Maslov index ($\mu$)

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.