  We formulate a refined version of the Ooguri-Strominger-Vafa (OSV)
conjecture. The OSV conjecture that Z_{BH} = |Z_{top}|^2 relates the BPS black
hole partition function to the topological string partition function Z_{top}.
In the refined conjecture, Z_{BH} is the partition function of BPS black holes
counted with spin, or more precisely the protected spin character. Z_{top}
becomes the partition function of the refined topological string, which is
itself an index. Both the original and the refined conjecture are examples of
large N duality in the 't Hooft sense. The refined conjecture applies to
non-compact Calabi-Yau manifolds only, so the black holes are really BPS
particles with large entropy, of order N^2. The refined OSV conjecture states
that the refined BPS partition function has a large N dual which is captured by
the refined topological string. We provide evidence that the conjecture holds
by studying local Calabi-Yau threefolds consisting of line bundles over a genus
g Riemann surface. We show that the refined topological string partition
function on these geometries is computed by a two-dimensional TQFT. We also
study the refined black hole partition function arising from N D4 branes on the
Calabi-Yau, and argue that it reduces to a (q,t)-deformed version of
two-dimensional SU(N) Yang-Mills. Finally, we show that in the large N limit
this theory factorizes to the square of the refined topological string in
accordance with the refined OSV conjecture.
