Crossing form - Knowlpedia

Let $V$ be a finite-dimensional real symplectic vector space, let $L_0$ be a fixed Lagrangian subspace, and let $L(t)$ be a smooth path of Lagrangian subspaces. If $t_0$ is a crossing, meaning $L(t_0)\cap L_0\ne\{0\}$ , choose a Lagrangian complement $L_1$ of $L(t_0)$ and write nearby $L(t)$ as the graph of $A(t):L(t_0)\to L_1$ , using $\omega$ to identify $L_1$ with $L(t_0)^*$ . Its crossing form is the quadratic form

\Gamma(L,L_0,t_0)(v)=\omega\bigl(v,\dot A(t_0)v\bigr),\qquad v\in L(t_0)\cap L_0.

Graph formula

This is the Robbin–Salamon convention for the graph chart; the resulting form is independent of the complement after the same symplectic identification is used.

The graph description in the core is local: the derivative $\dot A(t_0)$ is symmetric under the identification induced by $\omega$ , so its restriction to the crossing subspace is a well-defined quadratic form.

Regular crossings and index contributions

The crossing is regular when $\Gamma(L,L_0,t_0)$ is nondegenerate. A regular crossing is isolated, and its local signed contribution to the Maslov index is the signature $\operatorname{sign}\Gamma(L,L_0,t_0)$ . Endpoint crossings require an endpoint convention; the Robbin–Salamon convention assigns half the signature at each regular endpoint.

References

Joel Robbin and Dietmar Salamon, “The Maslov index for paths,” Topology 32 (1993), 827–844. DOI record. Relevant: §2, crossing forms and regular crossings.
Viktor 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 Maslov cycle and its coorientation.