Higgs bundle

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

A Higgs bundle can be thought of as a "simplified version" of a flat holomorphic connection on a holomorphic vector bundle, where the derivative is scaled to zero. The nonabelian Hodge correspondence (also known as the Corlette–Simpson correspondence) says that, under suitable stability conditions, the category of flat holomorphic connections on a smooth, projective complex algebraic variety, the category of representations of its fundamental group, and the category of Higgs bundles over this variety are actually equivalent, so one can learn a lot about gauge theory (connections) by working with the simplified objects, Higgs bundles.

See also

• Hitchin's equations
• Hitchin system
• Nonabelian Hodge correspondence

References

• Hitchin, Nigel 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, MR 0887284
• Corlette, Kevin (1988). "Flat G-bundles with canonical metrics". Journal of Differential Geometry. 28 (3): 361–382. doi:10.4310/jdg/1214442469. MR 0965220.
• Simpson, Carlos T. (1992), "Higgs bundles and local systems", Publications Mathématiques de l'IHÉS, 75: 5–95, doi:10.1007/BF02699491, MR 1179076
• 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, MR 2343296