A Microlocal Category Associated to a Symplectic Manifold

Abstract

For a symplectic manifod subject to certain topological conditions, a category enriched in A ∞ modules over the Novikov ring is constructed. The construction is based on the category of modules over Fedosov's deformation quantization algebra that have an additional structure, namely, an action of the fundamental groupoid up to inner automorphisms. Based in large part on the ideas of Bressler-Soibelman, Feigin, and Karabegov, it is motivated by the theory of Lagrangian distributions and and is related to other microlocal constructions of a category starting from a symplectic manifold, such as those due to Nadler-Zaslow and Tamarkin. In the case when our manifold is a flat two-torus, the answer is very close to both the microlocal category of Tamarkin and the Fukaya category as computed by Polishchuk and Zaslow.

Twisted complexes and simplicial homotopies

“…People who are familiar with A ∞ -categories may find that the definition of twisted complexes is similar to the construction of A ∞ -functors. Actually this is the approach taken by [Tsy18] and [AØ18]. In this paper we satisfy ourselves with Definition 1.3 and refer interested readers to [Tsy18, Section 16] and [AØ18, Section 4] for the A ∞ -approach.…”

“…In the late 1970's Toledo and Tong [TT78] introduced twisted complexes as a way to get their hands on perfect complexes of sheaves on a complex manifold. Twisted complex, which consists of locally defined complexes together with higher transition functions, soon plays an important role in the study of complex geometry, algebraic geometry, as well as dg-categories and A ∞ -categories, see [OTT81b], [OTT81a], [OTT85], [BK91], [Wei16], [BHW17], [Tsy18], and [AØ18].…”

The nonequivariant coherent-constructible correspondence for toric surfaces

“…In general, Kontsevich's conjecture are pursued by Nadler and Ganatra-Pardon-Shende [Nad14,GPS]. Along this line, Tamarkin and Tsygan are trying to construct microlocal categories expected to be equivalent to Fukaya categories of closed symplectic manifolds [Tam15,Tsy15].…”