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I am wondering whether correlation functions e.g. in 2D CFTs such as

$$\langle \phi\phi\phi\rangle=C\frac{1}{z_{12}^{h_1+h_2-h_3}z_{23}^{h_2+h_3-h_1}z_{31}^{h_3+h_1-h_2}}\frac{1}{\bar z_{12}^{\bar h_1+\bar h_2-\bar h_3}\bar z_{23}^{\bar h_2+\bar h_3-\bar h_1}\bar z_{31}^{\bar h_3+\bar h_1-\bar h_2}}$$

where the weights $h,\bar h$ are conformal dimensions $\pm$ spins of the fields

$$h=\frac{\Delta+s}{2}~~~,~~~\bar h=\frac{\Delta-s}{2}\,,$$

are always supposed to be single valued functions?

I am asking because of the following consideration. Writing e.g.

$$z_{12}=r e^{i\theta}~~~,~~~\bar z_{12}=r e^{-i\theta}$$

we have

$$z_{12}^{h_1+h_2-h_3}\bar z_{12}^{\bar h_1+\bar h_2-\bar h_3}=r^{\Delta_1+\Delta_2-\Delta_3}e^{i\theta(s_1+s_2-s_3)}$$

As long as $s_1+s_2-s_3$ is an integer the phase factor stays single valued under a full rotation

$$e^{i\theta(s_1+s_2-s_3)}=e^{i(\theta+2\pi)(s_1+s_2-s_3)}$$

However, the above assumes that all conformal dimensions $\Delta_i$ are real! In case if they are complex numbers (like in non-unitary CFTs) the part $r^{\Delta_1+\Delta_2-\Delta_3}$ could also become non-single valued! Is that OK, and the correlation function is just not single valued in such a case? Or perhaps we have to generalize the weights as

$$h=\frac{\Delta+s}{2}~~~,~~~\bar h=\frac{\Delta^*-s}{2}\,,$$

in such a case, where $\Delta^*$ is the complex conjugate of $\Delta$, to preserve single-valuedness? How to decide which one it is? Thanks for any suggestion!

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$r^\Delta$ is single-valued even if $\Delta$ is complex, so long $r$ is real positive. Moreover, there are plenty of non-unitary CFTs where all conformal dimensions are real, starting with minimal models.

Some CFTs have multivalued correlation functions, but this is typically because spins are fractional, as in parafermions.

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  • $\begingroup$ Now that you mention it, I can't think of a way to make $r^\Delta$ wander over its branch cut either. Thanks for pointing it out! $\endgroup$ – Kagaratsch Jun 21 '18 at 13:12

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