A First Course in String Theory

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2.7 A more general construction for cones?

Consider the plane and the complex coordinate . We have seen that the identification , with an integer greater than two, can be used to construct a cone.

Examine now the identification

where and are relatively prime integers (their greatest common divisor is one).

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Determine a fundamental domain for identification.

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Given two relatively prime numbers *a* and *b*, there exists integers *m* and *n* such that .

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[guess]

Since *M* and *N* are relatively prime numbers, there exists integers *m* and *n* such that .

So

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Therefore,

for some integers *m* and *n*.

As a result, a fundamental domain is provided by the points *z* that satisfy .

[guess]

— Me@2021-03-09 04:58:02 PM

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2021.03.09 Tuesday (c) All rights reserved by ACHK