Microlocal categories over Novikov rings

TL;DR

This paper introduces enhanced microlocal categories over Novikov rings for Liouville manifolds, enabling a generalized sheaf quantization theory for Lagrangian branes and exploring their structural properties.

Contribution

It defines new microlocal categories over Novikov rings and develops a sheaf quantization framework for Lagrangian branes, extending previous theories.

Findings

Categories exhibit intersection point estimates

Properties like interleaving distances and energy stability are established

Conjecture of equivalence with Fukaya categories over Novikov rings

Abstract

In this paper, we define a family of categories for each Liouville manifold, which is an enhanced version of the category first introduced by Tamarkin. Using our categories, for any (possibly non-exact immersed) Lagrangian brane, we develop a theory of sheaf quantization generalizing the previous researches. In particular, our theory involves the notion of a sheaf-theoretic bounding cochain, which is a conjectural counterpart of the theory of Fukaya--Oh--Ohta--Ono. We also study several structures of our categories for sufficiently Weinstein manifolds and properties known in the classical Tamarkin category; intersection points estimates, interleaving distances, energy stability with respect to Guillermou--Kashiwara--Schapira autoequivalence, and the completeness of the distance. We conjecture that our category is equivalent to a Fukaya category defined over the Novikov ring.