Monopoles and Landau-Ginzburg Models I

Abstract

The endpoint of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega )$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. In the first paper, we exploit the framework of the gauged Landau-Ginzburg models to address two model problems for the (perturbed) Seiberg-Witten moduli spaces on either $\mathbb{C}\times \Sigma$ or ${\mathbb{H}}_{+}^2\times \Sigma$, where $\Sigma$ is any compact Riemann surface of genus $\ge 1$. Our first result states that finite energy solutions to the perturbed equations on $\mathbb{C}\times \Sigma$ are necessarily trivial. The second states that small energy solutions on ${\mathbb{H}}_{+}^2\times \Sigma$ necessarily have energy decaying exponentially in the spatial direction. These results will lead eventually to the compactness theorem in the second paper.

arXiv comment58 pages. v3. The statement of the main results are revised to include higher genus surfaces. v4. Exposition improved as requested by referees; accepted by Comm. Anal. Geom

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Cited by in corpus (3)

• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Monopoles and Landau-Ginzburg Models II: Floer Homology
• Monopoles and Landau-Ginzburg Models III: A Gluing Theorem