Relative Maslov index - Imported Knowlpedia Content

Definition

Let $\Lambda_0,\Lambda_1:[a,b]\to\Lambda(V)$ be continuous paths of Lagrangian subspaces in a finite-dimensional real symplectic vector space. Their relative Maslov index is the Maslov index

\mu(\Lambda_0,\Lambda_1) =\mu\bigl(\Delta,\Lambda_0\times\Lambda_1\bigr)

in $(V\oplus V,-\omega\oplus\omega)$ , where $\Delta=\{(v,v):v\in V\}$ is the diagonal Lagrangian. This convention orders the two paths and fixes the sign. For smooth pairs with only regular crossings, where $\Lambda_0(t)\cap\Lambda_1(t)\neq0$ , the index is the sum of signatures of the relative crossing forms, with one-half of the endpoint signatures. Continuous pairs are defined by regular perturbation or homotopy.

Relative crossing form

At a crossing $t$ , each moving plane determines a quadratic form on the common subspace. The relative crossing form is their difference

\Gamma(\Lambda_0,\Lambda_1,t) =\Gamma(\Lambda_0,\Lambda_1(t),t) -\Gamma(\Lambda_1,\Lambda_0(t),t).

When it is nondegenerate, the crossing is regular. The signature formula is

\mu(\Lambda_0,\Lambda_1) =\tfrac12\operatorname{sign}\Gamma(a) +\sum_{a<t<b}\operatorname{sign}\Gamma(t) +\tfrac12\operatorname{sign}\Gamma(b).

Robbin and Salamon prove that this agrees with the diagonal construction Robbin–Salamon, Theorem 3.290052-W).

Properties

The relative index is natural under a common path of symplectic transformations, additive under concatenation and direct sum, and antisymmetric:

\mu(\Lambda_1,\Lambda_0)=-\mu(\Lambda_0,\Lambda_1).

If $\Lambda_1(t)\equiv L_0$ is constant, it reduces to the path index relative to $L_0$ . When both endpoint pairs are transverse, it is integer-valued and invariant under homotopies that preserve endpoint transversality Robbin–Salamon, Corollary 3.390052-W).

Example and near-miss

In $\mathbb R^2$ , keep $\Lambda_1=\mathbb R$ fixed and rotate $\Lambda_0$ counterclockwise through one transverse crossing in the interior. The relative index is $+1$ . If a crossing occurs at an endpoint instead, the Robbin–Salamon value may be $+\tfrac12$ , so it is incorrect to assert integer-valuedness without transverse endpoints or another endpoint convention.

Relation to spectral flow

For suitable paths of self-adjoint first-order operators with Lagrangian boundary data, spectral flow is expressed by a relative Maslov index. Cappell, Lee, and Miller compare several definitions and develop this relation systematically Cappell–Lee–Miller, 1994. Analytic applications may reverse the order of the two boundary-data paths, which reverses the sign.

References

Joel Robbin and Dietmar Salamon, “The Maslov index for paths,” Topology 32 (1993), 827–844. DOI record90052-W). Relevant: §3, especially Theorem 3.2 and Corollary 3.3.
Sylvain E. Cappell, Ronnie Lee, and Edward Y. Miller, “On the Maslov index,” Communications on Pure and Applied Mathematics 47 (1994), 121–186. DOI record. Relevant: equivalent definitions and the relation to spectral flow.