The Duality Cascade

TL;DR

This work surveys the duality cascade in ${\cal N}=1$ gauge theories, weaving Seiberg dualities, the Klebanov–Witten conifold CFT, and their gravity dual on warped deformed conifolds into a coherent RG-flow story. It demonstrates how a cascade of exact dual descriptions emerges from a single underlying flow, with two holomorphic couplings mapping to three physical gauge couplings and a rich fixed-point structure that includes twice as many approximate fixed points as naively expected. The analysis connects field-theoretic RG dynamics to a smooth supergravity flow at large $k$, elucidating confinement and chiral-symmetry breaking, the appearance of a massless Goldstone multiplet, and the deformation of the conifold into its warped/deformed counterpart. Overall, the paper provides a concrete, calculable framework where confinement and IR physics can be studied through both gauge-theory dualities and their string-theory holographic description, with implications for QCD-like dynamics and beyond-Standard-Model phenomenology.

Abstract

The duality cascade, and its dual description as string theory on the warped deformed conifold, brings together several sophisticated topics, some of which are not widely known. These lectures, which contain a number of previously unpublished results, and are intended for experts as well as students, seek to explain the physics of duality cascades. Seiberg duality is carefully introduced, with detailed attention to the physical implications of duality away from the far infrared. The conifold is briefly introduced and strings on the conifold (the Klebanov-Witten model) are discussed. Next, fractional branes are introduced. The duality cascade is then constructed in field theory and in its dual supergravity description. Among the newly published results: it is shown why supergravity sees the cascade as smooth; how the two holomorphic couplings (dilaton and integrated two-form in supergravity) are related to the three physical couplings in the gauge theory; that there are actually twice as many approximate fixed points in the cascade as might be naively expected. These notes are based on lectures given at TASI 2003 and at the 2003 PIMS Summer School on Strings, Gravity & Cosmology.

Figures (42)

• Figure 1: The effect of a mass term grows in the infrared.
• Figure 2: For $N_f$ in the conformal window, the gauge coupling has a stable infrared fixed point at $g=g_*$.
• Figure 3: For $N_f>2N$ the coupling $\eta$ is irrelevant both at $g=0$ and at $g=g_*$.
• Figure 4: For $N_f<2N$ the coupling $\eta$ is relevant at $g_*$; its initial decrease is reversed once $g$ is sufficiently large.
• Figure 5: The $M$ fields are decoupled on the $y=0$ axis; the theory reaches a Seiberg SQCD fixed point. If $y\neq0$, the Seiberg SQCD fixed point is destabilized; what happens next?