Abstract

ZUNIGA-ROJAS, Ronald A.. A brief survey of higgs bundles. Rev. Mat [online]. 2019, vol.26, n.2, pp.197-214. ISSN 1409-2433.  http://dx.doi.org/10.15517/rmta.v26i2.38315.

Considering a compact Riemann surface of genus greater or equal 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 because in the context of physics and gauge the- ory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle to- gether with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror sym- metry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

Keywords : Higgs bundles; Hodge bundles; moduli spaces; stable triples; vector bundles.

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