# Hypercharge Quantisation and Fermat's Last Theorem

### Submission summary

 As Contributors: David Tong Arxiv Link: https://arxiv.org/abs/1907.00514v1 Date submitted: 2019-08-06 Submitted by: Tong, David Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics

### 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.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 1907.00514v1 on 6 August 2019

## Reports on this Submission

### Report

In this paper the authors look at the problem of cancelling anomalies in the Standard Model. A long time ago it was realized that by including the cancellation of the mixed gauge-gravitational anomalies along with the usual gauge anomalies in the Standard Model, then there are two possible solutions for the hypercharges [4,3], one of which is the solution found in nature. Both solutions are rational.

The authors here assume that the hypercharge gauge symmetry is $U(1)$ and not $\mathbb{R}$ and so the rationality of the hypercharges is an initial condition. They drop the requirement that the mixed anomalies cancel and instead only assume cancellation of the gauge anomalies. After a change of variables they are able to turn the anomaly equation for the hypercharges into Fermat's cubic equation, which was proven by Euler to only have trivial solutions in the rationals. Those solutions correspond to the same ones found in [4] and [3]. Very nice.

### Requested changes

The requested changes only concern references.

1. A reference to Georgi and Glashow, Phys.Rev. D6 (1972) 429 should be included in [1]. While not explicitly spelled out in the paper, the choice of the spectrum in the second unnumbered equation line has no $SU(3)^3$ anomalies.
2. The reference for the mixed gauge-gravity anomaly in [2] should also include Delbourgo and Salam, Phys.Lett. 40B (1972) 381-382 and Eguchi and Freund, Phys.Rev.Lett. 37 (1976) 1251
3. Since [4] preceded [3] by 7 years, shouldn't that reference go first? It is also hard to say how well known the result was at the time. Is there any reference before [4] discussing this?

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