  We study the moduli dependence of the chiral ring in N = (4,4) superconformal
field theories, with special emphasis on those CFTs that are dual to type IIB
string theory on AdS3xS3xX4. The chiral primary operators are sections of
vector bundles, whose connection describes the operator mixing under motion on
the moduli space. This connection can be exactly computed using the constraints
from N = (4,4) supersymmetry. Its curvature can be determined using the tt*
equations, for which we give a derivation in the physical theory which does not
rely on the topological twisting. We show that for N = (4,4) theories the
chiral ring is covariantly constant over the moduli space, a fact which can be
seen as a non-renormalization theorem for the three-point functions of chiral
primaries in AdS3/CFT2. From the spacetime point of view our analysis has the
following applications. First, in the case of a D1/D5 black string, we can see
the matching of the attractor flow in supergravity to RG-flow in the boundary
field theory perturbed by irrelevant operators, to first order away from the
fixed point. Second, under spectral flow the chiral primaries become the Ramond
ground states of the CFT. These ground states represent the microstates of a
small black hole in five dimensions consisting of a D1/D5 bound state. The
connection that we compute can be considered as an example of Berry's phase for
the internal microstates of a supersymmetric black hole.
