/sci/ - Science & Math

Let [math](V,\omega)[/math] be a symplectic vector space and let [math]\Gamma(V)[/math] be the set of all Lagrangian spaces [math]\lambda \in \Gamma(V)[/math] such that [math]\lambda ^{\perp} = \lambda[/math], where [math]A^\perp = \{v \in V \mid \forall a \in A,\omega(a,v) = 0\}[/math].
Put [math]\lambda_1,\lambda_2,\lambda_3\in \Gamma(V)[/math], and let [math]\langle \cdotp,\cdotp\rangle[/math] be a (symmetric) bilinear form on [math](\lambda_1 + \lambda_2) \cap \lambda_3[/math] defined such that, for [math]a = a_1 + a_2, b \in (\lambda_1 + \lambda_2) \cap \lambda_3[/math], [math]\langle a,b\rangle = \omega(a_2,b)[/math].
The Maslov index [math]\mu(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{Z}[/math] of the triple [math](\lambda_1 ,\lambda_2, \lambda_3)[/math] is defined to be the signature of the bilinear form [math]\langle \cdotp,\cdotp\rangle[/math] on [math](\lambda_1 + \lambda_2) \cap \lambda_3[/math], i.e.
[eqn]
\mu(\lambda_1,\lambda_2,\lambda_3) = \#\{v \in (\lambda_1 + \lambda_2) \cap \lambda_3 \mid \langle v,v\rangle > 0\} - \#\{v \in (\lambda_1 + \lambda_2) \cap \lambda_3 \mid \langle v,v\rangle < 0\}.
[/eqn]
Let [math]N: V_1 \implies V_2[/math] be a Lagrangian relation in the sense that [math]N[/math] is Lagrangian and [math]N \subset (-V_1)\oplus V_2[/math], then [math]N[/math] induces a map [math]N_*: \Gamma(V_1) \rightarrow \Gamma(V_2)[/math] such that [math]N_*(\lambda) = \{v_2 \in V_2 \mid \exists v_1 \in \lambda, (v_1,v_2) \in N\}.[/math] Let [math]N^*:\Gamma(V_2) \rightarrow \Gamma(V_1)[/math] denote the dual map.
The Maslov index then satisfies [eqn] \mu(\lambda_1,\lambda_2,N^*(\lambda_1')) - \mu(\lambda_1,\lambda_2,N^*(\lambda_2')) + \mu(N_*(\lambda_1),\lambda_1',\lambda_2') - \mu(N_*(\lambda_2),\lambda_1',\lambda_2') = 0, [/eqn]
where [math]\lambda_1,\lambda_2 \in \Gamma(V_1)[/math] and [math]\lambda_1',\lambda_2' \in \Gamma(V_2)[/math].