Lagrangian Grassmannian

Short description: Type of vector space in mathematics

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

U(n)/O(n),

where U(n) is the unitary group and O(n) the orthogonal group. Following Vladimir Arnold it is denoted by Λ(n). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V.

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 ⁠1/2⁠n(n + 1)

Sp(n)/U(n),

where Sp(n) is the compact symplectic group.

As a homogeneous space

To see that the Lagrangian Grassmannian Λ(n) can be identified with U(n)/O(n), note that ${\mathbb{ℂ}}^n$ is a 2n-dimensional real vector space, with the imaginary part of its usual inner product making it into a symplectic vector space. The Lagrangian subspaces of ${\mathbb{ℂ}}^n$ are then the real subspaces $L\subseteq {\mathbb{ℂ}}^n$ of real dimension n on which the imaginary part of the inner product vanishes. An example is ${\mathbb{ℝ}}^n\subseteq {\mathbb{ℂ}}^n$. The unitary group U(n) acts transitively on the set of these subspaces, and the stabilizer of ${\mathbb{ℝ}}^n$ is the orthogonal group $\mathrm{O}(n)\subseteq \mathrm{U}(n)$. It follows from the theory of homogeneous spaces that Λ(n) is isomorphic to U(n)/O(n) as a homogeneous space of U(n).

It is a compact manifold of dimension $n(n+1)/2$. It is a (real, nonsingular) projective algebraic variety. Given a Lagrangian subspace A, the set of Lagranigian subspaces complementary to A is affine. Given an arbitrary complementary subspace B, this affine space consists of the graphs of symmetric linear operators $u:B\to A$, $G(u)=\{b+u(b)|b\in B\}$. This is an affine space of dimension $n(n+1)/2$ since the dimensions of A and B are both n. Symmetry here means that the form $\omega (b,u(b^{\prime }))$ is a symmetric form on B. Likewise, the tangent space at a lagrangian subspace A is the space of symmetric opeators $A\to A^{*}$.

From the fibration $$1\to O(n)\to U(n)\to \mathrm{\Lambda }(n)$$ the fundamental group may be inferred from the long exact homotopy sequence: $${\pi }_1(\mathrm{\Lambda }(n))=\mathbb{\mathbb{Z}}.$$

Topology

The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: $\mathrm{\Omega }(\mathrm{S}\mathrm{p}/\mathrm{U})\simeq \mathrm{U}/\mathrm{O}$, and $\mathrm{\Omega }(\mathrm{U}/\mathrm{O})\simeq \mathbb{\mathbb{Z}}\times \mathrm{B}\mathrm{O}$ – they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension).

In particular, the fundamental group of $U/O$ is infinite cyclic. 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. 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

$M\to \mathrm{\Lambda }(n)$

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

$H^1(M,\mathbb{\mathbb{Z}})$

of the distinguished generator of

$H^1(\mathrm{\Lambda }(n),\mathbb{\mathbb{Z}})$.

Maslov index

Given a fixed Lagrangian subspace $L$ in the Lagrangian Grassmannian $\mathrm{\Lambda }$, the subset $$M_L=\{V\in \mathrm{\Lambda }\mid V\cap L\ne 0\}$$ is the Maslov cycle, a singular hypersurface in $\mathrm{\Lambda }$. For a generic path of Lagrangian subspaces in $\mathrm{\Lambda }$ whose endpoints are transverse to $L$, the Maslov index is the signed intersection number of the path with $M_L$. The Maslov index is invariant under homotopies of paths through Lagrangian subspaces, provided the endpoints remain transverse to the chosen reference Lagrangian $L$. For a loop, it depends only on the homotopy class of the loop and is independent of the choice of $L$.

The Maslov index is important in the study of caustics and semiclassical asymptotics. Roughly speaking, when a family of Lagrangian subspaces crosses the Maslov cycle, one encounters a caustic relative to the chosen reference Lagrangian $L$; in the theory of Fourier integral operators and the WKB approximation, such crossings produce phase corrections governed by the Maslov index.

References

• V. I. Arnold, Characteristic class entering in quantization conditions, Funktsional'nyi Analiz i Ego Prilozheniya, 1967, 1,1, 1-14, doi:10.1007/BF01075861.
• V. P. Maslov, Théorie des perturbations et méthodes asymptotiques. 1972
• Ranicki, Andrew, The Maslov index home page, http://www.maths.ed.ac.uk/~aar/maslov.htm, retrieved 2009-10-23  Assorted source material relating to the Maslov index.