Deformations of Toric Singularities and Fractional Branes

TL;DR

This paper presents a method to analyze how fractional branes affect the IR behavior of gauge theories at toric Calabi-Yau singularities, linking geometric deformations to gauge theory properties.

Contribution

It introduces a simple algorithm based on Altmann's rule to compute anomaly-free rank distributions for fractional branes in toric singularities.

Findings

Algorithm successfully computes rank distributions for various examples.

Matches gauge theory moduli space with deformed geometry in key cases.

Classifies IR behaviors via weights on (p,q) web legs.

Abstract

Fractional branes added to a large stack of D3-branes at the singularity of a Calabi-Yau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann's rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional branes can be classified by looking at suitable weights associated with the external legs of the (p,q) web. We check the proposal on many examples and match in some interesting cases the moduli space of the gauge theory with the deformed geometry.