Communications in Analysis and Geometry

Volume 28 (2020)

Number 8

The Second of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

Orthogonal Higgs bundles with singular spectral curves

Pages: 1895 – 1931

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n8.a6

Authors

Steve Bradlow (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)

Lucas Branco

Laura P. Schaposnik (Department of Mathematics, University of Illinois, Chicago, Il., U.S.A.)

Abstract

We examine Higgs bundles for non-compact real forms of $SO(4,\mathbb{C})$ and the isogenous complex group $SL(2,\mathbb{C}) \times SL(2,\mathbb{C})$. This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral data.

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The authors are thankful for financial support from from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 RNMS: GEometric structures And Representation varieties (the GEAR Network) which financed several research visits during which the paper was written. The authors are thankful to J. Heinloth, N. Hitchin, S. Rayan, R. Rubio, and A. Thompson for useful conversations. L.P. Schaposnik is partially supported by the NSF grant DMS-1509693, the NSF CAREER Award DMS-1749013, and by the Alexander von Humboldt Foundation. This material is also based upon work supported by the National Science Foundation under Grant No. DMS- 1440140 while SB and LPS were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2019 semester.

Received 6 September 2019

Accepted 4 August 2020

Published 8 January 2021