Anomaly cancellation for a $U(1)$ factor

TL;DR

This paper employs arithmetic geometry to analyze anomaly cancellation equations in U(1) gauge theories, revealing infinitely many solutions for hypercharges in models like the Standard Model, and discusses the potential for finding all solutions.

Contribution

It introduces a geometric approach to solving anomaly cancellation equations, characterizing solutions as rational points on a cubic hypersurface, and demonstrates the existence of infinitely many solutions in Standard Model-like scenarios.

Findings

Infinitely many anomaly-free hypercharge assignments exist.

Solutions form a projective cubic hypersurface over rationals.

Geometric methods enable parameterization of solutions.

Abstract

We use methods of arithmetic geometry to find solutions to the abelian local anomaly cancellation equations for a four-dimensional gauge theory whose Lie algebra has a single $\mathfrak{u}_1$ summand, assuming that a non-trivial solution exists. The resulting polynomial equations in the integer $\mathfrak{u}_1$ charges define a projective cubic hypersurface over the field of rational numbers. Generically, such a hypersurface is (by a theorem of Koll{\'a}r) unirational, making it possible to find a finitely-many-to-one parameterization of infinitely many solutions using secant and tangent constructions. As an example, for the Standard Model Lie algebra with its three generations of quarks and leptons (or even with just a single generation and two $\mathfrak{su}_3\oplus\mathfrak{su}_2$ singlet right-handed neutrinos), it follows that there are infinitely many anomaly-free possibilities for the $\mathfrak{u}_1$ hypercharges. We also discuss whether it is possible to find all solutions in this way.