Theory Update 13

In a new comment on the neutrino braids, kneemo recalls that the braid group on three strands extends the famous modular group. He matches the four neutrino braids to group generators $A$ and $B$:


Here $A$ and $B$ are related to the usual generators $S$ and $T$ by $A = T$ and $S = ABA$. If we had a copy of the modular group for each generation, we would have (roughly speaking) the M theory duality group $SL(2,Z) \times SL(2,Z) \times SL(2,Z)$, famously discussed in connection with three qubits. So as we hoped, the generations may indeed be associated with a tripling of spin. By vertically composing three copies of a neutrino braid, we obtain a (not quite the) identity braid.