Varieties of four-dimensional gauge theories

TL;DR

This paper uses algebraic geometry to classify anomaly-free fermion representations in 4D gauge theories, revealing a rich geometric structure and dominance of chiral representations for certain gauge groups.

Contribution

It introduces a geometric framework for classifying anomaly-free fermion representations and provides an efficient algorithm for finding explicit solutions.

Findings

Equivalence classes of representations correspond to rational points on a dense projective variety.

Chiral representations dominate non-chiral ones for gauge groups with n ≥ 5.

An algorithm is provided for explicitly constructing anomaly-free irreducible representations.

Abstract

We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras $\mathfrak{su}_n$ for $n\geq 3$. We show that there exist equivalence classes of such representations that are in bijection with the rational points on a projective variety that are dense in a region of the underlying real variety diffeomorphic to $\mathbb{R}^{n-3}$. It follows that the chiral ones overwhelm the non-chiral ones for $n \geq 5$. We present an efficient algorithm to find explicit anomaly-free irreducible representations and discuss the generalization to reducible representations.