Para-complex geometry and cyclic Higgs bundles
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913716334755840 |
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| author | Rungi, Nicholas Tamburelli, Andrea |
| author_facet | Rungi, Nicholas Tamburelli, Andrea |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Para-complex geometry and cyclic Higgs bundles Rungi, Nicholas Tamburelli, Andrea Differential Geometry Geometric Topology 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. |
| title | Para-complex geometry and cyclic Higgs bundles |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2503.01615 |