occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Friday, June 20, 2008

### M Theory Lesson 200

Now imagine that $\omega$ is an $N$th root of unity for some initially arbitrary $N$. Then our generators $\sigma_{1}$ and $\sigma_{2}$ obey ordinary matrix relations of the form and $\sigma_{1} \sigma_{2} \sigma_{1} = \sigma_{2} \sigma_{1} \sigma_{2}$ holds. We also have $(\sigma_{1} \sigma_{2} \sigma_{1})^{2} = \omega^{6} \cdot 1$, so if $\omega$ is a 6th root of unity the modular relation holds. One also has that $(\sigma_{1} \sigma_{2})^{3} = \omega^{6} \cdot 1$. This is the operator usually chosen to represent $ST$ in the modular group.