Lagrangian subspace

A Lagrangian subspace of a $2n$-dimensional symplectic vector space $(V,\omega )$ is an $n$-dimensional subspace $L$ on which $\omega$ vanishes.

The Lagrangian Grassmannian is the space of all Lagrangian subspaces of a symplectic vector space. The unitary group acts transitively on $\mathrm{LGr}({\mathbb{R}}^{2n})$, and the stabilizer of ${\mathbb{R}}^n$ is the orthogonal group $O(n)$, giving

The Maslov index assigns an integer to a loop of Lagrangian subspaces. Under $\mathrm{LGr}({\mathbb{R}}^{2n})\cong U(n)/O(n)$, it is the degree of the map induced by $A\mapsto \det (A{)}^2$.

The Maslov map is well defined because every matrix in $O(n)$ has determinant $\pm 1$. The loopmaps to $e^{2\pi it}$ and therefore has Maslov index one.

For a rank-$2n$ symplectic vector bundle $E\to B$, the Lagrangian Grassmannian bundle has fiber $\mathrm{LGr}(E_b)$ over $b$. A symplectic trivialization of $E$ induces a trivialization $\mathcal{L}Gr(E)\cong B\times \mathrm{LGr}({\mathbb{R}}^{2n})$.

If $E\to B$ is an oriented real plane bundle, then $\mathcal{L}Gr(E)=\mathbb{P}(E)$ is an oriented circle bundle. Its transition rotations have twice the angle of those of $E$, so