2.10 A spacetime orbifold in two dimensions, 4

A First Course in String Theory

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(b) … the family of curves

\displaystyle{x^+ x^- = a^2},

where \displaystyle{a > 0}

Show that the identification (2) relates points on each separate curve.

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Identification (2):

\displaystyle{(x^+, x^-) \sim \left( e^{-\lambda} x^+, e^{\lambda} x^- \right)}, where \displaystyle{e^\lambda \equiv \sqrt{\frac{1+\beta}{1-\beta}}}.

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\displaystyle{   \begin{aligned}  (x^+)' &= e^{- \lambda} x^+ \\  (x^-)' &= e^{\lambda} x^- \\  \end{aligned}  }

\displaystyle{   \begin{aligned}  (x^+)' (x^-)'   &= e^{- \lambda} x^+ e^{\lambda} x^- \\  &= a^2 \\  \end{aligned}  }

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— Me@2021-12-15 04:18:27 PM

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