# 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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