Two Dimensional Yang-Mills, Black Holes and Topological Strings

We show that topological strings on a class of non-compact Calabi-Yau threefolds is equivalent to two dimensional bosonic U(N) Yang-Mills on a torus. We explain this correspondence using the recent results on the equivalence of the partition function of topological strings and that of four dimensional BPS black holes, which in turn is holographically dual to the field theory on a brane. The partition function of the field theory on the brane reduces, for the ground state sector, to that of 2d Yang-Mills theory. We conjecture that the large N chiral factorization of the 2d U(N) Yang-Mills partition function reflects the existence of two boundaries of the classical AdS_2 geometry, with one chiral sector associated to each boundary; moreover the lack of factorization at finite N is related to the transformation of the vacuum state of black hole from a pure state at all orders in 1/N to a state which appears mixed at finite N (due to O(e^{-N}) effects).