  We study the RG flow of N=1 world-volume gauge theories of D3-brane probes on
certain singular Calabi-Yau threefolds. Taking the gauge theories out of
conformality by introducing fractional branes, we compute the NSVZ
beta-function and follow the subsequent RG flow in the cascading manner of
Klebanov-Strassler. We study the duality trees that blossom from various
Seiberg dualities and encode possible cascades. We observe the appearance of
duality walls, a finite limit energy scale in the UV beyond which the
dualization cascade cannot proceed. Diophantine equations of the Markov type
characterize the dual phases of these theories. We discuss how the
classification of Markov equations for different geometries into families
relates the RG flows of the corresponding gauge theories.

High Energy Physics - Theory