Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary Depth and Deformations of Symplectic Cohomology

Yoel Groman

TL;DR

This work develops a formal deformation theory relating ambient symplectic cohomology $SC^*_M(D)$, defined over the Novikov field, to intrinsic symplectic cohomology $SC^*_{ heta}(D)$, defined from a local Liouville primitive. A central construction is an $R$-filtration on $SC^*_M(D)$ whose associated graded is equivalent to the intrinsic theory, yielding a locality spectral sequence that tracks how embedding data deform the local invariant. A quantitative deformation theory is introduced via boundary depth $eta(D)$, with $eta(D)< ext{ħ}$ guaranteeing controlled deformations and enabling homological perturbation techniques. The paper also defines the $ au$ invariant, capturing embedding torsion and flux data, and proves rigidity results for BV-algebras and related structures, which underpin reconstruction results for SYZ mirrors in holomorphic and tropical settings. Applications to SYZ mirror symmetry are emphasized: under spectral sequence collapse and undeformedness, one can recover affinoid analytic mirrors from intrinsic local models, illuminating how local pieces assemble into a global non-Archimedean mirror and enabling local-to-global mirror constructions.

Abstract

We study the relation between two versions of symplectic cohomology associated to a Liouville domain $D$ embedded in a symplectic manifold $M$: the ambient version $SC^*_M(D)$ defined over the Novikov field and depending on the embedding, and the intrinsic version $SC^*_θ(D)$ depending on the choice of a local Liouville form and defined over the ground field. We show that when $D$ has sufficiently small boundary depth, the ambient version can be viewed as a deformation of the intrinsic one. This is achieved by constructing a filtration whose associated graded reproduces the intrinsic theory, and developing quantitative tools to control the deformation. We apply our results to constructing local pieces of the SYZ mirror.

Boundary Depth and Deformations of Symplectic Cohomology

TL;DR

This work develops a formal deformation theory relating ambient symplectic cohomology , defined over the Novikov field, to intrinsic symplectic cohomology , defined from a local Liouville primitive. A central construction is an -filtration on whose associated graded is equivalent to the intrinsic theory, yielding a locality spectral sequence that tracks how embedding data deform the local invariant. A quantitative deformation theory is introduced via boundary depth , with guaranteeing controlled deformations and enabling homological perturbation techniques. The paper also defines the invariant, capturing embedding torsion and flux data, and proves rigidity results for BV-algebras and related structures, which underpin reconstruction results for SYZ mirrors in holomorphic and tropical settings. Applications to SYZ mirror symmetry are emphasized: under spectral sequence collapse and undeformedness, one can recover affinoid analytic mirrors from intrinsic local models, illuminating how local pieces assemble into a global non-Archimedean mirror and enabling local-to-global mirror constructions.

Abstract

We study the relation between two versions of symplectic cohomology associated to a Liouville domain embedded in a symplectic manifold : the ambient version defined over the Novikov field and depending on the embedding, and the intrinsic version depending on the choice of a local Liouville form and defined over the ground field. We show that when has sufficiently small boundary depth, the ambient version can be viewed as a deformation of the intrinsic one. This is achieved by constructing a filtration whose associated graded reproduces the intrinsic theory, and developing quantitative tools to control the deformation. We apply our results to constructing local pieces of the SYZ mirror.
Paper Structure (74 sections, 85 theorems, 173 equations, 4 figures)

This paper contains 74 sections, 85 theorems, 173 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Formal deformation results
  3. Theorems A
  4. Proof strategy of Theorems A
  5. The locality spectral sequence
  6. A first application
  7. Quantitative deformation theory
  8. Deformation of the attending structures
  9. Remarks about the proof of Theorems \ref{['mainTmA']}-\ref{['tmProductDeformation']}
  10. Neck stretching and deformation
  11. Flux and the $\tau$ invariant
  12. Collapse in SYZ Mirror Symmetry
  13. Examples
  14. The hypothesis $\tau(L)=\infty$ in general
  15. The local to global method in Floer theory
  16. ...and 59 more sections

Key Result

Theorem A1

Fix a locally defined Liouville primitive $\theta$ on $D$. There exists an ${\mathbb R}$-filtration an $\hbar=\hbar(M,D,\theta)>0$, and a weak homotopy equivalence defined over ${\mathbb Z}$ Here $\mathop{\mathrm{gr}}\nolimits_{\hbar}(SC^*_M(K))$ denotes the subquotient of elements with valuation $\geq 0$ modulo those with valuation $\geq\hbar$The notation $\Lambda_{[0,\hbar)}$ clashes with the n

Figures (4)

  • Figure 1: The local-to-global method.
  • Figure 2: The spectral sequence from intrinsic to ambient symplectic cohomology.
  • Figure 3: The relation between ambient and intrinsic symplectic cohomology. Theorem A: The associated graded is weakly homotopy equivalent to the intrinsic complex tensored with $\Lambda_{[0,\hbar)}$. Theorem B: When boundary depth $\beta(D) < \hbar$ the arrow on the right lifts to a homotopy equivalence of $SC^*_M(D)$ with field coefficients to the deformed complex.
  • Figure 4: The trichotomy of Floer solutions for S-shaped Hamiltonians.

Theorems & Definitions (235)

  • Theorem A1
  • Theorem A2
  • Remark 1.1
  • Theorem A3
  • Remark 1.2
  • Theorem A4
  • Example 1.3
  • Theorem 1.4: Reconstruction for regular fibers
  • Remark 1.5
  • Proposition 1.6
  • ...and 225 more