Problem 2.6

A First Course in String Theory

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).