Higgs bundle

In mathematics , Higgs bundles are an aid in the representation theory of surface groups and fundamental groups of complex manifolds. They were introduced by Nigel Hitchin and named after Peter Higgs because of the analogy to Higgs bosons .

definition

A Higgs bundle is a pair consisting of a holomorphic vector bundle over a Riemann surface and a Higgs field, i.e. H. a -valent holomorphic 1-form . ${\displaystyle (V,\phi )}$ ${\displaystyle V}$ ${\displaystyle \Sigma }$ ${\displaystyle \operatorname {End} (V)}$ ${\displaystyle \phi }$

Stability, polystability

A Higgs bundle is called stable if the inequality for all -invariant holomorphic sub-bundles ${\displaystyle (V,\phi )}$ ${\displaystyle \phi }$ ${\displaystyle W\subset V}$

${\displaystyle {\frac {\deg(W)}{\operatorname {rank} (W)}}<{\frac {\deg(V)}{\operatorname {rank} (V)}}}$

applies. Here denotes the degree of a vector bundle and its rank, i.e. the dimension of its fibers . (Note that the inequality should only apply to -invariant sub-bundles, i.e. a stable Higgs bundle does not necessarily have to be a stable vector bundle .) ${\displaystyle \operatorname {deg} }$ ${\displaystyle \operatorname {rank} }$ ${\displaystyle \phi }$

A Higgs bundle is called polystable if it is a direct sum ${\displaystyle (V,\phi )}$

${\displaystyle (V,\phi )=\bigoplus _{i=1}^{l}(V_{i},\phi _{i})}$

stable Higgs bundle with

${\displaystyle {\frac {\deg(V_{i})}{\operatorname {rank} (V_{i})}}={\frac {\deg(V)}{\operatorname {rank} (V)}}}$

for is. ${\displaystyle i=1,\ldots ,l}$

Representation theory

Building on the results of Corlette and Donaldson, Hitchin and Simpson proved the following equivalences for Riemann surfaces : ${\displaystyle \Sigma }$

${\displaystyle \left\{{\begin{array}{c}{\mbox{Stabile}}\ {\mbox{Higgs-Bündel}}\\(V,\Phi )\ {\mbox{über}}\ \Sigma \end{array}}\right\}=\left\{{\begin{array}{c}{\mbox{Irreduzible}}\ {\mbox{Darstellungen}}\\\rho \colon \pi _{1}\Sigma \to GL(n,\mathbb {C} )\end{array}}\right\}}$

${\displaystyle \left\{{\begin{array}{c}{\mbox{Polystabile}}\ {\mbox{Higgs-Bündel}}\\(V,\Phi )\ {\mbox{über}}\ \Sigma \end{array}}\right\}=\left\{{\begin{array}{c}{\mbox{Reduktive}}\ {\mbox{Darstellungen}}\\\rho \colon \pi _{1}\Sigma \to GL(n,\mathbb {C} )\end{array}}\right\}}$

Higher-dimensional generalization

A Higgs bundle is defined for higher-dimensional complex manifolds as a pair of a holomorphic vector bundle over and a -valent holomorphic 1-form that satisfies the equation . ${\displaystyle X}$ ${\displaystyle (V,\phi )}$ ${\displaystyle V}$ ${\displaystyle X}$ ${\displaystyle \operatorname {End} (V)}$ ${\displaystyle \phi }$ ${\displaystyle \phi \wedge \phi =0}$

(In the case of Riemann surfaces, this equation is trivially fulfilled.)

literature

• Corlette, Kevin: Flat G-bundles with canonical metrics. J. Differential Geom. 28 (1988) no. 3, 361-382.
• Donaldson, Simon: Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3) 55 (1987) no. 1, 127-131.
• Hitchin, Nigel: The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987) no. 1, 59-126.
• Simpson, Carlos: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1 (1988) no. 4, 867-918.

Web links

• Joseph Le Potier: Fibers de Higgs et systèmes locaux (Séminaire Bourbaki)
• Steven B. Bradlow, Oscar García-Prada, Peter B. Gothen: WHAT IS ... a Higgs bundle? (Notices of the AMS)
• Richard A. Wentworth: Higgs bundles and local systems
• Peter B. Gothen: Surface group representations and Higgs bundles
• William M. Goldman: Higgs bundles and geometric structures on surfaces
• Jan Swoboda: Higgsbündel and display varieties (Yearbook of the Max Planck Society )
• Oliver Guichard: An introduction to the differential geometry of flat bundles and of Higgs bundles
• Laura Schaposnik: Higgs bundles - recent applications