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Polygons in Minkowski three space and parabolic Higgs bundles of rank two on CP^1

Indranil Biswas, Carlos Florentino, Leonor Godinho, Alessia Mandini

Abstract

Consider the moduli space of parabolic Higgs bundles (E,Φ) of rank two on CP^1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by (E,Φ)\mapsto (E,-Φ). We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.

Polygons in Minkowski three space and parabolic Higgs bundles of rank two on CP^1

Abstract

Consider the moduli space of parabolic Higgs bundles (E,Φ) of rank two on CP^1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by (E,Φ)\mapsto (E,-Φ). We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.

Paper Structure

This paper contains 8 sections, 12 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Hyperpolygon spaces
  3. A circle action
  4. Spaces of Parabolic Higgs bundles
  5. An involution
  6. Polygons in Minkowski $3$-space
  7. Back to the involution
  8. An Example

Key Result

Theorem \oldthetheorem

The fixed-point set of the involution in eq:0.1 of the space of parabolic Higgs bundles $\mathcal{H}(\beta)$ is with $\alpha_i=\beta_2(x_i)-\beta_1(x_i)$, where $\mathcal{M}_{\beta,2,0}$ is the space of rank two degree zero parabolic vector bundles over $\mathbb C{\mathbb P}^1$, and where $\mathcal{Z}_S\,\subset\, \mathcal{H}(\beta)$ is formed by parabolic Higgs bundles $\mathbf{E}\,=\,(E,\Phi)\i

Theorems & Definitions (17)

  • Theorem \oldthetheorem
  • Remark 1.1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: K2
  • Remark 2.1
  • Proposition \oldthetheorem: K2
  • Theorem \oldthetheorem: K2
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • ...and 7 more