Automorphisms of Higgs bundle moduli spaces for real groups

• Autores: Manuel Jesús Pérez García
• Directores de la Tesis: Óscar García Prada (dir. tes.)
• Lectura: En la Universidad Autónoma de Madrid ( España ) en 2018
• Idioma: español
• Número de páginas: 146
• Tribunal Calificador de la Tesis: Andre Gama Oliveira (presid.) , Mario García Fernández (secret.) , Alfonso Zamora (voc.)
• Enlaces
  • Tesis en acceso abierto en: Biblos-e Archivo
• Resumen

  • Let $G$ be a connected real form of a complex semisimple Lie group $G^{\CC}$ with Lie algebra $\gg$. Let $H$ be a maximal compact subgroup of $G$ and let $\theta$ be a Cartan involution of $\gg$ such that it induces a decomposition into $\pm 1$-eigenspaces $\gg=\hh\oplus\mm$, where $\hh$ is the Lie algebra of $H$. A $(G,\theta)$-Higgs bundle over a compact Riemann surface $X$ is a pair consisting on a holomorphic principal $H^{\CC}$-bundle $E$ and a holomorphic section $\varphi$ of $E(\mm^{\CC})\otimes K$ where $E(\mm^{\CC})$ is the bundle associated to $E$ via the isotropy representation $\iota_{\CC}:H^{\CC}\rightarrow\mathrm{GL}(\mm^{\CC})$ and $K$ is the canonical bundle over $X$. Consider the moduli space $\MM(G,\theta)$ of isomorphism classes of polystable $(G,\theta)$-Higgs bundles over $X$. In this thesis we study the action of finite order automorphisms of $\MM(G,\theta)$ defined by combining the multiplication of the Higgs field by an $n$th-root of unity and the action of an element in $(H^1(X,Z(H^{\CC})\cap\mathrm{Ker}(\iota_{\CC}))\rtimes\mathrm{Out}(g,\theta))_n$, where $\mathrm{Out}(G,\theta)$ is the group of outer automorphisms of $G$ that commute with $\theta$. In addition, we describe its fixed points subvarieties and, through non-abelian Hodge correspondence, we translate these results to the moduli space $\mathcal{R}(G,\theta)$ of representations of the fundamental group of $X$ in $G$. A parabolic $(G,\theta)$-Higgs bundles is, roughly speaking, the extension of the notion of $(G,\theta)$-Higgs bundles to Higgs bundles over punctured Riemann surface. In the last chapter of this thesis we study involutions defined on the moduli space of isomorphism classes of polystable parabolic $(G,\theta)$-Higgs bundles over a punctured Riemann surface $X$ and we describe its fixed points subvarieties.