We can now see an important reason why our programme requires the inclusion of the rank one ${\mathbin{\mathbb{F}}}$-local systems $E\rightarrow L$ in the objects $(L,E,b)$ of $D^{b}{\mathbin{\mathscr{F}}}(M)$, as mentioned in Remark 3.7. We can also justify our definition of Lagrangian branes in Definition 2.18.

Firstly, note that if the initial local systems $E^{t}$ for $t<T_{1}$ above are trivial, the local systems $E^{t}$ for $t>T_{1}$ may not be trivial, as across the ‘neck’ region $E^{t}$ for $t>T_{1}$ has holonomy $a_{0}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{T_{1}}|_{p^{T_{1}}},E_{-}^{T_{1}}|_{p^{T_{1}}}\bigr)\cong{\mathbin{\mathbb{F}}}$, and we need not have $a_{0}=1$. So this surgery can pass from trivial to nontrivial local systems $E^{t}$. If we omitted local systems $E$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$, then the data $a_{0}$ in $b^{T_{1}}$ would be lost under the surgery, and $L^{t}$ for $t>T_{1}$ might have $HF^{*}$ obstructed.

Secondly, we take ${\mathbin{\mathbb{F}}}$ to be a field (rather than say a commutative ring) so that $0\neq a_{0}\in{\mathbin{\mathbb{F}}}$ implies that $a_{0}$ is an isomorphism.

Thirdly, observe that the argument above would not work for higher rank local systems $E\rightarrow L$, which is why we restrict to rank one. If $E^{T_{1}}$ has different ranks $n_{\pm}$ on $L_{\pm}^{T_{1}}$, then it cannot extend across the ‘neck’ to make $E^{t}$ for $t>T_{1}$. If $E^{T_{1}}$ has the same rank $n>1$ on $L_{+}^{T_{1}},L_{-}^{T_{1}}$, then $a_{0}\neq 0$ no longer implies that $a_{0}$ is an isomorphism, so we cannot use $a_{0}$ to extend $E^{T_{1}}$ across the ‘neck’.