M Theory Lesson 311

Under the correspondence between ribbon twists and roots of unity, as in there is a matching of particle ribbon diagrams to matrices, giving in particular Observe how two copies of each $1$-circulant set arise from the choice of crossings in the braid diagram. Complex conjugacy is a flip of crossings. Given these matrices, a twisted Fourier transform of the form $F C_{a} F^{\dagger} = DC_{b}$, for $D$ a phased diagonal, takes MUB type operators $C_{a}$ to particle circulants $C_{b}$. For example, Let us not forget the quarks! Quarks do not correspond to basis circulants, so under the inverse twisted transform they come from more general matrices. For example, the $C_a$ phases of an up quark are given by which is clearly not a MUB operator. Note, however, that this is a magic matrix, with all rows and columns summing to $- \pi / 6$.