Orientation of a virtual vector bundle

Definition 10.4.5 (Orientation). Let \(V\) be a virtual vector bundle of rank \(d\) on an anima \(A\). Its associated local system of \(R\)-lines is \[ \xi _{V,R}\colon A \xrightarrow {\;\xi _V[-d]\;} \Line _{\S } \xrightarrow {-\otimes R} \Line _R. \] Here \(\xi _V[-d]\) lands in the virtual-rank-zero component \(\Line _{\S }\subseteq \Pic (\S )\). An \(R\)-orientation of \(V\) is a trivialization \(\xi _{V,R} \cong R_A\).