Spectral data for G-Higgs bundles

Abstract

We develop a new geometric method of understanding principal G-Higgs bundles through their spectral data, for G a real form of a complex Lie group. In particular, we consider the case of G a split real form, as well as G = SL(2,R), U(p,p), SU(p,p), and Sp(2p,2p). Further, we give some applications of our results, and discuss open questions.

Oxford University DPhil Thesis, 170 pages

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• Mirror symmetry, Langlands duality, and the Hitchin system
• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations
• Quotient groups of the fundamental groups of certain strata of the moduli space of quadratic differentials
• A Special Subgroup of the Surface Braid Group

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• Singular Geometry and Higgs Bundles in String Theory
• An introduction to spectral data for Higgs bundles
• Cameral data for SU(p+1,p)-Higgs bundles
• Lectures on Higgs moduli and abelianisation
• Orthogonal Higgs bundles with singular spectral curves
• Higgs bundles and applications
• Triality for Homogeneous Polynomials
• Higgs bundles, abelian gerbes and cameral data
• Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system
• Higgs bundles over elliptic curves for real groups
• Spectral Data for SU(1,2) Higgs Bundles
• The Norm map on the compactified Jacobian, the Prym stack and Spectral data for G-Higgs pairs