Presymplectic geometry and Liouville sectors with corners and its monoidality

TL;DR

This paper characterizes Liouville sectors with corners using presymplectic geometry, introduces a monoid structure for these sectors, and explores their automorphisms, advancing the understanding of wrapped Fukaya categories.

Contribution

It extends Liouville sector theory to corners via presymplectic geometry and establishes a monoid structure, enabling new functorial and bundle constructions.

Findings

Liouville sectors with corners form a canonical monoid.

Automorphism group of Liouville sectors identified.

Affirms the optimality of the original Liouville sector definition.

Abstract

We provide a presymplectic characterization of Liouville sectors introduced by Ganatra-Pardon-Shende in terms of the characteristic foliation of the boundary, which we call Liouville $\sigma$-sectors. We extend this definition to the case with corners using the presymplectic geometry of null foliations of the coisotropic intersections of transverse coisotropic collection of hypersurfaces which appear in the definition of Liouville sectors with corners. We show that the set of Liouville $\sigma$-sectors with corners canonically forms a monoid which provides a natural framework of considering the K\"unneth-type functors in the wrapped Fukaya category. We identify its automorphism group which enables one to give a natural definition of bundles of Liouville sectors. As a byproduct, we affirmatively answer to a question raised in Question 2.6 in [GPS20], which asks about the optimality of their definition of Liouville sectors [GPS20].