Black holes, instanton counting on toric singularities and q-deformed two-dimensional Yang-Mills theory

We study the relationship between instanton counting in View the MathML source Yang–Mills theory on a generic four-dimensional toric orbifold and the semi-classical expansion of q-deformed Yang–Mills theory on the blowups of the minimal resolution of the orbifold singularity, with an eye to clarifying the recent proposal of using two-dimensional gauge theories to count microstates of black holes in four dimensions. We describe explicitly the instanton contributions to the counting of D-brane bound states which are captured by the two-dimensional gauge theory. We derive an intimate relationship between the two-dimensional Yang–Mills theory and Chern–Simons theory on generic Lens spaces, and use it to show that the known instanton counting is only reproduced when the Chern–Simons contributions are treated as non-dynamical boundary conditions in the D4-brane gauge theory. We also use this correspondence to discuss the counting of instantons on higher genus ruled Riemann surfaces.