A New Supersymmetry Index for the D1-D5 CFT

We propose a new supersymmetry index for the D1-D5 CFT for $T^4$, relevant to the AdS$_3$/CFT$_2$ correspondence. In a novel formulation of symmetric orbifold CFTs based on the Schur-Weyl duality, we show how this index can be naturally described and its protection is argued based on the detailed nature of exactly marginal operators in these theories. This index is a one-parameter generalization of the standard index and gives more fine-grained information about the structure of microstates than previously available. We demonstrate precise matching of the new index between supergravity and CFT below the black-hole threshold, where the standard index -- the modified elliptic genus -- is trivial. Above the threshold, we uncover a decomposition of black-hole microstates into distinct sectors, invisible to the modified elliptic genus.