Lagrangian Grassmannian

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To see that the Lagrangian Grassmannian Λ(n) can be identified with U(n)/O(n), note that ${\displaystyle \mathbb {C} ^{n}}$ 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 ${\displaystyle \mathbb {C} ^{n}}$ are then the real subspaces ${\displaystyle L\subseteq \mathbb {C} ^{n}}$ of real dimension n on which the imaginary part of the inner product vanishes. An example is ${\displaystyle \mathbb {R} ^{n}\subseteq \mathbb {C} ^{n}}$. The unitary group U(n) acts transitively on the set of these subspaces, and the stabilizer of ${\displaystyle \mathbb {R} ^{n}}$ is the orthogonal group ${\displaystyle \mathrm {O} (n)\subseteq \mathrm {U} (n)}$. 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 ${\displaystyle n(n+1)/2}$. 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 ${\displaystyle u:B\to A}$, ${\displaystyle G(u)=\{b+u(b)|b\in B\}}$. This is an affine space of dimension ${\displaystyle n(n+1)/2}$ since the dimensions of A and B are both n. Symmetry here means that the form ${\displaystyle \omega (b,u(b'))}$ is a symmetric form on B. Likewise, the tangent space at a lagrangian subspace A is the space of symmetric opeators ${\displaystyle A\to A^{*}}$.

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

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

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

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

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

of the distinguished generator of

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

Given a fixed Lagrangian subspace ${\displaystyle L}$ in the Lagrangian Grassmannian ${\displaystyle \Lambda }$, the subset $${\displaystyle M_{L}=\{V\in \Lambda \mid V\cap L\neq 0\}}$$ is the Maslov cycle, a singular hypersurface in ${\displaystyle \Lambda }$.^[1] For a generic path of Lagrangian subspaces in ${\displaystyle \Lambda }$ whose endpoints are transverse to ${\displaystyle L}$, the Maslov index is the signed intersection number of the path with ${\displaystyle M_{L}}$.^[2] The Maslov index is invariant under homotopies of paths through Lagrangian subspaces, provided the endpoints remain transverse to the chosen reference Lagrangian ${\displaystyle L}$. For a loop, it depends only on the homotopy class of the loop and is independent of the choice of ${\displaystyle L}$.^[1][2]

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 ${\displaystyle L}$; in the theory of Fourier integral operators and the WKB approximation, such crossings produce phase corrections governed by the Maslov index.^[3][4]