The chiral ring of AdS3/CFT2 and the attractor mechanism

We study the moduli dependence of the chiral ring in N = (4, 4) superconformal field theories, with special emphasis on those CFT's that are dual to type IIB string theory on AdS 3 ×S 3 ×X 4. 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 AdS 3 /CFT 2. 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.