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Presymplectic geometry and Liouville sectors with corners and its monoidality

Yong-Geun Oh

TL;DR

This work provides an intrinsic presymplectic characterization of Liouville sectors, including those with corners, by analyzing the boundary characteristic foliation and its leaf space, and establishes a canonical smoothing procedure to obtain smooth sections. It shows that Liouville σ-sectors with corners form a monoid under product and identifies their automorphism group, enabling a natural notion of Liouville automorphisms and bundles of sectors. The results connect presymplectic boundary data to Liouville geometry at infinity, prove equivalences with GPS’s sector framework, and supply canonical splitting data via integrable-system techniques. These constructions yield a robust framework for monoidal structures in wrapped Fukaya categories and pave the way for Künneth-type functors and descent results in the sectorial setting, with concrete answers to questions raised by GPS about optimality and convexity at infinity.

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 $σ$-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 which provides a natural framework of considering the Künneth-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].

Presymplectic geometry and Liouville sectors with corners and its monoidality

TL;DR

This work provides an intrinsic presymplectic characterization of Liouville sectors, including those with corners, by analyzing the boundary characteristic foliation and its leaf space, and establishes a canonical smoothing procedure to obtain smooth sections. It shows that Liouville σ-sectors with corners form a monoid under product and identifies their automorphism group, enabling a natural notion of Liouville automorphisms and bundles of sectors. The results connect presymplectic boundary data to Liouville geometry at infinity, prove equivalences with GPS’s sector framework, and supply canonical splitting data via integrable-system techniques. These constructions yield a robust framework for monoidal structures in wrapped Fukaya categories and pave the way for Künneth-type functors and descent results in the sectorial setting, with concrete answers to questions raised by GPS about optimality and convexity at infinity.

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 -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 -sectors with corners canonically forms a monoid which provides a natural framework of considering the Künneth-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].

Paper Structure

This paper contains 31 sections, 52 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Presymplectic characterization of Liouville sectors
  3. Interpolation of presymplectic and Liouville geometry at infinity
  4. Transverse coisotropic collections and Liouville $\sigma$-sectors with corners
  5. Automorphism group of Liouville $\sigma$-sectors with corners
  6. Liouville manifolds with boundary and orientations
  7. Null foliation
  8. Convexity of $H_\infty =\partial_\infty M \cap H$ and contact vector field
  9. Sectional characterization of sectorial hypersurfaces
  10. The leaf space is a topological manifold
  11. Smooth structure on the leaf space
  12. Construction of a smooth section: smoothing
  13. Symplectic structure on the leaf space
  14. Induced Liouville structure on the leaf space
  15. Geometry of transverse coisotropic collections
  16. ...and 16 more sections

Key Result

Theorem 1.5

Under the above definition of Liouville $\sigma$-sector, the following holds:

Theorems & Definitions (116)

  • Remark 1.1
  • Definition 1.3: Liouville $\sigma$-sectors
  • Remark 1.4
  • Theorem 1.5: Theorem \ref{['thm:equivalence-H']} for $H = \partial M$
  • Remark 1.6
  • Proposition 1.7
  • Theorem 1.8: Theorem \ref{['thm:GPS-question']}; Question 2.6 gps
  • Proposition 1.9
  • Theorem 1.10
  • Definition 1.11: $\sigma$-sectorial hypersurface
  • ...and 106 more