[Paper Review] Polyfold and SFT Notes II: Local-Local M-Polyfold Constructions

Joel W. Fish, Helmut Hofer|arXiv (Cornell University)|Aug 15, 2018

Geometric and Algebraic Topology18 references3 citations

TL;DR

This paper develops local-local M-polyfold constructions for symplectic field theory (SFT), focusing on sc-smoothness in nodal and periodic orbit settings. It establishes the smoothness of key maps like the $ abla$-map and $ ilde{B}_T$-map, proving that $ ilde{B}_T$ is a smooth diffeomorphism with derivative 1 at 0, enabling controlled gluing and stretching constructions in polyfold theory.

ABSTRACT

Some early chapters of the upcoming book "Polyfold Constructions: Tools, Techniques, and Functors"

Motivation & Objective

• To develop a foundational framework for constructing M-polyfolds and strong bundles in symplectic field theory using local-local building blocks.
• To establish sc-smoothness criteria for maps arising in nodal Riemann surface and periodic orbit constructions.
• To prove that key transition maps, such as $ ilde{B}_T$, are smooth diffeomorphisms with derivative 1 at 0, enabling consistent gluing and stretching in polyfold constructions.
• To provide a toolkit for recycling analysis in a checkable, systematic way via the 'LEGO'-type system of polyfold constructions.

Proposed method

• Uses the fundamental lemma to characterize sc-smoothness via $C^1$-regularity and extension of derivatives to lower levels of the sc-Banach space hierarchy.
• Applies the $ abla$-map and $ ilde{B}_T$-map constructions to model stretching near periodic orbits and nodal degenerations.
• Defines and analyzes the map $ ilde{B}_T(x) = rac{x}{1 + x \ abla(x)}$, where $ abla(x) = \ln(T + D e^{-1/x})$, proving its smoothness and derivative 1 at 0.
• Constructs the map $B(r,c) = \varphi^{-1}(\varphi(r) + c)$ on $\Omega$, showing smoothness and vanishing higher-order derivatives at $r=0$, crucial for local models.
• Uses the composition $\mathsf{C}_T(x,c) = \mathsf{B}_T \circ B(x, c/T)$ to prove smoothness of the stretched map $\mathsf{C}_T$ on $\Omega_T$, with $\partial_x \mathsf{C}_T(0,c) = 1$.
• Applies results from [38] and DM-theory to ensure compatibility and universality of local constructions in the polyfold framework.

Experimental results

Research questions

• RQ1 How can sc-smoothness be characterized in terms of $C^1$-regularity and derivative extensions across levels of the sc-Banach space hierarchy?
• RQ2 What conditions ensure that a map defined via a logarithmic scaling, such as $\tilde{B}_T(x) = \varphi^{-1}(T \cdot \varphi(x))$, is smooth at $x=0$?
• RQ3 How can the behavior of maps near $x=0$ be controlled to ensure $\partial_x \mathsf{C}_T(0,c) = 1$ in the context of stretching near periodic orbits?
• RQ4 What is the role of the $\nabla$-map in ensuring smoothness of transition maps in nodal and periodic orbit constructions?
• RQ5 How can local-local M-polyfold constructions be composed into global Fredholm theory frameworks via the 'LEGO'-type system?

Key findings

• The map $\tilde{B}_T(x) = \varphi^{-1}(T \cdot \varphi(x))$ is a smooth diffeomorphism on $[0,1)$ with $\tilde{B}_T(0) = 0$ and $\tilde{B}_T'(0) = 1$, ensuring compatibility with gluing structures.
• The function $g(x) = x \cdot \ln(T + D e^{-1/x})$ is smooth at $x=0$ with $g(0) = 0$, $g'(0) = \ln(T)$, and $g^{(n)}(0) = 0$ for $n \geq 2$, which underpins the smoothness of $\tilde{B}_T$.
• The map $B(r,c) = \varphi^{-1}(\varphi(r) + c)$ is smooth on $\Omega \subset [0,1) \times \mathbb{R}$, with $\partial_r B(0,c) = 1$, $\partial_c B(0,c) = 0$, and all higher-order derivatives at $r=0$ vanishing.
• The composition $\mathsf{C}_T(x,c) = \mathsf{B}_T \circ B(x, c/T)$ is smooth on $\Omega_T$, and satisfies $\partial_x \mathsf{C}_T(0,c) = 1$ for all $c$, which is essential for consistent stretching near periodic orbits.
• The map $\mathsf{C}_T$ provides a smooth transition in the context of M-polyfold constructions, ensuring that the derivative at $x=0$ remains 1 under rescaling, which is critical for Fredholm theory.
• The results confirm that the local-local constructions are compatible with the global polyfold framework, enabling the systematic building of M-polyfolds and strong bundles for SFT.