Higgs bundle
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In mathematics, a Higgs bundle is a pair (E,\varphi) consisting of a
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
''E'' and a
Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
\varphi, a holomorphic 1-form taking values in the bundle of endomorphisms of ''E'' such that \varphi \wedge \varphi=0. Such pairs were introduced by , who named the field \varphi after
Peter Higgs Peter Ware Higgs (born 29 May 1929) is a British theoretical physicist, Emeritus Professor in the University of Edinburgh,Griggs, Jessica (Summer 2008The Missing Piece ''Edit'' the University of Edinburgh Alumni Magazine, p. 17 and Nobel Prize ...
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 surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s) was introduced later by
Carlos Simpson Carlos Tschudi Simpson (born 30 June 1962) is an American mathematician, specializing in algebraic geometry. Simpson received his Ph.D. in 1987 from Harvard University, where he was supervised by Wilfried Schmid; his thesis was titled ''Systems of ...
. 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 In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundame ...
says that, under suitable stability conditions, the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
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.


History

Higgs bundles were first introduced by Hitchin in 1987, for the specific case where the holomorphic vector bundle ''E'' is over a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
. Further, Hitchin's paper mostly discusses the case where the vector bundle is rank 2 (that is, the fiber is a 2-dimensional vector space). The rank 2 vector bundle arises as the solution space to
Hitchin's equations In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written ...
for a principal
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
bundle. The theory on Riemann surfaces was generalized by Carlos Simpson to the case where the base manifold is compact and Kähler. Restricting to the dimension one case recovers Hitchin's theory.


Stability of a Higgs bundle

Of particular interest in the theory of Higgs bundles is the notion of a stable Higgs bundle. To do so, \varphi-invariant subbundles must first be defined. In Hitchin's original discussion, a rank-1 subbundle labelled ''L'' is \varphi-invariant if \varphi(L) \subset L \otimes K with K the canonical bundle over the Riemann surface ''M''. Then a Higgs bundle (E, \varphi) is stable if, for each \varphi invariant subbundle L of E, \text L < \frac\text(\wedge^2 E), with \text being the usual notion of degree for a complex vector bundle over a Riemann surface.


See also

*
Hitchin system In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the th ...


References

* * Vector bundles Complex manifolds {{topology-stub