### M Theory Lesson 127

The Lie group $E8$ (which appeared in Lisi's TOE) is just a classical manifold, so at some point we will want to consider deformations of it. Returning to a much simpler group to begin with, recall that the $S_3$ permutation $(231)$ generates a basis for the $3 \times 3$ Fourier transform. That is, a general circulant is expressed in the form

$a_0 + a_1 (231) + a_2 (312)$

where $(312) = (231)^{2}$. The usual non-commutative replacement for $S_3$ is the braid group on three strands, $B_3$. The elements $(231)$ and $(312)$ are replaced by $\sigma_2 \sigma_{1}^{-1}$ and $\sigma_{1}^{-1} \sigma_2$ respectively. Thus a squaring of the element $(231)$ is replaced by a switching of the order of $B_3$ generators. Note also that an example of a $(231)$ braid is the Bilson-Thompson diagram for a left electron, whereas the right electron is represented by $(312)$. This suggests that a mass circulant is indeed associated to an operator ordering for left and right handed particles, if one takes seriously the utility of a braid analogue to the vector $(a_0, a_1, a_2)$.

$a_0 + a_1 (231) + a_2 (312)$

where $(312) = (231)^{2}$. The usual non-commutative replacement for $S_3$ is the braid group on three strands, $B_3$. The elements $(231)$ and $(312)$ are replaced by $\sigma_2 \sigma_{1}^{-1}$ and $\sigma_{1}^{-1} \sigma_2$ respectively. Thus a squaring of the element $(231)$ is replaced by a switching of the order of $B_3$ generators. Note also that an example of a $(231)$ braid is the Bilson-Thompson diagram for a left electron, whereas the right electron is represented by $(312)$. This suggests that a mass circulant is indeed associated to an operator ordering for left and right handed particles, if one takes seriously the utility of a braid analogue to the vector $(a_0, a_1, a_2)$.

## 1 Comments:

Z_3 permutes fermion generations in dynamical symmetry in which handle numbers g=0,1,2 define SU(3) triplet and bosons with wormhole throats having genera g_1and g_2 define SU(3) octet and singlet with ordinary bosons identifiable as singlets. LHC might find octets with neutral flavour changing currents serving as a signature, one might hope.

CKM matrix and U and D type mixing matries defining mixing of genera define matrices of this dynamical family-SU(3).

If I understand correctly, circulant property means that the rows of mass matrix are obtained by Z_3 action. This might make sense for neutrinos but for quarks and charged leptons the topological mixing matrices U and D cannot be circulants.

Rationality of U and D matrices is very powerful property inspired by algebraic universality and fixes U and D to very high degree. Also algebraic variants might be considered.

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