Problem 2.6

A First Course in String Theory

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2.5 Constructing \displaystyle{T^2/\mathbb{Z}_3} orbifold

(a) A fundamental domain, with its boundary, is the parallelogram with corners at \displaystyle{z = 0, 1} and \displaystyle{e^{i \pi/3}}. Where is the fourth corner? Make a sketch and indicate the identifications on the boundary. The resulting space is an oblique torus.

(b) Consider now an additional \displaystyle{\mathbb{Z}_3} identification

\displaystyle{z \sim R(z) = e^{2 \pi i/3} z}

To understand how this identification acts on the oblique torus, draw the short diagonal that divides the torus into two equilateral triangles. Describe carefully the \displaystyle{{Z}_3} action on each of the two triangles (recall that the action of \displaystyle{R} can be followed by arbitrary action with \displaystyle{T_1}, \displaystyle{T_2}, and their inverses).


[guess]

(a)

\displaystyle{z = 1 + e^{\frac{i \pi}{3}}}

(b)

[guess]

— Me@2021-02-11 06:03:36 PM

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