In mathematics, a Higgs bundle is a pair consisting of a holomorphic vector bundle E and a Higgs field , a holomorphic 1-form taking values in the bundle of endomorphisms of E such that . Such pairs were introduced by Nigel Hitchin (1987),[1] who named the field after Peter Higgs because of an analogy with Higgs bosons. The term 'Higgs bundle', and the condition (which is vacuous in Hitchin's original set-up on Riemann surfaces) was introduced later by Carlos Simpson.[2]
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 says that, under suitable stability conditions, the category of flat holomorphic connections on a smooth projective complex algebraic variety, the category of representations of the fundamental group of the variety, and the category of Higgs bundles over this variety are actually equivalent. Therefore, one can deduce results about gauge theory with flat connections by working with the simpler Higgs bundles.
^Hitchin, Nigel (1987). "The self-duality equations on a Riemann surface". London Mathematical Society. 55 (1): 59–126. doi:10.1112/plms/s3-55.1.59. Retrieved 10 November 2022.
^Simpson, Carlos (1992). "Higgs bundles and local systems" (PDF). Publications Mathématiques de l'IHÉS. 75 (1): 5–95. doi:10.1007/BF02699491. S2CID 56417181. Retrieved 10 November 2022.
mathematics, a Higgsbundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle...
X} , called the Higgs field. Additionally, the Higgs field must satisfy Φ ∧ Φ = 0 {\displaystyle \Phi \wedge \Phi =0} . A Higgsbundle is (semi-)stable...
physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all...
sector Higgs bundle, a vector bundleHiggs prime, a class of prime numbers Alan Higgs (died 1979), English businessman and philanthropist Alf Higgs (born 1904)...
bundles generalises to these objects by requiring the reductions of structure group are compatible with the Higgs field of the principal Higgsbundle...
notion of system of Hodge bundles, which can be seen as a special case of the higher dimensional generalization of Higgsbundles introduced earlier by Nigel...
canonical bundle, so a pair ( F , Φ ) {\displaystyle (F,\Phi )} called a Hitchin pair or Higgsbundle, defines a point in the cotangent bundle. Taking Tr...
quotient (of Hitchin, Anders Karlhede, Ulf Lindström and Martin Roček); Higgsbundles, which arise as solutions to the Hitchin equations, a 2-dimensional...
Examples of hyper-Kähler manifolds include ALE spaces, K3 surfaces, Higgsbundle moduli spaces, quiver varieties, and many other moduli spaces arising...
(mathematics), a connection over a principal bundle G with a section (the Higgs field) of the associated adjoint bundle Monopole, the first term in a multipole...
Spaces of Framed G–HiggsBundles and Symplectic Geometry'. With Indranil Biswas and Ana Peón Nieto. July 2019. Paper 'On HiggsBundles on Nodal Curves'...
the quotient bundle P / H → X. These sections are treated as classical Higgs fields. The idea of the pseudo-Riemannian metric as a Higgs field appeared...
these moduli spaces and the moduli spaces of principal bundles, vector bundles, Higgsbundles, and geometric structures on topological spaces, given generally...
Yunfeng (2019). "On the construction of moduli stack of projective Higgsbundles over surfaces". arXiv:1911.00250 [math.AG]. Examples of Stacks Notes...
geometric structures, including moduli spaces of vortices, moduli spaces of Higgsbundles and their relation to character varieties of surface groups and higher...
gauge theory. Only in the late 1980s, when QFT was reformulated in fiber bundle language for application to problems in the topology of low-dimensional...
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