) = HH (D π F uk(x)) H (X, T X ) = HKR HH (D b Coh(X)) F uk(x) and Coh(X) turns out to be very important mathematical objects (Heegaard Floer theory, SeidelSmith Khovanov homology, representation theory, birational geometry,...)
42 Why is that AWESOME Can now understand the conjecture is simple cases like surfaces (CY 3folds are complicated!) Can (finally!) compute things on the Aside, find autoequivalences on the Bside,... Openstring mirror symmetry implies closedstring mirror symmetry by taking Hochschild cohomology! (GPS,2016): QH (X) = HH (D π F uk(x)) H (X, T X ) = HKR HH (D b Coh(X)) F uk(x) and Coh(X) turns out to be very important mathematical objects (Heegaard Floer theory, SeidelSmith Khovanov homology, representation theory, birational geometry,...)
43 Why is that AWESOME Can now understand the conjecture is simple cases like surfaces (CY 3folds are complicated!) Can (finally!) compute things on the Aside, find autoequivalences on the Bside,... Openstring mirror symmetry implies closedstring mirror symmetry by taking Hochschild cohomology! (GPS,2016): QH (X) = HH (D π F uk(x)) H (X, T X ) = HKR HH (D b Coh(X)) F uk(x) and Coh(X) turns out to be very important mathematical objects (Heegaard Floer theory, SeidelSmith Khovanov homology, representation theory, birational geometry,...)
44 StormingerYauZaslow conjecture (1996) The crucial missing bit: what does it mean for two manifolds to be mirror to each other geometrically? and when does it happen?? A Lagrangian is special if Im(Ω L ) = 0. SYZ (very roughly): X, ˆX are mirrors they carry mutually dual special Lagrangian torus fibrations T n X B, ˆT n ˆX B, ˆT n = Hom(π 1 (T ), U(1)). over a common base (=moduli space of special Lagrangians).
45 StormingerYauZaslow conjecture (1996) The crucial missing bit: what does it mean for two manifolds to be mirror to each other geometrically? and when does it happen?? A Lagrangian is special if Im(Ω L ) = 0. SYZ (very roughly): X, ˆX are mirrors they carry mutually dual special Lagrangian torus fibrations T n X B, ˆT n ˆX B, ˆT n = Hom(π 1 (T ), U(1)). over a common base (=moduli space of special Lagrangians).
46 Motivation from HMS ˆX is the moduli space of points of ˆX. Every p ˆX yields a skyscraper sheaf O p D b Coh( ˆX). By HMS, this corresponds to a Lagrangian L p, such that HF (L p, L p ) = Ext (O p, O p ) = C n Thus, as graded vector spaces, H (L p ) = H (T n ). ˆX is the moduli space of T n (+...) in the F uk(x). Not quite true as stated... (near large complex structure limit, slag have singularities need instanton corrections etc)
47 Motivation from HMS ˆX is the moduli space of points of ˆX. Every p ˆX yields a skyscraper sheaf O p D b Coh( ˆX). By HMS, this corresponds to a Lagrangian L p, such that HF (L p, L p ) = Ext (O p, O p ) = C n Thus, as graded vector spaces, H (L p ) = H (T n ). ˆX is the moduli space of T n (+...) in the F uk(x). Not quite true as stated... (near large complex structure limit, slag have singularities need instanton corrections etc)
48 Motivation from HMS ˆX is the moduli space of points of ˆX. Every p ˆX yields a skyscraper sheaf O p D b Coh( ˆX). By HMS, this corresponds to a Lagrangian L p, such that HF (L p, L p ) = Ext (O p, O p ) = C n Thus, as graded vector spaces, H (L p ) = H (T n ). ˆX is the moduli space of T n (+...) in the F uk(x). Not quite true as stated... (near large complex structure limit, slag have singularities need instanton corrections etc)
49 Questions?
Topics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:
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