In this thesis, we consider the categories of sheaves with singular support on certain Lagrangians and the categories of microlocal sheaves with support on certain Lagrangians obtained by microlocalization, and study properties of functors between these categories.First, we study one class of the microlocal restriction functor for open inclusions, namely microlocalization on the conical Lagrangian ends. We show a duality and exact triangle arising from the microlocalization functor. Using that, we describe the adjoint functors of microlocalization, and prove that they form a spherical adjunction when the Legendrian at infinity is a full stop or swappable stop. Using the description of the adjoint functors of microlocalization, we prove sheaf quantization theorems, constructing right inverses to the microlocalization functor for noncompact Lagrangian submanifolds, generalizing previous works of Guillermou and Jin-Treumann. In particular, we show a sheaf quantization theorem for Lagrangian cobordisms of Arnol’d and a conditional quantization theorem for Lagrangian cobordisms in symplectic field theory. Then, we study the microlocal specialization functor on closed subdomain embeddings of Weinstein sectors, which is right adjoint to the Viterbo restriction functor, constructed by Nadler-Shende. We show that the specialization functor is natural with respect to compositions of embeddings. Using that, we give another description of Lagrangian cobordism functor in symplectic field theory, which is compatible with the sheaf quantization functor.

Creator

• Li, Wenyuan

DOI

• https://doi.org/10.21985/n2-v333-8f39

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:16549
• etdadmin_upload_985220

Date created

• 2023-01-01

Resource type

• Dissertation

Rights statement

• In Copyright