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

x 3

+

y 3

=

z 3

.

If hypercharge is quantised, then x

x

,

y

y

and

z

z

must be integers. The trivial (and only) solutions, with

x=0

x = 0

or

y=0

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.