Varieties of four-dimensional gauge theories

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 ${\mathfrak{s}\mathfrak{u}}_n$ for $n\ge 3$. 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 $n\ge 5$. We present an efficient algorithm to find explicit anomaly-free irreducible representations and discuss the generalization to reducible representations.