Higgs bundle

In mathematics, a Higgs bundle is a pair (E,φ) consisting of a holomorphic vector bundle E and a Higgs field φ, a holomorphic 1-form taking values in End(E) such that φ ∧ φ = 0. Such pairs were introduced by Hitchin (1987), who named the field φ after Peter Higgs because of an analogy with Higgs bosons. The term 'Higgs bundle', and the condition φ ∧ φ = 0 (which is vacuous in Hitchin's original set-up on Riemann surfaces) was introduced later by Simpson.

See also

• Hitchin system
• Simpson correspondence

References

• Gothen, Peter B.; García-Prada, Oscar; Bradlow, Steven B. (2007), "What is... a Higgs bundle?" (PDF), Notices of the American Mathematical Society, 54 (8): 980–981, ISSN 0002-9920, MR 2343296
• Hitchin, N. J. (1987), "The self-duality equations on a Riemann surface", Proceedings of the London Mathematical Society, Third Series, 55 (1): 59–126, CiteSeerX 10.1.1.557.2243, doi:10.1112/plms/s3-55.1.59, ISSN 0024-6115, MR 0887284
• Simpson, Carlos T. (1992), "Higgs bundles and local systems", Publications Mathématiques de l'IHÉS, 75 (75): 5–95, doi:10.1007/BF02699491, ISSN 1618-1913, MR 1179076