Pure spinor superstring in AdS4×CP3 with unconstrained ghosts

We construct the action for the pure spinor superstring in the coset description of AdS4×CP3 superspace, using the variables which solve the pure spinor condition. As a test of the consistency of the approach, we use the background field method to verify the absence of central charge at the second order in the expansion and to show the one-loop finiteness of the effective action