Duality Walls, Duality Trees and Fractional Branes

TL;DR

The paper develops a framework to study non-conformal quiver gauge theories from D-brane probes on toric singularities by combining NSVZ beta functions with $a$-maximization to determine exact anomalous dimensions. It uses duality trees and Picard-Lefschetz monodromies to analyze cascades, uncovering Duality Walls—finite-energy scales where Seiberg duality halts and the degrees of freedom proliferate—along with a Z2 symmetry that mirrors T-duality in the holographic dual. The analysis applied to the conifold, $F_0$, and $dP_1$ reveals closed cycles, toric islands, and wall-structured RG flows, with implications for horizon volumes via the AdS/CFT correspondence. The work culminates in a general methodology for toric and del Pezzo geometries and provides horizon-volume results for several examples, connecting field-theoretic data to geometric invariants.

Abstract

We compute the NSVZ beta functions for N = 1 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 non-zero in the presence of fractional branes. As a result there is a non-trivial 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.

Figures (20)

• Figure 1: Generic quiver for any of the Seiberg dual theories in the duality tree corresponding to a D3-brane probing $\mathbb{C}^3/\mathbb{Z}_3$, the complex cone over $dP_0$.
• Figure 2: Tree of Seiberg dual theories for $dP_0$. Each site of the tree represents a gauge theory, and the branches between sites indicate how different theories are related by Seiberg duality transformations.
• Figure 3: Some first cases of the Seiberg dual phases in the duality duality tree for the theory corresponding to a D3-brane probing $\mathbb{C}^3/\mathbb{Z}_3$, the complex cone over $dP_0$.
• Figure 4: Quiver diagram for the gauge theory on $N$ D3-branes probing the conifold.
• Figure 5: The "duality tree" of the conifold. Its single site represents the standard $SU(N) \times SU(N)$ theory. The closed link coming out the site and returning to it represents the fact that the theory, being self-dual, transforms into itself under Seiberg duality.