Bosonization, cocycles, and the D1-D5 CFT on the covering surface

We consider the D1-D5 CFT near the orbifold point, specifically the computation of correlators involving twist sector fields using covering surface techniques. As is well known, certain twists introduce spin fields on the cover. Here we consider the bosonization of fermions to facilitate computations involving the spin fields. We find a set of cocycle operators that satisfy constraints coming from various $SU(2)$ symmetries, including the $SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ R-symmetry. Using these cocycles, we consider the correlator of four spin fields on the cover, and show that it is invariant under all of the $SU(2)$ symmetries of the theory. We consider the mutual locality of operators, and compute several three-point functions. These computations lead us to a notion of radial ordering on the cover that is inherited from the original computation before lifting. Further, we note that summing over orbifold images sets certain branch-cut ambiguous correlators to zero.