Lagrangian Grassmannian

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In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V whose dimension is n(n+1)/2 (where the dimension of V is 2n). It may be identified with the homogeneous space

U(n)/O(n)

where U(n) is the unitary group and O(n) the orthogonal group. After Vladimir Arnold it is denoted by Λ(n).

A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space V of dimension 2n. It may be identified with the homogeneous space of complex dimension n(n+1)/2

Sp(n)/U(n)

where Sp(n) is the complex symplectic group.

[edit] Topology

The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: \Omega(Sp/U) \simeq U/O, and \Omega(U/O) \simeq Z\times BO.

In particular, the fundamental group of U / O is infinite cyclic, with a distinguished generator given by the square of the determinant of a unitary matrix, as a mapping to the unit circle. Its first homology group is therefore also infinite cyclic, as is its first cohomology group. Arnold showed that this leads to a description of the Maslov index, introduced by V. P. Maslov.

For a Lagrangian submanifold M of V, in fact, there is a mapping

M → Λ(n)

which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in

H1(M, Z)

of the distinguished generator of

H1(Λ(n), Z).

[edit] Maslov index

A path of symplectomorphisms of a symplectic vector space may be assigned a Maslov index; it will be an integer if the path is a loop, and a half-integer in general.

If this path arises from trivializing the symplectic vector bundle over a periodic orbit of a Hamiltonian vector field on a symplectic manifold or the Reeb vector field on a contact manifold, it is known as the Conley-Zehnder index. It computes the spectral flow of the Cauchy-Riemann-type operators that arise in Floer homology.

It appeared originally in the study of the WKB approximation and appears frequently in the study of quantization and in symplectic geometry and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.

[edit] References

  • V. I. Arnold, Une classe charactéristique intervenant dans les conditions de quantification, in V. P. Maslov, Théorie des perturbations et methods asymptotiques. French translation 1972