Notes on anomalies, elliptic curves and the BS-D conjecture

Abstract

We consider anomaly cancellation for $SU(N)\times SU(2)\times U(1)$ gauge theories where the left-handed chiral multiplets are in higher $SU(2)$ representations. In particular, if the left-handed quarks and leptons transform under the triplet representation of $SU(2)$ and if the $U(1)$ gauge group is compact then up to an overall scaling there is only one possible nontrivial assignment for the hypercharges if $N=3$, and two if $N=9$. Otherwise there are infinitely many. We use the Mordell-Weil theorem, Mazur's theorem and the Cremona elliptic curve database which uses Kolyvagin's theorem on the Birch Swinnerton-Dyer conjecture to prove these statements.

arXiv comment17 pages; v2: References added, matches published version

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Cited by in corpus (1)

• Varieties of four-dimensional gauge theories