Maslov index
E860126
The Maslov index is a topological invariant that assigns an integer to loops or paths of Lagrangian subspaces in symplectic geometry, capturing their phase change or winding behavior.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
integer-valued invariant
ⓘ
symplectic invariant ⓘ topological invariant ⓘ |
| additiveUnder |
concatenation of paths
ⓘ
direct sum of symplectic vector spaces ⓘ |
| appearsIn |
Bohr–Sommerfeld quantization conditions
NERFINISHED
ⓘ
phase of oscillatory integrals ⓘ |
| appliesTo |
loops of Lagrangian subspaces
ⓘ
paths of Lagrangian subspaces ⓘ |
| associatedWith |
Maslov cycle
NERFINISHED
ⓘ
fundamental group of the Lagrangian Grassmannian ⓘ universal covering of the Lagrangian Grassmannian ⓘ |
| captures |
phase change
ⓘ
winding behavior ⓘ |
| codomain | integers ⓘ |
| context |
complex symplectic manifolds
ⓘ
real symplectic vector spaces ⓘ |
| definedVia |
crossings with a reference Lagrangian
ⓘ
degree of a map to the circle ⓘ intersection number with Maslov cycle ⓘ |
| dependsOn | homotopy class of the path ⓘ |
| domain | Lagrangian Grassmannian NERFINISHED ⓘ |
| field |
mathematical physics
ⓘ
symplectic geometry ⓘ |
| generalizes | winding number ⓘ |
| hasVariant |
Conley–Zehnder index
NERFINISHED
ⓘ
Robbin–Salamon index ⓘ |
| introducedIn | 1960s ⓘ |
| invariantUnder | homotopy with fixed endpoints ⓘ |
| namedAfter | Vladimir Pavlovich Maslov NERFINISHED ⓘ |
| relatedConcept |
Maslov class
ⓘ
phase of a Lagrangian submanifold ⓘ |
| relatedTo |
Lagrangian Grassmannian
NERFINISHED
ⓘ
Lagrangian subspace ⓘ index theory ⓘ intersection theory ⓘ spectral flow ⓘ symplectic vector space ⓘ |
| takesValuesIn | integers ⓘ |
| usedIn |
Floer theory
NERFINISHED
ⓘ
Gromov–Witten theory NERFINISHED ⓘ Hamiltonian dynamics ⓘ Lagrangian intersection theory NERFINISHED ⓘ WKB approximation NERFINISHED ⓘ microlocal analysis ⓘ quantization ⓘ semiclassical analysis ⓘ |
| usedToDefine |
Maslov class of a Lagrangian submanifold
NERFINISHED
ⓘ
grading in Lagrangian Floer homology ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.