A landscape of contact manifolds via rational SFT

TL;DR

The paper constructs a RSFT-derived hierarchy ${ m H_{cx}}$ by passing from the exact symplectic cobordism category ${rak Con}$ through a BL$_inite$–algebra framework, yielding three invariant levels: algebraic planar torsion ${ m APT}$, planarity ${ m P}$, and order of semi-dilation ${ m SD}$. Finite planarity and nontrivial SD are shown to obstruct exact cobordisms and provide new obstructions and classifications for contact manifolds, including iterated planar open books, affine/divisor-complement boundaries, and links of singularities. The approach unifies RSFT with a tree-graph calculus and virtual techniques (Pardon's VFC/polyfolds) to define counts of genus-0 holomorphic curves and associated algebraic structures, while establishing functorial behavior and surjectivity of the invariants. Concrete computations are given for affine varieties, Brieskorn links, and quotient singularities, with implications for Weinstein conjecture and embeddings; the framework also connects to uniruledness and Gromov–Witten data via relative invariants. Overall, the work provides a computable, augmentation-aware RSFT framework to gauge contact complexity and cobordism relations across dimensions, illuminating when exact cobordisms can or cannot exist and highlighting rich geometric structures behind RSFT invariants.

Abstract

We define a hierarchy functor from the exact symplectic cobordism category to a totally ordered set from a $BL_\infty$ (Bi-Lie) formalism of the rational symplectic field theory (RSFT). The hierarchy functor consists of three levels of structures, namely algebraic planar torsion, order of semi-dilation and planarity, all taking values in $\mathbb{N}\cup \{\infty\}$, where algebraic planar torsion can be understood as the analogue of Latschev-Wendl's algebraic torsion in the context of RSFT. The hierarchy functor is well-defined through a partial construction of RSFT and is within the scope of established virtual techniques. We develop computation tools for those functors and prove all three of them are surjective. In particular, the planarity functor is surjective in all dimension $\ge 3$. Then we use the hierarchy functor to study the existence of exact cobordisms. We discuss examples including iterated planar open books, spinal open books, affine varieties with uniruled compactification and links of singularities.

Figures (16)

• Figure 1: forest of labeled trees
• Figure 2: Gluing forests $\Leftrightarrow$ applying operations
• Figure 3: If we switch the output order, it still represents the same glued forest as Figure \ref{['fig:gluing']}.
• Figure 4: A component of $\widehat{\phi}$
• Figure 5: An admissible partition

Theorems & Definitions (207)

• Theorem 1
• Theorem 2
• Theorem 3: Theorem \ref{['thm:main']} + Corollary \ref{['cor:product']}
• Corollary 4
• Corollary 5: Corollary \ref{['cor:noIP']}
• Theorem 6: Theorem \ref{['thm:CP']}
• Remark 1.1
• Theorem 7: Theorem \ref{['thm:embedding']}
• Remark 1.2
• Theorem 8: Theorem \ref{['thm:generalization']}