Lifting of superconformal descendants in the D1-D5 CFT

We consider D1-D5-P states in the untwisted sector of the D1-D5 orbifold CFT where we excite one copy of the seed CFT with a left-moving superconformal descendant. When the theory is deformed away from this region of moduli space these states can `lift', despite being BPS at the orbifold point. For descendants formed from the supersymmetry $G^{\alpha}_{\!\dot{A},-s}$ and R-symmetry $J^a_{-n}$ current modes we obtain explicit results for the expectation value of the lifts for various subfamilies of states at second order in the deformation parameter. A smooth $\sim\sqrt{h}$ behaviour is observed in the lifts of these subfamilies for large dimensions. Using covering space Ward identities we then find a compact expression for the lift of the above $J^a_{-n}$ descendant states valid for arbitrary dimensions. In the large-dimension limit this lift scales as $\sim\sqrt{h}\,$, strengthening the conjecture that this is a universal property of the lift of D1-D5-P states. We observe that the lift is not simply a function of the total dimension, but depends on how the descendant level is partitioned amongst modes.