Presymplectic characterization of Liouville sectors with corners, and its monoidality

We provide a presymplectic characterization of Liouville sectors introduced by Ganatra–Pardon–Shende in [10, 12] in terms of the characteristic foliation of the boundary, which we call Liouville σ-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 σ-sectors with corners canonically forms a monoid that provides a natural framework for considering the Künneth-type functors in the wrapped Fukaya category. We identify its automorphism group that enables one to give a natural definition of bundles of Liouville sectors . As a byproduct, we affirmatively answer a question raised in [10, Question 2.6], which asks about the optimality of their definition of Liouville sectors in [10].