Refined black hole ensembles and topological strings

We formulate a refined version of the Ooguri-Strominger-Vafa (OSV) conjecture. The OSV conjecture that ZBH = |Ztop|2 relates the BPS black hole partition function to the topological string partition function Ztop. In the refined conjecture, ZBH is the partition function of BPS black holes counted with spin, or more precisely the protected spin character. Ztop 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 N2. 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.