We consider states of the D1-D5 CFT where only the left-moving sector is
excited. As we deform away from the orbifold point, some of these states will
remain BPS while others can `lift'. We compute this lifting for a particular
family of D1-D5-P states, at second order in the deformation off the orbifold
point. We note that the maximally twisted sector of the CFT is special: the
covering surface appearing in the correlator can only be genus one while for
other sectors there is always a genus zero contribution. We use the results to
argue that fuzzball configurations should be studied for the full class
including both extremal and near-extremal states; many extremal configurations
may be best seen as special limits of near extremal configurations.Comment: 51 pages, 6 figure
