On Conformal Theories in Four Dimensions

TL;DR

This paper explores orbifold constructions of 4D super-Yang-Mills theories, proposing a method to obtain conformal theories with various supersymmetries by embedding discrete groups into the gauge group, and verifies their conformality through beta function calculations.

Contribution

It introduces a canonical embedding approach for discrete subgroups in gauge groups to produce conformal theories, extending previous work and including explicit beta function analyses.

Findings

Existence of conformal theories for every discrete subgroup of SU(4).

Construction of superconformal theories with N=1 and N=2 supersymmetry.

Beta functions vanish at two loops for N=1 theories.

Abstract

Extending recent work of Kachru and Silverstein, we consider ``orbifolds'' of 4-dimensional $\mathcal{N}=4$ SU(n) super-Yang-Mills theories with respect to discrete subgroups of the SU(4) $R$-symmetry which act nontrivially on the gauge group. We show that for every discrete subgroup of SU(4) there is a canonical choice of imbedding of the discrete group in the gauge group which leads to theories with a vanishing one-loop beta-function. We conjecture that these give rise to (generically non-supersymmetric) conformal theories. The gauge group is $\otimes_i SU(Nn_i)$ where $n_i$ denote the dimension of the irreducible representations of the corresponding discrete group; there is also bifundamental matter, dictated by associated quiver diagrams. The interactions can also be read off from the quiver diagram. For SU(3) and SU(2) subgroups this leads to superconformal theories with $\mathcal{N}=1$ and $\mathcal{N}=2$ respectively. In the $\mathcal{N}=1$ case we prove the vanishing of the beta functions to two loops.