  The D1D5 CFT has a large set of states that are supersymmetric at the `free'
orbifold point in moduli space. When we perturb away from this point, some of
these states join into long multiplets and lift in energy, while others remain
supersymmetric. The count of unlifted states can be bounded below by an index,
but the index does not yield the pattern of lifting; i.e., which states join
into a long multiplet and how much this multiplet lifts. In this paper we
consider the simple case of the D1D5 CFT where the orbifold CFT is a sigma
model with targets space $(T^4)^2/S_2$ and consider states at energy level $1$.
There are $2688$ states at this level. The lifted states form a triplet of long
multiplets, and we compute their lift at second order in perturbation theory.
Half the members of the long multiplet are in the untwisted sector and half are
in the twisted sector. This and other similar studies should help in the
understanding of fuzzball states that describe extremal holes, since CFT
sectors with low twist describe shallow throats in the dual gravity solution
while sectors with high twist describe deep throats.
