Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold. A Lagrangian brane in $M$ is a pair $(L,E)$, where $L$ is a compact, spin, graded Lagrangian in $M$, and $E\rightarrow L$ is a rank one ${\mathbin{\mathbb{F}}}$-local system on $L$, for ${\mathbin{\mathbb{F}}}$ as in Definition 2.17. That is, $E$ is a locally constant rank one ${\mathbin{\mathbb{F}}}$-vector bundle over $L$, so that if $p\in L$ then $E|_{p}$ is a dimension one ${\mathbin{\mathbb{F}}}$-vector space, which is locally independent of $p$.

In this section we take $L$ to be embedded, but in §2.6 $L$ can be immersed, and in §3 we will (conjecturally) allow $L$ to have certain kinds of singularities.