‘Lagrangian branes’ are the objects for which we will define Lagrangian Floer cohomology and Fukaya categories; the term is used in the same way by Seidel [64, §12a] and Haug [28, §3.1], for instance, although with different definitions. Our definition is designed to try to make the programme of §3 work. The precise details of Definition 2.18 will be important in Remark 3.7 and §3.4, and are discussed in Remark 3.13.

If we take ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{C}}}$ then $E\rightarrow L$ is a complex line bundle on $L$ with a flat connection $\nabla_{E}$, which is determined up to isomorphism by its holonomy $\mathop{\rm Hol}\nolimits(\nabla_{E}):\pi_{1}(L)\rightarrow{\mathbin{\mathbb{C}}}^{*}$. In String Theory and Mirror Symmetry it is natural to suppose that $\nabla_{E}$ preserves a unitary metric on $E$, so that $\mathop{\rm Hol}\nolimits(\nabla_{E})$ takes values in ${\rm U}(1)\subset{\mathbin{\mathbb{C}}}^{*}$. One can also allow $E$ to be an ${\mathbin{\mathbb{F}}}$-local system of higher rank. Kontsevich [44] and Fukaya [18, §2.1] include a unitary local system $E\rightarrow L$ of arbitrary rank in objects of their Fukaya categories.

We need to restrict to $E$ of rank one, and not to impose the unitary condition.

Much of the literature on Lagrangian Floer cohomology and Fukaya categories including [20, 64] omits the local system $E\rightarrow L$, which is equivalent to taking $E$ to be trivial, $E={\mathbin{\mathbb{F}}}\times L\rightarrow L$. As in §3.4, we cannot do this, since in the programme of §3.2 involving families $(L^{t},E^{t},b^{t})$ for $t\in[0,\infty)$, starting with $E^{0}$ trivial, after a surgery at $t=T_{i}$, we can have $E^{t}$ nontrivial for $t>T_{i}$.