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In 2d CFT, restricting to the holomorphic part of the theory, the torus partition function is given by $$Z(q)=\text{Tr}q^{L_0-c/24}$$ where $L_0$ is the element of the Virasoro algebra, i.e. mode of energy-momentum tensor, $c$ is the central charge, and the trace is over the Hilbert space in radial quantization.

This is analogous to the partition function in the statistical mechanics $$Z(q)=\text{Tr}e^{-\beta H}$$ where $\beta=1/T$ and $H$ is the Hamiltonian.

I can see that the $L_0$ is can be associated with the Hamiltonian in the statistical mechanics. However, what does the temperature (i.e. $\beta$) in statistical mechanics correspond to in the 2d CFT?

Should I think of it as a partition function with external thermal bath that sets temperature fixed?

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    $\begingroup$ The $q=e^{2\pi i \tau}$ insert the analogue of $\beta=\frac{1}{T}$ as well as a "twisting" parameter since $\tau$ is a complex number. $\endgroup$
    – Nogueira
    Dec 9, 2021 at 20:36
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    $\begingroup$ Exactly, the temperature is part of $\tau$. Also, keep in mind that most theories don't holomorphically factorize. $\endgroup$ Dec 9, 2021 at 20:43
  • $\begingroup$ @Nogueira Thank you for your answer. I understand that $\tau$ is $\beta$ here. Do you mean twisting by 'multiplied by $i$'? Or do you mean 'complex temperature'? $\endgroup$
    – Nugi
    Dec 9, 2021 at 21:24
  • $\begingroup$ @ConnorBehan Thank you. So the inverse temperature is the complex structure, i.e. magnitude of momentum (or distance in momentum space) is temperature? Also, I am thinking about four-dimensional theories. (More specifically, 4d $\mathcal{N}=2 chiral algebras constructed by Beem, Rastelli, et.al.) Except the even dimensions, what other information can we immediately use to determine criteria of holomorphic factorization? $\endgroup$
    – Nugi
    Dec 9, 2021 at 21:28
  • $\begingroup$ @Nugi $\tau$ is the modular parameter of the torus. The torus basically compute the partition function of the CFT in a $S^{1}$, where the "size" of the torus is the inverse of the temperature. The modular parameter is complex is because we can also twist the $S^{1}$ along the torus, i.e. rotate the $S^{1}$ by acting with the generator $L_0-\overline L_0$. $\endgroup$
    – Nogueira
    Dec 9, 2021 at 21:50

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