Hypercharge Quantisation and Fermat's Last Theorem

Abstract

What values of the Standard Model hypercharges result in a mathematically consistent quantum field theory? We show that the constraints imposed by the lack of gauge anomalies can be recast as the equation x^3 + y^3 = z^3. If hypercharge is quantised, then x, y and z must be integers. The trivial (and only) solutions, with x=0 or y=0, reproduce the hypercharge assignments seen in Nature. This argument does not rely on the mixed gauge-gravitational anomaly, which is automatically vanishing if hypercharge is quantised and the gauge anomalies vanish.

2.25 pages. v2: References added. v3: More references, and comment on the right handed neutrino

References in corpus (5)

• Exploring 2-Group Global Symmetries
• An Anomaly-free Atlas: charting the space of flavour-dependent gauged $U(1)$ extensions of the Standard Model
• General solution to the U(1) anomaly equations
• Anomaly-free Model Building with Algebraic Geometry
• Number-Theory Dark Matter

Cited by in corpus (4)

• Solving local anomaly equations in gauge-rank extensions of the Standard Model
• Anomaly-free chiral $U(1{)}_D$ and its scotogenic implication
• Varieties of four-dimensional gauge theories
• Integer solutions to the anomaly equations for a class of chiral gauge theories