Maslov class of the Lagrangian Grassmannian - Imported Knowlpedia Content

Definition

For $n\geq1$ , identify the real Lagrangian Grassmannian with $\Lambda(n)=U(n)/O(n)$ . Since $\det(AQ)^2=\det(A)^2$ for $Q\in O(n)$ , the map

\det^2:\Lambda(n)\longrightarrow S^1,\qquad [A]\longmapsto\det(A)^2

is well defined. The Maslov class of the Lagrangian Grassmannian is

\mu_\Lambda=(\det^2)^*\eta\in H^1(\Lambda(n);\mathbb Z),

where $\eta$ is the positively oriented generator of $H^1(S^1;\mathbb Z)$ . The map $\det^2$ induces an isomorphism on fundamental groups, so $\mu_\Lambda$ is a generator of $H^1(\Lambda(n);\mathbb Z)$ Arnol'd, 1967. The sign is fixed by declaring counterclockwise rotation of a Lagrangian line through a half-turn to have value $+1$ .

Evaluation on loops

If $\ell:S^1\to\Lambda(n)$ is a loop and $A(t)$ is a path of unitary representatives, then

\langle\mu_\Lambda,[\ell]\rangle =\operatorname{wind}\bigl(\det(A(t))^2\bigr).

Although $A(1)$ may differ from $A(0)$ by an orthogonal matrix, squaring the determinant makes the phase close. Robbin and Salamon give the equivalent formula using a unitary frame and its determinant phase Robbin–Salamon, Remark 2.690052-W).

Duality with the Maslov cycle

Fix a reference Lagrangian $L_0$ . With its standard coorientation, the Maslov cycle $\Sigma(L_0)$ represents the Poincaré dual of $\mu_\Lambda$ in the intersection-theoretic sense. Therefore a generic loop evaluates on $\mu_\Lambda$ by its signed number of crossings with $\Sigma(L_0)$ . Changing $L_0$ changes the representative cycle but not the cohomology class Arnol'd, 1967.

Normalization in dimension one

For $\Lambda(1)=U(1)/O(1)\cong\mathbb RP^1$ , write a line as $e^{i\theta}\mathbb R$ , with $\theta$ defined modulo $\pi$ . Then $\det^2$ sends it to $e^{2i\theta}$ . The loop $\theta:0\to\pi$ winds once around $S^1$ , so its Maslov number is $1$ . Conventions assigning $2$ to this loop use twice the generator defined here.

Conventions and scope

The generator of an infinite cyclic group is determined only up to sign until a positive rotation convention is chosen. Some symplectic-path indices use a doubled normalization, and some endpoint-inclusive path indices take half-integer values. Those choices do not alter the integral universal class defined here but do alter formulas that compare different index conventions.

References

V. I. Arnol'd, “On a characteristic class entering into conditions of quantization,” Functional Analysis and Its Applications 1 (1967), 1–14. DOI record. Relevant: construction of the characteristic class and its cycle.
Joel Robbin and Dietmar Salamon, “The Maslov index for paths,” Topology 32 (1993), 827–844. DOI record90052-W). Relevant: Remark 2.6 and §§2–3.