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?
[1]: