  We study a limit in which a relativistic CFT reduces to conformal quantum
mechanics, and relate the partition functions of the two theories. When the
initial CFT is holographic, our limit coincides with an ultra-spinning limit in
the gravity dual. We therefore propose that ultra-spinning black holes are dual
to an appropriate ensemble in finite-dimensional conformal quantum mechanics.
The limit is studied in detail for SCFTs in four and six dimensions. These
theories have a superconformal index which can be computed by gluing together
two or more blocks. Applying our limit to the index effectively isolates a
single such block. Our results therefore suggest that ultra-spinning black
holes play the role of blocks in the gravitational dual of holomorphic
factorisation.
