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Title: Koszul duality and categorical dynamics
Author: Li, Yin
Awarding Body: King's College London
Current Institution: King's College London (University of London)
Date of Award: 2020
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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 non-vanishing 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 so-called exact Calabi-Yau 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 6-dimensional Milnor fibers Mp,q,r which are affine hypersurfaces in C4. They are Milnor fibers of stabilizations of cusp and simple elliptic singularities in C3. Explicit computations enable us to identify their compact Fukaya categories F(Mp,q,r) with the cyclic completions of certain directed quiver algebras Ap,q,r, and their wrapped Fukaya categories W(Mp,q,r) with the Calabi-Yau completions of the same quiver algebras Ap,q,r, therefore showing that the Fukaya categories F(Mp,q,r) and W(Mp,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 Mp,q,r, and show that the compact Fukaya category F(Mp,q,r) is split-generated by vanishing cycles. In particular, the manifolds Mp,q,r provide interesting examples of Liouville manifolds whose wrapped Fukaya categories are exact Calabi-Yau in the sense of Keller. An exact Calabi-Yau 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 Calabi-Yau structures in the sense of Kontsevich-Vlassopoulos [69]. For a Weinstein manifold M, the existence of an exact Calabi-Yau structure on its wrapped Fukaya category W(M) imposes strong restrictions on its symplectic topology. Under the cyclic open-closed map constructed by Ganatra [43], an exact Calabi-Yau structure on W(M) induces a class ̃b in the degree one equivariant symplectic cohomology SH1S1(M). Using ̃b ∈ SH1 S1(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 HF0(L, L), and non-trivially on HFn(L, L). This enables us to prove that for many Weinstein manifolds with exact Calabi-Yau 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 quasi-dilation has an exact Calabi-Yau wrapped Fukaya category. Finally, using Koszul duality, we prove that there are examples of Weinstein manifolds whose wrapped Fukaya categories are exact Calabi-Yau, but which do not admit quasi-dilations.
Supervisor: Lekili, Yanki Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available