From toric geometry to quiver gauge theory: The equivalence of a-maximization and Z-minimization

AdS/CFT predicts a precise relation between the central charge a, the scaling dimensions of some operators in the CFT on D3-branes at conical singularities and the volumes of the horizon and of certain cycles in the supergravity dual. We review how this quantitative check can be performed for all toric singularities. We discuss how these results are related to the recently discovered map between toric singularities and tilings; in particular, we discuss how to find the precise distribution of R-charges in the quiver gauge theory using dimers technology. (Abstract Copyright [2006], Wiley Periodicals, Inc.)