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Morse theory and Seiberg-Witten moduli spaces of 3-dimensional cobordisms, I

Yi-Jen Lee

TL;DR

This work studies moduli spaces of Seiberg-Witten equations on a 3D cobordism with cylindrical ends under large perturbations, $\mu_r = r\,df + w$, where $f$ is a harmonic Morse function and $r\ge 1$. The authors establish a framework in which admissible configurations exist and, for large $r$, the moduli space $\mathscr{Z}_{r,w}(Y,{\mathfrak{s}};f)$ is either empty when the Spin$^c$ degree $d<0$ or a smooth orientable manifold of dimension $d_- + d_+$, with an end-point map to vortex moduli spaces on the ends that is a Lagrangian immersion. They develop key analytic results: existence and genericity of admissible $f$, the vortex correspondence on the ends, and important a priori estimates plus exponential decay for SW solutions along the ends. Together these results lay groundwork for a large-perturbation Seiberg-Witten Floer theory on cobordisms and connect gauge-theoretic invariants to symplectic/monopole frameworks, informing potential Atiyah-Floer-type correspondences in dimension three.

Abstract

Motivated by a variant of Atiyah-Floer conjecture proposed in \cite{L2} and its potential generalizations, we study in this article and its sequel as a first step properties of moduli spaces of Seiberg-Witten equations on a 3-dimensional cobordism with cylindrical ends (CCE) \(Y\), perturbed by closed 2-forms of the form \(r*d\ff+w\), where \(r\geq 1\), where \(\ff\) is a harmonic Morse function with certain linear growth at the ends of \(Y\), and \(w\) is a certain closed 2-form.

Morse theory and Seiberg-Witten moduli spaces of 3-dimensional cobordisms, I

TL;DR

This work studies moduli spaces of Seiberg-Witten equations on a 3D cobordism with cylindrical ends under large perturbations, , where is a harmonic Morse function and . The authors establish a framework in which admissible configurations exist and, for large , the moduli space is either empty when the Spin degree or a smooth orientable manifold of dimension , with an end-point map to vortex moduli spaces on the ends that is a Lagrangian immersion. They develop key analytic results: existence and genericity of admissible , the vortex correspondence on the ends, and important a priori estimates plus exponential decay for SW solutions along the ends. Together these results lay groundwork for a large-perturbation Seiberg-Witten Floer theory on cobordisms and connect gauge-theoretic invariants to symplectic/monopole frameworks, informing potential Atiyah-Floer-type correspondences in dimension three.

Abstract

Motivated by a variant of Atiyah-Floer conjecture proposed in \cite{L2} and its potential generalizations, we study in this article and its sequel as a first step properties of moduli spaces of Seiberg-Witten equations on a 3-dimensional cobordism with cylindrical ends (CCE) , perturbed by closed 2-forms of the form , where , where is a harmonic Morse function with certain linear growth at the ends of , and is a certain closed 2-form.
Paper Structure (2 sections, 1 theorem, 7 equations)

This paper contains 2 sections, 1 theorem, 7 equations.

Table of Contents

  1. Introduction
  2. Notation and Conventions

Key Result

Theorem 1.1

Let $Y\colon \Sigma _-\to \Sigma _+$ be a CCE, and $\hbox{\itshapef}$ is an admissible function. Fix a $\mathop{\mathrm{Spin}}\nolimits^c$ structure ${\mathfrak{s}}$ on $Y$ with degree $d$, and let $d_\pm:=d+\frac{\chi(\Sigma _{\rm min})-\chi (\Sigma _\pm)}{2}$. Let $\hat{{\mathscr{W}}}$ be the spac which is a Lagrangian immersion. In the above, ${\mathscr{V}}_{r,d}(\Sigma )$ denotes the moduli sp

Theorems & Definitions (3)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1