Para-complex geometry and cyclic Higgs bundles

2025/03/03 by Nicholas Rungi, Andrea Tamburelli, Rungi, Nicholas +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2503.01615

openalex publication_date 2025/03/03 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

We introduce para-complex and pseudo-Riemannian geometric methods for the study of representations of surface groups in SL(2m+1,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 SL(2m+1,ℝ)-Higgs bundles (not necessarily in the Hitchin component) and a special class of surfaces, which we call isotropic P-alternating, in the para-complex hyperbolic space 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.

Cited by

• Equiaffine immersions and pseudo-Riemannian space forms
• On Geometric Structures in the Einstein Universe for SO₀(p,p+1)-Hitchin Representations

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