Maslov class of a Lagrangian submanifold - Imported Knowlpedia Content

Definition

Let $f:L^n\to(V^{2n},\omega)$ be a Lagrangian immersion into a real symplectic vector space. Its Maslov class is

\mu_L=\gamma_f^*\mu_\Lambda\in H^1(L;\mathbb Z),

where $\gamma_f:L\to\Lambda(V)$ is the Lagrangian Gauss map and $\mu_\Lambda$ is the universal Maslov class. More generally, the same formula applies after choosing a symplectic trivialization of $f^*TM$ for an immersion into a symplectic manifold $M$ , or after supplying equivalent Maslov-covering data. For a loop $c:S^1\to L$ , the integer $\langle\mu_L,[c]\rangle$ is the Maslov index of the loop of tangent Lagrangian planes $df_{c(t)}(T_{c(t)}L)$ .

Geometric interpretation

Fix a reference Lagrangian plane $L_0\subset V$ . For a generic immersion and loop, the Maslov number counts, with signs, the points where the tangent plane $\gamma_f(c(t))$ fails to be transverse to $L_0$ . In other words, it is the intersection number of $\gamma_f\circ c$ with the Maslov cycle. This is the characteristic-class interpretation introduced by Arnol'd Arnol'd, 1967.

Examples

An affine Lagrangian plane has constant Gauss map, hence zero Maslov class. The zero section in a cotangent bundle likewise has zero class in its canonical local model. For a Lagrangian curve in $\mathbb R^2$ , the class records the winding of the unoriented tangent line: one positive half-turn of the tangent line evaluates to $1$ under the normalization used here.

Ambient-manifold caveat

For a general $(M,\omega)$ , the canonical Gauss map is a section of the Lagrangian-Grassmannian bundle, not automatically a map to one fixed $\Lambda(n)$ . An integral grading obstruction can therefore depend on an ambient Maslov covering or on a trivialization such as one induced by a chosen squared canonical-volume form. Seidel formulates this dependence in terms of Maslov coverings Seidel, §2. Omitting this datum can make the phrase “the Maslov class in $H^1(L;\mathbb Z)$ ” ambiguous.

Distinction from disk indices

The class $\mu_L$ evaluates on loops in $L$ . The Maslov index of a relative disk $u:(D,\partial D)\to(M,L)$ instead measures the loop of Lagrangian boundary conditions $T_{u(e^{it})}L$ after trivializing $u^*TM$ . It defines a homomorphism on $\pi_2(M,L)$ and need not be determined by $\mu_L$ alone in a nontrivial ambient manifold. These invariants agree only after the relevant trivializations and boundary-loop identifications are specified.

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: the Gauss-map definition of the Maslov characteristic class.
Paul Seidel, “Graded Lagrangian submanifolds,” Bulletin de la Société Mathématique de France 128 (2000), 103–149. arXiv record. Relevant: §2, Maslov coverings, gradings, and their obstruction classes.