Theory Update 102

Recall that the twisted Fourier transform $T$ took the quark braids to phase matrices. Letting $\omega$ be the primitive cubed root of unity and expressing $12$th roots in terms of integers in the set $(-5,6)$, an example product of phase matrices is

Similarly, one obtains the other $3 \times 3$ particle matrices from a phase product. Note that the down quark phases differ from the up phases only by a column permutation (color index) and complex conjugation. This is quark lepton/boson complementarity. We cannot ignore bosons in Fourier supersymmetry. For example, the identity photon matrix is obtained from the product of $T(\overline{d}_{R})$ and its entrywise complex conjugate. But complex conjugates do not always define inverses! For example, the product of $T(\overline{u}_{L})$ with its conjugate is the $e_{L}^{+}$ matrix, which equals $\omega \cdot (231)$.