A Trio of Dualities: Walls, Trees and Cascades

TL;DR

The paper analyzes non-conformal RG flows of ${\cal N}=1$ quiver gauge theories on D3-brane probes of singular Calabi–Yau threefolds, using fractional branes to trigger cascades. It derives NSVZ beta-functions in the $M/N$-small limit, identifies Seiberg-duality-driven duality trees, and uncovers duality walls that cap the UV extent of cascades. By studying the conifold and the zeroth Hirzebruch surface $F_0$, it demonstrates a rich spectrum of cascades, including a UV-terminating wall and a Z2 symmetry related to T-duality, as well as a sensitivity to initial conditions. A central insight is the correspondence between dual phases and Markov-type Diophantine equations, which classify cascades into families and suggest how cascades in one geometry imply analogous cascades in related geometries, linking RG dynamics to algebraic geometry and number theory.

Abstract

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.