Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, either compact or suitably convex at infinity, and $D^{b}{\mathbin{\mathscr{F}}}(M)$ the derived Fukaya category of $M$, enlarged as in Conjecture 3.2 to include immersed Lagrangians, and maybe also some classes of singular Lagrangians. As in Remark 2.24, we may take all objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ to be of the form $(L,E,b)$, we do not need twisted complexes.

Suppose $\alpha\in{\mathbin{\mathbb{R}}}$ is such that $[\Omega]\cdot[L]\notin e^{i\pi\alpha}\cdot(0,\infty)$ for all $(L,E,b)$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$, where $[\Omega]\in H^{m}(M;{\mathbin{\mathbb{C}}})$ and $[L]\in H_{m}(M;{\mathbin{\mathbb{Z}}})$. As there are only countably many such homology classes $[L]$, this holds for generic $\alpha\in{\mathbin{\mathbb{R}}}$. Write ${\mathbin{\cal A}}_{\alpha}$ for the full subcategory of $D^{b}{\mathbin{\mathscr{F}}}(M)$ with objects $(L,E,b)$ such that the phase function $\theta_{L}$ of $L$ maps $L\rightarrow(\pi\alpha,\pi(\alpha+1))$. Write ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ for the full subcategory of objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ isomorphic to an object of ${\mathbin{\cal A}}_{\alpha}$, so that ${\mathbin{\cal A}}_{\alpha},{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ are equivalent categories with ${\mathbin{\cal A}}_{\alpha}\subset{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}\subset D^{b}{\mathbin{\mathscr{F}}}(M)$.

We have ${\mathbin{\cal A}}_{\alpha}[1]\!=\!{\mathbin{\cal A}}_{\alpha+1}$ and ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}[1]\!=\!{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha+1}$. The condition on $\alpha$ is to avoid taking phases in a half-open interval $(\pi\alpha,\pi(\alpha+1)]$, which could cause problems. If $(L,E,b)\in{\mathbin{\cal A}}_{\alpha}$, then $L$ is almost calibrated (has phase variation less than $\pi$).

Using the almost calibrated condition, we see that every $(L,E,b)\in{\mathbin{\cal A}}_{\alpha}$ has a unique global phase $\phi(L)\in(\pi\alpha,\pi(\alpha+1))$ with $\int_{L}\Omega=Re^{i\phi(L)}$ for $R>0$, as in Thomas [69, §3]. If $(L^{\prime},E^{\prime},b^{\prime})\in{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ then $(L^{\prime},E^{\prime},b^{\prime})\cong(L,E,b)$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ for some $(L,E,b)\in{\mathbin{\cal A}}_{\alpha}$, and $\int_{L^{\prime}}\Omega=\int_{L}\Omega=Re^{i\phi(L)}$, where $\phi(L)$ is independent of the choice of $(L,E,b)$. Thus we may define $\phi(L^{\prime})=\phi(L)$ for $(L^{\prime},E^{\prime},b^{\prime})\in{\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$.

In a similar way to Thomas [69, Def. 5.1], we say that a nonzero object $(L,E,b)$ in ${\mathbin{\cal A}}_{\alpha}$ or ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ is stable (or semistable) if there is no distinguished triangle

in $D^{b}{\mathbin{\mathscr{F}}}(M)$ with $(L_{1},E_{1},b_{1}),(L_{2},E_{2},b_{2})$ nonzero objects in ${\mathbin{\cal A}}_{\alpha}$ or ${\mathbin{\smash{\,\overline{\!\mathcal{A}}}}}_{\alpha}$ such that $\phi(L_{1})\geqslant\phi(L_{2})$ (or $\phi(L_{1})>\phi(L_{2})$).