Speaker
Description
Given a gauge Lie algebra, it is natural to seek representations for four-dimensional spacetime fermions that are anomaly-free and chiral. However, this problem is difficult to solve in full generality. Even for irreducible representations, where we only have to study $\mathfrak{su}(n)$ for $n\geq3$, solutions would seem to be few and far between: a trial-and-error scan by Eichten, Kang and Koh found only three for $\mathfrak{su}(5)$, for example. I will explain how concepts in rational algebraic geometry show that there are in fact infinitely many anomaly-free chiral representations of semisimple algebras that are either irreducible or are products of irreducible ones. In particular, equivalence classes of these representations are in bijection with rational points on projective varieties, and all such points can be found in the above two cases. When the gauge algebra contains one extra $\mathfrak{u}_1$ summand, I will show how, while it is still possible to parameterize infinitely many solutions, it is not necessarily the case that all are found. Finally, I will give an overview of the problem of anomaly cancellation with multiple $\mathfrak{u}_1$ summands.
This talk is based on work done with Ben Gripaios in 2409.15430, 2501.09860 and 2508.11583.