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I'm reading Carlip's approach to Cardy Formula (https://arxiv.org/abs/hep-th/9806026).

He considers the partition function on the torus of modulus $\tau$ to be

$$\mathcal{Z}(\tau, \bar{\tau})= \text{Tr}~e^{2\pi i \tau L_0}e^{-2\pi i \bar{\tau}\bar{L}_0}\tag{2.3}$$ and a "modular invariant quantity" $$\mathcal{Z}_0= \text{Tr}~e^{2\pi i \tau (L_0 - \frac{c}{24})}e^{-2\pi i \bar{\tau}(\bar{L}_0 - \frac{\bar{c}}{24})}\tag{2.2}$$ which is used in the rest of that section to relate the entropy to the central charge.

The main point is about this "invariant quantity", because other lectures/books, e.g. D. Tong (http://www.damtp.cam.ac.uk/user/tong/string/six.pdf , p. 23) and A.N Schellekens (https://www.nikhef.nl/~t58/CFT.pdf, p. 56), goes trough the construction of the partition function on the torus and get $\mathcal{Z}_0$ as the result.

Since we are moving things from the plane to the cilinder/torus and viceversa, I am pushed to think that $\mathcal{Z}$ and $\mathcal{Z}_0$ are indeed the same object, but looked in different ways. But maybe I'm completely wrong, so I decided to ask here looking for some help.

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