### Lieven's Trinities

Lieven Le Bruyn has an absolutely wonderful post about Arnold's trinities. Examples include the Platonic groups, the exceptional triple $(E_6, E_7, E_8)$ and the fields $\mathbb{C}$, $\mathbb{H}$ and the octonions. Lieven asks, do you have other trinities you like to worship?

In M Theory we have all of these and lots more! The Riemann surface moduli triple $(M(0,6), M(1,3), M(2,0))$ of twistor dimension. Idempotent triples for the particle generations. Three kinds of being in ternary logic. The three squares on an associahedron in dimension 3. Three parity cubes for the exceptional Jordan algebra over the octonions. The three states of Peirce's Hegelian philosophy. The three crossings on a trefoil knot and the braid group $B_3$. The triple $(B_{3}, PSL(2, \mathbb{Z}), S_3)$ of braids, modular group and hexagon (or triangle).

Update: A pdf version of Arnold's paper has kindly been provided by Lieven.

Aside: I just installed the latest version of Firefox and it has ruined some of the maths fonts. Is this problem going to be fixed?

In M Theory we have all of these and lots more! The Riemann surface moduli triple $(M(0,6), M(1,3), M(2,0))$ of twistor dimension. Idempotent triples for the particle generations. Three kinds of being in ternary logic. The three squares on an associahedron in dimension 3. Three parity cubes for the exceptional Jordan algebra over the octonions. The three states of Peirce's Hegelian philosophy. The three crossings on a trefoil knot and the braid group $B_3$. The triple $(B_{3}, PSL(2, \mathbb{Z}), S_3)$ of braids, modular group and hexagon (or triangle).

Update: A pdf version of Arnold's paper has kindly been provided by Lieven.

Aside: I just installed the latest version of Firefox and it has ruined some of the maths fonts. Is this problem going to be fixed?

## 2 Comments:

do you have a link to material where i can read up on the trinity of moduli spaces?

i do like your (S3,PSL,B3) trinity!

Hi Lieven! I often discuss this trinity of moduli, because it is very important in quantum gravity, but I know of no properly published references on it. I will try to write more. Of course, there is a lot of string literature and there may be some stringy papers that focus on these particular moduli, but kneemo would know better than me what these references are.

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