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

Transcription

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, non-degenerate 2-form. 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, non-degenerate 2-form. 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 1-parameter family of symplectomorphism { φ t H}. The time-1 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 1-parameter family of symplectomorphism { φ t H}. The time-1 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 1-parameter family of symplectomorphism { φ t H}. The time-1 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 K-vector 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 J-holomorphic 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 J-holomorphic 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 Donaldson-Fukaya 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 Donaldson-Fukaya category can be obtained from the Fukaya category by taking cohomology.

19 Triangle product As a result, the L i are objects of the Donaldson-Fukaya 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 Donaldson-Fukaya category can be obtained from the Fukaya category by taking cohomology.

20 Triangle product As a result, the L i are objects of the Donaldson-Fukaya 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 Donaldson-Fukaya category can be obtained from the Fukaya category by taking cohomology.

21 Chain-level refinement We define the morphism spaces as the chain-complexes 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 chain-homotopic 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 Chain-level refinement We define the morphism spaces as the chain-complexes 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 chain-homotopic 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 Chain-level refinement We define the morphism spaces as the chain-complexes 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 chain-homotopic 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 Chain-level refinement We define the morphism spaces as the chain-complexes 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 chain-homotopic 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 Meta-theorem (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 -quasi-isomorphism. F uk(m) is the right setting to discuss various relations between Lagrangians, or rather, the split-closure 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 book-keeping 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 space-time span a world-sheet (=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 space-time (=manifolds). The theory is required to be supersymmetric and conformal. Most common example is a nonlinear σ-model - where the input data is a Calabi-Yau 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 = 1-supersymmetric TFT s: the A- and B-model. There exists super-symmetric 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 = 1-supersymmetric TFT s: the A- and B-model. There exists super-symmetric 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. Gromov-Witten 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 three-fold: Number of conics (d = 2) on a quintic three-fold: Gromov (1985). Gromov-Witten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count J-holomorphic curves: solutions to a perturbed Cauchy-Riemann PDE (with a possibly non-integrable J!)

34 Counting is fun! Let (M, ω, J) be a Kähler manifold. Gromov-Witten 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 three-fold: Number of conics (d = 2) on a quintic three-fold: Gromov (1985). Gromov-Witten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count J-holomorphic curves: solutions to a perturbed Cauchy-Riemann PDE (with a possibly non-integrable J!)

35 Counting is fun! Let (M, ω, J) be a Kähler manifold. Gromov-Witten 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 three-fold: Number of conics (d = 2) on a quintic three-fold: Gromov (1985). Gromov-Witten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count J-holomorphic curves: solutions to a perturbed Cauchy-Riemann PDE (with a possibly non-integrable J!)

36 Counting is fun! Let (M, ω, J) be a Kähler manifold. Gromov-Witten 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 three-fold: Number of conics (d = 2) on a quintic three-fold: Gromov (1985). Gromov-Witten invariants are symplectic invariants! We can actually define GW invariants for any symplectic manifold... But as before, we must count J-holomorphic curves: solutions to a perturbed Cauchy-Riemann PDE (with a possibly non-integrable 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 Gauss-Manin 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 three-fold predict Gromov-Witten invariants! (agreed with known results for d = 1, 2, 3). Proved mathematically by Givental and Lian-Li-Yau in many improtant cases (including all CY and Fano complete intersections in toric varieties)... and so a new field in mathematics was born!

39 D-Branes Open strings propagate the worldsheet is a surface with boundary. Constraints on the values of the fields at the boundary are called D-branes. Field theory axioms there is a category of D-branes. A-model branes are Lagrangian manifolds + flat bundles. B-model 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 3-folds are complicated!) Can (finally!) compute things on the A-side, find auto-equivalences on the B-side,... Open-string mirror symmetry implies closed-string 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, Seidel-Smith Khovanov homology, representation theory, birational geometry,...)

41 Why is that AWESOME Can now understand the conjecture is simple cases like surfaces (CY 3-folds are complicated!) Can (finally!) compute things on the A-side, find auto-equivalences on the B-side,... Open-string mirror symmetry implies closed-string 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, Seidel-Smith Khovanov homology, representation theory, birational geometry,...)

42 Why is that AWESOME Can now understand the conjecture is simple cases like surfaces (CY 3-folds are complicated!) Can (finally!) compute things on the A-side, find auto-equivalences on the B-side,... Open-string mirror symmetry implies closed-string 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, Seidel-Smith Khovanov homology, representation theory, birational geometry,...)

43 Why is that AWESOME Can now understand the conjecture is simple cases like surfaces (CY 3-folds are complicated!) Can (finally!) compute things on the A-side, find auto-equivalences on the B-side,... Open-string mirror symmetry implies closed-string 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, Seidel-Smith Khovanov homology, representation theory, birational geometry,...)

44 Storminger-Yau-Zaslow 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 Storminger-Yau-Zaslow 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?