Wrapped Fukaya categories

Transcription

1 Wrapped Fukaya categories Mohammed Abouzaid Clay / MIT July 9, 2009 Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

2 Outline 1 Wednesday: Construction of the wrapped category 2 The Pontryagin Category Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

3 Goals: Setup: Compute the wrapped Fukaya category for cotangent bundles, i.e. find a model from algebraic topology. Give evidence for a restriction functor in wrapped Floer homology. (M, λ) is a Liouville domain. Q M is an exact compact Lagrangian submanifold. The case M = T Q is already interesting. Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

4 The Pontryagin Category Moore paths Q is a smooth compact manifold. Given a pair (q 0, q 1 ) of points in Q, let Ω(q 0, q 1 ) denote the space of (Moore) paths from q 0 to q 1. Definition The Pontryagin Category P(Q) consists of the following data: Objects are points of Q. Morphism spaces are the chain complexes Hom (q 0, q 1 ) = C (Ω(q 0, q 1 )) Composition is given by concatenation of paths. To make this into a differential graded category, it s convenient to use normalised cubical chains. After a few sign twists, we obtain an A category with vanishing higher products. Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

5 The Pontryagin Category The string category A generalisation of the string topology of Chas and Sullivan should produce a category String(Q) whose objects are all submanifolds of Q, morphisms are chains on the spaces of paths between these submanifolds, and multiplication are obtained by combining the concatenation of paths with the intersection product along the submanifolds. Blumberg, Cohen, and Teleman have constructed a chain-level model for this category (in fact, they construct a model in spectra). The outcome of their construction is Theorem The category String(Q) is generated by any basepoint q Q. Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

6 The Pontryagin Category Twisted complexes: Objects Twisted complexes give an explicit model for the derived category of an A category. Objects are pairs T = ( [m i ], D = (δ i,j )) with are objects of P(Q), m i is a sequence of integers. D is an upper triangular matrix of degree 1. δ i,j Hom 1 mj m i (, ) δ i,j 0 if i j. D satisfies a homotopy version of d 2 = 0: µ P 2 (D, D) = ±µ P 1 (D). Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

7 Twisted complexes : Morphisms The Pontryagin Category Morphism are direct sums Hom (T 1, T 2 ) i,j Hom (q 1 i, q 2 j )[m 2 j m 1 i ] with differential µ Tw(P) 1 (A) = µ P 1 (A) ± µ P 2 (D 2, A) ± µ P 2 (A, D 1 ). Composition is defined as matrix multiplication µ Tw(P) 2 (A 2, A 1 ) = µ P 2 (A 2, A 1 ) Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

8 The Pontryagin Category Twisted complexes from Morse theory According to the result of Blumberg, Cohen, and Teleman, there should be a twisted complex corresponding to Q itself. Given a Morse function f on Q consider a twisted complex ( ) T f = xi Crit(f )x i [m i ], D f = (δi,j) f The critical points are ordered by the value of f, with m i begin the Morse index. The matrix coefficients of D f are δ f i,j = ev ([T (x j ; x i )]) where T (x j ; x i ) is the moduli space of gradient flow lines from x i to x j. Must choose a compatible system of fundamental classes and parametrisations for the moduli spaces of gradient lines. Uses the fact that T (x j ; x i ) is a (topological) manifold with boundary. Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

9 Leray-Serre spectral sequence The Pontryagin Category Lemma There is a chain equivalence Idea of proof. Hom (T f, T f ) C n (Q) The endomorphism algebra of T f is a chain model for the homology of the total space of the path space fibration: Ω(q, q ) P(Q) Q Q. Starting with any Morse function on a submanifold N Q, the same strategy produces a twisted complex over P(Q) whose endomorphism algebra is a chain model for H (Ω(N, N)). Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

10 Setup for Restriction (M, λ) is a Liouville domain. Q is a smooth compact Lagrangian (+ technical conditions for signs and grading to make sense). Theorem There exists an A functor W(M) TwP(Q). Starting from L an exact Lagrangian submanifold, we must first construct a twisted complex T L = ( qi L Q [m i ], D L ) Follow the strategy from Morse theory. Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

11 Functor on objects The intersection points are ordered by action: The homological shift m i agree with the Maslov index of. The component δ i,j of D L is given by evaluating the moduli space of strips along the edge mapping to Q: ev ([R(, )]). L Q Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

12 Functor on objects The equation follows from Gromov compactness: µ P 2 (D L, D L ) = ±µ P 1 (D L ) (1) L L Q q k Q q k δ i,j µ 1 (δ i,j ) = µ 2 (δ k,j, δ i,k ) In order for (1) to hold at the chain level, we must make compatible choices of fundamental classes and parametrisations. Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

13 Linear part of the Functor The next step is to define a chain map F : CW (L) Hom (T L, T L ) = i,j C (Ω(, )). The left hand side is given by an infinite telescope δ δ δ CF (L;H) CF (L; 2H) CF (L; 3H) κ κ κ id id id q CF (L; H) q CF (L; 2H) q CF (L; 3H) δ δ δ Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

14 Homotopy commutativity So we must construct a homotopy commutative diagram CF (L; kh) CF (L; (k + 1)H) id κ q CF (L; kh) i,j C (Ω(, )) Using the isomorphism CF (L; kh) = CF (L; φ k L), we expect that the map CF (L; kh) C (Ω(, )) i,j uses holomorphic triangles with boundary conditions L, Q, and φ k (L). More convenient to consider solutions to the equation on the half-strip [0, + ) [0, 1]. (du kx dt) 0,1 = 0 (2) Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

15 Image of the top row Given x CF (L; kh), and, in L Q, denote the moduli space of solutions to (2) with these boundary conditions by The image of x is H(, x, ) F(x) = ev [H(, x, )] C (Ω(, )). The evaluation map assigns to each solution of (2) its restriction to the boundary segment mapping to Q, which yields a path from to. kdt L Q L Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

16 Half Popsicles Consider a family of 1-form γ on [0, + ) [0, 1] which agree with kdt near + and (k + 1)dt near {0} [0, 1]. We shall call this the moduli space of half popsicles, and denote it H 1,{1}. Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

17 Image of the bottom row We denote by H 1,{1} (, x, ) the moduli space of solutions to the equation (du X γ) 0,1 = 0 with appropriate endpoints. And use evaluation along the edge mapping to Q to define F(qx) = i,j ev [H 1,{1} (, x, )] C (Ω(, )). The condition of being a chain map is that F(qx) + µ 2 (D L, F(qx)) + µ 2 (F(qx), D L ) = F(qµ 1 (x)) + F(x) + F(κ(x)) Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18

18 x x kdt x kdt x kdt L L (k + 1)dt q k F(qx) = x µ 2 (D L, F(qx)) x (k + 1)dt L kdt L (k + 1)dt q k F(κx) F(q x) F(x) µ 2 (F(qx), D L ) Mohammed Abouzaid (Clay / MIT) Wrapped Fukaya Categories July 9, / 18