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On the wrapped Fukaya category and based loops. (English) Zbl 1298.53092

Given a Liouville domain \(M\), one can define the ‘wrapped’ Fukaya category \(\mathcal{W}(M)\), which is a version of the Fukaya category incorporating non-compact Lagrangians; it is an open-string version of symplectic cohomology. Given a Liouville subdomain \(M^{in} \subset M\), one can define an \(A_{\infty}\) restriction functor \[ \widetilde{\mathcal{W}}(M) \rightarrow \mathcal{W}(M^{in}), \] where \(\widetilde{\mathcal{W}}(M)\) is a subcategory of \(\mathcal{W}(M)\) consisting of Lagrangians which intersect the boundary of \(M^{in}\) in a controlled way [M. Abouzaid and P. Seidel, Geom. Topol. 14, No. 2, 627–718 (2010; Zbl 1195.53106)]. In particular, if \(Q \subset M\) is an exact Lagrangian submanifold, then there is an embedding of the disk cotangent bundle of \(Q\) into \(M\) as a Weinstein neighbourhood, so there is a restriction functor \[ \widetilde{\mathcal{W}}(M) \rightarrow \mathcal{W}(T^*Q). \]
In the same situation (\(Q \subset M\) an exact Lagrangian submanifold of a Liouville domain \(M\)), this paper constructs a restriction functor \[ \mathcal{W}(M) \rightarrow Tw(\mathcal{W}(T^*Q)), \] where ‘\(Tw\)’ denotes the category of twisted complexes. The main strength of this result compared with the previously-mentioned one is that one no longer has to restrict oneself to Lagrangians intersecting the boundary of the Weinstein neighbourhood nicely.
The functor is constructed via a topological model for \(\mathcal{W}(T^*Q)\), denoted \(\mathcal{P}(Q)\), the ‘Pontryagin category’ of \(Q\). Briefly, it is a differential graded category whose objects are points of \(q\), morphism spaces are chains on the space of paths between two points, composition is induced by concatenation of paths. The paper defines an \(A_{\infty}\) functor \(\mathcal{W}(M) \rightarrow Tw(\mathcal{P}(Q))\); the key point is the construction of some moduli spaces of pseudo-holomorphic curves with one boundary component mapped to \(Q\), with evaluation maps from the moduli spaces to the space of paths in \(Q\), given by restricting the curve to its boundary component.
In particular, if \(M = T^*Q\), for any \(q \in Q\) one obtains an \(A_{\infty}\) morphism \[ CW^*(T^*_qQ) \rightarrow C_{-*}(\Omega_q Q). \] A morphism in the other direction was constructed on the level of cohomology (and shown to be an isomorphism) in [A. Abbondandolo and M. Schwarz, ibid. 14, No. 3, 1569–1722 (2010; Zbl 1201.53087)]. It is proven that these maps are inverses on cohomology.
These results were used in the author’s subsequent paper [Adv. Math. 228, No. 2, 894–939 (2011; Zbl 1241.53071)] to show that a cotangent fibre \(T^*_qQ\) generates the wrapped Fukaya category of the cotangent bundle, and therefore that the induced functor \[ Tw(\mathcal{W}(T^*Q)) \rightarrow Tw(\mathcal{P}(Q)) \] is an equivalence of categories; this was a key component of the author’s subsequent proof [Invent. Math. 189, No. 2, 251–313 (2012; Zbl 1261.53077)] that nearby Lagrangians with vanishing Maslov class are homotopy equivalent.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
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