[Paper Review] Counting Trivial Curves in Rational Symplectic Field Theory

Oliver Fabert|arXiv (Cornell University)|Sep 20, 2007

Geometric and Algebraic Topology5 citations

TL;DR

This paper generalizes Taubes' obstruction bundle technique from Gromov-Witten theory to symplectic field theory (SFT), enabling computation of multiple cover contributions in moduli spaces with boundary. It proves that the differential in rational SFT and contact homology is strictly decreasing with respect to the action filtration, resolving a key structural question in the theory.

ABSTRACT

Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov-Witten theory to the case of moduli spaces with boundary. Our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration.

Motivation & Objective

• To extend Taubes' obstruction bundle technique to moduli spaces with boundary in symplectic field theory.
• To compute multiple cover contributions in rational SFT where Fredholm index one curves are inherently non-regular.
• To establish the action filtration behavior of the differential in rational SFT and contact homology.
• To resolve the structure of the differential in SFT by proving it is strictly decreasing under the natural action filtration.

Proposed method

• Generalizes Taubes' obstruction bundle technique to handle holomorphic curves in SFT with boundary, particularly branched covers of orbit cylinders.
• Applies the obstruction bundle method to moduli spaces of curves with Fredholm index one, which are known to be non-regular for any cylindrical almost complex structure.
• Uses the structure of branched covers of orbit cylinders as foundational examples in SFT to build the obstruction theory.
• Imposes a natural action filtration on the chain complex and analyzes the differential's behavior with respect to this filtration.
• Relies on the geometry of holomorphic curves and the properties of cylindrical almost complex structures to derive the filtration result.
• Establishes that the differential strictly decreases action by analyzing contributions from multiple covers via the generalized obstruction bundle.

Experimental results

Research questions

• RQ1 How can Taubes' obstruction bundle method be extended to moduli spaces with boundary in symplectic field theory?
• RQ2 What are the contributions of multiple covers to the differential in rational SFT when index one curves are non-regular?
• RQ3 Does the differential in rational SFT and contact homology strictly decrease with respect to the action filtration?
• RQ4 Can the structure of the differential in SFT be fully characterized using obstruction bundle techniques in the presence of boundary?

Key findings

• The differential in rational symplectic field theory is strictly decreasing with respect to the natural action filtration.
• The obstruction bundle technique is successfully generalized to moduli spaces with boundary, enabling computation of multiple cover contributions in SFT.
• Multiple cover contributions are determined via the generalized obstruction bundle method, even in the absence of regularity for index one curves.
• The result confirms a fundamental structural property of the SFT differential, supporting its role in contact homology.
• The action filtration strictly decreases under the differential, implying no action-preserving terms in the differential.
• The method provides a systematic way to compute contributions from branched covers of orbit cylinders in SFT.