Superconformal Block Quivers, Duality Trees and Diophantine Equations

TL;DR

This paper extends the understanding of superconformal block quiver gauge theories by revealing an algebraic structure linking their fixed points to root systems and interpreting dualities as Weyl group actions.

Contribution

It introduces a new algebraic framework for superconformal quiver theories, connecting Diophantine equations to root systems and Weyl group symmetries.

Findings

Derived Diophantine equations for low block numbers

Reorganized equations for arbitrary block numbers

Linked superconformal conditions to imaginary roots and Weyl group actions

Abstract

We generalize previous results on N=1, (3+1)-dimensional superconformal block quiver gauge theories. It is known that the necessary conditions for a theory to be superconformal, i.e. that the beta and gamma functions vanish in addition to anomaly cancellation, translate to a Diophantine equation in terms of the quiver data. We re-derive results for low block numbers revealing an new intriguing algebraic structure underlying a class of possible superconformal fixed points of such theories. After explicitly computing the five block case Diophantine equation, we use this structure to reorganize the result in a form that can be applied to arbitrary block numbers. We argue that these theories can be thought of as vectors in the root system of the corresponding quiver and superconformality conditions are shown to associate them to certain subsets of imaginary roots. These methods also allow for an interpretation of Seiberg duality as the action of the affine Weyl group on the root lattice.