### M Theory Lesson 125

The $D_4$ triality in Garrett's paper (slide 16) leaves invariant the $W^{3}$. Deleting this component of the $4 \times 4$ rotation matrix reduces it to the matrix

010

001

100

which is the $(231)$ Fourier basis $3 \times 3$ circulant familiar to M theorists. Observe that this matrix permutes the $\frac{1}{2} \omega_{R}^{3}$, $\frac{1}{2} \omega_{L}^{3}$ and $B_{1}^{3}$ gravity fields.

010

001

100

which is the $(231)$ Fourier basis $3 \times 3$ circulant familiar to M theorists. Observe that this matrix permutes the $\frac{1}{2} \omega_{R}^{3}$, $\frac{1}{2} \omega_{L}^{3}$ and $B_{1}^{3}$ gravity fields.

## 2 Comments:

The story is continually exciting and adds to the M-theory lessons. Many features appear to correspond to the natural world.

Thanks, Louise. Yes, surely particle masses are one of the most basic things we can measure. It is time we understood where they came from.

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