Varieties of four-dimensional gauge theories

Given a gauge Lie algebra, it is natural to seek representations for four-dimensional spacetime fermions that are anomaly-free and chiral. Even for irreducible representations, where the problem reduces to studying su(n) for n≥3, solutions seem to be few and far between: a trial-and-error scan by Eichten, Kang and Koh found only three for su(5), for example. In this talk, I will explain how concepts in algebraic geometry show that there are in fact infinitely many anomaly-free chiral representations that are either irreducible or are products of irreducible ones. In particular, I will show that equivalence classes of these representations are in bijection with rational points on projective varieties and that there is an efficient algorithm to find all these points; this construct will show in one way how the chiral representations "dominate" the non-chiral ones. I will also discuss the implications of Manin's conjecture on the distribution of such representations and show how, in another way, this means that the chiral ones are "rarer" than the non-chiral ones.

This talk is based on 2409.15430 and 2501.09860 with Ben Gripaios.