Title:

Koszul duality and categorical dynamics

This thesis is essentially a combination of my papers [74] and [75]. We study two closely related topics in symplectic topology: Koszul duality and categorical dynamics. Koszul duality is a duality phenomenon between homologically smooth and proper A_{∞}algebras. When applied in the geometric context, it requires that the wrapped Fukaya category W(M) of a Liouville manifold to be recoverable from its full subcategory F(M) of compact Lagrangian submanifolds, which in particular implies the isomorphism HH*(W(M)) ≅ HH*(F(M)) between the Hochschild cohomologies. Categorical dynamics, on the other hand, studies (infinitesimal) symmetries on the Fukaya category F(M). When such symmetries have geometric origins, they define cohomology classes in the symplectic cohomology SH*(M), i.e. Hochschild cohomology HH*W(M) of the wrapped Fukaya category. In particular, the restriction morphism HH*(W(M)) → HH*(F(M)) is nonvanishing on the classes defining the symmetries on F(M). Thus both of the theories can be regarded as imposing restrictions on the symplectic topology of M so that the Fukaya categories F(M) and W(M) are closely related. To explore the relationships between these two theories, we show that the method of constructing Liouville manifolds whose Fukaya categories F(M) carry dilating C*actions can also be applied to obtain examples of Liouville manifolds whose Fukaya categories are related by A_{∞}Koszul duality. On the other hand, we show that for many Liouville manifolds whose Fukaya categories are Koszul dual, there is a generalized theory of categorical dynamics, so that infinitesimal symmetries exist on A_{∞}subcategory of F(M) consisting of mutually orthogonal spherical objects, and these infinitesimal symmetries can be patched together to give rise to a cyclic homology class in HC_{n+1}(W(M)). This is the socalled exact CalabiYau structure to be mentioned below. As a consequence, this thesis is naturally divided into two parts. In the first part of this thesis (Chapters 3, 4, 5, 6), we consider a family of 6dimensional Milnor fibers M_{p,q,r} which are affine hypersurfaces in C^{4}. They are Milnor fibers of stabilizations of cusp and simple elliptic singularities in C^{3}. Explicit computations enable us to identify their compact Fukaya categories F(M_{p,q,r}) with the cyclic completions of certain directed quiver algebras A_{p,q,r}, and their wrapped Fukaya categories W(M_{p,q,r}) with the CalabiYau completions of the same quiver algebras A_{p,q,r}, therefore showing that the Fukaya categories F(M_{p,q,r}) and W(M_{p,q,r}) are related to each other via A_{∞}Koszul duality. As applications, we prove the formality of the Fukaya A_{∞}algebra of a basis of vanishing cycles in M_{p,q,r}, and show that the compact Fukaya category F(M_{p,q,r}) is splitgenerated by vanishing cycles. In particular, the manifolds M_{p,q,r} provide interesting examples of Liouville manifolds whose wrapped Fukaya categories are exact CalabiYau in the sense of Keller. An exact CalabiYau structure on a homologically smooth A_{∞}category, being the key notion for our study in the second part of this thesis (Chapters 7, 8, 9), is a special kind of smooth CalabiYau structures in the sense of KontsevichVlassopoulos [69]. For a Weinstein manifold M, the existence of an exact CalabiYau structure on its wrapped Fukaya category W(M) imposes strong restrictions on its symplectic topology. Under the cyclic openclosed map constructed by Ganatra [43], an exact CalabiYau structure on W(M) induces a class ̃b in the degree one equivariant symplectic cohomology SH^{1}_{S}^{1}(M). Using ̃b ∈ SH^{1} _{S}^{1}(M), we construct an endomorphism on the Floer cohomology HF*(L, L) of a Lagrangian sphere L €⊂ M with dimension n ≥ 3, which acts trivially on HF^{0}(L, L), and nontrivially on HF^{n}(L, L). This enables us to prove that for many Weinstein manifolds with exact CalabiYau wrapped Fukaya categories, there is an upper bound on the number of disjoint Lagrangian spheres, and the rational homology classes of these spheres are linearly independent. These results generalize those of Seidel in [96] since any Weinstein manifold admitting a quasidilation has an exact CalabiYau wrapped Fukaya category. Finally, using Koszul duality, we prove that there are examples of Weinstein manifolds whose wrapped Fukaya categories are exact CalabiYau, but which do not admit quasidilations.
