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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hitchin system canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10829551 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hitchin system Context triple: [Nigel Hitchin, notableFor, Hitchin system]
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A.
Beilinson–Drinfeld Grassmannian
The Beilinson–Drinfeld Grassmannian is a geometric object in algebraic geometry and representation theory that generalizes the affine Grassmannian to configurations of multiple points, playing a central role in the geometric Langlands program.
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B.
Hurwitz space
A Hurwitz space is a moduli space that parametrizes branched covers of Riemann surfaces (or algebraic curves) with specified branching data.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
D.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
E.
Lefschetz pencil
A Lefschetz pencil is a geometric structure on an algebraic variety given by a one-parameter family of hyperplane sections with only isolated, well-controlled singularities, fundamental in the study of its topology and geometry.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hitchin system Target entity description: 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.
-
A.
Beilinson–Drinfeld Grassmannian
The Beilinson–Drinfeld Grassmannian is a geometric object in algebraic geometry and representation theory that generalizes the affine Grassmannian to configurations of multiple points, playing a central role in the geometric Langlands program.
-
B.
Hurwitz space
A Hurwitz space is a moduli space that parametrizes branched covers of Riemann surfaces (or algebraic curves) with specified branching data.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
D.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
E.
Lefschetz pencil
A Lefschetz pencil is a geometric structure on an algebraic variety given by a one-parameter family of hyperplane sections with only isolated, well-controlled singularities, fundamental in the study of its topology and geometry.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hitchin system Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.