Problem 2.7

A First Course in String Theory

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2.7 A more general construction for cones?

Consider the \displaystyle{(x, y)} plane and the complex coordinate \displaystyle{z = x + iy}. We have seen that the identification \displaystyle{z \sim e^{\frac{2 \pi i}{N}} z}, with \displaystyle{N} an integer greater than two, can be used to construct a cone.

Examine now the identification

\displaystyle{z \sim e^{2 \pi i \frac{M}{N}} z, ~~~ N > M \ge 2,}

where \displaystyle{M} and \displaystyle{N} are relatively prime integers (their greatest common divisor is one).

Determine a fundamental domain for identification.

Given two relatively prime numbers a and b, there exists integers m and n such that \displaystyle{m a + n b = 1}.

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[guess]

Since M and N are relatively prime numbers, there exists integers m and n such that \displaystyle{m M + n N = 1}.

So

\displaystyle{\frac{M}{N} m = \frac{1 - nN}{N}}.

Therefore,

\displaystyle{\left[e^{2 \pi i \frac{M}{N}}\right]^m = e^{2 \pi i \frac{1 - nN}{N}}} for some integers m and n.

As a result, a fundamental domain is provided by the points z that satisfy \displaystyle{0 \le \arg(z) < 2 \pi \frac{1}{N}}.

[guess]

— Me@2021-03-09 04:58:02 PM

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2021.03.09 Tuesday (c) All rights reserved by ACHK