Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings

We count the number of bound states of BPS black holes on local Calabi-Yau three-folds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a q-deformed Yang-Mills theory on the Riemann surface. Following the recent connection between the black hole entropy and the topological string partition function, we find that for a large black hole charge N, up to corrections of O(e−N), ZBH is given as a sum of a square of chiral blocks, each of which corresponds to a specific D-brane amplitude. The leading chiral block, the vacuum block, corresponds to the closed topological string amplitudes. The subleading chiral blocks involve topological string amplitudes with D-brane insertions at (2g−2) points on the Riemann surface analogous to the Ω points in the large N 2d Yang-Mills theory. The finite N amplitude provides a non-perturbative definition of topological strings in these backgrounds. This also leads to a novel non-perturbative formulation of c=1 non-critical string at the self-dual radius.