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In Conformal Field Theory Philippe by Di Francesco, Pierre Mathieu David Sénéchal

Sec 10.l. Conformal Field Theory on the Torus

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eq.10.9 says the modular transformation $\mathcal{T}$ and $\mathcal{S}$ satisfies

1. $$ (\mathcal{S}\mathcal{T})^3=\mathcal{S}^2=+1 $$

  1. But I also heard that given chiral central charge $c$ $$ (\mathcal{S}\mathcal{T})^3=e^{2 \pi i c/8}\mathcal{S}^2 $$

  2. I also heard that $$ (\mathcal{S}\mathcal{T})^6 =\mathcal{S}^4=+1 $$

How can these three statements all be true? Please kindly correct any mistake any of us made above and provide an explanation?

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    $\begingroup$ Heard where? In which context? $\endgroup$
    – Qmechanic ♦
    Commented Dec 12, 2023 at 5:28
  • $\begingroup$ For the second definition, are there certain constraints on what $c$ should be or what values it can be? Or is it that the RHS must be equal to one, so that puts constraints on $c$ in order to give unity? Also, where did statement 3 come from since this just seems like statement number 1 but with higher powers. $\endgroup$
    – MathZilla
    Commented Jan 6 at 1:05

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