A Brief Survey of Higgs Bundles
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Publication:5132913
DOI10.15517/RMTA.V26I2.38315zbMATH Open1451.14110arXiv1803.08072OpenAlexW2980943218MaRDI QIDQ5132913
Publication date: 12 November 2020
Published in: Revista de Matemática: Teoría y Aplicaciones (Search for Journal in Brave)
Abstract: Considering a compact Riemann surface of genus greater than two, a Higgs~bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin's work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as "Higgs fields" beacuse in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name "Higgs bundle" for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief construction of the moduli space of Higgs bundles.
Full work available at URL: https://arxiv.org/abs/1803.08072
Research exposition (monographs, survey articles) pertaining to algebraic geometry (14-02) Variation of Hodge structures (algebro-geometric aspects) (14D07) Vector bundles on curves and their moduli (14H60) Hypersurfaces and algebraic geometry (14J70)
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