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

Sec 10.l. Conformal Field Theory on the Torus

see https://physics.stackexchange.com/q/792144/386735

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

1.

$$

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

$$

2. But I also heard that given chiral central charge $c$

$$

(\mathcal{S}\mathcal{T})^3=e^{2 \pi i c/8}\mathcal{S}^2

$$

3. 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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