In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of V is 2n).
It may be identified with the homogeneous space of complex dimension 1/2n(n + 1) where Sp(n) is the compact symplectic group.
of real dimension n on which the imaginary part of the inner product vanishes.
The unitary group U(n) acts transitively on the set of these subspaces, and the stabilizer of
The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem:
Its first homology group is therefore also infinite cyclic, as is its first cohomology group, with a distinguished generator given by the square of the determinant of a unitary matrix, as a mapping to the unit circle.
For a Lagrangian submanifold M of V, in fact, there is a mapping which classifies its tangent space at each point (cf.
The Maslov index is the pullback via this mapping, in of the distinguished generator of A path of symplectomorphisms of a symplectic vector space may be assigned a Maslov index, named after V. P. Maslov; it will be an integer if the path is a loop, and a half-integer in general.
It computes the spectral flow of the Cauchy–Riemann-type operators that arise in Floer homology.
[1] It appeared originally in the study of the WKB approximation and appears frequently in the study of quantization, quantum chaos trace formulas, and in symplectic geometry and topology.
It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.