Fortuity in the D1-D5 system

We reformulate the lifting problem in the D1-D5 CFT as a supercharge cohomology problem, and enumerate BPS states according to the fortuitous/monotone classification. Working in the deformed T^4 T 4 symmetric orbifold theory, we give precise definitions of monotone and fortuitous cohomology classes generalizing the definitions in [C.-M. Chang and Y.-H. Lin, arXiv: 2402.10129] and illustrate them in the N=1 N = 1 theory. For N=2 N = 2 , we construct the cohomology explicitly and match it to the exact BPS partition function. We further describe how to assemble BPS states at smaller N N into BPS states at larger N N , and interpret their holographic duals as black hole bound states and massive stringy excitations on smooth horizonless (e.g. Lunin-Mathur) geometries.