Duality walls, duality trees and fractional branes

In this paper, we compute the NSVZ beta functions for [Formula: see text] four-dimensional quiver theories arising from D-brane probes on singularities, complete with anomalous dimensions, for a large set of phases in the corresponding duality tree. While these beta functions are zero for D-brane probes, they are nonzero in the presence of fractional branes. As a result there is a nontrivial RG behavior. We apply this running of gauge couplings to some toric singularities such as the cones over Hirzebruch and del Pezzo surfaces. We observe the emergence in string theory, of “Duality Walls,” a finite energy scale at which the number of degrees of freedom becomes infinite and beyond which Seiberg duality does not proceed. We also identify certain quiver symmetries as T-duality-like actions in the dual holographic theory.