Hitchin system
E886933
The Hitchin system is an influential integrable system in algebraic geometry and mathematical physics, arising from the study of Higgs bundles on Riemann surfaces and playing a key role in areas such as mirror symmetry and the geometric Langlands program.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic completely integrable system
ⓘ
integrable system ⓘ |
| arisesFrom |
Yang–Mills theory
NERFINISHED
ⓘ
self-duality equations on a Riemann surface ⓘ |
| associatedTo |
complex reductive Lie algebra
ⓘ
complex reductive Lie group ⓘ |
| definedOn |
moduli space of semistable Higgs bundles
ⓘ
moduli space of stable Higgs bundles NERFINISHED ⓘ |
| field |
algebraic geometry
ⓘ
mathematical physics ⓘ |
| hasApplication |
quantization of integrable systems
ⓘ
study of moduli of curves ⓘ topological quantum field theory ⓘ |
| hasBase |
compact Riemann surface
ⓘ
smooth projective curve over complex numbers ⓘ |
| hasBaseSpace | Hitchin base NERFINISHED ⓘ |
| hasDimensionProperty |
base dimension equals number of independent Hamiltonians
ⓘ
generic fiber is an abelian variety ⓘ |
| hasFiber |
Jacobian of spectral curve
ⓘ
Prym variety ⓘ |
| hasMap |
Hitchin map
NERFINISHED
ⓘ
characteristic polynomial map ⓘ |
| introducedBy | Nigel Hitchin NERFINISHED ⓘ |
| property |
Lagrangian fibration
ⓘ
algebraic ⓘ completely integrable ⓘ holomorphic ⓘ |
| publication | Stable bundles and integrable systems NERFINISHED ⓘ |
| publicationYear | 1987 ⓘ |
| relatedTo |
S-duality in gauge theory
ⓘ
Seiberg–Witten integrable systems NERFINISHED ⓘ character varieties ⓘ flat connections ⓘ geometric Langlands program NERFINISHED ⓘ mirror symmetry ⓘ non-abelian Hodge theory ⓘ opers ⓘ |
| specialCase |
Hitchin system for GL(n)
NERFINISHED
ⓘ
Hitchin system for SL(n) NERFINISHED ⓘ Hitchin system for classical groups ⓘ |
| usesConcept |
Hamiltonian system
ⓘ
Higgs bundle NERFINISHED ⓘ Hitchin fibration NERFINISHED ⓘ Poisson structure ⓘ Riemann surface NERFINISHED ⓘ holomorphic symplectic form ⓘ moduli space of Higgs bundles ⓘ moment map ⓘ spectral curve ⓘ symplectic geometry ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.