Para-complex geometry and cyclic Higgs bundles

TL;DR

This work develops a para-complex, para-Kähler framework to study surface-group representations into $\mathrm{SL}(2m+1,\mathbb{R})$, establishing a one-to-one correspondence between stable cyclic $\mathrm{SL}(2m+1,\mathbb{R})$-Higgs bundles and isotropic $\mathbf{P}$-alternating surfaces in the para-complex hyperbolic space $\mathbb{H}_{\tau}^{2m}$. It proves that the Gauss map of these surfaces yields a conformal harmonic map into the symmetric space $\mathrm{SL}(2m+1,\mathbb{R})/\mathrm{SO}(2m+1)$, providing a geometric bridge between Higgs-bundle data and minimal maps, including the $m=1$ Labourie–Loftin/Hitchin picture and Guichard–Wienhard domains. A para-complexification of cyclic Higgs bundles is developed, enabling a constructive link from $\mathrm{SL}(2m+1,\mathbb{R})$ representations to equivariant immersed surfaces in $\mathbb{H}_{\tau}^{2m}$, with detailed moduli-space descriptions (stratified by degrees) and aHolomorphic data-driven parameterization in many cases. The paper also situates these results within higher rank Teichmüller theory, showing immersion Hol maps as entities encoding representation data and recovering known structures for $m=1$ while extending to non-Hitchin components for $m\ge 2$, including geometric structures on projectivized tangent bundles that align with Guichard–Wienhard's domains.

Abstract

We introduce para-complex and pseudo-Riemannian geometric methods for the study of representations of surface groups in $\mathrm{SL}(2m+1,\mathbb{R})$. For $m=1$ our techniques allow to recover several known results for Hitchin representations without any reference to convex projective geometry or hyperbolic affine spheres. In particular, we describe analytically the Guichard-Wienhard domain of discontinuity in the flag variety and the corresponding concave foliated flag structure of Nolte-Riestenberg. In higher rank, we obtain a one-to-one correspondence between stable cyclic $\mathrm{SL}(2m+1,\mathbb{R})$-Higgs bundles (not necessarily in the Hitchin component) and a special class of surfaces, which we call isotropic $\mathbf{P}$-alternating, in the para-complex hyperbolic space $\mathbb{H}^{2m}_τ$. As a result, we give a geometric interpretation to the holomorphic differential $q_{2m+1}$ in the Hitchin base in terms of harmonic sequences for immersions in para-complex manifolds.

Theorems & Definitions (83)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 2.1
• Proposition 2.2: RT_bicomplex
• Remark 2.3
• Proposition 2.4