Hitchin fibration
E886932
The Hitchin fibration is a fundamental geometric structure in the theory of Higgs bundles that organizes their moduli space into an algebraically completely integrable system with deep connections to representation theory and the geometric Langlands program.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraically completely integrable system
ⓘ
fibration ⓘ geometric structure ⓘ |
| actsOn | moduli space of Higgs bundles ⓘ |
| baseDimension | half the dimension of the Higgs moduli space ⓘ |
| codomain | Hitchin base identified with space of invariant polynomials valued differentials ⓘ |
| constructedFrom |
characteristic polynomial of the Higgs field
ⓘ
invariant polynomials of a complex reductive Lie algebra ⓘ |
| context |
complex reductive algebraic group G
ⓘ
smooth projective algebraic curve over complex numbers ⓘ |
| definedOn |
moduli space of semistable Higgs bundles
ⓘ
moduli space of stable Higgs bundles ⓘ |
| describedIn | Stable bundles and integrable systems (Hitchin, 1987) NERFINISHED ⓘ |
| domain | moduli space of G-Higgs bundles on a smooth projective curve ⓘ |
| dualTo | Hitchin fibration for Langlands dual group NERFINISHED ⓘ |
| field |
algebraic geometry
ⓘ
differential geometry ⓘ mathematical physics ⓘ representation theory ⓘ |
| genericFiberDimension | half the dimension of the Higgs moduli space ⓘ |
| hasBase | Hitchin base NERFINISHED ⓘ |
| hasComponent | discriminant locus where spectral curve is singular ⓘ |
| hasFiber |
generic fiber is a Jacobian of a spectral curve
ⓘ
generic fiber is a Prym variety in the non-simply connected case ⓘ generic fiber is an abelian variety ⓘ |
| hasGeneralization |
parabolic Hitchin fibration
ⓘ
wild Hitchin fibration ⓘ |
| hasProperty |
algebraically completely integrable
ⓘ
base is an affine space ⓘ completely integrable Hamiltonian system ⓘ defines an integrable system on the moduli of Higgs bundles ⓘ fibers are Lagrangian with respect to natural symplectic form ⓘ generic fibers are torsors under abelian varieties ⓘ |
| hasSymmetry | action of the Picard stack of the spectral curve ⓘ |
| inspired | subsequent work on integrable systems from moduli spaces ⓘ |
| introducedBy | Nigel Hitchin NERFINISHED ⓘ |
| introducedIn | 1987 ⓘ |
| mapType |
Lagrangian fibration
ⓘ
algebraic map ⓘ proper map ⓘ |
| namedAfter | Nigel Hitchin NERFINISHED ⓘ |
| relatedTo |
Dolbeault moduli space
NERFINISHED
ⓘ
Higgs bundle ⓘ S-duality in gauge theory ⓘ geometric Langlands program NERFINISHED ⓘ mirror symmetry ⓘ moduli of local systems ⓘ non-abelian Hodge theory ⓘ |
| usedIn |
construction of Hecke eigensheaves
ⓘ
proofs and formulations of geometric Langlands duality ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.