Lifting at higher levels in the D1D5 CFT

The D1D5P system has a large set of BPS states at its orbifold point. Perturbing away from this 'free' point leads to some states joining up into long supermultiplets and lifting, while other states remain BPS. We consider the simplest orbifold which exhibits this lift: that with $N=2$ copies of the free $c=6$ CFT. We write down the number of lifted and unlifted states implied by the index at all levels upto $6$. We work to second order in the perturbation strength $\lambda$. For levels upto $4$, we find the wavefunctions of the lifted states, their supermultiplet structure and the value of the lift. All states that are allowed to lift by the index are in fact lifted at order $O(\lambda^2)$. We observe that the unlifted states in the untwisted sector have an antisymmetry between the copies in the right moving Ramond ground state sector, and extend this observation to find classes of states for arbitrary $N$ that will remain unlifted to $O(\lambda^2)$.