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The moduli space of solutions to the Extended Bogomolny equations on $Σ\times \mathbb{R_+}$

Panagiotis Dimakis

Abstract

We study moduli spaces of solutions to the extended Bogomolny equations on $Σ\times \mathbb{R_{+,y}}$ with gauge group $\operatorname{SL}(2,\mathbb{C})$ satisfying the generalized Nahm pole boundary condition as $y\to 0$ and limiting to complex flat connections as $y\to \infty$. Refining the Kobayashi-Hitchin correspondence of \cite{MH2}, we identify these moduli spaces with certain holomorphic lagrangian sub-manifolds inside the moduli space of Higgs bundles.

The moduli space of solutions to the Extended Bogomolny equations on $Σ\times \mathbb{R_+}$

Abstract

We study moduli spaces of solutions to the extended Bogomolny equations on with gauge group satisfying the generalized Nahm pole boundary condition as and limiting to complex flat connections as . Refining the Kobayashi-Hitchin correspondence of \cite{MH2}, we identify these moduli spaces with certain holomorphic lagrangian sub-manifolds inside the moduli space of Higgs bundles.
Paper Structure (14 sections, 9 theorems, 60 equations)

This paper contains 14 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The geometry of Hitchin's moduli space
  4. The Higgs bundle moduli space
  5. The $\mathbb{C}^{\star}$-action on $\mathcal{M}_H$
  6. The extended Bogomolny equations
  7. Hermitian geometry
  8. Boundary conditions
  9. The generalized Nahm pole boundary condition
  10. The holomorphic line bundle
  11. Moduli structure of solutions to EBE on $\Sigma\times\mathbb{R}_+$
  12. The tangent space
  13. Holomorphic Lagrangians
  14. Further directions

Key Result

Theorem 1.1

Given an even effective divisor $\mathcal{D}$ denote the moduli space of solutions to the EB equations with divisor $\mathcal{D}$ as $\underline{\operatorname{EBE}}_{\mathcal{D}}$. We define $\mathcal{L} := K(-\mathcal{D})^{1/2}$ where $K$ is the canonical bundle of $\Sigma$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Nonabelian Hodge correspondence
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Lemma 3.1
  • Remark 3.2
  • ...and 13 more