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A lot of books are concerned with CFT on a torus (Euclidean signature), for instance in Polchinski's String Theory volume 1, the propagator of a free massless boson on a torus is

$$ G(z,\bar{z};w,\bar{w})_{T^2}=-\frac{\alpha'}{2}\log\left|\theta_1\left(\tau \middle| \frac{z-w}{2\pi} \right)\right|^2 + \alpha'\frac{[\Im(z-w)]^2}{4\pi\tau_2}. $$

Now, in principle, it is possible to recover this expression by canonically quantizing the boson on the torus and compute the VEV:

$$\tag{*} G(z,\bar{z};w,\bar{w})_{T^2}= \langle 0|\phi(z,\bar{z})\phi(w,\bar{w})|0\rangle_{T^2}. $$

QUESTION

Is there a clever way to get the mode expansion just looking at the corresponding CFT on flat space which is well known?

I was thinking that since $\partial\phi$ is a primary field, in principle his Laurent expansion should be severely constrained by CFT.

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  • $\begingroup$ On a torus correlators are not VEV's, but thermal averages (time is periodic). You can use OPE to relate them to torus 1pt functions. The latter can be computed by writing thermal average as a sum over states and relating that to sphere 3pt functions. $\endgroup$ – Peter Kravchuk Dec 8 '17 at 17:48
  • $\begingroup$ Could you give me some formulas or reference? $\endgroup$ – MaPo Dec 11 '17 at 16:02

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