Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, and $L,L^{\prime}$ graded Lagrangians in $M$, with phase functions $\theta_{L},\theta_{L^{\prime}}$, which intersect transversely at $p\in M$. By a kind of simultaneous diagonalization, we may choose an isomorphism $T_{p}M\cong{\mathbin{\mathbb{C}}}^{m}$ which identifies $J|_{p},g|_{p},\omega|_{p}$ on $T_{p}M$ with the standard versions (2.2) on ${\mathbin{\mathbb{C}}}^{m}$, and identifies $T_{p}L,T_{p}L^{\prime}$ with the Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ respectively, where

$$\Pi_{0}=\bigl\{(x_{1},\ldots,x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},\;\>\Pi_{\boldsymbol{\phi}}=\bigl\{({\rm e}^{i\phi_{1}}x_{1},\ldots,{\rm e}^{i\phi_{m}}x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},$$

(2.12)

for $\phi_{1},\ldots,\phi_{m}\in(0,\pi)$. Then $\phi_{1},\ldots,\phi_{m}$ are independent of choices up to order. Define the degree $\mu_{L,L^{\prime}}(p)\in{\mathbin{\mathbb{Z}}}$ of $p$ by

$$\mu_{L,L^{\prime}}(p)=(\phi_{1}+\cdots+\phi_{m}+\theta_{L}(p)-\theta_{L^{\prime}}(p))/\pi.$$

This an integer as $\theta_{L^{\prime}}(p)=\theta_{L}(p)+\phi_{1}+\cdots+\phi_{m}\mod\pi{\mathbin{\mathbb{Z}}}$. Exchanging $L,L^{\prime}$ replaces $\phi_{1},\ldots,\phi_{m}$ by $\pi-\phi_{1},\ldots,\pi-\phi_{m}$, so that $\mu_{L,L^{\prime}}(p)+\mu_{L^{\prime},L}(p)=m$. Since $\phi_{1},\ldots,\phi_{m}\in(0,\pi)$, we see that

$$(\theta_{L}(p)-\theta_{L^{\prime}}(p))/\pi<\mu_{L,L^{\prime}}(p)<(\theta_{L}(p)-\theta_{L^{\prime}}(p))/\pi+m.$$

(2.13)